CFD Online Discussion Forums - Noise and Turbulence

If you are solving the compressible Navier-Stokes equations, you have to compute the thermodynamic pressure and hence acoustics are a part of the solution. The accuracy of the acoustic solution depends on the closure model used in case of turbulent flows. If are using a DNS approach, closure is a non-issue and the acoustic solution should be very accurate. You can look into the CTR homepage for some papers (especially those by Prof. Lele).

However, DNS using the compressible Navier-Stokes equations can be very expensive and has to be used only in case of highly compressible (Mach number, M > 0.5) flows. At low Mach numbers, flow dynamics (velocity fields) affect the acoustics but not vice-versa. So the acoustics can be decoupled from the velocity fields and can be computed separately. Also, at low M, acoustics are weak and have a relatively low amplitude. Then a linear solution should be sufficient in most cases.

If a linear acoustic solution suffices, you can use the classical Lighthill approximation discussed in many fluid mechanics books. The source term for the acoustic equation (for the acoustics waves) should include a component arising from turbulent fluctuations of velocity in case of turbulent flows. This is the sequence of steps in this method.

* Compute the velocity and density (or temperature) fields using the incompressible N-S equations.

* Compute the source terms for the acoustic equations.

* Solve the acoustic equations.

There is a paper by Karen Pao and Phil Collela that discusses this approach. I am not sure where it is published but you can find it in Pao's homepage (which has a link from Collela's homepage at LBNL). Note that the pressure waves have very high speeds (relative to fluid velocities) and have to integrated using small time-steps. The above approach circumvents this problem by using a mesh coarser than one used to compute velocities for the pressure. Grid spacing for pressure equations, in general, should depend on the highest acoustic wavenumber that needs to be resolved.

I am not sure you can use dimensional arguments here since acoustics are associated with compressible flows which is not necessarily the case with turbulence. If you use dimensional analysis, any dependence on \gamma (ratio of specific heats) and Mach number, quantities that relate to the compressibility of the medium, can not be captured. This is due to the fact that both quantities are non-dimensional.