The Boundary Layer Integral Equations

Control volume used in deriving the boundary layer integral equation. 

The boundary layer integral equations have been derived above without recourse to the partial differential equations for boundary layer flow. They can, however, be determined directly from these equations. Consider, for example, the laminar momentum equation (2.140). Integrating this equation across the boundary layer to some distance from the wall, i being greater than the boundary layer thickness, gives because du/dy is zero outside the boundary layer and because dp/dx is independent of y ... 

The quite extensive attention given above to the boundary layer integral equations must not be taken as indicating that these equations are, today, widely used. A study of the derivations does help in gaining an understanding of the meaning of the equations governing convective heat transfer and are, to some extent, the basis... 

Attention was then turned to developing duct flows. A numerical solution for thermally developing flow in a pipe was first considered. Attention was then turned to plane duct flow when both the velocity and temperature fields are simultaneously developing. An approximate solution based on the use of the boundary layer integral equations was discussed. 

Using the slug-flow model, show that the boundary-layer energy equation reduces to the same form as the transient-conduction equation for the semi-infinite solid of Sec. 4-3. Solve this equation and compare the solution with the integral analysis of Sec. 6-5. 

It may be noted that no assumptions have been made concerning the nature of the flow within the boundary layer and therefore this relation is applicable to both the streamline and the turbulent regions. The relation between ux and y is derived for streamline and turbulent flow over a plane surface and the integral in equation 11.9 is evaluated. 

The integral in equation 11.55 clearly has a finite value within the thermal boundary layer, although it is zero outside it. When the expression for the temperature distribution in the boundary layer is inserted, the upper limit of integration must be altered from /... 

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 ... 

Taking account of the boundary conditions, this equation can be integrated by elementary methods at each given instant and in a given layer. This determines the function late stage the plane field may be represented in the form H2 = curl (n ), where n = (0,0,1). After this the function (which now coincides with the vector potential component Az) is also subject to an equation of the heat conduction type. Consequently, H2 decays asymptotically. 

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. 

Integrate x-momentum equation with respect to y over the boundary layer thickness 8(x). Eliminate velocity component v(x,y) in the equation by means of the continuity equation, resulting in the momentum integral equation. 

The limiting forms of the equations that result from the application of these conservation principles to this control volume as dx -+ 0 give a set of equations governing the average conditions across the boundary layer. These resultant equations are termed the boundary layer momentum integral and energy integral equations. 

In order to illustrate how these integral equations are derived, attention will be given to two-dimensional, constant fluid property flow. First, consider conservation of momentum. It is assumed that the flow consists of a boundary layer and an outer inviscid flow and that, because the boundary layer is thin, the pressure is constant across the boundary layer. The boundary layer is assumed to have a distinct edge in the present analysis. This is shown in Fig. 2.20. 

This is termed the boundary layer momentum integral equation. As previously mentioned, it is equally applicable to laminar and turbulent flow. In laminar flow, u is the actual steady velocity while in turbulent flow it is the time averaged value. 

The way in which the momentum integral equation is applied will be discussed in detail in the next chapter. Basically, it involves assuming the form of the velocity profile, i.e., of the variation of u with y in the boundary layer. For example, in laminar flow a polynomial variation is often assumed. The unknown coefficients in this assumed form are obtained by applying the known condition on velocity at the inner and outer edges of the boundary layer. For example, the velocity must be zero at the wall while at the outer edge of the boundary layer it must become equal to the freestream velocity, u. Thus, two conditions that the assumed velocity profile must satisfy are ... 

In writing this equation, it has been noted that since be lies in the freestream where the temperature is constant, there can be no heat transfer into the control volume through it. Longitudinal conduction effects have also been ignored because the boundary layer is assumed to be thin. This is consistent with the neglect of the effects of longitudinal viscous forces in the derivation of the momentum integral equation. 

This is termed the boundary layer energy integral equation. 

The energy integral equation can be derived in a similar way from the energy equation (2,145). When this equation is integrated across the boundary layer it gives if the dissipation term is ignored ... 

Consider the continuity equation (3.1). Because the boundary conditions given in Eq. (3.6) give v as zero at the wall, this continuity equation can be integrated with respect to y leading to the following expression for v at any point in the boundary layer. 

Since the variation of /" with tj is given by the solution for the velocity profile, this equation is easily integrated to give the variation of 6 with rj. It will be noted that /", which is equal to d(u/ui)ldrjy tends to zero outside the boundary layer, i.e., at large values of 17. The actual value of the upper limit of the integrals in Eq. (3.36) will not, therefore, affect the result provided it is sufficiently large. 

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. 

Some of the commonly used methods for obtaining solutions to problems involving laminar external flows have been discussed in this chapter. Many such problems can be treated with adequate accuracy using the boundary layer equations and similar ity integral and numerical methods of solving these equations have been discussed. A brief discussion of the solution of the full governing equations has also been presented. 

The integral equation analysis given in Chapter 6 solved for the boundary layer momentum thickness, 62, which is related to the displacement thickness by the form factor, H, which is defined by ... 

Using this, equation, derive the boundary layer momentum integral equation for this type of flow. 

The approximate integral equation method that was discussed in Chapters 2 and 3 can also be applied to the boundary layer flows on surfaces in a porous medium. As discussed in Chapters 2 and 3, this integral equation method has largely been superceded by purely numerical methods of the type discussed above. However, integral equation methods are still sometimes used and it therefore appears to be appropriate to briefly discuss the use of the method here. Attention will continue to be restricted to two-dimensional constant fluid property forced flow. 

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 ... 

This is the integral momentum equation of the boundary layer. If the pressure is constant throughout the flow,... 

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