Velocity profile in turbulent flow

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 ... 

Typical average velocity profiles in laminar and turbulent flow are also given in Fig. 6-12. Note that the velocity profile in turbulent flow is much fuller than that in laminar flow, with a sharp drop near the surface. The turbulent boundary ... 

Because of the relatively flat velocity profile in turbulent flow, the channel geometry has only a small influence on the friction factor (as discussed in the previous section) and the Sherwood and Nusselt numbers. The turbulent Sherwood and Nusselt numbers of rod bundles can therefore be related to those of circular tubes. The few experimental data, as compiled by Ref. 6, suggest that for relative pitches between 1.1 and 2.0 (which... 

An interesting point is the dependence of the turbulent Nu numbers on the boundary condition. For laminar flow it was shown that the dependence of Nu on the boundary condition rapidly fades away when the relative pitch is increased The difference in Nu between the two limiting boundary conditions is less than 30% for relative pitches larger than 1.1. Because of the flatter velocity profile in turbulent flow, the dependence on the boundary conditions is weak in turbulent flow, except for small relative pitches. An estimate of the maximum influence of the boundary condition on the turbulent Nu number can be obtained from the respective values for laminar flow. 

Figure 10.3. von Karman model for the velocity profile in turbulent flow Also shown is the... 

Velocity profiles in turbulent flow of power-law fluids... 

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. 

The mean profiles of velocity, temperature and solute concentration are relatively flat over most of a turbulent flow field. As an example, in Figure 1.24 the velocity profile for turbulent flow in a pipe is compared with the profile for laminar flow with the same volumetric flow rate. As the turbulent fluxes are very high but the velocity, temperature and concentration gradients are relatively small, it follows that the effective diffusivities (iH-e), (a+eH) and (2+ed) must be extremely large. In the main part of the turbulent flow, ie away from the walls, the eddy diffusivities are much larger than the corresponding molecular diffusivities ... 

If the velocity profile is known together with the distribution of e then, for any assumed relationship between the distributions of c and e, Eq. (7.49) can be used to deduce the temperature profile. Once this has been obtained the relation between the Nusselt and Reynolds numbers can be derived. Before illustrating this procedure, there is a simplifying assumption that can be introduced without any significant loss of accuracy. Because the velocity profile in turbulent pipe flow is relatively flat over a large portion of the pipe cross-section, it is usually sufficiently accurate to replace u in the integral in the expression for I by the constant value um, i.e., to write ... 

The developed velocity profile for turbulent flow in a tube will appear as shown in Fig. 5-15. A laminar sublayer, or film, occupies the space near the surface, while the central core of the flow is turbulent. To determine the heat transfer analytically for this situation, we require, as usual, a knowledge of the temperature distribution in the flow. To obtain this temperature distribution, the... 

Perhaps the first thought that comes to mind is to determine the shear stress in an analogous manner to laminar flow from t - -p, duldr, where u(r) is the average velocity profile for turbulent flow. But the experimental studies show that this is not the case, and the shear stress is much larger due to the turbulent fluctuations. Therefore, it is convenient to think of the turbulent shear stress as consisting of two parts the laininai component, wlticli accounts for... 

A model to account for roughness in a BSR may be derived from Eq. (7). The part of Eq. (7) within parentheses represents the universal velocity profile for turbulent flow along hydraulically smooth surfaces. For nonsmooth surfaces, the same expression for the velocity profile has been proved experimentally to be adequate, but then the second constant is smaller than 5.5 the first constant, 2.5, appears not to depend on the surface roughness. From this it can be made plausible that the roughness in rod bundles could be described with an empirical roughness function, R(h ), implemented in Eq. (7) ... 

The turbulent flow velocity profile for Newtonian fluids is arbitrarily divided into three regions the viscous sublayer, the buffer layer, and the turbulent core. To represent velocity profiles in pipe flow, friction velocity defined as... 

Unfortunately, it is not possible to derive an analogue velocity profile for turbulent flow in an anal dical manner based on the generalized momentum equations. However, a number of entirely empirical relations of similar simplicity exist for the velocity profile in turbulent pipe flow. One such relation often found in introductory textbooks on engineering fluid flow is the power law velocity profile. ... 

Ivanyuta,Yu.P. Chekalova,L.A.t Investigation of velocity profile of turbulent flows of weak polymer solutions in a... 

Figure 3.10-4. Universal velocity profile for turbulent flow in smooth circular tubes. 

Laminar Flow Although heat-transfer coefficients for laminar flow are considerably smaller than for turbulent flow, it is sometimes necessary to accept lower heat transfer in order to reduce pumping costs. The heat-flow mechanism in purely laminar flow is conduction. The rate of heat flow between the walls of the conduit and the fluid flowing in it can be obtained analytically. But to obtain a solution it is necessary to know or assume the velocity distribution in the conduit. In fully developed laminar flow without heat transfer, the velocity distribution at any cross section has the shape of a parabola. The velocity profile in laminar flow usually becomes fully established much more rapidly than the temperature profile. Heat-transfer equations based on the assumption of a parabolic velocity distribution will therefore not introduce serious errors for viscous fluids flowing in long ducts, if they are modified to account for effects caused by the variation of the viscosity due to the temperature gradient. The equation below can be used to predict heat transfer in laminar flow. 

Bogue, D. C. and Metzner, A. B. 1963. Velocity profiles in turbulent pipe flow, Newtonian and non-Newtonian fluids. Industrial and Engineering Chemistry Fundamentals 2 143-149. 

Nicodemo, L., Aciemo, D., and Astarita, G. 1969. Velocity profiles in turbulent pipe flow of drag-reducing hquids. Chemical Engineering Science 24 1241-1246. 

Velocity profiles in turbulent boundary layer are determined experimentally since solutions to the flow equations are not available. Based on experimental data, universal velocity profile is proposed in which the boundary layer is divided into four regions—laminar sublayer, buffer zone, turbulent core, and turbulent wake. The turbulent velocity profile is given in terms of dimensionless velocity defined as... 

Theoretical relationships between fluid flow rate and pressure drop for turbulent flow in pipes of interest similar to that established for laminar flows are not readily available. However, empirical relations based on experiments have been fitted long ago by pioneers such as Blasius and others to provide quantitative assessment of frictional resistances in terms of Re (Schlichting, 1979). The velocity profile for turbulent flow in large pipes can be characterized as... 

BOGUE D.C., "Velocity profiles in turbulent non-newtonian pipe flow". Thesis presented... 

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. 

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,... 

Hao et al. (2007) investigated the water flow in a glass tube with diameter of 230 Lim using micro particle velocimetry. The streamwise and mean velocity profile and turbulence intensities were measured at Reynolds number ranging from 1,540 to 2,960. Experimental results indicate that the transition from laminar to turbulent flow occurs at Re = 1,700—1,900 and the turbulence becomes fully developed at Re > 2,500. 

The flow of jets becomes turbulent at much lower Re numbers than channel flows. Calculating the stress from the mean velocity profiles does not reflect the true situation in turbulent flow. As in the case in most bioreactors, the maximum turbulent stress is determined by the turbulence, which can be calculated using Eqs. (2)-(4). It occurs in free jets after the nozzle, at the edge of the mixing zone. The following is generally valid ... 

---