5'2 FiniteVolume Method

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5.2 Finite-Volume Method

• Closely related to Subdomain Method
• But without explicit introduction of trial or
  interpolation function
• Approximate the flux terms directly (rather than
  the function itself)
• Use the integral form of PDEs (instead of
  weighted residuals)
• Numerical Heat Transfer and Fluid Flows, S.V.
  Patankar, McGraw-Hill, 1980.

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Navier-Stokes Equations

• 2D Compressible N-S equations
• General Form

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Greens Theorem

• 3D Volume integral ltgt Surface integral
• 2D Surface integral ltgt Line integral

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Triangular elements 5-sided or 6-sided control
volumes
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5.2.1 First Derivatives
NE
N
C
n
NW
E
D
e
P
w
W
B
s
SE
A
S
SW
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First Derivatives

• Evaluate surface integrals for arbitrary control
  surfaces

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First Derivatives
? dy
? dx
C
dx
dy
D
? dy
? dx
B
A
dx
B
dy
A
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First Derivatives
dx
dy
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Finite-Volume Method
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Linear Interpolation
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Uniform Cartesian Grids
vn
C
D
ue
uw
A
B
vs
or
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General Curvilinear Coordinates

• For curvilinear grids, FVM provides a
  discretization in Cartesian coordinates without
  explicit (body-fitted) coordinate transformation
• Grid curvature terms are ignored
• Alternatively, one may apply FVM in transformed
  plane

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5.2.2. Second Derivatives

• Convective Transport Equations

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Convective Transport Equations
Line integral for element centered at P
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Second Derivatives
NE
Line integral for element centered at s
N
C
n
NW
E
D
e
P
w
B
W
s
A
SE
B?
S
A?
SW
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Second Derivatives

• -- First derivatives centered
  at s
• -- came from second derivatives centered at
  P
• Use Greens theorem again for elements centered
  at s, e, n, and w (not centered at P)

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NE
Line integral for element centered at s
N
C
n
NW
E
D
e
P
w
B
W
s
A
SE
B?
S
A?
SW
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Bilinear Interpolation

• Ignore grid expansion or contraction
• Bilinear interpolation

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Flux Evaluation
Diffusive and convective fluxes
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Convective Transport Equation
Line integral for elements centered at s, e, n,
w four overlapped surface (line) integrals
NE
N
n
NW
E
e
P
w
W
s
SE
B
S
A
SW
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Convective Transport Equations
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Convective Transport Equations
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Uniform Cartesian Grid
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Central Difference
Identical to the finite-volume method
Unphysical solutions may occur if cell Reynolds
numbers gt 2
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Exponential Scheme

• Ref Numerical Heat Transfer and Fluid Flow, by
  S.V. Patankar, 1980

Steady, 1-D
Fw
Fe
P
W
E
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Exponential Scheme

• Cell Peclet number Pe

Gradual shift to upwind
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Exponential Scheme

• Cartesian grids only

Gn
?xn
Fe
Fw
?ye
?yw
?xs
Gs
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Exponential Scheme

• Nonuniform Cartesian grids
• Uniform Cartesian grid (and ue uw u, vn vs
  v)

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1D Convective Transport Equation

• Uniform Cartesian grid (and ue uw u)
• Pe Pw Px u?x/a
• Exponential Scheme
• Linear FVM (Central Difference)

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FVM Pure Diffusion

• Uniform Cartesian Grid

0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
Linear
Exponential
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FVM Convection-Diffusion

• Uniform Cartesian Grid

0.25
0.249896
0.2625
0.2375
0.264473
0.237609
0.25
0.249896
Linear
Exponential
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FVM Convection-Diffusion

• Uniform Cartesian Grid

0.25
0.240156
0.375
0.125
0.379922
0.139765
0.25
0.240156
Linear
Exponential
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FVM Convection-Dominant

• Uniform Cartesian Grid

0.25
0.083327
1.5
-1.0
0.833308
0.000038
0.25
0.083327
Linear
Exponential
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FVM Convection-Dominant

• Uniform Cartesian Grid

0.25
0.000000
12.75
-12.25
1.000000
0.000000
0.25
0.000000
Linear
Exponential
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FVM Skewed Upwind

• Uniform Cartesian Grid

0.2375
0.237510
0.2625
0.2375
0.262490
0.237510
0.2625
0.262490
Linear
Exponential
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FVM Skewed Upwind

• Uniform Cartesian Grid

0.125
0.134471
0.375
0.125
0.365529
0.134471
0.375
0.365529
Linear
Exponential
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Finite-Volume Method

• Uniform Cartesian Grid

-1.0
0.000023
1.5
-1.0
0.499977
0.000023
1.5
0.499977
Linear
Exponential
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FVM Convection-Dominant

• Uniform Cartesian Grid

-12.25
0.000000
12.75
-12.25
0.500000
0.000000
12.75
0.500000
Linear
Exponential
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AJM
Error in textbook
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Successive Overrelaxation (SOR)
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Grid Refinement