Universal realizable anisotropic prestress closure for the normalized Reynolds stress.

摘要

The Reynolds averaged Navier-Stokes (RANS-) equation is an exact, albeit unclosed, equation that relates the mean velocity field to the mean pressure field and the Reynolds stress. The continuity equation and a Reynolds stress model provide a low-order statistical closure for the RANS-equation. This research has developed a new algebraic closure model for the Reynolds stress that is realizable for all turbulent flows. In the new theory, the normalized Reynolds (NR-) stress is a solution to an implicit, non-linear, dyadic-valued, algebraic equation that depends on the relative importance of a local turbulent time scale, a local viscous time scale, a local time scale related to the mean field velocity gradient, and a time scale associated with the frame of reference. The theory stems from an analysis of the dynamic equation governing the fluctuating velocity field of a constant property Newtonian fluid in a rotating frame of reference. Therefore, the resulting closure can be applied in either inertial or non-inertial frames regardless of the class of benchmark flows used to determine the phenomenological closure parameters.;The foregoing low-order closure model for the RANS-equation generalizes earlier research by Parks (1997) and Weispfennig (1997) based on an integral analysis of turbulent fluctuating velocity fields and the physical assumption that all space-time turbulent correlations have finite memories. In this research, the Parks-Weisfennig approach is extended to non-inertial frames. A preclosure equation shifts the turbulence closure problem from the NR-stress to a normalized prestress. The prestress is caused by pressure fluctuations and fluctuations in the instantaneous Reynolds stress. A self-consistent hypothesis, similar to the one for the pressure/strain rate correlation, is used to relate the prestress to the NR-stress. In the present research, a closure for the prestress is developed and combined with the preclosure equation for the NR-stress to produce a universal realizable anisotropic prestress (URAPS-) closure for the NR-stress. A critical review of other algebraic closure models in the literature indicates that the URAPS-closure provides an answer to one of the key questions in turbulence modeling: Can a low-order closure model for the NR-stress be formulated that is realizable for all turbulent flows independent of the specific benchmark flows used for calibration?;The URAPS-closure is formulated as a mapping of a non-negative operator into itself. The mapping depends on the rotational operator W associated with the frame of reference, a local scalar-valued turbulent transport time scale tauR, and an operator F&parl0;≡1 u+2W &parr0;: ℜ&parl0; R,tR F&parr0;= R, R≡u 'u' tru' u' . The URAPS-closure is used to predict the components of the NR-stress for three benchmark flows: rotating homogeneous decay, rotating homogeneous shear, and spanwise rotating fully-developed channel flows. The URAPS-predictions are consistent with complementary direct numerical simulations of these flows and, thereby, partially supports its use as a closure model for the RANS-equation.