The turbulent portion

Blasius has given the following approximate expression for the shear stress at a plane smooth surface over which a fluid is flowing with a velocity Us, for conditions 

the shear stress is expressed as a function of the boundary layer thickness and it is therefore implicitly assumed that a certain velocity profile exists in the fluid. As a first assumption, it may be assumed that a simple power relation exists between the velocity and the distance from the surface in the boundary layer, or  

If the velocity profile is the same for all stream velcK ities, the shear stress must be defined by specifying the velocity Ux at any distance y from the surface. The boundary layer thickness, determined by the velocity profile, is then no longer an independent variable so that the index of 8 in equation 11.25 must be zero or  

Equation 11.26 is sometimes known as the Prandtl seventh power law. Differentiating equation 11.26 with respect to y gives  

Putting the constant equal to zero, implies that 5 = 0 when j = 0, that is that the turbulent boundar 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 surluce is only slightly longer than the critical distance x,. for the laminar-turbulent transition. 

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The Reynolds analogy is equivalent to setting the turbulent Prandtl number as defined in Eq. (24) equal to unity. Figure 13 shows the effect of Reynolds number upon a space average value of the turbulent Prandtl number (C3, P3, S7). Information presented in Fig. 13 is open to uncertainty since it is based upon measurements for air and represents only the space average value of this ratio throughout the turbulent portion of the stream. The turbulent Prandtl number is undoubtedly a function of position as well as of the Reynolds number for a given stream (PI). 

In the discussion of the use of the Reynolds analogy for the prediction of the heat transfer rate from a flat plate it was assumed that when there was transition on the plate, the x-coordinate in the turbulent portion of the flow could be measured from the leading edge. Develop an alternative expression based on the assumption that the momentum thickness before and after transition is the same. This assumption allows an effective origin for the x-coordinate in die turbulent portion of the flow to be obtained. 

The turbulent portion of the fully developed frictional pressure drop from Chapter 3 is written as ... 

For turbulent flow of a fluid past a solid, it has long been known that, in the immediate neighborhood of the surface, there exists a relatively quiet zone of fluid, commonly called the Him. As one approaches the wall from the body of the flowing fluid, the flow tends to become less turbulent and develops into laminar flow immediately adjacent to the wall. The film consists of that portion of the flow which is essentially in laminar motion (the laminar sublayer) and through which heat is transferred by molecular conduction. The resistance of the laminar layer to heat flow will vaiy according to its thickness and can range from 95 percent of the total resistance for some fluids to about I percent for other fluids (liquid metals). The turbulent core and the buffer layer between the laminar sublayer and turbulent core each offer a resistance to beat transfer which is a function of the turbulence and the thermal properties of the flowing fluid. The relative temperature difference across each of the layers is dependent upon their resistance to heat flow. 

Equations (22-86) and (22-89) are the turbulent- and laminar-flow flux equations for the pressure-independent portion of the ultrafiltra-tion operating curve. They assume complete retention of solute. Appropriate values of diffusivity and kinematic viscosity are rarely known, so an a priori solution of the equations isn t usually possible. Interpolation, extrapolation, even precuction of an operating cui ve may be done from limited data. For turbulent flow over an unfouled membrane of a solution containing no particulates, the exponent on Q is usually 0.8. Fouhng reduces the exponent and particulates can increase the exponent to a value as high as 2. These equations also apply to some cases of reverse osmosis and microfiltration. In the former, the constancy of may not be assumed, and in the latter, D is usually enhanced very significantly by the action of materials not in true solution. 

Two trains coming from opposite directions approached the area where the cloud was present. Each consisted of an electrically powered locomotive and 19 coaches constructed of metal and wood. The turbulence of the trains probably mixed up the vapor and mist with overlying air to form a flammable cloud portion. Either train could have ignited the cloud, most likely at catenary wires which powered the locomotives. 

Fuel from a fiilly unobstructed jet would be diluted to a level below its lower flammability limit, and the flammable portion of the cloud would be limited to the jet itself. In practice, however, jets are usually somehow obstructed by objects such as the earth s surface, surrounding structures, or equipment. In such cases, a large cloud of flammable mixture will probably develop. Generally, such a cloud will be far from stagnant but rather in recirculating (turbulent) motion driven by the momentum of the jet. 

Tubular Anodes Tubular anodes are supplied in diameters between 12-5 and 32 mm and have been designed for installations where water conditions on the plant under protection are known to be turbulent. The tubular anode has a number of holes drilled in the active portion of the anode and the nonactive portion is filled with sand to act as a damping agent. As in the case of rod anodes they are supplied complete with mounts ready for installation in the prepared bosses on the plant under protection. They are particularly suitable for internal protection of pump casings and internal protection of pipelines, carrying salt water or other low resistivity liquids. 

It may be assumed that the fully turbulent portion of the boundary layer starts at y+ — 30, that the ratio of the mixing length to the distance y from the surface, Af/v = 0.4, and that for a smooth surface u+ = I4 at v 30. 

Finally, there must be a flame acceleration mechanism, such as congested areas, within the flammable portion of the vapor cloud. The overpressures produced by a vapor cloud explosion are determined by the speed of flame propagation through the cloud. Objects in the flame pathway (such as congested areas of piping, process equipment, etc.) enhance vapor and flame turbulence. This turbulence results in a much faster flame speed which, in turn, can produce significant overpressures. Confinement that limits flame expansion, such as solid decks in tnulti-lcvc process structures, also increases flame speed. Without flame acceleration, a large fireball or flash fire can result, but not an explosion. 

Figure 7.4 Representations of hydrodynamic flow, showing (a) laminar flow through a smooth pipe and (b) turbulent flow, e.g. as caused by an obstruction to movement in the pipe. The length of each arrow represents the velocity of the increment of solution. Notice in (a) how the flow front is curved (known as Poiseuille flow ), and in (b) how a solution can have both laminar and turbulent portions, with the greater pressure of solution flow adjacent to the obstruction.

In Example 8.5, we saw how the diffusive boundary layer could grow. The boundary never achieves 1 m in thickness or even 5 cm in thickness, because of the interaction of turbulence and the boundary layer thickness. The diffusive boundary layer is continually trying to grow, just as the boundary layer of Example 8.5. However, turbulent eddies periodically sweep down and mix a portion of the diffusive boundary layer with the remainder of the fluid. It is this unsteady character of the turbulence... 

It is probable that ignition at bright spots occurs due to shock compression, however combustion of the remaining portion of the substance may be related to turbulent combustion. Satisfaction of the Chapman-Jouguet rule in spin detonation appears probable and is in agreement with experiment, however a rigorous proof of this assertion is lacking. 

The tubes in Figs. 10.66 and 10.6c are flowing full, and, because of the turbulence, the jets issuing from them will have a broomy appearance. Because of the contraction of the jet at the entrance to these tubes, the local velocity in the central portion of the stream will be... 

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