Aerodynamics Txt Book

Aerodynamics for Students : A Web Site dedicated to Theoretical Aerodynamics

Table of Contents Introduction Part 1. Fundamentals of Fluid Mechanics Properties of Fluids Fluid Statics Control Volume Analysis, Integral Methods. Potential Flow Dimensional Analysis Introduction to Boundary Layers Viscous Flow in Pipes Part 2. Introduction to Theory of Flight Visit the Australian Ultralight Federation's Flight Theory Pages Aircraft Instruments Part 3. Aerodynamics Properties of the Atmosphere Joukowski Flow Mapping & Aerofoils 2-D Flow Aerofoil Section Geometry Thin Aerofoil Theory (2-D Sections) 2-D Panel Method Solutions Lifting Line Theory (3-D Wings) Vortex Lattice Method (3-D Wings) Part 4. Gasdynamics Supersonic Flow Measurement Simulation of Rarefied Gas Flow Part 5. Aircraft Performance Aircraft Performance (Simple numerical predictions) Aircraft Performance (Exact analytical optimisation approach) Mission Performance based Optimisation Part 6. Propulsion Blade Element Propeller Analysis Ideal Cycle Gas Turbine Analysis Part 7. Data Sheets

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Aerodynamics for Students : A Web Site dedicated to Theoretical Aerodynamics

Conversion Factors Density of Liquids Standard Atmosphere Compressible Subsonic Flow Compressible Supersonic Flow Normal Shock Wave Data Oblique Shock Wave Data Part 8. Assignment/Tutorial Resources MATLAB files EXCEL files FORTRAN files Part 9. Experimental Data Library Wind Tunnel Experimental Data The SNR Wind Tunnel

Copyright © 1995-2006, Aerospace, Mechanical & Mechatronic Engineering, University of Sydney. Authors : D.J. Auld, K. Srinivas with contributions from the students and friends of AMME

Links to Other Aerodynamics/Fluid Mechanics/Flight Theory Web Pages Software for Aerodynamic Design, (W.H.Mason, Virginia Tech) Aerospace Engineering Software, (Java Applets)(W.Davenport, Virginia Tech) Compressible Aerodynamics Calculator. (W.Davenport Virginia Tech) XFOIL Aerofoil section Analysis and Design (Marc Drela, MIT) Aircraft/Wing VLM based Analysis and Design (Marc Drela, MIT) Aerofoil and Propeller Design and Analysis (Martin Hepperle) Australian Ultralight Federation Theoretical Aerodynamics (Desktop Aeronautics) NASA Aeronautics Education Pages (NASA Lewis) Simulation and Explanations of Science (Fluids Mechanics etc.) Advanced Topics in Aerodynamics

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Aerodynamics for Students : A Web Site dedicated to Theoretical Aerodynamics

WHAT IS AERODYNAMICS? Aerodynamics is the branch of dynamics that treats the motion of air (and other gaseous fluids) and the resulting forces acting on solids moving relative to such fluid.

Aerodynamic results will fall into different categories of behaviour depending on velocity range (slow speed, high speed, supersonic, hypersonic), depending on size and shape of the object (large, small, complex 3D solid) and the physical properties of the fluid (dense, rarefied, viscous, inviscid). Many different aerodynamic situations can be analysed using a range of available theories. The important steps of ● ● ● ●

flow field definition, calculation of velocity field around the object, calculation of flow pressure and shear distribution, integration of these distributions on the surface of the body

are the tools used for most theoretical aerodynamic prediction. The aim is to be able to predict the lift, drag, thrust and moments acting on objects or vehicles in motion.

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Aerodynamics for Students : A Web Site dedicated to Theoretical Aerodynamics

WHAT IS LIFT? Lift is the aerodynamic force acting at right angles to the direction of motion of the object. It is produced by the interaction of the moving object and the fluid. This interaction typically leads to a pressure differential being set up between upper surface and lower surface of the object. The nett effect of high pressure below and low pressure above will produce a force which sustains the object against descent due to gravity. The physical mechanisms in the fluid/body interaction that create lift are very complex. The laws of conservation of mass and momentum (including the effect of fluid rotation) result in fluid flow paths, velocity and pressure distributions which can significantly change the magnitude of lift due to small changes in flow angle or surface curvature. It is hoped that by studying the following chapters on the theoretical basis of fluid flow, students will begin to understand these mechanisms.

WHAT IS DRAG? Drag is the aerodynamic force resisting the motion of the object through the fluid. It is produced by front/rear pressure differences, shearing between fluid and solid surface, compression of the gas at high speed and residual lift components induced by 3D flow rotation.

WHAT IS THRUST? Thrust is the aerodynamic force produced in the direction of motion and is required to overcome drag and thus sustain the forward flight of the vehicle. It is produced by mechanical means (an engine) which effectivily transfers energy into the flow, in the form of increased fluid momentum. Thrust is the forward reaction to this fluid momentum change.

WHAT IS MOMENT? Moment is the aerodynamic torque produced by out of balance forces. An object or vehicle has no solid structure to support it in the air. A balance is required and all forces must act through the same point (typically the center of gravity of the object). Any variation of the point of application for aerodynamic forces will produce a couple, leading to a moment which will cause the vehicle to start to rotate. The study of these moments and the effect they have on the stability of motion of the vehicle is covered in more detail in the Flight Mechanics courses.

WHAT IS CONTAINED IN THESE PAGES? In the chapters of this web text we try to explain methods of analysis that can be used to predict the behaviour of various flight vehicles and components. The aim is to provide introductory engineering tools that will help in the undestanding of fundamental aerodynamics. The primary system of units to be used in this text is the SI System.

Contents Page © AMME, University of Sydney, 2006

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propsoffluids

Next: Properties of Fluids

●

Properties of Fluids ❍ What is a Fluid? ❍ Continuum Hypothesis ❍

❍

Viscosity, ■

Formulas for Viscosity

■

Kinematic Viscosity,

Density, ■

Specific Volume, v

■

Specific Weight,

❍

Specific Gravity, SG Pressure, Temperature and Velocity Ideal Gas Law

❍

Bulk Modulus,

❍

Bulk Modulus for Gases Vapour Pressure

■

❍

■

❍

●

Surface Tension, ■ Pressure inside a Drop of Fluid ■ Capillary Tube

Table of Contents.

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Fluids

Next: What is a Fluid? Up: propsoffluids Previous: propsoffluids

Fluids Fluids are very familiar to us. Our body itself is mostly water while what surrounds us is largely air which again is a fluid. In fact, Greeks and Indians in the past worshiped earth, fire, water and air; three of these being fluids. We encounter motion of fluids almost everywhere- inside our bodies, in our daily activity such as taking a shower, cleaning, swimming etc. We also fly and sail which are nothing but motion of an object through a fluid. Fluid Mechanics is a science that studies the behaviour of fluids and its effect on other bodies. It comprises of fluid statics, which is a study of fluids at rest and fluid dynamics, which concerns fluids in motion. Then we also classify them further as aerodynamics, which specialises in flow of air. If we concentrate on water, we have hydrostatics and hydrodynamics.

Figure 1.1:Three Approaches to fluid Mechanics There are three approaches to Fluid Mechanics – Experimental, Theoretical and Computational. Experimental approach is the oldest approach, perhaps also employed by Archimedes when he was to investigate a fraud. It is a very popular approach where you will make measurements using a wind tunnel or a similar equipment. But this is a costly venture and is becoming costlier day by day. Then we have the theoretical approach where we employ the mathematical equations that govern the flow and try to capture the fluid behaviour within a closed form solution i.e., formulas that can be readily used. This is perhaps the simplest of the approaches, but its scope is somewhat limited. Not every fluid flow renders itself to such an approach. The resulting equations http://www.aeromech.usyd.edu.au/aero/fprops/propsoffluids/node1.html (1 of 3)24/2/2006 13:34:47

Fluids

may be too complicated to solve easily. Then comes the third approach- Computational. Here we try to solve the complicated governing equations by computing them using a computer. This has the advantage that a wide variety of fluid flows may be computed and that the cost of computing seems to be going down day by day. With the result the emerging discipline Computational Fluid Dynamics, CFD, has become a very powerful approach today in industry and research. It is worth noting that any theoretical calculation or a numerical computation has to be validated. For this it is usual to rely on experiments. After all seeing is believing. Theory guides and experiment decides. In this course, we concentrate mainly on the theoretical approach as it gives us a good insight into fluid mechanics. Of course, we will be referring to the information provided by the experimental studies. Computational approach becomes too involved for a first course in fluid mechanics.

Subsections

●

What is a Fluid? Continuum Hypothesis

●

Viscosity,

●

●

❍

Formulas for Viscosity

❍

Kinematic Viscosity,

Density, ❍

Specific Volume, v

❍

Specific Weight,

❍

Specific Gravity, SG

●

Pressure, p Ideal Gas Law

●

Bulk Modulus,

●

❍

● ●

Bulk Modulus for Gases

Vapour Pressure Surface Tension, ❍ Pressure inside a Drop of Fluid ❍ Capillary Tube

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What is a Fluid?

Next: Continuum Hypothesis Up: Fluids Previous: Fluids

What is a Fluid? It is well known that matter is divided into solids and fluids. Fluids can be further divided into Liquids and Gases. It is taught in schools, rightly so, that solids have a definite shape and a definite size, while the liquids have a definite size, but no definite shape. They assume the shape of the container they are poured into. Gases on the other hand have neither a shape nor a size. They can fill any container fully and assume its shape. But we are engineers. We need a more precise definition. This comes when we consider the response of a solid or a fluid to a shear force. A solid resists a shear force while a fluid deforms continuously under the action of a shear force.

Figure 1.1 Experiment to define a Fluid

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What is a Fluid?

A thought experiment is carried out to explain this further. Consider two flat plates of infinite length placed a distance h apart as shown in Fig.1.2. The lower plate is fixed while the upper plate is allowed to move. Let us fill the gap in between the plates first with a solid substance. If now a shear force is applied to the upper plate the solid block deforms as shown. Line ab assumes a new position ab'and the upper plate is displaced by a distance bb'. The deformation produced is proportional to the applied shear stress F/A, where A is the area of the solid surface in contact with the plate. Now let us fill the gap with a fluid, say water. What happens when a shear force is applied to the top plate? We find that it moves continuously ie., point b keeps moving and occupies positions b1, b2, b3, b4 etc at different instants of time. The fluid block between the plates deforms and continues to deform as long as the force is applied. This experiment shows that a fluid at rest cannot resist shear stress. Such an experiment also helps us to define viscosity, which we will take up later.

Figure 1.2, Deformation of a Solid(a) and a Fluid (b) under the action of a shear force

Next: Continuum Hypothesis Up: Fluids Previous: Fluids (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Continuum Hypothesis

Next: Viscosity, Up: Fluids Previous: What is a Fluid?

Continuum Hypothesis We know very well that all matter is made up of molecules, which are in random motion. Any fluid we consider has molecules bombarding each other and the boundaries, i.e. the walls of the container. There is no guarantee whatever that molecules are present at that point at a given instant of time. But still we say that fluid velocity at a point is so many meters per second or that density is so many kgs per square meter. Where is the justification for this? Of course, we can say that we define density or velocity at point in an average sense. That is as an average of velocities (or densities) of the molecules that pass through a small volume surrounding that point. The size of this small volume has to meet with certain criteria. It must be smaller than the physical dimensions of the region under consideration like the wing of an aircraft or the pipe in a hydraulic system. At the same time it must be sufficiently large to accommodate a large number of molecules to make any averaging meaningful. It seems that there is a lower limit to the size of this volume.

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Continuum Hypothesis

Figure 1.3: Definition of Density

The existence of this limit is established by considering the definition of density as mass per unit volume (

). Consider a small volume

around the point P (Fig. 1.3) within

the region of interest, R. Let us calculate density at P by considering different sizes of

.

Values of density so calculated are plotted in the same figure. It is clear that the size has an enormous influence on the calculated value of density. Too small a calculated density fluctuates because the number of molecules within significantly with time. Too big a

, the value of is varying

might mean that density itself is varying significantly

within the region of interest. As seemed before it is clear that there is a limit

below

which molecular variations assume importance and above which one finds a macroscopic variation of density within the region. Therefore it appears that density is best defined as a limit -

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Continuum Hypothesis

(1.1)

At Standard Temperature and Pressure conditions (STP) the limit (

)is around

and air at this tiny volume has about x number of molecules. This is a large enough number to give a constant value of density despite the rigorous molecular motion within it. For many of the applications in Fluid Mechanics, this volume is smaller than the overall dimensions of the regions of interest considered such as an aeroplane, wing of an aeroplane, ship or the parts of an engine etc. These considerations do not hold good when we go to greater altitudes. For example, at an altitude of 130 km the molecular mean free path is about 10.2 m and there are only (Molecular Mean Path,

x

molecules in a cubic meter of air

is defined as the average distance a molecule has to travel before

it collides with another molecule. At STP conditions its value is

). Under these

conditions it becomes necessary to consider the effect of every molecule or groups of molecules, as in calculations concerning re-entry vehicles. That branch of fluid mechanics is called Rarefied Gasdynamics.

Next: Viscosity, Up: Fluids Previous: What is a Fluid? (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Viscosity,

Next: Formulas for Viscosity Up: Fluids Previous: Continuum Hypothesis

Viscosity, We all have a feel for viscosity. More viscous a fluid more difficult for it to flow. Oils flow at a slower rate than water. We understand viscosity as a property that tends to retard fluid motion. But we do have a more rigorous definition of viscosity, which can be developed from the thought experiment described before.

