Hypersonic flow introduction

The last model in the campaign is the most unusual: a small-scale model of a blunt cargo-return capsule, 2.5 m in diameter at full scale, from a NASA study of returning experiments and samples from low Earth orbit. the 10 by 10 cannot reach the Mach numbers the capsule meets during entry, which exceed Mach 20, but it can measure the capsule's stability and drag at the supersonic end of its descent, before the parachutes open. Kenji Watanabe uses the test to introduce hypersonic flow, the regime above roughly Mach 5, where the methods of the previous lessons begin to fail and new physics dominates. This lesson is introductory: it gives the main ideas and the simple methods engineers use for first estimates, and it points to the specialist courses and tools needed for real design.

What makes a flow hypersonic

There is no sharp boundary at Mach 5. Hypersonic flow is defined by the appearance of several effects that grow with Mach number until they control the flow. The first is that shocks lie very close to the body. Lesson 4 showed that the wave angle on a wedge approaches the deflection angle as the Mach number rises; at Mach 20 the shock on a 10-degree wedge sits only a few degrees off the surface, and the thin region between them, the shock layer, contains all the disturbed flow. The second is the entropy layer: around a blunt nose the shock is strongly curved, so streamlines passing near the nose cross a nearly normal shock and gain much more entropy than those farther out, leaving a layer of hot, low-density gas wrapped around the body.

The third effect is viscous interaction. At high Mach numbers the boundary layer is very thick, because the gas within it is hot and its viscosity high and its density low. The thick boundary layer displaces the outer flow, changing the shock shape and the surface pressure, which in turn changes the boundary layer. The fourth, and most important for re-entry vehicles, is high-temperature chemistry. The temperature behind a strong shock rises so high that the molecules of air vibrate, dissociate and, at entry speeds, ionize. The perfect-gas relations used throughout this course stop applying. The fifth is low density: at the altitudes where entry heating peaks, the air is so thin that, for small vehicles or sharp edges, the molecular mean free path approaches the body size, and the continuum assumption fails.

Newtonian theory

In the seventeenth century Isaac Newton imagined a fluid as a stream of particles that strike a surface, lose their normal momentum and then slide along it. The idea gives a very poor model of subsonic flow, but at hypersonic speed, where the shock layer is thin and the flow behind the shock runs almost parallel to the surface, it becomes surprisingly good. Applying momentum conservation to particles striking a surface inclined at angle \(\theta\) to the flow gives the Newtonian pressure coefficient.

Lester Lees proposed in the 1950s that the constant 2 be replaced by the pressure coefficient at the stagnation point behind a normal shock, \(C_{p,\text{max}}\), giving modified Newtonian theory. That value follows from the Rayleigh pitot formula of Lesson 3. It is 1.66 at Mach 2, 1.81 at Mach 5 and approaches 1.84 for air at very high Mach numbers. Modified Newtonian theory predicts the pressure on blunt bodies to within a few percent over most of their faces, and it is still used for first estimates of the aerodynamics of capsules and the windward surfaces of lifting entry vehicles.

Newtonian theory has one other striking prediction. Surfaces that face away from the flow, in the shadow of the body, have \(C_p = 0\): they receive no impacts. Real leeward pressures are slightly below the free-stream value, so the pressure coefficient is small and negative, but compared with the windward pressures they are negligible. The aerodynamics of a hypersonic body is dominated by its windward surfaces.

Mach number independence

As the Mach number rises, the pressure coefficient and the shock shape around a body approach limits that no longer depend on the Mach number. The program shows the stagnation pressure coefficient changing by less than 2 percent between Mach 5 and Mach 50. This Mach number independence principle, demonstrated by Oswatitsch, means that the force coefficients of a blunt capsule measured at Mach 8 or 10 in a tunnel can be applied, with modest corrections, at much higher flight Mach numbers. It is one reason the capsule test in the 10 by 10 at the tunnel's top Mach number still gives useful data for the entry phase, although the high-temperature effects that also matter in flight cannot be reproduced.

