Low-frequency approximation in unsteady aerodynamics.

Abstract

Fig. 1 Refraction of a Mach wave (compression wave) in a supersonic, isobaric flow exhibiting a certain Mach number profile.compression wave reaches the ground. This procedure requires the numerical evaluation of a series of n products.Instead of doing such numerical calculations, the problem can also be described by a simple, analytic equation that relates the intensity of the transmitted wave to the local Mach number M of the freest-ream (Fig. 1). If the intensity of the wave is expressed in terms of the deflection angle 6 of each originally horizontal streamline, this relation isMQ and 0 0 are initial values. Equation (1) has been reported by several authors 4 "" 6 in different connection. For the sonic boom problem, however, one is interested in determining the pressure change Ap across the wave. Again under the same assumptions (i.e., with a two-dimensional, parallel, supersonic flow of constant static pressure, but exhibiting a certain Mach number profile), one derives from Eq. (1)(2) M Q and Ap 0 now play the role of initial values. Ap as a function of M (see Fig. 2) has a minimum for M = (2) 1/2 and goes to infinity for M = 1; Eqs. (1) and (2) are not valid for values of M very close to 1.fWith the aid of Fig. 2, the problem described in the beginning of this Note can be solved as follows. The strength of the sonic boom on the ground is determined with one of the well-known methods for a so-called standard atmosphere. This corresponds to one particular point on the curve in Fig. 2 or to a set of initial values M 0 , Apo. Then, the deviations from the standard atmosphere in the weather layer due to wind (tail-wind or head-wind) and to temperature changes are converted to changes in Mach number. AM may be either positive or negative depending on the directions of the temperature gradient and wind. Having the new Mach number M = M 0 + AM, one reads directly the corresponding pressure change Ap of the sonic boom.This procedure, which can be handled much easier than the numerical methods mentioned before, yields the same quantitative results, e.g., curves such as those in Fig. 4 of Ref. 3. These changes cf the sonic boom strength are usually not higher than 6 to 7%, unless one approaches the local Mach number M = 1 in the modified reference system. Here, however, one approaches also the limits of validity of the governing equations. Nomenclature= aspect ratio ,2/i) = distribution function of surface singularities = unit step function = col/U m , reduced frequency = reference length = Mach number = wing surface = freestream velocity nondimensional coordinates, moving with flight velocity in negative x direction parameter in x integration |1 -M" 2 ! 1 / 2 velocity potential, referred to U m l frequency