Turbulent dissipation rate definition

As an example, for steady, incompressible, and isothermal turbulent flows using the k- model, the independent equations are (1) the continuity equation, Eq. (5.61) (2) the momentum equation, Eq. (5.65) (3) the definition of the effective viscosity, /xeff (combination of Eq. (5.64) and Eq. (5.72)) (4) the equation of turbulent kinetic energy, Eq. (5.75) and (5) the equation for the dissipation rate of turbulent kinetic energy, Eq. (5.80). Thus, for a three-dimensional model, the total number of independent equations is seven. The corresponding independent variables are (1) velocity (three components) (2) pressure (3) effective viscosity (4) turbulent kinetic energy and (5) dissipation rate of turbulent kinetic energy. Thus, the total number of independent variables is also seven, and the model becomes solvable. 

Consider the extension of this definition for the dissipation rate to the case of turbulence. We can introduce the mean dissipation rate by averaging both sides of (1.402), and introducing the decomposition of the velocity field into mean and fluctuating variables. The result is ... 

If a flow in the tank is turbulent, either because of high power levels or low viscosity, then a typical velocity pattern at a point would be illustrated by Fig. 3. The velocity fluctuation i can be changed into a root mean square value (RMS), which has great utility in estimating the intensity of turbulence at a point. So in addition to the definitions above, based on average velocity point, we also have the same quantities based on the root mean square fluctuations at a point. We re interested in this value at various rates of power dissipation, since energy dissipation is one of the major contributors to a particular value of RMS v. ... 

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... 

We have seen in Chapters 2 and 4 that turbulence dramatically strengtherrs the dissipation of kinetic energy. This property results from the definition of the instantaneous rate of kinetic energy dissipation per unit volume ... 

The last term is always negative. It corresponds to dissipation. Regarding the equation for k, the dissipation term is simply the definition of the rate of dissipation. Regarding the equation for e, dissipation takes place with the characteristic time of turbulence [8.12], since k = s Xt. 

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