Transients in Navier-Stokes flows

Hello everybody,

This is my first question in this forum. Talking with colleagues we were not able to reach a clear conclusion, so I post the question here.

Let’s assume a steady-state Navier-Stokes flow as obtained, for instance with the LB method.
Now we apply a small local perturbation to the pressure field in one point of the fluid for a small time and let the LB system relax back to its original steady state.

Question: does the “numerical” transient velocity field observed during the relaxation process bears any relationship to the equivalent “physical” transient velocity field? Under which conditions? Can you provide me please with any references on this particular subject?

Thanks in advance for your time!


Andrea Cortis


In some limits it is physical. I guess you are talking about an acoustic perturbation or some thing similar. If yes you may be interested in the following paper.




Thanks Orestis for your answer.

I should have said earlier that what I had in mind was creeping flow in a porous medium such as the example in:


so, relatively small Reynolds numbers.
I played a bit with that example and looked at the evolution of the x-component of the velocity field at one particular fluid location in the middle of the domain

and you can observe prominent high-frequency oscillations in the early stages of the transient, which are reproduced even when I drop the driving pressure back to zero in the middle of the simulation. The questions is how physical these oscillations are. Any thoughts?

Personally I have always believed these sorts of oscillations to be due to fast changes in the system. For instance, If you simply apply a pressure drop between your inlet and outlet through prescribed pressure boundaries, but your fluid begins at rest, a ‘shock’ occurs as the model begins where between the ‘zero’ iteration and the first iteration the pressure drop is suddenly applied.

The same could be said for the case where you drop the driving pressure back to zero between one iteration and the next. Have you tried ramping the driving pressure up/down over a number of iterations?

I am not sure how physical this behaviour is.

In my opinion, as long as the “shock” is not too strong (we remain in a range where compressibility effects remain small, because in our case we are interested in the incompressible limit) these oscillations are physical. As you pointed out one is applying a very strong discontinuity in the constrains on the fluid. Therefore it has to “react” to counterbalance this change of state. This transient regime, as long as one remains in a weakly compressible regime should be consistent with the macroscopic equations of fluids.

First of all thanks everybody for what is becoming a very interesting discussion!

I tried the suggestion of ramping up the change in the inlet pressure with a smooth “erfc(t)”-like function: some early-times high-frequency components seem to dampen, but the large-frequency oscillations do persist. So perhaps these latter ones are physical indeed.

I am very interested in this topic, so I would like to run now some simulations with parameters characteristic of real fluids, for instance water at 25C in a pore space where the grains are 1mm of radius. I was wondering if you could point me at the documentation that deals with lattice-to-realworld conversion: I have been poking around but couldn’t find it yet. I have found the documents at:

If these oscillations are real, the frequency must be dictated by the physical properties of the fluid.

Many thanks in advance!


Another thing that is very important for the “stationarity” of your flow, is a proper initialization of your distribution functions.

The Chapman-Enskog expansion of the velocity distribution functions is given by


This way maybe the oscillations could also be reduced…

Hello Orestis, I am not quite following what you are saying. I was assuming that in my simulation above, the distribution functions were already correct at iteration 1000 were correct, as they represent a steady state for the system. Am I missing something very obvious here?

Not at all.

What I’m saying is that you probably initialize your problem with f_i=f_i^eq. A more accurate initilization would be to use f_i^eq+f_^1 (add more terms of the CE expansion).