Shock-Expansion Theory: Applications to Supersonic Airfoils

Consider a flat plate of length c at an angle of attack a in a supersonic flow, as sketched in Figure 9.33. On the top surface, the flow is turned away from itself; hence, an expansion wave occurs at the leading edge, and the pressure on the top surface p2 is less than the freestream pressure p2 < p. At the trailing edge, the flow must return to approximately (but not precisely) the freestream direction. Here, the flow is turned back into itself, and consequently a shock wave occurs at the trailing edge. On the bottom surface, the flow is turned into itself; an oblique shock wave occurs at the leading edge, and the pressure on the bottom surface p2 is greater than the freestream pressure p2 > p. At the trailing edge, the flow is turned into approximately (but not precisely) the freestream direction by means of an expansion wave. Examining Figure 9.33, note that the top and bottom surfaces of the flat plate experience uniform pressure distribution of p2 and p2, respectively, and that p2 > p2. This creates a

net pressure imbalance which generates the resultant aerodynamic force R, shown in Figure 9.33. Indeed, for a unit span, the resultant force and its components, lift and drag, per unit span are

R’ = (рз – P2)c [9.46]

L’ — (p3 — p2)c cos a [9.47]

D’ = {рз — p2)c sin a [9.48]

In Equations (9.47) and (9.48), рз is calculated from oblique shock properties (Section 9.2), and p2 is calculated from expansion-wave properties (Section 9.6). Moreover, these are exact calculations; no approximations have been made. The inviscid, su­personic flow over a flat plate at angle of attack is exactly given by the combination of shock and expansion waves sketched in Figure 9.33.

The flat-plate case given above is the simplest example of a general technique called shock-expansion theory. Whenever we have a body made up of straight-line segments and the deflection angles are small enough so that no detached shock waves occur, the flow over the body goes through a series of distinct oblique shock and expansion waves, and the pressure distribution on the surface (hence the lift and drag) can be obtained exactly from both the shock – and expansion-wave theories discussed in this chapter.

As another example of the application of shock-expansion theory, consider the diamond-shape airfoil in Figure 9.34. Assume the airfoil is at 0° angle of attack. The supersonic flow over the airfoil is first compressed and deflected through the angle є by the oblique shock wave at the leading edge. At midchord, the flow is expanded through an angle 2e, creating an expansion wave. At the trailing edge, the flow is turned back to the freestream direction through another oblique shock. The pressure distributions on the front and back faces of the airfoil are sketched in Figure 9.34;

note that the pressures on faces a and c are uniform and equal to pi and that the pressures on faces b and d are also uniform but equal to pi, where pi < pi – In the lift direction, perpendicular to the freestream, the pressure distributions on the top and bottom faces exactly cancel (i. e., L’ = 0). In contrast, in the drag direction, parallel to the freestream, the pressure on the front faces a and c is larger than on the back faces b and d, and this results in a finite drag. To calculate this drag (per unit span), consider the geometry of the diamond airfoil in Figure 9.34, where l is the length of each face and t is the airfoil thickness. Then,

In Equation (9.49), P2 is calculated from oblique shock theory, and pi is obtained from expansion-wave theory. Moreover, these pressures are the exact values for supersonic, inviscid flow over the diamond airfoil.

At this stage, it is worthwhile to recall our discussion in Section 1.5 concerning the source of aerodynamic force on a body. In particular, examine Equations (1.1),

(1.2) , (1.7), and (1.8). These equations give the means to calculate L’ and D’ from the pressure and shear stress distributions over the surface of a body of general shape. The results of the present section, namely, Equations (9.47) and (9.48) for a flat plate and Equation (9.49) for the diamond airfoil, are simply specialized results from the more general formulas given in Section 1.5. However, rather than formally going through the integration indicated in Equations (1.7) and (1.8), we obtained our results for the simple bodies in Figures 9.33 and 9.34 in a more direct fashion.

