Oblique shocks

The second model scheduled for the trisonic tunnel is a generic external-compression inlet, part of a the National Research Council study of engines for future quiet supersonic airliners and business jets. Its front is a pair of ramps that turn the incoming Mach 2 air through two small angles before a final normal shock at the cowl lip. Kenji Watanabe wants to know why the designers bothered with ramps at all, and how much better they are than the simple pitot inlet whose loss Lesson 3 calculated. The answer lies in oblique shocks: shocks inclined to the flow, which compress and turn it at the same time, with far less loss than a normal shock of the same total compression.

Turning a supersonic flow

When a supersonic flow meets a wall that turns into it, a concave corner or the leading edge of a wedge, the flow cannot curve gradually as a subsonic flow would. Instead an oblique shock forms at the corner, inclined at a wave angle \(\beta\) to the upstream flow. Across it the flow turns through the deflection angle \(\theta\), so that it runs parallel to the new wall, and its pressure, density and temperature rise. If the wall turns away from the flow, the opposite happens: the flow expands smoothly through a fan of Mach waves, the subject of Lesson 5.

The key idea: only the normal component changes

An oblique shock can be analysed with no new physics. Resolve the upstream velocity into a component normal to the shock, \(u_1 = V_1\sin\beta\), and a component tangential to it, \(w_1 = V_1\cos\beta\). The momentum equation in the tangential direction has no pressure gradient, because the pressure does not change along the shock, so \(w_2 = w_1\). The normal component, on the other hand, obeys exactly the same conservation equations as a normal shock. An oblique shock is therefore a normal shock with a uniform sideways velocity added, and the normal-shock relations of Lesson 3 apply if the upstream Mach number is replaced by its normal component.

The pressure, density and temperature ratios are those of a normal shock at \(M_{n1}\). Because \(M_{n1}\) can be much smaller than \(M_1\), an oblique shock can be very weak even in a very fast flow. The downstream normal component is subsonic, but the total downstream velocity, with its unchanged tangential part, is usually still supersonic. The flow behind a weak oblique shock remains supersonic.

The theta-beta-M relation

Geometry links the deflection angle to the wave angle. The tangential velocities are equal and the normal velocities are in the ratio of the densities, so \(\tan\beta/\tan(\beta - \theta) = \rho_2/\rho_1\). Substituting the normal-shock density ratio gives the theta-beta-M relation, the single most used equation in supersonic aerodynamics.

Given \(M_1\) and \(\beta\), the deflection follows directly. The usual problem is the reverse: the wall angle \(\theta\) is known, and \(\beta\) is wanted. Plotting \(\theta\) against \(\beta\) for a fixed Mach number shows that \(\theta\) is zero when \(\beta\) equals the Mach angle, where the shock degenerates into a Mach wave, rises to a maximum, then falls to zero again at \(\beta = 90°\), the normal shock. Every deflection below the maximum therefore has two solutions: a weak shock with the smaller wave angle and a strong shock with the larger one.

In external flows the weak solution almost always occurs: a wedge in a free stream carries a weak oblique shock. The strong solution appears when the downstream pressure is forced to be high, for example inside a duct where a back pressure is imposed. If the wall angle exceeds the maximum deflection, no attached shock is possible. The shock then stands detached ahead of the body as a curved bow shock, normal on the stagnation streamline and weakening into a Mach wave far away. For Mach 2 in air the maximum deflection is about 23 degrees, so a 25-degree wedge has a detached bow shock.

# Oblique shocks: solve the theta-beta-M relation for the weak and strong waves
import math
G = 1.4
def theta_of(beta, M):                       # flow deflection for a wave angle beta (radians)
    t = 2 / math.tan(beta) * (M**2 * math.sin(beta)**2 - 1) / (M**2 * (G + math.cos(2 * beta)) + 2)
    return math.atan(t)
def max_deflection(M):
    mu = math.asin(1 / M); best = (0, mu)
    for i in range(1, 4000):
        b = mu + (math.pi / 2 - mu) * i / 4000
        th = theta_of(b, M)
        if th > best[0]: best = (th, b)
    return best
def beta_of(theta, M, strong=False):
    th_max, b_max = max_deflection(M)
    if theta > th_max: return None            # detached shock
    lo, hi = (b_max, math.pi / 2) if strong else (math.asin(1 / M), b_max)
    for _ in range(80):                       # bisection on the chosen branch
        mid = (lo + hi) / 2
        f = theta_of(mid, M) - theta
        if (f > 0) != strong: hi = mid
        else: lo = mid
    return (lo + hi) / 2
def after(M1, beta, theta):
    Mn1 = M1 * math.sin(beta)
    Mn2 = math.sqrt((1 + (G - 1) / 2 * Mn1**2) / (G * Mn1**2 - (G - 1) / 2))
    p21 = 1 + 2 * G / (G + 1) * (Mn1**2 - 1)
    return Mn2 / math.sin(beta - theta), p21
M1 = 2.0
for th_deg in (5, 10, 15, 20):
    th = math.radians(th_deg)
    bw, bs = beta_of(th, M1), beta_of(th, M1, True)
    M2w, pw = after(M1, bw, th); M2s, ps = after(M1, bs, th)
    print(f"theta {th_deg:2d} deg: weak beta {math.degrees(bw):5.2f} M2 {M2w:.3f} p2/p1 {pw:.3f} | strong beta {math.degrees(bs):5.2f} M2 {M2s:.3f} p2/p1 {ps:.3f}")
th_max, b_max = max_deflection(M1)
print(f"M {M1}: maximum deflection {math.degrees(th_max):.2f} deg at beta {math.degrees(b_max):.2f} deg")
print("25 deg wedge at M 2:", "detached bow shock" if beta_of(math.radians(25), M1) is None else "attached")
# two-ramp external compression inlet versus one normal shock at M 2.0
def p0_normal(M):
    p21 = 1 + 2 * G / (G + 1) * (M**2 - 1); r21 = (G + 1) * M**2 / ((G - 1) * M**2 + 2)
    return r21 ** (G / (G - 1)) * (1 / p21) ** (1 / (G - 1))
M, rec = M1, 1.0
for th_deg in (8, 8):
    th = math.radians(th_deg); b = beta_of(th, M)
    rec *= p0_normal(M * math.sin(b)); M, _ = after(M, b, th)
rec *= p0_normal(M)
print(f"two 8 deg ramps then a normal shock: total pressure recovery {rec:.3f} (normal shock alone {p0_normal(M1):.3f})")

