Thursday, November 16, 2017

Sub-grid Scale (SGS) Stress Models in Large Eddy Simulation

The simulation of turbulent flow has been a considerable challenge for many decades. There are three main approaches to compute turbulence: 1) the Reynolds averaged Navier-Stokes (RANS) approach, in which all turbulence scales are modeled; 2) the Direct Numerical Simulations (DNS) approach, in which all scales are resolved; 3) the Large Eddy Simulation (LES) approach, in which large scales are computed, while the small scales are modeled. I really like the following picture comparing DNS, LES and RANS.

DNS (left), LES (middle) and RANS (right) predictions of a turbulent jet. - A. Maries, University of Pittsburgh

Although the RANS approach has achieved wide-spread success in engineering design, some applications call for LES, e.g., flow at high-angles of attack. The spatial filtering of a non-linear PDE results in a SGS term, which needs to be modeled based on the resolved field. The earliest SGS model was the Smagorinsky model, which relates the SGS stress with the rate-of-strain tensor. The purpose of the SGS model is to dissipate energy at a rate that is physically correct. Later an improved version called the dynamic Smagorinsky model was developed by Germano et al, and demonstrated much better results.

In CFD, physics and numerics are often intertwined very tightly, and one may draw erroneous conclusions if not careful. Personally, I believe the debate regarding SGS models can offer some valuable lessons regarding physics vs numerics.

It is well known that a central finite difference scheme does not contain numerical dissipation.  However, time integration can introduce dissipation. For example, a 2nd order central difference scheme is linearly stable with the SSP RK3 scheme (subject to a CFL condition), and does contain numerical dissipation. When this scheme is used to perform a LES, the simulation will blow up without a SGS model because of a lack of dissipation for eddies at high wave numbers. It is easy to conclude that the successful LES is because the SGS stress is properly modeled. A recent study with the Burger's equation strongly disputes this conclusion. It was shown that the SGS stress from the Smargorinsky model does not correlate well with the physical SGS stress. Therefore, the role of the SGS model, in the above scenario, was to stabilize the simulation by adding numerical dissipation.

For numerical methods which have natural dissipation at high-wave numbers, such as the DG, SD or FR/CPR methods, or methods with spatial filtering, the SGS model can damage the solution quality because this extra dissipation is not needed for stability. For such methods, there have been overwhelming evidence in the literature to support the use of implicit LES (ILES), where the SGS stress simply vanishes. In effect, the numerical dissipation in these methods serves as the SGS model. Personally, I would prefer to call such simulations coarse DNS, i.e., DNS on coarse meshes which do not resolve all scales.

I understand this topic may be controversial. Please do leave a comment if you agree or disagree. I want to emphasize that I support physics-based SGS models.

1. I agree with DNS on coarse meshes ...

2. About clasic Central Difference scheme in LES. CD scheme is an unbounded scheme, isn't it? How do we deal with that? Why the oscillations that it produces are physical (turbulence) and not numerical?

1. Because of the nonlinearity of the Navier-Stokes equations, CD schemes will see an accumulation of high wave-number eddies, which is not physical, rather than numerical. Perhaps that is why CD schemes are called "unbounded"? I would prefer the use of a spatial filter to stabilize the simulation because one can add lesss artificial dissipation this way. The only sure way to know the oscillations gnenerated are physical is to perform a grid refinement study.

2. I prefer subgrid modelling too (rather than ILES), because in that way I think that you have more control in your modelling. What do you mean "will see an accumulation of high wave-number eddies, which is not physical"? If it is NOT physical, why do we use CD?

3. For nonlinear problems governed by the Burger's or Navier-Stokes equations, there is a cascading process where energy goes from the larger scales to the smaller scales. At the smallest scale, physical viscosity dissipates enough energy so that no smaller scales can be generated. At the resolution for LES, the physical dissipation is not capable of fully dissipate the high-wave number eddies, which must be damped by numerical dissipation or a SGS model. CD has the advantage of being non-dissipative. That's why it is often used in LES. But you do need a SGS model or filtering to maintain stability.

3. I think your example of LES here is very low resolution. These days I would expect it to look more like your DNS example. Especially with adaptive wake refinement schemes which can help manage model size whilst reducing the amount of sub grid modelling required.

1. I agree as this is just an illustration.