Predicting Pressure Distributions Using Cfd

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

Predicting Pressure Distributions on Surfaces of Arbitrary Geometry from CFD – a preliminary study

By I Gede Adi Susila

Thesis submitted to the Department of Civil Engineering University of Newcastle upon Tyne in partial fulfilment of requirement for the degree of Master of Science in Structural Engineering

APPROVED:

P. D. Gosling, Supervisor

August 2001 Newcastle upon Tyne Keywords: Wind Loads, Fabric Membrane Structure, Cable-suspended roof, published data, CFD method, LES (Large Eddy Simulation- Smagorinsky +Lilly model viscosity).

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

Predicting Pressure Distributions on Surfaces of Arbitrary Geometry from CFD – a preliminary study

Abstract Structural fabric membrane for many years have been applied and developed for large span enclosures for variety of purposes. More recently, structures that combine highly flexible cable integrated with fabric membranes have been designed as structural integral system. Accurate assessment of wind load distribution is important because the large surface area usually projected by a fabric membrane structure means that wind pressure is a significant load case. The requirement to predict wind loading on structures of complex geometry form is absolutely needed. Computational Fluidal Dynamic (CFD analysis) is highly pointed to solve a number of wind tunnel test problem on the computer simulation. Large-eddy simulation (LES) technique with the Smagorinsky eddy-viscosity model has been applied in order to predict pressure coefficients for 3-D domes and catenoid models. “Fluent” has been used to analyze the flows. Published data of Maher and the ASCE have been used as the basis guideline to enable wind loading to be applied appropriately.

mean external Cp

Table of Mean pressure coefficient around the sphere under LES simulation 1 0.621

0.5

0.318

0

-0.144 0

-0.5

30

60

90

120

-0.441

150

-0.744

180

-0.289

-1 -1.2

-1.5

Plan view: pressure coefficients for y/d = ½ (hemisphere) Maher’s Plan view of pressure coefficient contour y/d=h/D=1/2 on CFD

Angle, @ (degree)

Cp Around Wall

Result of LES computations are compared with those from laminar models as well as those from turbulent models based on Reynolds–average Navier-Stokes equation (RANS model) and those from experiment. The numerical experiment results for all models with various configurations to be exited by the turbulent wind forces were identified. The LES results from 3D computational agreed very well with the experimental or published data. For the dome case of h/d=1/2 ratio, the result can be sort it out into the maximum positive Cp=+0.621 and the maximum negative in the centre of dome is Cp = -1.2. The coefficient offered was quit similar to the published data of Cp=+0.6 and Cp=-1.0, respectively. In the limited study presented in this dissertation, CFD has been shown to a reasonable prediction of wind pressure distributions. Conceivably it could replace some wind tunnel tests. However, further study of CFD applied to the structural engineering problem is still needed in order to evaluate the reliability of the numerical results. Keywords: Wind Loads, Fabric Membrane Structure, Cable-suspended roof, published data, CFD method, LES (Large Eddy Simulation- Smagorinsky +Lilly model viscosity). Corresponding author: E-mail: [email protected] / [email protected] / [email protected] Present Address: 11 Sedgley Road, Crumpsall, Manchester, M8 5AG. UK.

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

Table of Contents List of Figure ............................................................................................................. vii List of Table .............................................................................................................. xv

Chapter 1. Introduction .......................................................................................... 1

Chapter 2. Literature Review................................................................................. 5 2.1 Introduction ................................................................................................ 5 2.2 Cable-Suspended Structures ....................................................................... 5 2.3 Fabric Membrane Structures ...................................................................... 17 2.4 Computational Fluid Dynamic (CFD) for Wind Loading .......................... 23 2.5 Wind Tunnel Test ....................................................................................... 35 2.5.1 Wind Tunnel Techniques................................................................... 37 2.5.2 Small Wind Tunnel............................................................................ 43 2.6 Conclusion .................................................................................................. 43

Chapter 3. Numerical Methods .............................................................................. 44 3.1 Introduction ................................................................................................ 44 3.2 Finite Element Theory and CFD Methods Reviews................................... 44 3.2.1 The CFD Code................................................................................... 51 Pre-processor .................................................................................... 51 Solver................................................................................................. 51 Post-processor.................................................................................... 52 3.2.2 Fluid Flow Problem and Governing Equations on CFD ................... 53 3.2.3 General Fluid Dynamic Background................................................. 60 3.3 General Strategies and Procedures ............................................................. 62 AutoCAD Reviews ............................................................................ 63 Pre-processor: GAMBIT Reviews .................................................... 64 Solver: Fluent Reviews...................................................................... 67 Post-processor.................................................................................... 68 3.4 Detail of Model Experimental .................................................................... 68

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

3.4.1 Single Cooling Tower Model ............................................................ 70 3.4.1.a Detail Procedure and Instruction ........................................... 71 3.4.1.b The Result of the Laminar Flows of Cooling Tower ............ 77 3.4.1.c The Result of the Turbulent Flows under Large Eddy Simulation (LES) of Cooling Tower ..................................... 82 3.4.2 Multiple Cooling Tower model ......................................................... 86 3.4.2.a The Result of the Turbulent Flows under Large Eddy Simulation (LES) of Multiple Cooling Tower ...................... 88 3.4.3 Single Sphere Model ......................................................................... 93 3.4.3.a The Result of the Turbulent Flows under Large Eddy Simulation (LES) of Single Sphere ....................................... 95 3.4.4 Multiple Sphere Model...................................................................... 97 3.4.4.a The Result of the Laminar of Multiple Sphere ...................... 99

Chapter 4. Experimental Methods......................................................................... 101 4.1 Introduction ................................................................................................ 101 4.2 Experimental Work Procedure ................................................................... 103 4.2.1 1:1000 Scale Model of Cooling Tower and Sphere Model ............... 103

4.2.2 Wind Tunnel Testing and Requirement............................................ 104 4.3 Published Experimental Data and Comparison with CFD result ............... 107

4.3.1 Published Data for Sphere/Domes problem ............................. 107 4.3.2 Published Data for Hyperbolic Cooling Tower problem ......... 108 4.3.3 Comparison and Discussion of Published data to the CFD result ................................................................................ 109 4.3.3.a Single Cooling Tower............................................................ 109 4.3.3.b Single Sphere......................................................................... 112

Chapter 5. Conclusions ........................................................................................... 115 5.1 Introduction ................................................................................................ 115 5.2 Wind Tunnel Testing.................................................................................. 115

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

5.3 Wind loading test on CFD method ............................................................. 115 5.3.1 Comparison reliability between Laminar and Turbulent problem flow model in CFD method ............................................................ 116 5.3.2 Comparison between published data and CFD method study of wind loading to fabric membrane structure................................... 116 5.4 General Conclusion and Recommendations

Bibliography............................................................................................................. 118

Appendix 1 ............................................................................................................... 119

Appendix 2 ............................................................................................................... 120

Appendix 3 ............................................................................................................... 123

Appendix 4 ............................................................................................................... 149

List of Figures Figure 1.1 Tent Model ............................................................................................... 1 Figure 1.2 Membrane Roof Model ............................................................................ 1 Figure 1.3 Illustration of wind acting on fabric structure.......................................... 2 Figure 1.4. Example contour of pressure coefficient ................................................ 3 Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

Figure 2.1.Static behaviour and various types of suspended roof ............................. 6 Figure 2.2 various types of suspended roof............................................................... 6 Figure 2.3 Behaviour of cable system ....................................................................... 6 Figure 2.4 Deflection of cable system ....................................................................... 7 Figure 2.5 Wind pressure distribution ....................................................................... 9 Figure 2.6 Antisymmetric wind load effect............................................................... 9 Figure 2.7. Circular plan referencing cable system. .................................................. 9 Figure 2.8. Stadium Detail by Irwin cs & Inc. Figure2.9-10..................................... 11 Figure 2.9. Mean force coefficient ............................................................................ 12 Figure 2.10. Mean deflection at φ=900 ...................................................................... 12 Figure 2.11. Structure Layout.................................................................................... 12 Figure 2.12. Wind Pressure Distribution, by Yasui, cs included Figure2.11 ............ 12 Figure 2.13. La-Plata Stadium by Rocha cs, & included Figure2.14-15.................. 13 Figure 2.14. Mean wind pressure, α=1800 .............................................................. 16 Figure 2.15 Standard Deviation of wind pressure, α=1800 ....................................... 16 Figure 2. 16 Hybrid double-layer system by Ando,cs............................................... 16 Figure 2.17. Wind pressure coefficient Distribution. ................................................ 16 Figure 2.18. Millennium Dome by Kronenburg, A & B ........................................... 17 Figure 2.19 Membrane in tension by Shaeffer .......................................................... 19 Figure 2.20 Hyperbolic surface of membrane .......................................................... 19 Figure 2.21The hangar structural scheme by Kazakevitch........................................ 20 Figure 2.22 Pressure distribution on the membrane roofing at any surface (a) the stage of erection, β=0; ε=0.5%- on upper surface, 2-the net values on the upper and lower surfaces;(b), (c) The completed stage (on the upper surface); 3,4 in section a; 3-β=900, ε=0.5%; 4-β=900 , ε=8%, etc.e by Kazakevitch ...... 20 Figure 2.23 Structural section, Park Dome Kumamoto (1999)................................. 22 Figure 2.24 General Approach Design Tensile Membrane Structure by Campbell (2000) 23 Figure 2.25 Visual Post-processing, Voogt (1990).................................................... 24 Figure 2.26 Stubwing and pressure measurement Voogt (1990)............................... 24 Figure 2.27 Calculation grid in a close vicinity of the cube, Mikkelsen & Livesey (1995) ...................................................................................... 25

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

Figure 2.28 Cp value shown as isobar for an angle of 00 Mikkelsen & Livesey (1995..................................................................................................... 25 Figure 2.29 Comparison between full-scale, model scale and numerically predicted Cp for h/z0= 180 ......................................................................................... 25 Figure 2.30 The computational grid in close vicinity of the obstacle, Lakehal (1998) .............................................................................................................. 26 Figure 2.31 Pressure coefficient distribution at the symmetry plane, Lakehal (1998) .............................................................................................................. 26 Figure 2.32 Comparison of pressure coefficient distribution at a horizontal plane z/H for different approach flow angles:/, Lakehal (1998) ............................................... 27 Figure 2.33 Unstructured hexahedral meshes around typical building configuration, Kim & Boysan (1999) ...................................................................................... 28 Figure 2.34 Flow over the curved two-dimensional hill- predictions using four different turbulence models, bottom left: pressure distribution and bottom right; skin-friction distribution. Kim & Boysan (1999) .......................................................................... 29 Figure 2.35 Distribution of pressure coefficient (Cp) on 1:1:0.5 building of conical vortex at the roof corner predicted by revised model k-ε ( k-ε−φ model by Kawamoto, 1995 30 Figure 2.36 Conical vortex at the roof corner predicted by LES, by Murakami, 1997

30

Figure2.37 Computational model of the AIJ project by Tamura,cs. ......................... 33 Figure 2.38 Kinetic turbulent energy: a). Smargorinsky model, b). Dynamic SGS model, by Tamura,cs ............................................................................................. 34 Figure 2.39 Mean pressure coefficient on the roof: a). Smargorinsky model, b). Dynamic SGS model, by Tamura,cs ............................................................................ 34 Figure 2.40 Heler-type dry-cooling tower, by Su,cs. ................................................ 35 Figure 2.41 Computational region and coordinate system, by Su, cs. ...................... 35 Figure 2.42 Contour of pressure in the horizontal plane (Z=9m, cross wind speed of 5 m/s), by Su, cs........................................................................................................ 35 Figure 2.43. Experimental planning and execution process diagram........................ 38 Figure 2.44. Representative data flow. ...................................................................... 39 Figure 2.45 Dome geometry and coordinate system, by Uematsu............................ 40

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

Figure 2.46 Distributions of the mean and rms pressure coefficient Cp and C’p H/D =1/4, by Uematsu................................................................. 40 Figure 2.47 Tapping arrangement and wind direction definition for single dome test, by Letchford,cs .......................................................................................... 42 Figure 2.48 Comparison of mean pressure coefficient along centerline of a smooth dome, by Letchford,cs .......................................................................................... 42 Figure3.1 Three-dimensional stress on an element, by Logan .................................. 45 Figure 3.2 Tetrahedral solid element, by Logan........................................................ 46 Figure 3.3 Mass flow in and out of fluid element, by Versteeg & Malalasekera...... 55 Figure 3.4 Stress components on three faces of fluid element, by Versteeg & Malalasekera 56 Figure 3.5 Stress components in the x-direction, by Versteeg & Malalasekera........ 56 Figure 3.6 (a) Boundary condition for an internal flow problem Versteeg & Malalasekera 58 Figure 3.6 (b) Boundary condition for external flow problem, by Versteeg & Malalasekera 59 Figure 3.7 Velocity profiles at different locations downstream of an obstacle, by Versteeg & Malalasekera......................................................................................... 59 Figure 3.8 by Potts (MMM336) ................................................................................ 60 Figure 3.9 by Potts (MMM336) ................................................................................ 60 Figure 3.10 Example mesh geometric in AutoCAD.................................................. 63 Figure 3.11 Arranged position of inlet, outlet and wall boundaries in AutoCAD .... 63 Figure 3.12 The geometry that will be exported from AutoCAD ............................. 63 Figure 3.13 Arranged model generated, domain, and floating element (tetrahedral) 65 Figure 3.14 Arranged position of inlet, outlet and wall boundaries in AutoCAD .... 66 Figure 3.15.a Sphere Elevation.................................................................................. 69 Figure 3.15.b Cooling Tower Elevation .................................................................... 69 Figure 3.16 Computational domain development ..................................................... 69

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

Figure 3.17 Sketch of Heler-type dry cooling tower (De1.igs of IGES file) ............ 70 Figure 3.18 Internal count space................................................................................ 70 Figure 3.19 Domain of Single Cooling Tower .......................................................... 71 Figure 3.20 Computational region and coordinate system. ....................................... 71 Figure 3.21. Surface mesh on rear of cooling tower ................................................. 72 Figure 3.22 Brick and Cooling tower ........................................................................ 73 Figure 3.23 Elements within a specified quality range of 0.6 upper and 0. 7 lower ratios 74 Figure 3.24 Plot the residual of laminar flow and number iteration converged at 118. 76 Figure 3.25 Plot the residual of turbulent flow and 427 number iteration converged 77 Figure 3.26.a Pressure coefficient contour of the whole body from the top of plan (Coded De1) ...................................................................................................... 77 Figure 3.26.b Pressure coefficient contour of the whole body from side elevation .. 78 Figure 3.26.c Diagram pressure coefficient in distance position of the model to the sources. ................................................................................................. 78 Figure 3.26.d Pressure coefficient contour occurred at z = 0.2 H ~ 32 m (Plane-5) . 79 Figure 3.26.e Diagram pressure coefficient at z =0.2 H ~ 32 m (Plane-5)................ 79 Figure 3.26.f Pressure coefficient contour occurred at z = 0.45 H ~ 72 m (Plane-6) 80 Figure 3.26.g Diagram pressure coefficient at z = 0.45 H ~ 72 m (Plane-6) ............ 80 Figure 3.26.h Pressure coefficient contour occurred at z = 0.7 H ~ 112 m (Plane-7) 81 Figure 3.26.i Diagram pressure coefficient at z = 0.7 H ~ 112 m (Plane-7) ............. 81 Figure 3.27.a Pressure coefficient contour of the whole body from the top of plan (Coded De11) .................................................................................................... 82 Figure 3.27.b Pressure coefficient contour of the whole body from side elevation .. 82 Figure 3.27.c Diagram pressure coefficient in distance position of the model to the sources 82

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

Figure 3.27.d Pressure coefficient contour occurred at z = 0.2 H ~ 32 m (Plane-5) . 83 Figure 3.27.e Diagram pressure coefficient at z =0.2 H ~ 32 m (Plane-5)................ 83 Figure 3.27.f Pressure coefficient contour occurred at z = 0.45 H ~ 72 m (Plane-6) 84 Figure 3.27.g Diagram pressure coefficient at z = 0.45 H ~ 72 m (Plane-6) ............ 84 Figure 3.27.h Pressure coefficient contour occurred at z = 0.7 H ~ 112 m (Plane-7) 85 Figure 3.27.i Diagram pressure coefficient at z = 0.7 H ~ 112 m (Plane-7) ............. 85 Figure 3.28 Sketch of Multiple Cooling Tower (De5.igs of IGES file).................... 86 Figure 3.29 Domain of Multiple Cooling Tower ...................................................... 86 Figure 3.30. Grid mesh generating of imported file IGES from AutoCAD in Gambit.

87

Figure 3.31. Surface mesh on rear of multiple cooling tower ................................... 87 Figure 3.32 Brick and Cooling tower ........................................................................ 87 Figure 3.33 Elements within a specified quality range of 0.6 upper and 0. 7 lower ratios

87

Figure 3.34.a Pressure coefficient contour of the whole body from the top of plan . 88 Figure 3.34.b Pressure coefficient contour of the whole body from side elevation .. 88 Figure 3.34.c Diagram pressure coefficient in distance position of the model to the sources. 89 Figure 3.34.d Pressure coefficient contour occurred at z = 0.2 H ~ 26 m (Plane-5) . 90 Figure 3.34.e Diagram pressure coefficient at z =0.2 H ~ 26 m (Plane-5)................ 90 Figure 3.34.f Pressure coefficient contour occurred at z = 0.45 H ~ 58.5 m (Plane-6)

91

Figure 3.34.g Diagram pressure coefficient at z = 0.45 H ~ 58.5 m (Plane-6) ......... 91 Figure 3.34.h Pressure coefficient contour occurred at z = 0.7 H ~ 91 m (Plane-7) . 92 Figure 3.34.i Diagram pressure coefficient at z = 0.7 H ~ 91 m (Plane-7) ............... 92 Figure 3.35 Sketch of Single Sphere (De3.igs of IGES file)..................................... 93 Figure 3.36 Domain of Single Sphere. ...................................................................... 93 Figure 3.37. Grid mesh generating of imported file IGES from AutoCAD in Gambit and already meshed on rear of sphere surface. ........................................................ 94

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

Figure 3.38. Brick and Sphere ................................................................................... 94 Figure 3.39 .The mesh developed on domain............................................................ 94 Figure 3.40 Elements within a specified quality range of 0.6 upper and 0. 7 lower ratios

94

Figure 3.41.a Pressure coefficient contour of the whole body from the top of plan (Coded De31) 95 Figure 3.41.b Pressure coefficient contour of the whole body from side elevation .. 96 Figure 3.41.c Diagram pressure coefficient in distance position of the model to the sources. 96 Figure 3.42 Sketch of Multiple Sphere (De4.igs of IGES file) ................................. 97 Figure 3.43 Domain of Multiple Sphere.................................................................... 97 Figure 3.44. Grid mesh generating of imported file IGES from AutoCAD in Gambit

98

Figure 3.45. Surface mesh on rear of multiple cooling tower ................................... 98 Figure 3.46. Brick and Sphere ................................................................................... 98 Figure 3.47 Elements within a specified quality range of 0.6 upper and 0. 7 lower ratios

98

Figure 3.48.a Pressure coefficient contour of the whole body from the top of plan (Coded De4) 99 Figure 3.48.b Pressure coefficient contour of the whole body from side elevation .. 100 Figure 3.48.c Diagram pressure coefficient in distance position of the model to the sources. 100 Figure 4.1a: 1:1000 Scale Model of Single Sphere .................................................. 101 Figure 4.1b: Sketch Model of Single Sphere............................................................. 101 Figure 4.2a: 1:1000 Scale Model of Single Sphere .................................................. 101 Figure 4.2b: Sketch Model of Multiple Sphere ........................................................ 101 Figure 4.3a: 1:1000 Scale Model of Single Cooling Tower..................................... 102 Figure 4.3b: Sketch Model of Single Cooling Tower .............................................. 102 Figure 4.4 a: 1:1000 Scale Model of Multiple Cooling Tower ................................ 102 Figure 4.4b: Sketch Model of Multiple Cooling Tower........................................... 102

Wind Loading on a Fabric Structure

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Figure 4.5 Typical open-circuit wind tunnel ............................................................. 104 Figure 4.6 Scheme the open-circuit of small wind tunnel......................................... 104 Figure 4.7.a – c Photo Small Wind Tunnel ............................................................... 105 Figure 4.8. Open or closed –throat wind tunnel. ....................................................... 106 Figure 4.9. Elevation of circular dome rising directly from the ground.................... 107 Figure 4.10. Plan view: pressure coefficients for y/d = ½ (hemisphere)................... 107 Figure 4.11. Plan view: pressure coefficient for y/d = ¼ .......................................... 107 Figure 4.12. Distribution of local mean pressure coefficient around the throat of the cooling tower (ASCE, 1987) ....................................................................................... 108 Figure 4.13. Distribution of local mean pressure coefficient at different height around the hyperbolic throat of the cooling tower (ASCE, 1987).......................... 108 Figure 4.14. Distribution of root – mean square pressure coefficient around throat of a hyperbolic cooling tower (ASCE, 1987) ................................................................ 109 Figure 4.15. Guiding the angle to describe the pressure coefficient around throat combine with various of a different height measurement. .......................................... 111 Figure 3.41.a Pressure coefficient contour of the whole body from the top of plan (Coded De31) 113 Figure 2.48 Comparison of mean pressure coefficient along centreline of a smooth dome, by Letchford, cs ......................................................................................... 113

Wind Loading on a Fabric Structure

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List of Tables Table 2.1 Details of model roof materials ................................................................... 21 Table 3.1 Laminar and turbulent flow model ............................................................ 61 Table 3.2 Turbulent flow equations for compressible flows, by Versteeg & Malalasekera...................................................................... 61 Table 4.1 Limiting values of Cpe and values CL for domes rising directly from the ground. ................................................................................................................. 108 Table 4.2 Distribution of local mean pressure coefficient around the hyperbolic Cooling Tower ro represented the fabric structure by CFD method under Large Eddy Simulation (LES-Smagorinsky &Lilly)................................................... 109 Table 4.3 Distribution of local mean pressure coefficient at different heights around the Cooling Tower to represented the fabric structure by CFD method under Large Eddy Simulation (LES-Smagorinsky & Lilly) ........................................ 110 Table 4.3.a. Distribution of local mean pressure coefficient at different heights around the Cooling Tower to represented the fabric structure by CFD method under Large Eddy Simulation (LES-Smagorinsky & Lilly)........................... 110 Table 4.3.b. Distribution of local mean pressure coefficient at different heights around the Cooling Tower to represented the fabric structure by CFD method under Large Eddy Simulation (LES-Smagorinsky & Lilly)........................... 111 Table 4.3.c. Distribution of local mean pressure coefficient at different heights around the Cooling Tower to represented the fabric structure by CFD method under Large Eddy Simulation (LES-Smagorinsky & Lilly)........................... 111 Table 4.4 Mean pressure coefficient around the sphere under LES simulation ........ 114

Wind Loading on a Fabric Structure

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Chapter 1 Introduction Fabric membrane structures utilize advanced technology that enables large span structure to be built as lightweight and easily deployable. Fabric structures include tents, pressurised and air supported, sails and inflatable that resist applied load by a combination of curvature and tension (prestress) roofing system. The tents models have been used with considered perfect advanced material of a membrane roof with predominantly tensile forces. A historical review of suspended roofs suggests that the tent is the earliest version of a tension roof, (Fig. 1.1). Fig.1.1 Tent Model

The tent structure has given inspiration to improve model structures. Fabric membrane structures have constructed for a large stadium, aircraft hangar, wide roofing on entertainment building and much more variety of purposed, (Fig.1.2). Fig.1.2 Membrane Roof Model

More recently, light fabric membrane structures combined with cable suspended roof are considered as integral system structures. However, a more accurate assessment of load distribution also considering extreme climate or weather condition effects is important for a complete and accurate understanding of behaviour of the system structure.