theta  5 deg: weak beta 34.30 M2 1.821 p2/p1 1.315 | strong beta 86.97 M2 0.584 p2/p1 4.487
theta 10 deg: weak beta 39.31 M2 1.641 p2/p1 1.707 | strong beta 83.70 M2 0.604 p2/p1 4.444
theta 15 deg: weak beta 45.34 M2 1.446 p2/p1 2.195 | strong beta 79.83 M2 0.644 p2/p1 4.355
theta 20 deg: weak beta 53.42 M2 1.210 p2/p1 2.843 | strong beta 74.27 M2 0.728 p2/p1 4.157
M 2.0: maximum deflection 22.97 deg at beta 64.66 deg
25 deg wedge at M 2: detached bow shock
two 8 deg ramps then a normal shock: total pressure recovery 0.935 (normal shock alone 0.721)

What the program shows

At Mach 2, a 10-degree deflection produces a weak shock at 39.3 degrees with a downstream Mach number of 1.64 and a pressure ratio of 1.71, or a strong shock at 83.7 degrees with a downstream Mach number of 0.60 and a pressure ratio of 4.44. The strong shock is almost a normal shock: its pressure rise is close to the 4.50 of the Mach 2 normal shock. As the deflection rises to 20 degrees, the weak shock steepens to 53.4 degrees and the downstream flow slows to Mach 1.21, close to sonic. Beyond 23 degrees the solution disappears.

The weak shock compresses much less than a normal shock at the same Mach number. That seems like a disadvantage until the entropy is considered. A weak shock's normal Mach number is only 1.27 for the 10-degree case, and its stagnation pressure loss is about 1.5 percent, compared with 28 percent for the normal shock. The inlet designer can therefore compress the air in several weak steps, each nearly isentropic, and finish with a weak normal shock at a much lower Mach number.

Shock-system inlets

That is exactly what the ramps of the the National Research Council inlet do. Two 8-degree ramps each produce a weak oblique shock, slowing the air from Mach 2.0 to about 1.71 and then to about 1.44. A final normal shock at the cowl lip takes it subsonic. The program multiplies the three stagnation pressure ratios and finds a total recovery of 0.935, compared with 0.721 for a normal shock alone. The difference, over 20 percent of stagnation pressure, is roughly 20 percent of engine thrust at the same fuel flow.

Klaus Oswatitsch showed in the 1940s that for a given number of shocks the recovery is best when all the oblique shocks have the same strength, and that the recovery improves steadily as more shocks are added, approaching isentropic compression in the limit of a smoothly curved ramp. Real inlets balance this against length, weight and the need to swallow the shock system smoothly when the engine demand changes. Supersonic airliners and fighters of the 1960s and 1970s used variable ramps that moved with flight Mach number to keep the shocks focused on the cowl lip, and modern low-boom concepts use carefully shaped axisymmetric or two-dimensional inlets with bleed systems that remove the boundary layer where the shocks meet the ramp surface.

Shock reflection and interaction

An oblique shock that strikes a solid wall cannot simply end, because the flow behind it is turned toward the wall. A reflected shock forms that turns the flow back parallel to the wall. The reflected shock is weaker than a normal shock but stronger than the incident one in pressure ratio, because the flow behind the incident shock is slower. If the required turning exceeds the maximum deflection for the flow behind the incident shock, a regular reflection is impossible and a Mach reflection forms, with a short normal-shock stem at the wall. Shocks striking boundary layers cause a further complication: the pressure rise can separate the layer, and the separation bubble changes the shock pattern. Shock-boundary-layer interaction is one of the main reasons inlets need bleed, and it reappears on transonic wings in Lesson 10.