SIMPLE algorithm-what can be relaxed if the scheme order is increased?
 

HI All

I want to learn what is known about the subject in the title .

I am using Openfoam

what could you recommend as "entry point reading" in order to understand what is known on this topic?
I am finding Google not very helpulf on this one
Thanks
JF

In what context do you want to increase the "scheme order"?

SIMPLE spatial accuracy of convection is commonly up to 3rd order in Fluent. There are structured codes (MFIX) that offer fifth-order upwind scheme for convection. There are also spectral codes that can be considered SIMPLE-type. One can rightly argue that Finitie Volume methods are formally second-order due to some of the assumptions they make about quadrature and treatment of pressure and diffusion terms. If you follow that line of thinking, then you need to look at Discontinuous Galerkin methods. But, for practical purposes, errors in convection...especially artificial diffusion...cause the most severe accuracy and effective resolution limits in CFD in general.

If you are talking about time accuracy, SIMPLE algorithms can easily be created that use higher-order Adams Moulton or Implicit Runge-Kutta schemes.

Hi
The context of my question is:

Can i increase boundary layer mesh size (at constant accuracy) if I increase velocity scheme order?

In my work I have a configuration with large solid-fluid contact surface.
Any increase in BL mesh thickness (at constant accuracy) can be used.

In FEM method, you can often do that because increased scheme order give you more control point per mesh.
Because FVM is face based, the same lne of reasoning does not apply, at least in a straightforward way.

I never thought of the fundamentals behind increasing scheme order in the context of finite volume methods.. so my question as I want to learn the math behind this.
Thanks
JF

FVM order is increased by increasing the stencil size--by including data from neighbor cells (and neighbors or neighbors, etc) into the face interpolations. This becomes a problem at boundaries, of course, because you don't have neighbors beyond the wall.

In principle, you could build one-sided interpolations and try to do p-type refinement at a boundary, but I am not sure that it will do a good job of capturing boundary layer behavior, even with Finite Elements. You can read Babuska paper:

http://www.caam.rice.edu/~jjy5/ShirinPaper.pdf

about hp-refinement and when to do h- and p- refinement. Boundary layers are not necessarily well interpolated by higher-order schemes. Even spectral methods use Chebyshev grids to try to put more collocation points near wall boundaries. That is because boundary layers have a characteristic length (the thickness) where the nature of the solution changes--near constant on the free stream side and rapidly changing near the wall. Fitting higher-order functions to that behavior can generate ringing, overshoots, etc. High-order methods are very good at capturing smooth data. But rapid variations like a boundary layer generally require h-type refinement...just more cell/elements in the wall-normal direction. Even with second-order schemes, one must be careful to not allow over/undershoots to ruin the physically of the solution. Without doing this, you could see unphysical flow reversals or even worse...say for a thermal boundary layer, you could have unbounded interpolation leading to negative temperatures!

So, I know that it is easy to be enticed by the allure of high-order schemes, but the phrase "high-order does not mean high-accuracy" bears repeating. Again, you may want to look at DG methods as they nicely span the Finite Element/Finite Volume approach and, in many ways, extract the best of both. There are recent efforts to build SIMPLE-type solvers using DG, but I haven't looked at them myself.

@mprinkey

Thanks a lot
this is the information I was looking for!

JF