Hypersonic Shock-Wave Relations and Another Look at Newtonian Theory
The basic oblique shock relations are derived and discussed in Chapter 9. These are exact shock relations and hold for all Mach numbers greater than unity, supersonic or hypersonic (assuming a calorically perfect gas). However, some interesting approximate and simplified forms of these shock relations are obtained in the limit of a high
Mach number. These limiting forms are called the hypersonic shock relations; they are obtained below.
Consider the flow through a straight oblique shock wave. (See, e. g., Figure 9.1.) Upstream and downstream conditions are denoted by subscripts 1 and 2, respectively. For a calorically perfect gas, the classical results for changes across the shock are given in Chapter 9. To begin with, the exact oblique shock relation for pressure ratio across the wave is given by Equation (9.16). Since Mn< = Mx sin/), this equation becomes
Exact: — = 1 + (M? sin2 /) – 1) [14.28]
P і У + 1
where /) is the wave angle. In the limit as M goes to infinity, the term M sin2 /) ;>> 1, and hence Equation (14.28) becomes
In a similar vein, the density and temperature ratios are given by Equations (9.15) and (9.17), respectively. These can be written as follows:
The relationship among Mach number M, shock angle and deflection angle в is expressed by the so-called в-fi-M relation given by Equation (9.23), repeated below:
This relation is plotted in Figure 9.7, which is a standard plot of the wave angle versus the deflection angle, with the Mach number as a parameter. Returning to Figure 9.7, we note that, in the hypersonic limit, where 9 is small, p is also small. Hence, in this limit, we can insert the usual small-angle approximation into Equation (9.23):
sin /J ~ p cos 2/3^1 tan 9 % sin 9 ~ 9
2 Г Mjp2 – 1 ‘
P [Mf(y + l) + 2_
Applying the high Mach number limit to Equation (14.33), we have
2 Г M2p2 ‘
~P _M2(y + 1)_
In Equation (14.34), M cancels, and we finally obtain in both the small-angle and hypersonic limits,
Note that, for у = 1.4,
It is interesting to observe that, in the hypersonic limit for a slender wedge, the wave angle is only 20 percent larger than the wedge angle—a graphic demonstration of a thin shock layer in hypersonic flow.
In aerodynamics, pressure distributions are usually quoted in terms of the nondimensional pressure coefficient Cp, rather than the pressure itself. The pressure coefficient is defined as
where p and q are the upstream (freestream) static pressure and dynamic pressure, respectively. Recall from Section 11.3 that Equation (14.37) can also be written as Equation (11.22), repeated below:
Combining Equations (11.22) and (14.28), we obtain an exact relation for Cp behind an oblique shock wave as follows:
In the hypersonic limit,
Pause for a moment, and review our results. We have obtained limiting forms of the oblique shock equations, valid for the case when the upstream Mach number becomes very large. These limiting forms, called the hypersonic shock-wave relations, are given by Equations (14.29), (14.31), and (14.32), which yield the pressure ratio, density ratio, and temperature ratio across the shock when Mx —> oo. Furthermore, in the limit of both M —> oo and small 9 (such as the hypersonic flow over a slender airfoil shape), the limiting relation for the wave angle as a function of the deflection angle is given by Equation (14.35). Finally, the form of the pressure coefficient behind an oblique shock is given in the limit of hypersonic Mach numbers by Equation (14.39). Note that the limiting forms of the equations are always simpler than their corresponding exact counterparts.
In terms of actual quantitative results, it is always recommended that the exact oblique shock equations be used, even for hypersonic flow. This is particularly convenient because the exact results are tabulated in Appendix B. The value of the relations obtained in the hypersonic limit (as described above) is more for theoretical analysis rather than for the calculation of actual numbers. For example, in this section, we use the hypersonic shock relations to shed additional understanding of the significance of newtonian theory. In the next section, we will examine the same hypersonic shock relations to demonstrate the principle of Mach number independence.
Newtonian theory was discussed at length in Sections 14.3 and 14.4. For our purposes here, temporarily discard any thoughts of newtonian theory, and simply recall the exact oblique shock relation for Cp as given by Equation (14.38), repeated below (with freestream conditions now denoted by a subscript oo rather than a subscript 1, as used earlier):
Equation (14.39) gave the limiting value of Cp as Mж -► oo, repeated below:
Now take the additional limit of у -► 1.0. From Equation (14.39), in both limits as Moo —»• oo and у —► 1.0, we have
Cp ^ 2 sin2 p [14.40]
Equation (14.40) is a result from exact oblique shock theory; it has nothing to do with newtonian theory (as yet). Keep in mind that p in Equation (14.40) is the wave angle, not the deflection angle.
Let us go further. Consider the exact oblique shock relation for the density ratio, P/Poc! given by Equation (14.30), repeated below (again with a subscript oo replacing the subscript 1):
Pi __ (y + l)Af^ sin2 p
Poo (y – l)Af^ sin2 p + 2
Equation (14.31) was obtained as the limit where Mx —► oo, namely,
Pi У + 1
Poo У 1
1, we find
that is, the density behind the shock is infinitely large. In turn, mass flow considerations then dictate that the shock wave is coincident with the body surface. This is further substantiated by Equation (14.35), which is good for Moo oo and small deflection angles:
In the additional limit as у —»■ 1, we have
that is, the shock wave lies on the body. In light of this result, Equation (14.40) is written as
Examine Equation (14.44). It is a result from exact oblique shock theory, taken in the combined limit of Мх —> oo and у —> 1. However, it is also precisely the newtonian results given by Equation (14.4). Therefore, we make the following conclusion. The closer the actual hypersonic flow problem is to the limits —> oo and у —> 1, the closer it should be physically described by newtonian flow. In this regard, we gain a better appreciation of the true significance of newtonian theory. We can also state that the application of newtonian theory to practical hypersonic flow problems, where у is always greater than unity, is theoretically not proper, and the agreement that is frequently obtained with experimental data has to be viewed as somewhat fortuitous. Nevertheless, the simplicity of newtonian theory along with its (sometimes) reasonable results (no matter how fortuitous) has made it a widely used and popular engineering method for the estimation of surface pressure distributions, hence lift – and wave-drag coefficients, for hypersonic bodies.