The large surface area usually projected by a fabric membrane structure means that wind pressure is a significant load case. It may also be could be strongly influenced by the basic structural form of the roof. However, a problem exist is predicting the applied loading to membrane roof structures estimates of wind loading to complex geometric form are required. Industry provided data of CFD (Computational Fluidal Dynamic) and relatively small number of wind tunnel test form the basis of an approach to enable wind loading to be applied appropriately.

Wind Loading on a Fabric Structure

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The aim of wind tunnel test and CFD is to provide information on local wind patterns, coefficient of wind pressure, and wind-induced structural vibration. The use of wind tunnels is to determine the response of a structure to wind forces and to determine the pattern of wind flow to leeward of a structure. Investigations are carried out on the eddy formation behind model membrane structures to find the frequency and strength of oscillatory forces on the structure of a turbulent air-stream, and on the simulation of natural boundary layer effect, (Fig.1.3). The objective of the present research is to develop procedures for accurate and efficient analysis, particularly to estimate wind loading on non-conventional structures and complex geometry with criteria to the design of tent structures or cable suspended roofs. In this study testing and CFD analysis of scale models are used to obtain a better understanding of how these structures behave under wind loading conditions. The specific contribution of qualitative observations and quantitative measurements of the behavior of the model could be used to supplement failure criteria in related design procedures and serve as a basis for analytical and physical comparisons. On CFD method, the initial design model structures were developed on AutoCAD in order to generate the complex geometric of shape model structures desired. Mesh model structure was exported into Gambit, which is a pre-processing CFD to assembled and associate with Fluent. A shape of wind tunnel model was also created on Gambit, which the model structure generation was placed in the middle

of

tunnel

model.

The

mesh

generation resulted by Gambit will then exported to the Fluent. The Fluent as a solver then associated a particularly need in which the model appropriately generated. Fig.1.3 Illustration of wind acting on fabric structure

Using facilities available, boundary conditions can be applied in order to specify in which condition approached. Fluent solver will process the mesh through running iteration for time period depend on number of element generation. The result of model generation can be obtained and displayed as contour pressure distribution or velocity distribution as well as data script, (Fig.1.4).

Wind Loading on a Fabric Structure

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Fig.1.4. Example contour of pressure coefficient

Several model structures were used in this particular cases such as Sphere, Cooling Tower, China hat model and tandem combination on each model in CFD. The same model tried to involve in wind tunnel too. In wind tunnel model, there were used lamp shade, small ball, and bowl and fiberglass resin as basic material to perform the shape model of fabric membrane structure. All models measured approximately in 150mm x 150mm each of plan area. The shape of structures model were constructed look like dome/sphere and China hat model represented cooling tower model. All of them were investigated under low-speed wind tunnel testing. When the airflow approaches a building, it is impinged around and over the surface of building. The force will create areas of pressure or suction on part of building such as facades, gables and roof. In this study, the material building is fabric membrane structure and cable suspended so that leads to the extraordinary buildings geometric developed. It is the significant requirement to evaluate wind loading by CFD method or wind tunnel test. Once model structure has been examined completely, the result can be combined to the standard method of wind loads in order to analysis a p structure related. The important result expected is local mean pressure coefficient ( Cp = 1 ) ρV 2 2 known as dimensionless pressure, it will then combined with the area pressure coefficient. The area pressure coefficient is integration the local mean pressure coefficient over a surface area such as the

Wind Loading on a Fabric Structure

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roof, gable, etc or part of building faced the wind and then divided by the area yields the area pressure coefficient, which can be used conveniently for determining the wind loads on specific area of building. The pressure coefficient being indicated by a positive Cp value and a negative value (suction pressure). The pressure counted at any point on the surface is also a fraction of the dynamic pressure (qs). Relationship to the building design is when value of dynamic pressure (qs) combined with pressure coefficient whether is external or internal. Since the combination between dynamic pressure and coefficient pressure occurred, the wind load can be obtained and then can be applied to the building design. Data from the computer simulation model and from wind tunnel testing or published data were collected, tabulated, and assessed. Comparable experimental results of CFD model simulations are discussed, conclusions are drawn, and recommendations for further research are presented. Procedures considering is intended to develop more relevant, efficiently, and effectively in order to know wind-loading distribution of the membrane structure presented.

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

Chapter 2 Literature Review 2.1 Introduction The use of permanently installed fabric membranes structure is increasing all the time. Typical structures constructed basically similar form to a normal building or integrated an extraordinary with whatsoever curvature form make it, however those have roof materials sheet replaced with fabric membranes (layer skin). This type structures under investigation, which has combined with cable suspended roof. Both of material structures are focused as large areas of research, using numerical and experimental methodologies. Many experimental and numerical researches have been established regarding wind loading using wind tunnel test and numerical methods on CFD (Computational Fluid Dynamic). However, a dearth of numerical as well as experimental research has been performed regarding wind load on variety of shape fabric membrane structures. Type of research can be found on pressurized arch, beam structure, and inflatable combined fabric membrane structures, these structural types have not been research to extent of many types of shape of structural supported membrane structure. Examples of research conducted on variety of shape of fabric membrane structure, and cablesuspended are presented. Consideration of various research methods and different aspect of the structure have been balanced. The research presented here is aimed to evaluate the need for further analysis and investigation of cable-suspended supported membrane structures.

2.2 Cable-Suspended Structures Cable-suspended roof has been used for many years ago. The ancient style of roofing system is tents, which are motivated to form an advance model structure of a membrane roof on the cable suspended. The historical of tent combined with cable is a review of suspended roofs suggest that the earliest version of tension roof. (Prem Krisnha 1978, p.1). The cable structure would be supported the membrane that is majority resisted pressure of wind load. The suspended roof was acting on which is the lower tension flange of the cable net and the upper compression flange is replaces by the edge ring or by the anchoring, (Fig.2.1b). Thus, it is consist of two cable rows that can only take two-dimensional tension so that is mean the structure will take a small compression only when pre-stressed without any shear taken or has no shear rigidity.

Wind Loading on a Fabric Structure

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Refer to these form that the original form of the roof

could

equilibrate a given load only with the addition

A

B Fig.2.1.Static behavior and various types of suspended roof

of shear, then the actual suspended roof is forced to change its shape into a new form that will be able to carry the load without shear. That is clear; the form adopted is one of funicular surface of the load. Since the cables are able to

Fig.2.2 various types of suspended roof.

take a compressive force; the suspended roof given a shape that it may be possible to pretension the cables. Hyperbolic surface shown in (Fig.2.2) that has an opposite curvature in two cable directions. Nonlinear behavior shown by cables when loaded, and there are varies of degree non-linearity with the types of cable structure and also the loading. A cable has to follow the funicular curve in order to sustain loads. It undergoes large geometric adjustments, particularly when the loading is concentrated or un-symmetric. Figure2.3. shows feature of the behavior of a cable, smaller

this or

to

a

larger

degree is applicable to cable-roof system and poses a serious problem in analysis. (Prem Krisnha 1978, p.3-5). Fig.2.3 Behavior of cable system

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

x

H=

ql2 8f

z

H +h =

1.1

(q + ∆q ) l 2 8( f + Wm )

1.2

Fig.2.4 Deflection of cable system

( Hf + hf ) + ( H Wm hWm ) =

(q + ∆q ) l 2 8

1.3

The characteristic of the behavior of the cable is that their deformation is relatively large, and affects the system of internal forces. Horizontal component of the cable force is due to the load f to let the cable sag, (Fig.2.4a) which can be magnitude on Eq.1.1. By increasing load ∆q, the cable elongation has been changed, which increased the sag on cable by Wm and resulted addition horizontal force by h, written on Eq.1.2. On an antisymmetric load qant case, the only matter has changed is its shape with no changed by arising force of H due to q, shown on, (Fig.7b). (Szabo & Kollar, 1984, p.16-18).

There are considered a cable segment as the basic structural element in suspended roof. Governing equations and the analysis of a freely hanging cable are important to understand, so that a special section on it would be written in Appendix 1. The cable is a strand or a rope made out of high-strength steel wire. The strand as well as the rope is protected with a uniform coating of pure zinc. Cables are available in standard size along with the appropriate fittings to facilitate cable connections with each other and also to other structural members. The breaking strength of cables is in the region of 1380 N/mm2 and the modulus of elasticity is of the order of 138 to 166 kN/mm2. (Prem Krisnha 1978, p.18-19).

In cable suspended roofs, the system of cables carries the roof load directly and as such has a primary structural function. The cable system also serves a false work for erection of cladding. The need for large-span roofs is quit often governed by functional and aesthetic requirement, rather than by economic or structural consideration. For the general design considerations that the problem encountered that suspended roof would be considered cause of the significantly in the design and construction of long span roofs, such as the need for a more accurate assessment of load distribution. Related to the suspension cable, there are

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

also have a problem of the provision of adequate static stiffness, avoiding the occurrence of flutter; anchorage and pre-tensioning of cables and the design of the supporting structure and etc., which assumed special importance for suspended roof. Site condition could be strong influence on basic structural form of the roof. If site condition doesn’t permit build individually, a large span suspended roof can be profitably planned between two buildings of adequate strength, which will serve as anchor for the cable.

The material cable roof suspended structure can be broadly subdivided into two-the supporting structure and the roof cladding. The supporting structure may be constructed of plain, reinforced, or pre-stressed concrete, steel, or a combination of the two. The final choice will be governed by aesthetic and structural considerations and economics. The cladding can be further subdivided into: (1) cables and their connections, (2) auxiliary framework that supports the decking and is placed over the cables, (3) roofing which is the external waterproofing skin, and (4) the insulating layer. The durability of cable can only be defined by design life expectation of the structure, which the materials are designed for permanent and temporary use. Application material cable supported roof was not involved in the specification of building codes so that is much more needing concern on fire protection due to the fire causes creep to the cable. Cable roof are generally classified as flexible material structures because the restriction of allowable deflection on these system doesn’t it same to the conventional of beam and slab structure. The necessary considerations of cable structures are deflection on the system and the limited slopes occurred. (Prem Krisnha, 1978, p.22-23). Many experimental and numerical researches have been done in order to manage the wind acting. The wind load acting to the light and widen the roof surface structures dominantly, which is measured per unit area of horizontal projection. The loading intensity can be specified in between two hanging cables and two bracing cable. The cables are anchored in boundary structures that may consist of column or anchored in arches. Simplified system is the cable net was composed of a system of simple tension members and the joint in the net are frictionless hinges. (See fig.2.1-2.2). However, the consideration static loading, temperature changes and support movements are enclosed, which the distributed load (Mollmann, 1974, p. 162-163). The load distribution would be supported in any direction by the suspended cable.

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

The cables structures are able to take a compression force in term of pre-stresses state the suspended roof has given a shape that it would be possible to pre-tension the cable. That is way the roof and the net should form a hyperbolic surface. The dynamic effect of the wind load action on suspended roof partly causes a tendency for the entire structure to vibrate and partially way give rise to local “flutter”. (Szabo & Kollar, 1984, p. 14). The cable specification would be more interesting to be defined in all aspect structures term with the reliability proved as primarily structures. Response structures of cable can be defined from the potential strength in tensile.

The basic of the cable behavior is that the changing their shape without elongation according to the balance the antisymmetric load with unchanged the cable force. Since that happened, the load distribution affected the cable defection that mostly due to the wind load. (Szabo & Kollar, 1984. p.30-31). In the case of antisymmetric load it is a significant change of the cable shape. It can be shown in figure 2.5.a.b, which is the change of the row of cables in one direction and the other case is that, the row of cables

Fig.2.5 Wind pressure distribution

Fig.2.6 Antisymmetric wind load effect

changed in both directions, shows in Tan α = 2.

fig. 2.5c. That is convenient taken as a basis those cable which intersect

0

α = 63 25’

each other at their quarter points. Figure 2.7 indicated the ratios of a

Sin α = 0.89.

circular ground plan. The affinity to the elliptic ground plan figure 2.5

Fig.2.7. Circular plan referencing cable system. Note: fig. 2.5-2.7 by Szabo, cs.

and 2.6, so that the same ratio can be retained.

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

The proportions of the loads (qx and qy) taken by the cables, including deflection wx and wy of the two perpendicular cables at their point of intersection (figure 2.6a). The two divisions load and the deflection of the two cables at the point intersection can be written into:

qx + q y = q

 0.89 lx  qx   2   wx = 8 nx

1.4 2

 0.89 ly  qy   2   wy = 8 ny

1.5 2

1.6

Explanation of figure 2.6b is somewhat more complicated due to only the y-direction cable is able to change its shape without elongation. Eq. 1.6 gives the deflection of the y-direction cable at the quarter point. On the other hand, the deflection of the x-directional cable can be determined by approximation of the static cable behavior: 3 l2 wk = h 16 f E A1 h=

1.7

∆q l 2 8f

1.8

At its quarter point, ¾ of the greatest deflection occurs 3 3 (0.89lx) 4 wx = q x 4 128 (0.79 fx) q ( EA1 ) x 9 lx 4 wx = qx q 512 fx ( EA1 ) x

1.9

1.10 (J Szabo & L. Kollar, 1984. p.31-47)

In order to gain design load of the fabric structure, that is very dominantly factor of wind load involved so that is needed to investigate wind behavior on the fabric structure shape in some cases. This kind of building type has been researched (e.g. Irwin & Wardlaw 1973; Marcelo Rocha, Sandro Cabral & Jorge Riena, 2000; Yasni, Marukawa, Katagiri, Katsumira, Tamura & Watanabe, 1999). Attempt to understand the structural characteristics have been undertaken with numerical and

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

experimental approaching methods. This research would be conducted to the appropriate design methodology research. Irwin and Wardlaw, 1973 performed wind tunnel test on retractable fabric roof for the Montreal Olympic Stadium in Canada, (Fig.2.8). The physical structure has a cable-supported membrane that is attached to a rigid structure. The potential roles on it have physical quantities of cable-supported membrane roof behavior under wind action in the following measurement: MF (mass of roof fabric per unit area), An appropriate Mc (mass of cable per unit length), set drawn are: ρ ( density of air), b (typical length of roof), ρUb U 2 , , KT (slope of tension versus strain curve for warp or weft direction), µ bg Ks (slope of shear loading versus shear strain curve for warp or weft direction), MF KT , , U (mean wind speed), ρb ρU 2b µ (viscosity of air), Ks M C E (Young’s modulus), , , 2 K ρ b T A (areas of roof), EAC D D (aerodynamic drag cable per unit length), , , 2 2 ρU b ρU 2b g (gravitational acceleration), ∆p ∆p (excess of internal over external pressure in zero wind), , γ, ζ, ζ (damping ration in vacou), ρU 2 VI (internal volume covered by roof), PI b3 γ PI , γ (ratio of specific heats for air), 2 ρ U VI ρ U 2 PI (absolute internal pressure). They have got internal pressure (pi) and the mean force coefficient as function of wind speed that is changed the deflection of the roof altering mean pressure distribution, (Fig.2.9 & 2.10) The experimental method also resulted the non-dimensional parameter (∆p/ρU2) described the relationship of an excess of internal pressure, density of air, and mean wind speed function.

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

From the conclusion drawn, added mass effect is significant and may in some case dominant for lightweight membrane. Fabric membranes are sensitive to wind tunnel noise and thorough knowledge of the acoustic environment

in

wind

tunnel

is

essential in interpreting the data. On present roof, the deflections due to Fig. 2.8. Stadium Detail by Irwin cs & Inc. fig.2.9-10

Fig. 2.9. Mean force coefficient

wind were significant.

Fig. 2.10. Mean deflection at φ=900

From the experimental above can be described that is significant influence of wind pressure distribution to the fabric membrane and the cable, which resulted the significant deflection also vibration measurement. Yasri, Marukawa, Katagiri, Katsumura, Tamura, and Watanabe (1999), was also performed wind tunnel testing on cable suspended roof of long span structures in Japan. Since the dead load of a long-span structure’s roof is relatively small, it is important to estimate wind-induced response to structure. They built two different models that are catenary’s-shape as a sag roof and wave-shape as a rise roof supported by cable, which the structures are combination between the cable and truss beam.

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

This

the

considered,

experiment they

method

developed

the

distribution of the mean wind pressure coefficient and fluctuating wind pressure coefficient. They utilized the Monte Carlo simulation for

producing

wind

pressure

simultaneously at multiple points. Fig.2.11. Structure Layout Fig.2.12. Wind Pressure Distribution, by Yasui, cs included Fig.2.11

Fluctuating wind pressure over the roof surface described of pi (t), (i=1,2, m) was a stationary Gaussian process with a cross spectrum density function Sij (ω), (i, j =1, 2, m) and mean value is 0. The fluctuating wind pressure can be estimated from:

pind =

1 N

N −1

∑X k =0

ik

 j 2πkn  exp    N 

(i = 1, 2, ..., m; n = 1, 2, ..., N ), j = − 1

1.11

where k is indicated the frequency and d is indicated an estimate value. Xk is an element of the complex vector defined by N Xk = Hk ζ k 2∆t

1.12

Wind Loading on a Fabric Structure

26

,

University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

 X 1k   H11k X  H  21k  2k   .   .     .   .  .  N  .  =  2 ∆t  H i1k  X ik   .  .      .  .   .  .      H m1k  X mk 

H 22 k . . . H i 2k . . . H m2k

    .  .   .  . . . H iik   . .  . .   . .  . . . H mik . . . H mmk 

ζ 1k  ζ   2k   .     .   .    ζ ik   .     .   .    ζ mk 

Hk is obtained by the LLT decomposing the cross spectrum density: S(ωk) = Η(ωk) Η∗Τ(ωk)

1.13

∗ ∗ ∗ 0  H 11 (ω k )   H 11 (ω k ) H 21 (ω k ) . . . H m 1 (ω k )     H (ω k ) H (ω k )  ∗ H αα (ω k ) . . . H m∗ 1 (ω k )  22  21     .   . . .  = ×  . . .   .     .   . . .     ∗ H mm (ω k )   H m 1 (ω k ) H m 2 (ω k ) . . . H mm (ω k )   0

where ζ ik is the complex number defined by ζ ik = ζ ik + jη ik. Here ζ ik and jη ik are mutually independent Gaussian probability variables. Using random variables below: E[ζ ik] = E[jη ik] = 0, E[ζ2 ik] = E[jη2 ik] = 0.5. The wind pressure ζ ik was obtained in terms of complex Fourier coefficient Xik , determined from i −1    X ik − ∑ H ijkζ jk    j =1  1.14 ζ ik =  H iik

They were drawn conclusion that the high sporadic negative pressure obtained in the wind tunnel test to observe the wind pressure at the edge of the roof. The power spectrum of the fluctuating wind pressure obtained from the simulation is in good agreement with the experimental done. From the experimental above can be described that is significant vertical displacement occur effected by wind load to the structure as well as resulted wind pressure distribution. Marcelo Rocha, Sandro Cabral, and Jorge Riera (2000) performed experimental research on wind tunnel testing as well as numerical methods, which is used Proper Orthogonal Decomposition (POD) method and Monte Carlo simulation of the tenstar cable roof structure of the La Plata Stadium,

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

in La Plata, Argentina. In addition, the impressive of shape roof model presented to cover the stadium, which is divided into two alternatives model

solution

complete

and

of partial

cover of the field.

(A)

From

the

experiment

conducted,

they

evaluated the mean wind pressure fluctuation, and standards deviations of the pressure field.

(B) Fig. 2.13. La-Plata Stadium by Rocha cs, & included fig.2.14-15

The POD method explained vector of wind pressure time series measured in n given point of a surface as p (t) = [p1 (t), p2 (t), pn (t)] and associated by µp = [µp, µp, µp]. Second statistical moments of wind pressures in the form of a covariance matrix: Cp = Sp Rp SpT, Where  σ1 0   . Sp =   .  .   0

1.17

σ2 .

.

. . 0

. .

.

0 1  ρ    21  .   , Rp =    .  .  .   . σ n   ρ n1

ρ12 . 1

.

.

.

. .

ρn2 .

. . ρ1n  . . ρ 2 n  .   . .  . .   . . 1 

are the diagonal matrix of wind pressure standard deviations and the symmetric matrix of correlation coefficient, respectively. Aware from the correlation coefficients, to define a zero time gap, that is,

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

Cp,ij=ρijσi σj = E {[pi (t) - µI][pj (t) - µj]} = E {pi (t) pj (t)} - µi µj

1.18

The correlation coefficient matrix can be subjected as Orthogonal Decomposition, which is accomplished by solving eigenvalue-eigenvector: Rpzj = λj zj

1.19

Where the n solutions (λj zj), j =1, 2, n. used to assemble the matrices     Λp =      

λ1 λ2

0 . .

