Velocity profile, turbulent

Figure 5.6. Illustration of the relationship between velocity profile, turbulent eddies, and mixing length.

At the particle scaie, dispersion is controlled by the local hydrodynamics, i.e. by velocity profiles, turbulence,... in the liquid films. At the bed scale, dispersion results from velocity distribution within the percolation structures. 

Clear evidence exists that beyond the linear regime the synthetic thermostat influences the results. The role of the heat removing mechanism becomes especially important at high shear rates where the assumption of the linear velocity profile (iyyi is incremented in the position equation of equations 27) of the SLLOD shear flow is unrealistic. To allow the formation of kink instabilities in the velocity profile (turbulent flow) the simple thermostating scheme of equations (27) mast be replaced by a profile unbiased thermostat which makes no assumption about the local streaming velocity. ... 

A low Reynolds number indicates laminar flow and a paraboHc velocity profile of the type shown in Figure la. In this case, the velocity of flow in the center of the conduit is much greater than that near the wall. If the operating Reynolds number is increased, a transition point is reached (somewhere over Re = 2000) where the flow becomes turbulent and the velocity profile more evenly distributed over the interior of the conduit as shown in Figure lb. This tendency to a uniform fluid velocity profile continues as the pipe Reynolds number is increased further into the turbulent region. 

Most flow meters are designed and caHbrated for use on turbulent flow, by far the more common fluid condition. Measurements of laminar flow rates may be seriously in error unless the meter selected is insensitive to velocity profile or is specifically caHbrated for the condition of use. 

In general, V For laminar Newtonian flow the radial velocity profile is paraboHc and /5 = 3/4. For fully developed turbulent flow the radial... 

Here, h is the enthalpy per unit mass, h = u + p/. The shaft work per unit of mass flowing through the control volume is 6W5 = W, /m. Similarly, is the heat input rate per unit of mass. The fac tor Ot is the ratio of the cross-sectional area average of the cube of the velocity to the cube of the average velocity. For a uniform velocity profile, Ot = 1. In turbulent flow, Ot is usually assumed to equal unity in turbulent pipe flow, it is typically about 1.07. For laminar flow in a circiilar pipe with a parabohc velocity profile, Ot = 2. 

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. 

Perforated Plates and Screens A nonuniform velocity profile in turbulent flow through channels or process equipment can be smoothed out to any desired degree by adding sufficient uniform resistance, such as perforated plates or screens across the flow channel, as shown in Fig. 6-38. Stoker Ind. Eng. Chem., 38, 622-624 [1946]) provides the following equation for the effect of a uniform resistance on velocity profile ... 

The universal turbulent velocity profile near the pipe wall presented in the preceding subsection Tncompressible Flow in Pipes and Channels may be developed using the Prandtl mixing length approximation for the eddy viscosity,... 

The dispersion that takes place in an open tube, as discussed in chapter 8, results from the parabolic velocity profile that occurs under conditions of Newtonian flow (i.e., when the velocity is significantly below that which produces turbulence). Under condition of Newtonian flow, the distribution of fluid velocity across the tube... 

The distribution of tracer molecule residence times in the reactor is the result of molecular diffusion and turbulent mixing if tlie Reynolds number exceeds a critical value. Additionally, a non-uniform velocity profile causes different portions of the tracer to move at different rates, and this results in a spreading of the measured response at the reactor outlet. The dispersion coefficient D (m /sec) represents this result in the tracer cloud. Therefore, a large D indicates a rapid spreading of the tracer curve, a small D indicates slow spreading, and D = 0 means no spreading (hence, plug flow). 

From Tolmin s theory and experimental data (e.g., Reichardtthe relationship between velocity profile and temperature profile in the jet cross-section can be expressed using an overall turbulent Prandtl number Pr = v /a, where Vf is a turbulent momentum exchange coefficient and a, is a turbulent heat exchange coefficient ... 

To calibrate larger sensors/instruments such as vane anemometers, a wind tunnel is required. A calibration wind tunnel consists of an open or closed tunnel, a fan to deliver the air, a nozzle to shape the velocity profile, and a mesh arrangement to uniform and reduce the flow turbulence. It may be necessary to control the air temperature in the tunnel by means of a heating/cooling sys-... 

The similarity of velocity and of turbulence intensity is documented in Fig. 12.29. The figure shows a vertical dimensionless velocity profile and a turbulence intensity profile measured by isothermal model experiments at two different Reynolds numbers. It is obvious that the shown dimensionless profiles of both the velocity distribution and the turbulence intensity distribution are similar, which implies that the Reynolds number of 4700 is above the threshold Reynolds number for those two parameters at the given location. 

Comparison of the velocity profiles for laminar and turbulent boundary layers. 

It is seen that it is important to be able to determine the velocity profile so that the flowrate can be calculated, and this is done in Chapter 3. For streamline flow in a pipe the mean velocity is 0.5 times the maximum stream velocity which occurs at the axis. For turbulent flow, the profile is flatter and the ratio of the mean velocity to the maximum... 

Figure 3.11. Shape of velocity profiles for streamline and turbulent flow...

On the assumption that the velocity profile in a fluid in turbulent flow is given by the Prandtl one-seventh power law, calculate the radius at which the flow between it and the centre is equal to that between it and the wall, for a pipe 100 mm in diameter,... 

Equation 11.12 does not fit velocity profiles measured in a turbulent boundary layer and an alternative approach must be used. In the simplified treatment of the flow conditions within the turbulent boundary layer the existence of the buffer layer, shown in Figure 11.1, is neglected and it is assumed that the boundary layer consists of a laminar sub-layer, in which momentum transfer is by molecular motion alone, outside which there is a turbulent region in which transfer is effected entirely by eddy motion (Figure 11.7). The approach is based on the assumption that the shear stress at a plane surface can be calculated from the simple power law developed by Blasius, already referred to in Chapter 3. 

The application to pipe flow is not strictly valid because u (= fRjp) is constant only in regions close to the wall. However, equation 12.34 appears to give a reasonable approximation to velocity profiles for turbulent flow, except near the pipe axis. The errors in this region can be seen from the fact that on differentiation of equation 12.34 and putting y = r, the velocity gradient on the centre line is 2.5u /r instead of zero. 

Approximate form of velocity profile In turbulent region... 

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 ... 

Show that, if the Blasius relation is used for the shear stress R at the surface, the thickness of the laminar sub-layer <5, is approximately 1.07 times that calculated on the assumption that the velocity profile in the turbulent fluid is given by PrandtFs one seventh power law. 

---