4--the finite volume method for convection-diffusion problems -2

8/10/2019 4--The Finite Volume Method for Convection-diffusion Problems -2

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2ndCentral differencing scheme

5 nodes 20 nodes

Pe=5 Pe=1.25


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The upwind differencing schemes

Flow direction is considered when computing at interface


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, always positiveW E

a a


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Example

Case 1:

Case 2:

5 nodes

5 nodes


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node: 1

node: 5

A

1 2

P

E

e

5

B

4

PW e


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Case 1

For all nodes: 1,2,3,4,5


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Exact solution: smooth

Good results


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Case 2

Large dissipation

Refine grid?

1storder high order?

Large gradient


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Extended to multi-dimensional problems

The flow is not aligned with the grid linesFalse diffusion

A pure convection process is considered

or numerical diffusion (dissipation)

Refine grid

1storder high order?


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The hybrid differencing scheme

Spalding(1972), hybrid: central + upwind scheme

upwind scheme: transportive----convection problem

central scheme: no direction----diffusion problem

piecewise formulae based on the local Peclet number at the cell faces.

Net flux through the west face:

PW e E

upwind

central


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General form of the discretised equation:


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Example

Case 1:

Case 2:

5 nodes

25 nodes

Pe=5

Pe=1

hybrid upwind

hybrid central


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Remarks

High Re flow

Upwind or hybrid..schemes

Exact solution: Large gradient

Pe2

good

dissipation

Under/overshoot

Large dissipation

2ndcentral

1stupwind

2ndcentral

1stupwind

may be fixed for the fluid

should be small

Low Re flow

Pe


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Higher-order differencing schemes

the 1stupwind scheme

the hybrid scheme(1stupwind+2ndcentral)

the power-law scheme(1stupwind+weighted)

the 2ndcentral scheme unstable

stable

large numerical diffusion

high-order

Leonard(1979), QUICK scheme

Quadratic UpstreamInterpolation for Convective Kinetics

Quadratic upwind differencing scheme

For high Pe


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Leonard(1979), QUICK scheme

Quadratic UpstreamInterpolation for Convective Kinetics

Quadratic upwind differencing scheme

ii-1i-2 i+1

i-1/2 i+1/2

i-1/2


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2nd

central3rd order for


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5 nodes

Example

For Node 1 or point P


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P,1 E,2

A ,w e

W,0

A , P, E

2nd

at A

1stat A

2ndcentral

For consistency


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Assessment of the QUICK scheme


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The power-law scheme

The hybrid scheme in the example satisfies the transportiveness requirement

Inadequacy: the accuracy is only first-order

Patankar(1980), the power-law scheme better than the hybrid scheme

Net flux through the west face:PW e E

upwind

4--the finite volume method for convection-diffusion problems -2

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