Boundary Layers over Arbitrary Bodies: Finite-Difference Solution

“Exact” solutions of the boundary layer equations. Equations (17.28)—(17.31), for the flow over bodies of arbitrary shape did not occur until the advent of the high-speed digital computer and ultimately not until the beginnings of computational fluid dy­namics. In this section we discuss a finite-difference technique for solving the general boundary layer equations; such finite-difference solutions represent the current state of the art in the analysis of boundary layers.

Let us set the perspective for our discussion. Equations (17.28)—(17.31) are the general boundary layer equations. For the special case of the flat plate, these equa­tions reduced to Equations (18.42) and (18.43), and for the stagnation region they reduced to Equations (18.63) and (18.64). In both special cases, these equations in terms of the transformed dependent and independent variables led to self-similar solu­tions (flow variations only in the transformed i] direction). For the general case of an arbitrary body, it is still useful to transform the full boundary-layer equations, Equa­tions (17.28)—(17.31), via the transformation given by Equations (18.59)-( 18.62). For a detailed derivation of these transformed equations, see Chapter 6 of Reference 55. The resulting form of the equation is:

x momentum:

[18.86]

where as before C = ррь/pe/ze, /’ = и/ие, and g = hj he. In Equations (18.84)—

(18.86) , the prime denotes the partial derivative with respect to q, that is, f = df/dq. Equations (18.84)—(18.86) are simply the transformed versions of Equations (17.28)-

(17.31) , with no loss of authority.

Examine Equations (18.84)—(18.86); they are the transformed compressible boundary layer equations. They are still partial differential equations, where both / and g are functions of § and r]. They contain no further approximations or assump­tions beyond those associated with the original boundary-layer equations. However, they are certainly in a less recognizable, somewhat more complicated-looking form than the original equations. However, do not be disturbed by this; in reality Equa­tions (18.84)—(18.86) are in a form that proves to be practical and useful.

The above transformed boundary-layer equations must be solved subject to the following boundary conditions. The physical boundary conditions were given imme­diately following Equations (17.28)—(17.31); the corresponding transformed bound­ary conditions are:

At the wall: q = 0 / = /’ = 0 g = gw (fixed wall temperature)

or g’ = 0 (adiabatic wall)

At the boundary-layer edge: q —>• oo /’ = 1 g=l

In general, solutions of Equations (18.84), (18.85), and (18.86) along with the appropriate boundary conditions yield variations of velocity and enthalpy throughout the boundary layer, via и = uef(%, q) and h = heg(£, q). The pressure throughout the boundary layer is known, because the known pressure distribution (or equivalently the known velocity distribution) at the edge of the boundary is given by pe = pe(^ ), and this pressure is impressed without change through the boundary layer in the lo­cally normal direction via Equation (18.85), which says that p = constant in the normal direction at any § location. Finally, knowing h and p throughout the bound­ary layer, equilibrium thermodynamics provides the remaining variables through the appropriate equations of state, for example, T = T{h, p), p = p(h, p), etc.

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