Complicated Homogeneous Shear DNS
 

My question is, is it possible to do a DNS of a set of coupled equations in which the flow remains homogeneous, but the mean velocity profile is affected (in an unknown way) through the coupling of the other equations?

I'm researching doing DNS of a homogeneous shear flow in which the linear momentum equation and another equation are coupled.

In the approaches in the literature, the shear rate is obviously constant, and the governing equations are solved for fluctuating terms only. E.g., Rogallo's method and Gerz et al.'s approach.

The problem is that the mean velocity in my set of equations will increase or decrease due to the coupling to the other equations. The increase or decrease in mean velocity will be uniform, so that the shear rate remains the same, (i.e., the slope of the velocity doesn't change). However, the magnitude of increase or decrease is dependant on the other equations. Is it possible using Rogallo's method of solving the equations that DNS will predict the magnitude of this decrease?

In the case with just the Navier-Stokes equations, the mean velocity is known before the problem is even solved. However, when the linear momentum equation is coupled to an equation that can affect velocity, the mean velocity becomes something that needs to be solved for (even though the shear rate remains the same). Since Rogallo's method only solves for fluctuating terms, Rogallo's method isn't applicable.

Thus, the difficulty seems to be finding the mean velocity profile so that you can do the coordinate transformation to get the variables in terms of fluctuating quantities only. This allows the Fourier-Spectral method to be applied since all boundary conditions are then periodic.

I've not done DNS so any replies are greatly appreciated.