15.8. Two-Dimensional Steady Compressible Flow
To this point, the emphasis in this chapter has been on one-dimensional flows in which flow properties vary only in the direction of flow. This section presents steady compressible flow results for more than one spatial dimension. To get started, consider a point source emitting infinitesimal pressure (acoustic) disturbances in a still compressible fluid in which the speed of sound is c. If the point source is stationary, then the pressure-disturbance wavefronts are concentric spheres. Figure 15.20a shows the intersection of these wavefronts with a plane containing the source at times corresponding to integer multiples of Δt.
When the source propagates to the left at speed U < c, the wavefront diagram changes to look like Figure 15.20b, which shows four locations of the source separated by equal time intervals Δt, with point 4 being the present location of the source. At the first point, the source emitted a wave that has spherically expanded to a radius of 3cΔt in the time interval 3Δt. During this time the source has moved to the fourth location, a distance of 3UΔt from the first point of wavefront emission. The figure also shows the locations of the wavefronts emitted while the source was at the second and third points. Here, the wavefronts do not intersect because U < c. As in the case of the stationary source, the wavefronts propagate vertically upward and downward, and horizontally upstream and downstream from the source. Thus, a body moving at a subsonic speed influences the entire flow field.
Now consider the case depicted in Figure 15.20c where the source moves supersonically, U > c. Here, the centers of the spherically expanding wavefronts are separated by more than cΔt, and no pressure disturbance propagates ahead of the source. Instead, the edges of the wavefronts form a conical tangent surface called the Mach cone. In planar two-dimensional flow, the tangent surface is in the form of a wedge, and the tangent lines are called Mach lines. An examination of the figure shows that the half-angle of the Mach cone (or wedge), called the Mach angle μ, is given by sinμ = (cΔt)/(UΔt), so that:
Figure 15.20 Wavefronts emitted by a point source in a still fluid when the source speed U is: (a) U = 0; (b) U < c; and (c) U > c. In each case the wavefronts are emitted at integer multiples of Δt. At subsonic source speeds, the wavefronts do not overlap and they spread ahead of the source. At supersonic source speeds, all the wavefronts lie behind the source within the Mach cone having a half angle sin–1(1/M).
(15.60)
The Mach cone becomes wider as M decreases and becomes a plane front (i.e., μ = 90°) when M = 1.
The situation depicted in Figure 15.20 has at least two interpretations. The point source could be part of a solid body, which sends out pressure waves as it moves through the fluid. Or, after a Galilean transformation, Figures 15.20b and c apply equally well to a stationary point source with a compressible fluid moving past it at speed U. From Figure 15.20c it is clear that in a supersonic flow an observer outside the source's Mach cone would not detect or hear a pressure signal emitted by the source, hence this region is called the zone of silence. In contrast, the region inside the Mach cone is called the zone of action, within which the effects of the disturbance are felt. Thus, the sound of a supersonic aircraft passing overhead does not reach an observer on the ground until its Mach cone reaches the observer, and this arrival occurs after the aircraft has passed overhead.
At every point in a planar supersonic flow there are two Mach lines, oriented at ±μ to the local direction of flow. Pressure disturbance information propagates along these lines, which are the characteristics of the governing differential equation. It can be shown that the nature of the governing differential equation is hyperbolic in a supersonic flow and elliptic in a subsonic flow. In addition, the method of characteristics may be applied to steady two-dimensional flow but this approach is not pursued here.
