Solution Procedures

Now that we are confronted by an intimidating set of partial-differential equations and complex boundary conditions, it is appropriate to discuss strategies for finding solutions. One method that is doomed to fail is to mount a frontal attack in an attempt to find a general solution to the problem. The student has no doubt already noticed that considerable effort has gone into examining special cases for the gov­erning equations as set forth in the tables in this chapter. For example, equations pertaining to steady, incompressible, or inviscid flows are carefully worked out. It should be obvious that our intention is to approach each problem from the stand­point of which simplifications can be made to reduce it to the most understandable and mathematically tractable form. In many cases, this results in a problem that can be solved fairly easily even without recourse to numerical means. Some of the most elegant and useful solutions were created using this approach.

We call this process the art of approximation. It was practiced in the past by the most innovative workers in aeronautics, and the various solutions are named after their creators. Confronted by the difficult nonlinear equations we now have derived, these investigators carefully determined where simplifying assumptions could be used to make the problem mathematically tractable. Students of these remarkable indi­viduals were required to learn this approach and often went on to improve the results or to create new methods of their own. As students study this textbook, they are pre­sented with numerous examples that show how this approach can be used to bring about extraordinary physical insight, as well as practical solutions, to difficult problems.

Many types of approximations can be introduced. These may take the form of appropriate physical assumptions such as the flow being incompressible or inviscid. They also may be based on linearization techniques such as those used in supersonic small-disturbance theory. These methods of approximation all receive careful atten­tion throughout the book.

Unfortunately, the analytical approach is not used as often at present as it was in the past. Some aerodynamicists believe that all of the “simple results have already been deduced.” Too often, we observe a tendency—when confronted by a diffi­cult aerodynamics problem—for an investigator to go directly to an experimental approach, or to what is equivalent to an experiment—the numerical solution by means of computational methods such as CFD. The latter approach is based on the ready availability of powerful and fast digital computers that can solve complex sets of differential equations (usually written by representing derivatives in an algebraic difference form) by iterative methods. In effect, this replaces the problem formu­lation by a general-purpose “black box.” In some complex situations, this is the only approach that can lead to the necessary information. However, it is important to understand that nothing can take the place of a thorough physical understanding of a problem. Therefore, it is of the utmost importance to practice solving problems in an approximate way by applying a set of appropriate simplifying assumptions. Then, even if the final solution requires relaxing some of those assumptions, and a brute – force numerical attack cannot be avoided, the means to check the numerical compu­tations are available. Many costly errors have been made because this principle was not clearly understood. Therefore, it is a major goal of this textbook to help students develop a thorough understanding of the available methods of solution, their rela­tive value in typical real-life situations, and their relationships to one another.

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