[Simon Smokes]

Hello everyone,

Again I'm doing my a thesis in numerical fluid mechanics and I have a

- transient
- advection problem

that can be described by the following equation:

d (rho*phi) \ dt + d (rho*phi*u) \ dx = q_0
(transient term) + advective/convective term = source term

As you can see, the Peclet number, defined as quotient of convective/advective and diffusive transport is "quite" high, due to the lack of diffusive mass transport. By discretising it with FVM it leads to:

transient term + [Flux_east * phi_east - Flux_west * phi_west) = source term


So far I tried several schemes to discretise the property phi on the cell face(s):

- 1. order upwind: very stable, but massiv numerical diffusion which increases with increasing number of time/iteration steps; starting with a sharp edge at time and nodal point 1, the edge 'melts away' to a long curve

- 2. order upwind (LUDS): very stable, less numerical diffusion, but finally the diffusion increases massivly again with increasing number of time/iteration steps; same problem like 1. order upwind

- QUICK: leads to oscillations, due to the high Peclet-Numbers

- QUICK in combination with Hayase et. al. and deferred correction: stability of Upwind in the coefficient matrix 'yes' but it does not work: the deferred correction does not converge, consequently the 'upwind-solution' is not that much refined

- analytical solution: not applicable because of the transient source term q_0 and transient boundary conditions

- other central differencing schemes: not applicable due to form of the governing equation



During my internet research I found several hints like using "AUSM" (Advection Upstream Splitting Method) or "Newton Raphson" as direct linearisation of phi.

But so far I could not find a really good comprehensible paper/journal article/thesis that clearly describes how to apply it on the property phi_east/west in the discretised equation.

I would be very thankful for any help and open for questions if anyone has the same problem or about the listed schemes above!

Best regards, Simon!

[FMDenaro]

Originally Posted by Simon Smokes

Hello everyone,

Again I'm doing my a thesis in numerical fluid mechanics and I have a

- transient
- advection problem

that can be described by the following equation:

d (rho*phi) \ dt + d (rho*phi*u) \ dx = q_0
(transient term) + advective/convective term = source term

As you can see, the Peclet number, defined as quotient of convective/advective and diffusive transport is "quite" high, due to the lack of diffusive mass transport. By discretising it with FVM it leads to:

transient term + [Flux_east * phi_east - Flux_west * phi_west) = source term


So far I tried several schemes to discretise the property phi on the cell face(s):

- 1. order upwind: very stable, but massiv numerical diffusion which increases with increasing number of time/iteration steps; starting with a sharp edge at time and nodal point 1, the edge 'melts away' to a long curve

- 2. order upwind (LUDS): very stable, less numerical diffusion, but finally the diffusion increases massivly again with increasing number of time/iteration steps; same problem like 1. order upwind

- QUICK: leads to oscillations, due to the high Peclet-Numbers

- QUICK in combination with Hayase et. al. and deferred correction: stability of Upwind in the coefficient matrix 'yes' but it does not work: the deferred correction does not converge, consequently the 'upwind-solution' is not that much refined

- analytical solution: not applicable because of the transient source term q_0 and transient boundary conditions

- other central differencing schemes: not applicable due to form of the governing equation



During my internet research I found several hints like using "AUSM" (Advection Upstream Splitting Method) or "Newton Raphson" as direct linearisation of phi.

But so far I could not find a really good comprehensible paper/journal article/thesis that clearly describes how to apply it on the property phi_east/west in the discretised equation.

I would be very thankful for any help and open for questions if anyone has the same problem or about the listed schemes above!

Best regards, Simon!

1) in your case Peclet is infinity
2) depending on the solution, you must use the integral form for non-regular solution (step solution)
3) It is known that linear schemes greater to first order accuracy are not monotone. In such cases you have to use some care (for example flux limiter) to make the scheme non-linear even for the linear equation
4) You can try the QUICKEST scheme, very accurate for regular solution. It can be supplied by a limiter for non regular solution

[robo]

Laney's text on computational gas dynamics covers this class of problems pretty comprehensively; and Pletcher, Anderson & Tannehill also cover gas dynamic problems, including the use of Ausm and similiar schemes. You're looking for FV methods for the Euler equation, in my experience it is not typical to discuss such problems in terms of a Peclet number. Try searching for Riemann solvers, flux limiters, TVD schemes etc.