Figure 1.4: Flow between parallel plates It was seen that when a shear force is applied to the top plate the fluid undergoes a continuous deformation ( What is a Fluid? Fig.1.4). As a result the block of fluid abcd deforms to ab'c'd after a time t. Let the speed of the top plate be U. It is an important property of fluids that the layer of fluid adjacent to a solid surface moves with the same velocity as the solid surface. This is called the "No Slip" condition. Accordingly fluid layer closer to the top plate moves with a speed U while that closer to the lower plate is at rest. Thus the velocity of the fluid varies continuously from zero on the lower plate to U at the upper plate. In other words a velocity gradient develops in the fluid. In the simple case of the flow between parallel plates this is a linear profile. The velocity gradient is given by

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Viscosity,

(1.2)

where h is the distance between the two plates. In a small instant of time equal to

we find that the upper plate has moved by a distance bb' which is

.

Now

(1.3)

Noting that for solids the shear stress proportional to rate of strain,

is proportional to strain

while for fluids it is

, which in turn is defined as

(1.4)

. Substituting for

we have

(1.5)

Since

is proportional to

, we have

or

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Viscosity,

(1.6)

It is found that for common fluids such as air, water and oil the relationship between shear stress and velocity gradient can be expressed as,

(1.7)

The constant of proportionality

is an important property of fluids in determining the flow

behaviour and is called Dynamic Viscosity or Absolute Viscosity. It is usual to refer to it as just Viscosity. It has the dimensions

and units of

in the SI system.

Fluids for which the shear stress varies linearly as rate of strain are called Newtonian Fluids. Many of the common fluids belong to this category- air, water. When the relationship between shear stress and rate of strain is not linear, the fluid is designated Non-Newtonian. Examples of this category are some of the industrial fluids such as plastics, sludge and biological fluids such as blood. Typical plots of shear stress vs rate of strain are shown in Fig.1.5. Rheology is the branch of fluid mechanics which specialises in these fluids. We consider primarily common fluids such as water and air and hence restrict ourselves to Newtonian fluids.

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Viscosity,

Figure 1.5 : Flow between parallel plates Viscosity of a fluid is strongly dependent on temperature and is a weak function of pressure. For example, when the pressure of air is increased from 1 atm to 50 atm, its viscosity increases only by about 10 percent allowing one to ignore its dependence on pressure. Fig.1.6 shows the manner of dependence of viscosity on temperature for some of the common fluids. It is seen that the viscosity of liquids deceases with temperature while that for the gases increases with temperature. This difference in behaviour is explained by the cohesive and intermolecular forces within the fluid. Liquids are characterized by strong cohesive forces and close packing of molecules. When temperature increases cohesive forces are weakened and there is less resistance to motion. Hence viscosity decreases. With gases, the cohesive forces are very weak and the molecules are spaced apart. Viscosity is due to the exchange of momentum between molecules as a result of random motion. As the temperature increases the molecular activity increases giving rise to an increased resistance to motion or in other words viscosity increases.

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Viscosity,

Figure 1.6 : Viscosity of Air and Water plotted against temperature

Subsections ●

Formulas for Viscosity

●

Kinematic Viscosity,

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Formulas for Viscosity

Next: Kinematic Viscosity, Up: Viscosity, Previous: Viscosity, Formulas for Viscosity A widely used formula for the calculation of viscosity of gases is the Sutherland Equation given by

(1.8)

where b and S are constants and T is temperature expressed in Eq. 1.8 . and

For air

.

Power Law is another approximation to calculate viscosity and is given by

(1.9)

where

is the value of viscosity at a reference temperature

which could be 273K. An empirical fit for the viscosity of liquids is

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,

Formulas for Viscosity

(1.10)

= 273.16K,

For water,

= 0.001792 kg/(m.s), a=-1.94, b = -

4.80 and c = 6.74. Another empirical formula for liquids is the Andrade equation namely,

(1.11)

where

and

are constants and

is the temperature in

Next: Kinematic Viscosity, Up: Viscosity, Previous: Viscosity, (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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K

Kinematic Viscosity,

Next: Density, Up: Viscosity, Previous: Formulas for Viscosity Kinematic Viscosity, In fluid flow problems viscosity often appears in combination with density in the form

(1.12)

One of the common examples is Reynolds Number, defined as VL/ important parameters in Fluid Dynamics.This term has the dimensions of

being one of the very

is referred to as Kinematic Viscosity and

. Figure 1.7 shows a plot of

temperature.

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for air and water against

Kinematic Viscosity,

Figure 1.7 :Kinematic Viscosity plotted against temperature plates

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Density,

Next: Specific Volume, v Up: Fluids Previous: Kinematic Viscosity,

Density, Density is defined as mass per unit volume of the substance. However, as discussed before, in the spirit of Continuum Hypothesis, it is defined as a limit. If

is the mass of a small volume

then

(1.13)

Unit of density in the SI system is the density of water is atmospheric pressure is

Under ordinary conditions while that for air at

C and

.

Density of liquids is somewhat insensitive to the changes in pressure and temperature. For gases there is a strong dependence of density on these quantities and is given by the equation of state of the particular gas.

Subsections ●

Specific Volume, v

●

Specific Weight,

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Density, ●

Specific Gravity, SG

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Specific Volume, v

Next: Specific Weight, Up: Density, Previous: Density, Specific Volume, v Specific Volume, v of a fluid is defined as the volume per unit mass and its numerical value is given by the reciprocal of density.

(1.14)

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Specific Weight,

Next: Specific Gravity, SG Up: Density, Previous: Specific Volume, v

Specific Weight,

Specific Weight,

of a fluid is defined as the weight per unit volume

and is related to density. (1.15)

where g is acceleration due to gravity. Its units (in SI units) are . Noting that the value of g is water at

is

pressure specific weight is

specific weight of

. For air at

and atmospheric

.

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Specific Gravity, SG

Next: Pressure, p Up: Density, Previous: Specific Weight, Specific Gravity, SG Specific Gravity, SG of a fluid is the ratio of its density to that of water under reference conditions, usually at

(i.e.,

.)

(1.16)

Specific Gravity being a ratio of densities is independent of units. Its value for mercury at is 13.55. There does not seem to a set standard for reference density. Sometimes for gases density of air under standard conditions

is also used as the reference.

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fluidstatics

Next: Fluid Statics

●

●

Fluid Statics ❍ Fluid Forces ❍ Pressure at a point within a fluid ❍ Equation for Pressure Field ■ Body forces ■ Total Force ■ Incompressible Fluids ❍ Compressible Fluids, Properties of Atmosphere ■ Standard Atmosphere ❍ Measurement of Pressure ■ Manometry ■ Mercury Barometer ■ Piezometer Tube ■ U-tube Manometer ■ Differential U-tube Manometer ❍ Hydrostatic Force on a submerged surface ■ Centre of Pressure ■ Geometrical Properties of Common Shapes ❍ Hydrostatic Force on a Curved Surface ❍ Buoyancy and Stability ■ Stability of Immersed and Floating Bodies Table of Contents

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Fluid Statics

Next: Fluid Forces Up: fluidstatics Previous: Contents Contents

Fluid Statics Fluid Statics and Fluid Dynamics form the two constituents of Fluid Mechanics. Fluid Statics deals with fluids at rest while Fluid Dynamics studies fluids in motion. In this chapter we discuss Fluid Statics. A fluid at rest has no shear stress. Consequently, any force developed is only due to normal stresses i.e, pressure. Such a condition is termed the hydrostatic condition. In fact, the analysis of hydrostatic systems is greatly simplified when compared to that for fluids in motion. Though fluid in motion gives rise to many interesting phenomena, fluid at rest is by no means less important. Its importance becomes apparent when we note that the atmosphere around us can be considered to be at rest and so are the oceans. The simple theory developed here finds its application in determining pressures at different levels of atmosphere and in many pressure-measuring devices. Further, the theory is employed to calculate force on submerged objects such as ships, parts of ships and submarines. The other application of the theory is in the calculation of forces on dams and other hydraulic systems. Specific topics developed in this chapter are1. 2. 3. 4. 5.

Pressure at a point within a fluid Equation for Pressure Field Manometry, measurement of pressure Force on an immersed surfaces Buoyancy and stability

Subsections ●

Fluid Forces

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Fluid Statics ● ●

●

●

●

● ●

Pressure at a point within a fluid Equation for Pressure Field ❍ Body forces ❍ Total Force ❍ Incompressible Fluids Compressible Fluids, Properties of Atmosphere ❍ Standard Atmosphere ❍ Troposphere ❍ Stratosphere Measurement of Pressure ❍ Manometry ❍ Mercury Barometer ❍ Piezometer Tube ❍ U-tube Manometer ❍ Differential U-tube Manometer Hydrostatic Force on a submerged surface ❍ Centre of Pressure ❍ Geometrical Properties of Common Shapes Hydrostatic Force on a Curved Surface Buoyancy and Stability ❍ Stability of Immersed and Floating Bodies

Next: Fluid Forces Up: fluidstatics Previous: Contents Contents (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Fluid Forces

Next: Pressure at a point Up: Fluid Statics Previous: Fluid Statics Contents

Fluid Forces In Fluid Mechanics we consider forces upon fluid elements. It is necessary to discuss the type of forces that could act on fluid elements. These Forces could be divided into two categories - Surface Forces, and Body Forces,

.

Figure 2.1 : Classification of forces

Surface forces are brought about by contact of fluid with another fluid or a solid body. The best example of this is pressure. The surface forces depend upon surface area of contact and do not depend upon the volume of fluid. On the other hand, body forces depend upon the volume of the substance and are distributed through the fluid element. Examples are weight of any substance, electromagnetic forces etc.

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Pressure at a point within a fluid

Next: Equation for Pressure Field Up: Fluid Statics Previous: Fluid Forces Contents

Pressure at a point within a fluid Consider a fluid at rest as shown in Fig. 2.2. From around the point of interest, P in the fluid let us pull out a small wedge of dimensions dx x dz x ds . Let the depth normal to the plane of paper be b. In some of the derivations we chose z to be the vertical coordinate. This is consistent with the use of z as the elevation or height in many applications involving atmosphere or an ocean. Let us now mark the surface and body forces acting upon the wedge.

Figure 2.2 : Pressure at a point

The surface forces acting on the three faces of the wedge are due to the pressures,

,

and

as

shown. These forces are normal to the surface upon which they act. We follow the usual convention that compression pressure is positive in sign. We again remind ourselves that since the fluid is at rest there is no shear force acting. In addition we have a body force, the weight, W of the fluid within the wedge acting vertically downwards.

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Pressure at a point within a fluid

Summing the horizontal and the vertical forces we have,

(2.1)

Noting that (2.2)

we have after simplification,

(2.3)

We note that the pressure in the horizontal direction does not change, which is a consequence of the fact that there is shear in a fluid at rest. In the vertical direction there is a change in pressure proportional to density of the fluid, acceleration due to gravity and difference in elevation. Now if we take the limit as the wedge volume decreases to zero, i.e., the wedge collapses to the point P, we have, (2.4)

This equation is known as Pascal's Law. It is important to note that it is valid only for a fluid at rest. In the case of a moving fluid, pressures in different directions could be different depending upon fluid accelerations in different directions. Hence, for a moving fluid pressure is defined as an average of the three normal stresses acting upon the fluid element.

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Equation for Pressure Field

Next: Body forces Up: Fluid Statics Previous: Pressure at a point

Contents

Equation for Pressure Field We have shown that for a fluid at rest pressure at any point is invariant with direction. But it does not prevent pressure itself varying from point to point within the fluid. In this section we try to establish a relationship for this variation of pressure.

Figure 2.3: Pressure at a point

Consider a rectangular element of fluid of dimensions dx x dy x dz with its centre at the point P(x,y,z) as shown in Figure 2.3. Let the pressure acting at the point P be equal to p. It is usually assumed that pressure varies continuously across the element. Consequently the pressures at the different faces of the element are calculated by expanding pressure in a Taylor series about the point P. Second and higher order terms are neglected. Accordingly,

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Equation for Pressure Field

A surface force balance in the x- direction gives

(2.5)

Similar force balancing is carried out in each of the other directions. Upon collecting terms we have for surface forces acting on the fluid element,

(2.6)

The terms within the parenthesis is called the pressure gradient. i.e.,

(2.7)

Thus, (2.8)

Thus it is seen that the net surface force upon the element is given by the pressure gradient and is not dependent upon the pressure level itself.

Subsections ●

Body forces

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Equation for Pressure Field ● ●

Total Force Incompressible Fluids

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Contents

Body forces

Next: Total Force Up: Equation for Pressure Field Previous: Equation for Pressure Field Contents Body forces The only body force that we consider is the weight of the fluid element or the gravity force. If this is designated

, we have

(2.9)

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Total Force

Next: Incompressible Fluids Up: Equation for Pressure Field Previous: Body forces Contents Total Force Adding the surface and body forces acting on the fluid element, we have total force,

(2.10)

On a unit volume basis this equation becomes

(2.11)

Thus we have obtained an expression for force upon a unit volume of a fluid element at rest. To extend this to the case of a moving fluid, one has to include normal and shear stresses due to viscosity in addition to the one given above. They are together balanced by inertia forces. Coming back to fluid at rest, the net force given by Eq.2.11 should be equal to zero. Accordingly, (2.12)

The above equation consists of three separate equations, one for each direction and each of them must be equal to zero. Thus for x, y and z directions we have,

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Total Force

(2.13)

If the coordinate system be so selected as to align one of x, y or z with acceleration due to gravity, g the equations simplify considerably. The natural selection is to have zdirection align with -g, such that

. Consequently,

. Then

we have,

(2.14)

The above equation shows that pressure in a static fluid does not vary in x or y direction. It varies only in the z-direction. This enables one to write,

(2.15)

The above equation is a fundamental equation in Fluid Statics. It defines the manner in which pressure varies with height or elevation and finds many applications. Mainly it enables one to determine atmospheric pressures at different elevations above the sealevel. Then we employ the same equation to determine pressure at various depths of an ocean. The other application is in Manometry , which forms the basis of a class of pressure measuring instruments. A close look at the equation reveals that the pressure gradient is a function of density, and acceleration due to gravity,

. The latter one,

is almost a constant and

therefore it is the variation in density with elevation that influences the pressure values. Density is constant for incompressible fluids and varies with pressure and temperature for compressible fluids. Therefore it is necessary to consider these two types of fluids separately.

