Mean velocity component

The mean value of the square of the velocity component is equal to the sum of the square of the mean velocity component and the mean of the square of the fluctuation. Corresponding relationships hold for the other... 

The mean velocity components are expressed as //, v, and ii>. We assume that the velocity components are stationary Gaussian random processes, so that, based on the preceding discussion, the autocovariances of u, V, and w can be written as (Papoulis, 1965, p. 397)... 

The resulting momentum balance for the mean velocity components U, V, and W becomes ... 

Consider, for example, assisting turbulent mixed convection over a vertical flat plate. This situation is schematically shown in Fig. 9.20. If it is assumed that the boundary layer assumptions apply, the governing equations for the mean velocity components and temperature are ... 

In order to assess the computational snapshot approach in more detail, predicted normalized mean velocity components and normalized turbulent kinetic energy were directly compared with the available data of Schafer et al. (1997). In the case of... 

The momentum equations (6.103 through 6.105) describing turbulent flow are the same as the momentum equations (6.43 through 6.45) describing laminar flow with additional terms including steady-state mean velocity components and time-averaged velocity components without body forces. 

Reasonable agreement is obtained between measured and calculated flow fields. Rotor creates a jet stream in the radial direction towards the cylindric wall. Two main flows circulates back to the impeller, one through the top side and second from the lower side of the stator. The comparisons of the measured and computed mean velocity components for radial direction as a function of cell height is showed in figure 10. 

Figure 3.30 shows nondimensionaUzed distributions of the axial mean velocity component. The data at three vertical positions above the attachment position, designated by the three open symbols, agree with one another. Consequently, the horizontal distributions of u are similar just like the horizontal distributions of... 

The liquid flow characteristics specified, for example, by the mean velocity components, the rms values of the turbulence components, and the Reynolds shear stresses are nearly independent of the mean diameter of bubbles provided that the gas flow rate is the same [43]. The same will be shown to be true for the merged bubbling jet. That is, the liquid flow characteristics in the merged jet are not dependent on the bubble diameters. Therefore, if the merging distance. He, is much smaller than the bath depth, Hi, dual nozzle gas injections would not be useful for the enhancement of mixing in the baths. 

As is widely known, mixing in a bath is governed mainly by large-scale recirculation and turbulent motion. The former is characterized by the mean velocity components in the three directions, while the latter is characterized by the root-mean-square (rms) values of the three turbulence components and the Reynolds shear stresses. Desirable mixing condition would be realized when the two kinds of motions are produced together. Unfortunately, these motions on the mixing time in a bath subjected to surface flow control are poorly understood. This chapter discusses these effects with reference to experiments in which three types of boundary conditions are imposed on the surface of a water bath stirred by bottom gas injection. 

The radial distributions of the axial and radial mean velocity components, u and V, are shown in Figs. 7.3 and 7.4, respectively. The distribution indicates that water moves upward for r < 20 x 10 m and downward for 20 x 10 m < r < 40 X10 m. The broken line in each figure denotes the location of the outer edge of the circular cylinder. Outside this region (r > 40 x 10 m), u and v nearly vanish. In other words, a dead water region develops for r > 40 x 10 m. 

Mixing in the bath is closely associated with the turbulence characteristics in addition to the mean velocity components presented above. For further understanding of the mixing condition, information is given below on the rms values of the axial and radial turbulence components, w rms and v rms, and the Reynolds shear stress V at three representative axial positions, z = 30, 50, and 70 x 10 m. In the presence of swirl motion, however, velocity measurements were impossible at z = 70 x 10-3 m due to the wave motion of the bath surface. 

Figure 8.4 shows the axial and vertical mean velocity components of water, u and v, measured along the x axis. The value of m decreases monotonically in the x direction for every gas flow rate. In particular, in the absence of gas flow, u is inversely proportional to X, as widely known for single-phase jets. This result is discussed in a subsequent section. Meanwhile, v for the two-phase jets becomes positive on the x axis, implying that upward water flow is induced by the bubbles. 

Accordingly, the vertical mean velocity component of water flow, v, is much smaller than the axial mean velocity component it. 

In a similar manner to Pratte and Baines [21, 22], this velocity ratio is used to correlate the upward deflection of water-air two-phase jets as well as the vertical distributions of the axial and vertical mean velocity components, u and v. 

The maximum value of the axial mean velocity component, Mm,sw, for the singlephase water jet, which occurs at the axis of the jet, is nondimensionalized by the water velocity at the pipe outlet, o. The results are plotted against x/rf i in Fig. 8.12. 

On the other hand, the maximum value of the vertical mean velocity component, Vm,sw does not appear on the axis of the jet but at a certain distance from the x axis. The relationship between Vm.sw and Mm,sw can be approximated by [22] ... 

The half-value radius of the vertical distribution of the axial mean velocity component is denoted by b. The measmed values shown in Fig. 8.15 can all be approximated by the solid line of the form ... 

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