# Hypersonic flow: Newtonian pressure, capsule drag and stagnation-point heating
import math
G = 1.4
def cp_max(M):                              # stagnation-point Cp behind a normal shock (modified Newtonian)
    p02_p1 = ((G + 1)**2 * M**2 / (4 * G * M**2 - 2 * (G - 1))) ** (G / (G - 1)) * (1 - G + 2 * G * M**2) / (G + 1)
    return (p02_p1 - 1) / (0.5 * G * M * M)
print(" M     Cp,max   (Newtonian value is 2)")
for M in (2, 5, 10, 20, 50):
    print(f"{M:3d} {cp_max(M):9.4f}")
# sphere-cone or spherical-segment capsule heat shield: integrate modified Newtonian Cp over the face
def capsule_cd(Rn_over_D, M):              # spherical cap of nose radius Rn, base diameter D, zero angle of attack
    cpm, D = cp_max(M), 1.0
    Rn = Rn_over_D * D; phi_max = math.asin(0.5 * D / Rn)   # half-angle of the cap
    n, drag = 400, 0.0
    for i in range(n):
        phi = (i + 0.5) / n * phi_max
        r = Rn * math.sin(phi)              # radius of the ring
        dA = 2 * math.pi * r * Rn * (phi_max / n)
        theta = math.pi / 2 - phi           # local surface inclination to the flow
        drag += cpm * math.sin(theta) ** 2 * dA * math.cos(phi)
    return drag / (math.pi * D * D / 4)
for k in (0.5, 1.0, 1.2, 2.0):
    print(f"cap with Rn/D = {k}: CD = {capsule_cd(k, 20):.3f}")
# Sutton-Graves stagnation-point convective heating for air: q = k sqrt(rho / Rn) V^3
k_sg = 1.7415e-4                            # SI units, W/m2 with rho in kg/m3, Rn in m, V in m/s
Rn = 3.0                                    # nose radius of a 2.5 m cargo capsule (blunt), m
for h, rho, V in ((80e3, 1.85e-5, 7400), (70e3, 8.3e-5, 7000), (60e3, 3.1e-4, 6000), (50e3, 1.03e-3, 4200)):
    q = k_sg * math.sqrt(rho / Rn) * V**3
    print(f"altitude {h / 1e3:3.0f} km  V {V} m/s  heating {q / 1e4:6.1f} W/cm2")
# why blunt: heating falls with the square root of nose radius
print(f"nose radius 0.3 m instead of 3.0 m: heating {math.sqrt(10):.2f} times higher")
# calorically perfect stagnation temperature (unrealistic at entry speeds)
T, M = 220.0, 25.0
print(f"perfect-gas T0 at M {M:.0f}: {T * (1 + (G - 1) / 2 * M * M):.0f} K (real air dissociates and stays far cooler)")

 M     Cp,max   (Newtonian value is 2)
  2    1.6573
  5    1.8088
 10    1.8317
 20    1.8374
 50    1.8391
cap with Rn/D = 0.5: CD = 0.919
cap with Rn/D = 1.0: CD = 1.608
cap with Rn/D = 1.2: CD = 1.678
cap with Rn/D = 2.0: CD = 1.780
altitude  80 km  V 7400 m/s  heating   17.5 W/cm2
altitude  70 km  V 7000 m/s  heating   31.4 W/cm2
altitude  60 km  V 6000 m/s  heating   38.2 W/cm2
altitude  50 km  V 4200 m/s  heating   23.9 W/cm2
nose radius 0.3 m instead of 3.0 m: heating 3.16 times higher
perfect-gas T0 at M 25: 27720 K (real air dissociates and stays far cooler)

Why entry vehicles are blunt

The capsule's heat shield is a shallow spherical cap, with a nose radius of about 1.2 times its diameter. Integrating modified Newtonian pressure over such a cap at Mach 20 gives a drag coefficient of about 1.68, close to the values measured on capsules of this shape. A hemisphere, with a nose radius half the diameter, gives only 0.92. The blunter face produces more drag, which slows the capsule higher in the atmosphere, where the air is thinner. More importantly, it reduces the heating.

