Aerodynmics 9

A practical application of these results is in the design of supersonic inlets for jet engines. A normal shock inlet is sketched in Figure 9.15a. Here, a normal shock forms ahead of the inlet, with an attendant large loss in total pressure. In contrast, an oblique shock inlet is sketched in Figure 9.15b. Here, a central cone creates an oblique shock wave, and the flow subsequently passes through a relatively weak normal shock at the lip of the inlet. For the same flight conditions (Mach number and altitude), the total pressure loss for the oblique shock inlet is less than for a normal shock inlet. Hence, everything else being equal, the resulting engine thrust will be higher for the oblique shock inlet. This, of course, is why most modern supersonic aircraft have oblique shock inlets. 9.3 SUPERSONIC FLOW OVER WEDGES AND CONES For the supersonic flow over wedges, as shown in Figures 9.12 and 9.13, the oblique shock theory developed in Section 9.2 is an exact solution of the flow field; no simplifying assumptions have been made. Supersonic flow over a wedge is characterized by an attached, straight oblique shock wave from the nose, a uniform flow downstream of the shock with streamlines parallel to the wedge surface, and a surface pressure equal to the static pressure behind the oblique shock p2. These properties are summarized in Figure 9.16a. Note that the wedge is a two-dimensional profile; in Figure 9.16a, it is a section of a body that stretches to plus or minus infinity in the direction perpendicular to the page. Hence, wedge flow is, by definition, two-dimensional flow, and our two-dimensional oblique shock theory fits this case nicely. In contrast, consider the supersonic flow over a cone, as sketched in Figure 9.16b. There is a straight oblique shock which emanates from the tip, just as in the case of a wedge, but the similarity stops there. Recall from Chapter 6 that flow over a three-dimensional body experiences a "three-dimensional relieving effect." That is, in comparing the wedge and cone in Figure 9.16, both with the same 20◦ angle, the flow over the cone has an extra dimension in which to move, and hence it more easily adjusts to the presence of the conical body in comparison to the two-dimensional wedge. One consequence of this three-dimensional relieving effect is that the shock wave on the cone is weaker than on the wedge; that is, it has a smaller wave angle, as compared in Figure 9.16. Specifically, the wave angles for the wedge and cone are 53.3 and 37◦, respectively, for the same body angle of 20◦ and the same upstream Mach number of 2.0. In the case of the wedge (Figure 9.16a), the streamlines are deflected by exactly 20◦ through the shock wave, and hence downstream of the shock the flow is exactly parallel to the wedge surface. In contrast, because of the weaker shock on the cone, the streamlines are deflected by only 8◦ through the shock, as shown in Figure 9.16b. Therefore, between the shock wave and the cone surface, the streamlines must gradually curve upward in order to accommodate the 20◦ cone. Also, as a consequence of the three-dimensional relieving effect, the pressure on the surface of the cone, pc, is less than the wedge surface pressure p2, and the cone surface Mach number Mc is greater than that on the wedge surface M2. In short, the main differences between the supersonic flow over a cone and wedge, both with the same body angle, are that (1) the shock wave on the cone is weaker, (2) the cone surface pressure is less, and (3) the streamlines above the cone surface are curved rather than straight. The analysis of the supersonic flow over a cone is more sophisticated than the oblique shock theory given in this chapter. The calculation of the supersonic flow over a cone is discussed in Section 13.6. For details concerning supersonic conical flow analysis, see Chapter 10 of Reference 21. However, it is important for you to recognize that conical flows are inherently different from wedge flows and to recognize in what manner they differ. This has been the purpose of the present section. (Note: The drag is finite for this case. In a supersonic or hypersonic inviscid flow over a two-dimensional body, the drag is always finite. D'Alembert's paradox does not hold for freestream Mach numbers such that shock waves appear in the flow. The fundamental reason for the generation of drag here is the presence of shock waves. Shocks are always a dissipative, drag-producing mechanism. For this reason, the drag in this case is called wave drag, and cd is the wave-drag coefficient, more properly denoted as cd,w.) 9.3.1 A Comment on Supersonic Lift and Drag Coefficients The result obtained in Example 9.6 is a stunning verification of the validity of the dimensional analysis discussed in Section 1.7. There we proved that, for a body of a given shape at a given angle of attack, the aerodynamic coefficients are simply a function of Mach number and Reynolds number [see Equations (1.42), (1.43),

Consider the oblique shock wave sketched in Figure 9.8. The angle between the shock wave and the upstream flow direction is defined as the wave angle, denoted by β. The upstream flow (region 1) is horizontal, with a velocity V1 and Mach number M1. The downstream flow (region 2) is inclined upward through the deflection angle θ and has velocity V2 and Mach number M2. The upstream velocity V1 is split into components tangential and normal to the shock wave, w1 and u1, respectively, with the associated tangential and normal Mach numbers Mt,1 and Mn,1, respectively. Similarly, the downstream velocity is split into tangential and normal components w2 and u2, respectively, with the associated Mach numbers Mt,2 and Mn,2. Consider the control volume shown by the dashed lines in the upper part of Figure 9.8. Sides a and d are parallel to the shock wave. Segments b and c follow the upper streamline, and segments e and f follow the lower streamline. Let us apply the integral form of the conservation equations to this control volume, keeping in mind that we are dealing with a steady, inviscid, adiabatic flow with no body forces. For these assumptions, the continuity equation, Equation (2.48), becomes ..... ............ ................................. .. .... S ρV · dS = 0 This surface integral evaluated over faces a and d yields −ρ1u1A1 + ρ2u2A2, where A1 = A2 = area of faces a and d. The faces b, c, e, and f are parallel to the velocity, and hence contribute nothing to the surface integral (i.e., V · dS = 0 for these faces). Thus, the continuity equation for an oblique shock wave is −ρ1u1A1 + ρ2u2A2 = 0 or ρ1u1 = ρ2u2 (9.2) Keep in mind that u1 and u2 in Equation (9.2) are normal to the shock wave. The integral form of the momentum equation, Equation (2.64), is a vector equation. Hence, it can be resolved into two components, tangential and normal to the shock wave. First, consider the tangential component, keeping in mind the type of flow we are considering: ..... ............ ................................ .. .... S (ρV · dS)w = − ..... ............ ................................ .. .... S (p dS)tangential (9.3) In Equation (9.3), w is the component of velocity tangential to the wave. Since dS is perpendicular to the control surface, then (pdS)tangential over faces a and d is zero. Also, since the vectors p dS on faces b and f are equal and opposite, the pressure integral in Equation (9.3) involves two tangential forces that cancel each other over faces b and f . The same is true for faces c and e. Hence, Equation (9.3) becomes −(ρ1u1A1)w1 + (ρ2u2A2)w2 = 0 (9.4) Dividing Equation (9.4) by Equation (9.2), we have w1 = w2 (9.5) Equation (9.5) is an important result; it states that the tangential component of the flow velocity is constant across an oblique shock. 622 PART 3 Inviscid, Compressible Flow The normal component of the integral momentum equation is, from Equation (2.64), ..... ............ ................................. .. .... S (ρV · dS)u = − ..... ............ ................................. .. .... S (pdS)normal (9.6) Here, the pressure integral evaluated over faces a and d yields the net sum −p1A1 + p2A2. Once again, the equal and opposite pressure forces on b and f cancel, as do those on c and e. Hence, Equation (9.6) becomes, for the control volume shown in Figure 9.8, −(ρ1u1A1)u1 + (ρ2u2A2)u2 = −(−p1A1 + p2A2) Since A1 = A2, this becomes p1 + ρ1u2 1 = p2 + ρ2u2 2 (9.7) Again, note that the only velocities appearing in Equation (9.7) are the components normal to the shock. Finally, consider the integral form of the energy equation, Equation (2.95). For our present case, this can be written as ..... ............ ................................. .. .... S ρ e + V2 2 V · dS = − ..... ............ ................................. .. .... S pV · dS (9.8) Again noting that the flow is tangent to faces b, c, f , and e, and hence V · dS = 0 on these faces, Equation (9.8) becomes, for the control volume in Figure 9.6, −ρ1 e1 + V2 1 2 u1A1 + ρ2 e2 + V2 2 2 u2A2 = −(−p1u1A1 + p2u2A2) (9.9) Collecting terms in Equation (9.9), we have −ρ1u1 e1 + p1 ρ1 + V2 1 2 + ρ2u2 e2 + p2 ρ2 + V2 2 2 = 0 or ρ1u1 h1 + V2 1 2 = ρ2u2 h2 + V2 2 2 (9.10) Dividing Equation (9.10) by (9.2), we have h1 + V2 1 2 = h2 + V2 2 2 (9.11) Since h + V2/2 = h0, we have again the familiar result that the total enthalpy is constant across the shock wave. Moreover, for a calorically perfect gas, h0 = cpT0; hence, the total temperature is constant across the shock wave. Carrying Equation (9.11) a bit further, note from Figure 9.8 that V2 = u2 +w2. Also, from Equation (9.5), we know that w1 = w2. Hence, V2 1 − V2 2 = u2 1 + w2 1 − u2 2 + w2 2 = u2 1 − u2 2 CHAPTER 9 Oblique Shock and Expansion Waves 623 Thus, Equation (9.11) becomes h1 + u2 1 2 = h2 + u2 2 2 (9.12) Let us now gather our results. Look carefully at Equations (9.2), (9.7), and (9.12). They are the continuity, normal momentum, and energy equations, respectively, for an oblique shock wave. Note that they involve the normal components only of velocity u1 and u2; the tangential component w does not appear in these equations. Hence, we deduce that changes across an oblique shock wave are governed only by the component of velocity normal to the wave. Again, look hard at Equations (9.2), (9.7), and (9.12). They are precisely the governing equations for a normal shock wave, as given by Equations (8.2), (8.6), and (8.10). Hence, precisely the same algebra as applied to the normal shock equations in Section 8.6, when applied to Equations (9.2), (9.7), and (9.12), will lead to identical expressions for changes across an oblique shock in terms of the normal component of the upstream Mach number Mn,1. Note that Mn,1 = M1 sin β (9.13) Hence, for an oblique shock wave, with Mn,1 given by Equation (9.13), we have, from Equations (8.59), (8.61), and (8.65), M2 n,2 = 1 + [(γ − 1)/2]M2 n,1 γ M2 n,1 − (γ − 1)/2 (9.14) ρ2 ρ1 = (γ + 1)M2 n,1 2 + (γ − 1)M2 n,1 (9.15) p2 p1 = 1 + 2γ γ + 1 (M2 n,1 − 1) (9.16) The temperature ratio T2/T1 follows from the equation of state: T2 T1 = p2 p1 ρ1 ρ2 (9.17) Note that Mn,2 is the normal Mach number behind the shock wave. The downstream Mach number itself, M2, can be found from Mn,2 and the geometry of Figure 9.8 as M2 = Mn,2 sin(β − θ) (9.18) Examine Equations (9.14) to (9.17). They state that oblique shock-wave properties in a calorically perfect gas depend only on the normal component of the upstream Mach number Mn,1. However, note from Equation (9.13) that Mn,1 depends on both M1 and β. Recall from Section 8.6 that changes across a normal shock wave depend on one parameter only—the upstream Mach number M1. 624 PART 3 Inviscid, Compressible Flow In contrast, we now see that changes across an oblique shock wave depend on two parameters—say, M1 and β. However, this distinction is slightly moot because in reality a normal shock wave is a special case of oblique shocks where β = π/2. Equation (9.18) introduces the deflection angle θ into our oblique shock analysis; we need θ to be able to calculate M2. However, θ is not an independent, third parameter; rather, θ is a function of M1 and β, as derived below. From the geometry of Figure 9.8, tan β = u1 w1 (9.19) and tan(β − θ) = u2 w2 (9.20) Dividing Equation (9.20) by (9.19), recalling that w1 = w2, and invoking the continuity equation, Equation (9.2), we obtain tan(β − θ) tan β = u2 u1 = ρ1 ρ2 (9.21) Combining Equation (9.21) with Equations (9.13) and (9.15), we obtain tan(β − θ) tan β = 2 + (γ − 1)M2 1 sin2 β (γ + 1)M2 1 sin2 β (9.22) which gives θ as an implicit function of M1 and β. After some trigonometric substitutions and rearrangement, Equation (9.22) can be cast explicitly for θ as tan θ = 2 cot β M2 1 sin2 β − 1 M2 1 (γ + cos 2β) + 2 (9.23) Equation (9.23) is an important equation. It is called the θ-β-M relation, and it specifies θ as a unique function of M1 and β. This relation is vital to the analysis of oblique shock waves, and results from it are plotted in Figure 9.9 for γ = 1.4. Examine this figure closely. It is a plot of wave angle versus deflection angle, with the Mach number as a parameter. The results