Velocity profile universal

For fully developed turbulent flow in a pipe, the whole of the flow may be regarded as lying within the boundary layer. The cross-section can then conveniently be divided into three regions  

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In turbulent flow, the velocity profile is much more blunt, with most of the velocity gradient being in a region near the wall, described by a universal velocity profile. It is characterized by a viscous sublayer, a turbulent core, and a buffer zone in between. 

Vfjp is the friction velocity and =/pVV2 is the wall stress. The friction velocity is of the order of the root mean square velocity fluctuation perpendicular to the wall in the turbulent core. The dimensionless distance from the wall is y+ = yu p/. . The universal velocity profile is vahd in the wall region for any cross-sectional channel shape. For incompressible flow in constant diameter circular pipes, = AP/4L where AP is the pressure drop in length L. In circular pipes, Eq. (6-44) gives a surprisingly good fit to experimental results over the entire cross section of the pipe, even though it is based on assumptions which are vahd only near the pipe wall. 

It is now possible to consider each of these regions in turn and to develop a series of equations to represent the velocity over the whole cross section of a pipe. Together, they constitute the Universal Velocity Profile. 

For a smooth pipe, therefore, the complete Universal Velocity Profile is given by ... 

It should be noted that equation 12.65a gives E = 12(p,/p) at y+ = 30, compared with 5(p./p) from equation 12.63. This arises because of the discontinuity in the Universal Velocity Profile at y4" = 30. 

In Table 12.1, the values of u+ calculated from equation 12.67 are compared with those given by the universal velocity profile (equations 12.37, 12.40 and 12.42). It will be seen that there is almost exact correspondence at y+ = 1000 and differences are less than 6 per cent in tire range 30 < y+ < 3000. 

Table 12.1. Comparison of values of calculated from equation 12.67 with those given by the universal velocity profile...

In the Taylor-Prandtl modification of the theory of heat transfer to a turbulent fluid, it was assumed that the heat passed directly from the turbulent fluid to the laminar sublayer and the existence of the buffer layer was neglected. It was therefore possible to apply the simple theory for the boundary layer in order to calculate the heat transfer. In most cases, the results so obtained are sufficiently accurate, but errors become significant when the relations are used to calculate heat transfer to liquids of high viscosities. A more accurate expression can be obtained if the temperature difference across the buffer layer is taken into account. The exact conditions in the buffer layer are difficult to define and any mathematical treatment of the problem involves a number of assumptions. However, the conditions close to the surface over which fluid is flowing can be calculated approximately using the universal velocity profile,(10)... 

In the Universal Velocity Profile , the laminar sub-iayer extends to values of y+ = 5 and the turbulent zone starts at y+ = 30 and the range 5 < y+ < 30, the buffer layer, is covered by a second linear relation between and In, y+. What is the maximum difference between the values of u+, in the range 5 < y4 < 30, using the two methods of representation of the velocity profile ... 

Explain the importance of the universal velocity profile and derive the relation between the dimensionless derivative of velocity trf, and the dimensionless derivative of distance from the surface y+, using the concept of Prandtl s mixing length kE. 

In the universal velocity profile a dimensionless velocity + is plotted against lnyf, where y+ is a dimensionless distance from the surface. For the region where eddy transport dominates (eddy kinematic viscosity kinematic viscosity), the ratio of the mixing length (Ag) to the distance (>>) from the surface may be taken as approximately constant and equal to 0.4. Obtain an expression for d +/dy+ in terms of y+. 

A reasonably consistent universal velocity profile is obtained by plotting (UJr - Ujb)/(Ujm - U.b) vs. r/r1/2 in Figs. 16-18, comparable with the Tollmien solution for a circular homogeneous jet in an infinite medium (Abramovich, 1963 Rajaratnam, 1976). 

Figure 16. Universal velocity profile for Run GSF-3—comparison with Tollmien s solution.

Equations 2.58, 2.70 and 2.71 enable the velocity distribution to be calculated for steady fully developed turbulent flow. These equations are only approximate and lead to a discontinuity of the gradient at y+ = 30, which is where equations 2.70 and 2.71 intersect. The actual profile is, of course, smooth and the transition from the buffer zone to the fully turbulent outer zone is particularly gradual. As a result it is somewhat arbitrary where the limit of the buffer zone is taken often the value y+ = 70 rather than j + = 30 is used. The ability to represent the velocity profile in most turbulent boundary layers by the same v+ - y+ relationships (equations 2.58, 2.70 and 2.71) is the reason for calling this the universal velocity profile. The use of in defining v+ and y+ demonstrates the fundamental importance of the wall shear stress. 

Universal velocity profile for turbulent flow Source N. Scholtz, VDI-Beriche6,7-12 (1955)... 

Prandtl-Nikuradse treatment and considered the liquid film to involve a laminar and a turbulent layer. Anderson and Mantzouranis work is an extension of the equations presented by Dukler and Bergelin (D5), and makes use of the von Karman universal velocity profile. In general, if any two of pressure drop, liquid flow rate, or film thickness are known, the third quantity can be calculated. 

Dukler and Bergelin (D16) used the universal velocity profile equations of Nikuradse ... 

Kutateladze and Styrikovich (K25) have presented a treatment very similar to that outlined above. Thomas and Portalski (T14) have also used the universal velocity profile concept, but for the case of countercurrent gas flow they have used instead of (81) the more exact form (for a vertical tube) ... 

Labuntsov (L2), 1957 Heat transfer to condensate films on vertical and horizontal surfaces. In laminar region, Nusselt equations are corrected for (a) inertia effects, (b) variation of physical properties with temperature, (c) effects of waves. In turbulent region various universal velocity profiles are used. Results compared with experimental data. 

Anderson and Mantzouranis (A5), 1960 Experimental and theoretical study of hold-up and film thicknesses for upward cocurrent gas/film flow in vertical tubes. Theory based on use of universal velocity profile. Numerous experimental data on friction factors, etc. 

Universal Velocity Profile. The Universal Velocity Profile is often used to describe the velocity near a wall. This profile is divided into three sections or layers and is shown in Figure 4. Here, dimensionless velocity, u+ = uyjp/tw, is plotted against dimensionless distance from wall, y+ = Vy/tfipJJi, where tw is the shear stress at the wall. 

Figure 4 Boundary layers and the universal velocity profile...

The importance of the universal velocity profile is discussed in Section 12.4. From equation 12.18, for isotropic turbulence, the eddy kinematic viscosity, EaXEuE where XE is the mixing length and uE is some measure of the linear velocity of the fluid in the eddies. The momentum transfer rate per unit area in a direction perpendicular to the surface at position y is then ... 

Obtain a dimensionless relation for the velocity profile in the neighbourhood of a surface for the turbulent flow of a liquid, using Prandtl s concept of a Mixing Length (Universal Velocity Profile). Neglect the existence of the buffer layer and assume that, outside the laminar sub-layer, eddy transport mechanisms dominate. Assume that in the turbulent fluid the mixing length Xe is equal to 0.4 times the distance y from the surface and that the dimensionless velocity u1 is equal to 5.5 when the dimensionless distance y+ is unity. 

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