Extracting statistics for turbulent flows directly from the Navier–Stokes equations poses a formidable challenge, particularly when dealing with unsteady or inhomogeneous flows. However, embracing Kolmogorov’s inertial range spectrum for isotropic turbulence as a dynamic equilibrium provides a conceptual starting point for perturbation theory. We review theoretical results, combining perturbation approaches, and phenomenological turbulence closures, which allow us to gain valuable insights into the statistics of unsteady and inhomogeneous turbulence. Additionally, we extend the ideas to the case of the mixing of a passive scalar.
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