Decay exponent of freely decaying homogeneous isotropic turbulence
In plain words
When stirring stops, turbulence in a box dies away with its energy falling as a power of time. Measured powers differ by up to a factor of two between experiments, and theory gives competing values that depend on how the largest swirls were set up.
Precise statement
Freely decaying homogeneous isotropic Navier-Stokes turbulence at high Re has kinetic energy $K(t) \sim t^{-n}$. Kolmogorov's argument with a conserved Loitsyansky integral gives $n = 10/7$ for an initial spectrum $E(k) \sim k^4$ at small $k$; Saffman's conserved integral gives $n = 6/5$ for $E(k) \sim k^2$. Determine whether n is fixed by the low-wavenumber exponent of the initial spectrum alone as $\mathrm{Re} \to \infty \text{ and } t \to \infty$ in a domain much larger than the integral scale, whether the Loitsyansky integral is conserved, and the value of n for each class.
What would settle it
DNS in domains much larger than the integral scale over long decay times at high Re, measuring n and the time dependence of the Loitsyansky and Saffman integrals for controlled initial spectra.
Status in the literature
2022 DNS observe both exponents approximately and over limited times for suitably arranged initial conditions, but the low-wavenumber spectrum drifts, so the invariants are not strictly conserved (Panickacheril John, Donzis, Sreenivasan, Phil. Trans. R. Soc. A 2022).