given in Figure 9.9 are plotted in some detail—this is a chart which you will need to use for solving oblique shock problems. Figure 9.9 illustrates a wealth of physical phenomena associated with oblique shock waves. For example: 1. For any given upstream Mach number M1, there is a maximum deflection angle θmax. If the physical geometry is such that θ>θmax, then no solution exists for a straight oblique shock wave. Instead, nature establishes a curved shock wave, detached from the corner or the nose of a body. This is illustrated in Figure 9.10. Here, the left side of the figure illustrates flow over a wedge and a concave corner where the deflection angle is less than θmax for the given upstream Mach number. Therefore, we see a straight oblique shock wave attached to the nose of the wedge and to the corner. The right side of Figure 9.10 gives the case where the deflection angle is greater than θmax; hence, there is no allowable straight oblique shock solution from the theory developed earlier in this section. Instead, we have a curved shock wave detached from the nose of the wedge or from the corner. Return to Figure 9.9, and note that the value of θmax increases with increasing M1. Hence, at higher Mach numbers, the straight oblique shock solution can exist at higher deflection angles. However, there is a limit; as M1 approaches infinity, θmax approaches 45.5◦ (for γ = 1.4). 2. For any given θ less than θmax, there are two straight oblique shock solutions for a given upstream Mach number. For example, if M1 = 2.0 and θ = 15◦, then from Figure 9.9, β can equal either 45.3 or 79.8◦. The smaller value of β is called the weak shock solution, and the larger value of β is the strong shock solution. These two cases are illustrated in Figure 9.11. 628 PART 3 Inviscid, Compressible Flow The classifications "weak" and "strong" derive from the fact that for a given M1, the larger the wave angle, the larger the normal component of upstream Mach number Mn,1, and from Equation (9.16) the larger the pressure ratio p2/p1. Thus, in Figure 9.11, the higher-angle shock wave will compress the gas more than the lower-angle shock wave, hence the terms "strong" and "weak" solutions. In nature, the weak shock solution usually prevails. Whenever you see straight, attached oblique shock waves, such as sketched at the left of Figure 9.10, they are almost always the weak shock solution. It is safe to make this assumption, unless you have specific information to the contrary. Note in Figure 9.9 that the locus of points connecting all the values of θmax (the curve that sweeps approximately horizontally across the middle of Figure 9.9) divides the weak and strong shock solutions. Above this curve, the strong shock solution prevails (as further indicated by the θ-β-M curves being dashed); below this curve, the weak shock solution prevails (where the θ-β-M curves are shown as solid lines). Note that slightly below this curve is another curve which also sweeps approximately horizontally across Figure 9.9. This curve is the dividing line above which M2 < 1 and below which M2 > 1. For the strong shock solution, the downstream Mach number is always subsonic M2 < 1. For the weak shock solution very near θmax, the downstream Mach number is also subsonic, but barely so. For the vast majority of cases involving the weak shock solution, the downstream Mach number is supersonic M2 > 1. Since the weak shock solution is almost always the case encountered in nature, we can readily state that the Mach number downstream of a straight, attached oblique shock is almost always supersonic. 3. If θ = 0, then β equals either 90◦ or μ. The case of β = 90◦ corresponds to a normal shock wave (i.e., the normal shocks discussed in Chapter 8 belong to the family of strong shock solutions). The case of β = μ corresponds to the Mach wave illustrated in Figure 9.4b. In both cases, the flow streamlines experience no deflection across the wave. 4. (In all of the following discussions, we consider the weak shock solution exclusively, unless otherwise noted.) Consider an experiment where we have supersonic flow over a wedge of given semiangle θ, as sketched in Figure 9.12. Now assume that we increase the freestream Mach number M1. As M1 increases, we observe that β decreases. For example, consider θ = 20◦ and M1 = 2.0, as shown on the left of Figure 9.12. From Figure 9.9, we find that β = 53.3◦. Now assume M1 is increased to 5, keeping θ constant at 20◦, as sketched on the right of Figure 9.12. Here, we find that β = 29.9◦. Interestingly enough, although this shock is at a lower wave angle, it is a stronger shock than the one on the left. This is because Mn,1 is larger for the case on the right. Although β is smaller, which decreases Mn,1, the upstream Mach number M1 is larger, which increases Mn,1 by an amount which more than compensates for the decreased β. For example, note the values of Mn,1 and p2/p1 given in Figure 9.12. Clearly the Mach 5 case on the right yields the stronger shock wave. Hence, in general for attached shocks with a fixed deflection angle, as the upstream Mach number M1 increases, the wave angle β decreases, and the shock wave becomes stronger. Going in the other direction, as M1 decreases, the wave angle increases, and the shock becomes weaker. Finally, if M1 is decreased enough, the shock wave will become detached. For the case of θ = 20◦ shown in Figure 9.12, the shock will be detached for M1 < 1.84. 5. Consider another experiment. Here, let us keep M1 fixed and increase the deflection angle. For example, consider the supersonic flow over a wedge shown in Figure 9.13. Assume that we have M1 = 2.0 and θ = 10◦, as sketched at the left of Figure 9.13. The wave angle will be 39.2◦ (from Figure 9.9). Now assume that the wedge is hinged so that we can increase its deflection angle, keeping M1 constant. In such a case, the wave angle will increase, as shown on the right of Figure 9.13. Also, Mn,1 will increase, 630 PART 3 Inviscid, Compressible Flow and hence the shock will become stronger. Therefore, in general for attached shocks with a fixed upstream Mach number, as the deflection angle increases, the wave angle β increases, and the shock becomes stronger. However, once θ exceeds θmax, the shock wave will become detached. For the case of M1 = 2.0 in Figure 9.13, this will occur when θ > 23◦. The physical properties of oblique shocks just discussed are very important. Before proceeding further, make certain to go over this discussion several times until you feel perfectly comfortable with these physical variations.

