Hi Seyed Ali Hosseini ,

I am not sure to understand what you mean by "artificially reduced acoustic modes propagation speeds"?

For LBMs, the isothermal weakly compressible limit flows from improper: (1) velocity discretizations of the Boltzmann equation (e.g., D2Q9, D3Q19 and D3Q27 lattices), and (2) discrete equilibrium.

Nevertheless, for finite values of the Mach number (Ma

Hi Christophe Coreixas ,

Thanks for the reply. Concerning your point on the consistency of acoustics in LBM, I would agree with your point of view if, and only if, we postulate that the reference temperature of the isothermal LB solver is always set so that the sound speed in the LB solver, i.e. cs (or cs*sqrt(theta) if you go for a non-dimensional temperature different from 1), is equal to the isothermal sound speed of the considered thermodynamic system. To illustrate what I mean let's consider for example the lid-driven cavity, with water at 300K as the working fluid with nu=1e-6 m2/s for simplicity, in a box of size 1 m and a lid moving at 1 mm/s (i.e. Re=1000). At this temperature, the sound speed is roughly 1400 m/s in water. For your typical LB simulation, you'd be setting dx = 0.01 and dt = 0.1 (therefore non-dimensional nu=1e-3). This choice of dt and dx would impose an isothermal sound speed of 0.1/sqrt(3) (assuming a D2Q9 or D3Q27) and an effective Mach number of app. 0.0014 instead of the physical Mach number of 7e-7. Am I right?

Best,

Ali

And for Chorin's ACM (Artificial Compressibility Method), it is just a weaker imposition of the divergence free velocity field, by modeling the space- and time-evolution of pressure using a parabolic PDE instead of the elliptical Poisson equation: dP/dt +beta*div(U) = 0.

Notice that the adjustable parameter beta has a role similar to that of cs in the LBM.

Best,

Ali

Well, in fact, to impose the correct sound speed, you need to compute the time step using the acoustic scaling,

c = cs dx/dt --> dt = dx c/cs

with c = 1400 [m/s] the physical sound speed, cs = 1/sqrt(3) the sound speed in lattice units, and dx = 0.01 [m] in your case. Once you have both dx and dt, you can compute u and nu in lattice units to impose the correct Mach and Reynolds numbers for your LB simulation.

This is what I did during my master thesis to perform computational aeroacoustics. Thanks to that scaling, you are even able to account for the specific heat ratio of air (1.4) via the definition of dt, so that you can properly compare your results with real-life experimental data.

Best, Christophe

Using dt = dx cs / c, for c = 1400, your time-step would be app. 1e-5 s. Which for a simulation intended for the incompressible regime is too much to pay right? And I think it might become even more expensive than a simulation with the Poisson equation with much larger time-steps(though I am not sure). Also having in mind that using the classical SRT collision operator with this choice of dt stability might be an issue.

And just to clarify, given that we are discussing applications in the incompressible regime (at least for the present discussion) correct acoustics are not what we are targeting anyways. I mention the acoustics just to establish the difference with the "real" incompressible formulation, and the relation to the ACM, and how one could pretend to have an efficient solver for the incompressible regime using the classical LBM.

Well, that problem is not LBM related, but instead, it arises when one tries to simulate an incompressible flow using a (weakly) compressible solver. Since the time step is related to the CFL number (which is in turn related to the velocity), then simulating flows with a small Mach number will lead to very small time steps.

Consequently, I am definitely not sure if in terms of wall-clock time, it is better to rely on a local (weakly) compressible approach (such as LBM) instead of using incompressible Navier-Stokes codes that solve the non-local Poisson equation.

Again I think we are missing the point by not acknowledging that for "incompressible" applications, the added value of an LB solver (or at least what is sold as the major advantage of LB solvers, and the majority of simulations targeting incompressible flows use this) over a compressible solver (that is fully local, given the parabolic nature of the coupled system of PDEs) is that you play with the acoustic scale and hence can use bigger time steps (imagine doing a turbulent channel flow simulation, with a choice of dt that would result in consistent acoustics).

In practice in such applications, the lattice sound speed is an adjustment parameter. By keeping the non-dimensional velocity below 0.1 (but not too small either) you (to a certain extent) guarantee separation of scales between acoustic and viscous modes and therefor minimise acoustic perturbations (that you are not interested in) and keep the time-step computationally acceptable.

There are other approximate solvers for the incompressible regime in that family such as the ACM, or the entropically damped artificial compressibility (EDAC) or the cryogenic formulation, or the grand potential based pseudo-compressible formulation (also known as the kinetically reduced local Navier Stokes) or the generalised pressure equation etc.

Eventually it all boils down to one question: which one is more efficient, a strict divergence-free model or weak formulations? the former is not local but filters out acoustics allowing for larger time-steps while the latter is local but admits compressibility errors (coming from acoustic modes propagating at the "lattice sound speed" velocity)?

