Journal of Fluid Mechanics (1997), 342:335-354. Cambridge University Press.

Copyright © 1997 Cambridge University Press

## On the decay of inhomogeneous turbulence

**J. R.** **CHASNOV**^{}

^{}Department of Mathematics,
The Hong Kong University of Science and Technology,
Clear Water Bay, Kowloon, Hong Kong

(Received January 7, 1997)

**Abstract**

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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