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Total Force

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Incompressible Fluids

Next: Compressible Fluids, Properties of Up: Equation for Pressure Field Previous: Total Force Contents Incompressible Fluids For incompressible fluids density is a constant. In addition as stated before, for most applications of practical interest acceleration due to gravity is also a constant. As a consequence the pressure equation is greatly simplified and Eq. 2.15 is readily integrated. Thus for incompressible fluids,

(2.16)

which on integration yields

(2.17)

A convenient form of the above equation is (2.18)

where

is the difference in elevation,

By rewriting the above equation, we have,

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.

Incompressible Fluids

(2.19)

These equations demonstrate that pressure difference between two points in an incompressible fluid is proportional to the difference in elevation or height between the two points. The term

is sometimes defined as the pressure head

and is the height of the fluid column of density supports a pressure difference of

(and specific weight,

) that

.

The above is used to determine pressures in atmosphere and ocean depths. For this, it is advisable to choose a convenient datum or reference. Depending upon the application as shown in the figure, sealevel(for atmospheric pressure) or free surface (for measurements in oceans and lakes) seems to be ideally suited. For an ocean or a lake, if at any depth

is the pressure acting on the free surface pressure

is given by

(2.20)

On the other hand for the atmosphere with

being the sealevel pressure, we

have, (2.21)

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Incompressible Fluids

Figure 2.4 : Measurement of pressure in atmosphere, oceans and lakes.

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Compressible Fluids, Properties of Atmosphere

Next: Measurement of Pressure Up: Fluid Statics Previous: Incompressible Fluids Contents

Compressible Fluids, Properties of Atmosphere The most common compressible fluid we know is air. Assuming air to behave like a perfect gas, Eqn.2.15 becomes,

or (2.22)

By integrating the above equation we obtain,

(2.23)

To solve the above equation we need to know how temperature T varies with altitude. For this we rely on the concept of Standard Atmosphere described next.

Subsections

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Measurement of Pressure

Next: Manometry Up: Fluid Statics Previous: Compressible Fluids Contents

Measurement of Pressure One of the direct applications of the equation of Fluid Statics we have derived is in the devices used to measure pressure. Now it is necessary to recall that we have an Absolute Pressure and a Gauge Pressure. We note that pressure is always measured as a difference or with respect to a datum or reference. Absolute pressure is measured relative to a perfect vacuum, whereas gauge pressure is measured relative to atmospheric pressure. Further, Absolute pressure is the sum of atmospheric pressure and the gauge pressure. See Fig. 2.8 below.

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Measurement of Pressure

Subsections ●

Manometry

●

Mercury Barometer

●

Piezometer Tube

●

U-tube Manometer

●

Differential U-tube Manometer

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Manometry

Next: Mercury Barometer Up: Measurement of Pressure Previous: Measurement of Pressure Contents Manometry We saw in previous sections that pressure is proportional to the height of a column of fluid. Manometry exploits this to measure fluid pressure. In other words we measure the height of a column of liquid supported by the pressure (actually the pressure difference). Barometer, piezometer and U-tube Manometer are some of the members of this class.

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Mercury Barometer

Next: Piezometer Tube Up: Measurement of Pressure Previous: Manometry Contents Mercury Barometer

Figure 2.9 : Mercury Barometer

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Mercury Barometer

Mercury Barometer (Fig.2.9) is the simplest device to measure atmospheric pressure at a location. It consists of a glass tube closed at one end immersed in a container filled with mercury. Because of the atmospheric pressure mercury rises in the tube as shown. If is the height of mercury above the fluid level in the container, we have (2.27)

where

is the pressure at A and will be equal to the vapour pressure

of mercury,

, which is around 0.16pa at a temperature of

It is usual to neglect

.

when the atmospheric pressure is given as

Sometimes atmospheric pressure is expressed as "mms of mercury" being equal to . At sealevel conditions where the pressure value is 101,327 Pascals and the specific weight of mercury is 133,100 N/m3, the barometric height is 761 mm Hg. Water could be used as the barometer fluid, but in that case the height of water will be around 10.36m!

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Piezometer Tube

Next: U-tube Manometer Up: Measurement of Pressure Previous: Mercury Barometer Contents Piezometer Tube

Figure 2.10 : Piezometer

Piezometer tube (Fig. 2.10) is perhaps the simplest of the pressure

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Piezometer Tube

measuring devices and consists of a vertical tube. In its application one end is connected to the pressure to be measured while the other end is open to the atmosphere as shown. Application of Eq.2.18 gives

2.28

or simply for the gauge pressure at A, 2.29

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U-tube Manometer

Next: Differential U-tube Manometer Up: Measurement of Pressure Previous: Piezometer Tube

Contents

U-tube Manometer

Figure 2.11 : U tube Manometer

Although the piezometer tube is simple in structure, it has a few practical disadvantages. In this regard U-tube Manometer (Fig. 2.11) seems to be a better alternative. It is a U-tube filled with what is called a gauge fluid. As before one end of the tube is exposed to the pressure to be measured while the other end is open to atmosphere. We have,

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U-tube Manometer

(2.28)

Considerable simplification is possible if the fluid A is a gas when

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is negligible, giving

Differential U-tube Manometer

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Differential U-tube Manometer

Figure 2.12 : Differential U-tube manometer

Differential U-tube manometer (Fig. 2.12) is very handy to measure the pressure difference directly and is basically similar to the U-tube manometer discussed above. What was the open end before is now connected to a different pressure, measure the difference

so that we

. Now we have,

so that (2.29)

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Hydrostatic Force on a submerged surface

Next: Centre of Pressure Up: Fluid Statics Previous: Differential U-tube Manometer

Contents

Hydrostatic Force on a submerged surface The other important utility of the hydrostatic equation is in the determination of force acting upon submerged bodies. Among the innumerable applications of this are the force calculation in storage tanks, ships, dams etc.

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Hydrostatic Force on a submerged surface

Figure 2.13 : Force upon a submerged object

First consider a planar arbitrary shape submerged in a liquid as shown in the figure. The plane makes an angle with the liquid surface, which is a free surface. The depth of water over the plane varies linearly. This configuration is efficiently handled by prescribing a coordinate frame such that the y-axis is aligned with the submerged plane. Consider an infinitesimally small area at a (x,y). Let this small area be located at a depth

from the free surface.

From Eq.2.18 we know that

(2.30)

where

is the pressure acting on the free surface. The hydrostatic force on the plane is given

by,

(2.31)

The integral,

is the first moment of surface area about x axis. If

coordinate of the centroid of the area we have,

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is the y-

Hydrostatic Force on a submerged surface

(2.32)

Consequently, Eq. 2.31 is rewritten as

(2.33)

and

where

is the pressure acting at the centroid.

Subsections ● ●

Centre of Pressure Geometrical Properties of Common Shapes

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Contents

Centre of Pressure

Next: Geometrical Properties of Common Up: Hydrostatic Force on a Previous: Hydrostatic Force on a Contents Centre of Pressure Force, given by Eq. 2.33 is the resultant force acting on the plane due to the liquid and acts at what is called the Center of Pressure (CP). It does not act at the centroid of the plane as it may seem. Let the coordinates of CP be

.

Noting that the moment of the resultant force is equal to the moment of the distributed force about the same axis, we have

(2.34)

Before substituting for pressure

in the above equation we note that the atmospheric

acting at the free surface also acts everywhere within the fluid and

also on both sides of the plane. As such it does not contribute to the net force upon the plane. So we drop term

from the equation for

. Eq.2.34 becomes

(2.35)

The term

is the well-known second moment of area about the x-axis

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Centre of Pressure

denoted by

leading to

(2.36)

is related to that about the x-axis passing through the centroid of the area, through the Parallel Axes Theorem given by

(2.37)

Consequently, we have

(2.38)

Similarly, taking moments about the y-axis, we obtain,

(2.39)

is the product of inertia with respect to x and y axes. Again on the application of the parallel axis theorem we have

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Centre of Pressure

(2.40)

where

is the product of inertia about the axes passing through the centroid.

The coordinates of the Centre of Pressure are thus given by

(Eqns.

2.40 and 2.38). The resulting force upon the immersed surface is therefore given by

(2.41)

The centre of pressure is given by

(2.42)

Expressions for the moments

,

etc for some of the common shapes are

given in the next section.

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Geometrical Properties of Common Shapes

Next: Hydrostatic Force on a Up: Hydrostatic Force on a Previous: Centre of Pressure Contents Geometrical Properties of Common Shapes

Figure 2.14 : Properties of some common shapes

Table 2.2 : Properties of Common Shapes

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Geometrical Properties of Common Shapes

Shape

A

A)Circle

B)Rectangle

0

bh

C)Triangle

0

-

D)Semicircle

Note that the determination of the resultant force

0

hinges on the

knowledge of the position of the centroid for the given shape. The , the Center of Pressure depends upon the moment of location of inertia and the product of inertia. These are functions of the geometry only and can be calculated once the shape is given. Table 2.2 along with Figure 2.14 gives these properties for some of the common shapes.

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Hydrostatic Force on a Curved Surface

Next: Buoyancy and Stability Up: Fluid Statics Previous: Geometrical Properties of Common Contents

Hydrostatic Force on a Curved Surface We encounter many arbitrarily shaped bodies immersed in liquids such as pipes and walls of containers. Forces on these may be calculated in the same manner as in the previous section. But the required integration of involved terms becomes very tedious. A more simplistic approach is to consider the forces resolved in the three coordinate directions separately. It may be noted that each component of force acts upon a projected area of the body. For example, force in x-direction will act normally on the area projected upon the yz plane.

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Hydrostatic Force on a Curved Surface

Figure 2.15 : Hydrostatic forces on a curved surface

Consider a curved surface as shown in Fig.2.15, immersed in a liquid. The resultant force

can be resolved into two components -

horizontal direction and

in the

in the vertical direction. We are considering a thin

body which is two-dimensional and as such there is no force in the direction normal to the paper. The configuration can be split into two parts for discussion purposes- one, the part between the body and the free stream , and two, the body itself,

. Consider each part separately.

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Hydrostatic Force on a Curved Surface

Block of fluid

is in equilibrium. The horizontal forces acting on it cancel

out. Thus, (2.43)

Similarly on the block

we have,

(2.44)

The vertical forces acting upon the fluid are (1) , the weight of block

and (3)

due to atmosphere, (2) , the weight of block

.

Consequently, (2.45)

Force

is given by the atmospheric pressure times the projected area

normal to it. The resultant force,

is given by,

(2.46)

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Buoyancy and Stability

Next: Stability of Immersed and Up: Fluid Statics Previous: Hydrostatic Force on a Contents

Buoyancy and Stability What is the vertical force acting on a body which is completely submerged in a fluid? Answer to such a question can be very well found in the theory developed in the previous section. Archimedes seems to have discovered the laws concerning submerged bodies as well as floating bodies. What is well known as Archimedes principle states 1. The vertical buoyant force experienced by a body immersed in a fluid is equal to the weight of the fluid displaced. 2. A floating body displaces its own weight of the fluid.

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Buoyancy and Stability

Figure 2.17 : Forces about a body immersed in a fluid-2

Proof is straight forward. Consider an elemental volume within the immersed body as shown in Fig.2.16 . Now the buoyant force is given by,

(2.47)

where

is the area of cross section of the elemental volume chosen. We have

from Eq. 2.48

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Buoyancy and Stability

(2.48)

passes through the centroid of the

It can be shown that the buoyant force,

displaced volume as shown in Fig.2.17. The point where this force acts is called "Center of Buoyancy", denoted as

.

The above result holds good even in the case of a partially submerged body i.e., a floating body. It is assumed that part of the body above the liquid level is in air. The weight of air displaced as a consequence is ignored. (Fig. 2.18). For this case as well,

(2.49)

Figure 2.18 : Partially Submerged Body

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Buoyancy and Stability

The theory developed so far does hold good in case of a fluid for which specific gravity

is not a constant, a layered fluid for example. However now the buoyant

force may not act at the centroid of the displaced volume. The theory developed is also applicable where the fluid involved is a gas, say air. Convection currents established in atmosphere depend upon the buoyant forces generated.

Subsections ●

Stability of Immersed and Floating Bodies

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Stability of Immersed and Floating Bodies

Next: Table of Contents Up: Buoyancy and Stability Previous: Buoyancy and Stability Contents Stability of Immersed and Floating Bodies Stability becomes an important consideration when floating bodies such as a boat or ferry is designed. It is an obvious requirement that a floating body such as a boat does not topple when slightly disturbed. We say that a body is in stable equilibrium if it is able to return to its position when slightly disturbed. Failure to do so denotes unstable equilibrium. What equilibrium a body enjoys is decided by the couple formed by the weight of the body and the buoyancy force. Consider the immersed body shown in Fig.2.19. In general, if the center of gravity of the body lies below the center of buoyancy stable equilibrium prevails. An overturning couple leading to unstable equilibrium results if the center of gravity is above the center of buoyancy (Fig.2.20).

Figure 2.19 : Stability of an immersed body

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Stability of Immersed and Floating Bodies

Figure 2.20: Instability of an immersed body It becomes more complicated when floating bodies are considered. Now as the body rotates responding to any disturbance the center of buoyancy can shift. This could render the body stable even though the center of gravity is above the center of buoyancy. This is particularly true of the bodies with a broader base such as a barge (Fig. 2.21). A slender body as shown in Fig. 2.22 is very susceptible for instability.

