m physical modeling / IICSM

PHYSICS / ENGINEERING / COMPUTING

Hypersonic vehicle design: civil aerothermodynamics

Study high-speed atmospheric flight through regime selection, compressible flow, aerodynamic loads, thermal protection and verification.

Subject library · 51 guides · derivations & worked examples

Fundamental physics

Space and flight

Materials and transport

Systems and mathematics

Marine and road vehicles

Electrical and computing

Companion site

Matter pathway: atom → solid → liquid → gas → plasma. Quantum mechanics and quantum field theory provide foundations across the pathway; they are not additional phases. This is a connected modeling route, not a universal heating curve. Actual phases depend on pressure, composition, and kinetics.

1. Choose the physical regime

Definitions & inputs. M Mach number,V velocity,a sound speed,Kn mean free path/body length,Re Reynolds number.

  1. Compare flow speed with acoustic propagation.

    M=V/a,a=γRTM=V/a,\quad a=\sqrt{\gamma RT}

  2. Viscous and rarefaction scales determine whether continuum Navier–Stokes is appropriate.

    Re=ρVL/μ,Kn=ℓ/LRe=\rho VL/\mu,\quad Kn=\ell/L

  3. Total enthalpy measures the available stagnation energy even when constant gamma fails.

    h0=h+V2/2h_0=h+V^2/2

Interpretation. High enthalpy may excite vibration, dissociate or ionize gas; use chemical nonequilibrium and rarefied-flow methods when the corresponding scales demand them.

↑ Return to definitions and contents

2. Compression and stagnation limits

Definitions & inputs. T0 stagnation temperature,γ heat capacity ratio,M1 upstream normal Mach.

  1. Combine h=cpT with total enthalpy conservation.

    T0/T=1+(γ−1)M2/2T_0/T=1+(\gamma-1)M^2/2

  2. Normal-shock momentum and energy balance produce a pressure jump.

    p2/p1=1+2γ(M12−1)/(γ+1)p_2/p_1=1+2\gamma(M_1^2-1)/(\gamma+1)

  3. Mass conservation and the equation of state give compression; entropy rises across the shock.

    ρ2/ρ1=(γ+1)M12/[2+(γ−1)M12]\rho_2/\rho_1=(\gamma+1)M_1^2/[2+(\gamma-1)M_1^2]

Interpretation. Oblique and curved shocks need local geometry and a multidimensional solution; shock interactions and transition can create localized heating.

↑ Return to definitions and contents

3. Loads and aerodynamic tradeoffs

Definitions & inputs. q dynamic pressure,S reference area,CL andCD coefficients,θ surface inclination.

  1. Reference area and coefficient definitions must match.

    q=12ρV2,L=qSCL,D=qSCDq=\tfrac12\rho V^2,\quad L=qSC_L,\quad D=qSC_D

  2. Newtonian momentum transfer is a rough pressure model on windward surfaces, not a full viscous solution.

    Cp≃2sin⁡2θC_p\simeq2\sin^2\theta

  3. Lift-to-drag ratio connects aerodynamic forces but does not determine heating or vehicle feasibility.

    L/D=CL/CDL/D=C_L/C_D

Interpretation. Shape, structure, thermal protection, stability and propulsion interact; CFD requires mesh, chemistry, turbulence and wind-tunnel validation.

↑ Return to definitions and contents

4. Thermal protection and model validation

Definitions & inputs. qdot incident heat flux,ε emissivity,σSB constant,Twall surface temperature,mc thermal capacity,k conductivity.

  1. Balance incoming heat, reradiation and conducted energy.

    mcpT˙=Aq˙in−AϵσT4−Q˙condmc_p\dot T=A\dot q_{in}-A\epsilon\sigma T^4-\dot Q_{cond}

  2. A radiative-equilibrium benchmark omits heat storage and conduction.

    Trad=(q˙in/(ϵσ))1/4T_{rad}=(\dot q_{in}/(\epsilon\sigma))^{1/4}

  3. Fourier conduction and diffusion time compare through-thickness response with exposure time.

    q˙cond=kΔT/L,tdiff∼L2/α\dot q_{cond}=k\Delta T/L,\quad t_{diff}\sim L^2/\alpha

Interpretation. Real TPS needs temperature-dependent properties, surface chemistry, recession and bond line limits. Compare analytical checks, mesh/time convergence, ground tests and uncertainty before interpreting vehicle predictions.

↑ Return to definitions and contents

Graphical worked example

Fixed free-stream density 0.02 kg/m³. Density changes with altitude in a real flight. X axis: Speed (m/s). Y axis: Dynamic pressure (kPa).
Fixed free-stream density 0.02 kg/m³. Density changes with altitude in a real flight. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. Sound speed

Definitions & inputs. γ1.4,R287J/(kgK),T220K.