Equation (9.32) relates the infinitesimal change in velocity dV to the infinitesimal deflection dθ across a wave of vanishing strength. In the precise limit of a Mach wave, of course dV and hence dθ are zero. In this sense, Equation (9.32) is an approximate equation for a finite dθ, but it becomes a true equality as dθ → 0. Since the expansion fan illustrated in Figures 9.2b and 9.26 is a region of an infinite number of Mach waves, Equation (9.32) is a differential equation which precisely describes the flow inside the expansion wave. iven by Equation (9.42) for a calorically perfect gas. The PrandtlMeyer function ν is very important; it is the key to the calculation of changes across an expansion wave. Because of its importance, ν is tabulated as a function of M in Appendix C. For convenience, values of μ are also tabulated in Appendix C. How do the above results solve the problem stated in Figure 9.26; that is how can we obtain the properties in region 2 from the known properties in region 1 and the known deflection angle θ? The answer is straightforward: 1. For the given M1, obtain ν(M1) from Appendix C. 2. Calculate ν(M2) from Equation (9.43), using the known θ and the value of ν(M1) obtained in step 1. 3. Obtain M2 from Appendix C corresponding to the value of ν(M2) from step 2. 4. The expansion wave is isentropic; hence, p0 and T0 are constant through the wave. That is, T0,2 = T0,1 and p0,2 = p0,1. From Equation (8.40), we have T2 T1 = T2/T0,2 T1/T0,1 = 1 + [(γ − 1)/2]M2 1 1 + [(γ − 1)/2]M2 2 (9.44) From Equation (8.42), we have p2 p1 = p2/p0 p1/p0 = 1 + [(γ − 1)/2]M2 1 1 + [(γ − 1)/2]M2 2 γ/(γ −1) (9.45) Since we know both M1 and M2, as well as T1 and p1, Equations (9.44) and (9.45) allow the calculation of T2 and p2 downstream of the expansion wav Comparing the results from this example and Example 9.10, we clearly see that the isentropic compression is a more efficient compression process, yielding a downstream Mach number and pressure that are both considerably higher than in the case of the shock wave. The inefficiency of the shock wave is measured by the loss of total pressure across the shock; total pressure drops by about 77 percent across the shock. This emphasizes why designers of supersonic and hypersonic inlets would love to have the compression process carried out via isentropic compression waves. However, as noted in our discussion on SCRAMjets, it is very difficult to achieve such a compression in real life; the contour of the compression surface must be quite precise, and it is a point design for the given upstream Mach number. At off-design Mach numbers, even the best-designed compression contour will result in shocks. 9.7 SHOCK-EXPANSION THEORY: APPLICATIONS TO SUPERSONIC AIRFOILS Consider a flat plate of length c at an angle of attack α in a supersonic flow, as sketched in Figure 9.36. 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 < p1. 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 p3 is greater than the freestream pressure p3 > p1. At the trailing edge, the flow is turned into approximately (but not precisely) the freestream direction by means of an expansion wave. Examining Figure 9.36, note that the top and bottom surfaces of the flat plate experience uniform pressure distribution of p2 and p3, respectively

In Chapter 8, we discussed normal shock waves, that is, shock waves that make an angle of 90◦ with the upstream flow. The behavior of normal shock waves is important; moreover, the study of normal shock waves provides a relatively straightforward introduction to shock-wave phenomena. However, examining Figure 7.5 and the photographs shown in Figure 7.6, we see that, in general, a shock wave will make an oblique angle with respect to the upstream flow. These are called oblique shock waves and are the subject of part of this chapter. A normal shock wave is simply a special case of the general family of oblique shocks, namely, the case where the wave angle is 90◦. In addition to oblique shock waves, where the pressure increases discontinuously across the wave, supersonic flows are also characterized by oblique expansion waves, where the pressure decreases continuously across the wave. Let us examine these two types of waves further. Consider a supersonic flow over a wall with a corner at point A, as sketched in Figure 9.2. In Figure 9.2a, the wall is turned upward at the corner through the deflection angle θ; that is, the corner is concave. The flow at the wall must be tangent to the wall; hence, the streamline at the wall is also deflected upward through the angle θ. The bulk of the gas is above the wall, and in Figure 9.2a, the streamlines are turned upward, into the main bulk of the flow. Whenever a supersonic flow is "turned into itself" as shown in Figure 9.2a, an oblique shock wave will occur. The originally horizontal streamlines ahead of the wave are uniformly deflected in crossing the wave, such that the streamlines behind the wave are parallel to each other and inclined upward at the deflection angle θ. Across the wave, the Mach number discontinuously decreases, and the pressure, density, and temperature discontinuously increase. In contrast, Figure 9.2b shows the case where the wall is turned downward at the corner through the deflection angle θ; that is, the corner is convex. Again, the flow at the wall must be tangent to the wall; hence, the streamline at the wall is deflected downward through the angle θ. The bulk of the gas is above the wall, and in Figure 9.2b, the streamlines are turned downward, away from the main bulk of the flow. Whenever a supersonic flow is "turned away from itself" as shown in Figure 9.2b, an expansion wave will occur. This expansion wave is in the shape of a fan centered at the corner. The fan continuously opens in the direction away from the corner, as shown in Figure 9.2b. The originally horizontal streamlines ahead of the expansion wave are deflected smoothly and continuously through the expansion fan such that the streamlines behind the wave are parallel to each other and inclined downward at the deflection angle θ. Across the expansion wave, the Mach number increases, and the pressure, temperature, and density decrease. Hence, an expansion wave is the direct antithesis of a shock wave. Oblique shock and expansion waves are prevalent in two- and threedimensional supersonic flows. These waves are inherently two-dimensional in nature, in contrast to the one-dimensional normal shock waves discussed in Chapter 8. That is, in Figure 9.2a and b, the flow-field properties are a function of x and y. The purpose of the present chapter is to determine and study the properties of these oblique waves. What is the physical mechanism that creates waves in a supersonic flow? To address this question, recall our picture of the propagation of a sound wave via molecular collisions, as portrayed in Section 8.3. If a slight disturbance takes place at some point in a gas, information is transmitted to other points in the gas by sound waves which propagate in all directions away from the source of the disturbance. Now consider a body in a flow, as sketched in Figure 9.3. The gas molecules which impact the body surface experience a change in momentum. In turn, this change is transmitted to neighboring molecules by random molecular collisions. In this fashion, information about the presence of the body attempts to be transmitted to the surrounding flow via molecular collisions; that is, the information is propagated upstream at approximately the local speed of sound. If the upstream flow is subsonic, as shown in Figure 9.3a, the disturbances have no problem working their way far upstream, thus giving the incoming flow plenty of time to move out of the way of the body. On the other hand, if the upstream flow is supersonic, as shown in Figure 9.3b, the disturbances cannot work their way upstream; rather, at some finite distance from the body, the disturbance waves pile up and coalesce, forming a standing wave in front of the body. Hence, the physical generation of waves in a supersonic flow—both shock and expansion waves—is due to the propagation of information via molecular collisions and due to the fact that such propagation cannot work its way into certain regions of the supersonic flow. Why are most waves oblique rather than normal to the upstream flow? To answer this question, consider a small source of disturbance moving through a stagnant gas. For lack of anything better, let us call this disturbance source a "beeper," which periodically emits sound. First, consider the beeper moving at subsonic speed through the gas, as shown in Figure 9.4a. The speed of the beeper is V, where V < a. At time t = 0, the beeper is located at point A; at this point, it emits a sound wave that propagates in all directions at the speed of sound, a. At a later time t this sound wave has propagated a distance at from point A and is represented by the circle of radius at shown in Figure 9.4a. During the same time, the beeper has moved a distance