Figure 2.21 : Stability of a floating body

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Stability of Immersed and Floating Bodies

Figure 2.22 : Instability of a floating body

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cvanalysis

Next: Integral Equations

●

Integral equations for the Control Volume analysis of Fluid Flow ❍ Basic Concepts ■ Velocity Field ■ Steady and Unsteady Flows ■ One, Two and Three Dimensional Flows ❍ Flow Description, Streamline, Pathline, Streakline and Timeline ■ Eularian and Lagrangian approaches ■ System and Control Volume ■ Differential and Integral Approach ❍ Integral Equations ■ Basic Laws for Fluid Flow ■ conservation of mass ■ Newton's Second Law of Motion ■ conservation of energy ■ Second Law of Thermodynamics ❍ Reynolds Transport Theorem ■ Derivation of the theorem for one-dimensional flow ❍ Conservation of Mass ■ Steady Flow ■ Incompressible Flow ■ Term V.dA ■ Application to an one-dimensional control volume ❍ Momentum Equation ❍ Bernoulli Equation ■ Assumptions ■ Application of Continuity Equation ■ Application of Momentum Equation

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cvanalysis

Terms on the Right Hand Side Application to moving Control Volumes Equation for Angular Momentum ■ Deformable Control Volumes and Control Volumes with non-inertial acceleration Energy Equation ■ Energy equation for a one-dimensional control volume ■ Low Speed Application Relationship between Energy Equation and Bernoulli Equation Bernoulli Equation for Aerodynamic Flow ■ Stagnation Pressure ■ Energy Grade Line ■ Kinetic Energy Correction Factor Applications of Bernoulli Equation ■ Flow through a Sharp-edged Orifice ■ Flow Through a Flow Nozzle ■ Flow through a Venturi Tube Important Applications of Control Volume Analysis ■ Measurement of Drag about a Body immersed in a fluid ■ Jet Impingement on a surface ■ Force on a Pipe Bend ■ Froude's Propeller Theory ■ Continuity Equation ■ Momentum Equation ■ Bernoulli Equation ■ Analysis of a Wind Turbine ■ Total Pressure Loss through a Sudden Expansion ■ Continuity Equation ■ Momentum Equation ■ Bernoulli Equation Measurement of Airspeed ■

❍ ❍

❍

❍

❍

❍

❍

❍

●

Table of Contents ...

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Integral equations for the Control Volume analysis of Fluid Flow

Next: Basic Concepts Up: cvanalysis Previous: cvanalysis

Integral Approach to the Control Volume analysis of Fluid Flow

Subsections ●

●

●

●

●

●

Basic Concepts ❍ Velocity Field ❍ Steady and Unsteady Flows ❍ One, Two and Three Dimensional Flows Flow Description, Streamline, Pathline, Streakline and Timeline ❍ Eularian and Lagrangian approaches ❍ System and Control Volume ❍ Differential and Integral Approach Integral Equations ❍ Basic Laws for Fluid Flow ■ conservation of mass ■ Newton's Second Law of Motion ■ conservation of energy ■ Second Law of Thermodynamics Reynolds Transport Theorem ❍ Derivation of the theorem for one-dimensional flow Conservation of Mass ❍ Steady Flow ❍ Incompressible Flow ❍ Term V.dA ❍ Application to an one-dimensional control volume Momentum Equation

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Integral equations for the Control Volume analysis of Fluid Flow ●

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Bernoulli Equation ❍ Assumptions ❍ Application of Continuity Equation ❍ Application of Momentum Equation ❍ Terms on the Right Hand Side Application to moving Control Volumes Equation for Angular Momentum ❍ Deformable Control Volumes and Control Volumes with noninertial acceleration Energy Equation ❍ Energy equation for a one-dimensional control volume ❍ Low Speed Application Relationship between Energy Equation and Bernoulli Equation Bernoulli Equation for Aerodynamic Flow ❍ Stagnation Pressure ❍ Energy Grade Line ❍ Kinetic Energy Correction Factor Applications of Bernoulli Equation ❍ Flow through a Sharp-edged Orifice ❍ Flow Through a Flow Nozzle ❍ Flow through a Venturi Tube Important Applications of Control Volume Analysis ❍ Measurement of Drag about a Body immersed in a fluid ❍ Jet Impingement on a surface ❍ Force on a Pipe Bend ❍ Froude's Propeller Theory ■ Continuity Equation ■ Momentum Equation ■ Bernoulli Equation ❍ Analysis of a Wind Turbine ❍ Total Pressure Loss through a Sudden Expansion ■ Continuity Equation ■ Momentum Equation ■ Bernoulli Equation Measurement of Airspeed

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Basic Concepts

Next: Velocity Field Up: Integral equations for the Previous: Integral equations for the

Basic Concepts

Subsections ● ● ●

Velocity Field Steady and Unsteady Flows One, Two and Three Dimensional Flows

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Velocity Field

Next: Steady and Unsteady Flows Up: Basic Concepts Previous: Basic Concepts Velocity Field

Figure 3.1 : Velocity Field

Velocity field implies a distribution of velocity in a given region say R (Fig.3.1). It is http://www.aeromech.usyd.edu.au/aero/fprops/cvanalysis/node5.html (1 of 2)24/2/2006 14:34:03

Velocity Field

denoted in a functional form as V(x,y,z,t) meaning that velocity is a function of the spatial and time coordinates. It is useful to recall that we are studying fluid flow under the Continuum Hypothesis which allows us to define velocity at a point. Further velocity is a vector quantity i.e., it has a direction along with a magnitude. This is indicated by writing velocity field as

(3.1)

Velocity may have three components, one in each direction, i.e, u,v and w in x, y and z directions respectively. It is usual to write

as

(3.2)

It is clear that each of u,v and w can be functions of x,y,z and t. Thus (3.3)

Each of the other variables involved in a fluid flow can also be given a field representation. We have temperature field, T(x,y,z,t), pressure field, p(x,y,z,t), density field,

etc.

Next: Steady and Unsteady Flows Up: Basic Concepts Previous: Basic Concepts (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Steady and Unsteady Flows

Next: One, Two and Three Up: Basic Concepts Previous: Velocity Field Steady and Unsteady Flows We have noted previously (see Velocity Field ) that velocity, pressure and other properties of fluid flow can be functions of time (apart from being functions of space). If a flow is such that the properties at every point in the flow do not depend upon time, it is called a steady flow. Mathematically speaking for steady flows,

(3.4)

where P is any property like pressure, velocity or density. Thus, (3.5)

Unsteady or non-steady flow is one where the properties do depend on time. It is needless to say that any start up process is unsteady. Many examples can be given from everyday life- water flow out of a tap which has just been opened. This flow is unsteady to start with, but with time does become steady. Some flows, though unsteady, become steady under certain frames of reference. These are called pseudosteady flows. On the other hand a flow such as the wake behind a bluff body is always unsteady. Unsteady flows are undoubtedly difficult to calculate while with steady flows, we have one degree less complexity.

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One, Two and Three Dimensional Flows

Next: Flow Description, Streamline, Pathline, Up: Basic Concepts Previous: Steady and Unsteady Flows One, Two and Three Dimensional Flows Term one, two or three dimensional flow refers to the number of space coordinated required to describe a flow. It appears that any physical flow is generally three-dimensional. But these are difficult to calculate and call for as much simplification as possible. This is achieved by ignoring changes to flow in any of the directions, thus reducing the complexity. It may be possible to reduce a three-dimensional problem to a two-dimensional one, even an onedimensional one at times.

Figure 3.2 : Example of one-dimensional flow

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One, Two and Three Dimensional Flows

Consider flow through a circular pipe. This flow is complex at the position where the flow enters the pipe. But as we proceed downstream the flow simplifies considerably and attains the state of a fully developed flow. A characteristic of this flow is that the velocity becomes invariant in the flow direction as shown in Fig.3.2. Velocity for this flow is given by

(3.6)

It is readily seen that velocity at any location depends just on the radial distance from the centreline and is independent of distance, x or of the angular position

. This represents a typical one-dimensional flow.

Now consider a flow through a diverging duct as shown in Fig. 3.3. Velocity at any location depends not only upon the radial distance distance. This is therefore a two-dimensional flow.

but also on the x-

Figure 3.3: Example of a two-dimensional flow

Concept of a uniform flow is very handy in analysing fluid flows. A uniform flow http://www.aeromech.usyd.edu.au/aero/fprops/cvanalysis/node7.html (2 of 3)24/2/2006 14:35:11

One, Two and Three Dimensional Flows

is one where the velocity and other properties are constant independent of directions. we usually assume a uniform flow at the entrance to a pipe, far away from a aerofoil or a motor car as shown in Fig. 3.4.

Figure 3.4 : Uniform Flow

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Flow Description, Streamline, Pathline, Streakline and Timeline

Next: Eularian and Lagrangian approaches Up: Integral equations for the Previous: One, Two and Three

Flow Description, Streamline, Pathline, Streakline and Timeline Streamline, pathline, streakline and timeline form convenient tools to describe a flow and visualise it. They are defined below.

Figure 3.5 : Streamlines

Figure 3.6: Streamline definition

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Flow Description, Streamline, Pathline, Streakline and Timeline

A streamline is one that drawn is tangential to the velocity vector at every point in the flow at a given instant and forms a powerful tool in understanding flows. This definition leads to the equation for streamlines.

(3.7)

where u,v, and w are the velocity components in x, y and z directions respectively as sketched.

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Flow Description, Streamline, Pathline, Streakline and Timeline

Figure 3.7 : Streamtube Hidden in the definition of streamline is the fact that there cannot be a flow across it; i.e. there is no flow normal to it. Sometimes, as shown in Fig.3.7 we pull out a bundle of streamlines from inside of a general flow for analysis. Such a bundle is called stream tube and is very useful in analysing flows. If one aligns a coordinate along the stream tube then the flow through it is one-dimensional.

Figure 3.8: Pathlines

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Flow Description, Streamline, Pathline, Streakline and Timeline

Figure 3.9: Streaklines Timeline

Figure 3.10:

Pathline is the line traced by a given particle. This is generated by injecting a dye into the fluid and following its path by photography or other means (Fig.3.8). Streakline concentrates on fluid particles that have gone through a fixed station or point. At some instant of time the position of all these particles are marked and a line is drawn through them. Such a line is called a streakline (Fig.3.9). Timeline is generated by drawing a line through adjacent particles in flow at any instant of time. Fig.3.10 shows a typical timeline. In a steady flow the streamline, pathline and streakline all coincide. In an unsteady flow they can be different. Streamlines are easily generated mathematically while pathline and streaklines are obtained through experiments. The following animation illustrates the differences between a streakline and a pathline.

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Flow Description, Streamline, Pathline, Streakline and Timeline

Figure 3.11: Animation to illustrate Streaklines and Pathlines. Subsections ● ● ●

Eularian and Lagrangian approaches System and Control Volume Differential and Integral Approach

Next: Eularian and Lagrangian approaches Up: Integral equations for the Previous: One, Two and Three (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Eularian and Lagrangian approaches

Next: System and Control Volume Up: Flow Description, Streamline, Pathline, Previous: Flow Description, Streamline, Pathline, Eularian and Lagrangian approaches Eularian and Lagrangian approaches seem to be the two methods to study fluid motion. The Eularian approach concentrates on fluid properties at a point P (x,y,z,t). Thus it is a field approach. In the Lagrangian approach one identifies a particle or a group of particles and follows them with time. This is bound to be a cumbersome method. But there may be situations where it is unavoidable. One such is the two phase flow involving particles.

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System and Control Volume

Next: Differential and Integral Approach Up: Flow Description, Streamline, Pathline, Previous: Eularian and Lagrangian approaches

System and Control Volume

Figure 3.12 : Piston cylinder arrangement

Terms system and control volume are very familiar to the one who has studied thermodynamics. The word system refers to a fixed mass with a boundary. However, with time the boundary of the system may change, but the mass remains the same. The usual example given is that of a piston-cylinder arrangement as shown in Fig.3.12. Consider a gas filled in the cylinder which is closed by a piston at the right hand end. Let us define gas as our system. If the piston is now operated by pushing or pulling it the gas compresses or expands. The boundary of our system moves. But the mass does not move out of the boundary since by definition system is a fixed mass. The definition does not prevent work or energy crossing the boundary.

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System and Control Volume

Figure 3.13 : System Approach

It is easy to analyse the system in the example of piston-cylinder arrangement that we have considered before. But the question is - Are all systems as simple as this? The answer is obviously a "no". In fluid dynamics we consider systems which are far more complicated. Take the flow about an aeroplane for example. If we define a system in such a flow and try to analyse it we find that it undergoes many changes as ilustrated in Fig. 3.13. The boundary changes rapidly and undergoes unmanageable distortions. The system approach is almost ruled out. The other examples are flow through turbomachinery, flow in hydraulic systems and many such. The other method we have is the Control Volume approach. Here we do not focus our attention on a fixed mass of fluid. Instead we establish a "window" for observation in the flow. This is what we call the control volume shown in Fig. 3.14. As against the system, a control volume has a fixed boundary. Mass, momentum and energy are allowed to cross the boundary. We perform a balance of mass, momentum and energy that flow across the boundary and deduce the changes that could take place to properties of flow within the control volume. The shape of the control volume does not change normally. It is easy to visualise that this is a convenient approach. in fact, it is the one that is commonly used in fluid dynamics.

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System and Control Volume

Figure 3.14 : Control Volume

We will consider a fixed control volume most of the time. But it is possible to have control volumes that change their boundary, those that deform etc. Obviously, these lead to more complicated equations. Examples of such control volumes are given in Fig.3.15.

Figure 3.15 : Moving and Collapsible Control Volumes

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System and Control Volume

The boundary of the control volume is referred to as control surface. From the above discussion it is clear that the system and control volume approaches are akin to Lagrangian and Euler approaches.

Next: Differential and Integral Approach Up: Flow Description, Streamline, Pathline, Previous: Eularian and Lagrangian approaches (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Differential and Integral Approach

Next: Integral Equations Up: Flow Description, Streamline, Pathline, Previous: System and Control Volume Differential and Integral Approach Differential approach aims to calculate flow at every point in a given flow field in the form P(x,y,z,t). When we determine the flow about an aerofoil using this approach we try to obtain the needed properties like everywhere within the region R surrounding the aerofoil as shown in Fig.3.16. From the detailed knowledge of the flow field we deduce features such as drag and lift. Aerofoil flow is complicated and we will have to solve the differential equations of motion. Obviously this is a costly approach calling for methods in computational fluid dynamics to be used.

