Prandtl-Meyer expansions

When the first schlieren photographs of the inlet model came back from the trisonic tunnel, Arjun noticed that the shocks were not the only features. Wherever the model surface turned away from the flow, at the shoulder of the second ramp and at the trailing edge of the cowl, there was a fan of faint lines spreading out from the corner. Those are expansion waves. A supersonic flow turning away from itself does not form a shock; it expands smoothly, accelerating and losing pressure through a continuous fan of Mach waves. The theory of that expansion was worked out by Ludwig Prandtl and his student Theodor Meyer in 1908, and it completes the set of tools needed to calculate two-dimensional supersonic flows.

Mach waves

Lesson 1 described the Mach cone of a small disturbance in supersonic flow. In two dimensions the cone becomes a pair of straight lines, Mach waves, inclined at the Mach angle \(\mu = \sin^{-1}(1/M)\) to the flow. A Mach wave is the limit of an oblique shock as the deflection goes to zero: an infinitely weak shock, across which the flow properties change by infinitesimal amounts and the entropy change is zero. Any smooth supersonic flow can be thought of as built from Mach waves. A wall turning slightly into the flow sends out compression Mach waves; a wall turning slightly away sends out expansion Mach waves.

Compression waves converge, because each one travels in flow that has already been slowed and has a larger Mach angle. They coalesce into an oblique shock. Expansion waves diverge, because each one travels in flow that has been accelerated and has a smaller Mach angle, so they spread into a fan. This is why compression in supersonic flow is sudden and irreversible, while expansion is gradual and isentropic.

The Prandtl-Meyer function

Consider a supersonic flow turning through a very small angle \(d\theta\) across a single Mach wave. Geometry and the momentum equation tangential to the wave give \(d\theta = \sqrt{M^2 - 1}\,dV/V\). Writing the velocity in terms of Mach number and the speed of sound, and using the energy equation to relate the speed of sound to the Mach number, allows this to be integrated from Mach 1. The result is the Prandtl-Meyer function \(\nu(M)\), the angle through which a sonic flow would have to turn to reach Mach number \(M\).

For a flow at Mach \(M_1\) turning away through an angle \(\theta\), the downstream Mach number follows from a single equation: \(\nu(M_2) = \nu(M_1) + \theta\). Because the expansion is isentropic, the stagnation pressure and temperature are unchanged, and the static pressure and temperature then follow from the isentropic relations of Lesson 2 at the two Mach numbers. The inverse problem, finding \(M_2\) from \(\nu(M_2)\), has no closed form, so the program uses bisection.

The function rises from zero at Mach 1 to 11.9 degrees at Mach 1.5, 26.4 degrees at Mach 2, 49.8 degrees at Mach 3 and 76.9 degrees at Mach 5. It approaches a limit of 130.45 degrees for air as the Mach number goes to infinity. That limit means a sonic flow can turn through at most 130 degrees, expanding to zero pressure and temperature, before it separates from the wall and leaves a vacuum behind it. In practice the gas condenses or the continuum assumption fails long before that point.

# Prandtl-Meyer expansions: nu(M), its inverse, and flow round a convex corner
import math
G = 1.4
def nu(M):                                   # Prandtl-Meyer function in degrees
    k = math.sqrt((G + 1) / (G - 1))
    return math.degrees(k * math.atan(math.sqrt((M**2 - 1) / k**2)) - math.atan(math.sqrt(M**2 - 1)))
def mach_from_nu(target):
    lo, hi = 1.0, 30.0
    for _ in range(80):
        mid = (lo + hi) / 2
        if nu(mid) < target: lo = mid
        else: hi = mid
    return (lo + hi) / 2
def p0_p(M): return (1 + (G - 1) / 2 * M * M) ** (G / (G - 1))
print(" M     nu deg   Mach angle deg")
for M in (1.0, 1.5, 2.0, 2.5, 3.0, 5.0):
    print(f"{M:4.1f}  {nu(M):7.3f}   {math.degrees(math.asin(1 / M)):7.3f}")
k = math.sqrt((G + 1) / (G - 1))
print(f"largest possible turn (expansion to M = infinity): {90 * (k - 1):.2f} deg")
# a M 2.0 stream turns 10 deg round a convex corner
M1, turn = 2.0, 10.0
M2 = mach_from_nu(nu(M1) + turn)
p21 = p0_p(M1) / p0_p(M2)
mu1, mu2 = math.degrees(math.asin(1 / M1)), math.degrees(math.asin(1 / M2))
print(f"\nM1 {M1} round a {turn:.0f} deg corner: M2 = {M2:.3f}, p2/p1 = {p21:.4f}")
print(f"fan from {mu1:.2f} deg (leading Mach line) to {mu2 - turn:.2f} deg (trailing line) from the upstream flow direction")
# underexpanded jet: a nozzle exit at M 2.0 and p = 2.0 pb expands at the lip to the ambient pressure
Me, pe_pb = 2.0, 2.0
M_after = math.sqrt(2 / (G - 1) * ((p0_p(Me) * pe_pb) ** ((G - 1) / G) - 1))
print(f"\nunderexpanded jet: M rises from {Me} to {M_after:.3f}; the flow turns outward by {nu(M_after) - nu(Me):.2f} deg at the lip")

