[FMDenaro]

Many of the answers can be deduced from the general LES theory as it is illustrated in the Sagaut's textbook.


In principle, the LES filter can be expressed both in space and in time but the temporal filter theory is less developed (see Sagaut). Generally, the spectrum of turbulence frequencies is assumed to be fully resolved by the time step since pi/dt (Nyquist cut-off) is higher than the highest physical component.

Then, the question becomes more of numerical analysis, that is the numerical stability constraint for the combination of dt,dx,dy,dx, velocity components, physical viscosity, eddy viscosity. Depending on the type of time and space integration you have a complex stability region in the hyperspace cx,cy,cz, Re_dx,Re_dy_Re_dz. The components of the CFL are strictly constrained by the physical and eddy viscosity.

[LuckyTran]

Your questions are so broad, and very general, I'm not sure how to even find a journal paper for any of them. It's not specific knowledge that belongs in any journal.

1. LES is spatially filtered and needs to be time-resolved. If it is time-resolved, this necessarily implies insensitivity to time-step. If you do proper time-resolved LES, then the time-step should have no impact. But if you do poorly time-resolved LES, well then it is not LES and there is no point discussing it because it's not LES. Proof by contrapositive. Yes of course you can apply a spatial filter and use an arbitrarily large time-step (because there's no one that can stop you). But if I run a URANS simulation with a time-step of 1 trillion years, I will be a clown.

2. What is implied here is that the Courant number of 1.5 is small enough to get you a time-resolved simulation. What's missing is a reference for specifically the 1.5. That you probably won't ever find, because it's too hard to prove for every single case. That's why people still run tests to show time-step inseitivity (just like mesh sensitivity studies). It's just a rule of thumb.

3. The CFL condition of 1 is derived from a 1D wave equation. Actually the CFL condition for 2D and 3D is even less. Any time you change the governing equation, you must start over and re-derive the analysis and come up with the proper CFL condition. See the heuristic explanation of the CFL condition.You are wondering what is happening to a perturbation as it moves across a discrete grid.. That's why the Courant number is what it is (i.e. Co=u*dt/dx) for the wave propagation problem. u*dt is the distance the perturbation moving at velocity u travels in time dt and dx is the cell size. At a Courant number of 1, you track the wave properly because you can see the disturbance propagate perfectly from one-cell to the next. But this is for an advective disturbance. A disturbance can also propagate via diffusion and even other means depending on the governing equation.

4. For a generalized flow, there is some length scale (usually around the integral length scale of your wind mill) that you want to resolve with your LES. This length scale is going to be LARGE, that's why we call it large eddy simulation. With this length scale and the expected velocity, there is a time-step size you need to perform your LES and have it be spatially and temporally resolved. For boundary layer problems (more specifically, wall-bounded flows), you need small cells next to walls because of the no-slip condition (the notorious y+~1 requirement). Now when you make this grid, you have really small cells. The time-step needed to keep the solution stable (for these small cells) is usually, several orders of magnitude smaller than the time-scale of the largest eddies that you actually care about. If you are not doing a no-slip problem, then this doesn't apply.

5. I'm not going to argue here. It is only a rule-of-thumb. You need to run a small enough time-step and prove (for every simulation) that it is indeed time-resolved. And each time, you'll get a slightly different Courant number. It may be 1 today, it may be 1.1 tomorrow. For sure, it will be some finite number less than infinity. The only way to know is to check it yourself and verify.

6. For a spatial cutoff of L and velocity U, the temporal time-scale that is being cut-off is roughly L/U. This comes from basic properties of waves. This is a bit oversimplified because you need to also include the Nyquist condition and account for the Heisenberg uncertainty (i.e. the Cauchy-Schwartz property).

Finally, keep in mind that not everybody calculates Courant number using the same definition. A Courant number of 2 for one person might be 4 for another. Also there are different Courant numbers for different physical phenomenon (acoustic waves, gravitational waves, etc.).