H. Julian Allen and Alfred Eggers at the NACA showed in the early 1950s that the convective heating at the stagnation point of a body varies inversely with the square root of its nose radius, and that a blunt body pushes most of the energy of entry into the air through its strong bow shock rather than into the vehicle. Their blunt-body principle overturned the intuition that a re-entry vehicle should be sharp and streamlined, and every crewed capsule since has followed it. The program uses the Sutton-Graves correlation, a simple form of the stagnation-point heating law, with a constant for air.

Along a representative entry the heating rises from about 18 W/cm² at 80 km to a peak near 38 W/cm² around 60 km, then falls as the capsule slows. A nose radius of 0.3 m instead of 3.0 m would raise the heating by a factor of 3.2. These are the convective heating rates; at the higher speeds of entry from the Moon or beyond, radiation from the extremely hot shock layer adds substantially to them. Heat shields absorb the energy by ablation, in which the surface material chars and erodes, carrying heat away, or, on reusable vehicles, by tiles and blankets that radiate most of the heat back to space.

High-temperature gas effects

The program's last line shows the problem with perfect-gas theory at entry speeds. At Mach 25 in air at 220 K, the perfect-gas stagnation temperature would be almost 28,000 K. Real air never gets that hot. Above about 800 K, energy goes into molecular vibration and the specific heats rise; above about 2,500 K, oxygen dissociates into atoms; above about 4,000 K, nitrogen dissociates; above about 9,000 K, the atoms begin to ionize. Each of these reactions absorbs energy, so the actual temperature behind the shock of an entering capsule is roughly 6,000 to 11,000 K, depending on speed and altitude. The effective value of \(\gamma\) falls toward 1.1 to 1.2, the shock stands closer to the body, and the density ratio across it rises well above the perfect-gas limit of 6.

The chemical reactions also take time. At high altitude, where collisions are rare, the gas behind the shock may not reach chemical equilibrium before it passes the vehicle, and nonequilibrium flow must be computed. The ionized gas around the capsule blocks radio signals for several minutes, the familiar communications blackout of entry. Surface catalysis, in which atoms recombine on the heat shield and release their energy there, can raise the heating noticeably. These effects are the domain of hypersonic aerothermodynamics, and they are why real entry vehicles are designed with specialized computational tools, arc-jet tests and flight data rather than the methods of this course.

Testing hypersonic flows

No single ground facility reproduces hypersonic flight. Conventional hypersonic wind tunnels heat their air to avoid liquefaction and reach Mach 6 to 10 or more, matching the Mach number but not the flight temperatures. Shock tunnels and expansion tubes use the strong shock from a sudden release of very high-pressure gas to produce flows with flight-like enthalpy for a few milliseconds. Arc-jet facilities heat a gas stream with an electric arc for minutes, reproducing the heating environment of a heat shield sample, though not the full aerodynamic flow. Engineers combine data from several facility types with computation and, ultimately, flight tests, to assemble a picture of the full flight environment.

Looking back over the course

This course has moved from the speed of sound to the edge of space. The isentropic relations, normal and oblique shocks and Prandtl-Meyer expansions are the building blocks of every supersonic and hypersonic calculation. Quasi-one-dimensional nozzle theory, with Fanno and Rayleigh flows, governs engines, tunnels and ducts. Linearized theory corrects subsonic results for compressibility and gives the lift and drag of thin supersonic wings. Transonic flow, the hardest regime to predict, is where the Swift 900 and almost every jet transport fly, and its design relies on correlations such as the Korn equation, on supercritical sections and on careful attention to buffet. The Swift 900 team will now take those choices into AERO-440, where computational fluid dynamics refines the wing and confirms the drag rise, and into AERO-340, where the engines that need those inlets and nozzles are designed.