The boundary layer equations

There have been a number of direct solutions in the form of Equation (9.65) for convective combustion. These have been theoretical - exact or integral approximations to the boundary layer equations, or empirical - based on correlations to experimental data. Some examples are listed below ... 

The boundary layer equations for an axisymmetric body, Eqs. (1-55), (10-17), and (10-18) have been solved approximately for arbitrary Sc (L4). For Sc oo the mean value of Sh can be computed from Eq. (10-20). Solutions have also been obtained for Sc oo for some shapes without axial symmetry, e.g., inclined cylinders (S34). Data for nonspherical shapes are shown in Fig. 10.3 for large Rayleigh number. The characteristic length in Sh and Ra is analogous to that used in Chapters 4 and 6 ... 

Given the particular circumstances of the flow in a long, narrow channel, explain the reduction of the governing equations to a boundary-layer form that accommodates the momentum and species development length. Discuss the essential characteristics of the boundary-layer equations, including implications for computational solution. 

It is important to emphasize that the similarity behavior is not the result of approximations or assumptions where certain physical effects have been neglected. Instead, these are situations where the full two-dimensional behavior can be completely represented by a one-dimensional description for special sets of boundary conditions. Of course, in all finite-dimensional systems there are edge effects that violate that similarity behavior. By way of contrast, however, one may consider the difference between the governing equations used here and the boundary-layer equations for flow parallel to a solid surface. The boundary-layer equations are approximations in which certain terms are neglected because they are small compared to other terms. Thus terms are dropped even though they are not exactly zero, whereas here the mathematical reduction is accomplished because certain terms vanish naturally over nearly all of the domain (excluding edge effects). 

Boundary-layer behavior is one of several potential simplifications that facilitate channel-flow modeling. Others include plug flow or one-dimensional axial flow. The boundary-layer equations, however, are the ones that require the most insight and effort to derive and to establish the ranges of validity. The boundary-layer equations retain a full two-dimensional representation of all the field variables as well as all the nonlinear behavior of Navier-Stokes equations. Nevertheless, when applicable, they provide a very significant simplification that can be used to great benefit in modeling. 

For the boundary-layer equations, where two-dimensional flow is retained, the continuity equation must retain both terms as order-one terms. Otherwise, a purely onedimensional flow would result. Certainly there are situations where one-dimensional flow... 

The boundary-layer equations represent a coupled, nonlinear system of parabolic partial-differential equations. Boundary conditions are required at the channel inlet and at the extremeties of the y domain. (The inlet boundary conditions mathematically play the role of initial conditions, since in these parabolic equations x plays the role of the time-like independent variable.) At the inlet, profiles of the dependent variables (w(y), T(y), and Tt(y)) must be specified. The v(y) profile must also be specified, but as discussed in Section 7.6.1, v(y) cannot be specified independently. When heterogeneous chemistry occurs on a wall the initial species profile Yk (y) must be specified such that the gas-phase composition at the wall is consistent with the surface composition and temperature and the heterogeneous reaction mechanism. The inlet pressure must also be specified. 

The method of lines is a computational technique that is particularly suited for solving coupled systems of parabolic partial-differential equations (PDE). The boundary-layer equations can be solved by the method of lines (MOL), although the task is facilitated considerably by casting the problem in a differential-algebraic setting [13]. As an introductory illustration, consider the heat equation... 

In the Von Mises form of the boundary-layer equations, it is r2 that must satisfy the consistent initial condition. Here, however, it is much easier to see that the initial profiles are not independent. Once the mesh in x/r has been set and the inlet profiles of u and T given, the consistent profiles in r2 follow easily from the solution of Eq. 7.60. 

Figure 7.9 shows a cylindrical duct whose radius varies as a function of position, R(z). As long as the radius varies smoothly and relatively smoothly, the channel flow may be treated as a boundary-layer problem. Discuss what, if any, changes must be made to the boundary-layer equations and the boundary-condition specifications to solve the variable-area boundary-layer problem. 

Scaling arguments are used to establish the circumstances where the boundary-layer behavior is valid. These arguments, which are usually made for external flows over surfaces, may be found in many texts on fluid mechanics (e.g., [350]). The essential feature of the boundary-layer approximation is that there is a principal flow direction in which the convective effects significantly dominate the diffusive behavior. As a result the flow-wise diffusion may be neglected, while the cross-flow diffusion and convection are retained. Mathematically this reduction causes the boundary-layer equations to have essentially parabolic characteristics, whereas the Navier-Stokes equations have essentially elliptic characteristics. As a result the computational simulation of the boundary-layer equations is much simpler and more efficient. 

These relationships have been used by Spalding in the dimensionless presentation both of theoretical values obtained in his approximate solution of the boundary layer equations (58) and of the experimental data (51, 55, 60). Emmons (3), who has solved the problem of forced convection past a burning liquid plane surface in a more rigorous fashion, shows graphically the rather close correspondence between values obtained from his exact solution and that of Spalding, and between the calculated values for flat plates and the experimental values for spheres. 

