Rarefied Hypersonic

Concept

Rarefied and Hypersonic Flows — Kinetic Theory, Nonequilibrium Gas Dynamics, and High-Enthalpy Effects

Scope: comprehensive treatment of flow regimes beyond the continuum limit. Includes molecular kinetic theory, slip and transition regimes, and hypersonic thermochemistry with nonequilibrium energy transfer.

1. Flow Regimes and the Knudsen Number

The degree of rarefaction is quantified by the Knudsen number: Kn=λLc,\text{Kn} = \frac{λ}{L_c}, where λλ is the molecular mean free path and LcL_c is a characteristic length scale (e.g. boundary layer thickness or body size).

Flow RegimeKnudsen numberDescription
Continuum (Navier–Stokes)Kn < 10⁻³Standard fluid mechanics valid
Slip Flow10⁻³ < Kn < 10⁻¹Velocity slip, temperature jump
Transition10⁻¹ < Kn < 10Continuum breakdown, kinetic corrections
Free MolecularKn > 10No intermolecular collisions; ballistic regime

2. Molecular Kinetic Theory of Gases

2.1 Boltzmann Equation

The molecular distribution function f(x,v,t)f(\mathbf{x},\mathbf{v},t) defines the probability density of finding a molecule with velocity v\mathbf{v}: DfDt=∂f∂t+v⋅∇f+Fm⋅∂f∂v=(∂f∂t)coll.\frac{Df}{Dt} = \frac{∂f}{∂t} + \mathbf{v}·∇f + \frac{\mathbf{F}}{m}·\frac{∂f}{∂\mathbf{v}} = \left(\frac{∂f}{∂t}\right)_{coll}.

Moments of ff yield macroscopic quantities: ρ=m∫fd3v,ρu=m∫vfd3v,p=13m∫c2fd3v.ρ = m \int f d^3v, \quad ρ\mathbf{u} = m \int \mathbf{v} f d^3v, \quad p = \frac{1}{3}m\int c^2 f d^3v.

2.2 Maxwell–Boltzmann Distribution

At equilibrium: f(v)=(m2πkT)3/2exp⁡(−m(v−u)22kT).f(v) = \left( \frac{m}{2πkT} \right)^{3/2} \exp\left(-\frac{m(v-u)^2}{2kT}\right).

Mean free path: λ=kT2πd2p.λ = \frac{kT}{\sqrt{2}πd^2p}.

2.3 Molecular Transport Properties

From Chapman–Enskog expansion: μ=516(mkT/π)1/2πd2Ωμ(T),μ = \frac{5}{16} \frac{(mkT/π)^{1/2}}{πd^2Ω_μ(T)}, k=154RMμ,Pr=cpμk≈2/3.k = \frac{15}{4} \frac{R}{M} μ, \quad Pr = \frac{c_pμ}{k} ≈ 2/3.

3. Slip and Temperature Jump Boundary Conditions

At Knudsen numbers near 10⁻³–10⁻¹, continuum boundary conditions fail.

3.1 Velocity Slip (Maxwell Model)

uwall−ugas=2−σvσvλ∂u∂n∣wall,u_{wall} - u_{gas} = \frac{2 - σ_v}{σ_v} λ \left.\frac{∂u}{∂n}\right|_{wall}, where σvσ_v is the tangential momentum accommodation coefficient (0 < σ_v ≤ 1).

3.2 Temperature Jump

Twall−Tgas=2−σTσT2γγ+1λ∂T∂n∣wall.T_{wall} - T_{gas} = \frac{2 - σ_T}{σ_T} \frac{2γ}{γ+1} λ \left.\frac{∂T}{∂n}\right|_{wall}.

These corrections are applied in high-altitude aerodynamics and microfluidic channels.

4. Breakdown of Continuum Assumptions

As Kn increases, the Navier–Stokes–Fourier equations lose validity because:

  • Molecular velocity distribution becomes non-Maxwellian.
  • Local thermodynamic equilibrium fails.
  • Stress and heat flux depend on higher-order gradients.

