(281f) The URAPS-Closure for the Normalized Reynolds Stress

The unclosed RANS-equation for a constant property Newtonian fluid is an exact equation that relates the mean velocity field to the mean pressure field and the Reynolds stress. An algebraic turbulence model for the Reynolds stress and the continuity equation for constant density fluids provide a low-order closure for the RANS-equation.

Koppula et al.(2009)have developed a new algebraic closure model for the normalized Reynolds stress that is realizable and universal for all turbulent flows. Universality stems directly from an analysis of the equation-of-motion in a non-inertial frame of reference. 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 closure model for the normalized Reynolds (NR-)stress is formulated as a non-negative mapping of the NR-stress into itself and is thereby realizable for all turbulent flows. The theory depends on the relative importance of four local time scales: a viscous time scale; a turbulent time scale; a time scale related to the mean field velocity gradient; and, a time scale associated with the non-inertial frame of reference.

The preclosure equation shifts the turbulence closure problem from the NR-stress to the normalized prestress. The prestress is caused by pressure fluctuations and fluctuations in the instantaneous Reynolds stress. A self-consistent hypothesis, similar to the classical Rotta-conjecture for the pressure/strain rate correlation, is used to relate the prestress to the NR-stress. The resulting anisotropic prestress (APS-) closure, which generalizes the isotropic prestress (IPS-) closure of Parks et al.(1998), is combined with the preclosure equation for the NR-stress to produce a universal realizable anisotropic prestress (URAPS-) closure. The URAPS-closure provides a resolution to one of the key questions in turbulence modeling: Can a low-order closure model for the NR-stress be formulated that is realizable and universal for all turbulent flows independent of the specific benchmark flows used for calibration?

The NR-stress predicted by the URAPS-closure for spanwise rotation of asymptotic homogeneous shear and for spanwise rotation of fully-developed channel flows are compared with available DNS results. The agreement between the URAPS-closure predictions and the DNS results justifies the use of the new closure as a low-order closure of the RANS-equation.

References

Koppula, S. K., A. Bénard, C. A. Petty, 2009; ?Realizable Algebraic Reynolds Stress Closure?, Chemical Engineering Science (in press).

Parks, S.M., K. Weispfennig, and C.A. Petty, 1998, ?An Algebraic Preclosure Theory for the Reynolds Stress?, Phys. Fluids, 10(3), 645-653.