Please check if you have taken nonlinearity (looks from the red curve) and dispersion induced phase delay in account.

Thank you for answer, Vikash

As I understand process, in CFD dispersion of the wave means that the phase difference between the signals at the different angles and its amplitudes (to axis of vibration) not equal to wave equation solution (graph 3 for the piston). Or the intensity directivety is different which can be easily demonstrated by increasing k*a. In the all graphs i have uploaded, k*a is small enough to properly resolve a wave on certain mesh (due to low order schemes it needs 20 cells per wave length) so dispersion is not a major factor.

What do you mean "have taken nonlinearity"? Just to cut data and adjust phase? So why when i simulate a segment of the pulsating sphere with identical as in the piston case oscillating velocity BC, the phase difference is smaller (see fig. 2)?

Fig 1. is for dipole (trembling sphere) and Fig. 3 for piston -- both have similar phase lag app 0.25 of the period.

IIya

I am talking about material-induced dispersion, where waves of different frequencies undergo different phase delay and also attenuation. I saw a constant delay in your plots so, I felt about it. Second, I thought about nonlinearity as I feel the peaks are somewhat truncated mildly. I am not 100% sure though. So, I guess for example if you are simulating wave propagation in water or blood or any other general medium; they are nonlinear. Change the amplitude of the wave (say, 1/3) and see if it changes the results. Nonlinearity manifests due to the interaction between nonlinear aspect of the medium and high amplitude of the wave.

Vikash,

I'm afraid that we're talking about same things but from a bit different points of view. With certain meshes material - induced dispersion is resolved good enough (i mean again directivety which can be represented by value of k*a = omega/c * a (c -- speed of sound in media, a -- radius).

The truncated peaks could be due to mesh motion solver in trembling sphere case. pimpleDyMFoam solver has it's own features and it was not investigated in context of such effects.

Anyway, you idea to change amplitude of the wave is what i did not before so i'll try it and write back later.

Vikash,

Despite fact, that half amplitude is calculating now, i've checked solution logs to control desity, temperature and sound speed fluctuations and conclude that there is no non-linearity in the investigated cases. It's obvious, I have reference pressure 101325 Pa, and in the extremal case when ka=5 pressure fluctuations is about 2125 Pa and usually they are smaller. The medium is air.

You can get dispersion error from your numerical scheme. Try decreasing your mesh spacing to see if the dispersion reduces. You could try using dispersion-relation-preserving (DRP) schemes. Another reason for inaccuracy is your boundary condition. When comparing against the analytical solution, do it at an instant when the wave has not reached the boundary, so you can isolate the effect of boundary conditions. 

Dear Anurag,

Yes, BC reflection was observed.  About mesh spacing I would say that reducing it has no effect on phase-difference.

Hello Ilya,

Actually, I got similar issues while running CFD simulaiton of different waves. If you already got the reasons why the phase differences happened, it would be much helpful if you could share the information.

Best,

Fangda