The results of this section illustrate a very important aspect of inviscid, supersonic flow. Note that Equation (9.48) for the flat plate and Equation (9.49) for the diamond airfoil predict a finite drag for these two-dimensional profiles. This is in direct contrast

to our results for two-dimensional bodies in a low-speed, incompressible flow, as discussed in Chapters 3 and 4, where the drag was theoretically zero. That is, in supersonic flow, d’Alembert’s paradox does not occur. In a supersonic, inviscid flow, the drag per unit span on a two-dimensional body is finite. This new source of drag is called wave drag, and it represents a serious consideration in the design of all supersonic airfoils. The existence of wave drag is inherently related to the increase in entropy and consequently to the loss of total pressure across the oblique shock waves created by the airfoil.

Finally, the results of this section represent a merger of both the left – and right- hand branches of our road map shown in Figure 9.5. As such, it brings us to a logical conclusion of our discussion of oblique waves in supersonic flows.

Calculate the lift and drag coefficiens for a flat plate at a 5° angle of attack in a Mach 3 flow. | Example 9.1 О Solution

Refer to Figure 9.33. First, calculate p2/pi on the upper surface. From Equation (9.43),

£i=Toi/£oi = 36ЛЗ= 0 668 P і P і / P2 55

Next, calculate рз/рі on the bottom surface. From the в-ji-M diagram (Figure 9.7), for Mi = 3 and в = 5°, p = 23.1 °. Hence,

M„,| = Mt sin fi = 3 sin 23.1° = 1.177

From Appendix B, for M„ і = 1.177, ръ/р = 1.458 (nearest entry).

Returning to Equation (9.47), we have

The lift coefficient is obtained from

From Equation (9.48),

D’ = (p} — pi )c sin a

seclusion, working on various improved methods of farming, including silkworm cultivation. Ernst’s mother, on the other hand, came from a family of lawyers and doctors and brought with her a love of poetry and music. Ernst seemed to thrive in this family atmosphere. Until the age of 14, his education came exclusively from instruction by his father, who read extensively in the Greek and Latin classics. In 1853, Mach entered public school, where he became interested in the world of science. He went on to obtain a Ph. D. in physics in 1860 at the University of Vienna, writing his dissertation on electrical discharge and induction. In 1864, he became a full professor of mathematics at the University of Graz and was given the title of Professor of Physics in 1866. Mach’s work during this period centered on optics—a subject which was to interest him for the rest of his life. The year 1867 was important for Mach—during that year he married, and he also became a professor of experimental physics at the University of Prague, a position he held for the next 28 years. While at Prague, Mach published over 100 technical papers—work which was to constitute the bulk of his technical contributions.

Mach’s contribution to supersonic aerodynamics involves a series of experiments covering the period from 1873 to 1893. In collaboration with his son, Ludwig, Mach studied the flow over supersonic projectiles, as well as the propagation of sound waves and shock waves. His work included the flow fields associated with meteorites, explosions, and gas jets. The main experimental data were photographic results. Mach combined his interest in optics and supersonic motion by designing several photographic techniques for making shock waves in air visible. He was the first to use the schlieren system in aerodynamics; this system senses density gradients and allows shock waves to appear on screens or photographic negatives. He also devised an interferometric technique which senses directly the change in density in a flow. A pattern of alternate dark and light bands are set up on a screen by the superposition of light rays passing through regions of different density. Shock waves are visible as a shift in this pattern along the shock. Mach’s optical device still perpetuates today in the form of the Mach-Zehnder interferometer, an instrument present in many aerodynamic laboratories. Mach’s major contributions in supersonic aerodynamics are contained in a paper given to the Academy of Sciences in Vienna in 1887. Here, for the first time in history, Mach shows a photograph of a weak wave on a slender cone moving at supersonic speed, and he demonstrates that the angle ц between this wave and the direction of flight is given by sin д = a/V. This angle was later denoted as the Mach angle by Prandtl and his colleagues after their work on shock and expansion waves in 1907 and 1908. Also, Mach was the first person to point out the discontinuous and marked changes in a flow field as the ratio Vla changes from below 1 to above 1.