. .

.

.

0

0

. . . .

.

.

0   z11   ρ   21    , Zp =  .   .   .     z n1 λn 

z12 z 22 . . . zn 2

. . z1n  . . . z 2 n  . .   . .  . .   . . . z nn  .

Λp is diagonal matrix of the square roots of the eigenvalues; λj and Zp are the corresponding orthonormal eigenvector, zj. Reconstitute of the correlation matrix as Rp = (Zp Λp) (Zp Λp)T

1.20

The optional covariance matrix as Cp = (Sp Zp Λp) (Sp Zp Λp)T

1.21

On the other hand, general derived equation of Monte Carlo simulation was also presented in this research, which is similar to the previous research above. They drawn discussion by mean of theoretical and practical example, which was depend on whether a global or a local structural response to wind pressure loads required. The POD method presents a fast convergence rate for global response, while for local response the effective correlation length of the pressure process taken into consideration on Computational Dynamic Fluid (CFD).

Fig.2.15 Standard Deviation of wind pressure,

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

Fig. 2.14. Mean wind pressure, α=1800

α=1800

Ando, Ishii, Suzuki, Masuda, Saito, (1999), was also performed wind tunnel test on construction of a double membrane air supported structure, which are consisted of cable suspended, membranes roof and few steel part. Since lightweight of long span structures can be built in Japan (1997), it is important to estimate wind-induced response to structure. They built a 1/500 scale model to validate the design of “Ukigumo Dome”, which resembles a floating cloud. The main roof of the dome is a cable reinforced double layer air-inflated as hybrid double-membrane air supported structure (fig.2.16). They were conducted wind tunnel test with velocity pressure setting at q= 301.8kgf/m2, which is designed load of return period of 500 years in Kumamoto. From the scale model, they obtained wind pressure coefficient around the roof, (fig.2.17). The coefficients of the windward, leeward and central sides are determined to be –1.1, +0.15 and –0.4, respectively.

Fig. 2. 16 Hybrid double-layer system by Ando,cs

Fig. 2.17. Wind pressure coefficient Distribution.

They were concluded that is the first building was constructed the movement roof with double membrane air-supported structure. It is obvious that wind loading was very important influence to the membrane structure since large area surface of roof exist and historical record of relatively highspeed wind occurred. Due to the cable-suspended structure is flexible, it is highly important to understand the response to wind loading for this research. Any prediction about behavior of cable suspended roof combined with membrane structure can be involved so that should be studied, in order to recognize the distribution pressure of passing wind on these structures.

2.3 Fabric membrane Structures

Wind Loading on a Fabric Structure

30

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

For many years fabric membrane structures has been used for a main component building structure. The roofing system would perhaps have used animal skin to form tents, which are considered perfect example of an advance structure of a membrane roof with dominantly tensile force. In between the tension members would have membrane attached or stretched over the boundary surface fitted. (Prem Krisnha 1978, p.1). The earlier structures (tents) built is not yet have improvement. It is because of the impermanent nature. The technology developed the fabric membrane and new material involved so that they have begun to be perceived as architecture and engineering structure. Sophisticated construction technique and complexity of requirement would be introduced as modern tensile membrane engineering system of portable as well as permanent building at the present day. “Perhaps the most high-profile building to be erected in UK this century is the Millennium Experience Dome,” which are using material cable, fabric membrane and several steel erections. The dome is an understatement of 320 meters in diameter, over 100 meters to the top of the mast, and more than 1000 meters around its circumference is using PTFE-coated fabric and galvanized cable.

Fig.2.18. Millennium Dome by Kronenburg,

(A)

(B)

The form of spherical tensioned fabric cap was taken by the enclosure. Tensioned steel cables arranged radially on the surface of structure is supported the skin-membrane, supported and braced from the columns by hanging and tie-down cables at 2 meter interval. (Kronenburg, 2000, p.13-14). Portable architecture using membrane structures are rose that is not only due to increased performance and longer lifetime but also because the adaptation of computer-aided design as well as computer program package to support the design structure. However, for the simple point, these structures are still designed to resist the basic loading criteria as conventional building. The original dead load of these structure relatively small, however

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

the imposed live load and wind pressure much more significant to influence the structural design. Wind pressure would be relatively difficult to generate on the building, which has complex geometries. The shape of structure basically governed by the physical principles that are begins with produced a stable structures with the membrane surface should have double curvature and defined mathematically as a hyperbolic paraboloid. The geometry of the membrane is established through a shape generation technique to ensure static equilibrium of the system. (Birdair Technical Info, 2001). The relationship between basic loading system and generated geometry of membrane roof should be involved in order to collect data loading on design structure. The advantage and appeal of fabric structures are because the lightweight efficiency in long span application and not easily constructed. The typical materials involved are PTFE (Teflon)-coated fiberglass, silicone coated fiberglass and vinyl-coated polyester that are inherently waterproof and require little maintenance. PTFE is chemically inert, resistant to moisture and microorganisms and has low deterioration. There is no bending and shear stiffness of cable combined with membrane occurred due to they rely on their form and internal pre-stress alone to perform the same function. Since they depend only on internal tensile forces, there are relatively simple equation would be under laid. (Shaeffer, 1996, ix). “Designer often attracted to fabric structure are intrigued with the wide range of forms which ca be built. Although the range of possible forms is extensive these are not ‘free-form’ structure.” Conforming to the physical principals must be governed is because behavior as limited characteristic of the material form. In tensile behavior, combining or weaving numerous linier tension components generally makes membranes. That is mostly using cable suspended with the membranes structurally and visually acts as tension surface. (Shaeffer, 1996, p.5.1-5.4).

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

Planar axial forces of membrane surface usually resist loads. Applying a pressure load at any point will tent to increase the tension in one direction and decrease it in the opposite. This will force the surface to deform until the axial force of the surface balance the applied load. Suction load will increase the membrane Fig. 2.19 Membrane in tension by Shaeffer

tension in other direction. (Figure 2.19)

Every point on a stable tension surface should be satisfied the axial equilibrium of: ∑ Fx = 0 , ∑ Fy = 0 and ∑ Fz = 0 Cone-like or hyperboloid surface are generated when a membrane is stretched between

two

vertically

displaced

concentric boundaries. The similar size and shape to the cooling tower form, or may be significantly different must supported tent forms to develop in order

Fig. 2.20 Hyperbolic surface of membrane

to associate to the membrane structure.

Simple physical model can be seen in figure 2.20, which the range of viable forms and proportions is significantly increased by the use of radial cable. Principle curvature generally follows meridional lines and an opposite sense of curvature was set as perpendicular to the meridional lines. The lightweight structure had been studied by Frei Otto (1967) that was mainly researched on model studies of shape of air-supported structures. There are wires tied over the membrane represent cable or net. The result of experimental research method is estimation of membrane stress that can be made on measuring the principle radii of curvature. Tension support can be introduced in the form of grid to reduce the radii of curvature of the membrane. The fabric membrane structure as much like a thin sheet that is light and has flexible nature. The wind load is dominantly affected to the whole of applying load on the structure so that is

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

necessary to gain the wind pressure acting to the structure. Kazakevitch, (1998) was performed the requirement of experimental investigation in the wind tunnel of wind load on the membrane roof.

Fig. 2.21 The hangar structural scheme by Kazakevitch. Fig. 2.22 Pressure distribution on the membrane roofing surface (a) the stage of erection, β=0; ε=0.5%on upper surface, 2-the net values on the upper and lower surfaces;(b), (c) The completed stage (on the upper surface); 3,4 in section a; 3-β=900, ε=0.5%; 4-β=900 , ε=8%,etc.e by Kazakevitch

The cylindrical membrane roof model at Riga Airport in Latvia determinate the wind pressure distribution and wind flow visualization over the roof surface in wind tunnel testing. Method to determine characteristic of natural dynamic of membrane system was described base on the complete Vlascov equations. The roof for hangar developed in the form of a cylindrical membrane roof of 108 m span and 60 m wide. (see fig. 2.21) The roof model was created on scale 1:250 and intended to determine wind pressure distribution along the upper and lower surface. The integral aerodynamic coefficients were calculated by means the integration over the appropriate surface:

C=

X 1 1 = p cos (n, x) d σ = ∫ qS qS σ Si

−

∑ pi cos (n1, x) ∆σ i

1.22

Wind Loading on a Fabric Structure

34

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

By analogy − 1 C = ∑ pi cos (ni , y )∆σ i S i − 1 C = ∑ pi cos (ni , z )∆σ i S i

1.23 1.24

where S is the area of the horizontal projection of the model roofing surface. It was measured of pi as the pressure mean aerodynamic coefficient at any point I, pi’ = pi/q defined as the net pressure, and the dynamic wind pressure; q=ρV/2,ρ the air flow density, V the velocity of the undisturbed flow, ∆σI the area of the element around point I, and ni the normal to the surface at point i. The result of the experimental was obtained on the wind pressure distribution along the upper and lower roofing surface. (see figure 2.22) Wind tunnel tests have proven useful to achieve design relevancy. Many kinds of membrane structures were also investigated to collect the relevant data. Irwin and Wardlaw, 1976 performed the experiment using fabric membrane of 1420 denier Kevlar 49 and 100 denier Kevlar 29 in wind tunnel testing that defined the independent of the elastic properties to the membrane (see figure 2.8-2.10). The Montreal Stadium was completed in 1987 using a polyurethane and PVC-coated Kevlar fabric. The details of full scale and model roof materials can be summarized in table below. Table. 2.1 Fabric

Full Scale

Model

1420 denier Kevlar 49 closely

100 denier Kevlar 29 woven as an open net

woven Airtight Coating

PVC on both side

High density polyethylene sheet on underside

Mass

Kevlar 1.1 kg/m2, Coating 1.1 kg/m2

Kevlar 0.009 kg/m2, Coating 0.006 kg/m2 Adhesive 0.005 kg/m2, Total 0.02 kg/m2

Total 2.2 kg/m2 Approximate KT

16 MN/m

0.14 MN/m

Approximate KS/KT

6.2 x 10-4

20 x 10-4

The result of experiment affected the natural frequency, and deflections under static load. The evidence released that the behavior of tensioned membrane in wind not sensitive to their elastic properties and very little strain of the membrane deflection. In addition, added mass is very

Wind Loading on a Fabric Structure

35

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

significant effect on the lightweight membrane and very sensitive to the effect of acoustic environment. Wind tunnel testing completed by Ando, Ishii, Suzuki, Masuda, Saito, (1999) on a double membrane air supported structure. The structural system involved is “Park Dome Kumamoto” which is the main roof of the dome is a cable reinforced double layer air-inflated membrane. The double membrane air supported structure is 107 m with conical trapezoide steel ring frame at the center maintains the thickness and shape of the air-supported structure. The 48 cables run radially at the upper and the lower regions between the ring frame and the exterior ring truss. The upper and lower rings have diameters of 10.6 m and 36.6m, respectively, and the ring frame is 14 m high. The

membrane

coated

glass

is

PTFE

fiber

fabric

covered those structure. The result of the experimental research on wind loading is distribution of wind pressure coefficient which they set the Fig. 2.23 Structural section, Park Dome Kumamoto (1999)

velocity pressure at q = 301.8 kgf/m2.

The wind pressure coefficient of the roof is collected. (See figure 2.16-2.17, page 13). To gain a better understanding of the behavior of these types of structure so that need more wind tunnel testing should be advocated. This kind structures was the first building with double-membrane air-supported structure, which is the original system applied as lightweight and built as a long span structure. That is mean large areas membrane surface roof will invoked by mostly wind loading that is need more investigation on every time want to build new structure. Thus, it will led the further study on wind load induced on membrane structures by such as procedure of experimental or numerical methods.

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

2.4 Computational Fluid dynamic (CFD) for Wind Loading Campbell (2000) on in

Boundary & Support Definition SHAPE (Form Finding)

his paper provided an overview

Pattern Boundary Definition

of the utilization of computing

SHAPE Architecture Evaluation

in the design and construction of tensile membrane structure. This paper pursued general methodology in the design and construction membrane

of structure

SHAPE (Pattern Surface)

Prestress Evaluation

PATTERN (Cutting Templates)

Support Structure Model Initial Element Size

ANALYSIS

tensile is

Presstres Shape

STRUCTURAL Evaluation

Element Size

illustrated in figure 2.21. It can be described from flow chart that is wind loading became

Erection/Stressing Sequence Difinition

Joint Design Detailing

DESIGN (Element Size)

Analysis

highly order demand to the analysis design structure, so that will be regarded in all

Typically Nonautomated Process

Typically automated Process

process design. Fig. 2.24. General Approach Design Tensile Membrane Structure by Campbell (2000)

A part from general approach design flowchart is may be prescribed boundary pre-stress to patterning of shape of structure generation. It is belief that would be easier to generate a shape module due to the ability of the digital computer as well as to analyze the system structure. It is highly recommended to pursue CFD method become a part of approaching design to predict wind loading to the structure, it is because the wind load basically very significant would applied mostly to those kind of structures. It is connected closely to the geometry modeling in CFD code with kinds of the design structures to develop related to fabric membrane. For the first time, CFD introduced in aircraft industry and on aerodynamic industry related. In ship design, CFD methods were mainly focused to integrate the geometry and analysis of ship. CFD methods to be used in close combination with wind tunnel testing by Voogt (1990) to investigate

Wind Loading on a Fabric Structure

37

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

during development phase of Fokker 50 &100 project Aircraft prototype and to analyze candidate shapes of airplane. The basis of CFD analysis can reduce significantly the number of wind tunnel model test. Furthermore, the analysis of flow in a free flight environment cannot be obtained on wind tunnel analysis. However, the combination of both methods is essential to reduce design cycle times and the potential development risk. Any attempts to gain a better understanding of critical flow phenomena using CFD methods may be extended predict wind behavior to fabric membrane structures. “The aerodynamic design process is aimed to get a number of aerodynamic requirements under certain geometric constraints. In the computational cycle a configuration is optimized for a set of selected parameters at the design condition”. Results of the computational can be visualized on graphic terminals, figure 2.25. In order to running the program, there are closely connected with CFD is geometry modeling. CFD has been a vital element in the design of the Fokker 100 wing, which one of example visual post-processing illustrated a changing isobar pattern on a wing. The relevant result of research to this study is indication of pressures computed as illustrated in figure 2.26.

Fig. 2.25 Visual Post-processing, Voogt (1990)

Fig. 2.26 Stubwing and pressure measurement Voogt (1990)

Prediction of wind effect on building surface was conducted by Mikkelsen and Livesey (1995) with concern on comparison of computed result to wind tunnel test and full-scale measurements. The research conducted evaluation on numerical K-ε model Kamaleon II, to predict wind pressure on structure surface. The 3D domain of 0.6x0.6x0.6 m3 corresponded to cube of 0.05x0.05x0.05 m3 was generated on computer program. The pressure distribution was calculated on

Wind Loading on a Fabric Structure

38

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

different wind angle approached of 00, 50, 150, 250, 350, & 450, and one sample pressure coefficient presented in figure 2.28 P − Po Cp = 1 ρU 2 2 o

1.25

where Cp = pressure coefficient, P=local pressure, Po=reference pressure, ρ=air density and Uo=free-stream velocity.

Fig. 2.27 calculation grid in a close vicinity of the cube, Mikkelsen & Livesey (1995)

Fig.2.28 Cp value shown as isobar for an angle of 00 Mikkelsen & Livesey (1995)

The calculation was a steady-state, which the isothermal flow condition consisting of 45 x 39 x27 cells (see figure 2.27). The density was higher in the vicinity of the house and a particular cell matched each pressure tap location in the full-scale model test. One example pressure distribution can be seen in figure 2.28. Comparison between full-scale and numerically

prediction

can

be

described in figure 2.29. The pressure distribution

Cp

result

indicated

closely match with the wind tunnel test and either numerical prediction of pressure distribution as well as the drag coefficient indicated slightly higher value than those by wind tunnel experimental. Fig. 2.29 Comparison between full-scale, model scale and numerically predicted Cp for h/z0= 180

Wind Loading on a Fabric Structure

39

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

It is significant to pursue the literature of CFD in order to understand of how the data obtained in the way of correlation to the real world condition. The attempt to qualify of wind-induced pressure on building, place in extraordinary geometry is crucial so that need to under take suitable model approach. One example research presented the global force acting on an obstacle to define the flow field at the symmetry plane by Lakehal (1998).

Fig. 2.30 The computational grid in close vicinity of the obstacle, Lakehal (1998)

Fig. 2.31 pressure coefficient distribution at the sysmetry plane, Lakehal (1998)

He admitted research on the Kupka building with reduced in model scaled of 1:200 with a sharp round-walled geometry provided by 1/10 building height. It can be seen in figure 2.30, the developed mesh generation of the Kupka building model and one of example result of coefficient pressure distribution can be seen in figure 2.31. Generating CFD codes on the model structure is used base on Reynolds-average Navier-Stokes (RANS) equations and large-eddy simulation (LES). In this research, the modeling app roach based on solving RANS equations, using standard version of k-ε turbulence model adapted for airflow simulation. The three dimensional steady-state flows are described by the Navier-Stoke equations, which express mass and momentum conservation. Reynolds averaging procedure and the eddy-viscosity concept, which obtained a system equation expressed nonorthogonal co-ordinate system ξi = (ξ ,η , χ ) ∂U m = 0, m = 1, 2, 3 ∂ξ m

1.26

(

)

 1 ∂U kξ j m   1 ∂ U iξ m  ∂  ∂ pβ im ξ β j  = − + Γ βi β k  U m U i − Γ   ξ ξ ξ ∂ξ m  J ∂ ∂ J ∂ i m j    

127

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

where U is the contravariant velocity expressed in term of the covariant components U ξ by Um= β ijU ξj and β ij the cofactor of

∂ξ jp

∂ξi

in the Jacobian (J) of the transformation ξ i ⇒ ξ i p , which

reads  ∂ξ mp     ∂ξ   ∂ξ p  J =  m , m = 1, 2, 3,  ∂η   ∂ξ p   m  ∂χ 

1.28

p=p/ρ + 2/3k represents the increased pressure (p=pressure, ρ= fluid density and k = turbulent kinetic energy), and Γ = v + vt the effective viscosity (laminar + turbulent). The eddy-viscosity is determined according to the algebraic expression vt =

Cµ k 2

ε which involves the turbulent scalars (k) and its rate

of dissipation (ε). In k-ε model, the turbulent quantities k and ε are obtained from transport equations and describing mean flow  1 ∂φ m  ∂  β j  = J ( Sφ+ − Sφ− ) , m=1,2,3 U mφ − Γφ   ∂ξ m   J ∂ξ j 

1.29

φ stands for either k or ε, where the net source/ink part are given. For k and ε equations by (G-ε) and (C1G –C2ε)ε/k, the diffusion coefficients Γφ by Γ/σφ. G represents the rate of production of turbulent kinetic energy resulting from the interaction of the turbulent motions and mean flow. ∂U ξj n   ∂U ξ v  ∂U ξ 1.30 G = 12  i β nj  i β nj + β i  , m=1, 2, 3  J  ∂ξ n ∂ξ n  ∂ξ n 

The empirical constants are assigned the standard values, so that Cµ=0.09; C1=1.44; C2=1.92; σk=1; and σε=1.3. Those are the basic formulation of Navier-Stokes applied to the model turbulent in CFD. Numerically prediction pressure distribution using CFD indicated fairly well performance pressure coefficient or non-dimensional data respected to the inflow wind profile. The result of research is pressure coefficient distribution (Cp= (p-po)/1/2ρUB2 , where po = reference pressure) on windward and leeward side of model which was compared between computational method and the experimental result, (see fig. 2.32) and also the example of pressure coefficient distribution can be seen in figure 2.31.

Wind Loading on a Fabric Structure

41

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

In their conclusion, the RANS equations predicted airflow features and induced loads on a threedimensional building-like model with complex boundaries. In the light of different results, the numerical procedure can be used to simulate realistically pressure-induced effect of a turbulent flow over a building. So that means, in this study, the behavior of wind load acting on the fabric membrane structure hopefully could be predicted. The promising aspect of CFD procedure to further investigation on wind behavior can be adapted. In order to achieve successful application on model used the CFD, it needed concerning on mesh and turbulence modeling. Regarding to the assessment, Kim and Boysan (1999), performed turbulence modeling, which is determined the fidelity of computational on the environmental application. Using CFD software FLUENT, they employing structured mesh for complex geometries that is often made a very difficult of adequate mesh structure applied. However, unstructured meshes have been generated over the typical building group using commercial preprocessors. The turbulence models to be discussed based on Reynolds-averaged Navier-Stokes (RANS) equation and large eddy simulation (LES) in lieu of increasingly important role it plays.

Wind Loading on a Fabric Structure

42

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

The issue of turbulence modeling of computational simulation for environment application presented, which was applied environmental flow such as atmospheric boundary layer to a smooth terrain and bluff bodies. They tried to explore the principle the needed of mean wind speed data and atmospheric turbulence data to accumulate accurately of presented atmospheric wind Fig. 2.33 Unstructured hexahedral meshes around typical building configuration, Kim & Boysan (1999)

and its effects on building and structure.

The complex model environmental such as topography around building, the flows of 3D, and the flows encountered in urban areas have tried to resolve. (Fig.2.33) Using the commercial CFD software FLUENT resulted prediction pressure and skin-friction distributions as ell as the periodic vortex shedding in turbulence flow over a square cylinder which was compared with the k-ε turbulence model on surface-mounted cube. The numerical approach predicted

pressure

distribution

on

surface

structures, prediction of flow

over

the

curve,

production of turbulent kinetic

energy and

to

predict periodic of vortex shedding Fig. 2.34 Flow over the curved two-dimensional hill- predictions using four different turbulence models, bottom left: pressure distribution and bottom right; skin-friction distribution. Kim & Boysan (1999)

in

turbulent

flow.

