Implicit Large-Eddy Simulations: lazy shortcut or numerical dark magic? Kinda both
Turbulence is a central topic for many different fields of applied sciences. For mathematicians it still represents an open problem in terms of its formalisation and even in practical applications, with the aid of powerful computers, we grasp its main characteristics but we are still far away from fully understanding and predicting every little detail of turbulent flows.
It is well known that any informal discussion about turbulence needs to start with a brief historical introduction. Such an introduction cannot begin with anyone different than Leonardo da Vinci, who was the first to address the beauty and mystery behind the motion of fluids. He was the first one to notice what Richardson and then Kolmogorov later formalised as the “kinetic energy cascade”. To directly quote Leonardo: “Nota il moto del vello dell’acqua, il quale fa uso de’ capelli, che hanno due moti, de’ quali l’uno attende al peso del vello, l’altro al liniamento delle sue volte; così l’acqua ha le sue volte vertiginose, delle quali una parte attende a l’impeto del corso principale, l’altra attende al moto incidente e refresso”.
With these few words Leonardo anticipated the Richardson’s theory by over 400 years: the flow is essentially subdivided in two main dynamics, one follows the main stream whereas another one is related to the small little variations around it. This is closer to Reynolds decomposition rather than kinetic energy cascade but hey, the guy discovered these with only his own eyes. Let’s go easy on him. In the following years, the phenomenology of turbulence grew and was later formalised by the holy-Graal of the fluid dynamics equation: the Navier-Stokes equations. Extremely beautiful but quite useless at the time: difficult to be solved with pen and paper and even more difficult to be formally studied via mathematical tools commonly used for other partial differential equations.
Hundreds of years later, we discovered a way to solve these equations using very powerful computers. In fact, nowadays, every single day, there’s a new, larger, stronger heavy-lifting supercomputer flexing its muscles with pure raw computational power. And even after all these years, after all these efforts, we are still quite far from actually solving anything related to turbulence.
There are three main ways to deal with turbulence from a numerical point of view: Reynolds-averaged Navier-Stokes, Large-eddy Simulations and Direct Numerical Simulations.
Let’s start from the obvious one: turbulence involves a lot of scales (like in Leonardo’s drawing) and if we want to get an accurate simulation we have to model ALL of them. We need to capture the large scales of the flow field and the tiny little vortices generated by turbulence. From a computational point of view this means that we have a large domain and we have to fill it with a lot of little tiny elements which represent the discrete form of the fluid. Consequently, we need to handle A LOT of data to solve the equations on these large number of tiny elements. I mean, for example, to solve all the scales involved in the flow of air around a plane in standard conditions you need many BILLIONS of tiny little elements and even on the largest supercomputer in the world this would take years to complete a single simulation. Boeing cannot wait that long for its results. This approach is definitely out of the question for practical flows.
The other side of the spectrum is RANS: let’s say we don’t really care about all the little tiny details of turbulence but at least we want to capture the mean dynamics. The mean here can be defined in different ways, it can be a temporal or spatial average if the domain has some type of periodicity. In this way we can reduce by quite a bit the computational cost because (big surprise) even if the flow field itself is chaotic and complex in turbulent flows, its mean is usually pretty nice looking and smooth. However, there’s no free lunch! By averaging the equations you get some nasty non-linear terms that you cannot write as a function of the averaged flow field. So, you gotta model them. And boy oh boy, there are A LOT of different models for these terms but a lot of them can lack of accuracy in some cases due to the very strong assumptions made in the averaging step.
Last but not least, there’s the Large-Eddy simulations approach which represents some kind of combination of RANS and DNS. In this framework you do not average the equations but you filter them. The main idea is that your simulation should solve accurately a certain range of scales which are affordable and then, the scales which are not resolved, should be modelled. The additional terms appearing in the filtered equations are similar to the ones in RANS but since you are solving much more physics, they are usually easier to be treated.
SO, LET’S TALK ABOUT LES.
What is actually interesting about them is that it is an approach which mixes numerics and modelling: the flow field is resolved down to a certain scale which is still far from the smallest possible scale in flow field. The remaining scales are modelled in a similar way with respect to the RANS approach. What is actually interesting about this stuff is that the numerical scheme itself, the fact itself that you are solving the equations on a discretised version of the fluid domain, already implies some kind of modelling. In other words, you are implicitly cutting certain scales. This approach is known as implicit LES: there is no analytical model for the SGS tensor. The numerics do it all. The other counterpart, instead, is the explicit LES approach where this term is modelled with some analytical formula based on the resolved scales.
In implicit LES an important concept is the idea of numerical dissipation: discretising the equation in a certain manner implicitly imposes a certain type of dissipation mechanism in the numerical solution. A classical easy example to present this is the discretisation of the transport equation using an upwind approach:
Where the transport them is discretised as:
which can be written also as:
Here we can recognise a centered finite difference scheme in the first term which is known to have null numerical dissipation and a discrete laplacian proportional to the grid size. In other words, an upwind discretisation is adding a dissipation proportional to the grid size.
This is a little stupid exercise but this concept also applies to more complex equations such as the Navier-Stokes equations and different discretisations are characterised by different numerical dissipation footprints. Sometimes it is too much to mimic the effects of turbulence, sometimes is not enough and the simulation becomes unstable. In other words, the problem of closure is not only about the explicit model you want to use (such as Smagorinsky model) but also about the discretisation of the equations themselves. So important that sometimes you can just use it for modelling turbulence altogether.
Again, here I am just sketching some ideas but there a lot of interesting works where it can be shown that certain discretisations naturally lead to additional terms which can be interpreted as closure terms in the LES framework (see for example all the magnificent work by Len Margolin [3,4]).
Long story short, implicit LES is a dangerous business. It’s like the Far West: you can get hurt pretty quickly and it is easy to trust the wrong guys but it is also an incredibly wide field with many many things yet to be discovered.
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Bibliography:
[1] Tonicello, N., Lodato, G., & Vervisch, L. (2021). LES of compression ramp using high-order dynamic SGS modeling. In AIAA Scitech 2021 Forum (p. 1947).
[2] Tonicello, N., Lodato, G., & Vervisch, L. (2022). Turbulence kinetic energy transfers in direct numerical simulation of shock-wave–turbulence interaction in a compression/expansion ramp. Journal of Fluid Mechanics, 935, A31.
[3] Grinstein, F. F., Margolin, L. G., & Rider, W. J. (Eds.). (2007). Implicit large eddy simulation (Vol. 10). Cambridge: Cambridge university press.
[4] Margolin, L. G., Rider, W. J., & Grinstein, F. F. (2006). Modeling turbulent flow with implicit LES. Journal of Turbulence, (7), N15.
[5] Tonicello, N., Lodato, G., & Vervisch, L. (2022). Analysis of high-order explicit LES dynamic modeling applied to airfoil flows. Flow, Turbulence and Combustion, 108(1), 77–104.