Boundary-layer thickness definition

Figure 1.25 shows the boundary layer that develops over a flat plate placed in, and aligned parallel to, the fluid having a uniform velocity upstream of the plate. Flow over the wall of a pipe or tube is similar but eventually the boundary layer reaches the centre-line. Although most of the change in the velocity component vx parallel to the wall takes place over a short distance from the wall, it does continue to rise and tends gradually to the value vx in the fluid distant from the wall (the free stream). Consequently, if a boundary layer thickness is to be defined it has to be done in some arbitrary but useful way. The normal definition of the boundary layer thickness is that it is the distance from the solid boundary to the location where vx has risen to 99 per cent of the free stream velocity v . The locus of such points is shown in Figure 1.25. It should be appreciated that this is a time averaged distance the thickness of the boundary layer fluctuates owing to the velocity fluctuations.

Figure 4-21 The concept of boundary layer and boundary layer thickness 5. (a) Compositional boundary layer surrounding a falling and dissolving spherical crystal. The arrow represents the direction of crystal motion. The shaded circle represents the spherical particle. The region between the solid circle and the dashed oval represents the boundary layer. For clarity, the thickness of the boundary layer is exaggerated, (b) Definition of boundary layer thickness 5. The compositional profile shown is "averaged" over all directions. From the average profile, the "effective" boundary layer thickness is obtained by drawing a tangent at x = 0 (r=a) to the concentration curve. The 5 is the distance between the interface (x = 0) and the point where the tangent line intercepts the bulk concentration.

While there is no distinct edge to the boundary layer, it is convenient to have some measure of the distance from the wall over which significant effects of viscosity exist. For this reason, it is convenient to arbitrarily define the boundary layer thickness, Su, as the distance from the wall at which u reaches to within 1% of the freestream velocity, i.e., to define 8 as the value of y at which u = 0.99uj. Using the result given in Fig. 3.4 then shows that u = 0.99wi, i.e., / = 0.99, when 17 = 5 which indicates, in view of the definition of 77, that 8U is approximately given by ... 

Equation 8.2 shows how the net flux density of substance depends on its diffusion coefficient, Dj, and on the difference in its concentration, Ac] 1, across a distance Sbl of the air. The net flux density Jj is toward regions of lower Cj, which requires the negative sign associated with the concentration gradient and otherwise is incorporated into the definition of Acyin Equation 8.2. We will specifically consider the diffusion of water vapor and C02 toward lower concentrations in this chapter. Also, we will assume that the same boundary layer thickness (Sbl) derived for heat transfer (Eqs. 7.10-7.16) applies for mass transfer, an example of the similarity principle. Outside Sbl is a region of air turbulence, where we will assume that the concentrations of gases are the same as in the bulk atmosphere (an assumption that we will remove in Chapter 9, Section 9.IB). Equation 8.2 indicates that Jj equals Acbl multiplied by a conductance, gbl, or divided by a resistance, rbl. 

Recall that we defined the boundary layer thickness as the distance from the surface for which utV 0.99. We observe from Table 6-3 that the value of tj corresponding to ulV = 0.99 is = 4.91. Substituting rj = 4.9J andy = B into the definition of the similarity variable (Eq. 6-43) gives 4.91 = 5 Vv/vx. Then the velocity boundary layer thickness becomes... 

Solving Eq. (S-58 numerically for the temperature profile for different Prandtl numbers, and using the definition of the thermal boundary layer, it is determined that 8/S, = Pr. Then the thermal boundary layer thickness becomes... 

The velocity boundary layer concept can be extended to define the temperature and concentration of a fluid. The temperature boundary layer thickness is the distance from the body to a layer at which the temperature is 99% of the temperature from an inviscid solution. The boundary layer thickness for the fluid concentration has the same definition. Their relationships are expressed by [29]... 

The boundary layer thickness, 8, is defined as the distance that is required for the flow to almost reach If. We might take an arbitrary number (say 99%) to define practically what we mean by nearly, but certain other definitions are used for convenience. The displacement and the momentum thicknesses are alternative measures of the boundary layer thickness and are used in the calculation of various boundary layer assets. 

This procedure for estimating the diffusion layer thickness was employed by Levich (1962), but recall that any definition of a boundary layer thickness is to some extent arbitrary. All of the results derived are seen to be consistent with the order-of-magnitude estimates given in the previous section. 

The Nernst boundary layer thickness is a simple characteristic of the mass transfer but its definition is formal since no boundary layer is in fact stagnant and least of all boundary layers on gas-evolving electrodes furthermore, the Schmidt number, known to influence mass transfer, is not incorporated in the usual dimensionless form. For this reason, lines representing data from gas evolution in two different solutions can be displaced from one another because of viscosity differences. Nevertheless, the exponent in the equation = aib (32)... 

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. 

The limitation of using such a model is the assumption that the diffusional boundary layer, as defined by the effective diffusivity, is the same for both the solute and the micelle [45], This is a good approximation when the diffusivities of all species are similar. However, if the micelle is much larger than the free solute, then the difference between the diffusional boundary layer of the two species, as defined by Eq. (24), is significant since 8 is directly proportional to the diffusion coefficient. If known, the thickness of the diffusional boundary layer for each species can be included directly in the definition of the effective diffusivity. This approach is similar to the reaction plane model which has been used to describe acid-base reactions. 

The exact solution of the problem leads to the same expression with a proportionality constant between 3 and 5, depending on the definition of the thickness of the boundary layer. In the following sections, the preceding evaluation procedure is applied to a large number of problems, particularly to complex cases for which limiting solutions can be obtained. As already noted in the introduction, the terms in the transport equations will be replaced by their evaluating expressions multiplied by constants. The undetermined constants will then be determined from solutions available for some asymptotic cases. 

Since the electro-osinotic flow is induced by the interaction of the externally applied electric field with the space charge of the diffuse electric double layers at the channel walls, we shall concentrate in our further analysis on one of these 0 1 2) thick boundary layers, say, for definiteness, at... 

As with the velocity boundary layer, the thermal boundary layer is assumed to have a definite thickness, dr, and outside this boundary layer the temperature is assumed to be constant. 

As discussed in Chapters 2 and 3, in the integral method it is assumed that the boundary layer has a definite thickness and the overall or integrated momentum and thermal energy balances across the boundary layer are considered. In the case of flow over a body in a porous medium, if the Darcy assumptions are used, there is, as discussed before, no velocity boundary layer, the velocity parallel to the surface near the surface being essentially equal to the surface velocity given by the potential flow solution. For flow over a body in a porous medium, therefore, only the energy integral equation need be considered. This equation was shown in Chapter 2 to be ... 

The drag coefficient is not only related to the shape and orientation of an object, as shown in Figure 6.28, it is also affected by the Reynolds number (Re) of the flow gas. This can be explained by inserting this statement into the definition of Reynolds number in Equation (2.58). The thickness of the boundary layer is decreased as the Reynold number increases, therefore reducing the contribution of the velocity boundary layer to the drag force. 

This definition merely represents a simplifying modef assumption as opposed to the conventional definition in which the thickness of boundary layer 6 is arbitrarily taken as the distance away from the surface where the velocity reaches 99% of the free stream velocity... 

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