They also defined that is significant improve the accuracy of numerical solutions for turbulent flows and the CFD demonstrated potential economical solver proposed for turbulent models. In their conclusion, the major issue that determine successful application of CFD to building aerodynamics that the unstructured mesh has a great potential to significantly save time and effort for mesh Wind Loading on a Fabric Structure

43

,

University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

generation. Large eddy simulation will play an increasingly more important role, especially in dealing with turbulence modeling issue. In this study case, LES would be delighted to try in generating unstructured mesh on membrane structure model. It is due to the LES shown more significantly improve the accuracy of numerical solution for turbulent flows. Muakami, (1997&1998) conducted research on CFD method, which the new trends in turbulence models for Computational Wind Engineering (CWE) is presented. Since CWE is known as a difficult problem to analyze of the flow around bluff body by CFD, he admitted research on current status and future trends in computational wind engineering with some reviewing another research to be compared and an overview of turbulence model applied in CWE-1997. One example from several research is the basic shape of the rectangular cylinder or bluff body that used model rectangular cylinder with D/B and H/D (B: breadth, H: height and D: depth). He performed research that belief about analysis of bluff body flows by LES (Large Eddy Simulation) can predicted the flow around it much more accurately than the k-ε model does. It can be seen in figure 2.36. In this term, a very confident appreciation given, which is decided the LES method shows the best reproduction of experimental data, next is the RSM (Reynolds Stress Model), and the k-ε model gives the poorest result. “In the recent LES computations the conventional Smagorinsky SGS (subgrid scale) model has replaced by the dynamic SGS model. The development of the dynamic SGS model is one of the most significant improved in the world of CFD. The appearance of dynamic LES makes it possible to predict the velocity and pressure field around a bluff body with higher accuracy.

Fig. 2.35 Distribution of pressure coefficient (Cp) on 1:1:0.5 building of conical vortex at the roof corner predicted by revised model k-ε ( k-ε−φ model by Kawamoto, 1995

Fig. 2.36 Conical vortex at the roof corner predicted by LES, by Murakami, 1997.

Wind Loading on a Fabric Structure

44

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

The new trends in LES were the improvement of a dynamic SGS model that was proposed recently by Germano with paper “A dynamic subgrid scale eddy viscosity model. 1991” and revised by Lilly “A proposed modification of the Gerrmano subgrid-scale closure method” 1992. These analytical models considered under realistic condition that the CFD estimation could gain sufficient result on wind-interaction problems. The standard Smagorinsky model (S model) has been widely used in the computation of LES. A simple eddy-viscosity type assumption is used for modeling the SGS stress: _

_

SGS stress τ ij =U iU j −U i U j ,

1.30

Eddy-viscosity model in S model: τ ij − 13 δ ijτ kk = −2v SGS S ij

1.31

v SGS = (C S ∆ ) 2 S

1.32

C S : 0.1 − 0.25 ,

1  ∂u ∂uj S ij =  i + 2  ∂x j ∂xi S = (2 S ij S ij )1 2

 ,  

1.33

1.34

In the standard dynamic SGS model based on the s model (DS), C (=CS2) is determined. The empirical model function fµ is required for damping vSGS in the area near the wall. v SGS = (C S ∆ f µ ) 2 S 1.35 +

f µ =1 − exp(− x n / 25)

1.36

The dynamic mixed model (DM) 2was proposed by Zang, 1993 and Vreman, 1994 as a linear combination of the DS model and the scale-similarity model revision . The basic DM model equations are shown in Eqs.1. 32 and 1.33 τ ij − 13 δ ijτ kk = − 2v SGS S ij + Bij − 13 δ ij Bkk = −2C∆2 S S ij + Bij − 13 δ ij Bkk Smagorinsky Model

1.37

scale − similarity mod el

Bij = u i u j − u i u j

1.38

In order to determine coefficient C in LES, the model of SGS stress τij and procedure for determining coefficient C can be seen below.  ∂ ui ∂ u j ∂ ui ∂ ui u j ∂ p ∂  + =− + + − τ ij + v ∂xt ∂x j ∂xi ∂x j  ∂ x ∂xi j 

   

1.39

Wind Loading on a Fabric Structure

45

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

(a) Base Model •

S model τ ij − 13 δ ijτ kk = −2C∆2 S S ij

1  ∂u ∂uj S ij =  i + 2  ∂x j ∂xi S = (2 S ij S ij )1 2 •

1.40

   

1.41 1.42

Scale-similarity (Bardina) model τ ij = Bij

1.43 1.44

Bij = u i u j − u i u j •

Filtered bardina model 1.45

τ ij = Lij + C ij + C B (u i − u i )(u j − u j )

•

Lij = u i u j − u i u j

1.46

C ij = u i (u j − u j ) + (u i − u i )u j

1.47

Mixed model τ ij − 13 δ ijτ kk = − 2C∆2 S S ij + Bij − 13 δ ij Bkk

1.48

(b) Procedure for determining C ( = C S2 ) •

tuning optimizing CS according to flow characteristics based on numerical experiment (standard S model; CS=0.1 (channel flow) ~ CS=0.25 (isotropic turbulence))

•

dynamic procedure with double filtering Germano identity $ ij = Tij − τˆij = u i u j − uˆ i uˆ j

•

$ ij : Re soled stress

Lilly’s least-square method (optimization of C at each point) 1 $ij M ij C=− : DS mod el 2 M kl2

1 M ij ($ij − H ij ) : DM mod el 2 M kl2 M ij = ∆ˆ 2 Sˆ Sˆij − ∆ˆ 2 S Sij

C=−

ˆ ˆ H ij = uˆiuˆ j − uˆi uˆ j − (ui u j − ui u j )

1.49

1.50

1.51 1.52 1.53

Wind Loading on a Fabric Structure

46

,

University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

•

Ghosal’s localization model C ( x) = ∫ K ( x, y )C ( y )dy + f ( x)

[

]

1.54

where [ ]+, denotes the positive part K and f are defined as functions of α = 2∆ˆ 2 Sˆ Sˆ and β = 2 ∆ˆ 2 S S ij

•

ij

ij

ij

Meneveau’s Lagrangian dynamic model 1 I LM C ( x, t ) = − : LDS mod el 2 I MM

C ( x, t ) = −

1 I LM − I HM 2 I MM

1.55

1.56

: LDM mod el

1.57

t

I LM = ∫ $ ij (t ' ) M ij (t ' )W (t − t ' )dt ' −∞ t

I MM = ∫ M ij (t ) M ij (t )W (t − t )dt '

'

'

1.58 '

−∞ t

1.59

−∞

1.60

I HM = ∫ M ij (t ' ) H ij (t ' )W (t − t ' )dt ' W(t-t’) : weighting function.

xi

three component of spatial coordinate (I=1, 2, 3; streamwise, lateral, vertical (or spanwise))

f

time-averaged value of f

S ij

strain-rate tensor

S

  1  ∂ u i ∂ u j scale of strain –rate  = + 2  ∂x j ∂xi  

ui

three component of velocity vector

p Uo

pressure time-averaged value

Cp

instantaneous pressure coefficient

   

2

    

of u1 at the inflow boundary for the case of 2D square cylinder.

Cp = ( p − < p o >) /( ρ U o2 / 2)

<po> reference static pressure vt eddy viscosity

k = u i' u 'j / 2

k

turbulent energy,

ε

dissipation rate of k

u i' u 'j

Reynolds stress

The S model is so simple and well designed that is has been applied to many flow fields and has attained great success. The conclusions are drawn that pointed at the advantages of dynamic LES over

Wind Loading on a Fabric Structure

47

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

the standard LES. It is very promising for accurately predicting the flow around a bluff body. Another conclusion are the CWE applications reviewed that the difficulties in applying CFD to wind engineering problem are caused by 1) large Reynolds number, 2) impinging at the front wall, 3) sharp edges of the bluff body, 4) remaining effect of flow obstacle at outflow around a bluff body, etc. An evaluation of problems is based of proper choice of turbulent particularly in CWE. The basis measures for the selection and evaluation of turbulence model are 1) prediction accuracy and 2) CPU time required. One disadvantage of using LES is that too much CPU time is required. Since that happened rapid evolution of CPU hardware needed to overcome the restriction and application of LES to CWE problems is realized in near future in widely. Another research has represented by Tamura, Kawai, Kawamoto, Nozawa, Sakamoto and Ohkuma, (1997) of numerical prediction of wind loading on buildings and structures using CFD related to large eddy simulation (LES) and the k-ε model for turbulent flows. In AIJ (Architectural Institute of Japan ) concerned on model of a low rise building with (breadth: depth: height = 1:1:0.5) have been computed by a member of working group measured the flows and the pressure around it. A half cube on a flat plate was adopted as a computational model of a low-rise building. (Fig.2.37) Regarding to the current status of CFD technology in wind engineering, they submitted a questionnaire to the wind and structural engineers in research institutes and private companies that they obtained conclusion of the CFD technique is widely

Fig.2.37 Computational model of the AIJ project by Tamura,cs.

.

Wind Loading on a Fabric Structure

48

,

University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

used for application in wind engineering, especially environmental problems and structural engineers are planning to use the

CFD

technique

for

wind-load

estimations. In this research, one of testing is concerned to the sub grid-scale model of LES. Fig 2.38 by Tamura,cs

One of the example case is adopted the standard Smagorinsky model and another one is adopted the dynamic SGS model.

Fig 2.39 by Tamura,cs

The result of the two LES cases has not deviated from each other, however they shown a different result in the space-time cross-correlation of fluctuating pressure on the roof.(Fig.2.38 and 2.39). The conclusion drawn that in the case of LES, the numerical scheme has an important role for the computed result so that CFD technique could have reliability to predict wind loading on building and structures from view points of numerical accuracy and computational costs. Su, Tang, & Fu (1997) were conducted research that analyzed fluid flow and thermal performance of a dry-cooling tower under cross wind condition. Numerical simulation using finite element volume (FVM) is 3-D structure shape generated on NSRT (Numerical Simulation in Turbulence Research) software, which has been developed by them of CFD method. The Heler’ cooling tower model invoked cross wind that resulted turbulence flow around and pressure contour. It is concerned to the temperature distribution, which air is played role over the tower. However, small portion of wind behavior to the surface of cooling tower have been presented. The horizontal crosswind acting to the tower affected that has resulted contour pressure distribution. It can be seen in figure 2.42.

Wind Loading on a Fabric Structure

49

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

Fig. 2.40 Heler-type tower, by Su,cs.

dry-cooling Fig. 2.41 Computational region and coordinate system, by Su, cs.

The contours of pressure in the horizontal plane Z=9m indicated lower

pressure

due

to

the

pressure close to the side surface of the tower and because of the large velocity of air.

Fig. 2.42 Contour of pressure in the horizontal plane (Z=9m, cross wind speed of 5 m/s), by Su, cs.

The results were compared with the respective experiment and the agreement is satisfactory. It is more concerned to the heat transfer on tower. Although the research is done for the heat transfer, however it can gave inspiration of general type of dry cooling tower model and has correlation to the shape of the membrane structure model.

2.5 Wind Tunnel Test A theoretical calculation of the wind load on a structure is quite difficult, which were the fundamental equations generated on the mechanics of airflow are very complicated and so many parameters in boundary conditions for the system of equations. Even nowadays, since the evolutions of advanced computer introduced, there are very few cases can be obtained on numerical calculations of wind loads on structures in turbulent flow. An actual and most accurate measurement for determining wind loads will be on full-scale structures that is belief impossible in practice. So the most appropriate method for that is using model tests in a wind tunnel. (Dyrbye, & Hansen, 1997, p. 177)

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

“The purpose of wind tunnel tests is to provide designers with information on local pattern, wind loads, and wind-induced structural vibration having an accuracy far exceeding that can be obtained from predictions based on other less expensive means such as theory, numerical analysis, expert judgment (consulting), and so on” (Liu, 1991, p.147). The use of wind tunnel tests steadily increased to improve design on many kind of structures shape. Due to the distributions of wind pressure and flow pattern around the structures may not be given by building code and standard as well as from any other source, so that the wind tunnel was the only way to generate the information. The wind tunnel test may be considered only use low-speed wind tunnels because it makes possible use model that can be prepared early in design cycles and to minimize cost. To gain a better understanding of wind loading on two model structures such as sphere/dome and cooling tower model, there are some references could be involved. Distribution of local mean pressure coefficient on circular cylinder and hyperbolic cooling tower represented on wind engineering by Liu, 1991, p.92-97, that explaining about the similarity of pressure distribution around both the structures later. The average value derived is the data from several studies (full scale measurement) described in ASCE (1987). As the basic consideration of wind force, that is defined a stream of air moving at velocity V exerts a force q per unit area, where q is dynamic head of air expressed below:

q = 12 ρ V 2 .

1.61

The total pressure remains constant at all the points that is stated by Bernoulli’s equation of 2

p1 + 12 ρ V1 = p2 + 12 ρ V2

2

1.62

where p1, p2 are the static pressures at two points in the air stream, ρ is the air density, and V1, V2 are the corresponding air velocities. (Sachs, 1978, p. 2) The flow pattern of incompressible flow around circular cylinder perpendicular to the flow depend on the Reynolds number that is expressed as Re = ρ VD/µ

1.63

where ρ is the density of the fluid, V is the velocity of the fluid relative to the cylinder, D is the cylinder diameter, and µ is the dynamic viscosity of the fluid. Local mean pressure coefficient can be derived by the pressure p at an arbitrary point on a structure in non-dimensional as follows: p 1.64 Cp = 1 2 ρ V 2

Wind Loading on a Fabric Structure

51

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

Where Cp is the dimensionless pressure (pressure coefficient). The pressure p is measured above ambient pressure. The velocity is that at a reference height, and ρ is the density of air. To understand the pressure fluctuation at various parts of a building, the good correlation of pressure occurred between windward external and the internal pressure produced by a windward opening. The pressure fluctuations on a building caused by free stream turbulence carried in approaching and the signature turbulence by the structure itself. The theory said that the building encounters a slowly varying large eddy in the wind, the flow around the building at any time t regarded quasi-steady. In this term, the pressure p at any location varies expressed: ρ V 2 (t ) p (t ) = Cp 2

1.65

Where p (t) is pressure at time t and V (t) is the free-stream velocity at t. (Liu, 1991, p.72-91.)

2.5.1 Wind Tunnel Techniques Wind tunnel test on a structural model are needed when the full-scale structure difficult to analyses. The use of wind tunnels is to determine the response of a structure to wind forces and to ascertain the pattern of wind flow to leeward of a structure. Investigation is carried out on the eddy formation behind bluff bodies to find the frequency and strength of oscillatory forces; on the structure of a turbulent air-stream, and on the simulation of natural boundary layer effect.

In general the wind tunnel test developed for aircraft work that is also suitable for structural model testing. Basically, two types of wind tunnel known that are open jet and closed jet. In the open jet tunnel the working section, where the model situated, has no side walls so that the air-stream is spilled out by model, and the force and pressure reading are artificially low. The closed jet tunnel has sided wall, constraint the air flow past the model, so that forces and pressure are artificially high. Measurements are made by conventional instrumentation, such as force and moment balances and pressure manometers and the stiffness and damping of flexible structures is either simulated in the model or by mounting a rigid model on springs with eddy-current damping. (Peter Sachs, 1978, p. 95) Guidelines for wind tunnel experiment (Barlow, Rae, & Alan Pope, 1999, p. 460-462) can be described that is a rather commonsense listing of requirement for initiating and executing a successful

Wind Loading on a Fabric Structure

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

aerodynamic experiment as the following step below. It was included block diagram adapted from AGRADAR 3043 in figure 2.43. 1. Clearly state the problem being addressed and define the purpose of the experiment. A clear statement of the problem being addressed will often critical in obtaining efficient application of their professional knowledge and skills or in avoiding a serious misunderstanding about what persons involved in the planning and execution of the experiment in sufficient time so that they can be mentally and physically prepared. The expected result from an experiment must have associated expected accuracy and precision that are the minimum goals in order that the objectives can be met. These accuracy and precision requirements should be a part of the problem statement. Maximum advantage must be taken of results from previous experiments, theories, and computations, as they are available in the professional literature or from corporate records. D e f in e d P u rp o s e s o f E x p e rim e n t & R e q u ire d A c c u r a c ie s o f th e R e s u lts

D e s ig n -R e q u ire d o u tp u t p a ra m e te r s -M e th o d s to e v a lu a te u n c e rta in ty -T e st m e th o d s -I n s tr u m e n ta tio n n e e d s

No

th e E x p e rim e n t - A c c u ra c ie s r e q u ir e d to m e e t n e e d s - M o d e l c o n fig u ra tio n - M e a s u r e m e n t re q u ir e d - C o r re c tio n s r e q u ir e d a n d m e th o d s

E n u m e r a te e r ro r s o u r c e s a n d e s tim a te e f fe c ts o f u n c e r ta iin tie s o n re s u lts Y es C an Im p ro v e m e n ts be m ade?

No

No

R e s u lts A c c e p ta b le ? Y es

S to p s e a r c h f o r A lte r n a tiv e to th is te s t!

P ro c e e d w ith P r e p a r a tio n s

S ta r t te s t a n d m o n ito r d a ta

No

M easu rem en t P ro b le m ?

No

Y es

Y es

S o lv e P ro b le m D o c u m e n t O u tc o m e I n c lu d e f in a l e v a lu a tio n o f u n c e rta in ty w ith ra n d o m a n d s y s te m a tic c o n trib u tio n s id e n tif ie d a n d q u a n tifie d .

R e s u lts A c c e p ta b le ?

C o n tin u e T est

Y es

P u rp o se A c h ie v e d ?

Fig. 2.43 Experimental planning and execution process diagram.

2. Identify the outcomes needed, including the ranges of values of parameters that will provide the information to resolve the problem. This will imply a range of operating states and

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configuration geometries. There will imply an accuracies and precisions associated with each variable or parameter that should be identified. 3. Identify feasible model provisions and compatible facilities. This will require conceptual and preliminary design of the models and fixtures. It will require identifying any wind tunnel boundary corrections to be applied along with tare, interference, and other data corrections. It will require assessment of the impact of these choices individually and in sum on the accuracy and precision of the outputs. 4. Prepare run schedules and configuration change implications. Embedded in these decisions will be the degree to which replication, randomization, and blocking can contribute to the enrichment of the data to be obtained. Compare the resources needed and resources available. Iterate step 1-4 until a match is obtained. Prepare a clear guide for the conduct of the experiment. Make sure all persons involved understand the required actions procedures. Make sure all persons, materials, models, instrumentation, and software will be available at the time and place for executing the experiment. 5. Initiate the experiment. Provide for monitoring of all processes and data gathering. Include process evaluation of achieved accuracies and precisions of measurements. 6. Conduct data analyses to provide quantitative evaluation of the achieved accuracies and precisions. This information should be provided to the aerodynamicists and other project personnel as a part of data package so that the product decisions can include appropriate consideration of outcome uncertainties. AGRAD AR 3043 contains an extensive example of an application to forces and pressure test that the data flow diagram included in figure 2.44.

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Experimental Aerodydnamic Coefficient

Model Deformation Corrections

Boundary & Support Corrections Flow Angle Calibrations Math Model

Buoyancy

Tares Prior Data

Positioning Systems

Encoders

Temperature

Inertial Forces

Model Forces

Balance System(s) Angle Sensors

Other Transducers

Temperature Sensors

Model Pressure

Pressure Reference

Data Acquisition

Wind Tunnel Environment

Fig. 2.44 Representative data flow.

There will very useful for a planning process an experiment with emphasis on the inclusion of uncertainty evaluation. The model would be developed in wind tunnel usually of ideal as small as possible, but there are obvious because of practical limitation. These are not always due to the difficulty of simulating fine detail such as no exactly sharp edges on our modeling. In practice, the model size is determined by sensitivity of the balance in force and moment measurements, and by the size of pressure tubing and it’s positioning in pressure test. For the flexible models the size is determined by comparative mechanical properties of the model and full-size materials. (Peter Sachs, 1978, p. 105)

Uematsu, Yamada, Inoue, and Hongo, (1997) performed a different type of model testing with the intention on wind-induced dynamic behavior to a rigidly jointed single–layer lattice dome with a long span. The dynamic response of nine latticed domes with a span of 120 m was analyzed in the time domain. They concerned the experimental method of characteristic of fluctuating wind pressure on domes and the mean pressure distribution as well as wind pressure coefficient (Cp & C’p). This type of structure has seldom been used due to many unsolved problem regarding the structural design and the wind-induced vibration is one of those problems.

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Since such a single-layer dome is a long span, the dynamic changes should be considered in the wind resistant. It is very different characteristic of material used from the study of fabric membrane structure, where the fabric membrane is usually light, flexible and tends to deflect and oscillate under turbulent wind force. The geometry of the wind tunnel models is schematically illustrated in figure 2.45 and one of example result of pressure distribution can be seen in figure 2.46.

Fig. 2.45 Dome geometry and coordinate system, by Uematsu.

Fig. 2.46 Distributions of the mean and rms pressure coefficient Cp and C’p H/D =1/4, by Uematsu.

They drawn conclusions of research, which the preset results, may give a reasonable basis for evaluation the dynamic response to the dome shape. The result of them can be used as a reference for further research of dome/sphere shape model structure. It very grateful to perusing the literature related to the wind loading with wind tunnel studies to collect information data experimental from the real structural modeled. However, in order to simplified, validation, comparison and many considerations that is important to pursue model experimental using computer program simulation. Letchford and Sarkar, (2000) performed wind tunnel test on rough and smooth parabolic domes, which simultaneous pressure measurement involved simulation atmospheric boundary layer flow. Mean and fluctuation pressure distribution have compared with earlier studies for similar shape and Reynolds number. The previous wind tunnel studies have undertaken by Maher’s classical study, Ogawa, and Taylor. The differences between them is that Ogawa and Taylor were presented measurement of fluctuating pressures and Taylor was the only one presented contour maps of maximum and minimum point pressure for hemispheres and truncated

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spheres. A complication of wind tunnel model studies of these types of structures is because the curved surface, which leads to Reynolds number effects. DU Reynolds numbers defined as ρ where ρ and µ are fluid properties, D is the base diameter

µ

and U is the mean velocity at the top of model height. The Reynolds numbers ware used in the range of each in dealt with Maher used R in the range of 6 x 105- 18x 105, Taylor dealt with R in the range from 1x105 to 3 x 105, and Ogawa also investigated a range of turbulent intensities with Reynolds numbers ranging from 1.2 x 105 to 2.1 x 105. The wind tunnel used is closed circuit, 1.8 m wide with a ceiling adjustable to ~1.8m. There is an upstream fetch of approximately 15m for developing appropriate simulations of the earth’s atmospheric boundary layers. A model dome was constructed with a base diameter (D) of 480 mm and height (h) of 150mm. The research was estimated pressure coefficient, which all pressure stated as non-dimensionalized by the mean dynamic pressure (1/2ρU2) at the top of the dome. ∆p is the instantaneous pressure difference between the surface pressure and a reference pressure in the wind tunnel.

