On model constants and second order closure for curved shear flows

The pressure-strain correlation model of Hanjalic and Launder is extended to rotating curved shear flows. Stationary plane flow data together with the requirement that the critical Richardson number should be predicted correctly are used to determine the model constants. The value of the constants are found to be the same as those determined by Hanjalic and Launder through computer optimization. It is also found that the pressure-strain correlation model of Hanjalic and Launder has negligible effects on the shear stress relation for curved shear flows compared to the simple return to isotropy model of Rotta. In view of this, significant improvement on second order closure of the mean flow equations for curved shear flows cannot be obtained by improving the simple Rotta model.Nomenclature a ₁, a ₂, a ₃ Constants defined in (11a...c) - A ₁...A ₆ Constants defined in (11d...i) - b _ij Reynolds stress matrix defined in (12)–(15) - c ₁ Model constant introduced by Rotta 2] and defined in (6) - c ₂ Model constant introduced by Rotta 2] and also in (3) - c _r, c ₃ Model constants introduced in (3) - F _c Curvature parameter defined in (11j) - F _R Rotation parameter defined in (11k) - k Longitudinal surface curvature - l ₁ Length scale introduced in (3) - l _b Mixing length for corner flow defined by Gessner and Po 13] - Fluctuating pressure - Twice the turbulence kinetic energy - Ri Gradient Richardson number for curved shear flows - Ri _cr Critical gradient Richardson number - u _i i-th component of fluctuating velocity field - U _i i-th component of mean velocity field - U Mean flow in x-direction - u, v, w Fluctuating velocity components along x, y, z axes, respectively - x _i i-th component of coordinate system - x, y, z Coordinate axes attached to surface - _i Coefficients defined in (20) - Dissipation length scale introduced in (5) - Fluid kinematic viscosity - Fluid density - _i i-th component of rotation vector - Speed of rotation of surface about oz axis