Turbulent dissipation rate derivation

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). 

A remaining unclosed term in (12.5.1-7) or (12.5.1-10) is the turbulence dissipation rate e. A transport equation for the latter can be rigorously derived, but it is difficult to close. An alternative semi-empirical transport equation is commonly used. It is derived in analogy with the turbulent kinetic energy transport equation and takes the form ... 

Applying a turbulent energy cascade model, Ertesvag and Magnussen [2000] derived expressions for nY and ft in terms of the turbulent kinetic energy k and the turbulence dissipation rate s ... 

The material covered in the appendices is provided as a supplement for readers interested in more detail than could be provided in the main text. Appendix A discusses the derivation of the spectral relaxation (SR) model starting from the scalar spectral transport equation. The SR model is introduced in Chapter 4 as a non-equilibrium model for the scalar dissipation rate. The material in Appendix A is an attempt to connect the model to a more fundamental description based on two-point spectral transport. This connection can be exploited to extract model parameters from direct-numerical simulation data of homogeneous turbulent scalar mixing (Fox and Yeung 1999). 

A spectral model similar to (3.82) can be derived from (3.75) for the joint scalar dissipation rate eap defined by (3.139), p. 90. We will use these models in Section 3.4 to understand the importance of spectral transport in determining differential-diffusion effects. As we shall see in the next section, the spectral interpretation of scalar energy transport has important ramifications on the transport equations for one-point scalar statistics for inhomogeneous turbulent mixing. 

The first factor occurs even in homogeneous flows with two inert scalars, and is discussed in Section 3.4. The second factor is present in nearly all turbulent reacting flows with moderately fast chemistry. As discussed in Chapter 4, modeling the joint scalar dissipation rate is challenging due to the need to include all important physical processes. One starting point is its transport equation, which we derive below. 

The transport equation for the rate of turbulent dissipation can also be derived as (Problem... 

Yim and Shamlou [22] provide a comprehensive list of energy dissipation rates in each unit process used in the manufacture of biomolecules (Table 4.1-1). Worden [23] showed that degradation commences when the size of this small-scale turbulent structure becomes equal to or less than the molecule s hydrodynamic diameter (Figure 4.1-2). The scale of these smallest eddies t) can be calculated based on the energy dissipation rate e and the diffusivity v using the relationship derived by Kolmogorov in 1941 [33] ... 

The power density Py is the characteristic quantity of turbulent flow. It determines the size of the smallest eddies and the intensity of microturbulence. In addition, it is a measure of the shear intensity in laminar flows or the intensity of cavitation in ultrasonic fields (see above). The power input P in the dispersion zone can be derived from the pressure drop (e.g. in pipes and nozzles) or can be measured caloricafly (e.g. for rotor-stator systems and ultrasonication Pohl 2005 Kuntzsch 2004). Additionally, P can be roughly approximated by the electric power consumption of the dispersion machine (e.g. for ultrasonication Mandzy et al. 2005 Sauter et al. 2008), even though the real values may be lower by a factor of 2 to 5. A further source of uncertainty is the volume of the dispersion zone (Vdisp). since the stress intensities are not uniformly distributed in dispersion apparatuses. In particular, this applies to agitated vessels, where the highest dissipation rates are obtained in the vicinity of the stirring instmment (Henzler and Biedermann 1996),... 

It was shown by Taylor [ 160] that an analysis of the dissipation term occurring within the turbulent kinetic energy balance equation (derived later in this chap) shows that in isotropic turbulence the energy dissipation rate is equal to ... 

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