Turbulent energy production

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 ... 

It was noted earlier that the Reynolds stress is negative when the mean velocity gradient is positive and vice versa, consequently turbulent energy production is always positive. Very close to the wall, the Reynolds stress... 

This equation contains three new terms, namely flux of scalar variance, production of variance and dissipation of scalar variance, which require further modeling to close the equation. The flux terms are usually closed by invoking the gradient diffusion model (with turbulent Schmidt number, aj, of about 0.7). This modeled form is already incorporated in Eq. (5.21). The variance production term is modeled by invoking an analogy with turbulence energy production (Spalding, 1971) ... 

From the above two figures it is evident that the turbulence in streamwise direction is increased as confirmed by others (See Rudd [15],for example). In spanwise direction however it is significantly decreased leading to increased anisotropy of normal Reynolds stresses,Maksimovid [9]. It is speculated that "decoupling" of fluctuations of these two velocity components is a direct consequence of different shape of vortices in the polymer flow. "Decoupled" velocity fluctuations decrease the turbulent energy production. (Khabakhpasheva et al [16]). 

The balanced equation for turbulent kinetic energy in a reacting turbulent flow contains the terms that represent production as a result of mean flow shear, which can be influenced by combustion, and the terms that represent mean flow dilations, which can remove turbulent energy as a result of combustion. Some of the discrepancies between turbulent flame propagation speeds might be explained in terms of the balance between these competing effects. 

The standard wall function is of limited applicability, being restricted to cases of near-wall turbulence in local equilibrium. Especially the constant shear stress and the local equilibrium assumptions restrict the universality of the standard wall functions. The local equilibrium assumption states that the turbulence kinetic energy production and dissipation are equal in the wall-bounded control volumes. In cases where there is a strong pressure gradient near the wall (increased shear stress) or the flow does not satisfy the local equilibrium condition an alternate model, the nonequilibrium model, is recommended (Kim and Choudhury, 1995). In the nonequilibrium wall function the heat transfer procedure remains exactly the same, but the mean velocity is made more sensitive to pressure gradient effects. 

The growth of axial turbulence in the flame zone of these ducted systems is attributed to the mean velocity gradient resulting from the combustion. The production of turbulence energy by shear depends on the product of the mean velocity gradient and the Reynolds stress. Such stresses provide the most plausible mechanism for the modest growth in turbulence observed. 

To understand the principal idea of Deacon s model we have to remember the key assumption of the film model according to which a bottleneck boundary is described by an abrupt drop of diffusivity, for instance, from turbulent to molecular conditions (see Fig. 19.3a). Yet, theories on turbulence at a boundary derived from fluid dynamics show that this drop is gradual and that the thickness of the transition zone from fully turbulent to molecular conditions depends on the viscosity of the fluid. In Whitman s film model this effect is incorporated in the film thicknesses, 8a and 8W (Eq. 20-17). In addition, the film thickness depends on the intensity of turbulent kinetic energy production at the interface as, for instance, demonstrated by the relationship between wind velocity and exchange velocity (Figs. 20.2 and 20.3). 

Now we turn our attention to flowing waters. Here the physics of the boundary is influenced by two kinds of motion, the motions induced by the wind and the water currents, respectively. The latter will be extensively discussed in Chapter 24. At this point it is sufficient to introduce the most important concept in fluid dynamics to quantify the intensity of turbulent motion and to assess the relative importance of several simultaneous processes of turbulent kinetic energy production. 

In this model, the velocity disturbance by the particle is from both the wake behind the particle (Rep > 20) and the vortex shedding (Rep > 400). Hence, the changes in the kinetic energy associated with the turbulence production are proportional to the difference between the squares of the two velocities and to the volume where the velocity disturbance originates. It is further assumed that the wake is half of a complete ellipsoid, with base diameter of dp (same as the particle diameter) and wake length of /w. Thus, the total energy production of the gas by the particle wake or vortex shedding is... 

The numerator is the rate of production of turbulence energy, and the denomenator is the rate of dissipation of mechanical energy by the mean field. The turbulent viscosity can therefore be described in terms of the rate of turbulence production. 

This form is appealing because the first term in F.-,/2 can be interpreted as a gradient diffusion of turbulent kinetic energy, and the second is negative-definite (suggestive of dissipation of turbulence energy). However, the rate of entropy production is proportional to... 

In the so-called log region of turbulent boundary layers, the turbulence energy is essentially determined by a delicate balance between the production and dissipation terms in Eq. (23). With q = u /k and I = ny [see Eqs. (8) and (42) ] a balance between the first two terms on the right in Eq. (40) gives... 

The value of Km depends on the properties of the mean flow at a particular location and time. To account for the contribution of thermal stratification (buoyancy) to the production or suppression of turbulent energy. Km is taken to be a function of the local value of the flux Richardson number, which expresses the ratio of the rate of generation of energy by buoyancy forces to the rate of generation of energy by the turbulent momentum fluxes. In this approach the influence of the past history of the turbulence on velocity field is not considered the approach is termed a local theory. 

Fig. 3 Impact of different feed positions on the precipitation of barium sulfate. The selectivity to by-product as a percent of reactant is shown for feed into zones of high and low turbulent energy dissipation. The impeller speed and reactant addition time were held constant. More by-product is formed at feed points where the local mixing is slow.

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