It is interesting to note that the ratio V/a was not denoted as Mach number by Mach himself. Rather, the term “Mach number” was coined by the Swiss engineer Jacob Ackeret in his inaugural lecture in 1929 as Privatdozent at the Eidgenossiche Technische Hochschule in Zurich. Hence, the term “Mach number” is of fairly recent usage, not being introduced into the English literature until the mid-1930s.

In 1895, the University of Vienna established the Ernst Mach chair in the philos­ophy of inductive sciences. Mach moved to Vienna to occupy this chair. In 1897 he suffered a stroke which paralyzed the right side of his body. Although he eventually

partially recovered, he officially retired in 1901. From that time until his death on February 19,1916 near Munich, Mach continued to be an active thinker, lecturer, and writer.

In our time, Mach is most remembered for his early experiments on supersonic flow and, of course, through the Mach number itself. However, Mach’s contempo­raries, as well as Mach himself, viewed him more as a philosopher and historian of science. Coming at the end of the nineteenth century, when most physicists felt com­fortable with newtonian mechanics, and many believed that virtually all was known about physics, Mach’s outlook on science is summarized by the following passage from his book Die Mechanik:

The most important result of our reflections is that precisely the apparently simplest mechanical theorems are of a very complicated nature; that they are founded on incomplete experiences, even on experiences that never can be fully completed; that in view of the tolerable stability of our environment they are, in fact, practically safeguarded to serve as the foundation of mathematical deduction; but that they by no means themselves can be regarded as mathematically established truths, but only as theorems that not only admit of constant control by experience but actually require it.

In other words, Mach was a staunch experimentalist who believed that the established laws of nature were simply theories and that only observations that are apparent to the senses are the fundamental truth. In particular, Mach could not accept the elementary ideas of atomic theory or the basis of relativity, both of which were beginning to surface during Mach’s later years and, of course, were to form the basis of twentieth – century modern physics. As a result, Mach’s philosophy did not earn him favor with most of the important physicists of his day. Indeed, at the time of his death, Mach was planning to write a book pointing out the flaws of Einstein’s theory of relativity.

Although Mach’s philosophy was controversial, he was respected for being a thinker. In fact, in spite of Mach’s critical outlook on the theory of relativity, Albert Einstein had the following to say in the year of Mach’s death: “I even believe that those who consider themselves to be adversaries of Mach scarcely know how much of Mach’s outlook they have, so to speak, adsorbed with their mother’s milk.”

Hopefully, this section has given you a new dimension to think about whenever you encounter the term “Mach number.” Maybe you will pause now and then to reflect on the man himself and to appreciate that the term “Mach number” is in honor of a man who devoted his life to experimental physics, but who at the same time was bold enough to view the physical world through the eyes of a self-styled philosopher.

9,10 Summary

The road map given in Figure 9.5 illustrates the flow of our discussion on oblique waves in supersonic flow. Review this road map, and make certain that you are familiar with all the ideas and results that are represented in Figure 9.5.

Some of the more important results are summarized as follows:

2. Consider an oblique shock wave with a wave angle of 30° in a Mach 4 flow. The upstream pressure and temperature are 2.65 x 104 N/m2 and 223.3 K, respec­tively (corresponding to a standard altitude of 10,000 m). Calculate the pressure, temperature, Mach number, total pressure, and total temperature behind the wave and the entropy increase across the wave.

3. Equation (8.80) does not hold for an oblique shock wave, and hence the column in Appendix В labeled pop/p cannot be used, in conjunction with the normal component of the upstream Mach number, to obtain the total pressure behind an oblique shock wave. On the other hand, the column labeled pop/po. i can be used for an oblique shock wave, using МП:. Explain why all this is so.

4. Consider an oblique shock wave with a wave angle of 36.87°. The upstream flow is given by Mi =3 and p — 1 atm. Calculate the total pressure behind the shock using

(a) ро г /Pd. і from Appendix В (the correct way)

(b) po i/pi from Appendix В (the incorrect way)

Compare the results.

5. Consider the flow over a 22.2° half-angle wedge. If Mi = 2.5, p — 1 atm, and 7) = 300 K, calculate the wave angle and p2, T2, and M2.

