Why is high order curved mesh important?

It is almost a year since my last blog! I really need to write something. Otherwise, whatever I wrote may become obsolete, soon.

Last week, the world held its breath to watch the independence referendum in Scotland. Because of personal history, I, too, followed the event with much interest. It is very remarkable that a very peaceful vote determined the fate and future of an entire nation. Yes or no, the Scots are to be congratulated on passionately exercising a sacred right  many other parts of the world can still only dream about.

Now back to high order CFD. Almost all finite volume (FV) CFD codes use linear control volumes (CV), cells or elements with straight edges. The geometry is approximated with linear triangles or quadrilaterals, and is thus "faceted". In a second order finite volume method, the solution is assumed linear within each CV, and discontinuous across the CV interfaces. This linear geometry representation is therefore compatible with the linear solution distribution without causing any significant accuracy loss as long as high curvature areas are properly resolved by the mesh.

As discussed earlier, high order methods are capable of achieving engineering accuracy on a much coarser mesh than the ones used in 2nd order FV methods. If a linear mesh is used in a high order simulation, the solution error caused by the linear boundary mesh may be 2nd order, thus destroying the high order accuracy. It is therefore critical to represent the geometry with quadratic or higher order elements supporting curved edges and faces. Both Gmsh and CGNS can handle high order elements.

Although high order elements have been used successfully in the finite element community, no commercial CFD mesh generators can produce high order meshes at present. The biggest challenge is the generation of highly clustered viscous meshes near high curvature regions. Additionally, high order meshes are usually much coarser than linear meshes. Robustness may become an issue in generating coarse meshes for complex geometries. I hope commercial mesh generators (Pointwise?) can conquer this challenge soon!

I believe high order mesh generation is now the biggest bottleneck in applying high order CFD in the design process (http://rsta.royalsocietypublishing.org/content/372/2022/20130318.abstract).