[motorbean]

Dear All,

I see that the viscous term in the momentum equation is



It does on fit exactly to the diffusion term in the general scalar transport equation, where the diffusion term is more like the Laplacian of phi, whereas the viscous term is not, especially for compressible flow:

For example, the diffusive term in the general transport equation for phi_k only contains the spatial derivative of phi_k, not the phi's of other transport equations. But (for example) in the x-momentum equation, the viscous term not only contains the spatial derivative of x-velocity, but also the spatial derivative of the transport variable of the y and z momentum equations, i.e. v and w.

In the UDF manual ("connectivity macros"), some hint is given about how the diffusion term is discretized for the general transport equation (GTE). However, since the viscous term does not resemble the GTE diffusion term well, Can anyone tell me how the viscous term in the momentum is discretized, please?

Thanks very much!

Best,
motorbean

[LuckyTran]

The viscous term is identical to the diffusion term in the general transport equation. I don't see the difference. The form of the viscous term shown here is more general than the form with a laplacian.

The discretization for the viscous term and the diffusive term is the same for both, which is central differencing. Integrate the governing equation over the control volume, apply divergence theorem to convert the volume integral of the divergence into a surface integral of fluxes. Then apply central difference to the face fluxes.

There are always cross terms in the transport equation. The only case where the cross-terms disappear is the trivial case of 1D diffusion.
Is the problem because you are thinking of the Laplacian? But even then, the x-component of the laplacian contains contributions from the y and z derivatives. Recall the Laplacian is the divergence of the gradient.

[motorbean]

Dear Tran, Thanks for your reply.

The problem is that, the general transport equation does not contain solution variable from other equations, e.g. the diffusion term is contains the derivatives of only its own solution variable phi_k, without the derivatives of the solution variables of the other transport equations.

However in 2D, the x-momentum equation for example, the viscous term not only contains the derivative of "u" (which is the solution variable of the x-mom eqn itself), but also contains derivatives of the solution variable "v" from the y-momentum equations, i.e. there is not only du/dx, du/dy, but also has dv/dx.

This is the difference I see between the viscous terms and the diffusion term: the present of dv/dx in x-mom eqn for example - the remnant stress terms that cannot be included in the diffusion terms in the general transport equation.

So my question is how to treat discretize the remnant terms? This is the issue not mention in many books on numerical methods, in which the discussion about the FVM discretization is based on the general transport equation.

[LuckyTran]

Actually, in the momentum transport equation of a Newtonian fluid (only momentum and its gradients appear which is equivalent to only terms in Phik's and gradients of Phik's appearing in the kth transport equation).

The difference is that momentum is a vector whereas the Phi's are usually scalars. The diffusion of momentum occurs by the gradient of the velocity, which is equivalent to diffusion of Phi occurring by gradient of Phi. If you try to refer to a particular component of momentum in a particular coordinate system (say x,y,z) then you end up with that many equations for momentum. i.e. momentum has components in the x y and z direction whereas Phi's do not. The extra terms arise from the conversion of the momentum vector into a scalar.

The reason for needing a discretization scheme is because in FVM the values are stored at cell centers, but face values are needed for faces. The "values" in this case are the velocity gradients or gradients of Phi.

Hence, one can appreciate that the diffusion term for both momentum transport and scalar transport equation can be discretized the same way. The issue is probably not often mentioned because it is only a matter of following the rules of FVM. The difference between momentum/scalar transport is the gradient variable (gradients of velocity, versus gradients of Phi). The discretization of the gradient could be different but in Fluent, diffusive terms are discretized using central differencing.