A second-order modelling of a stably stratified sheared turbulence submitted to a non-vertical shear

Abstract

In this work, the evolution of homogeneous stably stratified turbulence submitted to a non-vertical shear is studied using second-order closure models. Two cases of turbulent flows are considered. Firstly, the case of a purely horizontal shear is considered. In this case, the evolution of the turbulence is studied according to the Richardson number Ri which is varied from 0.2 to 2.0 when other parameters are kept constant. In the second case, two components of shear are present. The turbulence is submitted to a vertical component S _v  = ∂U ₁/∂x ₃ = S cos(θ) and a horizontal component S _h  = ∂U ₁/∂x ₂ = S sin(θ). In this case, we study the influence of shear inclination angle θ on the evolution of turbulence. In both cases, we are referred respectively to the recent direct numerical simulations of Jacobitz (2002 J. Turbulence 3 055) and Jacobitz and Sarkar (1998 Phys. Fluids 10 1158–68) which are, to our knowledge, the most recent results of the above-mentioned flows.

Transport equations of second-order moments

are derived. The Shih–Lumley (SL) (Shih T H 1996 Turbulence Transition and Modeling ed H D S Henningson, A V Johansson and P H Alfredsson (Dordrecht: Kluwer); Shih and Lumley J L 1989 27th Aerospace Meeting 9–12 January, Center of Turbulent Research, Nevada) and the Craft–Launder (CL) (Craft T J and Launder B E 1989 Turbulent Shear Flow Stanford University, USA, pp 12-1–12-6; Launder B E 1996 Turbulence Transition and Modeling ed H D S Henningson, A V Johansson and P H Alfredsson (Dordrecht: Kluwer)) second-order models are retained for the pressure–strain correlation φ _ij and the pressure–scalar gradient correlation φ _{i ρ}. The corresponding models are also retained for the dissipation ε of the turbulent kinetic energy and an algebraic model is retained for the dissipation ε_ρρ of the variance of the scalar. A fourth-order Runge–Kutta method is used for the numerical integration of the closed systems of non-linear dimensionless differential equations. A good agreement between the predictions of second-order models and values of direct numerical simulation of Jacobitz has been generally observed for the principal component of anisotropy b ₁₂. A qualitative agreement has been observed for the ratios K/E and K _ρ/E of the kinetic and potential energies to the total energy E.

Corresponding address: BP 12, PTT Tunis, Chabbi 1089, Mont-Fleury Tunis, Tunisia.

Notes

Corresponding address: BP 12, PTT Tunis, Chabbi 1089, Mont-Fleury Tunis, Tunisia.