Reynolds shear stress

BalHca and Ryu [158] correlated reductions in cell yield in Datura stramonium suspensions with the increased Reynolds stresses associated with higher aeration rates in a 1.2-1 ALR. A more recent study [159] of C. roseus suspensions cultivated in a 1.5-1 bubble column showed that the increased bubble sizes associated with both larger sparger pores and higher aeration rates caused a reduction in system performance. Here, also, it was postulated that the effects were due to increased Reynolds shear stresses in the flow field. However, it was not possible to rule out gas-stripping effects. 

With the aid of the two-color Laser-Doppler-Anemometry (LDA), Bewersdorff was able to measure the axial and the radial turbulence intensities simultaneously and also the Reynolds shear stresses. The injection of polymer results in a damping of both intensities in the region of their maxima. In his Reynolds shear stress measurements he showed that the polymer injection results in a drastic damping, and the stress maximum is shifted towards the center of the pipe. In a homogeneous polymer solution the maximum of the Reynolds shear stress remains in the same position-as for water. Only in the region of the buffer zone are the shear stresses reduced. 

When a fluid is in turbulent flow past a rigid surface, fluctuations of velocity in the direction normal to the surface are inhibited, and very close to the surface they may he negligible. Then the Reynolds shear stress is small compared with the viscous stresses, and it has been common to describe the region as a laminar sublayer. In fact, turbulent fluctuations of velocity in planes parallel to the wall are considerable in comparison with the mean velocity. 

Menzel, T, Weide, T., Staudacher, O., Wein, O. and Onken, U. (1990), Reynolds shear stress for modelling of bubble column reactors, Ind. Eng. Chem. Res., 29, 988-994. 

A three dimensional turbulent flow field in unbaffled tank with turbine stirrer or 6-paddle stirrer was numerically simulated by the method of finite volume elements [80], whereas in the case of free surface the vortex profile was also determined using iterative techniques. The prediction of the velocity and turbulence fields in the whole tank and the stirrer power was compared with literature data and their own results. Of the two simulation techniques used, turbulent eddy-viscosity/zc-e turbulence model and the DS model (differential 2. order shear stress), only the latter produced satisfactory results. In particular it proved that fluctuating Coriolis forces have to be taken into account by source terms in the transport equation for the Reynolds shear stress. 

In Cartesian coordinates the Reynolds shear stress pv Vy represents a flux of rr-momentum in the direction of y. Prandtl assumed that this momentum was transported by discrete lumps of fluid, which moved in the y direction over a distance I without interaction conserving the momentum and then mixed with the existing fluid at the new location. The mixing length, /, is supposed to be a variable analogous to the mean free path of kinetic theory in this process. 

Menzel T, in der Weide T, Staudacher O, Wein O, Onken U (1990) Reynolds Shear Stress for Modeling of Bubble Column Reactors. Ind Eng Chem Res 29 988-994... 

Figure 4. Time-averaged profiles of turbulence intensities, and , and Reynolds shear stress, , for the middle section of 15 cm bubble column at U up = I cm/s. (Reproduced with permission from reference 12.

Reynolds shear stresses arise out of the correlation of turbulent fluctuations... 

Turbulence in Microchannels, Fig. 1 (a) Rms velocity fluctuations and (b) Reynolds shear stress as measured by microPIV for turbulent flow in a 536 )im diameter glass microtube... 

Velocity Fluctuations and Reynolds Stresses Li and Olsen [9, 10] were the first researchers to measure profiles of velocity fluctuations in turbulent microchannel flow. They measured streamwise and transverse velocity fluctuations and Reynolds shear stresses for a range of Reynolds numbers spanning the laminar through fully turbulent regime. They found good agreement between their measured fluctuations and Reynolds shear stresses and values reported for macroscale turbulent duct flow. 

Natrajan et al. [12] collected velocity fluctuation and Reynolds shear stress data for turbulent flow in round microtubes for Re = 4,500 (well within the fully turbulent flow regime) and compared their experimental results with the results of a direct numerical simulation of turbulent pipe flow at a comparable Reynolds number. The comparison of these results is shown in Fig. 1,... 

