Test-field model for inhomogeneous turbulence

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Abstract
The test-field model for isotropic turbulence is restated in a form which is independent of the choice of orthogonal basis functions for representing the velocity field. The model is then extended to non-stationary inhomogeneous turbulence with a mean shearing velocity, contained by boundaries of arbitrary shape. A modification of the model is introduced which makes negligible changes in the numerical predictions but which greatly simplifies computations when the co-variance matrix and related statistical matrices are non-diagonal. The altered model may be regarded as a kind of generalization of Orszag's eddy-damped Markovian model, with the damping factors determined systematically, in representation-independent form, from dynamical equations. The final equations of the test-field model are presented in a sufficiently explicit form to serve as a starting point for numerical work. To facilitate comparison, the corresponding direct-interaction equations for inhomogeneous turbulence with mean shear are presented also, in a uniform notation. The test-field model is much faster to compute than the direct-interaction approximation because, in the former, only single-time statistical functions need be computed. This advantage is at the cost of a less rich and less faithful representation of the dynamics.