Figure 3.16: Differential and Integral approaches to calculate flow about an Aerofoil.

Not every flow is as complicated as an aerofoil flow. In addition there is not necessary always to get a detailed information of the flow. One may http://www.aeromech.usyd.edu.au/aero/fprops/cvanalysis/node11.html (1 of 2)24/2/2006 16:53:24

Differential and Integral Approach

establish a big control volume to encompass the region R and calculate the overall features like drag and lift by studying what happens at the boundary of the control volume, i.e., at the control surface.This procedure is called the Integral approach. Both these approaches are important to us. First we discuss the integral approach. When we are more familiar with the methods to analyse the flow we take up the differential approach.

Next: Integral Equations Up: Flow Description, Streamline, Pathline, Previous: System and Control Volume (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Integral Equations

Next: Basic Laws for Fluid Up: Integral equations for the Previous: Differential and Integral Approach

Integral Equations

Subsections ●

Basic Laws for Fluid Flow ❍ conservation of mass ❍ Newton's Second Law of Motion ❍ conservation of energy ❍ Second Law of Thermodynamics

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Basic Laws for Fluid Flow

Next: conservation of mass Up: Integral Equations Previous: Integral Equations Basic Laws for Fluid Flow What laws do govern a fluid flow? Surprisingly, these are the well known conservation principles and only a handful of them. They are the same ones as will calculate problems in solid mechanics. These laws could be introduced as follows. Consider a system and its surroundings. By definition everything outside of a system is the surrounding. The system is subject to a few laws.

Subsections ● ● ● ●

conservation of mass Newton's Second Law of Motion conservation of energy Second Law of Thermodynamics

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conservation of mass

Next: Newton's Second Law of Up: Basic Laws for Fluid Previous: Basic Laws for Fluid

conservation of mass Consider a system of a fixed mass, as shown in Fig.3.17. We know that this mass does not change and is conserved. This leads to the law of conservation of mass , namely,

Figure 3.17: A System

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conservation of mass

(3.8)

with

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Newton's Second Law of Motion

Next: conservation of energy Up: Basic Laws for Fluid Previous: conservation of mass

Newton's Second Law of Motion Newton's second law is the next one to be imposed upon fluid motion. It is known that the rate of change of momentum is proportional to the applied force. If F is the force upon a system,

(3.9)

where M is the linear momentum. Further, (3.10)

It is to be realised that momentum M and velocity V are vectors and each of a component in each of the coordinate directions. Accordingly, Eq. 3.10 represents three equations. The form of this equation holds good for angular momentum. If a torque T acts upon the system. We have,

(3.11) where

which again is a vector equation. Torque T can be due to body forces and/or surface forces. In addition there can also be torque directly introduced into the system such as that through a mechanical shaft connected to the system.

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conservation of energy

Next: Second Law of Thermodynamics Up: Basic Laws for Fluid Previous: Newton's Second Law of

conservation of energy The first law of thermodynamics which is a statement of the conservation of energy principle states, (3.12)

i.e.,

where dQ is the heat added to the system, dW is the work done by the system and dE is the consequent change in energy of the system. In addition, we have

(3.13)

Energy, e is a sum of internal energy, u, kinetic energy and potential energy. Thus

(3.14)

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Second Law of Thermodynamics

Next: Reynolds Transport Theorem Up: Basic Laws for Fluid Previous: conservation of energy

Second Law of Thermodynamics While the first law of thermodynamics states that energy is conserved, the second law establishes a direction in which a process can take place. If dS is the change in entropy and dQis the heat added and T the temperature,

(3.15)

In addition to the above relations we may need an equation of state,

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Reynolds Transport Theorem

Next: Derivation of the theorem Up: Integral equations for the Previous: Second Law of Thermodynamics

Reynolds Transport Theorem You may have already seen the dilemma we are in. First of all we favoured a control volume approach because it is easier and very relevant to study motion of fluids. Then we enunciated the basic laws that a fluid motion has to obey and hence lead to the equation of motion. But these are all valid for a system. The question is "How are we going to connect the basic laws for a system with a control volume approach for fluids?". This question has been foreseen by many already. The result is what is called the Reynolds Transport Theorem. The derivation of the Reynolds Transport Theorem may seem too involved. But when the basis of the theorem is understood, it is indeed easy to follow its derivation. We shall start with a system and the rate at at which an extensive property N changes in it. This we try to express in terms of a corresponding intensive property

associated with the

control volume, which to start with coincides with the system. To make the concept clear it seems beneficial to consider first an one-dimensional flow to derive the equation. As a second step we extend them to a general flow.

Subsections ●

Derivation of the theorem for one-dimensional flow

Next: Derivation of the theorem Up: Integral equations for the

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Derivation of the theorem for one-dimensional flow

Next: Conservation of Mass Up: Reynolds Transport Theorem Previous: Reynolds Transport Theorem Derivation of the theorem for one-dimensional flow Consider a stream tube in an one-dimensional flow as sketched. we remind ourselves that the flow takes place entirely through the stream tube and there is no flow across it, i.e., in a direction normal to it. Let is consider a system S in the flow. Let us prescribe a control volume CV coincident with it at time t0 (Fig.3.18). We recall that the system is an entity of fixed mass and is allowed to move and deform. On the other hand a control volume has fixed a boundary, which we denote as CS. In this analysis we keep it stationary. After the lapse of time

i.e., at time

we find that the control volume remains at the same position,

I+II while the system has moved to occupy the position II + III. We see that during the time interval mass contained in region I has entered the control volume and that in III has left the control volume.

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Derivation of the theorem for one-dimensional flow

Figure 3.18: Control Volume and system for an one-dimensional flow

Consider an extensive property N associated with the control volume. By definition we have,

(3.16)

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Derivation of the theorem for one-dimensional flow

where subscript denotes a system. Further we have at

(3.17)

On substituting these into Eq.3.16 and noting that at t0 the system and the control volume coincide, i.e.,

, we have

(3.18)

By readjusting the terms we have,

(3.19)

We can now take up each of the three limits on the RHS of the above equation. The first limit gives,

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Derivation of the theorem for one-dimensional flow

(3.20)

recalling that N is an extensive property and that

where m is the mass given by

is the corresponding intensive property such times volume, i.e,

.

The second limit, which gives the rate of change of N within III could be written as

(3.21)

The right hand side simply the rate at which N is going out of the control volume though the boundary, i.e., the control surface at right and is equal to

(3.22)

where A is the area of cross section of III, V is the velocity normal to the area. Similarly we have for I, i.e., the rate at which N enters the control volume through the boundary or control surface at left,

(3.23)

Upon substituting Eqns. 3.20,3.22 and 3.23 into Eqn. 3.19, we have

(3.24)

Eqn. 3.24 is the Reynolds Transport equation for the control volume considered. Each of the terms in the equation tells something significant. Putting the equation is words we have,

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Derivation of the theorem for one-dimensional flow

Rate of change of property N within the system =

Rate of change of property N within the control volume + Rate of outflow of property N through the control surface Rate of inflow of property through the control surface.

which seems very obvious. The above result can be generalised to any control volume of any shape, but fixed in space. Let us now consider such a general control volume as shown in Fig.3.19 . For such a control volume it is difficult to define an inlet boundary and an outlet boundary. It is best to consider the net flow of property N into the control volume. Accordingly, the above verbal equation is written as

Rate of change of property N within the system =

Rate of change of property N within the control volume + Net rate of change of property N through the control surface

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Derivation of the theorem for one-dimensional flow

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Derivation of the theorem for one-dimensional flow

Figure3.19: General Control Volume and System

Whether the flow at any small segment of control surface is an inflow or an outflow is decided by the direction of the velocity vector and that of the area vector at that segment. Consider a small area

at the control surface (Fig.3.19 ). Let the velocity acting upon it be

rate at which property N escapes or enters the control volume through the velocity component normal to

, i.e,

. The

depends upon

. In fact the rate of flow of N through

is given by

(3.25)

Integrating this for the entire control surface gives the net rate of flow of N into the control volume. I.e,

(3.26)

Consequently we can write the Reynolds Transport theorem for a general control volume as

(3.27)

Abstract as it seems, Eqn. 3.27 simplifies when we consider concrete control volumes and many times becomes self-evident. This will become clear as we consider many applications of the Reynolds Transport theorem.

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Conservation of Mass

Next: Steady Flow Up: Integral equations for the Previous: Derivation of the theorem

Conservation of Mass First we apply the Reynolds Transport theorem, Eq. 3.27 to derive an equation for conservation of mass. We note that in the equation, N is the extensive property of interest which now is mass m. The corresponding intensive property is

(3.28)

Accordingly we substitute for m and

in Eq.3.27. We have

(3.29)

By definition that a system is an entity of fixed mass, the left hand side of the above equation is zero, thus giving the equation for conservation of mass as

(3.30)

which expresses that the rate of accumulation of mass within a control volume is equal to the net rate of flow of mass into the control volume. http://www.aeromech.usyd.edu.au/aero/fprops/cvanalysis/node20.html (1 of 2)24/2/2006 16:55:52

Conservation of Mass

This equation is also called the Continuity Equation.

Subsections ● ● ● ●

Steady Flow Incompressible Flow Term V.dA Application to an one-dimensional control volume

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Steady Flow

Next: Incompressible Flow Up: Conservation of Mass Previous: Conservation of Mass Steady Flow For a steady flow the time derivative in the equation vanishes. As a result,

(3.31)

In addition if the flow is incompressible,

=constant and we have

(3.32)

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Incompressible Flow

Next: Term V.dA Up: Conservation of Mass Previous: Steady Flow Incompressible Flow The equation simplifies further when we consider an incompressible flow where density

is a constant. Consequently we have,

(3.33)

Dividing by density,

,

(3.34)

The first term is the rate of change of volume within a control volume, which for a fixed control volume is zero by definition. This gives a simple form of the equation for the conservation of mass for the control volume as

(3.35)

Thus for an incompressible flow the continuity equation is the same irrespective of whether the flow is steady or unsteady.

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Term V.dA

Next: Application to an one-dimensional Up: Conservation of Mass Previous: Incompressible Flow Term V.dA The term appears in almost all the equations for a control volume analysis - mass, momentum and energy. We need to understand it and its sign convention properly. Consider any part of a control surface and . We are let the area be dA. Let the velocity vector acting on it be interested in the velocity normal to the area that is convecting the mass, momentum or energy.This is given by terms of scalars as

. This term is given in

(3.36)

where as shown A negative positive

is the angle between area vector and velocity vector. suggests an inflow into the control volume while a is an outflow from the control volume.

(c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Application to an one-dimensional control volume

Next: Momentum Equation Up: Conservation of Mass Previous: Term V.dA Application to an one-dimensional control volume Consider an one-dimensional stream tube flow as shown in Fig.3.20. Let us mark a control volume bound by surface 1and surface 2. We know that there is any inflow/ outflow of mass only through these two surfaces. The remaining surface S being made up of streamlines does not allow any mass flow through it.

Figure 3.20 : Control Control Volume for an one-dimensional steady flow

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Application to an one-dimensional control volume

We assume that a uniform flow prevails at surfaces 1 and 2, V1 and V2 being the velocities. If the areas of cross section are A1 and A2, an application of the continuity equation 3.31 gives

(3.37)

simplifying to (3.38)

Next: Momentum Equation Up: Conservation of Mass Previous: Term V.dA (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Momentum Equation

Next: Bernoulli Equation Up: Integral equations for the Previous: Application to an one-dimensional

Momentum Equation Let us now derive the momentum equation resulting from the Reynolds = where is the Transport theorem, Eqn. 3.27. Now we have momentum. Note that momentum is a vector quantity and that it has a component in every coordinate direction. Thus,

(3.39)

Consider the left hand side of Eqn. 3.27. We have

which is

proportional to the applied force as per Newton's Second Law of motion. Thus,

(3.40)

where

is again a vector. It is necessary to include both body forces, and surface forces,

. Thus,

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Momentum Equation

(3.41)

Now we substitute for

in the right hand side of Eqn. 3.27 giving,

(3.42)

Writing this as three equations, one for each coordinate direction we have,

(3.43)

The term

represents the u momentum that is convected in/out

by the surface in a direction normal to it. In fact momentum in other direction can also be convected out from the same area. These are given by and

As stated before the term

.

is replaced by

The equation thus derived finds immense application in fluid dynamic

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.

Momentum Equation

calculations such as force at the bending of a pipe, thrust developed at the foundation of a rocket nozzle, drag about an immersed body etc. We consider some of these later.

Next: Bernoulli Equation Up: Integral equations for the Previous: Application to an one-dimensional (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Bernoulli Equation

Next: Assumptions Up: Integral equations for the Previous: Momentum Equation

Bernoulli Equation The momentum equation we have just derived allows us to develop the Bernoulli Equation after Bernoulli (1738). This equation basically connects pressure at any point in flow with velocity. It is one of the widely used equations in fluid dynamics to calculate pressure with the knowledge of velocity. We derive the equation for a stream tube and consider its generalisation , its applicability and limitations later. Since we are interested in the fluid behaviour at a point consider a differential stream tube within a flow and a small control volume within it as shown in Fig.3.21. Since we are considering a stream tube, any flow takes place only along it and through the ends of it. The flow is therefore one dimensional in nature and takes places in a direction s along the stream tube. Accordingly, we denote the velocity by Vs. There is no flow across the tube. Let the length of the stream tube be ds.