Fig. 2.47 Tapping arrangement and wind direction definition for single dome test, by Letchford,cs

Fig. 2.48 Comparison of mean pressure coefficient along centerline of a smooth dome, by Letchford,cs

They were governed equation as: ∆p Cp = 1 Mean pressure coefficient, ρU 2 2 ∆p Cprms = 1 rms Standard deviation pressure coefficient, ρU 2 2

∆pˆ Cˆ p = 1 ρU 2 2

Mean peak maximum pressure coefficient,

1.66 1.67

1.68

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( ( ∆p Cp = 1 ρU 2 2 Uˆ G= 3 U1 h

Mean peak minimum pressure coefficient,

1.69

Gust velocity ratio

1.70

The corresponding coefficients are Cˆ p C pˆ = Mean maximum pseudo-steady pressure coefficient, G2 ( ( Cp Mean minimum pseudo-steady pressure coefficient. Cp = G2

1.71 1.72

They specified conclusion that mean and fluctuating pressure distributions are well approximated by a spherical dome of the same height to diameter ratio. The pressure distributions were independent of Reynolds number in the range 2.3 x 105-4.6 x 105 defined by velocity at top of dome and base diameter. In this study, hemisphere is one of the structural geometry on fabric membrane structures under wind force acting investigated. Regarding to the previous research, the result can be referenced to further study to the similar shape geometry with variation on material structure involved. Published data by Maher’s classical study of the dome surface can be described in Chapter 4, which will be compared with the CFD result. Those tables and diagrams involved are based on the work of Maher in 1965 and some by Blessmann in 1971, both of the result of them that arrived in general conclusions. Mean and fluctuation pressure distribution will have compared with those earlier studies.

2.5.2 Small Wind Tunnel Regarding the impression of useful the large wind tunnel with a large jet and more speed, the smaller wind tunnel might be considered in order to minimize cost as the fundamental advantages with the economically in operation. Small tunnel is much less in cost to build and less run as well as carried out the smaller model generated and time consuming. “The key to successful experiments made in a small tunnel is to have a clear understanding of the likely role of Reynolds number on the objects of the experiments. That is a matter of whether the relevant effect of Reynolds number in small wind tunnel.” There are not exactly true, that has no effect of Reynolds number on such cases. However, small wind tunnel is often used for instruction in

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the methods of experimentation and the result has done well. “Pressure distribution measurement on airfoils can be instructive even at relatively low Reynolds numbers. For a given airfoil shape the distribution does not change drastically with Reynolds numbers so long as angle of attack is well below stall. Many experiment concerning wind tunnel wall corrections are suitable for small tunnel. These tend to be little affected by Reynolds numbers”. (Barlow, Rae, & Alan Pope, 1999, p. 665).

2.6 Conclusion “The quantitative analysis of the behavior of fabric structures under severe loadings, has developed to the point where they can now, be engineered in every way to the same performance criteria as a permanent structure” (Russell, 2000). More specific research need for the fabric membrane structure, which is considered the structure as an integrated whole without analyzed in separately structural components. By applied CFD methods and subjecting scale model of fabric membrane structure model to wind loading simulation, a more understanding of the behavioral of them will be pursued.

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Chapter 3 Numerical Methods 3.1 Introduction Numerical study was investigated wind loading on the fabric membrane structure that mostly occurred on roof surface. Using Computational Fluid Dynamics (CFD) methods, prediction and evaluation of wind loads impinging to the membrane structure is being assessed with concerned on developing models structure. The CFD can solve some problem in which allow the immediate solution of the flow field without advancing in time and space. High-speed digital computer instrumentation needed in order to advance the practical simulation. Some computers package contributions also required for complex problem might be developed. In this case, some programme computer package can be involved since they can be associated to the main program solver. Since the latest role of CFD in engineering, prediction problem became in confident to generate in three-dimensional fluid dynamics. The simple and the complex model immediately can be identified using the computer package. One of them is AUTOCAD package that possibly can develop the initial model generation. The reason this package involved is because the availability and the handling ability on it in order to maximise productivity in CFD. In term of CFD code, there are three components of the CFD codes have been used are GAMBIT as pre-processor programme and FLUENT as a programme solver. The post-processor is GSVIEW of postscript-based generation to translate the dynamic result display of model developed. Those are mostly governed by the finite element method, which have been translated into a computational programme, particularly on fluid dynamic. Such as the principle methods, understanding of how the computational simulation work is significant in which the theory involved, including the theory of finite element methods as the basic of computational fluid dynamic developed.

3.2 Finite Element Theory and CFD Method Reviews Computational Fluid Dynamics is the analysis of systems, which are fluid flow and other associated phenomena involved in computer-based simulation. The technique of this is applications of finite element methods for fluids where the wide range application area copes and very powerful pursued.

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Three dimensional or solid elements considered is useful for the stress analysis of general threedimensional bodies that require more-precise analysis than is possible through two dimensional or axisymmetric analyses. The basic three-dimensional element is tetrahedron, which is used in the development of the shape function, stiffness matrix and force matrices in term of a global coordinate system. Referencing the theory of basic development on three dimensional, Finite Element published by Logan (1986) derived to consider the three dimensional infinitesimal element in Cartesian coordinate with dimensions dx, dy, & dz, and normal and shear stress. Normal stress are perpendicular to the faces of the element, and are presented by σ x , σ y , and σ z . Shear

σ τ

τ τ τ

σ

stress act in the faces (planes) of

τ τ

σ

the element, and are presented by τ xy , τ yz , τ zx and etc. The moment equilibrium of element on Appendix 2 are given by

Fig.3.1 Three-dimensional stress on an element, by Logan

τ xy =τ yx , τ yz =τ zy τ zx =τ xz

2.1

The element strain/displacement relationship are obtained on Appendix 2 are given by

εx =

∂u ∂v ∂w ,εy = εz = ∂x ∂y ∂z

2.2

where u, v, and w are displacement associate with the x, y, and z directions. The shear strain γ are given by

γ xy =

∂u ∂v ∂v ∂w ∂w ∂u + = γ yx , γ yz = + = γ zy , γ zx = + = γ xz ∂y ∂x ∂x ∂z ∂z ∂y

2.3

where, only three independent shear strains exist. Representing the stress and strains by column matrices as

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σ x  ε x  σ  ε   y  y σ z  ε z  {σ } =   , {ε } =   τ xy  γ xy      τ yz  γ yz  τ zx  γ zx 

2.4

The stress/strain relationship for an isotropic material are given by

{σ } = [D ]{ε }

2.5

where {σ} and {ε} are defined by Eq. 2.4 and the constitutive matrix [D]is given by

v  1− v  1− v     E [D] = (1 + v) (1 − 2v)      Symmetry 

0  0 0 0  0 0 0   1 − 2v 0 0   2  1 − 2v 0  2  1 − 2v   2  0

v v 1− v

0

2.6

In this study, developing the tetrahedral element is focused that it is because in CFD method, the model and the domain approached by tetrahedral discretization. Tetrahedral element considered is shown in figure 3.2 with corner nodes 1, 2, 3, and 4.

Fig. 3.2 Tetrahedral solid element, by Logan

u1  v   1   w1    .  {d } =  .    .  u   4 v4     w4 

2.7

There is three degree of freedom per node, or twelve total degree of freedom per element. For a compatible displacement field, the element displacement functions u, v, and w must be linear along

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each edge because only two points (the corner nodes) exist along each edge and the functions must be linear in each plane side of the tetrahedron. The linear displacement function as u(x, y, z) = a1 + a2x + a3y + a4z

2.8

v(x, y, z) = a5 + a6x + a7y + a8z

2.9

w(x, y, z) = a9 + a10x + a11y + a13z

2.10

Skipping the straightforward but tedious detail, would be obtain

u ( x, y , z ) =

1 {(α 1 + β 1 x + γ 1 y + δ 1 z )u1 + (α 2 + β 2 x + γ 2 y + δ 2 z )u 2 6V (α 3 + β 3 x + γ 3 y + δ 3 z )u 3 + (α 4 + β 4 x + γ 4 y + δ 4 z )u 4 }

2.11

where 6V is obtained by evaluating the determinant

1 1 6V =  1  1

x1 y1 z1  x 2 y 2 z 2  x3 y 3 z 3   x4 y4 z4 

2.12

and V represents the volume of the tetrahedron. The coefficients αi, βi, γi, and δi (I=1, 2, 3, 4) in Eq. (2.11) are given by

 x2 y 2 z 2   1 y2 z2   1 x2 z 2  1       α 1 =  x3 y 3 z 3  β 1 = −  1 y 3 z 3  γ 1 =  1 x3 z 3  δ 1 = −  1  x 4 y 4 z 4   1 y 4 z 4   1 x 4 z 4   1

x2 y 2  x3 y 3  x 4 y 4 

2.13

x1 y1  x3 y 3  x 4 y 4 

2.14

x1 y1  x 2 y 2  x 4 y 4 

2.15

and

 x1 y1 z1   1 y1 z1   1 x1 z1  1       α 2 = −  x3 y3 z 3  β 2 =  1 y 3 z 3  γ 2 = −  1 x3 z 3  δ 2 =  1  x 4 y 4 z 4   1 y 4 z 4   1 x 4 z 4   1 and

 x1 y1 z1   1 y1 z1   1 x1 z1  1       α 3 =  x 2 y 2 z 2  β 3 = −  1 y 2 z 2  γ 3 =  1 x 2 z 2  δ 3 = −  1  x 4 y 4 z 4   1 y 4 z 4   1 x 4 z 4   1 and

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 x1 y1 z1   1 y1 z1   1 x1 z1  1       α 4 = −  x 2 y 2 z 2  β 4 =  1 y 2 z 2  γ 4 =  1 x 2 z 2  δ 4 = −  1  x3 y3 z 3   1 y 3 z 3   1 x3 z 3   1

x1 y1  x 2 y 2  x3 y 3 

2.16

Expressions for v and w are obtained by simply substituting vi’s for all ui’s and then wi’s for all ui’s in Eq. 2.11. The displacement expression for u given by Eq. 2.11, with similar expressions for v and w, can be written equivalently in expanded form in term of the shape functions and unknown nodal displacements as

u   N 1 0 0    v  =  0 N 1 0  w  0 0 N 1   

N2

0

0

N3

0

0

N2

0

0

N3 0

0

0

N2 0

0

0

N4

0

0

N4

N4 0

0

u1  v   1  w1    0  .    0   .  N 4   .    u 4    v 4  w4 

2.17

where the shape functions are given by

N1 =

(α 1 + β 1 x + γ 1 y + δ 1 z ) 6V

N3 =

(α 3 + β 3 x + γ 3 y + δ 3 z ) (α + β 4 x + γ 4 y + δ 4 z ) N4 = 4 6V 6V

N2 =

(α 2 + β 2 x + γ 2 y + δ 2 z ) 6V 2.18

The element strains for the three-dimensional stress state are given by ∂u  ε x   ∂x   ε  ∂v ∂ y y       ∂ w ε x   ∂z  {ε } =   =  ∂u ∂v  γ xy   ∂y + ∂x  γ   ∂v + ∂w   yz   ∂z ∂y  γ zx   ∂∂wx + ∂∂uz 

2.19

Using Eq. (2.17) in Eq. (2.19), would obtained

{ε } = [B]{d }

2.20

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where

[B] = [B1 B 2 B 3 B 4 ]

2.21

The sub matrix B1 in Eq. 2.21 is defined by

 N 1, x  0 0 B1 =   N 1, y  0 N  1, z

0

  0  N 1, x   0   N 1, y  N 1, x 

0

N 1, y 0 N 1, x N 1, z 0

2.22

Sub matrices B 2 , B 3 and B 4 are defined by simply indexing the subscript in Eq.2.22 from 1 to 2, 3, and 4, respectively. Substituting the shape functions from Eq. 2.18 into 2.22, B1 is expressed as

 β1 0  1 0 B1 =  6V γ 1 0  δ 1

0

γ1 0

β1 δ1 0

0  0  δ1   0  γ1   β 1 

2.23

with similar expressions for B 2 , B 3 and B 4 . The element stresses are related to the element strains by

{σ }= [D]{ε }

2.24

where the constitutive matrix for an elastic material is given by Eq. 2.23 The element stiffness matrix is given by

[k ]= ∫∫∫[B]T [D][B]dV

2.25

v

Since both matrices [B] and [D] are constant for the simple tetrahedral element, Eq.2.25 can be simplified to

[k ]= [B]T [D][B]V

2.26

where, V is the volume of the element. The element stiffness matrix is order of 12 x 12. The element body force matrix is given by

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[ fb]= ∫∫∫[N ]T {X }dV

2.27

v

where [N] is given by the 3 x 12 matrix in Eq. 2.17, and

X b  {X }= Yb  Z   b

2.28

For constant body forces, the nodal components of the total resultant body forces can be distributed to the nodes in four equal parts. The surface forces are given by

[ fs]= ∫∫ [N ]

T

s

 px    evaluated on  p y  surface1, 2 , 3    pz 

2.29

where px, py, and pz are the x, y, and z component, respectively, of p. Simplifying and integrating Eq. 2.29 that can be shown as

 px  p   y  pz     px  p   y  pz  { f s } = S123  3   px  p   y  pz    0  0    0 

2.30

where S123 is the area of the surface associated with nodes 1, 2, and 3. The formulation of tetrahedral element review is concerned to the developing finite element volume related to the CFD methods. “The cornerstone of computational fluid dynamics is the fundamental governing equations of fluid dynamics-the continuity, momentum and energy equations”. Anderson (1996). The basic equations of fluid motion are always to follow the philosophy of:

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•

Appropriate fundamental physical principle from the law physics, such as a. Mass is conserved. b. F=ma (Newton’s 2nd law) c. Energy is conserved.

•

Applied the physical principle to the suitable model of the flow.

•

To extract the mathematical equations which embody such physical principles.

The physical meaning of substantial derivative is important to establish a common notation in aerodynamics development.

3.2.1 The CFD Code From the reference “An Introduction to CFD -The Finite Volume Method” by Versteeg & Malalasekera, (1995), CFD codes and formulations are derived. CFD codes are structured around the numerical algorithms that can solve fluid flow problems. The CFD code contained three main elements that are: 1) a pre-processor, 2) a solver and 3) a post-processor. Brief examine the function of each of these elements derived: 1. Pre-processor Pre-processing consist of the input of a flow problem to a CFD program by means of an operatorfriendly interface and the subsequent transformation of this input into a form suitable for use by the solver. The user activities at the pre-processing stage involved •

Definition of the geometry of the region of interest: the computational domain.

•

Grid generation –the sub-division of the domain into a number of smaller, non-overlapping sub-domains: a grid (or mesh) of cells (or control volumes or element)

•

Selection of the physical and chemical phenomena that need to be modelled.

•

Definition of fluid properties.

•

Specification of appropriate boundary conditions at cells, which coincide with or touch the

domain boundary. The solution to a flow problem (velocity, pressure, temperature etc) is defined at nodes inside each cell. The accuracy of a CFD solution is governed by the number of cells in the grid, which is fineness of the grid generation, the more accurate result gain. It is dependent how good the computer hardware related to the iteration time assumption. At present it is still up to the skills of the CFD user to design a grid that is a suitable compromise between desired accuracy and solution cost. In order to maximise productivity of CFD personnel all major codes now include their own CAD-style interface and/or facilities to import data from proprietary surface modellers and mesh generators. 2. Solver In outline the numerical methods that form the basis of the solver perform the following steps:

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•

Approximation of the unknown flow variables by means of simple functions.

•

Discretisation by substitution of the approximations into the governing flow equations and subsequent mathematical manipulations.

• Solution of the algebraic equations. The theory of finite elements has been developed initially for structural stress analysis. A standard work for fluids applications is Zienkiewickz and Taylor (1991). The finite difference formulation was originally developed the finite volume method as basic of CFD technique established. The finite volume method concerned with most well-established and thoroughly validated general purpose CFD technique. It is central to four of the five main commercially available CFD codes: PHOENICS, FLUENT, FLOW3D and STAR-CD. The numerical algorithm consists of the following steps: •

Formal integration of the governing equations of fluid flow over all the (finite) control volumes of the solution domain.

•

Discretisation involves the substitution of a variety of finite-difference-type approximations for the terms in the integrated equation representing flow processes such as convection, diffusion and sources. This converts the integral equations into a system of algebraic equations.

•

Solution of the algebraic equations by an iterative method.

3. Post-processor As in pre-processing a huge amount of development work has recently taken place in the postprocessing field. Owing to the increase popularity of engineering workstations, many of with have outstanding graphics capabilities, the leading CFD package are now equipped with versatile data visualisation tools. These include: • Domain geometry and grid display • Vector plots • Line and shaded contour plots • 2D and 3D surface plots • Particle tracking • View manipulation (translation, rotation, scaling etc.) • Colour postscript More recently these facilities may also include animation for dynamic result display and in addition to graphics all codes produce trusty alphanumeric output and have data export facilities for further manipulation external to the code. As in many other branches of CAE the graphics output capabilities of CFD codes have revolutionised the communication of ideas to the non-specialist.

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3.2.2 Fluid Flow Problem and Governing Equations on CFD In solving fluid flow problems, it need to be aware that the underlying physics is complex and the results generated by a CFD code are at best as good as the physics embedded in it and at worst as good as bits operator. Elaborating on the latter issue first, the user of a code must have skills in a number of areas. Prior to setting up and running a CFD simulation there is a stage of identification and formulation of the flow problem in terms of the physical and chemical phenomena that need to be considered. Performing the actual CFD computation itself requires operator skills of a different kind. Specification of the domain geometry and grid design is the main tasks at the input stage and subsequently the user needs to obtain a successful simulation result. The two aspects that characterise such a result are convergence of the iterative process and girl independence. The solution algorithm is iterative in nature and in a converged solution the so-called residuals – measures of the overall conservation of the flow properties – are very small. Progress towards a converged solution can be greatly assisted by careful selection of the settings of various relaxation factors and accelerations devices. The only way to eliminate errors due to the coarseness of a grid is to perform a grid dependence study, which is a procedure of successive refinement of an initially coarse grid until certain key results do not change. Then the simulation is grid independent. Optimisation of the solution speed requires considerable experience with the code itself, which can only be acquired by extensive use. There is no formal way of estimating the errors introduced by inadequate grid design for a general flow. Good initial grid design relies largely on an insight into the expected properties of the flow. The only way to eliminate errors due to the coarseness of a grid is to perform a grid dependence study, which is procedure of successive refinement of an initially coarse grid until certain key results do not change. Then the simulation is grid independent. A systematic search for gridindependent result forms an essential part of all high quality CFD studies. Every numerical algorithm has its own characteristic error patterns. Well-known CFD euphemisms for the word error are terms such as numerical diffusion, false diffusion or even numerical flow. At the end of a simulation the user must make a judgement whether the results are ‘good enough’. It is impossible to assess the validity of the models of physics and chemistry embedded in a program as complex as a CFD code or the accuracy of its final result by any means other than comparison with experimental test work. Anyone wishing to use CFD in a serious way

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must realise that it is no substitute for experimentation, but a very powerful additional problemsolving tool. Validation of a CFD code requires highly detailed information concerning the boundary conditions of a problem and generates a large volume of results. To validate these in a meaningful way it is necessary to produce experimental data of similar scope CFD computation involves the creation of a set of numbers that (hopefully) constitutes a realistic approximation of a real-life system. One of the advantages of result, but in the prescient words of C. Hastings (1955), written in pre-IT days: ‘The purpose of computing is insight not number’. The underlying message is rightly cautionary. The main outcome of any CFD exercise is improved understanding of the behaviour of a system, but since there are no cast iron guarantees with regard to the accuracy of a simulation we need to validate our results frequently and stringently.

It is clear that there are guidelines for good operating practice, which can assist the user of CFD code and repeated validation plays a key role as the final quality control mechanism. However, the main ingredients for success in CFD are experience and a through understanding of the physics of fluid flows and the fundamentals of the numerical algorithms. Without these it is very unlikely that the user gets the best out of a code. The governing equations of fluid flow represent mathematical statements of the conservation laws of physics, as written by Aderson, (1996) in this paper, page 48. The first step in the derivation of the mass conservation equation is to write down a mass balance for the fluid element. Rate of increase of mass in fluid element

=

Net rate of flow of mass into fluid element

The rate of increase of mass in the fluid element is ∂ ∂p ( ρδxδyδz ) = δxδyδz ∂t ∂t

2.31

It is needed to account for the mass flow rate across a face of the element which is given by the product of density, area and the velocity component normal to the face. From Figure 3.3, it can be seen that the net rate of flow of mass into the element across its boundaries is given by  ∂ ( ρu ) 1  ∂( ρu ) 1  ∂( ρv) 1    . δx δyδz −  ρu + . δx δyδz +  ρv − . δy δxδz  ρu − δx 2  δx 2  δy 2    

 ∂ ( ρv ) 1  ∂ ( ρw) 1  ∂ ( ρw) 1    −  ρv + . δy δxδz +  ρw − . δz δxδy −  ρw + . δz δxδy δy 2  δz 2  δz 2    

2.32

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Flow, which are directed into the element produce an increase of mass in the element and get a positive sign and those flows that are leaving the element are given a negative sign. ρ

ρ δ

ρ

ρ

ρ δ

ρ δ

δ

δ

ρ δ

ρ

δ

ρ

ρ

ρ δ

ρ δ

δ

δ

δ

Fig.3.3Mass flow in and out of fluid element, by Versteeg & Malalasekera

The rate of increase of mass inside the element is equated to the net rate of flow of mass into the element across its face (Fig. 3.3). All terms of the resulting mass balance are arranged on the left hand side of the equals sign and the expression is divided by the element volume δxδyδz. This yield ∂p ∂ ( ρu ) ∂( ρv) ∂ ( ρw) + + + =0 ∂t ∂x ∂y ∂z

or in more compact vector notation

2.33

∂p + div( ρu ) = 0 ∂t

2.34

Equation (2.34) is the unsteady, three-dimensional mass conservation or continuity equation at a point in a compressible fluid. The first term on the left hand side is the rate of change in time of the density (mass per unit volume). The second term describes the net flow of mass out of the element across its boundaries and uts called the convective term. For an incompressible fluid (i.e.a liquid) the density ρ is constant and equation (2.34) becomes div u = 0 or in longhand notation ∂u ∂v ∂w + + =0 ∂x ∂y ∂z

2.35

2.36

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In term of momentum equation in three dimensions, the Newton’s second law states that the rate of change of momentum of a fluid particle equals the sum of the forces on the particle. Rate of increase of = Sum of forces on fluid particle momentum of fluid particle The rates of increase of x-,y- and z- momentum per unit volume of a fluid particle are given by Du Dv Dw ρ ρ ρ 2.37 Dt Dt Dt We distinguish two types of forces on fluid particle:

•

Surface forces - pressure forces and viscous forces

•

Body forces -gravity force, centrifugal forces, Coriolis & Electromagnetic force The state of stress of a fluid element is defined in terms of the pressure and the nine

viscous stress components show in figure 3.4.

