Prandtl velocity distribution

The relation between the mean velocity and the velocity at the axis is derived using this expression in Chapter 3. There, the mean velocity u is shown to be 0.82 times the velocity us at the axis, although in this calculation the thickness of the laminar sub-layer was neglected and the Prandtl velocity distribution assumed to apply over the whole cross-section. The result therefore is strictly applicable only at very high Reynolds numbers where the thickness of the laminar sub-layer is vety small. At lower Reynolds numbers the mean velocity will be rather less than 0.82 times the velocity at the axis. 

When Re > 10 , the following equation, derived by means of the iogarith-mic velocity distribution by Prandtl and the empirical research results of Nile uradse, is valid ... 

Velocity distribution profiles in Zone 3 of the jet were found to be simi-lar. They can be computed by applying momentum-transfer theory (Prandtl-Tollmein) and vorticity-transfer theory (Taylor-Goldstein). Modification of these theories with different assumptions has resulted in several equa-... 

There is a natural draw rate for a rotating disk that depends on the rotation rate. Both the radial velocity and the circumferential velocity vanish outside the viscous boundary layer. The only parameter in the equations is the Prandtl number in the energy equation. Clearly, there is a very large effect of Prandtl number on the temperature profile and heat transfer at the surface. For constant properties, however, the energy-equation solution does not affect the velocity distributions. For problems including chemistry and complex transport, there is still a natural draw rate for a given rotation rate. However, the actual inlet velocity depends on the particular flow circumstances—there is no universal correlation. 

Keulegan (Kl3), 1938 Extension of Prandtl-von KdrmSn turbulent flow theories to turbulent flow in open channels. Effects of wall roughness, channel shape, and free surface on velocity distribution are considered. 

It was noted that the velocity in a channel approaching a weir might be so badly distributed as to require a value of 1.3 to 2.2 for the kinetic energy correction factor. In unobstructed uniform channels, however, the velocity distribution not only is more uniform but is readily amenable to theoretical analysis. Vanonil has demonstrated that the Prandtl universal logarithmic velocity distribution law for pipes also applies to a two-dimensional open channel, i.e., one that is infinitely wide. This equation may be written... 

Water is flowing through a 150 mm diameter pipe and its flowrate is measured by means of a 50 mm diameter orifice, across which the pressure differential is 2.27 x 104 N/m2. The coefficient of discharge of the orifice meter is independently checked by means of a pitot tube which, when situated at the axis of the pipe, gave a reading of 100 mm on a mercury-under-water manometer. On the assumption that the flow in the pipe is turbulent and that the velocity distribution over the cross-section is given by the Prandtl one-seventh power law, calculate the coefficient of discharge of the orifice meter. 

The velocity distribution in turbulent flow is discussed in Section 3.3.6 where the Prandtl... 

For experimental determination the flow pattern produced by the stirrer was initially visualized using different photographic methods (e.g. [574, 497]), but hydraulic probes were also used to determine the pressure distribution (e.g. [135]) and velocity distribution (e.g. [437]). Also, convection probes (spherical probes) and pressure probes (Prandtl s Pitot tube) were used. Later constant temperature hotwire/hot-film anemometry was used. Currently contactless laser doppler velocim-ctry (LDV)/anemometry (LDA) is exclusively utilized. 

The Power Law Velocity Distribution. The solution for the power law velocity distribution is introduced in Prandtl [42] in the following form ... 

Figure 6.5 shows the velocity distributions in a boundary layer of a liquid with Pr , = 100 (e.g., sulfuric acid at room temperature). For this Prandtl number, the thermal boundary layer penetration into the liquid is much less than the flow boundary layer, and the regions where viscosity variations occur are confined close to the surface. The curve corresponding to p /pe = 1 is the Blasius solution (see Fig. 6.1). The curve labeled p ,/pe = 0.23 corresponds to a heated surface where the low viscosity near the surface requires steeper velocity gradients to maintain a continuity of shear with the outer portion of the boundary layer. The heated free-stream cases reveal the opposite effects. In general, the outer portions of the flow boundary layers act similarly to the velocity distribution of the Blasius case except for their being displaced in or out by the effects that have taken place in the thermal boundary layer.

Prandtl Number Equal to Unity. If Pr = 1, considerable simplification results. Equation 6.38 acquires a form identical with Eq. 6.37, / being analogous to A solution of the energy equation, therefore, is directly expressible in terms of the velocity distribution as... 