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Bernoulli Equation

Figure 3.21: Differential Control Volume for an onedimensional steady flow

Since it is a small stream tube any property changes only slightly along it. If the area, velocity, density and pressure at the left hand end i.e., the inlet end, (1) be

. Let us treat the flow as

incompressible (this restriction can be removed later). We assume that at the outlet end (2) the corresponding properties to be . Let us now apply the momentum equation to the differential control volume we have considered. In any derivations or while solving problems http://www.aeromech.usyd.edu.au/aero/fprops/cvanalysis/node26.html (2 of 3)24/2/2006 16:57:46

Bernoulli Equation

involving fluid flows it helps to list out the assumptions made. Accordingly we start with a listing of the assumptions.

Subsections ❍ ❍ ❍ ❍

Assumptions Application of Continuity Equation Application of Momentum Equation Terms on the Right Hand Side

Next: Assumptions Up: Integral equations for the Previous: Momentum Equation (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Assumptions

Next: Application of Continuity Equation Up: Bernoulli Equation Previous: Bernoulli Equation

Assumptions 1. A stream tube with no cross flow considered. 2. The flow is steady 3. Fluid is incompressible (

=constant).

It is necessary to note that any application of the momentum equation should be preceded by the Continuity Equation. We cannot obtain complete information about the flow by applying momentum equation alone.

(c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Application of Continuity Equation

Next: Application of Momentum Equation Up: Bernoulli Equation Previous: Assumptions

Application of Continuity Equation Equation 3.30 gives

The first term in the equation cancels out because of the steady flow assumption (2 see Assumptions). Since all the flow takes place through (1) and (2) only the remaining term reduces to

giving (3.44)

where

is the mass flow rate through the control volume.

(c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Application of Momentum Equation

Next: Terms on the Right Up: Bernoulli Equation Previous: Application of Continuity Equation

Application of Momentum Equation From Eqn. 3.43 we have the momentum Equation-

(3.45)

Since the flow is steady, the first term on the RHS drops out. We need to evaluate the body forces

and surface forces

acting on the control volume.

Body Force. The only body force acting is the weight of the fluid within the control volume. We need to consider the component of this in the

direction. Accordingly,

noting that sin q = dz, we have ,

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Application of Momentum Equation

Dropping terms such as dz x dA, we have,

(3.46)

Surface Forces The surface force is due to pressure acting upon the boundaries of the control surface. There are three terms that contribute - end (1), end (2) and the bounding surface of the stream tube. Force on each of these is given by the product of pressure and area. For the bounding surface we take this force to be the product of an average pressure, multiplied by the effective area, dA. Thus we have for the surface forces,

Cancelling out terms and neglecting products such as dp dA, we have (3.47)

Next: Terms on the Right Up: Bernoulli Equation Previous: Application of Continuity Equation (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Terms on the Right Hand Side

Next: Application to moving Control Up: Bernoulli Equation Previous: Application of Momentum Equation

Terms on the Right Hand Side we have on the RHS of Eqn. 3.45,

substituting for Vs r A VsFrom Eqn. 3.44 , we have RHS =

(3.48)

Now collecting terms for the LHS and RHS we have,

i.e.,

i.e.,

i.e.,

The above equation is readily integrated for an incompressible flow ( http://www.aeromech.usyd.edu.au/aero/fprops/cvanalysis/node30.html (1 of 2)24/2/2006 16:58:46

(3.49)

Terms on the Right Hand Side

=constant). As a result we have,

(3.50)

Equation 3.50 is called the Bernoulli Equation. Note that it connects pressure (p), elevation (z) and velocity (Vs). Once it is understood that the equation is valid along a streamline (i.e., within a stream tube) we can drop the subscript,s for velocity giving,

(3.51)

It may be pointed out that the equation is valid for steady flows only in absence of any friction such as the one due to viscosity. Further the flow is to be incompressible. We will derive the Bernoulli equation again but based on energy considerations.

Next: Application to moving Control Up: Bernoulli Equation Previous: Application of Momentum Equation

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Application to moving Control Volumes

Next: Equation for Angular Momentum Up: Integral equations for the Previous: Terms on the Right

Application to moving Control Volumes The Continuity and the Momentum Equations we have derived can be extended to cases where the control volume is not fixed in space. One such case is when the control volume is moving with a constant velocity, say an aeroplane or a ship moving at a constant speed. Note that the equations we have derived assume that the speeds are all referred to the control volume. So it becomes a simple matter to consider a control volume moving at a constant speed, Vcv. Define

(3.52) which now is the speed relative to the control volume. The equation for Reynolds Transport theorem, Eqn.3.27 gets altered as

(3.53)

(c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Equation for Angular Momentum

Next: Deformable Control Volumes and Up: Integral equations for the Previous: Application to moving Control

Equation for Angular Momentum Many of the flow devices and machinery involve rotating components. Examples are Centrifugal pumps, Turbines and Compressors. The analysis of such systems is facilitated by the Reynolds Transport theorem written for angular momentum. we have from Eqn.3.41,

where,

It becomes necessary now to calculate the angular momentum about some point, say O. Then we have,

(3.54)

Substitution into the equation for Reynolds theorem (Eqn.3.27) gives, (3.55)

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Equation for Angular Momentum

Further, the LHS of the above equation is the sum of all the moments about the point , ie.,

. Accordingly we have,

(3.56)

Subsections ●

Deformable Control Volumes and Control Volumes with non-inertial acceleration

Next: Deformable Control Volumes and Up: Integral equations for the Previous: Application to moving Control (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Deformable Control Volumes and Control Volumes with non-inertial acceleration

Next: Energy Equation Up: Equation for Angular Momentum Previous: Equation for Angular Momentum Deformable Control Volumes and Control Volumes with non-inertial acceleration It is possible to extend our analysis to the general cases of deformable control volumes and those that undergo acceleration. But these are not necessary in a first course in fluid mechanics. However, there are many textbooks that do cover such advanced topics.

(c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Energy Equation

Next: Energy equation for a Up: Integral equations for the Previous: Deformable Control Volumes and

Energy Equation We now apply the Reynolds Transport theorem (Eqn. 3.27) to derive an equation for energy conservation in a control volume. Now we have,

(3.57)

On the LHS we have

, which from the First Law of Thermodynamics is

(3.58)

Where

is the rate at which heat is added to the system and

is the rate at which work is done on/by the system.

Substituting in Eqn.3.27 we have

(3.59)

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Energy Equation

In the above equation, e should include all forms of energy - internal, potential, kinetic and others. The others category will include nuclear, electromagnetic and other sources of energy. But for simple fluid flows these are not important. Fields such as Magneto Hydrodynamics and Relativistic Fluid Dynamics will involve these forms of energy too. We have then

(3.60)

Concerning work, we have different kinds - shaft work, Ws, work done by pressure, Wp and work due to shear forces on the control surface. Shaft work includes any work that is directly added to the system by means of a pump, piston etc. Work done by pressure is calculated as

(3.61)

where dA ia an elemental area over the control surface, the velocity Vn is into the control volume (hence gets a negative sign). This equation is integrated over the control surface to obtain the total work due to pressure. Thus,

(3.62)

Work due to shear forces is small and is usually neglected. Heat added transfer. Upon substituting for various terms we have,

i.e.,

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becomes important only occasionally in problems involving heat

Energy Equation

(3.63)

where h is specific enthalpy given by

. Equation 3.63 is the general form of the Energy Equation for a control volume.

Subsections ● ●

Energy equation for a one-dimensional control volume Low Speed Application

Next: Energy equation for a Up: Integral equations for the Previous: Deformable Control Volumes and (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Energy equation for a one-dimensional control volume

Next: Low Speed Application Up: Energy Equation Previous: Energy Equation Energy equation for a one-dimensional control volume

Figure 3.22 : Control Control Volume for an onedimensional steady flow

Consider the one-dimensional control volume that we have analysed before and shown in Fig.3.22. If we interpret the velocity, density, pressure and other variables to be uniform across the ends or that they are the averaged values we have for a steady flow

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Energy equation for a one-dimensional control volume

(3.64)

Note that for continuity,

(3.65)

On division by

and denoting

by q and

by

we have after rearrangement of terms,

(3.66)

Note that the term

is equal to the Total enthalpy denoted by H0. Accordingly the

Eqn.3.66 becomes

(3.67) That is to say that the total enthalpy of a control volume is conserved unless heat or work is added to / taken out of the control volume.

Next: Low Speed Application Up: Energy Equation Previous: Energy Equation (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Low Speed Application

Next: Relationship between Energy Equation Up: Energy Equation Previous: Energy equation for a Low Speed Application In Low Speed application, especially in civil engineering, it is usual to express energy as a Head, with each of the terms in Eqn. 3.66 having the units of a Length, m. This is done by dividing the equation throughout by g. Thus,

or (3.68)

The term

is called the Pressure Head and

the velocity Head. Terms hq and hs represent the heat

added and shaft work converted to "head" units. If we consider a simple pipe flow without the shaft work then the equation becomes

(3.69)

The terms within the parenthesis is what is called the Total Head or Available Head. Clearly with the flow some available head is lost because of friction and heat transfer. It is a common practice to use the above equation in the following form-

(3.70)

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Low Speed Application

The losses that take place between "inlet" i.e.,1 and "outlet" i.e., 2 are obtained through measurements and correlations.

Next: Relationship between Energy Equation Up: Energy Equation Previous: Energy equation for a (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Relationship between Energy Equation and Bernoulli Equation

Next: Bernoulli Equation for Aerodynamic Up: Integral equations for the Previous: Low Speed Application

Relationship between Energy Equation and Bernoulli Equation An examination of Eqns. 3.70 and 3.51 brings out the connection between the Energy equation and the Bernoulli equation. It is clear that two equations become one when losses that occur between (1) and (2) are ignored. Bernoulli Equation can be used only when we are considering a frictionless flow along a streamline. Further it is required that the flow be incompressible without any addition of heat or shaft work.

(c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Bernoulli Equation for Aerodynamic Flow

Next: Stagnation Pressure Up: Integral equations for the Previous: Relationship between Energy Equation

Bernoulli Equation for Aerodynamic Flow In aerodynamics one deals with considerably higher speeds than in flows of interest to civil engineers. An aeroplane flies at speeds of the order of 500 kmph and more, while river flows or household pipe flows may involve 10 kmph or so. Consequently, the kinetic energy in aerodynamic flows is very large when compared to the potential energy. Accordingly, it is usual to neglect potential energy for such flows. The Bernoulli Equation as a consequence becomes,

(3.71)

Subsections ● ● ●

Stagnation Pressure Energy Grade Line Kinetic Energy Correction Factor

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Stagnation Pressure

Next: Energy Grade Line Up: Bernoulli Equation for Aerodynamic Previous: Bernoulli Equation for Aerodynamic Stagnation Pressure

Figure 3.23 : Stagnation Point on (a) Simple Body and (b) a complicated Body

Consider the application of the above form of Bernoulli equation for the flow about a body such as an aeroplane as shown in Fig.3.23. Let rs be a streamline that passes through the stagnation point of the flow, i.e., the point where the flow is brought to rest or where the velocity is zero. Applying the Bernoulli equation along rs we have,

(3.72)

where ps and Vs are the pressure and velocity at the point s. It is known that Vs= 0. Therefore,

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Stagnation Pressure

(3.73)

ps is referred to as Stagnation Pressure. Obviously it is the maximum pressure experienced by the fluid. It becomes a very convenient constant for the Bernoulli Equation for aerodynamics flows. It is the pressure experienced by the fluid when it is brought to rest. it is as if the kinetic energy of the flowing fluid is converted into pressure as a consequence of the fluid being brought to rest. The term "p" is the pressure seen by the moving fluid and is referred to as Static Pressure.

Next: Energy Grade Line Up: Bernoulli Equation for Aerodynamic Previous: Bernoulli Equation for Aerodynamic (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Energy Grade Line

Next: Kinetic Energy Correction Factor Up: Bernoulli Equation for Aerodynamic Previous: Stagnation Pressure Energy Grade Line Terms Energy Grade Line and Hydraulic Grade Line are frequently used by hydraulic engineers. Let us express each of the terms of the Bernoulli equation as a head. We have seen that in absence of work and heat transfer,

(3.74)

where term H is not to be mistaken for enthalpy and is to be taken as the total head. If the above equation is graphically represented we see that the total energy value being constant becomes a horizontal line as shown in Fig. 3.24 and is called the Energy Grade Line. One other line that is defined is the Hydraulic Grade Line, which is the Energy Grade Line take away the velocity head (i.e., V2/2g).

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Energy Grade Line

Figure 3.24: Energy Grade Line (EGL) and Hydraulic Grade Line (HGL) for an one-dimensional flow.

If the losses are taken into account the EGL will drop accordingly. Any work extraction along the path as with a turbine, will be seen as a sudden drop in the EGL. Any work addition will be reflected as a sharp rise. HGL follows similar trends.

Next: Kinetic Energy Correction Factor Up: Bernoulli Equation for Aerodynamic Previous: Stagnation Pressure (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Kinetic Energy Correction Factor

Next: Applications of Bernoulli Equation Up: Bernoulli Equation for Aerodynamic Previous: Energy Grade Line Kinetic Energy Correction Factor We have assumed in the derivation of Bernoulli equation that the velocity at the end sections (1) and (2) is uniform. But in a practical situation this may not be the case and the velocity can very across the cross section. A remedy is to use a correction factor for the kinetic energy term in the equation. If an end section then we can write for energy,

is the average velocity at

(3.75)

After simplification we find that

(3.76)

Consequently, Eqn.3.70 (Low Speed Application) is written as

(3.77)

where is the Kinetic Energy Factor. Its value for a fully developed laminar pipe flow is around 2, whereas for a turbulent pipe flow it is between 1.04 to 1.11. It is usual to take it is 1 for a turbulent flow. It should not be neglected for a laminar flow.

