Kolmogorov inertial range spectra for inhomogeneous turbulence

It is shown that the statistical orthogonality of the Karhunen–Loeve (KL) eigenfunctions with respect to both energy and dissipation makes them a particularly good basis for the definition of an energy spectrum in inhomogeneous fluid flows. An effective wave number is defined to characterize the KL eigenfunctions. The definition preserves the relationship between the dissipation and energy spectra that holds for Fourier spectra. With the spectrum and wave number so defined, the scale-similarity arguments that lead to the existence of a spectral inertial range apply. It is also shown that the existence of a spectral inertial range in the KL eigenspectrum is consistent with Kolmogorov’s scale-similarity formulation for structure functions. An example of the KL spectrum obtained from a numerically simulated plane channel flow is presented.