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.
The renormalization group is applied to compute anisotropic corrections to the scalar eddy diffusivity representation of turbulent diffusion of a passive scalar. The corrections are linear in the mean velocity gradients. All model constants are computed theoretically. A form of the theory valid at arbitrary Reynolds number is derived. The theory applies only when convection of the velocity–scalar correlation can be neglected. A ratio of diffusivity components, found experimentally to have a nearly constant value in a variety of shear flows, is computed theoretically for flows in a certain state of equilibrium. The theoretical value is well within the fairly narrow range of experimentally observed values. Theoretical predictions of this diffusivity ratio are also compared with data from experiments and direct numerical simulations of homogeneous shear flows with constant velocity and scalar gradients.
A renormalization group is developed for the Navier–Stokes equations driven by an anisotropically correlated random stirring force. The stirring force generates homogeneous turbulence with a preferred direction. The force correlation is the sum of a small anisotropic perturbation and an isotropic correlation chosen so that the fixed point of the renormalization group has a k−5/3 energy spectrum. Fixed points for the anisotropic correlation are found near this isotropic fixed point. Two types of anisotropy are analyzed. When the additional stirring is in the plane perpendicular to the preferred direction, the renormalized viscosity is increased. When it is aligned with the preferred direction, the viscosity is decreased. A possible connection with the inverse energy cascade of 2-D turbulence is discussed.
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.