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Canuto, V.M., M.S. Dubovikov, and G. Yu, 1999: A dynamical model for turbulence. IX. Reynolds stresses for shear-driven flows. Phys. Fluids, 11, 678-691, doi:10.1063/1.869939.
We present a new expression for the one-point Reynolds stress τij in terms of the strain and vorticity of the large scales. The τij are expressed in terms of only five basic orthogonal tensors rather than the traditional ten tensors. The expression for τij, Eq. (24), contains no adjustable parameters. The derivation of τij is based on the two-point closure dynamic equations for the spectral Reynolds stresses Rij(k) that were developed earlier and the results of which were validated on a wide variety of data comprising shear, buoyancy, two-dimensional (2-D) turbulence, rotation, etc. For the case of homogeneous turbulence, we also derive an expression for the empirical coefficients of the equation that depend on the invariants of the flow, the turbulent kinetic energy K and the production P. Examples for special flows are given. The new expressions for τij are shown to reproduce well data from Tavoularis and Corrsin, DNS data, stationary data (pipe flow, channel flow, and homogeneous flow), and the SmagorinskyLilly constant, which is shown to be a dynamical variable since it depends on the ratio P/ε and on the invariant {S3}S3.
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