Turbulent dissipation rate, transport equation

The Reynolds-averaged approach is widely used for engineering calculations, and typically includes models such as Spalart-Allmaras, k-e and its variants, k-co, and the Reynolds stress model (RSM). The Boussinesq hypothesis, which assumes pt to be an isotropic scalar quantity, is used in the Spalart-Allmaras model, the k-s models, and the k-co models. The advantage of this approach is the relatively low computational cost associated with the computation of the turbulent viscosity, fit. For the Spalart-Allmaras model, one additional transport equation representing turbulent viscosity is solved. In the case of the k-e and k-co models, two additional transport equations for the turbulence kinetic energy, k, and either the turbulence dissipation rate, s, or the specific dissipation rate, co, are solved, and pt is computed as a function of k and either e or co. Alternatively, in the RSM approach, transport equations can be solved for each of the terms in the Reynolds stress tensor. An additional scale-determining equation (usually for s) is also required. This means that seven additional transport equations must be solved in 3D flows. 

At the next level of complexity, a second transport equation is introduced, which effectively removes the need to fix the mixing length. The most widely used two-equation model is the k-e model wherein a transport equation for the turbulent dissipation rate is... 

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

The description is based on the previously defined single-particle (Lagrangian) or one-point (Eulerian) joint velocity-composition (micro-)PDF, /(r,yr). As mentioned in Section 12.4.1, in the one-point description no information on the local velocity and scalar (species concentrations, temperature,. ..) gradients and on the frequency or length scale of the fluctuations is included and the related terms require closure models. The scalar dissipation rate model has to relate the micro-mixing time to the turbulence field (see (12.2-3)), either directly or via a transport equation for the turbulence dissipation rate e. A major advantage is that the reaction rate is a point value and its behavior and mean are described exactly by a one-point PDF, even for arbitrarily complex and nonlinear reaction kinetics. 

In the RANS-approach, turbulence or turbulent momentum transport models are required to calculate the Reynolds-stresses. This can be done starting from additional transport equations, the so-called Reynolds-stress models. Alternatively, the Reynolds-stresses can be modeled in terms of the mean values of the variables and the turbulent kinetic energy, the so-called turbulent viscosity based models. In either way, the turbulence dissipation rate has to be calculated also, as it contains essential information on the overall decay time of the velocity fluctuations. In what follows, the more popular models based on the turbulent viscosity are focused on. A detailed description of the Reynolds-stress models is given in Annex 12.5.l.A which can be downloaded from the Wiley web-page. 

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 ... 

In whichever approach, the common denominator of most operations in stirred vessels is the common notion that the rate e of dissipation of turbulent kinetic energy is a reliable measure for the effect of the turbulent-flow characteristics on the operations of interest such as carrying out chemical reactions, suspending solids, or dispersing bubbles. As this e may be conceived as a concentration of a passive tracer, i.e., in terms of W/kg rather than of m2/s3, the spatial variations in e may be calculated by means of a usual transport equation. 

The turbulent kinetic energy and its dissipation rate are obtained from the adapted transport equations. 

The transport equations for the turbulent kinetic energy, k, and the turbulence dissipation, e, in the RNG k-s model are again defined similar to the standard k-s model, now utilizing the effective viscosity defined through the RNG theory. The major difference in the RNG k-s model from the standard k-s model can be found in the e balance where a new source term appears, which is a function of both k and s. The new term in the RNG k- s model makes the turbulence in this model sensitive to the mean rate of strain. The result is a model that responds to the effect of strain and the effect of streamline curvature,... 

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). 

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). 

For high-Reynolds-number homogeneous turbulent flows,22 the right-hand side of the dissipation-rate transport equation thus reduces to the difference between two large terms 23... 

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. 

Using (3.130), the transport equation for the scalar dissipation rate in high-Reynolds-number homogeneous turbulence becomes... 

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. 

In order to close the set of modeled transport equations, it is necessary to estimate turbulent viscosity or if the k-e model is used, the turbulent kinetic energy, k and turbulent energy dissipation rate, s. The modeled forms of the liquid phase k and s transport equations can be written in the following general format (subscript 1 denotes... 

The transport equation for the dissipation rate of the gas-phase turbulent kinetic energy is given by [122, 126, 8] ... 

The transport equation for the turbulent energy dissipation rate in cylindrical coordinates is written as ... 

The particular transport equation for the liquid phase turbulent energy dissipation rate is written ... 

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