  1. Choose the governing model and isolate the requested quantity.

    a=γRTa=\sqrt{\gamma RT}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1.4(287)(220)\sqrt{1.4(287)(220)}

  3. Evaluate the expression; the result uses the units shown.

    Result=297.3146 m/s\mathrm{Result}=297.3146\ \mathrm{m/s}

Interpretation. Calorically perfect gas.

↑ Return to definitions and contents
Example 02. Mach number

Definitions & inputs. V1500m/s,a300m/s.

  1. Choose the governing model and isolate the requested quantity.

    M=V/aM=V/a

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1500/3001500/300

  3. Evaluate the expression; the result uses the units shown.

    Result=5 \mathrm{Result}=5\ {}

Interpretation. Regime indicator, not a heating prediction.

↑ Return to definitions and contents
Example 03. Specific kinetic energy

Definitions & inputs. V1500m/s.

  1. Choose the governing model and isolate the requested quantity.

    ek=V2/2e_k=V^2/2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    15002/21500^2/2

  3. Evaluate the expression; the result uses the units shown.

    Result=1125000 J/kg\mathrm{Result}=1125000\ \mathrm{J/kg}

Interpretation. Add static enthalpy for total enthalpy.

↑ Return to definitions and contents
Example 04. Perfect-gas stagnation ratio

Definitions & inputs. γ1.4,M5.

  1. Choose the governing model and isolate the requested quantity.

    T0/T=1+.2M2T_0/T=1+.2M^2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1+.2(25)1+.2(25)

  3. Evaluate the expression; the result uses the units shown.

    Result=6 \mathrm{Result}=6\ {}

Interpretation. Chemistry and varying heat capacity omitted.

↑ Return to definitions and contents
Example 05. Perfect-gas stagnation temperature

Definitions & inputs. Static220K,ratio6.

  1. Choose the governing model and isolate the requested quantity.

    T0=6TT_0=6T

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    6(220)6(220)

  3. Evaluate the expression; the result uses the units shown.

    Result=1320 K\mathrm{Result}=1320\ \mathrm K

Interpretation. Illustrative benchmark, not wall temperature.

↑ Return to definitions and contents
Example 06. Dynamic pressure

Definitions & inputs. ρ.02kg/m³,V1500m/s.

  1. Choose the governing model and isolate the requested quantity.

    q=ρV2/2q=\rho V^2/2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .5(.02)(15002).5(.02)(1500^2)

  3. Evaluate the expression; the result uses the units shown.

    Result=22500 Pa\mathrm{Result}=22500\ \mathrm{Pa}

Interpretation. Local free-stream state.

↑ Return to definitions and contents
Example 07. Lift

Definitions & inputs. q22500Pa,S2m²,CL.2.

  1. Choose the governing model and isolate the requested quantity.

    L=qSCLL=qSC_L

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    22500(2)(.2)22500(2)(.2)

  3. Evaluate the expression; the result uses the units shown.

    Result=9000 N\mathrm{Result}=9000\ \mathrm N

Interpretation. Assumed aerodynamic coefficient.

↑ Return to definitions and contents
Example 08. Drag

Definitions & inputs. q22500Pa,S2m²,CD.1.

  1. Choose the governing model and isolate the requested quantity.

    D=qSCDD=qSC_D

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    22500(2)(.1)22500(2)(.1)

  3. Evaluate the expression; the result uses the units shown.

    Result=4500 N\mathrm{Result}=4500\ \mathrm N

Interpretation. Same reference area.

↑ Return to definitions and contents
Example 09. Lift-to-drag

Definitions & inputs. CL.2,CD.1.

  1. Choose the governing model and isolate the requested quantity.

    L/D=CL/CDL/D=C_L/C_D

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .2/.1.2/.1

  3. Evaluate the expression; the result uses the units shown.

    Result=2 \mathrm{Result}=2\ {}

Interpretation. At the specified operating point.

↑ Return to definitions and contents
Example 10. Drag power

Definitions & inputs. D4500N,V1500m/s.

  1. Choose the governing model and isolate the requested quantity.

    P=DVP=DV

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    4500(1500)4500(1500)

  3. Evaluate the expression; the result uses the units shown.

    Result=6750000 W\mathrm{Result}=6750000\ \mathrm W

Interpretation. Mechanical energy loss rate, not engine sizing.

↑ Return to definitions and contents
Example 11. Reynolds number

Definitions & inputs. ρ.02,V1500,L1m,μ1.5×10⁻⁵Pa·s.

  1. Choose the governing model and isolate the requested quantity.

    Re=ρVL/μRe=\rho VL/\mu

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .02(1500)/(.000015).02(1500)/(.000015)

  3. Evaluate the expression; the result uses the units shown.

    Result=2000000 \mathrm{Result}=2000000\ {}

Interpretation. Viscous regime indicator.

