Turbulent kinetic energy production term

The first-order closure models are all based on the Boussinesq hypothesis [19, 20] parameterizing the Reynolds stresses. Therefore, for fully developed turbulent bulk flow, i.e., flows far away from any solid boundaries, the turbulent kinetic energy production term is modeled based on the generalized eddy viscosity hypothesis , defined by (1.380). The modeled fc-equation is... 

A refined one-dimensional numerical ocean model of the southern Baltic Sea was used by Axell (2002) to investigate suitable parameterizations of unresolved turbulence and compared it with available observations. The turbulence model is a k- model that includes extra source terms of turbulent kinetic energy production by internal waves and Langmuir circulation due to unresolved, breaking internal waves and Langmuir... 

The balanced equation for turbulent kinetic energy in a reacting turbulent flow contains the terms that represent production as a result of mean flow shear, which can be influenced by combustion, and the terms that represent mean flow dilations, which can remove turbulent energy as a result of combustion. Some of the discrepancies between turbulent flame propagation speeds might be explained in terms of the balance between these competing effects. 

In turbulent flow, there is direct viscous dissipation due to the mean flow this is given by the equivalent of equation 1.98 in terms of the mean values of the shear stress and the velocity gradient. Similarly, the Reynolds stresses do work but this represents the extraction of kinetic energy from the mean flow and its conversion into turbulent kinetic energy. Consequently this is known as the rate of turbulent energy production ... 

It is of interest to note that if the convection and diffusion terms are negligible in the turbulence kinetic energy equation, i.e., if the rate of production of kinetic energy is just equal to the rate of dissipation of turbulence kinetic energy, Eq. (5.62) reduces to ... 

This form is appealing because the first term in F.-,/2 can be interpreted as a gradient diffusion of turbulent kinetic energy, and the second is negative-definite (suggestive of dissipation of turbulence energy). However, the rate of entropy production is proportional to... 

Analogous to the transport equation of turbulence kinetic energy, k, the first term on the right-hand side represents production or generation , the second represents diffusion and the final term represents dissipation . The third term of the right-hand side, which has no counterpart in the k equation, represents redistribution . It is the correlation between fluctuations in pressure and velocity gradients, which results in enhancement of velocity fluctuations in one direction at the expense of those in the other directions. It is necessary to model these terms in order to close the set of equations. 

A transport equation for the turbulent kinetic energy, or actually the momentum variance, can be derived by multiplying the equation for the fluctuating component v[, (1.389), by 2u, thereafter use the product rule of calculus to convert some of the terms in the provisional equation, and Anally time average the resulting equation [154]. 

D Production of turbulent kinetic energy by buoyancy (if this term is negative, it represents loss of kinetic energy by buoyancy). 

These are the basic equations used in the description of atmospheric turbulence. The key feature of interest in this discussion is the buoyant production of turbulent kinetic energy, that is, term (D. In order to have a means of assessing the importance of this term, let us consider the ratio of terms and (D,... 

The first three terms on the right-hand side of Eq. (5.5) account for turbulent diffusion, mean flow shear production, and decay of turbulence kinetic energy of phase i. The fourth term on the right-hand side of Eq. (5.5) accounts for production of turbulence energy from slip between phases. The coefficient j)y is given by... 

The first part of the production term corresponds with the production term of the standard / - model. Notice that the second production term is related to the time scale r. The introduction of this additional term enables the energy transfer to respond more efficiently to the mean strain than the standard k-e model does. Thus, r enables the development of a field of suppressing the well-known overshoot phenomenon of the turbulent kinetic energy k. This overshoot appears, when the standard k- model is applied to flow conditions with large values of mean strain [4,7,8]. 

In Eq. (1.6), the two terms on the left side represent, respectively, the increase in Reynolds stress with respect to time and coordinate (three-dimensional) the terms on the right side denote, respectively, the molecular diffusion, the turbulent diffusion, the stress production, the pressure-stain of the flow, and the dissipation of turbulent kinetic energy. 

The second and third terms, represented the production of turbulent kinetic energy, can be considered proportional to the gradient of t/,- and as follows ... 

The Sjtg term represents the production of turbulent kinetic energy due to the movement of the particles. This term is approximated by (4.236). 

This equation possesses production and dissipation terms that are similar to those in the kinetic energy transport equation, except that they are divided by the turbulence time scale of the energy containing eddies, Tf = K As for the fc-equation, the Rejmolds stresses are parameterized based on the eddy viscosity hypothesis. 

The first term on the LHS represents the rate of accumulation of mean kinetic energy within the control volume. The second term on the LHS describes the advection of mean kinetic energy by the mean velocity. The third term on the LHS represents the interaction between the mean flow and turbulence. The first term on the RHS represents the production of MKE when pressure gradients accelerate the mean flow. The second term on the RHS represents the molecular dissipation of mean motions. 

If we compare the k equation (1.407) with the mean kinetic energy equation (1.459) we see that they both contain a term describing the interaction between the mean flow and turbulence. We are of course referring to the velocity variance production term, which is the second last term in (1.459). The sign of this term differ in the two equations. Thus, the energy that is mechanically produced as turbulence is lost from the mean flow, and vice versa. 

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