Turbulent dissipation rate

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). 

In order to simplify the notation, we will first denote the fluctuating velocity gradient by 

Differentiating both sides of (2.99) with respect to x then yields 

We next denote the random dissipation rate by18 e = vgjigji, 

The first term on the right-hand side of (2.121) can be rewritten using the continuity equation as 

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As mentioned before in Eq. (3), the most common source of SGS phenomena is turbulence due to the Reynolds number of the flow. It is thus important to understand what the principal length and time scales in turbulent flow are, and how they depend on Reynolds number. In a CFD code, a turbulence model will provide the local values of the turbulent kinetic energy k and the turbulent dissipation rate s. These quantities, combined with the kinematic viscosity of the fluid v, define the length and time scales given in Table I. Moreover, they define the local turbulent Reynolds number ReL also given in the table. 

DEFINED IN TERMS OF THE TURBULENT KINETIC ENERGY k, AND TURBULENT DISSIPATION RATE , AND THE... 

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. 

In Chapter 2, we show that the turbulence integral time scale can be defined in terms of the turbulent kinetic energy k and the turbulent dissipation rate / by r = /[Pg.26]

The relationship between the various length scales can be best understood by looking at their dependence on the turbulence Reynolds number defined in terms of the turbulent kinetic energy k, the turbulent dissipation rate e, and the kinematic viscosity v by... 

Thus, like the turbulence dissipation rate, the scalar dissipation rate of an inert scalar is primarily determined by the rate at which spectral energy enters the scalar dissipation range. Most engineering models for the scalar dissipation rate attempt to describe (kd, t) in terms of one-point turbulence statistics. We look at some of these models in Chapter 4. 

Unlike the turbulence dissipation rate tensor, which is isotropic at high Reynolds number, the joint scalar dissipation rate tensor is usually highly anisotropic. Indeed, when r< = T, it is often the case for inert scalars that eap = eaa = , so that the joint scalar dissipation rate tensor is singular. 

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... 

One common difficulty when applying the E-model is the need to know the turbulent dissipation rate e for the flow. Moreover, because e will have an inhomogeneous distribution in most chemical reactors, the problem of finding e a priori is non-trivial. In most... 

Models conditioned on the turbulent dissipation rate attempt to describe non-stationary effects due to the fluctuating strain-rate field, and thus should be adequate for flamelet applications which require a model for the mixture-fraction dissipation rate at the stoichiometric surface. 

Studies on thermodynamic restrictions on turbulence modeling show that the kinetic energy equation in a turbulent flow is a direct consequence of the first law of thermodynamics, and the turbulent dissipation rate is a thermodynamic internal variable. The principle of entropy generation, expressed in terms of the Clausius-Duhem and the Clausius-Planck inequalities, imposes restrictions on turbulence modeling. On the other hand, the turbulent dissipation rate as a thermodynamic internal variable ensmes that the mean internal dissipation will be positive and the thermodynamic modeling will be meaningful. 

Dimensionless Dimensionless Turbulent dissipation rate J/(kg s) ft Ibftdbm s)... 

These relationships are valid for isolated bubbles moving under laminar flow conditions. In the case of turbulent flow, the effect of turbulent eddies impinging on the bubble surface is to increase the drag forces. This is typically accounted for by introducing an effective fluid viscosity (rather than the molecular viscosity of the continuous phase, yUf) defined as pi.eff = Pi + C pts, where ef is the turbulence-dissipation rate in the fluid phase and Cl is a constant that is usually taken equal to 0.02. This effective viscosity, which is used for the calculation of the bubble/particle Reynolds number (Bakker van den Akker, 1994), accounts for the turbulent reduction of slip due to the increased momentum transport around the bubble, which is in turn related to the ratio of bubble size and turbulence length scale. However, the reader is reminded that the mesoscale model does not include macroscale turbulence and, hence, using an effective viscosity that is based on the macroscale turbulence is not appropriate. 

In dilute particle-laden turbulent flow, the fluid shear rate acting on the particles is approximated as Gf = where f is the turbulence-dissipation rate in the continuous phase,... 

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