When pressure disturbances from the source are of finite amplitude, they may evolve into a shock wave that is not normal to the flow direction. Such oblique shock waves are commonly encountered in ballistics and supersonic flight, and differ from normal shock waves because they change the upstream flow velocity's magnitude and direction. A generic depiction of an oblique shock wave is provided in Figure 15.21 in two coordinate systems. Figure 15.21a shows the stream-aligned coordinate system where the shock wave resides at an angle σ from the horizontal. Here the velocity upstream of the shock is horizontal with magnitude V1, while the velocity downstream of the shock is deflected from the horizontal by an angle δ and has magnitude V2. Figure 15.21b shows the same shock wave in a shock-aligned coordinate system where the shock wave is vertical, and the fluid velocities upstream and downstream of the shock are (u1, v) and (u2, v), respectively. Here v is parallel to the shock wave and is not influenced by it (see Exercise 15.22). Thus an oblique shock may be analyzed as a normal shock involving u1 and u2 to which a constant shock-parallel velocity v is added. Using the Cartesian coordinates in Figure 15.21b where the shock coincides with the vertical axis, the relationships between the various components and angles are:
Figure 15.21 Two coordinate systems for an oblique shock wave. (a) Stream-aligned coordinates where the oblique shock wave lies at shock angle = σ and produces a flow-deflection of angle = δ. (b) Shock-normal coordinates which are preferred for analysis because an oblique shock wave is merely a normal shock with a superimposed shock-parallel velocity v.
The angle σ is called the shock angle or wave angle and δ is called the deflection angle. The normal Mach numbers upstream (1) and downstream (2) of the shock are:
Because u2 < u1, there is a sudden change of direction of flow across the shock and the flow is turned toward the shock by angle δ.
Superposition of the tangential velocity v does not affect the static properties, which are therefore the same as those for a normal shock. The expressions for the ratios p2/p1, ρ2/ρ1, T2/T1, and (s2 – s1)/cv are therefore those given by (15.39) and (15.41) through (15.43), if M1 is replaced by Mn1 = M1sinσ. For example:
(15.61, 15.62)
Thus, the normal-shock table, Table 15.2, is applicable to oblique shock waves when M1sinσ is used in place of M1.
The relation between the upstream and downstream Mach numbers can be found from (15.40) by replacing M1 by M1sinσ and M2 by M2sin (σ – δ). This gives:
(15.63)
An important relation is that between the deflection angle δ and the shock angle σ for a given M1, given in (15.62). Using the trigonometric identity for tan (σ – δ), this becomes:
(15.64)
A plot of this relation is given in Figure 15.22. The curves represent δ versus σ for constant M1. The value of M2 varies along the curves, and the locus of points corresponding to M2 = 1 is indicated. It is apparent that there is a maximum deflection angle δmax for oblique shock solutions to be possible; for example, δmax = 23° for M1 = 2. For a given M1, δ becomes zero at σ = π/2 corresponding to a normal shock, and at σ = μ = sin−1(1/M1) corresponding to the Mach angle. For a fixed M1 and δ < δmax, there are two possible solutions: a weak shock corresponding to a smaller σ and a strong shock corresponding to a larger σ. It is clear that the flow downstream of a strong shock is always subsonic; in contrast, the flow downstream of a weak shock is generally supersonic, except in a small range in which δ is slightly smaller than δmax.
Oblique shock waves are commonly generated when a supersonic flow is forced to change direction to go around a structure where the flow area cross-section is reduced. Two examples are shown in Figure 15.23 that show supersonic flow past a wedge of half-angle δ, or the flow past a compression bend where the wall turns into the flow by an angle δ. If M1 and δ are known, then σ can be obtained from Figure 15.22, and Mn2 (and therefore M2 = Mn2/sin(σ – δ)) can be obtained from the shock table (Table 15.2). An attached shock wave, corresponding to the weak solution, forms at the nose of the wedge, such that the flow is parallel to the wedge after turning through an angle δ. The shock angle σ decreases to the Mach angle μ1 = sin–1(1/M1) as the deflection δ tends to zero. It is interesting that the corner velocity in a supersonic flow is finite. In contrast, the corner velocity in a subsonic (or incompressible) flow is either zero or infinite, depending on whether the wall shape is concave or convex. Moreover, the streamlines in Figure 15.23 are straight, and computation of the field is easy. By contrast, the streamlines in a subsonic flow are curved, and the computation of the flow field is not as easy. The basic reason for this is that, in a supersonic flow, small pressure disturbances do not propagate upstream of Mach lines or shock waves, hence the flow field can be constructed step by step, proceeding downstream. In contrast, disturbances propagate both upstream and downstream in a subsonic flow so that all features in the entire flow field are related to each other.
