Theoretical (Analytical) Solutions

Students learning any field of physical science or engineering at the beginning are usually introduced to nice, neat analytical solutions to physical problems that are simplified to the extent that such solutions are possible. For example, when Newton’s second law is applied to the motion of a simple, frictionless pendulum, students in elementary physics classes are shown a closed form analytical solution for the time period of the pendulum’s oscillation, namely,

T = 2njq~g

where T is the period, і is the length of the pendulum, and g is the acceleration of gravity. However, a vital assumption behind this equation is that of small amplitude oscillations. Similarly, in studying the motion of a freely falling body in a gravitational field, the distance у through which the body falls in time t after release is given by

However, this result neglects any aerodynamic drag on the body as it falls through the air. The above examples are given because they are familiar results from elementary physics. They are examples of theoretical, closed-form solutions—straightforward algebraic relations.

The governing equations of aerodynamics, such as the continuity, momentum, and energy equations derived in Sections 2.4, 2.5, and 2.7, respectively, are highly non-linear, partial differential, or integral equations; to date, no general analytical solution to these equations has been obtained. In lieu of this, two different philoso­phies have been followed in obtaining useful solutions to these equations. One of these is the theoretical, or analytical, approach, wherein the physical nature of certain aerodynamic applications allows the governing equations to be simplified to a suffi­cient extent that analytical solutions of the simplified equations can be obtained. One such example is the analysis of the flow across a normal shock wave, as discussed in Chapter 8. This flow is one-dimensional, i. e., the changes in flow properties across the shock take place only in the flow direction. For this case, the у and z derivatives in the governing continuity, momentum, and energy equations from Sections 2.4, 2.5, and 2.7 are identically zero, and the resulting one-dimensional equations, which are still exact for the one-dimensional case being considered, lend themselves to a direct analytical solution, which is indeed an exact solution for the shock wave properties. Another example is the compressible flow over an airfoil considered in Chapters 11 and 12. If the airfoil is thin and at a small angle of attack, and if the freestream Mach number is not near one (not transonic) nor above five (not hypersonic), then many of the terms in the governing equations are small compared to others and can be neglected. The resulting simplified equations are linear and can be solved ana­lytically. This is an example of an approximate solution, where certain simplifying assumptions have been made in order to obtain a solution.

The history of the development of aerodynamic theory is in this category—the simplification of the full governing equations apropos a given application so that analytical solutions can be obtained. Of course this philosophy works for only a limited number of aerodynamic problems. However, classical aerodynamic theory is built on this approach and, as such, is discussed at some length in this book. You can expect to see a lot of closed-form analytical solutions in the subsequent chapters, along with detailed discussions of their limitations due to the approximations necessary to obtain such solutions. In the modem world of aerodynamics, such classical analytical solutions have three advantages:

1. The act of developing these solutions puts you in intimate contact with all the physics involved in the problem.

2. The results, usually in closed-form, give you direct information on what are the important variables, and how the answers vary with increases or decreases in these variables. For example, in Chapter 11 we will obtain a simple equation for the compressibility effects on lift coefficient for an airfoil in high-speed subsonic flow. The equation, Equation (11.52), tells us that the high-speed effect on lift coefficient is governed by just Мж alone, and that as increases, then the lift coefficient increases. Moreover, the equation tells us in what way the lift coefficient increases, namely, inversely with (1 — M^)1/2. This is powerful information, albeit approximate.

3. Finally, the results in closed-form provide simple tools for rapid calculations, making possible the proverbial “back of the envelope calculations” so important in the preliminary design process and in other practical applications.