There are also other considerations for pseudo-compressible formulations (such as the checkerboard effect for collocated nodes for models such as the ACM or LBM) that we can also discuss...

For people intrigued by this question, I thought this might be an interesting read (with a detailed comparative study of LBM and ACM):

https://mediatum.ub.tum.de/doc/1437209/document.pdf

Dear Seyed,

Thank you for your interesting question.

When considering LBM as a solver for incompressible fluid flows then I also share your understanding that LBM pertains to the class of artificial compressibility methods (ACMs). In line with ACM original idea, LBM also relaxes the strict enforcement of the divergence-free condition of the velocity field and tries to reproduce this eliptic condition in an hyperbolic way, i.e. by incorporating the solenoidal velocity field condition as an (approximated) outcome of a pressure ``transport equation''. Anyway, so far, I have not said anything new...

What I would like to add to the discussion is that when we talk about the ACM formulation we need to bear in mind that many ACMs exist and Chorin's classical ACM formulation is one within other possible alternatives within this framework.

The following paper surely explains this way better than me:

Article Entropically damped form of artificial compressibility for e...

Having clarified this point and trying to answer your question, I believe that the macroscopic balance equations effictively solved by LBM should not be considered as 100% equivalent to those found in the standard ACM form of Chorin's work. To me, the reason why this difference is often unnoticed is that, in the process of asymptotically deriving the form of the LBM macroscopic equations, we tend to introduce several scaling assumptions that often lead to the neglect of numerous terms, which we deem as higher-order in terms of either Kn or Ma. (One example of such a neglected term is the Laplacian of pressure in the mass balance equation, which I think may play an important role in the context of your question.) In my opinion the assumptions under these scalings (and therefore the neglect of those terms) may be questionable, at least under finite grid resolutions or finite time steps.

For that reason, I expect that the actual LBM possesses different numerical characteristics than Chorin's ACM. Loosely speaking, I expect that the LBM form of reproducing the incompressibility condition will be perhaps more in line with the ACMs equations shown in the link above (or other ACMs found in the literature).

Best,

Goncalo

Goncalo Silva,

Agree with your point on the existence of different classes of ACM and therefore different forms of the pressure hyperbolic/parabolic PDE. In fact this article Article Thermodynamic Theory of Incompressible Hydrodynamics

Now not having used the ACM myself, I am wondering whether the hyperbolic form of the pressure PDE in Chorin's formulation (as opposed to parabolic PDEs for the EDAC and KRLNS) means that acoustic modes are not damped (except for numerical dissipation coming from discretisation)? Or whether when re-written in terms of characteristics one would recover a dissipation term? And furthermore, if we assume that the damping is the major "improvement", would that mean that the weak performances of the original ACM (for unsteady flows) are due to non-dissipating acoustics? while those from LB disappear over time?

Ali

Dear Seyed,

That is true. Perhaps, Ansumali and Karlin were the pioneers in this field to try formulating different ACMs that improve Chorin's original idea. By the way, other work I also like is the one below from Toutant:

Article General and exact pressure evolution equation

Before answering to your question, I would also like to leave the disclaimer that I have also never used ACM for transient problems either. The only time I used it was in a simple steady-state problem, where the interesting features that we are talking about here have vanished.

Regarding your previous comment. I agree with your second paragraph that the key ingredient of those ``improved'' ACMs (where I would include LBM as well) is indeed the addition of the acoustic damping term, in the form of the pressure Laplacian, which parabolizes the otherwise hyperbolic pressure evolution equation. By the way, now that I writing this, I also remembered that Pietro Asinari has already performed some works in this fiedl, relating ACM to LBM:

Article Link-wise Artificial Compressibility Method

Article Artificial Compressibility Method and Lattice Boltzmann Meth...

Perhaps you are aware of these studies already.

On the advantageous role played by that dissipation term in the pressure evolution equation, I would again refer to Clausen work, namely Section 2 C, which links all these methods and discusses their behavior.

Finally, and without trying to divert too much from the main discussion, it is also perhaps worth saying that all ACM studies mentioned above, like Ansumali and Karlin, Clausen, Toutant, etc., should not be considered as numerical studies. Rather, in my opinion, their main value is that they aim to establish a physical backgroung to the inclusion of the acoustic damping terms within ACM.

Yet, if we seek for this idea in numerical-oriented studies, then the inclusion of such acoustic damping terms is very old, though with the sole justification of reaching improved numerics (either on stability or accuracy):

Article Conservation laws in discrete geometry

This last paragraph is just to reinforce the idea that these acoustic damping terms indeed appear to bring beneficial properties to the time-dependent evolution characteristics of the numerical schemes (and by implication to their solutions), since otherwise I could not comprehend very well why so many people have included them when tackling so many different problems.

Best,

Goncalo