τ

The pressure, a normal stress, is denoted by

τ τ

τ

τ τ τ τ

τ

p. Viscous stresses are denoted by ι. The

τ τ τ

usual suffix notation ιij is applied to

τ

indicate the direction of the viscous

τ

τ

stresses. The suffices

τ

direction on a surface normal to the i-

Fig.3.4Stress components on three faces of fluid element, by Versteeg & Malalasekera

τ

τ

τ

τ

δ

τ

τ

δ

direction.

δ

δ τ

and j in ιij indicate

that the stress component acts in the j-

τ

τ

I

First we consider the xδ δ

δ τ

δ

τ

τ

τ

δ

δ

Fig.3.5 Stress components in the x-direction, by Versteeg & Malalasekera

components of the forces

Forces

due to pressure p and

aligne

stress components ιxx, ι.yx

d with

ι.zx show in figure

the

3.4. The magnitude of a

directi

force resulting from a

on of

surface

a

and

stress

is

the

product of stress and area.

co-

ordina te axis

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get a positive sign and those in the opposite direction a negative sign. The net force in the xdirection is the sum of the force components acting in that direction on the fluid element. On the pair of faces (E, W) have    ∂p 1   ∂τ xx 1  ∂p 1   ∂τ xx 1   p − δx . 2 δx  − τ xx − δx . 2 δx δyδz + −  p + δx . 2 δx  + τ xx + δx . 2 δx δyδz           ∂p ∂τ xx  2.38 = − + δx δxδyδz  δx ∂x  The net force in the x-direction on faces (N,S) is ∂τ yx 1  ∂τ yx 1  ∂τ yx   − τ yx − . δy δxδz + τ yx + . δy δxδz = δxδyδz ∂y 2  ∂y 2  ∂y  

2.39

The final net force in the x direction on the T and B is given by ∂τ 1  ∂τ 1  ∂τ   − τ zx − zx . δz δxδy + τ zx + zx . δz δxδy = zx δxδyδz ∂z 2  ∂z 2  ∂z  

2.40

The total force per unit volume on the fluid due to these surface stresses is equal to the sum of Eq. 2.38, Eq. 2.39 and Eq. 2.40 divided by the volume δxδyδz: ∂(− p + τ xx ) ∂τ yx ∂τ zx + + ∂x ∂y ∂z Without considering the body force, overall effect can be included by defining a source SMx of xmomentum per-unit volume per unit time. The x-component of the momentum equation is: Du ∂(− p + τ xx ) ∂τ yx ∂τ zx = + + + S Mx ρ 2.41a Dt ∂x ∂y ∂z The y-component of the momentum equation is: Dv ∂τ xy ∂ (− p + τ yy ) ∂τ zy ρ = + + + S My Dt ∂x ∂y ∂z

2.41b

The z-component of the momentum equation is: Dz ∂τ xz ∂τ yz ∂(− p + τ zz ) = + + + S Mz ρ Dt ∂x ∂y ∂z

2.41c

Energy equation involved is derived from the first law of thermodynamics which states that he rate of change of energy of a fluid particle is equal to the rate of heat addition to the fluid particle plus the rate of work done on the particle. Rate of increase of energy of fluid particle

=

Net rate of heat added to fluid particle

+

Net rate of work done on fluid particle.

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The energy equation can be specified often as the sum of internal (thermal) energy i, kinetic energy ½ (u2+ v2+ w2) and gravitational potential energy. The energy E equation is:  ∂ (uτ xx ) ∂ (uτ xy ) ∂ (uτ zx ) ∂ (vτ xy ) ∂ (vτ yy ) ∂ (vτ zy ) DE = − div( pu ) +  + + + + + ρ Dt ∂y ∂z ∂x ∂y ∂z  ∂x ∂ ( wτ xz ) ∂ ( wτ yz ) ∂ ( wτ zz )  + + + + div(k grad T ) + S E ∂x ∂y ∂z 

2.42

The energy is became E = i +½ (u2+ v2+ w2). The motion of a fluid in three dimensions is described by a system of five partial differential equations: mass conservative Eq. 2.34, x-, y-, and z moment equations Eq. 2.41a-c and energy equation Eq. 2.42. The state of substance in thermodynamic equilibrium is used ρ and T as state variables for state equation of pressure p and specific internal energy i: p=p(ρ, Τ) and i=i(ρ, Τ)

2.43

For a perfect gas the following equations of state are useful: p=ρRT and i=CvT

2.44

Finally, a command differential form for all the flow equations identified as transport equation and developed integrated forms which are central to the finite element volume CFD method: for steady state processes derived as: ∫ n .( ρφu)dA = ∫ n . (Γ grad φ )dA + ∫ Sφ dV A

A

2.45

CV

and for time-dependent processes  ∂ ∫∆t ∂t  CV∫ ( ρφ )dV dt + ∆∫t ∫A n . ( ρφu)dA = ∆∫t ∫A n . (Γ grad φ )dAdt + ∆∫t CV∫ Sφ dVdt

2.46

Generating model on CFD method is difficult without knowing a great deal about flow before solving a problem.. It is very difficult to specify the precise number and nature of allowable boundary conditions on any fluid/fluid boundary in the far field. For the convenient attempt, the boundary condition Fig.3.6 (a). Boundary condition for an internal flow problem Versteeg & Malalasekera

for flow problem specified in figure below

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Fig.3.6 (b) Boundary condition for external flow problem, by Versteeg & Malalasekera

It is obvious that flow inside a CFD solution domain is driven by the boundary conditions. The difficulties are encountered in obtaining solution, therefore paramount importance that supplied physically realistic and well-posed boundary conditions are applied. The boundary conditions mostly affected the rapid of divergence of CFD simulation. A set of ‘best’ boundary condition for viscous fluid flows, which included the inlet, outlet and wall condition. The finite volume method implementation included three conditions, constant pressure, symmetry and periodicity, which are physically realistic and very useful I practical calculations. Some permissible state combination in subsonic flows: •

Walls only

•

Wall and inlet and at least one outlet

•

Wall and inlet and at least one constant pressure boundary

•

Wall and constant pressure boundaries

Position of outlet boundaries is became a significant matter to contribute in how the flow can behave effectively. “If outlet boundaries are placed too close to solid obstacles it s possible that the flow will not have reached a fully developed state which may lead to sizable

errors.

It

is

imperative that the outlet boundary is placed much more further downstream than 10 height downstream of the last obstacle to give accurate result” (fig. 3.7) Fig. 3.7 Velocity profiles at different locations downstream of an obstacle, by Versteeg & Malalasekera

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3.2.3 General Fluid Dynamic Background Definition of fluid is a substance, which cannot sustain shear stresses whilst at rest. The fluid problem generally interested in microscopic, rather than molecular scale behaviour that possibly assumed as a continuous and homogeneous substance. Physical of air justified of 1m cubes contains 27x1024 molecules and the viscosity of air is the resistance to continuous shearing due to a property of the fluid, which has value of µ =1.82x10-5 kg/m.s at 200 C known as the dynamic viscosity or absolute viscosity. Newton’s law of viscosity for fluid is the relationship between shear stress τ and rate of shear strain γ, with the constant of proportionality being the dynamic viscosity: d τ = µ (γ ) = µγ dt

2.47

or shear stress = dynamic viscosity x rate of shear strain. While the rate of shear strain is equal to the velocity gradient normal to the shear plane: dc τ =µ dn

2.48

τ µ

Fig. 3.8 by Potts (MMM336)

Fig. 3.9 by Potts (MMM336)

Considering steady flow of fluid along a duct of uniform cross sectional flow area (A) at velocity (c)(figure 3.9). The net volume of fluid crossing plane x-x is thus dV= A.c.dt, and the net mass is ρ.A.c.dt. Alternatively, volume flow rate Q (= dV/dt) = A.c and mass flow rate is m (=dm/dt) = ρ.A.c. In CFD method the consideration of the laminar and turbulent flows can be approached from the definition and also can be specified the behaviour of air by using pipe model in order to know the differences between them. A simple comparison of both flow features is discussed. The physical model for the laminar is described by the figure below in left part, on the other hand, the turbulent behaviour can be described in right part of the table below:

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Table 3.1 Laminar and Turbulent flow model

A turbulence model is more focused due to the more complicated problem faced. The turbulent model can be simplified as a computational procedure to close the system of mean flow equation described on table below: Table 3.2 Turbulent flow equations for compressible flows, by Versteeg & Malalasekera

Continuity ∂p + div ( ρU ) = 0 ∂t

Reynolds equations ∂ ( ρU ) ∂P + div ( ρUU ) = − + div ( µ grad U ) ∂t ∂x  ∂ ( ρ u ' 2 ) ∂ ( ρ u ' v ') ∂ ( ρ u ' w')  + − − −  + S Mx ∂x ∂y ∂z  

2.49

2.50a

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∂ ( ρV ) ∂P + div ( ρVU ) = − + div( µ grad V ) ∂t ∂x  ∂ ( ρ u ' v') ∂ ( ρ v' 2 ) ∂ ( ρ u ' w')  + − − −  + S My ∂x ∂y ∂z   ∂ ( ρW ) ∂P + div ( ρWU ) = − + div( µ grad W ) ∂t ∂x  ∂ ( ρ u ' w') ∂ ( ρ u ' w') ∂ ( ρ w' 2 )  + − − −  + S Mz ∂x ∂y ∂z  

2.50b

2.50c

Scalar transport equation  ∂ ( ρ u ' ϕ ') ∂ ( ρ v' ϕ ') ∂ ( ρ w' ϕ ')  ∂ ( ρΦ) + div ( ρΦU ) = div (ΓΦ gradΦ) + − − −  + SΦ ∂t ∂ x ∂ y ∂ z  

2.51

For most engineering purposes that is unnecessary to resolve the details of the turbulent fluctuations. The effect of turbulence on the mean flow is the only effect usually sought. Large eddy simulations are considered in order to investigate the effect to the model problems. Large eddy simulation (LES) are turbulence models where the time-equations are solved for the mean flow and the largest eddies and where the effect of smaller eddies are modelled. Large eddy simulations are at present at the research stage and the calculations are too costly considered in general purpose computational. Nowadays, anticipation may be already done the improvement in computer hardware or may change the perspective in the future. Some LES equations have derived in Eq. 1.30-1.60 in Chapter 2.

3.3 General Strategies and Procedures In CFD method, the computer hardware and software must be available in order to accommodate model data and then run the computer programme so that can be identified the prediction result of wind load acting on the fabric membrane building model. The Sun Ultra 5 series workstations in the University of Newcastle upon Tyne are all licensed to run Fluent version 5 and Gambit. The Fluent 5 is the latest version CFD software from Fluent Inc., and is a state-of-the-art commercial CFD solver, using the latest unstructured mesh approach. However, Fluent 5 does not have any internal facilities for mesh generation, and the necessary grids must be produced using a separate package of Pre-processor. The latest and (most powerful) preprocessor can be used is Gambit that is supplied by Fluent. Inc too.

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Creating grid model in Gambit need understanding to identify menu function of software and must be familiar wait. In order to gain a good and clean result, Gambit has Graphics User Interface or “GUI” as guide information and as tutor via Netscape. Creating mesh structure model to present fabric membrane structures are far more difficult due to the shape model of structure usually as a complex geometry. For the simple shape, the model can be generated directly in Gambit, or when has no choice that is need time consumed to generate the model. Due to the ability of Gambit to associate another programme computer package, the geometric mesh model can be developed in such as CAD programme, that is depend on how far can be used too those programme.

1. AutoCAD Reviews In this study, AutoCAD programme package has been used to generate grid model structure. In this commercial package, the three-dimensional model can easily be generated due to wide range of ability to specify interface in accurately. AutoCAD offers two methods for creating 3D model: surface modelling and solid modelling. Several kinds of simple models are presented below is 3D surface modelling generated that are concerned.

Fig.3.10 Example mesh geometric in AutoCAD

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Fig.3.11Arranged position of inlet, outlet and wall boundaries in AutoCAD

Fig.3.12 The geometry that will be exported from AutoCAD

Thus, is because the fabric membrane structure developed is based on surface with mostly curving. Using AutoCAD’s 3D capabilities, simple object can be created by manipulation of current 3D surface available. There are some basic shape available for 3D model such as cone, sphere/dome, torus, pyramid and also have surface developed that such as edge surface, 3D mesh, revolved surface, tabulated surface, and ruled surface. Generating model initially in AutoCAD is much more promised due to easier to arrange the shape of model and to decide easily the dimension of domain as well as the shape of tunnel/domain generation. It is important part in order to fulfil the requirement the position of outlet boundary, which can be seen in figure 3.7. The domain must be arranged in deal with the base point of coordinate (x,y,z = 0,0,0)It is because when importing done, the domain should created again if only model imported to Gambit with the same domain pattern in AutoCAD. It is unnecessary when importing included the domain depend on kind of domain developed.

Once the geometry of model structures completed as in example in figure 3.12, the model is ready to export into Gambit. Initially, it is important to create such as an IGES file before exporting, in order to associate between the programme consoles. IGES file is the file interface from the AutoCAD that can only be read by Gambit. The IGES files can be easily operated on Mechanical Desktop 4 as family of AutoCAD programme. Such as a simple save to IGES file or imported directly, IGES file has been created. Since that happened the IGES file needed to transfer to the Gambit as a pre-processor.

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2. Pre-processor: GAMBIT Reviews The model structures actually can be created directly in Gambit. However, sometime when a complicated structure model has been created, it will spend such along time in generation. This procedure is naturally on the way and relevant problem need time consumed. Developing such as model structure in Gambit is depend on how the user familiar with the programme, more experience on it much more helpful. Fortunately, when processing model still has a problem, CAD program is the alternative ways to produce model structures. In this study, creating initial model structure has been done in AutoCAD, the IGES file has been created and has already transferred into Gambit. Processing geometry in Gambit has need through the guideline or using the tutorial of importing and cleaning up the dirt geometry in order to have a right way in developing. In general term of Gambit procedure, it can be summarised about importing IGES file in the following item below: •

Importing an IGES file

•

Connecting edges, using manual and an automatic method

•

Merge face

•

Creating a triangular surface mesh, or others

•

Mesh a volume with a tetrahedral mesh or using different volume mesh

• Prepare the mesh to be read into Fluent 5. In term of importing and cleaning geometry, there are strategies of how to dealt with and passed through the pre-processor (Gambit) before solved by the Solver (Fluent 5). Creating a fully unstructured tetrahedral mesh around a China hat as an example problem in Gambit, firstly the model geometry imported as an IGES file. The tutorial will be guided the step that would typically follow to prepare an imported CAD geometry for meshing. It is the geometry “dirty” that is needed to clean up the geometry using the tool available in Gambit. A very obvious tutorial guided how to do the right thing fixed the gap automatically either during mesh importing or subsequently by means of the “connect edge” command.

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A

B

C

D Fig.3.13 Arranged model generated, domain, and floating element (tetrahedral)

The original CAD geometry is not modified during the fixing process; the modifications required to eliminate the gaps are made using ”virtual” geometry. Some edges in the original geometry are very short and will be eliminated using the “vertex connect” command. Other edges are not automatically connected, because they are farther apart than the specified tolerance, it is needed to connect such edges manually. The imported geometry includes a number of small surfaces, the edges of which may unnecessarily constrain the mesh generation process. Using the “merge faces” command, GAMBIT allows to easily combined these surfaces prior to meshing. It can then have GAMBIT automatically create a triangular mesh on the China hat model, it can be seen in figure 3.13 B. Since the imported geometry consists only of the China hat, it is need to create a suitable domain around the China hat model structure in order to conduct a CFD analysis (this is loosely equivalent to placing the structure in a wind tunnel)(fig.3.11). The remainder of the tutorial shown how to add a real box around the structure, use virtual geometry to create some missing faces, and finally stitch all faces together into a

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single volume. This volume can then be meshed (without any decomposition) using a tetrahedral meshing scheme or using another suitable mesh volume. (Fig. 3.13A-D) The next attempt in Gambit process is set a boundary type to the domain in which condition flow to be specified. In order for the mesh to be properly transferred to Fluent, the edges must be assigned boundary types, such as wall, inlet, outlet, etc. In this general example, velocity inlet Fig.3.14 Arranged position of inlet, outlet and wall boundaries in AutoCAD

created on the tunnel

where moving air while other side of tunnel as the outflow and mostly of all side of tunnel is wall or only two side of tunnel is wall with free edges on the top. Figure 3.13 show the boundary or vicinity requirement so that when model structure completely passed through the examination in this process, the mesh developed can be appreciated by the Fluent. Finally, once the boundary type has been set, the mesh is ready to transfer to the Fluent (solver). In the main Gambit window, the command is File
Export
Mesh
Accept.

3. Solver: Fluent Reviews Starting Fluent 5, firstly need to define as 3D base in order to specify the 3D environment identification, which is same orientation developed in the previous process. The next step is defining viscous model and fluid properties. In the main Fluent window, that is Define
Models
Viscous. Laminar is a default viscous model have got, if it was intending to solve a turbulent flow, then another turbulence models such as k-epsilon, Reynolds stress and Large Eddy simulation can be selected. Then, material fluids need to define whether is default or define as a special ‘custom’ of fluid material to be selected. Boundary condition of zone will be identified as default automatically by this programme. In the main Fluent window, that is Define
Boundary condition
Set. When it is continued the

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example problem above, boundary conditions will appear as zone identified that are velocity inlet zone, wall zone and outflow zone. In order to update the velocity inlet motion, it is allowed a value data entry to be set. The next term of Fluent solution is iterative solution for flow field, in the main Fluent window, the following command step is Solve
Controls
Solution. Iterative solution of the governing conservation equations is the important part due to the non-linearity of equations must be solved until the iterative process is converges. Final solution can be recorded by using residual monitor available in order to judge convergence behaviour on graph of residual against iteration numbers. During iteration, time to be consumed that are depend on how the simple or complex problem have. It is long time to be consumed when such as turbulent model solved due to more complex governing equation developed. Once the iteration has convergence, how the solution is progressing can be checked. In the main Fluent window, Display
Velocity Vectors/Contour
Pressure
Pressure coefficient
Display. The graphical display window will show the velocity vectors or contour of pressure coefficient, zooming needed to view the velocity field in more detail if desired. In Fluent, data resulted as graphical and diagram. Simple command direction File
Hardcopy, the data can be obtained. The default graphics display window on the screen shows plots with a black background and colored objects (foreground). At this point, to preview the hardcopy, Preview, which is the desired case for hardcopy printouts.

4. Post-processor As an explanation before by Versteeg and Malalaseker, other facilities may also included in order to manipulate the dynamic result display into an addition graphical. As in many other programme graphics output capabilities, thus can be connected to CFD codes, which have revolutionised the communication of ideas to the non-specialist. Once the graphical has been resulted on Fluent, its mean processing of the whole step development of the numerical methods are nearly finished. Graphical data and diagram can be simply obtained from Fluent. It is simple to command (File
Hardcopy), the data can be printed out directly to printer. There are several types of file association can arranged the dynamic graphical in order to present data properly. In this study, graphical and diagram data saved as a format file under GSview programme. In order to organise data graphical and diagram into a different format file, the graphical

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as well as diagram can be saved as EPS or PS format with colour postscript based development. It is as an alternative procedure to produce graphical format on project report or any journal. It is a procedure review done due to the availability of programme and in order to make smooth transfer into different programme rather than to printout directly from Fluent or Unix cluster. A difficult arrangement established on these programme during presenting graphical and diagram data on paper. However, there are impressive format graphical and diagram presented directly on computer monitor display.

3.4 Detail Of Model Experimental In this study, pursued numerical simulation of tensile membrane structures are developed in CFD simulation as an engineering tool presented wind tunnel. The model structures generated are placed on tunnel model of computational domain. The main structures models developed are sphere, and cooling tower. Those models are presented a basic shape of fabric membrane structures. The model represented variations of structures that consist of cable suspended roof that support fabric membranes. The structures shape models have chosen due to the needed to compare, the result of CFD methods to the published data available of the previous research. At the first stage of the project, it was planned to solve the common specified problem in wind engineering by CFD methods that adopted as a computational model of a low-rise building with sphere and cooling tower model developed. (Figure 3.15.a & b).

