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. Ranges of validity
Figure 8.11 diagrams the range of validity of the five potential equations considered here, versus |V0|aVg which is a measure of “non-slenderness,” and versus the freestream Mach number MTO.
The following observations can be made:
• At low speeds where ^ 1, all five equations are equally valid, even for non-slender bodies (viscous effects are not being considered here). The simplest Laplace equation is then the logical choice to use here.
• For low-subsonic Mach numbers, above MTO > 0.3 or so, compressibility effects become progressively more pronounced, in which case the PG equation becomes the logical choice to use.
• For flows sufficiently close to sonic, MTO ~ 1, specifically transonic flows, the PG equation becomes unsuitable because it cannot represent normal shock waves. In this case the simplest possible equation which can be used is TSD, since it can capture normal shock waves and their associated wave drag.
• For supersonic flows sufficiently far past MTO = 1, the PG equation again becomes valid. In this situation it becomes a form of the wave equation, and can represent weak oblique shocks for which the flow remains everywhere locally supersonic.
• For all but very low freestream Mach numbers, the PG or TSD equations become increasingly restricted to smaller body thicknesses and/or small angles of attack as MTO increases. The reason is that the leading terms which were dropped in the PG and TSD derivations were of the form M^фхфу, etc. Hence, for a fixed error from these terms, the upper limit on the tolerable | Vф|avg must decrease as MTO increases. [7]
equations for adequate accuracy. Solving PP2 is not any easier or less expensive than solving the FP equation, so PP2 is not used in practice (here it was only a stepping stone to TSD and PG).
• For all but low-speed flows, FP also has an upper limit on body slenderness, even though no small – disturbance approximations were used in its derivation. The reason is that high-speed non-slender flows will have strong shock waves and large shock-wake velocity defects, which invalidate the isen- tropy and irrotationality assumptions underlying the FP equation.