Figure 15.22 Plot of oblique shock solutions. The strong-shock branch is indicated by dashed lines on the right, and the heavy dotted line indicates the maximum deflection angle δmax. (From Ames Research Staff, 1953, NACA Report 1135.)
Figure 15.23 Two possible means for producing oblique shocks in a supersonic flow. In both cases a solid surface causes the flow to turn, and the flow area is reduced. The geometry shown in the right panel is sometimes called a compression corner.
As δ is increased beyond δmax, attached oblique shocks are not possible, and a detached curved shock stands in front of the body (Figure 15.24). The central streamline goes through a normal shock and generates a subsonic flow in front of the wedge. The strong-shock solution of Figure 15.22 therefore holds near the nose of the body. Farther out, the shock angle decreases, and the weak-shock solution applies. If the wedge angle is not too large, then the curved detached shock in Figure 15.24 becomes an oblique attached shock as the Mach number is increased. In the case of a blunt-nosed body, however, the shock at the leading edge is always detached, although it moves closer to the body as the Mach number is increased.
We see that shock waves may exist in supersonic flows and their location and orientation adjust to satisfy boundary conditions. In external flows, such as those just described, the boundary condition is that streamlines at a solid surface must be tangent to that surface. In duct flows the boundary condition locating the shock is usually the downstream pressure.
From the foregoing analysis, it is clear that large-angle supersonic flow deflections should be avoided when designing efficient devices that produce minimal total pressure losses. Efficient devices tend to be slender and thin, and their performance may be analyzed using a weak oblique shock approximation that can be obtained from the results above in the limit of small flow deflection angle, δ ≪ 1. To obtain this expression, simplify (15.64) by noting that as δ → 0, the shock angle σ tends to the Mach angle μ1 = sin−1(1/M1). And, from (15.61) we note that (p2 – p1)/p1 → 0 as M 1 2 sin 2 σ − 1 → 0 
(as σ → μ and δ → 0). Then from (15.61) and (15.64):
(as σ → μ and δ → 0). Then from (15.61) and (15.64):
Figure 15.24 A detached shock wave. When angle of the wedge shown in the left panel of Figure 15.23 is too great for an oblique shock, a curved shock wave will form that does not touch body. A portion of this detached shock wave will have the properties of a normal-shock wave.
(15.65)
As δ → 0 
, tanδ ≈ δ, cotμ = [M12 – 1]1/2, sinσ ≈ 1/M1 and:
, tanδ ≈ δ, cotμ = [M12 – 1]1/2, sinσ ≈ 1/M1 and:
(15.66)
The interesting point is that the relation (15.66) is also applicable to weak expansion waves and not just weak compression waves. By this we mean that the pressure increase due to a small deflection of the wall toward the flow is the same as the pressure decrease due to a small deflection of the wall away from the flow. This extended range of validity of (15.66) occurs because the entropy change across a weak shock may be negligible even when the pressure change is appreciable (see (15.44b) and the related discussion). Thus, weak shock waves can be treated as isentropic or reversible. Relationships for a weak shock wave can therefore be applied to a weak expansion wave, except for some sign changes. In the final section of this chapter, (15.66) is used to estimate the lift and drag of a thin airfoil in supersonic flow.
When an initially horizontal supersonic flow follows a curving wall, the wall radiates compression and expansion waves into the flow that modulate the flow's direction and Mach number. When the wall is smoothly curved these compression and expansion waves follow Mach lines, inclined at an angle of μ = sin–1(1/M) to the local direction of flow (Figure 15.25). In this simple circumstance where there is no upper wall that radiates compression or expansion waves downward into the region of interest, the flow's orientation and Mach number are constant on each Mach line. In the case of compression, the Mach number decreases along the flow, so that the Mach angle increases. The Mach lines may therefore coalesce and form an oblique shock as in Figure 15.25a. In the case of a gradual expansion, the Mach number increases along the flow and the Mach lines diverge as in Figure 15.25b.
Figure 15.25 Gradual compression and expansion in supersonic flow. (a) A gradual compression corner like the one shown will eventually result in an oblique shock wave as the various Mach lines merge, each carrying a fraction of the overall compression. (b) A gradual expansion corner like the one shown produces Mach lines that diverge so the expansion spreads to become even more gradual farther from the wall.
