In statistically stationary conditions, the turbulent energy spectrum in a high Reynolds number flow exhibits a k −5/3 (Kolmogorov) regime, with a faster decay at larger, dissipative wavenumbers. Here, we investigate how the energy spectrum of a turbulent flow evolves in time when turbulence decays freely, in the absence of forcing. Results from direct numerical simulation of decaying turbulence in a periodic box with several different initial conditions suggest a generalized self-similar spectrum,where η(t) and L(t) are, respectively, the small (Kolmogorov) and large scales of the flow. A closure method allows us to obtain an explicit form of the spectrum, which reproduces the deviations from the Kolmogorov spectrum at small k observed numerically. The solution can also be used to determine the second and third order structure functions in decaying turbulent flows, and to study their scaling behavior. Our results compare favorably with high-Reynolds number wind tunnel data. Moreover, our theoretical results provide support to the interesting empirical observation by Pearson et al (2002 Phys. Fluids 14 1288-90) that, independent of the large scale structure of the flow, the dimensionless energy dissipation rate is a universal constant when scaled in terms of the turbulent kinetic energy of the flow, and of the length scale corresponding to the peak of the compensated energy spectrum.3 1 4 . Non-stationary turbulent flows play a very important role in many practical problems, and as such, have received much attention [3, 14-17]. For these non-stationary flows, the situation is significantly more

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We derive from first principles analytic relations for the second- and third-order moments of $\boldsymbol{\mathsf{m}}$ , the spatial gradient of fluid velocity $\boldsymbol{u}$ , $\boldsymbol{\mathsf{m}} = \nabla \boldsymbol{u}$ , in compressible turbulence, which generalize known relations in incompressible flows. These relations, although derived for homogeneous flows, hold approximately for a mixing layer. We also discuss how to apply these relations to determine all the second- and third-order moments of the velocity gradient experimentally for isotropic compressible turbulence.