Mach Number Independence
Examine again the hypersonic shock-wave relation for pressure ratio as given by Equation (14.29); note that, as the freestream Mach number approaches infinity, the pressure ratio itself also becomes infinitely large. On the other hand, the pressure coefficient behind the shock, given in the hypersonic limit by Equation (14.39), is a constant value at high values of the Mach number. This hints strongly of a situation where certain aspects of a hypersonic flow do not depend on Mach number, as long as the Mach number is sufficiently high. This is a type of “independence” from the Mach number, formally called the hypersonic Mach number independence principle. From the above argument, Cp clearly demonstrates Mach number independence. In turn, recall that the lift- and wave-drag coefficients for a body shape are obtained by integrating the local Cp, as shown by Equations (1.15), (1.16), (1.18), and (1.19). These equations demonstrate that, since Cp is independent of the Mach number at high values of Mx, the lift and drag coefficients are also Mach number independent. Keep in mind that these conclusions are theoretical, based on the limiting form of the hypersonic shock relations.
Let us examine an example that clearly illustrates the Mach number independence principle. In Figure 14.13, the pressure coefficients for a 15° half-angle wedge and a 15° half-angle cone are plotted versus freestream Mach number for у = 1.4. The exact wedge results are obtained from Equation (14.38), and the exact cone results are obtained from the solution of the classical Taylor-Maccoll equation. (See Reference 21 for a detailed discussion of the solution of the supersonic flow over a cone. There, you will find that the governing continuity, momentum, and energy equations for a conical flow cascade into a single differential equation called the
Taylor-Maccoll equation. In turn, this equation allows the exact solution of this conical flow field.) Both sets of results are compared with newtonian theory, Cp = 2 sin2 0, shown as the dashed line in Figure 14.13. This comparison demonstrates two general aspects of newtonian results:
1. The accuracy of the newtonian results improves as Мж increases. This is to be expected from our discussion in Section 14.5. Note from Figure 14.13 that below Moo = 5 the newtonian results are not even close, but the comparison becomes much closer as M^ increases above 5.
2. Newtonian theory is usually more accurate for three-dimensional bodies (e. g., the cone) than for two-dimensional bodies (e. g., the wedge). This is clearly evident in Figure 14.13 where the newtonian result is much closer to the cone results than to the wedge results.
However, more to the point of Mach number independence, Figure 14.13 also shows the following trends. For both the wedge and the cone, the exact results show that, at low supersonic Mach numbers, Cp decreases rapidly as Мж is increased. However, at hypersonic speeds, the rate of decrease diminishes considerably, and Cp appears to reach a plateau as M<*, becomes large; that is, Cp becomes relatively independent of Moo at high values of the Mach number. This is the essence of the Mach number independence principle; at high Mach numbers, certain aerodynamic quantities such as pressure coefficient, lift – and wave-drag coefficients, and flow-field structure (such as shock-wave shapes and Mach wave patterns) become essentially independent of the Mach number. Indeed, newtonian theory gives results that are totally independent of the Mach number, as clearly demonstrated by Equation (14.4).
Another example of Mach number independence is shown in Figure 14.14. Here, the measured drag coefficients for spheres and for a large-angle cone cylinder are plotted versus the Mach number, cutting across the subsonic, supersonic, and hypersonic regimes. Note the large drag rise in the subsonic regime associated with the drag – divergence phenomenon near Mach 1 and the decrease in Сд in the supersonic regime beyond Mach 1. Both of these variations are expected and well understood. For our purposes in the present section, note, in particular, the variation of Co in the hypersonic regime; for both the sphere and cone cylinder, CD approaches a plateau and becomes relatively independent of the Mach number as M^ becomes large. Note also that the sphere data appear to achieve “Mach number independence” at lower Mach numbers than the cone cylinder.
Keep in mind from the above analysis that it is the nondimensional variables that become Mach number independent. Some of the dimensional variables, such as p, are not Mach number independent; indeed, p —>• oc and Mcc —»• oo.
Finally, the Mach number independence principle is well grounded mathematically. The governing inviscid flow equations (the Euler equations) expressed in terms of suitable nondimensional quantities, along with the boundary conditions for the limiting hypersonic case, do not have the Mach number appearing in them—hence, by definition, the solution to these equations is independent of the Mach number. See References 21 and 55 for more details.
Figure 14.14 Drag coefficient for a sphere and a cone
cylinder from ballistic range measurements; an example of Mach number independence at hypersonic speeds. (Source: Cox and Crabtree, Reference 61 .j