Next: Applications of Bernoulli Equation Up: Bernoulli Equation for Aerodynamic Previous: Energy Grade Line (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Applications of Bernoulli Equation

Next: Flow through a Sharp-edged Up: Integral equations for the Previous: Kinetic Energy Correction Factor

Applications of Bernoulli Equation Bernoulli Equation is one of the most important equations in Fluid Mechanics and finds many applications. One such is the measurement of flow by introducing a restriction within the flow. The restriction may take the form of an orifice plate or a converging-diverging nozzle. The required formula will be first derived for an orifice plate and will be extended to other devices.

Subsections ● ● ●

Flow through a Sharp-edged Orifice Flow Through a Flow Nozzle Flow through a Venturi Tube

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Flow through a Sharp-edged Orifice

Next: Flow Through a Flow Up: Applications of Bernoulli Equation Previous: Applications of Bernoulli Equation Flow through a Sharp-edged Orifice Consider an orifice plate placed in a pipe flow as shown in Fig.3.25 . We assume that the thickness of the plate is small in comparison to the pipe diameter. Let the orifice be sharp edged. The effect of a rounded plate is a matter of detail and will not be considered here.

Figure 3.25: Flow through an Orifice Body

A fully developed flow prevails at the upstream of the orifice. The presence of the orifice makes the flow accelerate through it thus increasing the velocity. It may appear that the flow fills the orifice completely and expands downstream of it. But this is not true. As the flow expands downstream it cannot fill the entire diameter of the pipe at http://www.aeromech.usyd.edu.au/aero/fprops/cvanalysis/node43.html (1 of 5)24/2/2006 18:26:02

Flow through a Sharp-edged Orifice

once. It requires a distance before it does. A recirculating flow develops immediate downstream of the nozzle. As a consequence the smallest diameter of flow is not equal to the orifice diameter, but smaller than it. The position of the smallest diameter occurs downstream of the orifice. We can deduce the flow rate through the pipe by measuring the pressure difference upstream of the nozzle and at the orifice. We make a few assumptions about the flow as follows 1. 2. 3. 4.

The flow is steady The flow is incompressible There is no friction or losses The velocities at sections (1) and (2) are uniform, i.e, they do not vary in a radial direction. 5. The pipe is horizontal, i.e., z is the same at (1) and (2), i.e., z1 = z2. This assumption can be relaxed easily. It is possible to have a fluid flowing through an inclined pipe. Then the gz term does not cancel out from LHS and RHS of the Bernoulli Equation. The equations we consider are (a) Continuity and (b) Bernoulli equations. a) Continuity Equation. (3.78)

b) Bernoulli Equation (3.79)

The Bernoulli Equation gives, (3.80)

Noting from Continuity Equation that

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Flow through a Sharp-edged Orifice

we have

(3.81)

Solving for V2 we have,

(3.82)

Consequently, the mass flow rate becomes,

(3.83)

The above equation gives the mass flow rate through the pipe in terms of the pressure drop and the areas. The equation gives only a theoretical value. In order to obtain a more realistic value one need to substitute the actual area at the minimum cross section or the Vena Contracta. This is not easy to measure. In addition losses may not be negligible as we have assumed. Extent of losses is a function of the Reynolds number. In practice, a Coefficient of Discharge is defined such that http://www.aeromech.usyd.edu.au/aero/fprops/cvanalysis/node43.html (3 of 5)24/2/2006 18:26:02

Flow through a Sharp-edged Orifice

(3.84)

Further if we define a ratio of diameters

such that

(3.85)

Sometimes the ratio

is referred to as Velocity of

Approach Factor. Again it is usual to combine this and the Discharge Coefficient to define a Flow Coefficient given by

(3.86)

Consequently the mass flow rate is given by, (3.87)

Thus the mass flow rate for a pipe can be calculated with the knowledge of pressure drop, the orifice diameter and the coefficient K. Extensive data exists in handbooks on the coefficient K. Pressure drop is usually measured by using a manometer as shown in Fig. 3.25. Now the pressure drop is obtained as h, the height of a liquid

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Flow through a Sharp-edged Orifice

column (which may be mercury). Accordingly the alternate form of Eqn.3.87 is

(3.88)

Next: Flow Through a Flow Up: Applications of Bernoulli Equation Previous: Applications of Bernoulli Equation (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Flow Through a Flow Nozzle

Next: Flow through a Venturi Up: Applications of Bernoulli Equation Previous: Flow through a Sharp-edged Flow Through a Flow Nozzle

Figure 3.26: Flow throug a Nozzle Though geometrically different from an orifice plate, a flow nozzle is conceptually similar to it. Fig.3.26 shows a flow nozzle which is just a converging nozzle placed in a pipe. The flow mechanism is similar to that for the orifice (see Flow through a Sharp-edged Orifice). Now the area A2 is the throat area of the nozzle.

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Flow through a Venturi Tube

Next: Important Applications of Control Up: Applications of Bernoulli Equation Previous: Flow Through a Flow Flow through a Venturi Tube

Figure 3.27: Venturi Tube

A Venturi Tube is a converging-diverging nozzle (Fig. 3.27) placed in a pipe. The principle of this was demonstrated by Giovanni Battista Venturi(1746-1822) in 1797. It was only later in 1887 that it was employed for flow measurement by Herschel. The mass flow rate is again given by Eqn. 3.84. ((see Flow through a Sharp-edged Orifice) http://www.aeromech.usyd.edu.au/aero/fprops/cvanalysis/node45.html (1 of 2)24/2/2006 18:26:32

Important Applications of Control Volume Analysis

Next: Measurement of Drag about Up: Integral equations for the Previous: Flow through a Venturi

Important Applications of Control Volume Analysis In this section we consider some of the important applications of the control volume analysis. Every analysis may or may not involve each of the equations we have derived- Continuity, Momentum, Bernoulli and Energy. These applications are important from a physical point of view.

Subsections ● ● ● ●

● ●

Measurement of Drag about a Body immersed in a fluid Jet Impingement on a surface Force on a Pipe Bend Froude's Propeller Theory ❍ Continuity Equation ❍ Momentum Equation ❍ Bernoulli Equation Analysis of a Wind Turbine Total Pressure Loss through a Sudden Expansion ❍ Continuity Equation ❍ Momentum Equation ❍ Bernoulli Equation

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Measurement of Drag about a Body immersed in a fluid

Next: Jet Impingement on a Up: Important Applications of Control Previous: Important Applications of Control Measurement of Drag about a Body immersed in a fluid Consider abody such as an aerofoil placed in a flow, which could be a in a wind tunnel. Far from the body the flow is uniform and inviscid. As the flow approaches the body many dramatic changes take place. The flow will start to depart from uniformity. But as the flow negotiates the body viscosity comes into play. Consequently, the velocity on the body surface is zero. The velocity catches up with the freestream speed as we move away from the body. In other words, a boundary layer develops. A boundary layer is not static. It grows as the flow moves downstream. When the flow leaves the body the centreline velocity is not zero anymore. It starts to build up slowly. This is the Wake region. If a velocity profile is measured across the wake by carrying out what is called a Wake Traverse, we see that it resembles that shown in Fig.3.28. The wake profile thus carries signatures of the viscous effect.

Figure 3.28: Measurement of Drag about an immersed body http://www.aeromech.usyd.edu.au/aero/fprops/cvanalysis/node47.html (1 of 4)24/2/2006 18:26:55

Measurement of Drag about a Body immersed in a fluid

If a force balance is conducted in a region surrounding the body/ aerofoil then a force imbalance is evident. This should be related to Drag. Consider the body/aerofoil placed in a wind tunnel. Let us prescribe a control volume ABCD surrounding it. The left and right hand boundaries AB and CD are far from the body. As a result the flow is uniform ( at a speed

) on AB. At the right hand boundary CD is the wake with the velocity profile as

sketched. We assume that the top and bottom boundaries of the control volume,AD and BC are far away from the body and the vertical component of velocity namely v is zero across them. subsubsectionAnalysis We make the following assumptions. 1. Steady Flow 2. Incompressible Flow 3. Static Pressure is same everywhere, which is actually a simplifying assumption. This could be relaxed. Continuity Equation. Since the flow is steady, we have,

i.e.,

since v component of velocity along BC and AD is zero, the equation reduces to

leading to

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(3.89)

Measurement of Drag about a Body immersed in a fluid

Momentum equation On applying the momentum equation to the control volume we have

(3.90)

i.e.,

(3.91)

since v = 0 on BC and AD, we have (3.92)

The body force Fbx on the control volume is zero. The surface forces are drag and that due to pressure. Since we have assumed that pressure is uniform, the latter is zero. Further length AB = length CD, allowing us to combine the integrals on the RHS. Thus we have,

(3.93)

In effect the velocities below C and that above D will be uniform and equal to

. Consequently the above equation could also be written as

(3.94)

A flaw in the above analysis should be apparent to you. Look at Eqn.3.89. This cannot be true. The mass flow going through AB at a uniform velocity cannot be equal to that across CDwhere the velocities are smaller than

. Some mass has to escape through AD and BC. In other words our

assumption of v = 0 on AD and BC is faulty. The equation for drag that we have obtained is inaccurate as a consequence. A more acceptable estimate for drag can be obtained by considering the v component of velocity on AD and BC. The other method is to make these boundaries streamlines of flow. Then . This is left as an exercise. http://www.aeromech.usyd.edu.au/aero/fprops/cvanalysis/node47.html (3 of 4)24/2/2006 18:26:55

Jet Impingement on a surface

Next: Force on a Pipe Up: Important Applications of Control Previous: Measurement of Drag about Jet Impingement on a surface Consider a jet with a cross section Aj at a speed vj impinging on a solid surface at an angle as shown in Fig.3.29. It is required to calculate the normal force exerted on the surface.

Figure 3.29: Jet impinging on a surface

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Jet Impingement on a surface

Let us consider the physics of the process first. As the jet impinges upon the surface, it splits into two parts. These move tangential to the surface. The normal component of the force however does act upon the surface and is to be countered for stability. Prescribe x and y axes parallel and perpendicular to the surface and chose a control volume as shown. At the entry to the control volume we have the momentum in the y-direction equal to

(3.95)

At the solid surface velocity normal is zero and as such there is no normal momentum acting. The normal force acting upon the surface is given by Eqn.3.95.

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Force on a Pipe Bend

Next: Froude's Propeller Theory Up: Important Applications of Control Previous: Jet Impingement on a Force on a Pipe Bend Consider a flow through a pipe bend as shown. The flow enters the bend with a speed V1 and leaves it a speed V2, the corresponding areas of cross section being A1 and A2 respectively. The velocities have components u and v in x and y directions. As the flow negotiates the bend it exerts a force upon it. This force is readily calculated by the momentum theorem.

Figure 3.30: Force on a Pipe Bend The continuity equation yields http://www.aeromech.usyd.edu.au/aero/fprops/cvanalysis/node49.html (1 of 2)24/2/2006 18:30:35

Force on a Pipe Bend

(3.96) Carrying out a force balance in x-direction, we have

(3.97)

In the y-direction we have,

giving

(3.98)

Thus the force components acting on the bend are

(3.99)

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Froude's Propeller Theory

Next: Continuity Equation Up: Important Applications of Control Previous: Force on a Pipe Froude's Propeller Theory Propellers are a mechanism for the propulsion of an aeroplane. In its generic form a propeller is pair of rotating blades mounted on a shaft that houses the engine as well. As the engine operates the propeller turns sucking a large amount of air. As this air passes through the rotating blades, it gets energised, its speed increases. In the process the required Thrust to propel the aircraft is produced.

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Froude's Propeller Theory

Figure 3.31: Froude analysis of a Propeller

The analysis we carry out follows William Froude (1810-1879). We consider the propeller as a thin disc rotating in air as shown in Fig.3.31. Let the pressure and velocity far away from the disc i.e., at section (1) be p1 and V1 respectively. The conditions just at (2)

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Froude's Propeller Theory

which is the front of the disc are p2 and V2. The disc imparts momentum and energy to the incoming air such that the pressure and velocity just behind the disc (3) are V3 and p3 respectively. At (4), far downstream the conditions are V4 and p4. We assume that the air which is influenced by the disc is confined to a slipstream as shown. Since the disc is thin and the area of cross section at (2) and (3) ar equal, we have (3.100) The pressures at (1) and (4) are equal to the freestream value. We consider the control volume formed by slipstream and the ends (1) and (4) and write the momentum equation.

Subsections ● ● ●

Continuity Equation Momentum Equation Bernoulli Equation

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Continuity Equation

Next: Momentum Equation Up: Froude's Propeller Theory Previous: Froude's Propeller Theory

Continuity Equation An application of the Continuity Equation gives, (3.101)

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Momentum Equation

Next: Bernoulli Equation Up: Froude's Propeller Theory Previous: Continuity Equation

Momentum Equation Considering first the forces the only force that acts upon the control volume is the net force on the disc or the Thrust, F. Pressures being equal at (1) and (4) does not contribute to the surface force. Since the flow takes place in a horizontal direction there is no body force to be considered. Accordingly, (3.102)

Noting that V2 = V3, this force F is equal to A(p3 - p2), where A is the area of cross section of the disc. As a consequence we have,

dividing by A, we have

noting that

we have (3.103)

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Bernoulli Equation

Next: Analysis of a Wind Up: Froude's Propeller Theory Previous: Momentum Equation

Bernoulli Equation It is easy to see that there is no addition of work or heat between sections (1) and (2) and also between (3) and (4). It is possible to apply Bernoulli equation between (1) and (2) and also between (3) and (4) but not between (2) and (3).

(3.104)

Since V2 = V3and p1 = p4 we have from the above equations

(3.105)

Eliminating p3 - p2 from Eqns.3.105 and 3.103 we have, (3.106)

If the velocities are referred to the freestream air speed, i.e., V1, we see

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Bernoulli Equation

that the propeller moves at a velocityV1 . The work done by the propeller on the air stream or the power output is then, power,

(3.107)

In addition some kinetic energy is added to the air stream, which goes as a waste. The power input therefore is given by

power input

(3.108)

From Eqns. 3.107 and 3.108 the efficiency of the propeller will be

(3.109)

The term

is called the Froude Efficiency.

