An Analytically Derived Shear Stress and Kinetic Energy Equation for One-Equation Modelling of Complex Turbulent Flows.

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 = q 2 / 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 = a 1 1 − AF Ri F q 2 / 2 is obtained for these flows, where F Ri F is a function of the gradient Richardson number Ri F , and a 1 is found to depend on turbulence models and their assumed constants. One set of constants yields a 1 = 0.378, while another gives a 1 = 0.328. With no body force, F ≡ 1 and the relation reduces to the Nevzgljadov equation with a 1 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 – ε model to a one-equation model that solves the k-equation, the shear stress and kinetic energy equation, and an ε evaluated by assuming isotropic eddy viscosity behavior. [ABSTRACT FROM AUTHOR]