V t and is now at point B in Figure 9.4a. Moreover, during its transit from A to B, the beeper has emitted several other sound waves, which at time t are represented by the smaller circles in Figure 9.4a. Note that the beeper always stays inside the family of circular sound waves and that the waves continuously move ahead of the beeper. This is because the beeper is traveling at a subsonic speed V < a. In contrast, consider the beeper moving at a supersonic speed V > a through the gas, as shown in Figure 9.4b. At time t = 0, the beeper is located at point A, where it emits a sound wave. At a later time t, this sound wave has propagated a distance at from point A and is represented by the circle of radius at shown in Figure 9.4b. During the same time, the beeper has moved a distance V t to point B. Moreover, during its transit from A to B, the beeper has emitted several other sound waves, which at time t are represented by the smaller circles in Figure 9.4b. However, in contrast to the subsonic case, the beeper is now constantly outside the family of circular sound waves; that is, it is moving ahead of the wave fronts because V > a. Moreover, something new is happening; these wave fronts form a disturbance envelope given by the straight line BC, which is tangent to the family of circles. This line of disturbances is defined as a Mach wave. In addition, the angle ABC that the Mach wave makes with respect to the direction of motion of the beeper is defined as the Mach angleμ. From the geometry of Figure 9.4b, we readily find that sinμ = at V t = a V = 1 M Thus, the Mach angle is simply determined by the local Mach number as μ = sin−1 1 M (9.1) Examining Figure 9.4b, the Mach wave, that is, the envelope of disturbances in the supersonic flow, is clearly oblique to the direction of motion. If the disturbances are stronger than a simple sound wave, then the wave front becomes stronger than a Mach wave, creating an oblique shock wave at an angle β to the freestream, where β>μ. This comparison is shown in Figure 9.5. However, the physical mechanism creating the oblique shock is essentially the same as that described above for the Mach wave. Indeed, a Mach wave is a limiting case for oblique shock (i.e., it is an infinitely weak oblique shock). This finishes our discussion of the physical source of oblique waves in a supersonic flow. Let us now proceed to develop the equations that allow us to calculate the change in properties across these oblique waves, first for oblique shock waves, and then for expansion waves. In the process, we follow the road map given in Figure 9.6.

Shock waves and boundary layers do not mix; bad things can happen when a shock wave impinges on a boundary layer. Unfortunately, shock-wave/boundarylayer interactions frequently occur in practical supersonic flows, and therefore we pay attention to this interaction in the present section. The fluid dynamics of a shock-wave/boundary-layer interaction is complex (and extremely interesting), and a detailed presentation is beyond the scope of this book. Here we give a brief qualitative discussion—just enough to acquaint you with the basic picture. Consider a supersonic flow over a surface wherein an oblique shock wave impinges on the surface, such as sketched in Figure 9.19. In this figure the flow is assumed to be inviscid, and the incident shock impinges at point B on the upper wall, giving rise to a reflected shock emanating from the same point. There is a discontinuous pressure increase at point B, a combination of the pressure increases across the incident and reflected shocks. Indeed, point B is a singular point where there is an infinitely large adverse pressure gradient. Imagine that suddenly we have a boundary layer along the wall in Figure 9.19. At point B, the boundary layer would experience an infinitely large adverse pressure gradient. In Section 4.12, we discussed what happens to a boundary layer when it experiences a large adverse pressure gradient—it separates from the surface. These are the basic elements of the shock-wave/boundary-layer interaction. The incident shock wave imposes a strong adverse pressure gradient on the boundary layer, which in turn separates from the surface, and the resulting flow field in the vicinity of the shock wave impingement becomes one of a mutual interaction between the boundary layer and the shock wave. This mutual interaction is sketched qualitatively in Figure 9.40. Here, for ease of presentation, we show a shock wave impinging on a lower wall rather than on the upper wall as in Figure 9.19. In Figure 9.40, we see a boundary layer growing along a flat plate. Because the external flow is supersonic, the boundarylayer velocity profile is subsonic near the wall and supersonic near the outer edge. At some downstream location an incident shock impinges on the boundary layer. The large pressure rise across the shock wave acts as a severe adverse pressure gradient imposed on the boundary layer, thus causing the boundary layer to locally separate from the surface. Because the high pressure behind the shock feeds upstream through the subsonic portion of the boundary layer, the separation takes place ahead of the theoretical inviscid flow impingement point of the incident shock wave. In turn, the separated boundary layer deflects the external supersonic flow into itself, thus inducing a second shock wave, identified here as the induced separation shock wave. The separated boundary layer subsequently turns back toward the plate, reattaching to the surface at some downstream location. Here again the supersonic flow is deflected into itself, causing a third shock wave called the reattachment shock. Between the separation and reattachment shocks, where the boundary layer is turning back toward the surface, the supersonic flow is turned away from itself, generating expansion waves shown in Figure 9.40. At the point of reattachment, the boundary layer has become relatively thin, the pressure is high, and consequently this becomes a region of high local aerodynamic heating. Further away from the plate, the separation and reattachment shocks merge to form the conventional reflected shock wave that is expected from the inviscid picture, as shown in Figure 9.19. The scale and severity of the interaction shown in Figure 9.40 depends on whether the boundary layer is laminar or turbulent. Since laminar boundary layers separate more readily than turbulent boundary layers (see Section 4.12), the laminar interaction usually takes place more readily with more severe attendant consequences than the turbulent interaction. However, the general qualitative aspects of the interaction shown in Figure 9.40 are the same for both cases. The shock-wave/boundary-layer interaction has a major effect on the pressure, shear stress, and heat-transfer distributions along the wall. Of particular consequence is the high local heat-transfer rate at the reattachment point, which at hypersonic speeds can peak to an order of magnitude larger than at neighboring locations. An example of the effect on the wall pressure distribution is shown in Figure 9.41a, patterned after the work of Baldwin and Lomax at the NASA Ames Research Center (B. S. Baldwin, and H. Lomax, "Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows," AIAA Paper No. 78-257, January 1978). Here, the pressure distribution along the wall in the interaction region is plotted versus distance along the wall, x, where x0 is the theoretical point of impingement for the incident shock in the inviscid flow case. The pressure distribution shows a steplike increase, with an intermediate plateau; this distribution is typical of the shock-wave/boundary-layer interaction. The external flow is at Mach 3 ahead of the incident shock, and the boundary layer is turbulent. The solid curve is a computational fluid dynamic (CFD) calculation, and the circles are experimental data. Notice that the pressure increase extends a distance equal to about four times the boundary-layer thickness ahead of the theoretical inviscid impingement point. The shear stress distribution is shown in Figure 9.41b. In the pocket of separated flow, τw becomes small and reverses its direction (negative values of τw) due to the low energy recirculating flow. Because of the consequent creation of separated flow, increased loss of total pressure, and high peak heat-transfer rates, shock-wave/boundary-layer interactions usually should be avoided as much as possible in the design of supersonic aircraft and flow devices. However, this is easier said than done. Shock-wave/ boundary-layer interactions are a fact of life in the practical world of supersonic flow, and that is why we have discussed their basic nature in this section. On the other hand, modern creative ideas have led to the beneficial use of the separated flow from a shock-wave/boundary-layer interaction to actually enhance the offdesign performance of jet-engine exhaust nozzles and for certain types of flow control. So the picture is not entirely black.

