Stresses turbulent

The quantity k is related to the intensity of the turbulent fluctuations in the three directions, k = 0.5 u u. Equation 41 is derived from the Navier-Stokes equations and relates the rate of change of k to the advective transport by the mean motion, turbulent transport by diffusion, generation by interaction of turbulent stresses and mean velocity gradients, and destmction by the dissipation S. One-equation models retain an algebraic length scale, which is dependent only on local parameters. The Kohnogorov-Prandtl model (21) is a one-dimensional model in which the eddy viscosity is given by... 

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. 

Meshalin, V. S. 1974. About turbulent stress in impinging jets. In Mechanics of (jases and Fluids. Moscow. 

The flow of jets becomes turbulent at much lower Re numbers than channel flows. Calculating the stress from the mean velocity profiles does not reflect the true situation in turbulent flow. As in the case in most bioreactors, the maximum turbulent stress is determined by the turbulence, which can be calculated using Eqs. (2)-(4). It occurs in free jets after the nozzle, at the edge of the mixing zone. The following is generally valid ... 

Since in the case of turbulent stress the ratio of particle diameter dp to length scale of turbulence qp is decisive for the stress regime (see Fig. 1) the model particle systems must have properties which guarantee dp/qp values which are in the same range as for the biological particle systems. 

Figure 11 shows the reference floe diameter for viscometers as a function of shear stress and also the comparison with the results for stirred tanks. The stress was determined in the case of viscosimeters from Eq. (13) and impeller systems from Eqs. (2) and (4) using the maximum energy density according to Eq. (20). For r > 1 N/m (Ta > 2000), the disintegration performance produced by the flow in the viscosimeter with laminar flow of Taylor eddies is less than that in the turbulent flow of stirred tanks. Whereas in the stirred tank according to Eq. (4) and (16b) the particle diameter is inversely affected by the turbulent stress dp l/T, in viscosimeters it was found for r > 1.5 N/m, independently of the type (Searle or Couette), the dependency dp l/ pi (see Fig. 11). 

Viscosimeter flow produces less stress than technical reactors (see Sect. 6.3.3). From the results with the floccular particle system it can be derived the following relationship (30). It estimates the turbulent stress of a technical, fully baffled stirred reactor which leads to the same damage of particles as the viscosimeter flow with the shear stress r. 

The possibility of correlating these fermentation parameters with the turbulent stress equation shows again that obviously similar relationships exist for both the biological systems and the model particle systems used here. 

Fig. 24. Influence of impeller type and working conditions on productivity P/Xt and growth rate p of Penicillin G batch fermentations with Penicillium chrysogenum (left hand diagram data from [60]) and correlation by using the turbulent stress Tt corresponding to Eq. (28) (right hand diagram) symbol explanations see Fig. 23...

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. 

Precisely owing to the continuum description of the dispersed phase, in Euler-Euler models, particle size is not an issue in relation to selecting grid cell size. Particle size only occurs in the constitutive relations used for modeling the phase interaction force and the dispersed-phase turbulent stresses. 

The anisotropy tensor is related to the turbulent stresses, of course, and is defined as... 

In these equations, Gk is the generation of turbulent kinetic energy, k, due to turbulent stress, and is defined by... 

For the buffer region, 5 [Pg.92]

Vedula, P., P. K. Yeung, and R. O. Fox (2001). Dynamics of scalar dissipation in isotropic turbulence A numerical and modeling study. Journal of Fluid Mechanics 433, 29-60. Verman, B., B. Geurts, and H. Kuertan (1994). Realizability conditions for the turbulent stress tensor in large-eddy simulations. Journal of Fluid Mechanics 278, 351-362. Vervisch, L. (1991). Prise en compte d effets de cinetique chimique dans lesflammes de diffusion turbulente par Tapproche fonction densite de probabilite. Ph. D. thesis, Universite de Rouen, France. 

