Renormalization group analysis of the Reynolds stress transport equation

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Abstract
The pressure‐gradient–velocity correlation and return to isotropy term in the Reynolds stress transport equation are analyzed using the Yakhot–Orszag renormalization group. The perturbation series for the relevant correlations, evaluated to lowest order in the ε expansion of the Yakhot–Orszag theory, are infinite series in tensor product powers of the mean velocity gradient and its transpose. Formal lowest‐order Padé approximations to the sums of these series produce a rapid pressure strain model of the form proposed by Launder et al. [J. Fluid Mech. 68, 537 (1975)], and a return to isotropy model of the form proposed by Rotta [Z. Phys. 129, 547 (1951)]. In both cases, the model constants are computed theoretically. The predicted Reynolds stress ratios in simple shear flows are evaluated and compared with experimental data. The possibility is discussed of deriving higher‐order nonlinear models by approximating the sums more accurately.