Oblique Shocks

The relationship between the conditions upstream and downstream of the shock is easily obtained from the analysis of Section 14.8, because viewing a normal shock in a moving frame of reference does not change the relationship between the upstream and downstream conditions in the original reference frame. The only modification is that the upstream Mach number is now defined as , rather than . Here, is the upstream sound speed. Thus, given that , we simply need to make the transformation in the previous analysis. Hence, Equations (14.107), (14.108), (14.109), and (14.113), yield

respectively. Here, , , , and are the upstream pressure, density, temperature, and specific entropy, respectively, , , , and are the corresponding downstream quantities, is the ratio of specific heats, the specific gas constant, and use has been made of Equation (14.59). As before (see Section 14.8), the second law of thermodynamics demands that , which implies that . This sets a minimum inclination of the upstream flow to the shock front for a given upstream Mach number. The maximum inclination is, of course, . Thus,

where

is termed the

According to Figure 15.1,

and

Eliminating , and then making use of Equation (15.2), we obtain

Now,

(15.11) |

so Equation (15.10) can be rearranged to give

Here, use has been made of the identity . The previous expression implies that at and , which are the limits of the range of allowed values for defined in Equation (15.5). Within this range, is positive, and, must, therefore, have a maximum value, . This is illustrated in Figure 15.2, where the relationship between and , for an ideal gas with , is plotted for various values of .

If then, for each value of and , there are two possible solutions, corresponding to two different values of . The larger value of corresponds to the stronger shock [because, according to Equation (15.1), the shock strength, , is a monotonically increasing function of ].

Also shown in Figure 15.2 is the locus of solutions for which . In the solution with the stronger shock, the downstream flow always becomes subsonic. On the other hand, in the solution with the weaker shock, the downstream flow remains supersonic, except for a small range of values of that are slightly smaller that .

Equation (15.10) can be rearranged to give

(15.13) |

which can be further reduced to

(15.14) |

For small deflection angles, , the previous expression can be approximated by