The Mach number is named in honor of Ernst Mach, an Austrian physicist and philosopher who was an illustrious and controversial figure in late nineteenthcentury physics. Mach conducted the first meaningful experiments in supersonic flight, and his results triggered a similar interest in Ludwig Prandtl 20 years later. 672 PART 3 Inviscid, Compressible Flow Who was Mach? What did he actually accomplish in supersonic aerodynamics? Let us look at this man further. Mach was born at Turas, Moravia, in Austria, on February 18, 1838. His father, Johann, was a student of classical literature who settled with his family on a farm in 1840. An extreme individualist, Johann raised his family in an atmosphere of 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 that 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 that 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 μ between this wave and the direction of flight is given by sinμ = 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 V/a 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 CHAPTER 9 Oblique Shock and Expansion Waves 673 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

The curved bow shock which stands in front of a blunt body in a supersonic flow is sketched in Figure 8.1. We are now in a position to better understand the properties of this bow shock, as follows. The flow in Figure 8.1 is sketched in more detail in Figure 9.23. Here, the shock wave stands a distance δ in front of the nose of the blunt body; δ is defined as the shock detachment distance. At point a, the shock wave is normal to the upstream flow; hence, point a corresponds to a normal shock wave. Away from point a, the shock wave gradually becomes curved and weaker, eventually evolving into a Mach wave at large distances from the body (illustrated by point e in Figure 9.23). A curved bow shock wave is one of the instances in nature when you can observe all possible oblique shock solutions at once for a given freestream Mach number M1. This takes place between points a and e. To see this more clearly, consider the θ-β-M diagram sketched in Figure 9.24 in conjunction with Figure 9.23. In Figure 9.24, point a corresponds to the normal shock, and point e corresponds to the Mach wave. Slightly above the centerline, at point b in Figure 9.23, the shock is oblique but pertains to the strong shock-wave solution in Figure 9.24. The flow is deflected slightly upward behind the shock at point b. As we move further along the shock, the wave angle becomes more oblique, and the flow deflection increases until we encounter point c. Point c on the bow shock corresponds to the maximum deflection angle shown in Figure 9.24. Above point c, from c to e, all points on the shock correspond to the weak shock solution. Slightly above point c, at point c , the flow behind the shock becomes sonic. From a to c , the flow is subsonic behind the bow shock; from c to e, it is supersonic. Hence, the flow field between the curved bow shock and the blunt body is a mixed region of both subsonic and supersonic flow. The dividing line between the subsonic and supersonic regions is called the sonic line, shown as the dashed line in Figure 9.23. The shape of the detached shock wave, its detachment distance δ, and the complete flow field between the shock and the body depend on M1 and the size and shape of the body. The solution of this flow field is not trivial. Indeed, the upersonic blunt-body problem was a major focus for supersonic aerodynamicists during the 1950s and 1960s, spurred by the need to understand the high-speed flow over blunt-nosed missiles and reentry bodies. Indeed, it was not until the late 1960s that truly sufficient numerical techniques became available for satisfactory engineering solutions of supersonic blunt-body flows. These modern techniques are discussed in Chapter 13. As illustrated by Example 9.8, the entropy is different along different streamlines behind a curved shock wave. For the case treated in Example 9.8, streamline a passes through the normal shock at point a and then flows downstream, wetting the surface of the body, as shown in Figure 9.25. This is the maximum entropy streamline. All other streamlines have smaller values of entropy; streamline b has a smaller entropy than streamline a because it passes through a weaker part of the curved shock wave at point b. Therefore, if you visualize a line that cuts through the flow field from point 1 to point 2 in Figure 9.25, the entropy decreases along this line from the body to the shock. That is, there exists an entropy gradient, ∇s, in the flow. For blunt-nosed hypersonic bodies the entropy gradient can be quite large, and is the source of the "entropy layer" that interacts with the boundary layer on hypersonic bodies (see, for example, Reference 52). The presence of entropy gradients in the flow behind a curved shock wave has another consequence—the production of vorticity in the flow. The physical connection between entropy gradients and vorticity is quantified by Crocco's theorem, a combination of the momentum equation and the combined first and second laws of thermodynamics: T ∇s = ∇ho − V × (∇ × V) Crocco's theorem In this equation, ∇s is the entropy gradient, ∇ho is the gradient in the total enthalpy, and ∇ ×V is the vorticity. For a derivation of Crocco's theorem, see, for example, Section 6.6 of Reference 21. For our discussion, we present Crocco's theorem simply to emphasize an important feature of the flow behind the curved shock shown in Figure 9.25. The flow is adiabatic, hence ∇ho is zero everywhere in the flow. However, ∇s is finite, and therefore from Crocco's theorem ∇ × V must be finite. Conclusion: The flow field behind a curved shock wave is rotational. As a result, a velocity potential with all its analytical advantages discussed earlier in this book cannot be defined for the blunt-body flow field. Consequently, the flow field behind a curved shock is computed by means of numerical solutions of the continuity, momentum, and energy equations. Such computational fluid dynamic solutions are discussed in Section 13.5. 9.6 PRANDTL-MEYER EXPANSION WAVES Oblique shock waves, as discussed in Sections 9.2 to 9.5, occur when a supersonic flow is turned into itself (see again Figure 9.2a). In contrast, when a supersonic flow is turned away from itself, an expansion wave is formed, as sketched in Figure 9.2b. Examine this figure carefully, and review the surrounding discussion in Section 9.1 before progressing further. The purpose of the present section is to develop a theory which allows us to calculate the changes in flow properties across such expansion waves. To this stage in our discussion of oblique waves, we have completed the left-hand branch of the road map in Figure 9.6. In this section, we cover the right-hand branch. The expansion fan in Figure 9.2b is a continuous expansion region that can be visualized as an infinite number of Mach waves, each making the Mach angle μ [see Equation (9.1)] with the local flow direction. As sketched in Figure 9.26, the expansion fan is bounded upstream by a Mach wave which makes the angle μ1 with respect to the upstream flow, where μ1 = arcsin(1/M1). The