Velocity fluctuations, turbulent

Turbulent velocity fluctuations ultimately dissipate their kinetic energy through viscous effects. MacroscopicaUy, this energy dissipation requires pressure drop, or velocity decrease. The ener dissipation rate per unit mass is usually denoted . For steady ffow in a pipe, the average energy dissipation rate per unit mass is given by... 

The neglect of a low turbulence effect and a laminar flow is not justified in regions close to solid surfaces where the turbulent velocity fluctuations... 

A very fine space resolution is required to measure the gradient of turbulent velocity fluctuations and calculate turbulent dissipation directly from the definition [5, 6]. 

Where the Reynolds stress formula (2) and the universal law of the theory of isotropic turbulence apply to the turbulent velocity fluctuations (4), the relationship (20) for the description of the maximum energy dissipation can be derived from the correlation of the particle diameter (see Fig. 9). It includes the geometrical function F and thus provides a detailed description of the stirrer geometry in the investigated range of impeller and reactor geometry 0.225derived from many turbulence measurements, correlation (9). 

Hydrodynamic effects on suspended particles in an STR may be broadly categorized as time-averaged, time-dependent and collision-related. Time-averaged shear rates are most commonly considered. Maximum shear rates, and accordingly maximum stresses, are assumed to occur in the impeller region. Time-dependent effects, on the other hand, are attributable to turbulent velocity fluctuations. The relevant turbulent Reynolds stresses are frequently evaluated in terms of the characteristic size and velocity of the turbulent eddies and are generally found to predominate over viscous effects. 

The term numerical diffusion describes the effect of artificial diffusive fluxes which are induced by discretization errors. This effect becomes visible when the transport of quantities with small diffusivities [with the exact meaning of small yet to be specified in Eq. (42)] is considered. In macroscopic systems such small diffusivities are rarely found, at least when being looked at from a phenomenological point of view. The reason for the reduced importance of numerical diffusion in many macroscopic systems lies in the turbulent nature of most macro flows. The turbulent velocity fluctuations induce an effective diffusivity of comparatively large magnitude which includes transport effects due to turbulent eddies [1]. The effective diffusivity often dominates the numerical diffusivity. In contrast, micro flows are often laminar, and especially for liquid flows numerical diffusion can become the major effect limiting the accuracy of the model predictions. 

The quantity G of the effective mixing mass flux is determined by the turbulent velocity fluctuations at the bubble-layer edge. The distance of the edge of the bubble layer from the wall is taken as the distance at which the size of the turbulent eddies is k times the average bubble diameter. Weisman and Pei have determined empirically that k equals 2.28. Only a fraction of the turbulent velocity fluctuations produced are assumed to be effective in reaching the wall. The effective velocity fluctuations are those in which the velocity exceeds the average velocity away from the wall produced by evaporation heat flux q"b. At the bubble layer-core interface, the effective mass flux to the wall is computed as... 

In real life, the parcels or blobs are also subjected to the turbulent fluctuations not resolved in the simulation. Depending on the type of simulation (DNS, LES, or RANS), the wide range of eddies of the turbulent-fluid-flow field is not necessarily calculated completely. Parcels released in a LES flow field feel both the resolved part of the fluid motion and the unresolved SGS part that, at best, is known in statistical terms only. It is desirable that the forces exerted by the fluid flow on the particles are dominated by the known, resolved part of the flow field. This issue is discussed in greater detail in the next section in the context of tracking real particles. With a RANS simulation, the turbulent velocity fluctuations remaining unresolved completely, the effect of the turbulence on the tracks is to be mimicked by some stochastic model. As a result, particle tracking in a RANS context produces less realistic results than in an LES-based flow field. 

Note that the correction terms are proportional to fT and result from turbulent velocity fluctuations (represented by a gradient-diffusion model). For the multi-environment model the composition vector is defined by... 

Chigiefl211 found that turbulence in a liquid jet has important disturbing influences throughout the liquid flow. At the liquid surface, turbulent velocity fluctuations directly cause protuberances and roughness that result in direct stripping by surrounding air flow. Large eddy structures in the air flow penetrate into the liquid and... 

