Hypersonics

Hypersonic flight involves speeds greater than Mach 5 (five times the speed of sound). At these extreme velocities, the physics of flight change dramatically, bringing about unique challenges in aerodynamics, materials, propulsion, and control systems. Hypersonics is critical for space access, long-range weapons, and future transportation systems.

Definition and Regimes

Speed Classifications

• Subsonic: M < 0.8
• Transonic: 0.8 ≤ M ≤ 1.2
• Supersonic: 1.2 < M < 5
• Hypersonic: M ≥ 5
• Orbital: M ≈ 25 (for Earth escape velocity)

Hypersonic Regimes

Hypersonic flow is further categorized by the hypersonic similarity parameter:

Where $M$ is Mach number and $\theta$ is the flow deflection angle.

Shock Wave Physics

Normal Shock Relations

For a normal shock wave, conservation of mass, momentum, and energy yield:

Where $\gamma = 1.4$ for air (at low temperatures).

Oblique Shocks

For oblique shock waves, the deflection angle $\theta$ relates to the shock angle $\beta$:

High-Temperature Gas Dynamics

Dissociation Effects

At hypersonic speeds, air molecules begin to dissociate:

• O₂ → 2O at ~3000K
• N₂ → 2N at ~6000K
• Electron detachment at ~9000K

This changes the gas properties and requires non-equilibrium thermodynamics.

Real Gas Effects

The ideal gas law becomes inadequate; instead:

Equations of state for high-temperature gases become complex, incorporating:

• Vibrational energy
• Electronic energy
• Dissociation reactions
• Ionization effects

Hypersonic Flow Equations

Newtonian Flow Theory

At very high Mach numbers, Newton's impact theory becomes accurate:

For the pressure coefficient on a surface inclined at angle $\theta$.

Hypersonic Small Disturbance Theory

For thin bodies at small angles of attack:

Where $v = \sqrt{M^2-1}$ for small perturbations.

Aerothermodynamics

Stagnation Point Heating

The heating rate at a stagnation point is given by:

Where $R_n$ is the nose radius and $\rho_w$ is the wall density.

Fay-Riddell Equation

For turbulent boundary layer heating:

Where $Pr_t$ is turbulent Prandtl number, $Re_x$ is Reynolds number, $k$ is thermal conductivity, $T_{aw}$ is adiabatic wall temperature, and $T_w$ is wall temperature.

Vehicle Design Considerations

Nose Blunting

Sharp leading edges will melt; noses must be blunt:

Where $q_{max}$ is peak heating rate and $R_n$ is nose radius.

Ablative Cooling

Materials that absorb heat and/or vaporize to protect the structure:

Where $h_{sub}$ is the enthalpy of sublimation.

Radiative Cooling

At high temperatures, radiation becomes significant:

Where $\varepsilon$ is emissivity, $\sigma$ is Stefan-Boltzmann constant, $T_s$ is surface temperature, and $T_\infty$ is ambient temperature.

Propulsion Challenges

Scramjet Operation

Supersonic combustion ramjet (scramjet) must:

• Compress air without creating normal shocks (which slow air to subsonic)
• Mix fuel with supersonic air
• Ignite and sustain combustion in supersonic flow
• Expand combustion products for thrust

Propulsion Regimes

Specific Impulse Variation

Where $T$ is thrust, $D$ is drag, $v_\infty$ is flight velocity, and $g_0$ is gravitational acceleration.

Materials Challenges

Thermal Protection Systems (TPS)

Materials must withstand extreme temperatures:

• Phenolic Impregnated Carbon Ablator (PICA): Used on SpaceX Dragon
• Reinforced Carbon-Carbon (RCC): Used on Space Shuttle nose and wing leading edges
• Ceramic Matrix Composites (CMC): Future high-temperature applications

Operating Temperature Ranges

• Aluminum: <600°F (315°C)
• Titanium: <1000°F (540°C)
• Superalloys: <2000°F (1095°C)
• CMCs: <3000°F (1650°C)
• Ablative materials: <5000°F (2760°C)

Control and Stability

Center of Pressure Shift

At hypersonic speeds, the center of pressure moves, affecting stability:

Control Effectiveness

Control surfaces become less effective at hypersonic speeds due to:

• Reduced dynamic pressure behind shock waves
• Flow separation
• Non-linear aerodynamics

Flight Environment Challenges

Altitude Effects

Performance varies with altitude:

• Higher altitude: Lower density, less heating, less drag
• Lower altitude: Higher density, more lift, more maneuverability

Atmospheric Reentry

Reentry corridor for spacecraft:

• Too shallow: Vehicle skips out of atmosphere
• Too steep: Excessive heating rate and deceleration
• Just right: Controlled descent with acceptable loads

Real-World Application: Reentry Heating Analysis

Consider a crew capsule reentering Earth's atmosphere after a mission to the International Space Station.

