Renormalization group analysis of the Reynolds stress transport equation

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.

Derivation of a second-order model for Reynolds stress using renormalization group analysis and the two-scale expansion technique

“…The model constants are calculated analytically. Compared with the evaluations of Rubinstein and Barton [12], the calculations are much simpler. Also, in the calculations, the value of the parameter y in Eq.…”

“…Also, in the calculations, the value of the parameter y in Eq. (11) is set to be one value y = 3 (in the calculations, the same parameter y is set to be the different values: y = −1 and y = 3, at the different places of the same calculation [8,11,12,[14][15][16][17]). In order to test the efficiency of the proposed model, we perform numerical simulations for two typical numerical examples: turbulent flow past a backward-facing step and fully developed flow in a rotating channel using the standard k-ε model, the GL model and the theoretical model proposed in this article.…”

“…Now, the RNG k-ε model [9,10] has been applied to the simulations of practical engineering problems, for instance, being included in CFD software such as FLUENT. Rubinstein and Barton [11,12] initially applied the YO RNG method to formulate the nonlinear Reynolds stress model and the secondorder model. In this article, a second-order closure is derived in an alternative way.…”

A novel implicit numerical treatment for non‐linear turbulence models using high and low Reynolds number formulations

“…Non-linear eddy-viscosity models originated in a general proposal was done by Pope [20]. However, only in the past two decades such models had greater progress, particularly with the works of Speziale [9], Nisizima and Yoshizawa [11], Rubinstein and Barton [12], Shih et al [13], among others. In these works, quadratic products were introduced involving the strain and vorticity tensors with different derivations and calibrations for each model.…”

“…To the best of the authors' knowledge, most of the published work on non-linear models [9,[11][12][13][14] are either written for the Cartesian coordinates and/or treat additional non-linear terms in a fully explicit manner. The recent literature has recognized the difficulty in obtaining convergence when running non-linear models in complex flows.…”