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Analysis of a Wind Turbine

Next: Total Pressure Loss through Up: Important Applications of Control Previous: Bernoulli Equation Analysis of a Wind Turbine

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Analysis of a Wind Turbine

Figure 3.32: Analysis of a Wind Turbine

A wind turbine (Fig.3.32) extracts energy from an air stream while a propeller adds energy to the air stream. The analysis follows the same lines. The wind turbines are smaller in size compared to the propellers. For the wind turbine too we have the result that

Power,

(3.110)

Now the efficiency is given by the Kinetic energy extracted divided by the kinetic energy in the free stream. Thus, (3.111)

The above expression has a maximum when V4/V1 = 1/3. The maximum theoretical efficiency is 59.3%.

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Total Pressure Loss through a Sudden Expansion

Next: Continuity Equation Up: Important Applications of Control Previous: Analysis of a Wind Total Pressure Loss through a Sudden Expansion

Figure 3.33: Losses through a Sudden Expansion, Borda-Carnot Equation

Consider a sudden expansion placed in a duct (Fig.3.33). The flow does not follow the area changes as suddenly as the geometry does. Any flow will find the sudden area increase difficult to negotiate. In fact a recirculating flow develops as was seen in case of the orifice flow. This gives to losses which are reflected in the total pressure at downstream being reduced. It is possible to calculate this loss from a control volume analysis.

Subsections ● ●

Continuity Equation Momentum Equation

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Total Pressure Loss through a Sudden Expansion ●

Bernoulli Equation

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Bernoulli Equation

Next: Measurement of Airspeed Up: Total Pressure Loss through Previous: Momentum Equation

Bernoulli Equation We remind ourselves that we cannot connect stations (1) and (2) with the Bernoulli Equation. But we just use the total pressure relation at (1) and (2). Accordingly,

(3.116)

Total pressure loss is hence equal to (3.117)

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Continuity Equation

Next: Momentum Equation Up: Total Pressure Loss through Previous: Total Pressure Loss through

Continuity Equation (3.112)

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Measurement of Airspeed

Next: Table of Contents ... Up: Integral equations for the Previous: Bernoulli Equation

Measurement of Airspeed Bernoulli equation readily allows one to determine the flow speed once the static and stagnation pressures are known. Rewriting Eqn.3.73 ( Stagnation Pressure)we have

(3.118)

It is therefore a matter of measuring the static and stagnation pressures at a given location. Static Pressure is conveniently measured by drilling a hole in the wall or the pipe, called the Pressure Tap (Fig. 3.34). A manometer or a pressure gauge is connected to the tap. During flow static pressure is communicated to the measuring device. Alternately one could use a Static Pressure probe shown in Fig. 3.35. This has holes which communicate the pressure to a measuring device. Measurement of stagnation pressure requires that the flow be brought to rest. A glass tube or a hypodermic needle aligned with the flow and facing upstream as shown in Fig. 3.36 will do the job. Alternately, what is called a Pitot Tube shown in Fig.3.37, with a hole facing upstream of the flow may be employed. The method shown in Fig. 3.38 suggests itself.

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Measurement of Airspeed

To Manometer or a gauge

To Manometer or a gauge Figure 3.34: Pressure Tap to measure Static Pressure Figure 3.35: Static Pressure Probe

Figure 3.36: Stagnation Tube

Figure 3.37: Pitot Tube

But for an accurate determination of flow speed, static and stagnation pressures are to be measured simultaneously . This is made possible by a Pitot-Staic tube shown in Fig. 3.39. This http://www.aeromech.usyd.edu.au/aero/fprops/cvanalysis/node59.html (2 of 3)24/2/2006 20:26:56

Measurement of Airspeed

combines the staic pressure probe and the pitot tube. The "staic holes" and the "stagnation hole" are as near to each other as possible.

Figure 3.38: Pitot tube used with a static pressure tap Figure 3.39: Pitot-static tube

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poten

Next: Elements of Potential Flow

DOWNLOAD SOFTWARE

●

Elements of Potential Flow ❍ General ❍ Conservation of Mass ■ Continuity Equation in Cylindrical Polar Coordinates ■ Continuity Equation for steady flow ■ Continuity Equation for an Incompressible flow ❍ Stream function ■ Properties of Stream Function

❍

❍ ❍

■

(a) Line

■

(b)

is a streamline between two streamlines is

proportional to the Volumetric Flow ■ Stream Function in Polar Coordinates Kinematics of Fluid Motion ■ Translation ■ Linear Deformation ■ Rotation ■ Meaning of Irrotationality ■ Angular Deformation Circulation Velocity Potential ■

Relationship between

■

Occurrence of Irrotational Flows

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and

poten ❍

❍

●

Simple Examples of Plane Potential Flows ■ Equations in Cartesian Coordinates ■ Equations in Polar Coordinates ■ Uniform Flow ■ Source and Sink ■ Vortex ■ Circulation around a Vortex ■ A Source-Sink Pair ■ Doublet Superposition of Elementary Flows ■ Uniform Flow and a Source ■ Rankine Oval ■ Flow Around a Circular Cylinder ■ Flow about a Lifting Cylinder ■ Stagnation Points for a lifting circular cylinder ■ Surface Pressure Distribution and Lift ■ Kutta-Joukowsky Theorem ■ Magnus Effect

Table of Contents ...

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Elements of Potential Flow

Next: General Up: poten Previous:

Elements of Potential Flow

Subsections ● ●

●

General Conservation of Mass ❍ Continuity Equation in Cylindrical Polar Coordinates ❍ Continuity Equation for steady flow ❍ Continuity Equation for an Incompressible flow Stream function ❍ Properties of Stream Function

❍

●

● ●

■

(a) Line

■

(b)

=constant is a streamline between two streamlines is proportional to

the Volumetric Flow Stream Function in Polar Coordinates

Kinematics of Fluid Motion ❍ Translation ❍ Linear Deformation ❍ Rotation ■ Meaning of Irrotationality ❍ Angular Deformation Circulation Velocity Potential ❍

Relationship between

and

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Elements of Potential Flow ❍

●

●

Occurrence of Irrotational Flows

Simple Examples of Plane Potential Flows ❍ Equations in Cartesian Coordinates ❍ Equations in Polar Coordinates ❍ Uniform Flow ❍ Source and Sink ❍ Vortex ■ Circulation around a Vortex ❍ A Source-Sink Pair ❍ Doublet Superposition of Elementary Flows ❍ Uniform Flow and a Source ❍ Rankine Oval ❍ Flow Around a Circular Cylinder ❍ Flow about a Lifting Cylinder ❍ Stagnation Points for a lifting circular cylinder ❍ Surface Pressure Distribution and Lift ❍ Kutta-Joukowsky Theorem ❍ Magnus Effect

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General

Next: Conservation of Mass Up: Elements of Potential Flow Previous: Elements of Potential Flow

General We now move to the Differential analysis of fluid motion. This is sharply different from the analysis done in the previous chapter using control volumes. Our interest now will be on a description of flow at any given point in the flow than on overall effects of flow on a control volume. Reasons why we require such an analysis are evident. There are numerous situations where one needs a point to point description of flow. One may require the distribution of shear stress on the surface of an aeroplane wing or the distribution of heat transfer coefficient in a piping in an air-conditioning system. This necessitates a more complete knowledge of the flow field than that provided by the Integral Approach of the previous chapter. Even here we appeal to the general laws of mechanics to give us the equations to solve. These are the conservation laws for mass, momentum and energy as before. The resulting equations now are Differential Equations (and hence the name Differential Approach) while the integral analysis gave us algebraic equations to solve. From this end, differential equations are more complicated. Remember we need to take into account all the forces acting when we write the momentum equation. We will see that the equations get very involved when viscous forces are considered along with other forces. This leads to what are called the Navier-Stokes Equations. We postpone the discussion of these equations to a later chapter and limit ourselves to simple flows which lend themselves to simple equations which are easily solved. As a result we consider only inviscid flows in this chapter. This leads to a simple analysis. In fact we will be solving only the continuity equation for mass to calculate velocity components. Pressure is obtained from Bernoulli equation. Of course, various assumptions will have to made to make the analysis easier. We discuss these in course of the text.

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General

We first derive the continuity equation which is a statement of the fact that mass is conserved. Then we introduce the stream function which is a powerful concept in fluid dynamics. By anaysing the kinematics of fluid motion we proceed to introduce concepts of Circulation and Irrotationality. Definition of Velocity Potential follows. We then write down the stream functions and velocity potentials for some of the simple flows like a uniform flow, source and sink flow and vortex flow. These flows are then superposed to arrive at solutions for complicated flows. Flow about a circular cylinder is then analysed in some detail.

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Conservation of Mass

Next: Continuity Equation in Cylindrical Up: Elements of Potential Flow Previous: General

Conservation of Mass

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Conservation of Mass

Figure 4.1: Differential Control Volume We derive the equation for mass conservation by considering a differential control volume at P(x,y, z)as shown in Fig.4.1. Let the dimensions of the volume be dx, dyand dz and velocity components at P be u,v and w. Assuming that the mass flow rate is continuous across the volume we can calculate the mass flow rates at the various faces of the cell by a Taylor Series expansion as we had done previously (Eqn. 2.5). Accordingly we have,

(4.1)

The net mass flow rate into the control volume as a consequence is given by, (4.2) Applying the Reynolds transport theorem for mass (Eqn. 3.30) will give,

(4.3)

From Eqn.4.1 and 4.2 we have,

(4.4)

Further in Eqn.4.3 noting that the control volume is tiny, the integral can be approximated as

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Conservation of Mass

(4.5)

The Reynolds Transport Theorem thus gives,

(4.6)

Cancelling out dx dy dz, we have,

(4.7)

Eqn. 4.7 is known as the Continuity Equation. Note that it is a very general equation with hardly any assumption except that density and velocities vary continually across the element we have considered. If we now bring in the gradient operator, namely,

(4.8)

and represent velocity as a vector,

(4.9)

Then the Continuity Equation can be written in a compact manner as (4.10)

Written in this form it enables one to consider any other system of coordinates with ease.

Subsections

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Conservation of Mass

● ● ●

Continuity Equation in Cylindrical Polar Coordinates Continuity Equation for steady flow Continuity Equation for an Incompressible flow

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Continuity Equation in Cylindrical Polar Coordinates

Next: Continuity Equation for steady Up: Conservation of Mass Previous: Conservation of Mass Continuity Equation in Cylindrical Polar Coordinates We have derived the Continuity Equation, 4.10 using Cartesian Coordinates. It is possible to use the same system for all flows. But sometimes the equations may become cumbersome. So depending upon the flow geometry it is better to choose an appropriate system. Many flows which involve rotation or radial motion are best described in Cylindrical Polar Coordinates. Let us now write equations for such a system. In this system coordinates for a point P are

and

, which

are indicated in Fig.4.2. The velocity components in these directions respectively are

and

. Transformation between the Cartesian

and the polar systems is provided by the relations, (4.11)

The gradient operator is given by,

(4.12)

As a consequence the continuity equation becomes, (4.13)

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Continuity Equation in Cylindrical Polar Coordinates

Figure 4.2: Cylindrical Polar Coordinate System

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Continuity Equation for steady flow

Next: Continuity Equation for an Up: Conservation of Mass Previous: Continuity Equation in Cylindrical Continuity Equation for steady flow For a steady flow the time derivative vanishes. As a result 4.7 becomes,

(4.14)

The equation in polar coordinates also undergoes the same simplification. (4.15)

These equations are the ones that are to be used for a compressible flow as we have kept density,

still variable.

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Continuity Equation for an Incompressible flow

Next: Stream function Up: Conservation of Mass Previous: Continuity Equation for steady Continuity Equation for an Incompressible flow For an incompressible flow density is a constant. Accordingly we have

(4.16)

and in polar coordinates we have, (4.17)

As noticed for the control volume analysis the continuity equation for an incompressible flow is the same whether the flow is steady or unsteady.

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Stream function

Next: Properties of Stream Function Up: Elements of Potential Flow Previous: Continuity Equation for an

Stream function Stream function is a very useful device in the study of fluid dynamics and was arrived at by the French mathematician Joseph Louis Lagrange in 1781. Of course, it is related to the streamlines of flow, a relationship which we will bring out later. We can define stream functions for both two and three dimensional flows. The latter one is quite complicated and not necessary for our purposes. We restrict ourselves to two-dimensional flows. Consider a two-dimensional incompressible flow for which the continuity equation is given by,

(4.18)

A stream function

is one which satisfies

(4.19)

Substituting these into Eqn.4.18, we have,

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Stream function

(4.20)

Thus the continuity equation is automatically satisfied. Thus if we can find a stream function

that meets with the eqn.4.19 the continuity

equation need not be solved. For the rest of the chapter we will be invariably describing flows with a stream function. Subsections ●

Properties of Stream Function ❍

(a) Line

❍

(b)

=constant is a streamline between two streamlines is proportional to the

Volumetric Flow ●

Stream Function in Polar Coordinates

Next: Properties of Stream Function Up: Elements of Potential Flow Previous: Continuity Equation for an (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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Properties of Stream Function

Next: (a) Line is a Up: Stream function Previous: Stream function Properties of Stream Function

Subsections

●

(a) Line

●

(b)

=constant is a streamline between two streamlines is proportional to the Volumetric

Flow

(c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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(a) Line is a streamline

Next: (b) between two streamlines Up: Properties of Stream Function Previous: Properties of Stream Function

(a) Line

= constant is a streamline

Let us consider a line given by

, a constant as shown in Fig. 4.3. We

have

i.e., giving -vdx + udy = 0 after substituting for

etc.

(4.21)

Going back to Eqn. 4.21, this is the equation to a streamline. What we have proved then is that

=constant line is a streamline of the flow. Alternately equation to a

streamline is given by

= constant.

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(a) Line is a streamline

Figure 4.3 : A Stream Line in A Flow

(c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney

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