Num Sol of Navier

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Numerical Solution of the

Incompressible Navier-Stokes Equations

Ae243 Biofluid Mechanics

Term Project

4 June 2004

Georgios Matheou


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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 lowReynolds numbers (creeping flows) up to complicated turbulent flows at

large Reynolds numbers.

The equations:

Continuity

Momentum

0u

Tuuuuu

p

t


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The Role of Pressure

Taking the divergence of the Navier-Stokes we get

The solution with initial and boundary condition

is =0 if, and only if, the right hand side is zero everywhere. Thus the pressure

satisfies the Poisson equation:

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.

i

j

j

i

x

u

x

up

Dt

D

22

1

u

0u

i

j

j

i

x

u

x

up

2


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The ProblemShear Driven Cavity

Some insects (dragonfly) have wings with welldefined 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:

LUlidRe

Flow visualization at

Re=0.01. (Taneda,

1979)


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Staggered Grid

Staggered Grid (Harlow andWelsh, 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 MethodExact Fractional Step Method (Chang, 2002)

Goal: satisfy

discreteincompressibility and eliminate the pressure equation

Incompressibility constraint:

Define volume fluxes as Ui=u Si and define the vector q that has the Uis in some

ordering. Then the above equation in matrix form is:

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

CV faces

iSudS 0nu

00111

11001:where

,

D

0Dq

1001

1100

0110

00110101

C


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Numerical MethodExact 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:

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

If we have a scalar quantity (like pressure), the discretized vector of which is,

then

is the discrete version of

Then:

which reproduces the continuous identity:

Csq

T

DG

G

p

0)( TTTT DCDCGC

0 p


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Finite Volume Formulation

x-momentum equation

Evaluate all integrals with the second order accurate

midpoint rule (uniform grid spacing in x and y):

In operator form:

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:

jiyxjixjiji Sny

un

x

uSpnSuV

t

u,,,, d

Re

1ddd

nu

21

21

21

21

21

21

21

21

21

21

,,,,

,1,,,1,,

2

,

2

,

,

Re

1

jijijiji

jijijijijijijiji

ji

y

u

y

uy

x

u

x

uyppyvuvuxuuy

dt

duyx

ji

u

jijii

u

ji

jiup

dt

du,

)(

,,

)(

,

,

Re

1

21 LH

brLqGq

M Re1

d

d

t


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Elimination of Pressure and Time Marching

Substituting q=Cs and premultipling the system by CT

the pressure iseliminated and the momentum equations are reduced to a single scalar

equation for s:

Using explicit Adams-Bashforth 2 for the convection terms and implicit

trapezoidal for the viscous we get the discrete system of equations:

)(d

dRe1 brCLCsC

sMCC

TTT t

brrCCsLMCCsLMC1TT1T

21

23

Re2Re2nnnn ttt


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VerificationRe=100

Comparison of steady state solution with data from Ghia

et al (1982).

Simulation atRe=100 with grid resolution of 100100.

Ghia resolution is 128128.

Computed main vortex center atx=0.6188 andy=0.7396

Ghia prediction atx=0.6172 andy=0.7344

Velocity along the midlines. Lines are

the current computation, circles are data

from Ghia.

Streamlines of steady state solution atRe=100


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Re=1000Grid: 200 - Velocity Field


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VorticityRe=1000

t=1.00

t=14.75

t=2.25

t=7.25


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References

W. Chang, F. Giraldo, and B. Perot. Analysis of an exact fractional stepmethod.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 incompressibleflow 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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