This term represents the spatial transport of e by turbulent velocity fluctuations. 

This relationship between the degree of diffusion by the turbulent velocity fluctuation and time is identical to the traditional general knowledge. 

On the other hand, the turbulent velocity fluctuations RMS v affect microscale processes, again variously estimated at 100 to 200 microns or less, on down to possibly the mixing length or possibly on down to the scale of molecular reaction. 

It since appears that the frequency and size of the turbulent velocity fluctuations are more compatible with phenomena on a molecular micron size scale In a mixing vessel. For example, in a publication Paul Treybal (4) shows the yield of a given reaction which had several alternate paths was determined by the RMS velocity fluctuations at the feed point. Paul used data from Schwartzberg Treybal (5) and Cutter to calculate RMS velocity at the feed point. 

Howarth (H15) developed an expression for collision efficiencies by assuming an analogy to bimolecular gas reactions. He assumed that a critical relative velocity IV exists along the lines of centers of two colliding drops which must be exceeded for a collision to result in a coalescence. By assuming that the three-dimensional Maxwell s equation describes the drop turbulent velocity fluctuations, he obtained the coalescence efficiency as the fraction of drops which have kinetic energy exceeding IV. Thus,... 

A new term is introduced, the so-called Reynolds stresses m-m). The overbar denotes a time average. This term is the correlation between the turbulent velocity fluctuations and uj, and it describes the transport of momentum in the mean flow due to turbulence. This term is difficult to model, and over the years a variety of turbulence models have been developed. Turbulence models are necessary for calculating time-averaged flow fields directly, without first having to calculate a fully time-dependent flow field and then doing time averaging. The use of turbulence models is therefore much more computationally efficient. A detailed discussion is beyond the scope of this entry, but it is important to note that not all turbulence models are equally suited for all types of flow. Table 1 summarizes the most common turbulence models and their properties. 

The critical Weber number We for the droplet disintegration can be obtained from the above relationship. It is the turbulent velocity fluctuations and not the shear rates, which are responsible for the droplet disintegration. Above a critical Weber number the droplet is unstable and breaks up. 

The Reynolds transport terms (i.e., the terms of the kind v Vy, T v y and c v y) are the mean rates of transport of momentum, heat, and mass across the corresponding CV-surfaces of the balanced element by the turbulent velocity fluctuations. This system of equations can not be solved because these terms are not known. We therefore need additional information regarding the Reynolds transport terms. We may postulate a model for the momentum transport terms, intending to relate the Reynolds transport stresses to the mean flow field variables. 

The drift velocity takes into account the dispersion effect due to the particle transport by the fluid turbulence. Prom the limiting case of particles with diameter tending towards zero, for which the drift velocity reduces to single turbulence correlation between the volumetric fraction of the dispersed phase and the turbulent velocity fluctuations of the continuous phase. The drift velocity v ritt is modeled as [33] ... 

Based on semi-empirical analysis, the fluid-particle turbulent dispersion tensor, Dgp, is expressed in terms of the covariance between the turbulent velocity fluctuations of the two phases and a fluid particle turbulent characteristic time ... 

Legisetty et al. [88] and Calabrese et al. [89] considered the effect of dispersed phase viscosity on drop breakage. In their models, attempts have been made to incorporate the effect of dispersed phase holdup on drop breakage by considering the turbulent velocity fluctuations to be damped by the dispersed phase drops as... 

If the continuous phase is a liquid, the main obstacle to coalescence is the drainage of the film of liquid in the small space in between the two particles. The efficiency is in these cases usually quantitied as a function (generally a negative exponential function) of the ratio of the characteristic time for droplet contact and film drainage. For example, in the case of small bubbles coalescing due to turbulent velocity fluctuations the coalescence kernel assumes the form (Buffo et al, 2012 Laakkonen et al, 2006 Petitti et al, 2010)... 

FIGURE 8-3 Three-dimensional distribution of root-mean-square turbulent velocity fluctuations upstream of a step (Kasagi and Matsunaga 1995). (Reprinted by permission of Butterworth/Heinemann.)... 

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