The legitimacy of employing Blasius type models for the shear stresses in stratified flows was checked in several studies. Kowalski made direct measurements of the Reynolds shear stress in the gas for horizontal stratified flow in pipes and found that the gas-wall friction factors are well approximated by the Blasius equation provided that the hydraulic diameter is utilized [64]. For the liquid phase, Andritsos and Hanratty [28] found that the use of the Blasius equation to calculate introduces some error. However, improvements achieved by using a more complicated model for which is based on velocity profile and eddy viscosity concepts, were found to be of mild effect on the integral flow characteristics. 

The relation of the general dispersion Equations 34 to dynamic waves is derived here by recalling that a pure dynamic wave occurs whenever the net force on the flowing fluids is produced only by concentration gradients (and is independent of the insitu concentration, Wallis [74]). In this case, the quasi-steady shear stress terms on the rhs of the combined momentum equation, which are functions of the insitu concentration, are ignored, whereby AF is considered as identically zero. However, the dynamic interfacial shear stress term, which is proportional to the concentration gradient, evolves from the Reynolds shear stresses in the turbulent field and is retained. The general dispersion Equation 34, with V = 0, becomes ... 

The diagonal components u, v, and w denote twice the average kinetic energy per imit mass of the respective velocity components. The nondiagonal terms u v, u w, and v w denote negative components of turbulent and Reynolds shear stress. 

Using the assumption that the fluid has Newtonian behavior, the fluctuation of Reynolds shear stress yields... 

The computed turbulent axial and radial normal stresses and shear stresses increase with increasing superficial gas velocity. Axial normal stresses are considerably higher than their radial counterpart and both exceed Reynolds shear stresses. The maximum in Reynolds shear stress increases remarkably as the gas velocity is raised from 6 cm/s to 10 cm/s and its location is in the neighborhood of the inversion point for the axial velocity profile. 

Reynolds shear stress, also variance of the spreading of particles operator (differential or other)... 

Compared with the bubble characteristics, the information on the liquid flow characteristics specified by the axial and radial mean velocities, u and v, the root-mean-square values of the axial and radial turbulence components, m and the Reynolds shear stress u V, and higher correlations of turbulence components, such as the skewness and flatness factors, are limited except for a water-air system [8-12]. 

The magnet probe is applicable to molten metal flows at a temperature lower than the Curie point of the permanent magnet. Previous magnet probe measurements for a Wood metal bath were carried out by Xie et al. [23] at bath temperatures lower than 150°C. The magnet probe used by Xie et al. cannot detect the axial and radial velocity components simultaneously. Accordingly, data on the Reynolds shear stress and higher order turbulence correlations, such as the skewness and flatness factors, have not been published. 

The measurement of the radial velocity component, v, is relatively more difficult than the axial velocity component, , because the magnitude of v is much smaller than that of u. The errors in measuring u are estimated to be 5%, while those for V are 10%. The time required for measuring the Reynolds shear stress is basically quite long. Thus, the measurement at only one axial position z = 4 cm was done. 

The radial distributions of the Reynolds shear stress u v are shown in Fig. 2.21. In the bubbling jet region, r/b < 1.5, the values for the Wood s metal-He system... 

Fig. 2.29 Radial distributions of Reynolds shear stress for each turbulent motion...

Figure 2.29 shows the radial distribution of the Reynolds shear stress for each turbulent motion. The absolute values for the inward interaction and ejection are large, as suggested from the turbulence kinetic energies. 

The measured values of for each turbulent motion remain almost unchanged in the radial direction, but in a strict sense, those for ejection and sweep changed trend around r/b = 1.0. This implies that the Fle-Wood s metal bubbling jet has two large-scale coherent structures, the boundary being located around r/by = 1.0. This result is consistent with the above-mentioned findings on the appearance frequency and the contributions of each turbulent motion to the turbulence kinetic energies and the Reynolds shear stress. 

Velocity measurements were carried out at four fixed vertical positions (z = 0.05, 0.10, 0.15, and 0.19m) for three gas flow rates = 41.4 x 10 , 100 x 10 , and 293 x 10 m /s just like the measurements of bubble characteristics [22]. The root mean square (rms) values of the axial and radial turbulence components, u rms and i/rms. and the Reynolds shear stress V were calculated from the following equations ... 

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