An analytically derived shear stress and kinetic energy equation for one-equation modelling of complex turbulent flows

Abstract

The Reynolds stress equations for two-dimensional and axisymmetric turbulent shear flows are simplified by invoking local equilibrium and boundary layer approximations in the near-wall region. These equations are made determinate by appropriately modelling the pressure–velocity correlation and dissipation rate terms and solved analytically to give a relation between the turbulent shear stress τ⁄ρ and the kinetic energy of turbulence (k = q2⁄2). This is derived without external body force present. The result is identical to that proposed by Nevzgljadov in A Phenomenological Theory of Turbulence, who formulated it through phenomenological arguments based on atmospheric boundary layer measurements. The analytical approach is extended to treat turbulent flows with external body forces. A general relation τ⁄ρ = a1[1 − AF(RiF)](q2⁄2) is obtained for these flows, where F(RiF) is a function of the gradient Richardson number RiF, and a1 is found to depend on turbulence models and their assumed constants. One set of constants yields a1 = 0.378, while another gives a1 = 0.328. With no body force, F ≡ 1 and the relation reduces to the Nevzgljadov equation with a1 determined to be either 0.378 or 0.328, depending on model constants set assumed. The present study suggests that 0.328 is in line with Nevzgljadov's proposal. Thus, the present approach provides a theoretical base to evaluate the turbulent shear stress for flows with external body forces. The result is used to reduce the k–e model to a one-equation model that solves the k-equation, the shear stress and kinetic energy equation, and an e evaluated by assuming isotropic eddy viscosity behavior.