Ferrari [96] also solved the von Mises form of the energy equation, but with the velocity distributions in the inner and outer portions of the boundary layer represented by polynomials of a velocity potential function. This analysis is applicable to gases in that it accounts for compressibility, but the solutions may be in error for high Prandtl numbers because of the rather approximate fit to the law of the wall. 

The numerical results of the various Reynolds analogy factors are compared in Fig. 6.36 for laminar Prandtl numbers ranging from those characteristic of gases to those of oils and for Re = 107. Results for very low laminar Prandtl numbers, characteristic of liquid metals, are not shown because the assumptions for the velocity distributions in the various analyses are... 

For the curves in Fig. 11.7, the value of n that gives the best representation of the experimental curves varies from for the lowest Reynolds number to jq for the highest Reynolds number. Prandtl selected j as the best average, deducing Prandtl s power velocity distribution rule. This is not an exact rule, because if it were a general rule, then all the curves in Fig. 11.7 would be identical. Furthermore, it cannot be correct very near the wall of the tube, because there it predicts that dVIdy is infinite and hence that the shear stress is infinite. Nonetheless, it is widely used because it is simple and, as we will see in Sec. 11.5, because it gives useful results. 

It is possible to find more complex correlations for the velocity distribution in a pipe which do not have the limitations of Prandtl s power rule. In Fig. 11.7 the Reynolds number appears as a parameter in the velocity distribution plot. In trying to produce a universal velocity distribution rule, it seems logical to change the coordinates in Fig. 11.7 so that the Reynolds number enters either explicitly or implicitly in one of the coordinates, in the hope of getting all the data onto one curve. 

Most of the best known theoretical fluid mechanic experts of the twentieth century have attempted to deduce a comprehensive theory of turbulence. The resulting theories each provide some insight into the relations that must exist between various quantities in turbulent flow, but all contain undefined constants which must be measured experimentally to make the theory fit the observations. The theories of Prandtl and von Karman are well summarized by Schlichting [9, chap. 19]. That of G. I. Taylor is summarized by Dryden [10]. The theories of Kolmogoroff are discussed by Hinze [1] and Corrsin [5]. The problem of calculating the velocity distribution in turbulent flow in a pipe from the various theories is discussed by Bird et al. [11]. 

Prandtl s one-seventh power law A relationship used to determine the velocity distribution for turbulent flow in pipes carrying fluids given as ... 

Generally speaking, the conventional numerical analysis with a k-e turbulence model and accurate treatment of thermophysical properties can successfully explain the unusual heat transfer phenomena of supercritical water. Heat transfer deterioration occurs due to two mechanisms depending on the flow rate. When the flow rate is large, viscosity increases locally near the wall by heating. This makes the viscous sublayer thicker and the Prandtl number smaller. Both effects reduce the heat transfer. When the flow rate is small, buoyancy force accelerates the flow velocity near the wall. This makes the flow velocity distribution flat and generation of turbulence energy is reduced. This type of heat transfer deterioration appears at the boundary between forced and natural convection. As the heat flux increases above the deterioration heat flux, a violent oscillation of wall temperature is observed. It is explained by the unstable characteristics of the steep boundary layer of temperature. 

Dukler Theory The preceding expressions for condensation are based on the classical Nusselt theoiy. It is generally known and conceded that the film coefficients for steam and organic vapors calculated by the Nusselt theory are conservatively low. Dukler [Chem. Eng. Prog., 55, 62 (1959)] developed equations for velocity and temperature distribution in thin films on vertical walls based on expressions of Deissler (NACA Tech. Notes 2129, 1950 2138, 1952 3145, 1959) for the eddy viscosity and thermal conductivity near the solid boundaiy. According to the Dukler theoiy, three fixed factors must be known to estabhsh the value of the average film coefficient the terminal Reynolds number, the Prandtl number of the condensed phase, and a dimensionless group defined as follows ... 

A comparison of Eqs. (3.52) and (3.53) and also of their boundary conditions as given in Eqs. (3.24) and (3.54) respectively, shows that these equations are identical in all respects. Therefore, for the particular case of Pr equal to one, the distribution of 9 through the boundary layer is identical to the distribution of uJu ). In this par-ticular case, therefore, Fig. 3.4 also gives the temperature distribution and the two boundary layer thicknesses are identical in this case. Now many gases have Prandtl numbers which are not very different from 1 and this relation between the velocity and temperature fields and the results deduced from it will be approximately correct for them. 

Figure 3.1. Velocity and temperature distributions in boundary layers at (a) extremely low Prandtl numbers and (b) very high Prandtl numbers...

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