If the wall has a sharp deflection (a corner) away from the approaching stream, then the pattern of Figure 15.25b takes the form of Figure 15.26 where all the Mach lines originate from the corner. In this case, this portion of the flow where it expands and turns, and is not parallel to the wall upstream or downstream of the corner, is known as a Prandtl-Meyer expansion fan. The Mach number increases through the fan, with M2 > M1. The first Mach line is inclined at an angle of μ1 to the upstream wall direction, while the last Mach line is inclined at an angle of μ2 to the downstream wall direction. The pressure falls gradually along any streamline through the fan. Along the wall, however, the pressure remains constant along the upstream wall, falls discontinuously at the corner, and then remains constant along the downstream wall. Figure 15.26 should be compared with Figure 15.25, in which the wall turns inward and generates an oblique shock wave. By contrast, the expansion in Figure 15.26 is gradual and isentropic away from the wall.
The flow through a Prandtl-Meyer expansion fan is calculated as follows. From Figure 15.22b, conservation of momentum tangential to the shock shows that the tangential velocity is unchanged, or:
We are concerned here with very small deflections, δ → 0 so σ → μ. Here, cosδ ≈ 1, sinδ ≈ δ, V1 ≈ V2(1 + δtanσ), so (V2 – V1)/V1 ≈ –δtanσ ≈ –δ/[M12 – 1]1/2, where tanσ ≈ 1/[M12 – 1]1/2. Thus, the velocity change dV for an infinitesimal wall deflection dδ can be written as dδ = –(dV/V)[M12 – 1]1/2 (first quadrant deflection). Because V = Mc, dV/V = dM/M + dc/c. With c = γ R T 
for a perfect gas, dc/c = dT/2T. Using (15.28) for adiabatic flow of a perfect gas, dT/T = –(γ – 1)M dM/[ 1 + ((γ – 1)/2)M2], then:
for a perfect gas, dc/c = dT/2T. Using (15.28) for adiabatic flow of a perfect gas, dT/T = –(γ – 1)M dM/[ 1 + ((γ – 1)/2)M2], then:
Integrating δ from 0 (radians) and M from 1 gives δ + ν(M) = const., where
(15.67)
is called the Prandtl-Meyer function. The sign of [M2 – 1]1/2 originates from the identification of tanσ = tanμ = [M12 – 1]–1/2 for a first quadrant deflection (upper half-plane). For a fourth quadrant deflection (lower half-plane), tanμ = –[M12 – 1]–1/2. For example, for Figure 15.25a or b with δ1, δ2, and M1 given, we would write:
would determine M2. In Figure 15.25a, δ1 – δ2 < 0, so ν2 < ν1 and M2 < M1. In Figure 15.25b, δ1 – δ2 > 0, so ν2 > ν1 and M2 > M1.
Example 15.8
A uniform flow at atmospheric pressure having M1 = 3.0 is deflected by 20°. What are the Mach number and pressure in the flow after the deflection if it occurs through (a) an oblique shock wave from a compression corner (Figure 15.23 right side panel), (b) an isentropic compression from a curved wall (Figure 15.25a), and (c) an isentropic expansion (Figure 15.25b).
Solution
For (a) an oblique shock wave must be considered. Using Figure 15.22, M1 = 3 and δ = 20°, leads to σ = 37.5°, so M1sinσ = 1.83. Thus, from (15.61) and (15.63):
For (b), the Prandtl-Meyer function may be used. Here the initial flow angle is 0° and ν(M1 = 3) = 49.76°. Thus, ν(M2) = 0 – 20° + 49.76 = 29.76°, for which M2 = 2.125. The downstream pressure can be recovered from the Table 15.1:
Here, both M2 and p2 are larger than those for (a) because this flow is isentropic while the oblique shock in (a) is not.
For (c), the Prandtl-Meyer function may again be used. Here again the initial flow angle is 0° and ν(M1 = 3) = 49.76°. Thus, ν(M2) = 0 + 20° + 49.76 = 69.76°, for which M2 = 4.31. The downstream pressure can be recovered from the Table 15.1:
