Dissipation rate, turbulent flow

VV( or q from equation (48) in the flame-sheet approximation. Since knowledge of the joint or conditioned functions is practically absent [27], statistical independence is often hypothesized or else it is merely assumed, less restric-tively, that the conditioned-mean dissipation equals the unconditioned mean. A small amount of data is available on unconditioned-average rates of scalar dissipation in turbulent flows (see discussions in [83]-[86]), and additional measurements are being made. These results allow estimates of Xc to be made, even though accurate calculations are beyond current capabilities. 

This response time should be compared to the turbulent eddy lifetime to estimate whether the drops will follow the turbulent flow. The timescale for the large turbulent eddies can be estimated from the turbulent kinetic energy k and the rate of dissipation e, Xc = 30-50 ms, for most chemical reactors. The Stokes number is an estimation of the effect of external flow on the particle movement, St = r /tc. If the Stokes number is above 1, the particles will have some random movement that increases the probability for coalescence. If St 1, the drops move with the turbulent eddies, and the rates of collisions and coalescence are very small. Coalescence will mainly be seen in shear layers at a high volume fraction of the dispersed phase. 

Figures 12 and 13 show the effects of agitation and time of exposure on suspensions of biological materials in bioreactors [61]. In turbulent flow the energy dissipation rate per unit mass, e, of a stirred bioreactor is normally expressed by the following equation ...

A survey of the published literature indicates that the ratio of the maximum to mean energy dissipation rate in the vessel, Smax/ m can vary substantially but typically in the range 10 to 100 [85]. Recent measurements [100] of the turbulent flow properties with a range of impellers and vessel configurations indicate that the differences between the reported ratios of Smax/Cm re partly due to differences in the geometrical variables. For example, detailed factorial designs of experiments showed significant effects of impeller diameter to tank diameter ratio and off-bottom clearance to impeller diameter ratio on the value of emax/Cm-... 

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. 

Fig. 12. Snapshot from a two-phase DNS of colliding particles in an originally fully developed turbulent flow of liquid in a periodic 3-D box with spectral forcing of the turbulence. The particles (in blue) have been plotted at their position and are intersected by the plane of view. The arrows denote the instantaneous flow field, the colors relate to the logarithmic value of the nondimensional rate of energy dissipation.

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. 

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 turbulent flow, there is direct viscous dissipation due to the mean flow this is given by the equivalent of equation 1.98 in terms of the mean values of the shear stress and the velocity gradient. Similarly, the Reynolds stresses do work but this represents the extraction of kinetic energy from the mean flow and its conversion into turbulent kinetic energy. Consequently this is known as the rate of turbulent energy production ... 

Table 2.1. The principal length and time scales, and Reynolds numbers characterizing a fully developed turbulent flow defined in terms of the turbulent kinetic energy k, turbulent dissipation rate e, and the kinematic viscosity v. 

Thus, only the normal Reynolds stresses (i = j) are directly dissipated in a high-Reynolds-number turbulent flow. The shear stresses (i / j), on the other hand, are dissipated indirectly, i.e., the pressure-rate-of-strain tensor first transfers their energy to the normal stresses, where it can be dissipated directly. Without this redistribution of energy, the shear stresses would grow unbounded in a simple shear flow due to the unbalanced production term Vu given by (2.108). This fact is just one illustration of the key role played by 7 ., -in the Reynolds stress balance equation. 

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... 

We are essentially assuming that the small scales are in dynamic equilibrium with the large scales. This may also hold in low-Reynolds-number turbulent flows. However, for low-Reynolds-number flows, one may need to account also for dissipation rate anisotropy by modeling all components in the dissipation-rate tensor s j. 

In a fully developed turbulent flow,22 the scalar spectral transfer rate in the inertial-convective sub-range is equal to the scalar dissipation rate, i.e., T k) = for /cei < < Kn. Likewise, when Sc 1, so that a viscous-convective sub-range exists, the scalar trans-... 

Equation (3.82) illustrates the importance of the scalar spectral energy transfer rate in determining the scalar dissipation rate in high-Reynolds-number turbulent flows. Indeed, near spectral equilibrium, 7 (/cd, 0 (like Tu(kDi, 0) will vary on time scales of the order of the eddy turnover time re, while the characteristic time scale of (3.82) is xn <[Pg.99]

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. 

As for the scalar dissipation rate, the remaining terms in (3.151) will be large in high-Reynolds-number turbulent flows. The vortex-stretching term Vf is defined by... 

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