Fig. 3.15. a Sphere Elevation

Fig. 3.15. b Cooling Tower Elevation

In this study, position of the velocity inlet, the wall and the outflow are arranged such as described below. These pattern of outflow/outlet boundaries are placed at 20 times the height of the obstacle (20 H) which was the same as in experiments performed by Tamura, cs. This distant is long enough in order to avoid the possibilities that the flow reached the range across a wake region with recalculation which my lead to measurement error. The typical velocity inlet was

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placed at 10 times the height of model (10H), while the wall can be arranged between 7H and 10H that are depended on the amount of volume element generation needed. It is important because of the limitation of space disk on the computer availability. The bigger domain have, the bigger quota space disk needed. The top wall was also placed at 10 times height of model (10H). Somewhat further conditions that the outlet boundary is placed at least at 10H as the typical literature reviewed before (fig. 3.7). However the outlet boundary placed much longer than 10H downstream of the last obstacle to give accurate result.

Fig. 3.16 Computational domain development

The model detail preparation was adopted as a computational model of a low-rise building with sphere and cooling tower model developed. Each of the basic models has been modified as a multiple model. There are several model involved including combination of each member that is described below:

3.4.1 Single Cooling Tower Model

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This model was adopted from the original Heler type dry-cooling tower. The model developed is simple shape rather than the real one with other component such as the water pipes, radiator, support beam, etc. It is because the shape of it is nearly the same as the requirement of the fabric membrane

Fig. 3.17 Sketch of Heler-type dry cooling tower (De1.igs of IGES file)

structures shape. The other reason is the needed of published data availability to compare with the further result. In this particular case, the model was arranged to the same shape and dimensional to the model developed in the experimental method (wind tunnel). It is because much easier to make model in AutoCAD and exported to Gambit. Providing model in the experimental method more difficult or need time to spend. Once the model has developed in the experimental or in wind tunnel test, the shape model can be are developed the same as the model in wind tunnel test immediately. The cooling tower model was measured as height (H = 16 cm), the top diameter (D2 = 10.5 cm), and the bottom structures (D1= 20.5 cm). Creating geometry in CFD was selected as a default of measurement in meter. Again, it is more useful to make model in small scale in order to reduce the space disk consumed. This model was modified into smaller scale in CFD, that was made as H=1.6 m, D2=1.05m and D2=2.05m, respectively. The cooling tower has divided into 6 (six) surfaces with every connection have a rib. This model was used 6 ribs and 1cover on the top, while there is no surface on the bottom. (Figure 3.17). A part of cooling tower geometry includes a number of small surfaces will be generated as constrain the mesh generation process as figure 3.18. Fig. 3.18 count space

Internal

Using the “merge faces” command, GAMBIT allows to easily combining these surfaces prior to meshing. In GAMBIT, then automatically create a triangular mesh on every surface of the model, in figure 3.18.

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In this particular case, there are domain developed, which is based on the height of z y

x

model situated in the suitable place. The dimensional of domain developed is 48 x 22.4 x 16 with unit a long x direction as the

Fig. 3.19 Domain of Single Cooling Tower

length, y direction as

wide and z direction as the height of domain respectively. For the study of cooling tower, the computational region developed is shown in figure 3.20. Beside of that is only half of the field simulation can present the whole simulation. However, in this study the whole body of model simulation involved in order to know the pressure pattern

distribution

three-dimensional. method capability Fig. 3.20 Computational region and coordinate system.

the

in In

capacity

of

fully CFD and

computer

is

absolutely needed. In this case,

because of the limitation capacity of disk space quota at about 128-140 MB and the speed limit of 128 MB RAM, relatively small grid number of mesh generated. The grid number for the cooling tower simulation is 30 x 10 x 20, i.e. 30 grids in the main flow direction, 10 grids along the circumference of the tower and 20 grids in wide direction.

3.4.1. Detail Procedure and Instruction: Instruction: In this report is a procedure that enables to solve 3-D of cooling tower in flow problem with the CFD program, Fluent. In this stage, the notation should already be familiar with used in this module, or can be described in the learning module, Fluent and Gambit-General Information.

Log on and launch Fluent:

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1. Log onto one of the UNIX cluster computers. It is useful completed the Gambit learning module for generating the grid. 2. To begin Gambit and Fluent from the UNIX % command prompt: Fluent. nit 3. To start working in Gambit, type: gambit De1-dev x11 Note: De1 is the example case of the cooling tower problem. 4. From main menu, specified Fluent 5 is the base of solver. Read the grid points and geometry of the cooling tower (in Gambit); used importing and cleaning up model design procedure.

1. Selected a solver (Fluent 5) 2. Imported the IGES file from AutoCAD as the import source of the developing cooling tower geometric: File
Import
IGES 3. Selected the cooling tower model (De1.igs) in the files list. 4. Checked the connectivity-based in colouring geometry: Specify colour mode. Note: Since the models were already arranged on AutoCAD that is can be reduced and eliminated the short edges depend on the knowledge on the model developed. If it is needed to eliminate very short edges, there are connection facilities available: Geometry
Edge
Connect/Disconnect edges. Since, there is no problem the connection between vertex and the sort edges, the step process can be continued. 5. Created a surface mesh on the face of the cooling tower body: Mesh
Face
Mesh Faces. In this case, mash faces developed on model is triangular element of pave type generation with 10 interval count spacing, shown in figure 3.18. Note:

the

surface

mesh of model body can be removed from display in order to make easier to see in the next steps.

Fig. 3.21. Surface mesh on rear of cooling tower

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6. Created a brick around the cooling tower body: Geometry
Volume
Create volume. The width (x) =48, depth (y) =22.4, and height (z) =16, shown in figure 3.19.and 3.22.

Fig. 3.22 Brick and Cooling tower

7. Removed unwanted geometry: Geometry
Volume
Delete Volumes. 8. Created straight edges on one of the line nearly to the bottom of model: Geometry
Edges


Split/merge edges. Two times performed split of the nearly line in order to make another surface connection between the model and the domain developed. Create straight edges between two point/vertex: Geometry
Edge
Create Edge (Fig.3.22) 9. Created faces at the bottom wall plane, where the model is mounted: Geometry
Face
Form face. Once the face can be made, the face creations need to be verified: Geometry
Face
Summarize/query faces/total entities. 10. Created volume: Geometry
Volume
Form volume. It can be applied in the stitch faces form to accept the selection of the faces to create volume. 11. Created mesh the edges: Mesh
Edge
Mesh Edges. The mesh created on the faces of the cooling tower is used a fine mesh and for the volume, more coarse mesh created. This can be done by instructed the Gambit to gradually change the mesh density between the coarse and the fine meshes. Its mean, to specify the distribution of nodes along the some edges in the geometry. 12. Created mesh the volume: Mesh
Volume
Mesh Volumes. The tetrahedral or hybrid from the elements option menu under schema in the mesh volume form of Tgrid was selected with interval count at 30. 13. Examined the volume mesh: Examine mesh. It can be identified mesh volume created, aspect ratio, how many nodes created and the skew of floating element volume created. The 3D Wind Loading on a Fabric Structure

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element or mesh volume can be evaluated of 111841 mesh volume and creating 23738 nodes developed with 1: 4 of aspect ratios.

Fig. 3.23 Elements within a specified quality range of 0.6 upper and 0. 7 lower ratios

14. Set Boundary Types: Zones
Specify Boundary Types. It is defined the velocity inlet zone in the surface of entry boundary, the outflow zone in the surface of exit boundary and the rest is wall or free boundary. (Fig. 3.20.) 15. Exported the mesh and saved the session problem: File
Export
Mesh
Accept. To save the current session: File
Exit (Gambit asked whether the session will be saved or not)

Read the grid points and geometry of the cooling tower in flow domain (in Fluent): 1. Selected File
Read
Case. In Select File, select De1.msh from the listing of available files shown, then OK. Fluent will read in the grid geometry and mesh that was previously created by Gambit. Some information is displayed on the main screen. If all went well, it should give no errors, and the word Done should appear. 2. Verified the integrity of the grid: Grid
Check. Look for any error messages that indicate a problem with the grid. If the grid is not valid, it will have to return to Gambit and regenerate the grid. 3. Look at the grid: Display
Grid
Display. A new window opens up showing the grid. If this window is too big, rescale it by dragging the edges of the window. It is best if the graphical display window is small enough that both it and the Fluent window are both visible simultaneously.

4. The graphical display can be zoomed-in or zoomed-out with the middle mouse button.

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Define the boundary conditions: 1. In Fluent, Define
Models
Viscous. Laminar flow is the default and the further investigation will also provided Turbulent flow calculations, where both of model flows are specified in Fluent.
OK. 2. The boundary conditions need to be specified. In Gambit, the boundary conditions were declared, i.e. wall, velocity inlet, etc., but actual values for inlet velocity, etc. were never defined. This must be done in Fluent. In Fluent: Define
Boundary
Conditions, and a new Boundary Conditions window will pop up. 3. In Boundary Conditions, selected name of velocity inlet or whatever named, which is the left side of the computational domain.
Set. 4. In Velocity Inlet, change Velocity Specification Method to Magnitude and Direction. Change Velocity Magnitude to 1 m/s.
OK. 5. The fluid needs to be defined. In Boundary Conditions, select fluid, and Set. The default fluid is air, which is the fluid we desire in this problem. Select air as the Material Name in the Fluid window, and
OK. Note: Defining
Material is the air as the default material name with default the density of 1.225 (kg/m3) and viscosity of 1.7894 x 10 –5 (kg/m-s). Leave the Materials window. 6. Return to the Boundary Conditions window. The default boundary conditions for the wall and the outflow are okay, so nothing needs done to those 7. Finally, Close the Boundary Conditions window. Set up some parameters and initialize: 1. In Fluent: Solve
Initialize
Init. The default initial values of velocity and gage pressure are all zero. The convergence can be sped up slightly be giving more realistic values of the initial velocity distribution. Apply, Init and Close. 2. As the code to monitor iterates, "residuals" are calculated for each flow equation. These residuals represent a kind of average error in the solution - the smaller the residual, the more converged the solution. In the main window, Solve
Monitors
Residual. In Residual Monitors, turn on Plot in the Options portion of the window. The Print option should already be on by default. Here, Print refers to text printed in Fluent, and Plot causes the code to plot the residuals on the screen while the code is iterating.

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3. The convergence criteria need to be set. In Fluent, Solve
Iterate to open up the Iterate window. The Number of Iterations can be predicted into small or big numbers depend on the model developed. For Laminar flow problem, iteration set up to 250, and for the Turbulent flow problem the iteration set until 1600 and
Iterate. The main screen will listed the number of the residuals after every iteration, while the graphics display window will plot the residuals as a function of iteration number. It can be seen in figure 3.24 (laminar) and 3.25 (turbulent). 4. Once the convergence criteria of the iteration has archived, the graphical and diagram data can be exploited in order to collect the target data. Since there are several measurements can be obtained, the only criteria of suitable data has been selected, and collected as a report project. It is because the relevant issue such as of pressure coefficient is the significant data targeted. In Fluent: Contour
Pressure
Pressure Coefficient
Display. In this stage, graphical contour and plot of diagram can be collected. The result selected can be printed out directly to the printer or can be saved as a variety of file format desired. In this study, the pressure coefficient contour as well as the diagram has saved in .EPS format in order to link with the other post-processor programme.

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Fig. 3.24 Plot the residual of laminar flow and number iteration converged at 118.

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Fig. 3.25 Plot the residual of turbulent flow and 427 number iteration converged

3.4.1. B The Result of the Laminar Flows of Cooling Tower

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Fig. 3.26.a Pressure coefficient contour of the whole body from the top of plan (Coded De1)

Fig. 3.26.b Pressure coefficient contour of the whole body from side elevation

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Fig. 3.26.c Diagram pressure coefficient in distance position of the model to the sources.

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Fig. 3.26.d Pressure coefficient contour occurred at z = 0.2 H ~ 32 m (Plane-5)

Fig. 3.26.e Diagram pressure coefficient at z =0.2 H ~ 32 m (Plane-5)

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Fig. 3.26.f Pressure coefficient contour occurred at z = 0.45 H ~ 72 m (Plane-6)

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Fig. 3.26.g Diagram pressure coefficient at z = 0.45 H ~ 72 m (Plane-6)

Fig. 3.26.h Pressure coefficient contour occurred at z = 0.7 H ~ 112 m (Plane-7)

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Fig. 3.26.i Diagram pressure coefficient at z = 0.7 H ~ 112 m (Plane-7)

3.4.1. C The Result of the Turbulent Flows under Large Eddy Simulation (LES) of Cooling Tower

Fig. 3.27.a Pressure coefficient contour of the whole body from the top of plan (Coded De11)

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Fig. 3.27.b Pressure coefficient contour of the whole body from side elevation

Fig. 3.27.c Diagram pressure coefficient in distance position of the model to the sources.

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Fig. 3.27.d Pressure coefficient contour occurred at z = 0.2 H ~ 32 m (Plane-5)

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Fig. 3.27.e Diagram pressure coefficient at z =0.2 H ~ 32 m (Plane-5)

Fig. 3.27.f Pressure coefficient contour occurred at z = 0.45 H ~ 72 m (Plane-6)

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Fig. 3.27.g Diagram pressure coefficient at z = 0.45 H ~ 72 m (Plane-6)

Fig. 3.27.h Pressure coefficient contour occurred at z = 0.7 H ~ 112 m (Plane-7)

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Fig. 3.27.i Diagram pressure coefficient at z = 0.7 H ~ 112 m (Plane-7)

3.4.2 Multiple Cooling Tower model This model was also adopted from the original Heler type dry-cooling tower, which was tight together of four

cooling

tower

with

connection in between, figure 3.28.

This

shape

is

developed is because the inspiration of the shape of tent that nearly has the same curve as the domination of

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the

fabric

membrane

structures, figure 1.2.

Fig. 3.28 Sketch of Multiple Cooling Tower (De5.igs of IGES file)

The cooling tower model was measured as height (H = 13 cm), the top diameter (D2 = 7 cm each), and the bottom structures (D1= 22 cm). This model geometry was modified in CFD with was selected as a default of measurement in meter. It is smaller scale developed that was made as H=1.3 m, D2=0.7m and D2=2.2m, respectively. This multiple cooling tower has also divided into several surfaces with every connection has a rib. This model was used 6 ribs and 1cover on the top, while there is no surface on the bottom. (Figure 3.17). A part of cooling tower geometry includes a number of small surfaces will be generated as constrain the mesh generation process as figure 3.18.

The domain developed was based on the

height of model situated

in

the

suitable

place.

The

dimensional of domain developed is 39 x 18.2 x 13 in unit with along of x direction as z y

x

the length, y direction as wide and z direction

as

the

height

of

domain

respectively. Fig. 3.29 Domain of Multiple Cooling Tower

For this investigation, the computational region developed is typically as shown in figure 3.20. The grid number for the cooling tower simulation is 35 x 10 x 25, i.e. 35 grids in the main flow direction, 10 grids along the circumference of the tower and 25 grids in wide direction. Detail procedure and instruction was typical information to be described start from page 71 above.

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Recapitulation to read the grid points and geometry of the multiple cooling tower (in Gambit) can be described in figure scheme below with the same procedure in importing and cleaning up model design.




Fig. 3.30. Grid mesh generating of imported file IGES from AutoCAD in Gambit.

Fig. 3.31. Surface mesh on rear of multiple cooling tower

Creating a brick around the multiple cooling tower body of the width (x) =39, depth (y) =18, 2, and height (z) =13, can be described below in figure 3.32.

Fig. 3.32 Brick and Cooling tower

Fig. 3.33 Elements within a specified quality range of 0.6 upper and 0. 7 lower ratios

Examining the volume mesh is part of evaluating the reliability of element developed. It can be identified mesh volume created, aspect ratio, how many nodes created and the skew of floating element volume created. The 3D element or mesh volume can be evaluated of 34683 meshes volume and creating 162937 nodes developed with 1: 7 of aspect ratios, which domain developed as 4 wall, 1 velocity inlet and 1 outflow. There is different meshes volume developed when the domain is as 3 walls, 1 velocity inlet and 2 outflows. The meshes volume developed is 29765 elements and 15042 nodes with aspect ratio of 1:7. By defining Velocity Inlet Magnitude typically at 1 m/s and the material air with default the density of 1.225 (kg/m3) and viscosity of 1.7894 x 10 –5 (kg/m-s), then the Fluent will processed the flow problem Typical process has been applied as well to process iteration prediction into a numbers Wind Loading on a Fabric Structure108

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depend on the model developed. Typical Laminar flow problem, iteration set up to 250, and the Turbulent problem iteration set until 1600. After process, numbers of iteration of the laminar is 104 and 308 iteration for turbulent The graphical and diagram data can be exploited, since the convergence criteria of the iteration have archived. There are several data and graphical measurement obtained with the only criteria of suitable data selected, and collected. The relevant issue to wind loading is pressure coefficient is the significant data targeted, that described below:

3.4.2. A The Result of the Turbulent Flows under Large Eddy Simulation (LES) of Multiple Cooling Tower Since the turbulent flows have the opportunity to present more promises result of the relevant pressure coefficient issue, so that the result of turbulent flows under Large Eddy Simulation (LES) presented. In this case, all the result as graphical

Fig. 3.34.a Pressure coefficient contour of the whole body from the top of plan

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Fig. 3.34.b Pressure coefficient contour of the whole body from side elevation

Fig. 3.34.c Diagram pressure coefficient in distance position of the model to the sources.

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Fig. 3.34.d Pressure coefficient contour occurred at z = 0.2 H ~ 26 m (Plane-5)

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Fig. 3.34.e Diagram pressure coefficient at z =0.2 H ~ 26 m (Plane-5)

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Fig. 3.34.f Pressure coefficient contour occurred at z = 0.45 H ~ 58.5 m (Plane-6)

Fig. 3.34.g Diagram pressure coefficient at z = 0.45 H ~ 58.5 m (Plane-6)

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Fig. 3.34.h Pressure coefficient contour occurred at z = 0.7 H ~ 91 m (Plane-7)

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Fig. 3.34.i Diagram pressure coefficient at z = 0.7 H ~ 91 m (Plane-7)

3.4.3 Single Sphere Model This

model

was

adopted from the original domes as the arched roof based on the Maher and Blessmann.

The

shape

developed is the basic of domes structures inspiration, which is used to be a fabric membrane structures, figure 3.35. This model is a circular dome rising directly from the ground with y/d = ½.

Fig. 3.35 Sketch of Single Sphere (De3.igs of IGES file)

The sphere model was measured as height, y or H = 7.5 cm, and diameter of bottom structures is D = 15 cm. Another model of circular dome formatted as y/d =1/4 and y/d =1/6 are also developed, which are presented in Appendix 3. The model geometry was modified in CFD with was selected as a default of measurement in meter. A smaller scale developed of H=0.75 m, and D=1.5m, respectively. The single sphere has divided into 8 (eight) surfaces with every connection have a rib. The surface of the sphere is known as smooth domes in the field of researcher. In this term, the geometry developed is involved small interval (10 interval meshes) of surfaces will be generated as constrain the mesh generation process as similar prospect in figure 3.18. The domain developed was based on the height of model, which is situated in the suitable place. The

dimensional

of

domain

developed is 22.5 x 10.5 x 7.5 in unit, which are long x direction as the length, y direction as wide and z direction as the height of domain

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respectively.

z y

x

Fig. 3.36 Domain of Single Sphere.

The computational region developed is typically as shown in figure 3.20. The grid number for the single sphere simulation is 35 x 8 x 20, i.e. 35 grids in the main flow direction, 8 grids along the circumference of the tower and 20 grids in wide direction. Detail procedure and instruction was typical information to be described start from page 71 above. Recapitulation to read the grid points and geometry of the single sphere (in Gambit) can be described in figure scheme below with the same procedure in importing and cleaning up model design.




Fig. 3.37. Grid mesh generating of imported file IGES from AutoCAD in Gambit and already meshed on rear of sphere surface.

Fig. 3.38. Brick and Sphere

Creating a brick around the multiple cooling tower body of the width (x) =22.5, depth (y) =10.5, and height (z) =7.5, can be described above in figure 3.38.

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Fig. 3.39 .The mesh developed on domain.

Fig. 3.40 Elements within a specified quality range of 0.6 upper and 0. 7 lower ratios

To evaluate the reliability of element developed the volume meshes need to be examined. The examining identified the mesh volume created, aspect ratio, how many nodes created and the skew of floating element volume created. The 3D element or mesh volume can be evaluated of 120838 meshes volume and creating 25504 nodes developed with 1: 4 of aspect ratios, which domain developed as 4 wall, 1 velocity inlet and 1 outflow. There is different meshes volume developed when the domain is as 3 walls, 1 velocity inlet and 2 outflows. The meshes volume developed is 25519 elements and 120909 nodes with aspect ratio of 1:4. Typical process has been applied as well to process iteration prediction set up to 250 for laminar flow and set up 1600 iteration for turbulent problem flows. The result is 88 iteration for laminar and 470 iterations for turbulent problem. Velocity Inlet Magnitude defined at 1 m/s and the material air with default the density of 1.225 (kg/m3) and viscosity of 1.7894 x 10 –5 (kg/m-s), then the Fluent will processed the flow problem The graphical and diagram data can be obtained since the convergence has been archived. Several data and graphical measurement collected that is described below:

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3.4.3. A The Result of the Turbulent Flows under Large Eddy Simulation (LES) of Single Sphere

Fig. 3.41.a Pressure coefficient contour of the whole body from the top of plan (Coded De31)

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Fig. 3.41.b Pressure coefficient contour of the whole body from side elevation

Fig. 3.41.c Diagram pressure coefficient in distance position of the model to the sources.

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3.4.4 Multiple Sphere Model This arranged

of

spheres

that

model

was

four

single

are

tight

together. In this particular case, the geometry of multi sphere developed is inspirited by the tensile structure and portable structure published literature.

The

shape

developed is among of the four sphere is model of circular dome rising directly from the ground with y/d = ½. That is created as fabric membrane structures, (figure

Fig. 3.42 Sketch of Multiple Sphere (De4.igs of IGES file)

3.42) The multiple sphere model was measured as height, y or H = 7.5 cm, and diameter of bottom structures is D = 25 cm. The model geometry was modified in CFD with was selected as a default of measurement in meter. A smaller scale developed of 1/10 of the real model that is H=0.75 m, and D=2.5m, respectively. On every single sphere has divided into 8 (eight) surfaces with every connection has a rib. In Gambit process, the geometry developed is small interval (10 interval meshes), which the surfaces generated as constrain the mesh generation process as similar potential in figure 3.18. The domain developed was based on the height of model, which is situated in the suitable place. The dimensional of domain developed is the same as the single sphere due to the same height that is 22.5 x 10.5 x 7.5 in unit, which are long x direction as the length, y

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direction as wide and z direction as the height of domain respectively.

z y

x

Fig. 3.43 Domain of Multiple Spheres.