expansion fan is bounded downstream by another Mach wave which makes the angle μ2 with respect to the downstream flow, where μ2 = arcsin(1/M2). Since the expansion through the wave takes place across a continuous succession of Mach waves, and since ds = 0 for each Mach wave, the expansion is isentropic. This is in direct contrast to flow across an oblique shock, which always experiences an entropy increase. The fact that the flow through an expansion wave is isentropic is a greatly simplifying aspect, as we will soon appreciate. An expansion wave emanating from a sharp convex corner as sketched in Figures 9.2b and 9.26 is called a centered expansion wave. Ludwig Prandtl and his student Theodor Meyer first worked out a theory for centered expansion waves in 1907-1908, and hence such waves are commonly denoted as Prandtl-Meyer expansion waves. The problem of an expansion wave is as follows: Referring to Figure 9.26, given the upstream flow (region 1) and the deflection angle θ, calculate the downstream flow (region 2). Let us proceed. Consider a very weak wave produced by an infinitesimally small flow deflection dθ as sketched in Figure 9.27. We consider the limit of this picture as dθ → 0; hence, the wave is essentially a Mach wave at the angle μ to the upstream flow. The velocity ahead of the wave is V. As the flow is deflected downward through the angle dθ, the velocity is increased by the infinitesimal amount dV, and hence the flow velocity behind the wave is V + dV inclined at the angle dθ. Recall from the treatment of the momentum equation in Section 9.2 that any change in velocity across a wave takes place normal to the wave; the tangential component is unchanged across the wave. In Figure 9.27, the horizontal line segment AB with length V is drawn behind the wave. Also, the line segment AC is drawn to represent the new velocity V + dV behind the wave. Then line BC is normal to the wave because it represents the line along which the change in velocity occurs. Examining the geometry in Figure 9.27, from the law of sines applied to triangle ABC, we see that

The result of Example 9.5 shows that the final total pressure is 76 percent higher for the case of the multiple shock system (case 2) in comparison to the single normal shock (case 1). In principle, the total pressure is an indicator of how much useful work can be done by the gas; this is described later in Section 10.4. Everything else being equal, the higher the total pressure, the more useful is the flow. Indeed, losses of total pressure are an index of the efficiency of a fluid flow—the lower the total pressure loss, the more efficient is the flow process. In this example, case 2 is more efficient in slowing the flow to subsonic speeds than case 1 because the loss in total pressure across the multiple shock system of case 2 is actually less than that for case 1 with a single, strong, normal shock wave. The physical reason for this is straightforward. The loss in total pressure across a normal shock wave becomes particularly severe as the upstream Mach number increases; a glance at the p0,2/p0,1 column in Appendix B attests to this. If the Mach number of a flow can be reduced before passing through a normal shock, the loss in total pressure is much less because the normal shock is weaker. This is the function of the oblique shock in case 2, namely, to reduce the Mach number of the flow before passing through the normal shock. Although there is a total pressure loss across the oblique shock also, it is much less than across a normal shock at the same upstream Mach number. The net effect of the oblique shock reducing the flow Mach number before passing through the normal shock more than makes up for the total pressure loss across the oblique shock, with the beneficial result that the multiple shock system in case 2 produces a smaller loss in total pressure than a single normal shock at the same freestream Mach number.

and that p3 > p2. This creates a net pressure imbalance that generates the resultant aerodynamic force R, shown in Figure 9.36. Indeed, for a unit span, the resultant force and its components, lift and drag, per unit span are R = (p3 − p2)c (9.46) L = (p3 − p2)c cos α (9.47) D = (p3 − p2)c sin α (9.48) In Equations (9.47) and (9.48), p3 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 9.36. 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 9.37. Assume the airfoil is at 0◦ angle of attack. The supersonic flow over the airfoil is first compressed and deflected through the angle ε by the oblique shock wave at the leading edge. At midchord, the flow is expanded through an angle 2ε, 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 9.37; note that the pressures on faces a and c are uniform and equal to p2 and that the pressures on faces b and d are also uniform but equal to p3, where p3 < p2. 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 9.37, where l is the length of each face and t is the airfoil thickness. Then, D = 2(p2l sin ε − p3l sin ε) = 2(p2 − p3) t 2 Hence, D = (p2 − p3)t (9.49) In Equation (9.49), p2 is calculated from oblique shock theory, and p3 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 9.36 and 9.37 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. CHAPTER 9 Oblique Shock and Expansion Waves 663 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 9.6. As such, it brings us to a logical conclusion of our discussion of oblique waves in supersonic flows. Expanding on the comments made in Section 9.3.1, reflect again on the result obtained in Example 9.6, where the drag coefficient was calculated for a 15◦ halfangle wedge in a Mach 5 flow. Reflect also on the result obtained in Example 9.12, where the lift and drag coefficients were calculated for a flat plate at a 5◦ angle of attack in a Mach 3 flow. Note that to calculate these coefficients we did not need to know the freestream pressure, density, or velocity. All we needed to know was: 1. The shape of the body 2. The angle of attack 3. The freestream Mach number These examples are clear-cut illustrations of the results of dimensional analysis discussed in Section 1.7, and are totally consistent with Equations (1.42) and (1.43), which emphasize that lift and drag coefficients for a body of given shape are functions of only Reynolds number, Mach number, and angle of attack. For the examples in this chapter, we are dealing with an inviscid flow, so Re is not relevant—only M∞ and α. 9.9 THE X-15 AND ITS WEDGE TAIL Examine the photograph of the X-15 hypersonic research vehicle shown in Figure 9.18. The viewpoint in this photograph is looking down at the top of the vehicle. Concentrate on the vertical tail at the rear of the airplane. The cross section of the vertical tail is clearly seen—it is a wedge cross section in contrast to the type of thin symmetric airfoil sections usually employed for vertical tails on airplanes. The wedge shape is further emphasized by examining Figure 9.38,