For the case of two-dimensional wavy film flow, Levich (L9) has shown that Eqs. (4) and (5) reduce to the familiar form of the boundary layer equations ... 

The boundary layer equations may be obtained from the equations provided in Tables 6.1-6.3, with simplification and by an order-of-magnitude study of each term in the equations. It is assumed that the main flow is in the x direction. The terms that are too small are neglected. Consider the momentum and energy equations for the two-dimensional, steady flow of an incompressible fluid with constant properties. The dimensionless equations are given by Eqs. (6.46) to (6.48). The principal assumption made in the boundary layer is that the hydrodynamic boundary layer thickness 8 and the thermal boundaiy layer thickness 8t are small compared to a characteristic dimension L of the body. In mathematical terms,... 

Write the boundary layer equations in dimensional form. 

In the derivation of the boundary layer equations, two of the assumptions made are as follows ... 

Several approximate methods exist for solving the boundary layer equations. The momentum-integral method of analysis is an important method. The principal steps of the method are listed below. 

The boundary layer equations were derived in a previous chapter, or may be deduced from the general convection equations in the early part of this chapter. For two-dimensional, steady flow over a flat plate of an incompressible, constant-property fluid, the continuity, x-momentum and the energy equations are as follows ... 

Because they do not contain the pressure as a variable. Eqs. (2.76) and (2.77) have been used quite extensively in solving problems for which the boundary layer equations (see later) cannot be used. For this purpose, instead of solving the Navier-Stokes and energy, simultaneously with the continuity equation, it is convenient to introduce the stream function, ip, which is defined such that... 

In order to illustrate how the boundary layer equations are derived, [7],[9],[13], [ 14],[ 15], consider two-dimensional constant fluid property flow over a plane surface which is set parallel to the x axis. The following are then defined ... 

There are some cases where this approach fails. One such case is that in which significant regions of separated flow exist. In this case, although the boundary layer equations are adequate to describe the flow upstream of the separation point, the presence of the separated region alters the effective body shape for the outer inviscid flow and the velocity outside the boundary layer will be different from that given by the inviscid flow solution over the solid surface involved. For example, consider flow over a circular cylinder as shown in Fig. 2.16. Potential theory gives the velocity, ui, on the surface of the cylinder as ... 

This equation then relates the pressure gradient term in the boundary layer equations to the freestream velocity distribution. 

Now, the rest of the terms retained in the boundary layer equations have the order of magnitude of unity and, therefore, for the boundary layer equations to apply, the dimensionless turbulence terms (u 2lu ) and (u v /u ), which are assumed to have the same order of magnitude, will have the order of magnitude of (8/L) at most. The first term in Eq. (2.154) is, therefore, negligible compared to the rest of the terms in the boundary layer equations. Therefore, the x-wise momentum equation for turbulent boundary layer flow is ... 

While the boundary layer equations that were derived in the preceding sections are much simpler than the general equations from which they were derived, they still form a complex set of simultaneous partial differential equations. Analytical solutions to this set of equations have been obtained in a few important cases. For the majority of flows, however, a numerical solution procedure must be adopted. Such solutions are readily obtained today using modest modem computing facilities. This was, however, not always so. For this reason, approximate solutions to the boundary layer equations have in the past been quite widely used. While such methods of solution are less important today, they are still used to some extent. One such approach will, therefore, be considered in the present text. 

Discuss the assumptions used in deriving the boundary layer equations. 

In all the solutions given in the present chapter, the fluid properties will be assumed to be constant and the flow will be assumed to be two-dimensional. In addition, dissipation effects in the energy equation will be neglected in most of this chapter, these effects being briefly considered in a last section of this chapter. Also, solutions to the full Navier-Stokes and energy equations will be dealt with only relatively briefly, the majority of the solutions considered being based on the use of the boundary layer equations. 

Similarity and integral equation methods for solving the boundary layer equations have been discussed in the previous sections. In the similarity method, it will be recalled, the governing partial differential equations are reduced to a set of ordinary differential equations by means of a suitable transformation. Such solutions can only be obtained for a very limited range of problems. The integral equation method can, basically, be applied to any flow situation. However, the approximations inherent in the method give rise to errors of uncertain magnitude. Many attempts have been made to reduce these errors but this can only be done at the expense of a considerable increase in complexity, and, therefore, in the computational effort required to obtain the solution. 

Another way of solving the boundary layer equations, involves approximating the governing partial differential equations by algebraic finite-difference equations [11]. The main advantages of this type of solution procedure are ... 

There are a number of schemes for numerically approximating the boundary layer equations and many different solution procedures based on these various schemes have been developed. In the present section, one of the simpler finite-difference schemes will be described. The solution procedure based on this scheme should give quite acceptable results for most problems. The scheme is easily extended to deal with turbulent ftnws as will he Hi rncce/t i AH i[Pg.123]

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