Computing Kolmogorov/Energy spectrum for turbulent boundary layer

Question

Previously, I have calculated energy spectrum for 3D DNS data obtained for isotropic turbulence which is equally spaced in all three directions and then to compute the energy spectrum, one performs Fourier transform and then accumulates energy located in different wavenumber bins and then gets the ($k^{-5/3}$) slope and it all works fine.

Now if I want to extend the same for a turbulent boundary layer case, how should I modify the technique? I cannot take a Fourier transform in the wall normal direction anymore and I am not sure how to compute the energy spectrum as such. There is no problem if I compute the spectrum for streamwise and spanwise plane since they are both defined using Fourier basis.

So my question is two-fold:

1. How do we compute energy spectrum for a 3D turbulent boundary layer?
2. If we cannot compute it for a 3D case i.e. if the argument is valid only for a 2D planar cut at various wall normal locations, then how do we define a scale for turbulent boundary layer cases, particularly along wall normal direction?

Answer 1

1) One usually chooses one or two representative wall-normal distances and presents the spectra for <u'u'>, <v'v'>, etc. over the homogeneous directions. For example, see Figure 11 within Spalart 1988 (http://dx.doi.org/10.1017/s0022112088000345).

2) By "scale" do you mean a lengthscale? If so, a lengthscale quantifying what aspect of the flow?

Backing up, what's your goal? To compare against someone's published result? To convince yourself that you have adequate resolution? To look at some aspect of the physics in more detail? Something else?

Answer 2

Generally from what I have seen, people either compute 1D spectra (in span wise and stream wise directions) or compute 2D spectra of horizontal slices in the vertical. Depending on what you are trying to look at, these horizontal spectra can then be averaged in the vertical.

Im not entirely sure what scale you are looking for, but let me add that in boundary layers it is not obvious that a Kolmogorov inertial range should exist because of the anisotropic nature of the flow and I believe that this is still an active research area (e.g. convective boundary layers). I think what has been found is that at small enough scales pressure fluctuations do result in "isotropotizing" the flow thus resulting in an inertial range but this is not obvious from the 3D triply periodic theory. One important statistic that is often looked at is the ratio of vertical-to-horizontal spectra. For example in convective boundary layers this ratio should be a constant 4/3 in regions of local isotropy.