The turbulent boundary layer

Another concept sometimes used as a basis for comparison and correlation of mass transfer data in columns is the Clulton-Colbum analogy (35). This semi-empirical relationship was developed for correlating mass- and heat-transfer data in pipes and is based on the turbulent boundary layer model... 

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. 

Putting the constant equal to zero, implies that <5 = 0 when x = 0, that is that the turbulent boundary layer extends to the leading edge of the surface. An error is introduced by this assumption, but it is found to be small except where the surface is only slightly longer than the critical distance xc for the laminar-turbulent transition. 

Thus the turbulent boundary layer is thickening at about four times the rate of the streamline boundary layer at the transition point. 

The relations between ux and y have already been obtained for both streamline and turbulent flow. A relation between and y for streamline conditions in the boundary layer is now derived, although it is not possible to define the conditions in the turbulent boundary layer sufficiently precisely to derive a similar expression for that case. 

It is found that the velocity at a distance y from the surface may be expressed as a simple power function (u oc y" for the turbulent boundary layer at a plane surface. What is the value of n if the ratio of the momentum thickness to the displacement thickness is 1.78 ... 

For turbulent flow on a rotating sphere or hemisphere, Sawatzki [53] and Chin [22] have analyzed the governing equations using the Karman-Pohlhausen momentum integral method. The turbulent boundary layer was assumed to originate at the pole of rotation, and the meridional and azimuthal velocity profiles were approximated with the one-seventh power law. Their results can be summarized by the... 

Integrating Eq. (6-24) over the turbulent boundary layer (from jq, the edge of the buffer layer, to y) gives... 

Equation (6-31) applies to the laminar sublayer region in a Newtonian fluid, which has been found to correspond to 0 < y+ < 5. The intermediate region, or buffer zone, between the laminar sublayer and the turbulent boundary layer can be represented by the empirical equation... 

Derive the relation between the friction factor and Reynolds number in turbulent flow for smooth pipe [Eq. (6-34)], starting with the von Karman equation for the velocity distribution in the turbulent boundary layer [Eq. (6-26)]. 

As the fluid s velocity must be zero at the solid surface, the velocity fluctuations must be zero there. In the region very close to the solid boundary, ie the viscous sublayer, the velocity fluctuations are very small and the shear stress is almost entirely the viscous stress. Similarly, transport of heat and mass is due to molecular processes, the turbulent contribution being negligible. In contrast, in the outer part of the turbulent boundary layer turbulent fluctuations are dominant, as they are in the free stream outside the boundary layer. In the buffer or generation zone, turbulent and molecular processes are of comparable importance. 

This part of the turbulent boundary layer is rich in coherent structures, ie the flow exhibits features that are not random. Flow visualization studies [Kline et al (1967), Praturi and Brodkey (1978), Rashidi and Banerjee (1990)] have revealed a fascinating picture. The first observations indicated the occurrence of fluid motions called inrushes and eruptions or bursts in the fluid very close to the wall. During eruptions, fluid... 

Equation 2.40 is an empirical equation known as the one-seventh power velocity distribution equation for turbulent flow. It fits the experimentally determined velocity distribution data with a fair degree of accuracy. In fact the value of the power decreases with increasing Re and at very high values of Re it falls as low as 1/10 [Schlichting (1968)]. Equation 2.40 is not valid in the viscous sublayer or in the buffer zone of the turbulent boundary layer and does not give the required zero velocity gradient at the centre-line. The l/7th power law is commonly written in the form... 

Conditions in the fully turbulent outer part of the turbulent boundary layer are quite different. In a turbulent fluid, the shear stress f is given by equation 1.95. As illustrated in Example 1.10, outside the viscous sublayer and buffer zone the eddy kinematic viscosity e is much greater than the molecular kinematic viscosity v. Consequently equation 1.95 can be written as... 

The changing character of the flow in the different regions of the turbulent boundary layer explains certain aspects of the friction factor chart. If the absolute roughness of the pipe wall is smaller than the thickness of the viscous sublayer, flow disturbances caused by the roughness will be damped out by viscosity. The wall is subject to a viscous shear stress. Under these conditions, the line on the friction factor chart... 

Apart from the nature of the bulk flow, the hydrodynamic scenario close to the surfaces of drug particles has to be considered. The nature of the hydrodynamic boundary layer generated at a particle s surface may be laminar or turbulent regardless of the bulk flow characteristics. The turbulent boundary layer is considered to be thicker than the laminar layer. Nevertheless, mass transfer rates are usually increased with turbulence due to the presence of the viscous turbulent sub-layer. This is the part of the (total) turbulent boundary layer that constitutes the main resistance to the overall mass transfer in the case of turbulence. The development of a viscous turbulent sub-layer reduces the overall resistance to mass transfer since this viscous sub-layer is much narrower than the (total) laminar boundary layer. Thus, mass transfer from turbulent boundary layers is greater than would be calculated according to the total boundary layer thickness. 

A more realistic description that avoids the countercurrent flow in the region between two successive paths is represented in Fig. 6b, but we will assume that the regions 2 have a much smaller extent than the regions 1. It is appropriate to cite here references [59,60] in which the Danckwerts renewal idea was used to describe the turbulent boundary layer near a wall, as well as the 1969 paper of Black [61] in which a model similar to that of Ruckenstein [58] is considered. 

The cause of this initial smooth zone and the subsequent fairly sudden transition to wavy flow is not completely clear. Working on a much larger scale, with mostly turbulent flow of the liquid layers on dam spillways, Bauer (Bl) has shown that the length of the smooth initial region is the same as the distance required for the turbulent boundary layer, which... 

Saltation of solids occurs in the turbulent boundary layer where the wall effects on the particle motion must be accounted for. Such effects include the lift due to the imposed mean shear (Saffman lift, see 3.2.3) and particle rotation (Magnus effect, see 3.2.4), as well as an increase in drag force (Faxen effect). In pneumatic conveying, the motion of a particle in the boundary layer is primarily affected by the shear-induced lift. In addition, the added mass effect and Basset force can be neglected for most cases where the particle... 

An extended version of the hybrid technique of PIV/LIF/SIT is reported by Kitagawa et al. (2005), in which the PTV technique is employed to measure the velocity field in liquid phase and track the velocity distribution of dispersed bubbles, in addition to the SIT measurement of bubbles shape and location in a microbubble-laden turbulent channel flow. It is well known that microbubbles injected into the turbulent boundary layer developing on a solid wall have a significant skin friction reduction effect. To investigate the interactions between the injected microbubbles (the void fraction is actually low but... 

U. Grigull [14] pointed out that the correspondence h v0 5 means the predominance of the laminar boundary layer, whereas h v0 8 refers to the prevailing of the turbulent boundary layer. 

The mean velocity distribution in the outer portion of the turbulent boundary layer on a flat plate is approximately given by ... 

The program assumes the flow is turbulent from the leading edge and that 62 = 0 when x = 0. The program can easily be modified to use a laminar boundary layer equation solution procedure to provide initial conditions for the turbulent boundary layer solution which would then be started at some assumed transition point. 

NUMERICAL SOLUTION OF THE TURBULENT BOUNDARY LAYER EQUATIONS... 

To determine the turbulent-boundary-layer thickness we employ Eq. (5-17) for the integral momentum relation and evaluate the wall shear stress from the empirical relations for skin friction presented previously. According to Eq. (5-52),... 

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