On the decay of inhomogeneous turbulence

The decay of high-Reynolds-number inhomogeneous turbulence in an unbounded domain is considered. The turbulence may be initially localized in one to three spatial directions and the fluid is assumed to be at rest at infinity in those directions. Previous arguments used to determine the decay laws of homogeneous turbulence are extended to the decay of inhomogeneous turbulence by integrating the turbulence statistics over the inhomogeneous directions. Dimensional arguments based on the invariance or near-invariance of low-wavenumber spectral coefficients associated with the integrated mean-square velocity are used to determine asymptotic decay laws for inhomogeneous turbulence. These decay laws depend on the number of inhomogeneous directions of the flow field and reduce to the well-known decay laws of homogeneous turbulence when this number is zero. Different decay laws are determined depending on the spectral behaviour at low wavenumbers. Asymptotic similarity states of the spectrum during the decay and of the distribution of the mean-square velocity along the inhomogeneous directions are also determined. An analytical result for the decay of the mean-square velocity at the centre of the initial disturbance is found, and the decay proceeds more rapidly with increasing number of inhomogeneous directions due to the transport of energy along those directions.

Large-eddy simulations of decaying turbulence homogeneous in a plane and localized in a single direction are performed to test the theoretical scaling laws. The numerically determined asymptotic decay laws of the integrated mean-square velocity agree well with the theoretical predictions. A self-similar decay of the spectra and mean-square velocity distributions is also observed. The simulation results suggest that when the low-wavenumber spectral coefficient is an exact invariant, a unique similarity state depending only on the initial value of this invariant and independent of all other aspects of the initial conditions is attained asymptotically.

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