Nonlinear Reynolds stress models and the renormalization group

Abstract

The renormalization group is applied to derive a nonlinear algebraic Reynolds stress model of anisotropic turbulence in which the Reynolds stresses are quadratic functions of the mean velocity gradients. The model results from a perturbation expansion that is truncated systematically at second order with subsequent terms contributing no further information. The resulting turbulence model applies to both low and high Reynolds number flows without requiring wall functions or ad hoc modifications of the equations. All constants are derived from the renormalization group procedure; no adjustable constants arise. The model permits inequality of the Reynolds normal stresses, a necessary condition for calculating turbulence-driven secondary flows in noncircular ducts.

On explicit algebraic stress models for complex turbulent flows

“…(66) is also of the same form as the anisotropic eddy viscosity models of Yoshizawa (1984) and Rubinstein and Barton (1990) …”

“…In these older empirical models, which typically were obtained by simple tensor invariance arguments, the Reynolds stresses were taken to be nonlinear polynomial functions of the mean velocity gradients. In recent years, nonlinear Reynolds stress models of this type have been obtained within the context of two-equation turbulence modeling by more formal expansion techniques incorporating, for example, the Direct Interaction Approximation (DIA) and the Renormalization Group (RNG) (see Yoshizawa 1984, Speziale 1987, Rubinstein and Barton 1990, and Yakhot et al 1992. These models, which are characterized by an explicit relationship between the Reynolds stress tensor and the mean velocity gradients (and possibly their time derivatives) have come to be referred to as "anisotropic eddy viscosity models.…”

The Navier–Stokes-alpha model of fluid turbulence

“…For example, this sort of approach is taken in deriving Reynolds stress algebraic equation models [36]. Rate-dependent closure models of mean turbulence have also been obtained by the two-scale DIA approach [37] and by the renormalization group method [38].…”

A Navier-Stokes solver for complex three-dimensional turbulent flows adopting non-linear modelling of the Reynolds stresses

“…quadratic or cubic, relationship. Several quadratic stress -strain relationships have been proposed in recent years, by Baker and Orzechowski [6], Speziale [7], Nosizima and Yoshizawa [11], Rubinstein and Barton [12], Myong and Kasagi [13], and Shih et al [14]. Craft et al [15] presented a cubic stress-strain relationship, claiming that no quadratic form is able to correctly account for the effects of streamline curvature on the turbulent stresses.…”