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=a11−AFRiFq2/2 is obtained for these flows, where FRiF 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–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.
Highlights
Many turbulence models are available to date
In view of the firm theoretical backing given to the one-equation model in the current study, the present approach could facilitate the extension of the one-equation model to cover a wide range of flows where external body force effects are present and are instrumental in changing the turbulent characteristics of the flow
By correlating experimental measurements of τ/ρ and q2 /2 obtained from turbulent boundary layers and free mixing flows, the constant was determined to be 0.3 by Harsha and Lee [8] and Bradshaw et al [9]
Summary
Many turbulence models are available to date. Some of the more popular ones are the zero-equation models, the two-equation models and the full Reynolds stress models [1]. The turbulent kinetic energy equation, or k-equation for short, can be derived from the Reynolds stress equations with suitable modelling assumptions invoked for the pressure–velocity correlation and the dissipation rate terms. There is no need to solve the mean field equations and Equation (1) simultaneously with a full set of modelled Reynolds stress equations This feature is quite attractive, especially for numerical simulation of turbulent thermal flows where simultaneous solutions of four more equations are required; these additional equations are the mean temperature equation, two modelled turbulent heat fluxes equations and a temperature variance equation in any 2-D flows. This one-equation model is simple, yet it has a slight drawback, i.e., assuming Equation (1) to be independent of flow type, and can be used without modifications to model simple and complex turbulent flows without a valid analytical foundation to support this assumption
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