6. Consider a flat plate at an angle of attack a to a Mach 2.4 airflow at 1 atm pressure. What is the maximum pressure that can occur on the plate surface and still have an attached shock wave at the leading edge? At what value of a does this occur?

7. A 30.2° half-angle wedge is inserted into a freestream with Mx — 3.5 and Poo = 0.5 atm. A Pitot tube is located above the wedge surface and behind the shock wave. Calculate the magnitude of the pressure sensed by the Pitot tube.

8. Consider a Mach 4 airflow at a pressure of 1 atm. We wish to slow this flow to subsonic speed through a system of shock waves with as small a loss in total pressure as possible. Compare the loss in total pressure for the following three shock systems:

(a) A single normal shock wave

(b) An oblique shock with a deflection angle of 25.3°, followed by a normal shock

(c) An oblique shock with a deflection angle of 25.3°, followed by a second oblique shock of deflection angle of 20°, followed by a normal shock

From the results of (a), (b), and (c), what can you induce about the efficiency of the various shock systems?

9. Consider an oblique shock generated at a compression corner with a deflection angle 9 = 18.2°. A straight horizontal wall is present above the comer, as shown in Figure 9.17. If the upstream flow has the properties Mi = 3.2, p = 1 atm and T = 520°R, calculate М3, p3, and T2 behind the reflected shock from the upper wall. Also, obtain the angle Ф which the reflected shock makes with the upper wall.

10. Consider the supersonic flow over an expansion comer, such as given in Figure 9.23. The deflection angle 9 = 23.38°. If the flow upstream of the corner is

given by Mi =2, p = 0.7 atm, 7j = 630°R, calculate M2, p2, T2, p2, P0.2, and 7І) 2 downstream of the comer. Also, obtain the angles the forward and rearward Mach lines make with respect to the upstream direction.

11. A supersonic flow at M] = 1.58 and p — 1 atm expands around a sharp corner. If the pressure downstream of the corner is 0.1306 atm, calculate the deflection angle of the corner.

12. A supersonic flow at M1 = 3, 7j = 285 K, and p = 1 atm is deflected upward through a compression corner with 9 = 30.6° and then is subsequently expanded around a corner of the same angle such that the flow direction is the same as its original direction. Calculate М3, p2, and 7) downstream of the expansion comer. Since the resulting flow is in the same direction as the original flow, would you expect М3 = Mi, p = p, and 7з = 7У? Explain.

13. Consider an infinitely thin flat plate at an angle of attack a in a Mach 2.6 flow. Calculate the lift and wave-drag coefficients for

(a) a = 5° (b) a = 15° (c) a = 30°

{Note: Save the results of this problem for use in Chapter 12.)

14. Consider a diamond-wedge airfoil such as shown in Figure 9.34, with a half-angle є = 10°. The airfoil is at an angle of attack a = 15° to a Mach 3 freestream. Calculate the lift and wave-drag coefficients for the airfoil.

15. Consider sonic flow. Calculate the maximum deflection angle through which this flow can be expanded via a centered expansion wave.

16. Consider a circular cylinder (oriented with its axis perpendicular to the flow) and a symmetric diamond-wedge airfoil with a half-angle of 5° at zero angle of attack; both bodies are in the same Mach 5 freestream. The thickness of the airfoil and the diameter of the cylinder are the same. The drag coefficient (based on projected frontal area) of the cylinder is 4/3. Calculate the ratio of the cylinder drag to the diamond airfoil drag. What does this say about the aerodynamic performance of a blunt body compared to a sharp-nosed slender body in supersonic flow?

17. 9.17 Consider the supersonic flow over a flat plate at angle of attack, as sketched in Figure 9.33. As stated in Section 9.7, the flow direction downstream of the trailing edge of the plate, behind the trailing edge shock and expansion waves, is not precisely in the free stream direction. Why? Outline a method to calculate the strengths of the trailing edge shock and expansion waves, and the direction of the flow downstream of the trailing edge.