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michujo June 15, 2012 08:20

Coefficients discretized momentum equation
 
Dear CFD-onliners,

I came across a question when trying to solve the momentum equation following Patankar's book (1980) indications.
When solving a general transport equation for a certain variable \phi, the discretized form takes the form:
a_P \phi_P=a_E \phi_E+a_W \phi_W+a_N \phi_N+a_S \phi_S+b

, where P,E,W,N,S stand for point, east face, west face, north face and south face, respectively. The coefficients depend on the type of numerical scheme used (central differences, upwind, exponential, power law...).

My question: how do we calculate the discretization coefficients for the momentum equations u and v? can we just assume that the momentum equations are another set of convection-diffusion equations, where the variable \phi are u and v? Then we could just use the same method as before (upwind, power law, ...).

Any help will be welcome.

Thanks in advance :).

cdegroot June 18, 2012 15:07

Quote:

Originally Posted by michujo (Post 366651)
Dear CFD-onliners,

I came across a question when trying to solve the momentum equation following Patankar's book (1980) indications.
When solving a general transport equation for a certain variable \phi, the discretized form takes the form:
a_P \phi_P=a_E \phi_E+a_W \phi_W+a_N \phi_N+a_S \phi_S+b

, where P,E,W,N,S stand for point, east face, west face, north face and south face, respectively. The coefficients depend on the type of numerical scheme used (central differences, upwind, exponential, power law...).

My question: how do we calculate the discretization coefficients for the momentum equations u and v? can we just assume that the momentum equations are another set of convection-diffusion equations, where the variable \phi are u and v? Then we could just use the same method as before (upwind, power law, ...).

Any help will be welcome.

Thanks in advance :).

The u and v momentum equations are not exactly like a convection-diffusion equation because of the pressure term. The convection and diffusion terms are discretized in the same way as described for a general convection-diffusion equation, however. Don't forget to think about pressure-velocity coupling as well.

michujo June 18, 2012 17:49

Hi Chris, I found later in Patankar's book that we could just use any differentiation scheme for the diffusive and convective terms in the momentum equation, as it is done for general convection-diffusion equations (I also found the coefficients I was looking for in Versteeg&Malalasekera book).
Yes, I will treat the pressure gradient term differently don't worry.

I appreciate your reply anyway.

Cheers.

leflix June 19, 2012 19:16

Quote:

Originally Posted by cdegroot (Post 367070)
The u and v momentum equations are not exactly like a convection-diffusion equation because of the pressure term. The convection and diffusion terms are discretized in the same way as described for a general convection-diffusion equation, however. Don't forget to think about pressure-velocity coupling as well.

Hi cdegroot,

The momentum equations are EXACTLY convection-diffusion equations with a source term. In this case the source term is the pressure gradient (dp/dx_i) + cross derivative of velocity (du_j/dx_i). (here d is partial derivative)

It can be discretized using the same schemes as the ones used for any scalar transport equation. The scalars here are the componnents of the velocity vector u and v.

cdegroot June 20, 2012 01:33

Quote:

Originally Posted by leflix (Post 367338)
Hi cdegroot,

The momentum equations are EXACTLY convection-diffusion equations with a source term. In this case the source term is the pressure gradient (dp/dx_i) + cross derivative of velocity (du_j/dx_i). (here d is partial derivative)

It can be discretized using the same schemes as the ones used for any scalar transport equation. The scalars here are the componnents of the velocity vector u and v.

Yes, I agree.


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