3--the finite volume method for convection-diffusion problems -1

8/10/2019 3--The Finite Volume Method for Convection-diffusion Problems -1

1/23

The finite volume method forconvection-diffusion problems


2/23

General transport equations

( )div( ) div( grad )U S

t

convective term diffusive term source term

0 div( grad ) S

Pure diffusion:

Steady convection-diffusion:

div( ) div( grad )U S


3/23

div( )d div( grad )d dCV CV CV

U V V S V

n ( )d n ( grad )d dA A CV

U A A S V

Integration over a control volume

Gauss divergence theorem


4/23

Steady 1D convection and diffusion

Assume no sources

n ( )d n ( grad )d dA A CV

U A A S V

n n


5/23

Steady 1D convection and diffusion

continuity equation

Define

1

(1a)

(1b)

(1a)

(1b)


6/23

Central differencing scheme

(1a)

at cell face


7/23



8/23


Example

Case 1:

Case 2:

Case 3:

5 nodes

5 nodes

20 nodes


9/23


Internal nodes: 2,3,4

node: 1

A

1 2

P

E

e


10/23


node: 5

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

Case 1:


11/23


Case 1:


12/23



13/23


Case 2:


14/23



15/23


Case 3:


16/23

Properties of discretisation of schemes

Case 2: F/D =5

Case 3: F/D =1.25F/D: Convective flux / diffusion

fai lure

Conservative equations integrated over CVs

Flux leaving

Flux enteringFlux leaving = Flux entering


17/23



18/23



19/23


Scarborough(1958)


20/23


Necessary condition

p p W W E Ea a a and P,W,Eare very close

p W E

, , same sign, andp W E p W Ea a a a a a

0, 0,


21/23



22/23


23/23

Properties of central schemes

may be fixed for the fluid

should be small

Low Re flow


High Re flow ?

Upwind or hybrid..schemes ?

3--the finite volume method for convection-diffusion problems -1

Documents