Turbulent energy dissipation spectrum

The second term on the right-hand side of (2.61) can be rewritten in terms of the turbulent energy dissipation spectrum as... 

The final phase of mixing during which chemical reactions can occur and before complete molecular homogeneity is achieved may be visualized as the molecular diffusion-controlled mixing of the smallest eddies in the turbulence energy dissipation spectrum. The smallest eddy size can vary over several orders of... 

From the experimentally determined energy spectrum we can obtain the mean turbulent energy per unit mass of fluid MTKE t), the turbulent energy dissipation rate per unit mass of fluid e t), and the integral length scale... 

The source terms on the right-hand sides of Eqs. (25)-(29) are defined as follows. In the momentum balance, g represents gravity and p is the modified pressure. The latter is found by forcing the mean velocity field to be solenoidal (V (U) = 0). In the turbulent-kinetic-energy equation (Eq. 26), Pk is the source term due to mean shear and the final term is dissipation. In the dissipation equation (Eq. 27), the source terms are closures developed on the basis of the form of the turbulent energy spectrum (Pope, 2000). Finally, the source terms... 

As discussed in Section 2.1, in high-Reynolds-number turbulent flows the scalar dissipation rate is equal to the rate of energy transfer through the inertial range of the turbulence energy spectrum. The usual modeling approach is thus to use a transport equation for the transfer rate instead of the detailed balance equation for the dissipation rate derived from (1.27). Nevertheless, in order to understand better the small-scale physical phenomena that determine e, we will derive its transport equation starting from (2.99). 

As in Section 2.1 for the turbulent energy spectrum, a model scalar energy spectrum can be developed to describe lop(n). However, one must account for the effect of the Schmidt number. For Sc < 1, the scalar-dissipation wavenumbers, defined by19... 

Note that as Re/, goes to infinity with Sc constant, both the turbulent energy spectrum and the scalar energy spectrum will be dominated by the energy-containing and inertial/inertial-convective sub-ranges. Thus, in this limit, the characteristic time scale for scalar variance dissipation defined by (3.55) becomes... 

The mean energy dissipation rate can be calculated directly from the energy spectrum using (1.332). Moreover, for isotropic turbulence, the energy dissipation rate and the micro scale are related as expressed by (1.323). Caution is required using the dissipation rate calculated from a turbulence model like the k-e model as in this model the dissipation quantity is merely a tuning variable for the shear stresses in a pipe and not necessarily a true physical dissipation rate. The latter approach requires validation for any applications. 

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