Plane channel turbulent flow

Schumann U (1975) Subgrid Scale Model for Einite Difference Simulations of Turbulent Flows in Plane Channels and Annuli. J Comput Phys 18 376-404... 

Formulas (1.6.1) and (1.6.2) have formed the basis of most theoretical investigations on the determination of the average fluid velocity and the drag coefficient in the stabilized region of turbulent flow in a circular tube (and a plane channel of width 2a). The corresponding results obtained on the basis of Prandtl s relation (1.1.21) and von Karman s relation (1.1.22) for the turbulent viscosity can be found in [276,427]. In what follows, major attention will be paid to empirical and semiempirical formulas that approximate numerous experimental data quite well. 

Qualitatively, the picture of stabilized turbulent flow in a plane channel is similar to that in a circular tube. Indeed, in the viscous sublayer adjacent to the channel walls, the velocity distribution increases linearly with the distance from the wall V(Y)/U = y+. In the logarithmic layer, the average velocity profile can be described by the expression [289]... 

More detailed information about the structure of turbulent flows in a circular (noncircular) tube and a plane channel, as well as various relations for determining the average velocity profile and the drag coefficient, can be found in the books [138, 198, 268, 276, 289], which contain extensive literature surveys. Systematic data for rough tubes and results of studying fluctuating parameters of turbulent flow can also be found in the cited references. 

Let us discuss qualitative specific features of convective heat and mass transfer in a turbulent flow through a circular tube and plane channel in the region of stabilized flow. Experimental evidence indicates that several characteristic regions with different temperature profiles can be distinguished. At moderate Prandtl numbers (0.5 < Pr < 2.0), the structure and sizes of these regions are similar to those of the wall layer and the core of the turbulent stream considered in Section 1.6. 

Classical theories of suspended solids in open channels are based on two-dimensional turbulent flow. Consider a two-dimensional turbulent flow with a velocity Uin the horizontal AT-direction and V in the vertical /-direction. Reynolds (1895) defined the shear stress parallel to jron a plane normal to / as... 

Zhaphashaev, U.K., Isakhanova, G.Z., 1998. Developed turbulent flow in a plane channel with simultaneous injection through one porous wall and suction through the other. J. Appl. Mech. Tech. Phys. 39, 53-59. 

Sandler SI (1999) Chemictil and engineering thermodynamics, 3rd edn. Wiley, New York Schlichting H, Gersten K (2000) Boundary-layer theory. Springer, Berlin Schumann U (1975) Subgrid settle model for finite difference simulations of turbulent flows in plane channels and annuli. J Comput Phys 18 376-404... 

Figure 13 plots an example of the processed PIV frame. The turbulent velocity field and its boundaries, solid wall, and liquid-free surface are simultaneously shown in Figure 13. The turbulence structures such as the coherent vortical structure near the bottom wall and its modification after release from the no-slip boundary condition near the free surface of the open-channel flow, and the evolvement of the free-surface wave can be seen in Figure 13. This simultaneous measurement technique for free-surface level and velocity field of the liquid phase using PIV has been successfully applied to the investigation of wave-turbulence interaction of a low-speed plane liquid wall-jet flow (Li et al., 2005d), and the characteristics of a swirling flow of viscoelastic fluid with deformed free surface in a cylindrical container driven by the constantly rotating bottom wall (Li et al., 2006c).

Fig. 13. Hussain and Reynolds MVFN calculation of the dispersion relationship for plane waves in turbulent channel flow.

Lateral Concentration Profiles in Horizontal Pipes. Previous theoretical studies for horizontal pipe and channel flow have concentrated on the variation of solids concentration in the vertical direction. In this case, gravity (including buoyancy), fluid-particle drag, turbulent diffusion, and particle-particle interaction effects must occur. In this section, the variation of solids concentration in a horizontal plane through the pipe axis is examined. 

The results were obtained from studies on plane mixing layers produced by allowing two parallel streams to join at the entrance to a rectangular channel about one metre in length. Some experiments were carried out in a channel with a cross-section of 40 x 20 mm, initial flow velocities (U and U2) of 2 and 1 m/s and free stream turbulence levels of about 0.5%, and others in a 40X40 mm channel with initial velocities of 1 and 0.5 m/s and turbulence levels of about 3%. Figure 1 shows a schematic representation of the formation and development of the mixing layer. The dashed lines f = 0.9 and 0.1 represent the positions at which the main stream velocity has the values U2 + 0.9 AU and + 0.1 AU respectively, where AU = - U2. These lines... 

The modification of hydrodynamic aspects is exploited in the falling-film cell [12], where the electrolyte flows as a thin fllm in the channel between an inclined plane plate and a sheet of expanded metal which work as electrodes. Other proposal is to include turbulence promoters in the interelectrode gap in conventional parallel plate electrochemical reactors [13-16], or the use of expanded metal electrodes immersed in a fluidized bed of small glass beads, called Qiemelec cell [17]. Likewise, the Metelec cell [18] incorporates a cylindrical foil cathode concentric arranged around an inner anode, with a helical turbulent electrolyte flow between the electrodes. The electrochemical hydrocyclone cell [19] makes use of the good mass-transfer conditions due to the helical downward accelerated flow in a modified conventional hydrocyclone. 

Where H is the charmel height (the smaller dimension in a rectangular channel), tw,av the average wall shear stress, V the kinematic viscosity, and p the density of the fluid. In internal flows, the laminar to turbulent transition in abrupt entrance rectangular ducts was found to occur at a transition Reynolds number Ret = 2200 for an aspect ratio ac = I (square ducts), to Ret = 2500 for flow between parallel planes with = 0 [4]. For intermediate channel aspect ratios, a linear interpolation is recommended. For circular tubes. Ret = 2300 is suggested. These transition Reynolds number values are obtained from experimental observations in smooth channels in macroscale applications of 3 mm or larger hydraulic diameters. Their applicability to microchannel flows is still an open question. 

The experimental apparatus has been described in Sect. 3.2.2. The electroresistivity probe was removed and a two-channel laser Doppler velocimeter was set up to measure the three velocity components of water flow in the bath. The origin of the cylindrical coordinates (z, r, 9) was placed at the center of the bath, as shown in Fig. 3.3. The velocity components were designated by u, v, and w, respectively. The components, u and v, were measured in the z — r plane including the centerline of the bath and the center of the nozzle exit [21,23]. Digitized velocity data were decomposed into the mean velocity and turbulence components as follows ... 

The bubble characteristics, represented by the bubble frequency /b, gas holdup a, mean bubble rising velocity b, and mean bubble chord length Lb, were measured mainly on the y—z plane with a two-needle electroresistivity probe. Water velocity was measured with a two-channel laser Doppler velocimeter [25], The locations at which measurements were made were chosen by reference to the results for the bubble characteristics. The characteristics of water flow were represented by the vertical mean velocity u, the root-mean-square value of the vertical turbulence component w rms, and the skewness and flatness factors of the vertical turbulence component, and F . Data aquisition time was 600 s for each position. 

Plane waves and structures in turbulent channel flow. Phys. Fluids A, 2, 2217-2226. 

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