Reentry Scenario

A spherical-cone reentry vehicle with:

• Nose radius: 0.3 m
• Base radius: 1.2 m
• Mass: 8,000 kg
• Reentry velocity: 7.8 km/s (orbital velocity)
• Reentry angle: -1.5° (shallow reentry corridor)

Newtonian Heating Calculation

import math

# Vehicle parameters
nose_radius = 0.3      # meters
velocity = 7800        # m/s (orbital velocity)
density_sea_level = 1.225  # kg/m³
altitude = 80000       # meters (beginning of significant heating)

# Calculate atmospheric density at altitude
# Exponential atmosphere model
scale_height = 7000    # meters (simplified)
density = density_sea_level * math.exp(-altitude / scale_height)

# Newtonian stagnation point heat rate (simplified)
# q = k * rho^0.5 * V^3 / R_n^0.5
k_factor = 1.793e-4   # Empirical constant (W·s^0.5/(m^2.5·kg^0.5))

heat_rate = k_factor * (density**0.5) * (velocity**3) / (nose_radius**0.5)

# Calculate total heat load
reentry_time = 300     # seconds (5 minutes of peak heating)
total_heat_load = heat_rate * reentry_time  # J/m²

print(f"Atmospheric density at {altitude:,} m: {density:.6f} kg/m³")
print(f"Stagnation point heat rate: {heat_rate/1000:.0f} kW/m²")
print(f"Total heat load: {total_heat_load/1e6:.0f} MJ/m²")

# Heat shield requirement
heat_shield_thickness = 0.1  # meters (simplified)
material_density = 300       # kg/m³ (low-density ablator)
material_specific_heat = 1200  # J/(kg·K)

# Calculate temperature rise capability
heat_capacity = material_density * heat_shield_thickness * material_specific_heat  # J/m³
temperature_rise = total_heat_load / heat_capacity

print(f"Heat shield temperature rise potential: {temperature_rise:.0f} K")
print(f"Required heat shield margin: {'Adequate' if temperature_rise < 1500 else 'Insufficient'}")

Deceleration Loads

During reentry, the vehicle experiences deceleration loads:

$a = \frac{\text{drag force}}{\text{mass}} = \frac{0.5\rho V^2 S C_D}{m}$

Your Challenge: Hypersonic Vehicle Design Optimization

Design a hypersonic cruise vehicle that balances performance, heating, and structural requirements.

Goal: Calculate the optimal flight conditions to minimize total mission time while staying within thermal limits.

Vehicle Parameters

# Hypersonic vehicle specifications
vehicle_gross_weight = 100000  # kg
fuel_fraction = 0.35           # 35% of gross weight is fuel
empty_weight_ratio = 0.25      # 25% of gross weight is structure/avionics

# Propulsion characteristics (scramjet)
tsfc_mach_5 = 2.5    # lbm/hr/lbf at Mach 5
tsfc_mach_10 = 1.8   # lbm/hr/lbf at Mach 10 (better efficiency at higher speed)

# Thermal constraints
max_heat_rate = 1000  # kW/m² (max allowable for TPS)
nose_radius = 0.5     # m (blunt nose for heat protection)

# Cruise conditions to evaluate
cruise_machs = [5, 6, 7, 8, 9, 10]
altitudes = [30000, 35000, 40000]  # meters

For each Mach number and altitude combination, calculate:

1. Heat rate on the vehicle
2. Propulsion efficiency
3. Flight time for a 1000 km mission
4. Fuel consumption for the mission

Hint:

• Heat rate scales approximately with: $\rho^{0.5} \cdot V^{3} / R^{0.5}$
• Density decreases exponentially with altitude
• Flight time = distance / (Mach number × speed of sound)
• Fuel consumption = flight time × TSFC × thrust

# TODO: Calculate flight performance for each condition
best_mach_altitude = (5, 30000)  # Determine optimal cruise condition
min_mission_time = 0  # seconds
max_fuel_efficiency = 0  # km/kg

# Print results
print(f"Optimal cruise Mach number: {best_mach_altitude[0]}")
print(f"Optimal cruise altitude: {best_mach_altitude[1]} m")
print(f"Minimum mission time: {min_mission_time:.1f} seconds")
print(f"Maximum fuel efficiency: {max_fuel_efficiency:.2f} km/kg")

# Check thermal constraints
thermal_constraint_satisfied = True  # Determine if thermal limits are met
print(f"Thermal constraints satisfied: {thermal_constraint_satisfied}")

What additional technologies would be needed to make sustained hypersonic flight practical for commercial applications?