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8/14/2019 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.