Asymptotic approximation boundary layers

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. ... 

Solution to the nondimensional axisymmetric stagnation-flow problem is plotted in Fig. 6.3. Since the viscous boundary layer merges asymptotically into the inviscid potential flow, there is not a distinct edge of the boundary layer. By convention, the boundary-layer thickness is defined as the point at which the radial velocity comes to 99% of its potential-flow value. From Fig. 6.3 it is apparent that the boundary-layer thickness S is approximately z 2. In addition to the boundary-layer thickness, a displacement thickness can be defined. The displacement thickness is the distance that the potential-flow field appears to be displaced from the surface due to the viscous boundary layer. If there were no viscous boundary layer (i.e., the inviscid flow persisted right to the surface), then the axial velocity profile would have a constant slope du/dz = —2. As shown in Fig. 6.3, projecting the constant axial-velocity slope to the surface obtains an intercept of u = 0 at approximately z = 0.55. Since the inviscid flow would have to come to zero velocity at the surface, z = 0.55 is the distance that the potential flow is displaced due to the viscous boundary layer. Otherwise, the potential flow is unaltered by the boundary layer. 

Equations for each of the perturbation functions xu yh Xu Yl are derived by substituting the asymptotic expansions into the initial differential system, by matching terms with the same power in e, and finally by writing the proper initial and boundary layer conditions. The zeroth-order outer approximation is the solution to the system... 

Near the point where the two streams first meet the chemical reaction rate is small and a self-similar frozen-flow solution for Yp applies. This frozen solution has been used as the first term in a series expansion [62] or as the first approximation in an iterative approach [64]. An integral method also has been developed [62], in which ordinary differential equations are solved for the streamwise evolution of parameters that characterize profile shapes. The problem also is well suited for application of activation-energy asymptotics, as may be seen by analogy with [65]. The boundary-layer approximation fails in the downstream region of flame spreading unless the burning velocity is small compared with u it may also fail near the point where the temperature bulge develops because of the rapid onset of heat release there,... 

Introductory note Most transport and/or fluids problems are not amenable to analysis by classical methods for linear differential equations, either because the equations are nonlinear (or simply too comphcated in the case of the thermal energy equation, which is linear in temperature if natural convection effects can be neglected), or because the solution domain is complicated in shape (or in the case of problems involving a fluid interface having a shape that is a priori unknown). Analytic results can then be achieved only by means of approximations. One approach is to simply discretize the equations in some way and turn on the computer. Another is to use the family of approximations methods known as asymptotic approximations that lead to useful concepts such as boundary layers, etc. This course is about the latter approach. However, it is not just a... 

Let us now return to the solution of our problem for Rr 1. Although the arguments leading to (4-25) were complex, the resulting equation itself is simple compared with the original Bessel equation. Our objective here is an asymptotic approximation of the solution for the boundary-layer region. In general, we may expect an asymptotic expansion of the form... 

Of course, the solution (4-181) is only the first approximation in the asymptotic series (4 175). In writing (4-177), we neglected certain smaller terms in the nondimensionalized equation, (4-170), because they were small compared with the terms that we kept. To obtain the governing equation for the second term in the boundary-layer region, we formally substitute the expansion, (4-175), into the governing equation, (4-170) ... 

The asymptotic formulation of the previous subsection has led not only to the important result given by (9-230) but also to a very considerable simplification in the structure of the governing equation in the thermal boundary-layer region. As a consequence, it is now possible to obtain an analytic approximation for 0. 

It should be remembered that (9-225) and its solution (9-240) represent only the first term in an asymptotic expansion for Pe - oo in the boundary-layer regime, and (9-203) and its solution (9-207) is the leading-order term in a corresponding expansion for the outer region. To obtain the next level of approximation, it is necessary to calculate an additional term in both of these expansions. We do not pursue this calculation here because of the... 

A potential advantage of the physical approach to boundary-layer theory is that it forces an emphasis on the underlying physical description of the flow. However, unlike the asymptotic approach presented here, the physically derived theory provides no obvious means to improve the solution beyond the first level of approximation. Provided that the physical picture underlying the analysis is properly emphasized, the asymptotic approach can incorporate the principal positive aspect of the earlier theories within a rational framework for systematic improvement of the approximation scheme. 

In spite of this, we shall see that potential-flow theory plays an important role in the development of asymptotic solutions for Re i>> 1. Indeed, if we compare the assumptions and analysis leading to (10-9) and then to (10-12) with the early steps in analysis of heat transfer at high Peclet number, it is clear that the solution to = 0 is a valid first approximation lor Re y> 1 everywhere except in the immediate vicinity of the body surface. There the body dimension, a, that was used to nondimensionalize (10-1) is not a relevant characteristic length scale. In this region, we shall see that the flow develops a boundary layer in which viscous forces remain important even as Re i>> 1, and this allows the no-shp condition to be satisfied. 

It has been stated repeatedly that the boundary-layer and potential-flow equations apply to only the leading term in an asymptotic expansion of the solution for Re F> 1. This is clear from the fact that we derived both in their respective domains of validity by simply taking the limit Re -= oc in the appropriately nondimensionalized Navier-Stokes equations. Frequently, in the analysis of laminar flow at high Reynolds number, we do not proceed beyond these leading-order approximations because they already contain the most important information a prediction of whether or not the flow will separate and, if not, an analytic approximation for the drag. Nevertheless, the reader may be interested in how we would proceed to the next level of approximation, and this is described briefly in the remainder of this section.13... 

Problem 10-12. Higher-Order Approximations for the Blasius Problem. The classical boundary-layer theory represents only the first term in an asymptotic approximation for Re 1. However, in cases involving separation, we do not seek additional corrections because the existence of a separation point signals the breakdown of the whole theory. When the flow does not separate, we can calculate higher-order corrections, and these provide useful insight and results. In this problem, we reconsider the familiar Blasius problem of streaming flow past a semi-infinite flat plate that is oriented parallel to a uniform flow. 

Although the boundary layer in this statement of the problem is asymptotic, that is, extends infinitely along the coordinate Y, one can approximately estimate its thickness if we adopt the convention that the velocity on the boundary of the layer differs from the undisturbed flow velocity at most by 1%. Then the boundary layer thickness is... 

Diffusion boundary layer approximation. Now let us take into account the fact that common fluids are characterized by large Schmidt numbers Sc. Obviously, by substituting the leading term of the expansion of v as z -4 0 into (3.2.8) and (3.2.9), one can readily obtain the asymptotics of these formulas as Sc -4 oo. By using (3.2.5) and (3.2.8) and carrying out some transformation, we obtain the dimensionless concentration... 

It is convenient to seek the asymptotics of the function Prandtl numbers starting from Eq. (3.3.3) with the extended variable 77 = C/Pr. As a result, we obtain the equation T"c + /((/Pr)T = 0. As Pr - 0, the argument of the function /(C/Pr) tends to infinity, which corresponds to a constant velocity inside the thermal boundary layer and f(j]) 77. In the other limit case as Pr —F 00, the argument of the function /(C/Pr) tends to zero, which corresponds to the linear approximation of the velocity inside the boundary layer and f rf) O.I66772. Substituting the above-mentioned leading terms of... 

Diffusion boundary layer approximation. The concentration mostly varies on the initial interval x = 0(1), that is, in the diffusion boundary layer near the film surface. In this region, the asymptotic solution can be obtained by substituting the expanded coordinate... 

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