ave illustrated in Figure 9.2a. In this picture, we can imagine the shock wave extending unchanged above the corner to infinity. However, in real life this does not happen. In reality, the oblique shock in Figure 9.2a will impinge somewhere on another solid surface and/or will intersect other waves, either shock or expansion waves. Such wave intersections and interactions are important in the practical design and analysis of supersonic airplanes, missiles, wind tunnels, rocket engines, etc. A perfect historical example of this, as well as the consequences that can be caused by not paying suitable attention to wave interactions, is a ramjet flight-test program conducted in the early 1960s. During this period, a ramjet engine was mounted underneath the X-15 hypersonic airplane for a series of flight tests at high Mach numbers, in the range from 4 to 7. (The X-15, shown in Figure 9.18, was an experimental, rocket-powered airplane designed to probe the lower end of hypersonic manned flight.) During the first high-speed tests, the shock wave from the engine cowling impinged on the bottom surface of the X-15, and because of locally high aerodynamic heating in the impingement region, a hole was burned in the X-15 fuselage. Although this problem was later fixed, it is a graphic example of what shock-wave interactions can do to a practical configuration. The purpose of this section is to present a mainly qualitative discussion of shock-wave interactions. For more details, see Chapter 4 of Reference 21. First, consider an oblique shock wave generated by a concave corner, as shown in Figure 9.19. The deflection angle at the corner is θ, thus generating an oblique shock at point A with a wave angle β1. Assume that a straight, horizontal wall is present above the corner, as also shown in Figure 9.19. The shock wave generated at point A, called the incident shock wave, impinges on the upper wall at point B. Question: Does the shock wave simply disappear at point B? If not, what happens to it? To answer this question, we appeal to our knowledge of shock-wave properties. Examining Figure 9.19, we see that the flow in region 2 behind the incident shock is inclined upward at the deflection angle θ. However, the flow must be tangent everywhere along the upper wall; if the flow in region 2 were to continue unchanged, it would run into the wall and have no place to go. Hence, the flow in region 2 must eventually be bent downward through the angle θ in order to maintain a flow tangent to the upper wall. Nature accomplishes thi downward deflection via a second shock wave originating at the impingement point B in Figure 9.19. This second shock is called the reflected shock wave. The purpose of the reflected shock is to deflect the flow in region 2 so that it is parallel to the upper wall in region 3, thus preserving the wall boundary condition. The strength of the reflected shock wave is weaker than the incident shock. This is because M2 < M1, and M2 represents the upstream Mach number for the reflected shock wave. Since the deflection angles are the same, whereas the reflected shock sees a lower upstream Mach number, we know from Section 9.2 that the reflected wave must be weaker. For this reason, the angle the reflected shock makes with the upper wall is not equal to β1 (i.e., the wave reflection is not specular). The properties of the reflected shock are uniquely defined by M2 and θ; since M2 is in turn uniquely defined by M1 and θ, then the properties in region 3 behind the reflected shock as well as the angle are easily determined from the given conditions of M1 and θ by using the results of Section 9.2 as follows: 1. Calculate the properties in region 2 from the given M1 and θ. In particular, this gives us M2. 2. Calculate the properties in region 3 from the value of M2 calculated above and the known deflection angle θ. An interesting situation can arise as follows. Assume that M1 is only slightly above the minimum Mach number necessary for a straight, attached shock wave at the given deflection angle θ. For this case, the oblique shock theory from Section 9.2 allows a solution for a straight, attached incident shock. However, we know that the Mach number decreases across a shock (i.e., M2 < M1). This decrease may be enough such that M2 is not above the minimum Mach number for the required deflection θ through the reflected shock. In such a case, our oblique shock theory does not allow a solution for a straight reflected shock wave. The regular reflection as shown in Figure 9.19 is not possible. Nature handles this situation by creating the wave pattern shown in Figure 9.20. Here, the originally straight incident shock becomes curved as it nears the upper wall and becomes a normal shock wave at the upper wall. This allows the streamline at the wall to continue parallel to the wall behind the shock intersection. In addition, a curved reflected shock branches from the normal shock and propagates downstream. This wave pattern, shown in Figure 9.20, is called a Mach reflection. The calculation of the wave pattern and general properties for a Mach reflection requires numerical techniques such as those to be discussed in Chapter 13. Another type of shock interaction is shown in Figure 9.21. Here, a shock wave is generated by the concave corner at point G and propagates upward. Denote this wave as shock A. Shock A is a left-running wave, so-called because if you stand on top of the wave and look downstream, you see the shock wave running in front of you toward the left. Another shock wave is generated by the concave corner at point H, and propagates downward. Denote this wave as shock B. Shock B is a right-running wave, so-called because if you stand on top of the wave and look downstream, you see the shock running in front of you toward the right. The picture shown in Figure 9.21 is the intersection of right- and left-running shock waves. The intersection occurs at point E. At the intersection, wave A is refracted and continues as wave D. Similarly, wave B is refracted and continues as wave C. The flow behind the refracted shock D is denoted by region 4; the flow behind the refracted shock C is denoted by region 4 . These two regions are divided by a slip line EF. Across the slip line, the pressures are constant (i.e., p4 = p4), and the direction (but not necessarily the magnitude) of velocity is the same, namely, parallel to the slip line. All other properties in regions 4 and 4 are different, most notably the entropy (s4 = s4). The conditions which must hold across the slip line, along with the known M1 θ1, and θ2, uniquely determine the shock-wave interaction shown in Figure 9.21. (See Chapter 4 of Reference 21 for details concerning the calculation of this interaction.) Figure 9.22 illustrates the intersection of two left-running shocks generated at corners A and B. The intersection occurs at pointC, at which the two shocks merge and propagate as the stronger shock CD, usually along with a weak reflected wave CE. This reflected wave is necessary to adjust the flow so that the velocities in regions 4 and 5 are in the same direction. Again, a slip line CF trails downstream of the intersection point. The above cases are by no means all the possible wave interactions in a supersonic flow. However, they represent some of the more common situations encountered frequently in practic

Aerodynmics 9

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