Turbulent Reynolds Stresses

FIGURE 9.14 Solids turbulent Reynolds stresses in a fluidized bed for 500- xm glass particles at ujumf - 2 

It may be of interest to note that summation of the normal components of the velocity correlations v iv J represents the solids kinetic energy, or granular temperature, which is a primary parameter in the kinetic theory of granular particles. 

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The flow pattern is ealeulated from eonservation equations for mass and mometum, in eombination with the Algebraie Stress Model (ASM) for the turbulent Reynolds stresses, using the Fluent V3.03 solver. These equations ean be found in numerous textbooks and will not be reiterated here. Onee the flow pattern is known, the mixing and transport of ehemieal speeies ean be ealeulated from the following model equation ... 

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. 

These turbulent momentum flux components are also called Reynolds stresses. Thus, the total stress in a Newtonian fluid in turbulent flow is composed of both viscous and turbulent (Reynolds) stresses ... 

Although Eq. (6-18) can be used to eliminate the stress components from the general microscopic equations of motion, a solution for the turbulent flow field still cannot be obtained unless some information about the spatial dependence and structure of the eddy velocities or turbulent (Reynolds) stresses is known. A classical (simplified) model for the turbulent stresses, attributed to Prandtl, is outlined in the following subsection. 

To render Eq. (5.62) solvable, it is necessary to provide an expression for the turbulent Reynolds stress. For isotropic turbulent flows, similar to the transport processes in laminar flows, a scalar turbulent viscosity /xT is defined using the Boussinesq formulation... 

To simulate the turbulent Reynolds stress, first we examine the dimensions of the Reynolds stress, the rate of strain, and the turbulent viscosity as follows... 

Figure 11.8 shows that the flow can be divided conceptually into three zones a laminar sublayer nearest the pipe wall, in which the shear stress is principally due toj viscous shear a turbulent core in the middle of the pipe, in which the shear stress is principally due to turbulent. Reynolds stresses and a layer between them, called the buffer layer, in which both viscous and Reynolds stresses are of the same order of magnitude. Good experimental measurements are difficult to make in the laminar sublayer and buffer layer, so there is some controversy over the best location for the boundaries shown in Fig. 11.8. Deissler [4] and coworkers place the buffer layer at a of 5 to 26, and Schlichting arid coworkers place it at a of 5 to about 70. Furthermore, current work seems to indicate that the location of the edge of these layers is not fixed in place but fluctuates up and down so these values indicate only the mean locations of these edges [5]. Thus, Fig. 11.8 may be too simple a picture of the actual behavior. Nonetheless, it provides a reasonable conceptual model and is able to correlate most of the available data with reasonable accuracy.

Later works by Miles [91-94]and Benjamin [84] showed that the evolution of wind stress component in phase with the wave slope arises from interaction of the surface perturbations and the mean turbulent airflow characterized by boundary-layer type velocity profiles. Their models are considered quasi-laminar, as the perturbations in turbulent Reynolds stresses induced by the surface perturbations were ignored. 

Reynolds Stress Models. Eddy viscosity is a useful concept from a computational perspective, but it has questionable physical basis. Models employing eddy viscosity assume that the turbulence is isotropic, ie, u u = u u = and u[ u = u u = u[ = 0. Another limitation is that the... 

Closure Models Many closure models have been proposed. A few of the more important ones are introduced here. Many employ the Boussinesq approximation, simphfied here for incompressible flow, which treats the Reynolds stresses as analogous to viscous stresses, introducing a scalar quantity called the turbulent or eddy viscosity... 

These extra turbulent stresses are termed the Reynolds stresses. In turbulent flows, the normal stresses -pu, -pv, and -pw are always non-zero beeause they eontain squared veloeity fluetuations. The shear stresses -pu v, -pu w, -pv w and are assoeiated with eorrelations between different veloeity eomponents. If, for instanee, u and v were statistieally independent fluetuations, the time average of their produet u v would be zero. However, the turbulent stresses are also non-zero and are usually large eompared to the viseous stresses in a turbulent flow. Equations 10-22 to 10-24 are known as the Reynolds equations. 

Turbulence modeling capability (range of models). Eddy viscosity k-1, k-e, and Reynolds stress. k-e and Algebraic stress. Reynolds stress and renormalization group theory (RNG) V. 4.2 k-e. low Reynolds No.. Algebraic stress. Reynolds stress and Reynolds flux. k- Mixing length (user subroutine) and k-e. 

An appropriate model of the Reynolds stress tensor is vital for an accurate prediction of the fluid flow in cyclones, and this also affects the particle flow simulations. This is because the highly rotating fluid flow produces a. strong nonisotropy in the turbulent structure that causes some of the most popular turbulence models, such as the standard k-e turbulence model, to produce inaccurate predictions of the fluid flow. The Reynolds stress models (RSMs) perform much better, but one of the major drawbacks of these methods is their very complex formulation, which often makes it difficult to both implement the method and obtain convergence. The renormalization group (RNG) turbulence model has been employed by some researchers for the fluid flow in cyclones, and some reasonably good predictions have been obtained for the fluid flow. 

Consequently, six additional unknowns, the Reynolds stresses obtained and the equations for turbulent flow beeome... 

Using turbulenee models, this new system of equations ean be elosed. The most widely used turbulenee model is the k-e model, whieh is based on an analogy of viseous and Reynolds stresses. Two additional transport equations for the turbulent kinetie energy k and the turbulent energy dissipation e deseribe the influenee of turbulenee... 

More advanced models, for example the algebraic stress model (ASM) and the Reynolds stress model (RSM), are not based on the eddy-viscosity concept and can thus account for anisotropic turbulence thereby giving still better predictions of flows. In addition to the transport equations, however, the algebraic equations for the Reynolds stress tensor also have to be solved. These models are therefore computationally far more complex than simple closure models (Kuipers and van Swaaij, 1997). 

DNS results are usually considered as references providing the same level of accuracy as experimental data. The maximum attainable Reynolds number (Re) in a DNS is, however, too low to duplicate most practical turbulent reacting flows, and hence, the use of DNS is neither to replace experiments nor for direct comparisons— not yet at least. However, DNS results can be used to investigate three-dimensional (3D) features of the flow (coherent structures, Reynolds stresses, etc.) that are extremely difficult, and sometimes impossible, to measure. One example of such achievement for nonreacting... 

Ekambra etal. [21] compared the results from ID, 2D, and 3D simulations of a bubble column with experimental results. They obtained similar results for holdup and axial velocity, while eddy viscosity, Reynolds stresses, and energy dissipation were very different in the three simulations as shown in Figure 15.7. This example also illustrates the importance of selecting the right variables for model vahdation. A 2D model will yield good results for velocity but will predict all variables based on turbulent characteristics poorly. 

In the velocity field of the determining eddies, which is characterized by the turbulent fluctuation velocity the particles experience a dynamic stress according to the Reynolds stress Eq. (2) ... 

To avoid gas-liquid mass transfer Hmitation, which would have a negative influence on productivity, in correctly operated bioreactors there are turbulent flow conditions with more or less pronounced turbulence, for which the Reynolds stress formula (Eq. (2)) can be used. Whereas, as a rule there is fully developed turbulent flow in technical apparatuses (see condition (6) and explanations in Sect. 8), this is frequently not the case in laboratory fermenters. Equations (3) and (4) are then only valid to a limited extent. 

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). 

For certain conditions, the hydrodynamic stresses generated by the very rapid fluctuations in turbulent flow, the so-called Reynolds stresses, can be estimated as [70] ... 

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