 M     nu deg   Mach angle deg
 1.0    0.000    90.000
 1.5   11.905    41.810
 2.0   26.380    30.000
 2.5   39.124    23.578
 3.0   49.757    19.471
 5.0   76.920    11.537
largest possible turn (expansion to M = infinity): 130.45 deg

M1 2.0 round a 10 deg corner: M2 = 2.385, p2/p1 = 0.5480
fan from 30.00 deg (leading Mach line) to 14.79 deg (trailing line) from the upstream flow direction

underexpanded jet: M rises from 2.0 to 2.444; the flow turns outward by 11.41 deg at the lip

Expansion at a corner and on a wing

A Mach 2 stream turning through 10 degrees round a convex corner has \(\nu_1 = 26.38°\), so \(\nu_2 = 36.38°\) and \(M_2 = 2.385\). The pressure falls to 0.548 of its upstream value. The fan begins with a wave at 30.0 degrees to the incoming flow and ends with a wave at 24.8 degrees to the turned flow, which is 14.8 degrees to the original direction. Across the fan the flow accelerates, the Mach angle shrinks, and each streamline bends smoothly.

Expansions are as important as shocks on supersonic bodies. On a diamond-shaped airfoil at Mach 2, the flow is compressed by oblique shocks at the leading edge, expanded at the shoulder at mid-chord, and recompressed at the trailing edge. The low pressure on the rear faces, produced by the expansions, is what gives the airfoil its wave drag in Lesson 11. On the upper surface of a wing at incidence, the leading-edge expansion produces the suction that carries much of the lift. In both cases, combining oblique shocks with Prandtl-Meyer expansions, called shock-expansion theory, gives exact results for two-dimensional bodies with straight faces, as long as the shocks remain attached.

Supersonic jets and nozzle plumes

A nozzle whose exit pressure differs from the surrounding pressure produces a jet with a characteristic diamond pattern. If the exit pressure is higher than ambient, the jet is underexpanded: the flow turns outward at the lip through an expansion fan until its pressure matches ambient. The program shows a Mach 2.0 jet at twice the ambient pressure accelerating to Mach 2.44 and turning outward by 11.4 degrees. The expansion waves reflect from the free jet boundary as compression waves, which coalesce into shocks, which reflect again as expansions, producing the bright repeating cells seen in the exhaust of rocket engines and afterburning engines. If the exit pressure is lower than ambient, the jet is overexpanded and the pattern starts with oblique shocks at the lip. Lesson 6 shows how the nozzle pressure ratio decides which happens.

Reflection and cancellation

An expansion fan striking a solid wall reflects as another expansion fan, because the wall must turn the flow back parallel to itself. An expansion striking a free boundary, where the pressure is fixed, reflects as a compression. These rules are the basis of the method of characteristics, a graphical and later numerical technique that builds a complete two-dimensional supersonic flow out of a network of Mach waves. It was the standard way of designing supersonic wind tunnel nozzles for decades. A nozzle designed by characteristics turns the flow outward to accelerate it, then turns the wall back so that each expansion wave arriving at the wall is cancelled, rather than reflected, leaving a uniform parallel stream at the exit. Lesson 7 uses that idea when it discusses the nozzle of the trisonic tunnel.

Expansions on the inlet model

The inlet model gives a concrete example of how expansions and shocks work together. Behind the second ramp the air is at about Mach 1.44, and the ramp surface then turns away from the flow toward the throat region, a few degrees at a time, so that the air can enter the subsonic diffuser. Each turn sends out a small expansion fan that accelerates the flow and lowers its pressure. Those fans travel outward and meet the oblique shocks coming down from the ramps, weakening them where they cross. If the shoulder is placed too far forward, its expansion reaches the cowl lip, accelerates the air just ahead of the terminal normal shock and makes that shock stronger, costing pressure recovery. If it is too far aft, the boundary layer arriving at the throat is thicker. The designers placed the shoulder so that its expansion fan passes just inside the cowl lip, and the schlieren photographs from the tunnel let the team check that the fan sits where the calculations put it.

The same reasoning applies at the trailing edge of the cowl, where the outer surface turns inward and the external flow expands round it. The low pressure there acts on the rear-facing parts of the cowl and adds to the external drag of the nacelle, a component that inlet designers try hard to keep small. A thin, gently boat-tailed cowl turns the flow through smaller angles and therefore produces weaker expansions and less drag than a blunt one, which is one reason supersonic nacelles are long and slender. The calculation is a direct use of the Prandtl-Meyer function: the turning angle of each surface gives the change in \(\nu\), and the pressure on that surface follows from the isentropic relations.

Where expansions cause trouble

Expansions are isentropic, but the flow behind them has a lower pressure, and the recompression that inevitably follows is not. On a transonic wing the flow expands round the leading edge into a supersonic pocket, then must recompress, usually through a shock, before reaching the trailing edge. Strong expansion followed by strong recompression is the pattern the supercritical wing designs of Lesson 10 work to avoid. On the inlet model, the expansion at the ramp shoulder accelerates the flow just before the cowl shock, which increases the strength of that shock if the shoulder is placed badly. The design of every supersonic shape is a balance between the compressions and expansions its surfaces create.