[lorbekl]

Ok, imagine for example a 1D flow for which the momentum equation (or in general any sort of convection only equation) has only the convective term whereas the diffusive term is absent. The Peclet number is then undefined and since the problem is uni-dimensional no numerical diffusion is present.



Am I right in assuming that in such a case differencing schemes like the hybrid, exponential and power law scheme don't really have an advantage over a simple upwind scheme?



I suppose higher order differencing schemes like QUICK would however still apply?

[FMDenaro]

Originally Posted by lorbekl

Ok, imagine for example a 1D flow for which the momentum equation (or in general any sort of convection only equation) has only the convective term whereas the diffusive term is absent. The Peclet number is then undefined and since the problem is uni-dimensional no numerical diffusion is present.



Am I right in assuming that in such a case differencing schemes like the hybrid, exponential and power law scheme don't really have an advantage over a simple upwind scheme?



I suppose higher order differencing schemes like QUICK would however still apply?

The best and simple model you can start working on is the linear advection equation


df/dt + u*df/dx=0


You have the exact solution f(x,t)=f(x-u*t,0). When you introduce a discretization, you numerical solution can be seen as an exact (in analytical meaning) of the "modified equation", that is the original PDE plus the terms of the local truncation error. Depending on the chosen discretization, the modified equation can show "numerical diffusion", "numerical dispersion", "violation of monotonicity".

The QUICK discretization is not suitable for unsteady solution, you should consider the QUICKEST scheme. However, they do not ensure monotonicity of the solution and numerical oscillations appears, especially for solution with high gradients.
On the other hand the FTUS is monotone but poorly accurate, producing a lot of numerical diffusion. Using the FTCS still not improve the solution of the problem, the scheme has no stability.

[LuckyTran]

You still have numerical diffusion. Is it important that Peclet number is or is not defined?

Originally Posted by lorbekl

Am I right in assuming that in such a case differencing schemes like the hybrid, exponential and power law scheme don't really have an advantage over a simple upwind scheme?

These schemes are only applied anyway only to the advective term (i.e. central differencing is used on the diffusive term). What you have accomplished by solving a pure advection equation is you have removed a different term (the diffusion term), which is discretized a different way. Ignoring non-linearities, you haven't changed anything about the way the advection terms are discretized and solved.

[lorbekl]

Is it important that Peclet number is or is not defined?

Well how I understood these differencing schemes is that for example hybrid scheme switches to upwind scheme when Pe>2 and power law switches to upwind at Pe>10. So if there is no diffusion the Pe number goes to infinity and all these schemes would automatically revert to the upwind scheme would they not?