Next: Vectors and Vector Fields Up: Two-Dimensional Compressible Inviscid Flow Previous: Flat Lifting Wings

# Exercises

1. Show that Equation (15.12) can be written in the form

 (15.207)

where

Let

Demonstrate that the oblique shock solution only exists for [i.e., when Equation (15.207) possesses three real roots.] Show that the strong shock solution, , and the weak shock solution, , are given by

where

Here, if , and if .

2. Assuming that information propagates with respect to a two-dimensional supersonic flow pattern at the local sound speed, show that, in order for the flow at some point to affect the flow at some other point , the latter point must lie between the and characteristics that pass through .

3. Show that for a weak oblique shock with ,

where , et cetera, and

Here, is the wave angle, the Mach angle, the deflection angle, the ratio of specific heats, the specific gas constant, and the upstream Mach number. Furthermore, , , , and are the upstream pressure, density, temperature, and specific entropy, respectively, whereas , , , and are the corresponding downstream quantities. Show, also, that

4. Show that for a weak oblique shock

where . Here, and are the Mach angles upstream and downstream of the shock front, respectively. Moreover, is the wave angle, and the deflection angle. Hence, deduce that the shock front subtends the same angle, , with the Mach lines upstream and downstream of it. In other words, the shock position is the average'' of the Mach line positions on either side of it. Consider supersonic flow incident on a wedge of small nose angle, with an afterbody, as illustrated in Figure 15.9(a). Assume that the shock front is attached to the apex of the wedge, and that the flow downstream of the shock is supersonic. Use the result just proved to show that the shape of the shock front in the region of attenuation by expansion (i.e., in the region in which the shock front is intersected by Mach lines emanating from the shoulder) is parabolic. [Hint: Use the well-known optical result that a parabolic mirror perfectly focuses a parallel beam of light rays.] (Leipmann & Roshko 1957.)

5. Show that, to second order in the deflection angle, , the relative change in pressure across a weak oblique shock is written

where . Here, is the upstream pressure, the downstream pressure, the upstream Mach number, and the ratio of specific heats. (Leipmann & Roshko 1957.)

6. An ideal gas of pressure, density, temperature, and Mach number , , , and , respectively, flows over a convex corner that turns through an angle , as shown in Figure 15.16. Consider a particular Mach line in the Prandtl-Mayer expansion fan that subtends an angle with the continuation of the upstream wall, as shown in the figure. Let , , , , , , and be the pressure, density, temperature, Mach number, magnitude of the deflection angle, Mach angle, and Prandtl-Mayer function, respectively, on the Mach line in question. Furthermore, let

where is the ratio of specific heats. Show that, inside the fan,

where . Here, is defined implicitly by . Demonstrate that the fan extends over the range of angles , where

Show that , , , ,

and

in the limit that . Hence, deduce that if then the the fan only extends over the region , and the region is occupied by a vacuum (i.e., a gas with zero pressure and density).

Assuming that , show that

and

where , et cetera. Of course, the previous four relations are the same as those for a weak shock. (See Exercise iii.) Why is this not surprising? (Hint: The jump in specific entropy across a weak shock is third order in the deflection angle.) Deduce that to second order in the deflection angle,

for a weak Prandtl-Mayer fan. (See Exercise v.)

7. If an oblique shock is intercepted by a wall then it is reflected, as illustrated in Figure 15.17. Calculate , assuming that the shocks are sufficiently weak that the approximate expressions of Section 15.4 can be used. Demonstrate that, in this limit,

where is the deflection angle of the incident shock, and the ratio of specific heats. Show that if and if . Demonstrate that and , where

8. Figure 15.18 shows a situation in which two oblique shocks of the same family [in this case, the family], produced by successive concave corners in a wall, merge together to form a single stronger shock [of the family]. Assuming that the shocks are sufficiently weak that the approximate expressions of Section 15.4 can be used (which implies that and ), demonstrate that

where

and is the ratio of specific heats. Show that

and also that and , where

Demonstrate that the strength of the merged shock is approximately the sum of the strengths of the two component shocks, and, hence, that the pressures on either side of the slipstream shown in the figure are equal (at least, to first order in and ). Finally, show that

where is specific entropy, and the specific gas constant. It is, thus, clear that the specific entropy is not quite the same on either side of the slipstream.

9. If two shocks of opposite families intersect then they pass through one another, but are slightly bent in the process, as illustrated in Figure 15.19. Assuming that the shocks are sufficiently weak that the approximate expressions of Section 15.4 can be used (which implies that , , et cetera), show that

where denotes pressure. Hence, deduce that

Show, that the respective strengths of the two shocks are unaffected by the intersection (at least, to first order in the deflection angles).

10. Show that Equation (14.66) can be written in the form

where is the Mach angle, the ratio of specific heats, the sound speed at the sonic point, and the flow speed. Deduce that

where

and, hence, that

Finally, demonstrate that

(Leipmann & Roshko 1957.)

11. Show that for a thin, symmetrical airfoil, with zero angle of attack, whose profile is a lens defined by two circular arcs, the drag coefficient is

where is the upstream Mach number, the maximum thickness, and the chord-length. Demonstrate that for a given thickness ratio, , the airfoil with the minimum drag is a symmetric diamond profile. (Leipmann & Roshko 1957.)

12. Prove that on a supersonic swept-back wing of infinite span the thin-airfoil pressure coefficient, (15.50), is multiplied by the sweepback factor,

where

Here, is the sweepback angle, and the upstream Mach number. [Hint: Resolve into components normal and parallel to the leading edge. The flow may then be studied in the plane normal to the leading edge using standard thin-airfoil theory.] (Leipmann & Roshko 1957.)

13. Consider the problem of subsonic flow past a wave-shaped wall that was discussed in Section 15.13. Show that if the flow is bounded by a second wall that lies at (where ) then

Show, on the other hand, that if the flow is bounded by a free surface at (where ) then

(Leipmann & Roshko 1957.)

Next: Vectors and Vector Fields Up: Two-Dimensional Compressible Inviscid Flow Previous: Flat Lifting Wings
Richard Fitzpatrick 2016-03-31