Num Sol Of Navier

Numerical Solution of the Incompressible Navier-Stokes Equations Ae243 Biofluid Mechanics Term Project 4 June 2004

Georgios Matheou

The Incompressible Navier-Stokes Equations • Why Care? – Life can not exist without fluids. – All biological flows are incompressible, i.e. no bird or fish flies/swims faster than M=0.3. – Internal flows are mostly laminar (makes things easier).

• In spite of their simplicity the Navier-Stokes describe flows at very low Reynolds numbers (creeping flows) up to complicated turbulent flows at large Reynolds numbers. • The equations:

u  0

– Continuity

– Momentum



u   u  u  p     u  u T t



 2

The Role of Pressure •

Taking the divergence of the Navier-Stokes we get

ui u j 1 2 D 2      u    p     x j xi  Dt  •

•

•

The solution with initial and boundary condition     u  0 is Δ=0 if, and only if, the right hand side is zero everywhere. Thus the pressure satisfies the Poisson equation: ui u j 2  p   x j xi The satisfaction of this Poisson equation is a necessary and sufficient condition for a divergence free velocity field to remain divergence free. The role of pressure is to enforce continuity, it is more a mathematical variable than a physical one. This observation leads to a strategy of solving the Navier-Stokes equations that imposes continuity by inverting a Poisson equation for a pressure-like variable.

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The Problem – Shear Driven Cavity •

• • •

Some insects (dragonfly) have wings with well defined cross-sectional corrugation (Kesel, 2000). Vortices develop in the valleys of the profile. The flow in the cavity is driven by shear. For a square cavity there is only one parameter that characterizes the flow, the Reynolds number:

Re 

U lid L



Flow visualization at Re=0.01. (Taneda, 1979)

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Staggered Grid • Staggered Grid (Harlow and Welsh, 1965) – Pressure is defined at the cell centers – Velocities are normal to the cell faces

• Attractive mathematical and physical properties – Do not display spurious pressure oscillations – Low memory requirements – Computationally efficient – Conservation properties (mass, momentum, kinetic energy, vorticity)

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Numerical Method – Exact Fractional Step Method (Chang, 2002) • •

Goal: satisfy discrete incompressibility and eliminate the pressure equation Incompressibility constraint:  u  nˆ dS   u Si  0 CV

•

Define volume fluxes as Ui=u Si and define the vector q that has the Ui’s in some ordering. Then the above equation in matrix form is: Dq  0, where :

•

faces

 1 0 0 1 1 D    1 1 1 0 0

We can construct a matrix C which is the null space of D, that is D C=0

 1 0 1 0   1 1 0 0    C   0 1 1 0    0 0 1  1    1 0 0 1  6

Numerical Method – Exact Fractional Step Method (Chang, 2002) (cont.) •

C is a discrete curl operator that allows us to define a discrete streamfunction s at the vertices of the mesh:

q  Cs

•

A discrete gradient operator G can be defined as the transpose of D:

G  DT •

•

If we have a scalar quantity (like pressure), the discretized vector of which is φ, then G is the discrete version of Then:

p

CTG  CT DT  (DC)T   0

which reproduces the continuous identity:

  p  0 7

Finite Volume Formulation •

x-momentum equation u 1  u u  ˆ  d V  u u  n d S  pn d S  n  i, j  t i, j   x i, j Re  x x y ny dSi, j

•

Evaluate all integrals with the second order accurate midpoint rule (uniform grid spacing in x and y): x y

•

•

dui , j dt



 



 y ui2 1 , j  ui2 1 , j  x ui , j  1 vi  1 , j 1  ui , j  1 vi  1 , j  y  pi , j  pi 1, j   2

2

In operator form:

2

2

dui , j

2

2

 u   1  u u   u     y    y       Re   x i  12 , j x i  12 , j  y i , j  1 y i , j  1  2 2    

1 (u ) Li , j ui , j 2 dt Re Changing variables from velocity u to volume fluxes U, normalizing in order to clear the denominator of the pressure gradient, the two momentum equations for the vector of fluxes q become: dq 1 M  G  Re Lq  r  b dt  H i(,uj)   i  1 pi , j 

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Elimination of Pressure and Time Marching • Substituting q=Cs and premultipling the system by CT the pressure is eliminated and the momentum equations are reduced to a single scalar equation for s: CTMC

ds 1 T  Re C LCs  CT (r  b) dt

• Using explicit Adams-Bashforth 2 for the convection terms and implicit trapezoidal for the viscous we get the discrete system of equations: t  n1 t  n 1   3  CT  M  L Cs  CT  M  L Cs  t CT  r n  r n1  b  2 Re  2 Re  2   2 

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Verification – Re=100 •Comparison of steady state solution with data from Ghia et al (1982). •Simulation at Re=100 with grid resolution of 100×100. •Ghia resolution is 128×128. •Computed main vortex center at x=0.6188 and y=0.7396 •Ghia prediction at x=0.6172 and y=0.7344

Velocity along the midlines. Lines are the current computation, circles are data from Ghia. Streamlines of steady state solution at Re=100 10

Re=1000 – Grid: 200² - Velocity Field

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Vorticity – Re=1000

t=1.00

t=2.25

t=7.25

t=14.75 12

References • W. Chang, F. Giraldo, and B. Perot. Analysis of an exact fractional step method. J. Comput. Phys., 180:183-199, 2002. • J. H. Ferziger and M. Peric. Computational Methods for Fluid Dynamics. Springer, 2002. • U. Ghia, K. N. Ghia, and C. T. Shin. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J. Comput. Phys., 8:387, 1982. • F. H. Harlow and J. E. Welch. Numerical calculations of time dependent viscous incompressible flow of fluid with a free surface. Phys. Fluids, 8(12):2182, 1965. • A. B. Kesel. Aerodynamic characteristics of dragonfly wing sections compared with technical aerofoils. J. Exp. Biol., 203:3125, 2000. • S. B. Pope. Turbulent flows. Cambridge, 2000. • S. Taneda. Visualization of separating flows. J. Phys. Soc. Jpn, 46:1935, 1979.

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