Boundary layer velocity profiles

While the engineer may frequently be interested in the heat-transfer characteristics of flow systems inside tubes or over flat plates, equal importance must be placed on the heat transfer which may be achieved by a cylinder in cross flow, as shown in Fig. 6-7. As would be expected, the boundary-layer development on the cylinder determines the heat-transfer characteristics. As long as the boundary layer remains laminar and well behaved, it is possible to compute the heat transfer by a method similar to the boundary-layer analysis of Chap. 5. It is necessary, however, to include the pressure gradient in the analysis because this influences the boundary-layer velocity profile to an appreciable extent. In fact, it is this pressure gradient which causes a separated-flow region to develop on the back side of the cylinder when the free-stream velocity is sufficiently large. 

Figure 12-6 shows the boundary-layer velocity profiles which result from various injection rates in a laminar boundary layer. The injection parameter... 

Figure 4.7-1. Boundary-layer velocity profile for natural convection heat transfer from a heated, vertical plate.

The calculations showed that there is no single boundary layer velocity profile in the DMR region instead, the profiles depend on i because of gradients in the mean flow. They vary systematically from the clean n = 1/7 power function near the foot of the Mach stem to the DG dusty boundary... 

The limit Sc - oo implies that the normal velocity V(x,0) is asymptotically small regardless of the size of B. Thus the change in V(x, 0) is too small to affect the leading-order boundary-layer velocity distributions. Nevertheless, we shall see that the mass transferrate is still changed. The other two limits B -> oo or B - -1 both correspond to V(x, 0) —> oo, which means that the velocity profiles will change and there is an intimate coupling between the momentum and mass transfer equations. 

Czarske J, Biittner L, Razik T, Muller H (2002) Boundary layer velocity measurements by a laser Doppler profile sensor with micrometre spatial resolution. Meas Sci Technol 13(12) 1979-1989... 

Entrance flow is also accompanied by the growth of a boundary layer (Fig. 5b). As the boundary layer grows to fill the duct, the initially flat velocity profile is altered to yield the profile characteristic of steady-state flow in the downstream duct. For laminar flow in a tube, the distance required for the velocity at the center line to reach 99% of its asymptotic values is given by... 

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

The thickness of the boundary layer may be arbitrarily defined as the distance from the surface at which the velocity reaches some proportion (such as 0.9, 0.99, 0.999) of the undisturbed stream velocity. Alternatively, it may be possible to approximate to the velocity profile by means of an equation which is then used to give the distance from the surface at which the velocity gradient is zero and the velocity is equal to the stream velocity. Difficulties arise in comparing the thicknesses obtained using these various definitions, because velocity is changing so slowly with distance that a small difference in the criterion used for the selection of velocity will account for a very large difference in the corresponding thickness of the boundary layer. 

At the distant edge of the boundary layer it is assumed that the velocity just equals the main stream velocity and that there is no discontinuity in the velocity profile. 

This relation for the thickness of the boundary layer has been obtained on the assumption that the velocity profile can be described by a polynomial of the form of equation 11.10 and that the main stream velocity is reached at a distance 8 from the surface, whereas, in fact, the stream velocity is approached asymptotically. Although equation 11.11 gives the velocity ux accurately as a function of v, it does not provide a means of calculating accurately the distance from the surface at which ux has a particular value when ux is near us, because 3ux/dy is then small. The thickness of the boundary layer as calculated is therefore a function of the particular approximate relation which is taken to represent the velocity profile. This difficulty cat be overcome by introducing a new concept, the displacement thickness 8. ... 

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. 

Thus, the shear stress is expressed as a function of the boundary layer thickness S 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 velocities, 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 < in equation 11.25 must be zero or ... 

At the inlet to the pipe the velocity across the whole section is constant. The velocity at the pipe axis will progressively increase in the direction of flow and reach a maximum value when the boundary layers join. Beyond this point the velocity profile, and the velocity at the axis, will not change. Since the fluid at the axis has been accelerated, its kinetic energy per unit mass will increase and therefore there must be a corresponding all in its pressure energy. 

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)... 

Obtain the momentum equation for an element of boundary layer. If the velocity profile in the laminar region may be represented approximately by a sine function, calculate the boundary-layer thickness in terms of distance from the leading edge of the surface. 

Explain why it is necessary to use concepts, such as the displacement thickness and the momentum thickness, for a boundary layer in order to obtain a boundary layer thickness which is largely independent of the approximation used for the velocity profile in the neighbourhood of the surface. 

Show that the velocity profile in the neighbourhood of the surface may be expressed as a sine function which satisfies the boundary conditions at the surface and at the outer edge of the boundary layer. 

Obtain the boundary layer thickness and its displacement thickness as a function of the distance from the leading edge of Ihe surface, when the velocity profile is expressed as a sine function. 

Figure 2.6 shows a typical temperature profile.t l The temperature boundary layer is similar to the velocity layer. The flowing gases heat rapidly as they come in contact with the hot surface of the tube, resulting in a steep temperature gradient. The average temperature increases toward downstream. 

The boundary layers for these three variables (gas velocity, temperature, and concentration) may sometimes coincide, although in slow reactions, the profiles of velocity and temperature may be fully developed at an early stage while the deposition reaction is spread far downstream the tube. 

Boundary layer similarity solution treatments have been used extensively to develop analytical models for CVD processes (2fl.). These have been useful In correlating experimental observations (e.g. fi.). However, because of the oversimplified fiow description they cannot be used to extrapolate to new process conditions or for reactor design. Moreover, they cannot predict transverse variations In film thickness which may occur even In the absence of secondary fiows because of the presence of side walls. Two-dimensional fully parabolized transport equations have been used to predict velocity, concentration and temperature profiles along the length of horizontal reactors for SI CVD (17,30- 32). Although these models are detailed, they can neither capture the effect of buoyancy driven secondary fiows or transverse thickness variations caused by the side walls. Thus, large scale simulation of 3D models are needed to obtain a realistic picture of horizontal reactor performance. 

Another device that finds frequent use is the stirred cell shown in Fig. 20-54. This device uses a membrane coupon at the bottom of the reservoir with a magnetic stir bar. Stirred cells use low fluid volumes and can be used in screening and R D studies to evaluate membrane types and membrane properties. The velocity profiles have been well defined (Schlichting, Boundary Layer Theory, 6th ed., McGraw-Hill, New York, 1968, pp. 93-99). 

Fig. 4.5.16 Schematic drawing of a boundary layer mixing mechanism. It is proposed that a thin layer with thickness 8 has a linear velocity profile with average velocity V/2. Material with bulk droplet volume fraction ( >in is drawn into the creamed layer (area Ac) and material with average creamed layer volume fraction (j)ou, is swept out. The remainder of the emulsion (inside the dashed circle) is stagnant.

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