The description of small scale turbulent fields in confined spaces by fundamental approaches, based on statistical methods or on the concept of deterministic chaos, is a very promising and interesting research task nevertheless, at the authors knowledge, no fundamental approach is at the moment available for the modeling of large-scale confined systems, so that it is necessary to introduce semi-empirical models to express the tensor of turbulent stresses as a function of measurable quantities, such as geometry and velocity. Therefore, even in this case, a few parameters must be adjusted on the basis of independent measures of the fluid dynamic behavior. In any case, it must be underlined that these models are very complex and, therefore, well suited for simulation of complex systems but neither for identification of chemical parameters nor for online control and diagnosis [5, 6],... 

The difference between this equation for turbulent flow and the Navier-Stokes equation for laminar flow is the Reynolds stress/turbulent stress term —pujuj appears in the equation of motion for turbulent flow. This equation of motion for turbulent flow involves non-linear terms, and it is impossible to be solved analytically. In order to solve the equation in the same way as the Navier-Stokes equation, the Reynolds stress or fluctuating velocity must be known or calculated. Two methods have been adopted to avoid this problem—phenomenological method and statistical method. In the phenomenological method, the Reynolds stress is considered to be proportional to the average velocity gradient and the proportional coefficient is considered to be turbulent viscosity or mixing length ... 

All of these relations contains terms involving statistical correlations among various products of fluctuating velocity, pressure, and stress terms. This renders them considerably more complex than their laminar flow counterparts. Reynolds succeeded in partially sol ving this dilemma by the expedient of introducing the turbulent stress tensor f, defined by... 

The presence of these turbulent stress terms can be derived using a more physical approach. To do this, consider, as before, a control volume of the type shown in Fig. 2.11 through which fluid is flowing, the flow being turbulent. 

Thus, the presence of the fluctuating turbulent velocity components causes the momentum transfer rate to be different from p X (mean velocity)2 x dA. But in applying the momentum conservation principle to the control volume the presence of additional momentum transfer is the equivalent of an additional force on the face of the control volume in the opposite direction to the momentum transfer. Thus, the additional momentum transfer due to the fluctuating velocity leads to an equivalent stress, i.e., force per unit area, of value pu 2 and this is what is termed the turbulent stress on the face. 

Therefore, since (-kdT/dx) and (-k3T/dy) are the time-averaged heat conduction rates per unit area in the x- and y-directions respectively, it will be seen that the effects of the additional turbulence terms are the same as an increase in the heat transfer rate. For this reason, these extra terms pcpu T and pcpv V are often termed the turbulent heat transfer terms. Their presence can be demonstrated in a more physical maimer using the same line of reasoning as was adopted in the discussion of the turbulent stresses. For examole. considering rhe plpm nt chnum in... 

Lastly, it is assumed that turbulent stress is proportional to, vi.e., that ... 

If the turbulent stress in the production term is written in terms of the eddy viscosity this equation becomes ... 

The presence of the solid wall has a considerable influence on the turbulence structure near the wall. Because there can be no flow normal to the wall near the wall, v decreases as the wall is approached and as a result the turbulent stress and turbulent heat transfer rate are negligible in the region very near the wall. This region in which the effects of the turbulent stress and turbulent heat transfer rate can be neglected is termed the sublayer or, sometimes, the laminar sublayer [1],[2], [26],[27],[28],[29]. In this sublayer ... 

For y > 5 the turbulence stress and heat transfer rate become important However, near the wall the total shear stress and total heat transfer rate will remain effectively constant and equal to the wall shear stress and wall heat transfer rate, respectively. 

Thus, Eq. (5.85) applies from the wall out to y+ =5 while Eq. (5.89) applies for y+ > 30. Between y+ = 5 and y+ — 30, where both the molecular and the turbulent stresses are important, experiments indicate that the velocity distribution is given by ... 

It is thus necessary to determine the value of rjpu. To do thh, it is noted that the turbulent stress is kero at the wall and, therefore, that ... 

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