The computational region developed is typically as shown in figure 3.20. The grid number for the single sphere simulation is 35 x 10x 25, i.e. 35 grids in the main flow direction, 10 grids along the circumference of the tower and 25 grids in wide direction. Detail procedure and instruction was typical information to be described start from page 71 above. Recapitulation of the grid points and geometry of the multiple spheres (in Gambit) with the same procedure in importing and cleaning up model design can be described in figure scheme below:




Fig. 3.44. Grid mesh generating of imported file IGES from AutoCAD in Gambit

Fig. 3.45. Surface mesh on rear of multiple cooling tower

Creating a brick around the multiple cooling tower body of the width (x) =48, depth (y) =22.4, and height (z) =16, can be described below in figure 3.39.

Fig. 3.46. Brick and Sphere

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Fig. 3.47 Elements within a specified quality range of 0.6 upper and 0. 7 lower ratios

The examining identified of the mesh volume created were evaluated 163452 meshes volume and 34763 nodes developed with 1: 4 of aspect ratios, which domain developed as 4 wall, 1 velocity inlet and 1 outflow. There is different meshes volume developed when the domain is as 3 walls, 1 velocity inlet and 2 outflows. The meshes volume developed is 162937 elements and 34683 nodes with aspect ratio of 1:4. Typical process has been applied as well to process iteration prediction set up to 250 for laminar flow and set up 1600 iteration for turbulent problem flows. The result is 99 iterations for laminar and error result in iterations for turbulent problem. Velocity Inlet Magnitude defined at 1 m/s and the material air with default the density of 1.225 (kg/m3) and viscosity of 1.7894 x 10

–5

(kg/m-s), then

the Fluent will processed the flow problem The graphical and diagram data can be obtained since the convergence has been archived. Several data and graphical measurement collected that is described below:

3.4.4 .a The Result of the Turbulent Flows under Large Eddy Simulation (LES) of Multiple Spheres

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Fig. 3.48.a Pressure coefficient contour of the whole body from the top of plan (Coded De4)

Fig. 3.48.b Pressure coefficient contour of the whole body from side elevation

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Fig. 3.48.c Diagram pressure coefficient in distance position of the model to the sources.

In this particular case, the result of laminar flows problem can be described in Appendix 3. In addition, one example of turbulent flow problem (RAN) has also been investigated. For further detail, all result from CFD method can be described in Appendix 3 and any other reason for this decision can also drawn in the summary of this study.

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Chapter 5. Conclusion 5.1 Introduction Four scale models were constructed, and wind load testing of these models have been observed regarding the behaviour of fabric membrane structure including two models as an additional test. The conclusion drawn from observations will be summarized below, along with recommendations for further research.

5.2 Wind Tunnel Testing Unfortunately, wind tunnel testing cannot be done; it is because a non-popular reasons that the technician and specific tools are not available during the period time. Those models promised to test in wind tunnel at that time, then such as a problem came up before testing. 1:1000 scale model was already constructed, however it doesn’t complete with the requirement of model such as maintaining the pressure taps. The technician was not available is a real problem to avoid testing. Fortunately, published data available led to a number of observations regarding the behaviour of wind load to these structures. In this particular case, data of wind tunnel test has completely been replaced and the published data available represented wind tunnel study.

5.3 Wind loading test by CFD method The numerical investigation of wind load testing on CDF method indicated that the general nature of the pressure and suction distributions on the model were obtained. In this particular case, the advantages of CFD method applied wind load to fabric membrane structure rather than to solve the dynamic fluid behaviour of wind problems. However, various flow model problems have been tried such as laminar and turbulence to the model structures. These current statuses of flow model applied in order to recognise a better result can be obtained. Laminar flow problem has been applied to the all model structures as a default in Fluent or solver CFD. Reynolds Number simulation (RANS) has been tried in sphere model only in order to compare to other simulation model. The latest model simulation has been tried is the dynamic Large Eddy simulation (LES).

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5.3.1 Comparison reliability between Laminar and Turbulent problem flow model in CFD method The results of these methods are more concern in predicting pressure coefficient, which is useful for the wind load to the structure. Eventually, many parameters can be observed in this area related to the result of CFD method such as velocity, dynamic and static pressure, and other behaviour of wind load to the structures. The result has been compared between the laminar and the turbulent model. More specific comparison has been done to the laminar simulation model, RANS model and the LES model. From the three models simulation involved, the dynamic Large Eddy simulation (LES by Smagorinsky & Lilly’s) model indicated has a better result obtained. The laminar model offered

the worst result in every stage investigation, however there is not take for along time to get the result when it running the iteration. On average, RANS and LES model need time consumed longer than in laminar model. It is depend on the capacity of computer has been used. This result can be proofed and observed in many other area studies. The results are indicated have agreement to earlier studies and indeed are supported by the researcher whom concerned to this problem.

5.3.2 Comparison between published data and CFD method study of wind loading to fabric membrane structure. Fabric membrane structures model has been developed in several shape model that are sphere, shape model as a cooling tower and tandem model or combination on each the basic model. The single sphere and single cooling tower shape model are similar to the model developed on the earlier study. Published data by Maher on domes model and ASCE 1987 on cooling tower replaced data wind tunnel testing on model fabric membrane structure. The numerical experiment on CFD method gives good opportunity to predict pressure distribution of wind loading to fabric structure and other parameter required. It is because wide range capabilities belong to the CFD and it is quite easy to develop model desired. It is also more application programme has already link with the CFD such as AutoCAD program, in order to make it easy to develop model. The pressure coefficient distribution that obtained from the CFD methods has agreed to the published data available, particularly on domes and cooling tower model. Very close value of maximum positive Cp = + 0.621 by dynamic LES turbulent method to the Cp = +0.6 (Maher’s

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dome study of h/D =½). Maximum Cp at the centre of dome of by CFD is Cp = –1.2 and Maher has Cp = -1. Pressure coefficient on various h/D values such as ¼ and 1/6 has also been observed that indicated similar argue to the published data. Mean and fluctuating pressure distribution on cooling tower by the CFD has also indicated good agreement to the ASCE published. The maximum value of pressure coefficient (positive Cp = + 0.965) occurred around the throat of cooling tower compare favourably to the ASCE at Cp =+1.0. The maximum negative Cp=- 1.4 to –1.7 were obtained by CFD that compared to the Cp = -1.5 by ASCE. This result has smoothly promised prediction on the study, which led to a number of observations regarding the behaviour of fabric membrane structures.

5.4 General Conclusion and Recommendations It was good opportunity of CFD method to predict wind loading to the fabric membrane structures. The result is promised to solve a problem such as wind load acting to the fabric structure or to other structures. The model developed in CFD is more flexible depend on the ability of user to make it. Many kinds model structure can be developed easily regarding to the aims of research study. All of them are such as the advantage of CFD methods presented, however it is depend on the capability and availability of computer hardware and the software. As an individual conclusion that this study would probably be replaced the wind tunnel testing to predict pressure coefficient and other parameters intended. It also may be concluded that many advantages can be used for other study related to structural engineering. This investigations have been conducted indicate that additional study of this type needs to be executed in order to obtain a better understanding to fabric membrane structures. One aspect that future research should address is the real wind tunnel testing to this type model structure. It is belief that more confidence study will be presented when wind tunnel studies can be established.

Appendix 1 In general, fundamental aspect cable mechanics can be illustrated by the treatment of a weightless string supported at A and B and loaded as shown in figure 1. The moment at a point X in the cable is given by:

Mx = Mex − Hz

1

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Moment at any point on cable = 0

T1

T5

z= H

B

A

2

H

H

SB SA X P4 P1 P2

Mex

P3

Fig. A1.1Funicular curve of a cable loaded with point loads

where Mex is the simple bending moment at x. Consider a string stretched between A and B figure A1. 2. a When load P to a central , it deflects By an amount W1, and the value of the tension in it changes from T1 to T1+∆ T1.

P

For the equilibrium of the deflected string,

P = 4(T1 + ∆T1 )

W1 L

3

P+P

Fig. A1.2 Load vs. deflection for a taut string or W 1 =

PL 4T2

4

where T2 = T1+ ∆ T1 Further loaded with an additional load P, it undergoes a change in tension equal to ∆T2 and a deflection W2 as shown in figure A1.2 b. For the equilibrium of final deflected shape.

2 P = 4(T2 + ∆T2 )(W 1 + W 2)

1 L

5

or, on substitution of the value of W1 from Eq. 3-4 and rearranging,

W2 =

PL − 4∆T2W 1

4(T2 + ∆T2 )

6

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Appendix 2

1. Differential Equations of equilibrium Initial consideration to the equilibrium of a plane element subjected to normal stresses σ x and

σ y , in plane shear stress τ xy (in unit of force per unit volume), and body forces Xb and Yb (in units of force per unit volume), as shown in figure A2.1

σ

σ

Firstly, the stresses are assumed to

τ

τ

be constant as they act on the width of each face, however the stresses

σ

are assumed to vary from one face

σ

σ

to the opposite. σ x acting on the

τ σ

τ

τ

τ

Fig.A2.1 Plane differential element subjected to stresses, by Logan



vertical face, whereas σ ∂   σ x +  x dx act on the right.  ∂x 

Summing forces in the x direction,

∂σx  dx dy (1) − σxdy (1) + Xb dx dy (1) + ∂x  ∂τ yx  + dy dx(1) − τ yx dx(1) = 0  ∂y 

∑ Fx =0 =  ∂x +  τ  yx 

left

1

Simplifying and canceling term in Eq. 1, obtained

∂σ x ∂τ yx + + Xb = 0 ∂x ∂y

2

Summing forces in the y direction

∂σ y ∂τ xy + + Yb = 0 ∂y ∂x

3

Three equilibrium equations must be satisfied, when considering only planar element. The third equation is equilibrium of moments about an axis normal to the x-y plane; taking moment about point C in figure A2.1.

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∂τ  dx dx  dy dy (1) + τ xy + xy dx  − τ yx dx(1) − ∂x 2  2 ∂τ 2  dy  τ yx + yx dy  ∂y  2 

∑ Mx =0 = τ

xy

4

Simplifying Eq.4 and neglecting higher –order term yields

τ xy = τ yx

5

Considering the three-dimensional state of stress (figure A2.2), which shows the additional stresses σ z , τ xz and τ yz . Extended the two dimensional equations 2, 3 and 5 to three dimensions, esulting total set of equilibrium equations is

∂σ x ∂τ xy ∂τ xz + + + Xb = 0 ∂x ∂y ∂z ∂τ xy ∂σ y ∂τ yz + + + Yb = 0 ∂x ∂y ∂z ∂τ xz ∂τ yz ∂σ z + + + Zb = 0 ∂x ∂y ∂z

6 The simplified equation is

σ

τ

τ σ

τ xy = τ yx

τ

τ τ

τ

σ

τ xz = τ zx

7

τ yz = τ zy

Fig.A2.2 Three-dimensional stress element, by Logan

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2. Strain/Displacement and Compatibility Equations Considering the differential element shown in figure A2.3, the un-deformed state is represented

by

the

dotted

lines

and

deformed shape is represented by the solid lines. Considering line element AB in the x direction, and A’B’ after deformation, where u an v represented the displacement in the x and y directions. Fig.A2.3 Differential element before and after deformation, by Logan

A' B'− AB AB

εx =

1.1 AB = dx 1.2 2

∂u   ∂v   2 ( A' B' ) =  dx + dx  +  dx  ∂x   ∂x   Evaluating

A’B’

2

using

the

binomial

2

1.3 theorem

and

neglecting

the

higher-order

term

2

 ∂u   ∂v    and   , it has  ∂x   ∂x  A' B ' = dx +

∂u dx ∂x

1.4

Using Eqs. 1.2 and 1.4 in Eqs. 1.1, obtained

εx =

∂u ∂x

1.5

Similarly, line element AD in y direction is

εy =

∂u ∂y

1.6

The shear strain is defined to be the changed in the angle between two lines, such as AB and AD. From figure A2.2 that the shear strain γ xy is the sum of two angles and is given by

γ xy =

∂u ∂v + ∂y ∂x

1.7

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Equations 1.5, 1.6 and 1.7 represent the strain/displacement relationship for in-plane behaviour. For the three dimensional, a displacement w in the z direction, than becomes straightforward the additional strain/displacement equation as

εz =

∂w ∂z

1.8

γ xz =

∂u ∂w + ∂z ∂x

1.9

γ yz =

∂v ∂w + ∂z ∂y

1.10

For the planar-elastic case, the compatibility equation by differentiating γ xy with respect to both x and y, and then using the definitions for ε x and ε y given by Eqs. 1.5 and 1.6 so that:

∂ 2γ xy

2 2 ∂ 2 ∂u ∂ 2 ∂v ∂ ε x ∂ ε y = + = + ∂x∂y ∂x∂y ∂y ∂x∂y ∂x ∂y 2 ∂x 2

1.11

This equation is called the condition of compatibility.

Appendix 3 1. The result of simulation models under laminar flows problem A. Single Cooling Tower (4Wall)

111841 mesh volume Fig. A3.1. a Pressure coefficient contour of the whole body from the top of plan (Coded De1)

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23758 nodes element Fig. A3.1. b Pressure coefficient contour of the whole body from side elevation

Fig. A3.1. c Diagram pressure coefficient in distance position of the model to the sources

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Fig. A3.1.d Pressure coefficient contour occurred at z = 0.2 H ~ 32 m (Plane-5)

Fig. A3.1.e Diagram pressure coefficient at z =0.2 H ~ 32 m (Plane-5)

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Fig. A3.1.f Pressure coefficient contour occurred at z = 0.45 H ~ 72 m (Plane-6)

Fig. A3.1.g Diagram pressure coefficient at z =0.45 H ~ 72 m (Plane-6)

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Fig. A3.1.h Pressure coefficient contour occurred at z = 0.7 H ~ 112 m (Plane-7)

Fig. A3.1.i Diagram pressure coefficient at z =0.7 H ~ 112 m (Plane-7)

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B. Single Sphere h/D = ¼ (4Wall)

86506 mesh volume Fig. A3.2.a Pressure coefficient contour of the whole body from the top of plan (Coded De2)

18603 nodes element Fig. A3.2.b Pressure coefficient contour of the whole body from side elevation

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Fig. A3.2.c Diagram pressure coefficient in distance position of the model to the sources.

C. Single Sphere h/D = ½ (4Wall)

120838 mesh volume Fig. A3.3.a Pressure coefficient contour of the whole body from the top of plan (Coded De3)

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25504 nodes element Fig. A3.3.b Pressure coefficient contour of the whole body from side elevation

Fig. A3.3.c Diagram pressure coefficient in distance position of the model to the sources.

D. Multiple Sphere (4Wall)

163452 mesh volume Fig. A3.4.a Pressure coefficient contour of the whole body from the top of plan (Coded De4)

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34762 nodes element Fig. A3.4.b Pressure coefficient contour of the whole body from side elevation

Fig. A3.4.c Diagram pressure coefficient in distance position of the model to the sources.

D. Multiple Sphere (4Wall)

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163452 mesh volume Fig. A3.4.a Pressure coefficient contour of the whole body from the top of plan (Coded De4)

34762 nodes element Fig. A3.4.b Pressure coefficient contour of the whole body from side elevation

Fig. A3.4.c Diagram pressure coefficient in distance position of the model to the sources.

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E. Multiple Cooling Tower (4Wall)

229257 mesh volume Fig. A3.5. a Pressure coefficient contour of the whole body from the top of plan (Coded De5)

48695 nodes element Fig. A3.5. b Pressure coefficient contour of the whole body from side elevation

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Fig. A3.5. c Diagram pressure coefficient in distance position of the model to the sources

Fig. A3.5.d Pressure coefficient contour occurred at z = 0.2 H ~ 26 m (Plane-5)

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Fig. A3.5.e Diagram pressure coefficient at z =0.2 H ~ 26 m (Plane-5)

Fig. A3.5.f Pressure coefficient contour occurred at z = 0.45 H ~ 58.5 m (Plane-6)

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Fig. A3.5.g Diagram pressure coefficient at z =0.45 H ~ 58.5 m (Plane-6)

Fig. A3.5.h Pressure coefficient contour occurred at z = 0.7 H ~ 91 m (Plane-7)

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Fig. A3.5.i Diagram pressure coefficient at z =0.7 H ~ 91 m (Plane-7)

F. Single Sphere h/D = 1/6 (4Wall)

93135 mesh volume Fig. A3.6.a Pressure coefficient contour of the whole body from the top of plan (Coded De6)

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19135 nodes element Fig. A3.6.b Pressure coefficient contour of the whole body from side elevation

Fig. A3.2.c Diagram pressure coefficient in distance position of the model to the sources.

2. The result of simulation models under turbulent flows problem (LESSmagorinsky &Lilly) A. Multiple Cooling Tower (3Wall)

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112827 meshes volume Fig. A31.1. a Pressure coefficient contour of the whole body from the top of plan (Coded LESDe11)

2393158 nodes element Fig. A31.1. b Pressure coefficient contour of the whole body from side elevation

Fig. A31.1. c Diagram pressure coefficient in distance position of the model to the sources

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Fig. A31.1.d Pressure coefficient contour occurred at z = 0.2 H ~ 32 m (Plane-5)

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Fig. A31.1.e Diagram pressure coefficient at z =0.2 H ~ 32 m (Plane-5)

Fig. A31.1.f Pressure coefficient contour occurred at z = 0.45 H ~ 72 m (Plane-6)

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Fig. A31.1.g Diagram pressure coefficient at z =0.45 H ~ 72 m (Plane-6)

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Fig. A31.1.h Pressure coefficient contour occurred at z = 0.7 H ~ 112 m (Plane-7)

Fig. A31.1.i Diagram pressure coefficient at z =0.7 H ~ 112 m (Plane-7)

B. Single Sphere h/D = ¼ (3Wall)

85392 meshes volume Fig. A31.2.a Pressure coefficient contour of the whole body from the top of plan (Coded LESDe21)

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18413 nodes element Fig. A31.2.b Pressure coefficient contour of the whole body from side elevation

Fig. A31.2.c Diagram pressure coefficient in distance position of the model to the sources.

C. Single Sphere h/D = ½ (3Wall)

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120909 meshes volume Fig. A31.3.a Pressure coefficient contour of the whole body from the top of plan (Coded LESDe31)

25519 nodes element Fig. A31.3.b Pressure coefficient contour of the whole body from side elevation

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Fig. A31.3.c Diagram pressure coefficient in distance position of the model to the sources.

D. Multiple Sphere (3Wall)

162937 meshes volume Fig. A31.4.a Pressure coefficient contour of the whole body from the top of plan (Coded LESDe41)

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34683 nodes element Fig. A31.4.b Pressure coefficient contour of the whole body from side elevation

Fig. A3.4.c Diagram pressure coefficient in distance position of the model to the sources.

E. Multiple Cooling Tower (3Wall)

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29765 mesh volume Fig. A31.5. a Pressure coefficient contour of the whole body from the top of plan (Coded LESDe51)

15042 nodes element Fig. A31.5. b Pressure coefficient contour of the whole body from side elevation

Fig. A31.5. c Diagram pressure coefficient in distance position of the model to the sources

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Fig. A31.5.d Pressure coefficient contour occurred at z = 0.2 H ~ 26 m (Plane-5)

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Fig. A31.5.e Diagram pressure coefficient at z =0.2 H ~ 26 m (Plane-5)

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Fig. A31.5.f Pressure coefficient contour occurred at z = 0.45 H ~ 58.5 m (Plane-6)

Fig. A31.5.g Diagram pressure coefficient at z =0.45 H ~ 58.5 m (Plane-6)

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Fig. A31.5.h Pressure coefficient contour occurred at z = 0.7 H ~ 91 m (Plane-7)

Fig. A31.5.i Diagram pressure coefficient at z =0.7 H ~ 91 m (Plane-7)

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F. Single Sphere h/D = 1/6 (3Wall)

92894 meshes volume Fig. A31.6.a Pressure coefficient contour of the whole body from the top of plan (Coded De6)

19290 nodes element Fig. A31.6.b Pressure coefficient contour of the whole body from side elevation

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Fig. A31.6.c Diagram pressure coefficient in distance position of the model to the sources.

3. The result of simulation models under turbulent flows problem (LESRANS) A. Single Sphere h/D = ½ (3Wall)

120909 meshes volume Fig. A32.3.a Pressure coefficient contour of the whole body from the top of plan (Coded RANSDe31)

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25519 nodes element Fig. A32.3.b Pressure coefficient contour of the whole body from side elevation

Fig. A31.3.c Diagram pressure coefficient in distance position of the model to the sources.

Diagram Pressure Coefficient of Single sphere h/D=1/2 Model

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Pressure Coefficient of Single sphere h/D=1/2 Model Convert to Angle Data

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Diagram Pressure Distribution of Single Cooling Tower Model

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Pressure Coefficient of Single Cooling Tower Model Convert to Angle Data

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Diagram Pressure Distribution of Single Cooling Tower Model around throat of Plane 5 =Z1/H=0.2 =32m

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Pressure Coefficient of Single Cooling Tower around throat of Plane 5 =Z1/H=0.2 =32m Convert to Angle Data

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Diagram Pressure Distribution of Single Cooling Tower Model around throat of Plane 6 =Z2/H=0.45 =72m

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Pressure Coefficient of Single Cooling Tower around throat of Plane 6 =Z2/H=045 =72m Convert to Angle Data

Wind Loading on a Fabric Structure174

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

Diagram Pressure Distribution of Single Cooling Tower Model around throat of Plane 7 =Z3/H=0.70 =112m

Wind Loading on a Fabric Structure175

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

Wind Loading on a Fabric Structure176

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University of Newcastle upon Tyne, Dept. of Civil Eng. Structural Eng.

Wind Loading on a Fabric Structure177