High Vs. Low Order Numerical Methods in CFD

Commercial codes frequently invoke second order implicit numerical models along with RANS simulation. They are the dominating work horse for most industry related applications. Nonetheless, as computational resources available to the industry have increased substantially in the last few decades in accordance with “Moore’s Law”, and due to RANS inherent limitations in modelling flows of which large instabilities and unsteadiness occur, much of the attention and effort has shifted to methods such as hybrid RANS-LES and even pure LES.

In LES the large energetic scales are resolved while the effect of the small unresolved scales is modeled using a subgrid-scale (SGS) model and tuned for the generally universal character of these scales. LES has severe limitations in the near wall regions, as the computational effort required to reliably model the innermost portion of the boundary layer (sometimes constituting more than 90% of the mesh) where turbulence length scale becomes very small is far from the resources available to the industry. Anecdotally, best estimates speculate that a full LES simulation for a complete airborne vehicle at a reasonably high Reynolds number will not be possible until approximately 2050.

Nonetheless, for highly unsteady, vortex dominating flows of which the physical phenomena is mainly derived by the large eddies, LES might be affordable and prevails.

For the sake of definition I shall refer to methods of third order and above as high order methods and for first and second order as low order methods. Furthermore a method is said to be to the n-th order if the solution error is proportional to the mesh size in the power of n.

Why are second order upwind methods so dominating today?

Well, some of the credit goes to inertia… There has been so much effort in the past few decades in the advancement of robustness and efficiency in second order numerical models so that they seem as optimal for most applications.

Is that all there is? Inertia?

Certainly not. Higher order methods are more complicated than low order methods and as such are harder to implement. They are also slower to converge to steady state due to much reduced numerical dissipation (we shall return to that in the following paragraphs). Furthermore, high order methods computational memory requirement is much higher.

So higher order methods are more expensive?

Not at all. It is most important to understand that while comparing high and low order models we can not evaluate a method efficiency based on the computational cost, on the same mesh. Yes, obtaining a converged steady solution with a high order method takes longer than with a low order method on the same mesh. On the other hand, a higher order method may achieve an error threshold much faster than a low order method because it can achieve it on a much coarser mesh.

Then when and for what engineering application shall we choose which method?

Returning one paragraph above it is now understood that obtaining the same accuracy level shall require a finer mesh for low order methods. For flows dominated by vortices which must be resolved for a long distance obtaining an engineeringly sufficient accuracy could prove very problematic with low order methods. Such an example may be found in the calculation of helicopter loads which are strongly dependent on the vortices generated by the tip of the rotor. As low order methods strongly dissipate the unsteady vortical structures, a remarkably high and impossible to solve (on computational resources level) mesh  should be generated to capture the phenomenon. Such is the case in general for vortex dominated flows and for problems of wave propagation. So in order to solve problems like the former or to directly conduct LES aeroacoustic calculations when they are possible, as for the case of jet noise, without having to invoke analogies to propagate sound, higher order methods are mandatory.

DNS at Re=400000 of NACA4412 applying high order methods (D. Hennigson et al. – KTH)

If one method is not decisively better than the other than what is the ideal route?

The ideal route should be to let the flow field dependent local order of accuracy using h-adaptation (refining the mesh) for regions of discontinuity and p-adaptation (increasing the order an element) for the smooth ones. For that to happened possibly some effort needs to be done on the implementation of robust higher order methods with suitable error estimation procedures for hp-adaptation methodologies in commercial CFD codes.

Which industry oriented CFD codes make use of high order methods?

There are not many OpenFOAM developments for such models, at least not so much as one should hope. Most high order schemes are found in the academy where it is widespread for the conduction of LES and DNS. Nonetheless there are some interesting codes to be found such as the LCS FAST code, implementing high order FEM along with hp-adaptation.

IMG_0593LCS Fast 5th order accurate transient simulation of the Elemental Rp1 track car

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