↑ Return to definitions and contents
Example 12. Knudsen number

Definitions & inputs. Mean free path10⁻⁵m,L.1m.

  1. Choose the governing model and isolate the requested quantity.

    Kn=ℓ/LKn=\ell/L

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    10−5/.110^{-5}/.1

  3. Evaluate the expression; the result uses the units shown.

    Result=0.0001 \mathrm{Result}=0.0001\ {}

Interpretation. Continuum applicability also depends on local gradients.

↑ Return to definitions and contents
Example 13. Normal shock pressure ratio

Definitions & inputs. M5,γ1.4.

  1. Choose the governing model and isolate the requested quantity.

    p2/p1=1+2γ(M2−1)/(γ+1)p_2/p_1=1+2\gamma(M^2-1)/(\gamma+1)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1+2.8(24)/2.41+2.8(24)/2.4

  3. Evaluate the expression; the result uses the units shown.

    Result=29 \mathrm{Result}=29\ {}

Interpretation. Ideal-gas normal shock.

↑ Return to definitions and contents
Example 14. Normal shock density ratio

Definitions & inputs. Same upstream.

  1. Choose the governing model and isolate the requested quantity.

    ρ2/ρ1=2.4M2/(2+.4M2)\rho_2/\rho_1=2.4M^2/(2+.4M^2)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    60/1260/12

  3. Evaluate the expression; the result uses the units shown.

    Result=5 \mathrm{Result}=5\ {}

Interpretation. Strong-shock perfect-gas limit is6 forγ1.4.

↑ Return to definitions and contents
Example 15. Normal shock temperature ratio

Definitions & inputs. Pressure29,density5.

  1. Choose the governing model and isolate the requested quantity.

    T2/T1=(p2/p1)/(ρ2/ρ1)T_2/T_1=(p_2/p_1)/(\rho_2/\rho_1)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    29/529/5

  3. Evaluate the expression; the result uses the units shown.

    Result=5.8 \mathrm{Result}=5.8\ {}

Interpretation. Same gas constant on both sides.

↑ Return to definitions and contents
Example 16. Newtonian pressure coefficient

Definitions & inputs. Inclination30°.

  1. Choose the governing model and isolate the requested quantity.

    Cp=2sin⁡2θC_p=2\sin^2\theta

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    2sin⁡230∘2\sin^230^\circ

  3. Evaluate the expression; the result uses the units shown.

    Result=0.5 \mathrm{Result}=0.5\ {}

Interpretation. Rough windward pressure approximation.

↑ Return to definitions and contents
Example 17. Stored-heat rise

Definitions & inputs. Heat100kJ,m10kg,cp1000J/(kgK).

  1. Choose the governing model and isolate the requested quantity.

    ΔT=Q/(mcp)\Delta T=Q/(mc_p)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    100000/(10(1000))100000/(10(1000))

  3. Evaluate the expression; the result uses the units shown.

    Result=10 K\mathrm{Result}=10\ \mathrm K

Interpretation. No concurrent cooling.

↑ Return to definitions and contents
Example 18. Radiative equilibrium

Definitions & inputs. Heat flux100000W/m²,ε.8,σ5.670374419×10⁻⁸.

  1. Choose the governing model and isolate the requested quantity.

    T=[q/(ϵσ)]1/4T=[q/(\epsilon\sigma)]^{1/4}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    [105/(.8(5.670374419×10−8))]1/4[10^5/(.8(5.670374419\times10^{-8}))]^{1/4}

  3. Evaluate the expression; the result uses the units shown.

    Result=1218.497 K\mathrm{Result}=1218.497\ \mathrm K

Interpretation. No conduction or heat storage in this estimate.

↑ Return to definitions and contents
Example 19. Conduction heat flux

Definitions & inputs. k.1W/(mK),ΔT500K,L.05m.

  1. Choose the governing model and isolate the requested quantity.

    q=kΔT/Lq=k\Delta T/L

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .1(500)/.05.1(500)/.05

  3. Evaluate the expression; the result uses the units shown.

    Result=1000 W/m2\mathrm{Result}=1000\ \mathrm{W/m^2}

Interpretation. Constant-property planar conduction.

↑ Return to definitions and contents
Example 20. Diffusion timescale

Definitions & inputs. L.05m,α10⁻⁷m²/s.

  1. Choose the governing model and isolate the requested quantity.

    t∼L2/αt\sim L^2/\alpha

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .052/10−7.05^2/10^{-7}

  3. Evaluate the expression; the result uses the units shown.

    Result=25000 s\mathrm{Result}=25000\ \mathrm s

Interpretation. Order-of-magnitude diffusion time, not exact TPS transient.

↑ Return to definitions and contents

Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.