The Burnett and Super-Burnett equations extend hydrodynamics by including second-order gradient corrections: τ=−μ(∇u+∇uT−23(∇⋅u)I)−β1λ2∇2(∇u)+⋯ .\mathbf{τ} = -μ(∇u + ∇u^T - \frac{2}{3}(∇·u)I) - β_1 λ^2∇^2(∇u) + \cdots.

5. Direct Simulation Monte Carlo (DSMC)

Numerical method for solving Boltzmann equation statistically:

  1. Divide domain into cells smaller than mean free path.
  2. Track representative molecules.
  3. Alternate free-molecular motion and probabilistic collisions.

Converges to Navier–Stokes in low-Kn limit; exact in free molecular limit.

Applications:

  • Atmospheric reentry
  • Vacuum systems
  • Micro-electro-mechanical systems (MEMS)

6. Hypersonic Flow Fundamentals

6.1 Definition

Hypersonic regime typically M > 5, characterized by:

  • High total enthalpy (air temperatures > 2000 K)
  • Strong normal shocks with large entropy rise
  • Significant temperature nonequilibrium and dissociation

6.2 Energy Modes in Gases

Molecules possess multiple energy modes: e=etrans+erot+evib+eelec+echem.e = e_{trans} + e_{rot} + e_{vib} + e_{elec} + e_{chem}.

ModeTypical activation (K)Description
Translational—Random molecular motion
Rotational50–300Excited in diatomic gases
Vibrational1000–5000Quantum vibrational states
Electronic>10000Ionization, radiation

At hypersonic speeds, rotational and vibrational modes lag behind translational energy — producing thermal nonequilibrium.

7. Nonequilibrium Thermochemistry

7.1 Dissociation and Ionization

Energy balance includes chemical reactions: N2+M⇌2N+M,O2+M⇌2O+M,\text{N}_2 + \text{M} \rightleftharpoons 2\text{N} + \text{M}, \quad \text{O}_2 + \text{M} \rightleftharpoons 2\text{O} + \text{M}, N+O⇌NO,N+e−⇌N++2e−.\text{N} + \text{O} \rightleftharpoons \text{NO}, \quad \text{N} + e^- \rightleftharpoons \text{N}^+ + 2e^-.

Equilibrium constant: Kp=exp⁡(−ΔG°RT).K_p = \exp\left(-\frac{ΔG^°}{RT}\right).

7.2 Rate Equations (Arrhenius Form)

ω˙=kf∏iCiνi−kb∏jCjνj,kf=ATne−Ea/RT.\dot{ω} = k_f \prod_i C_i^{ν_i} - k_b \prod_j C_j^{ν_j}, \quad k_f = A T^n e^{-E_a/RT}.

Relaxation time for vibrational energy (Millikan–White correlation): log⁡10τv=A+BT−1/3.\log_{10} τ_v = A + B T^{-1/3}.

8. Shock Layer Structure

8.1 Translational–Rotational–Vibrational Nonequilibrium

Temperature separation behind hypersonic shock: Ttrans>Trot>Tvib.T_{trans} > T_{rot} > T_{vib}.

8.2 Energy Conservation Across Shock Layer

htotal=htrans+hrot+hvib+hchem+u22.h_{total} = h_{trans} + h_{rot} + h_{vib} + h_{chem} + \frac{u^2}{2}.

Species mass, momentum, and energy equations must be solved simultaneously, often with finite-rate chemistry.

8.3 Electron Energy Equation

ρecveDTeDt=−∇⋅qe+J⋅E+Qionization.ρ_e c_{ve} \frac{DT_e}{Dt} = -∇·q_e + J·E + Q_{ionization}.

9. High-Temperature Gas Models

ModelValidityNotes
Calorically Perfect Gas< 800 Kc_p constant
Thermally Perfect Gas< 2000 Kc_p(T), single T
Chemically Reacting Gas2000–6000 Kdissociation, recombination
Ionized Gas> 6000 Kplasma regime

Internal energy with curve-fit polynomials: cp(T)=a+bT+cT2+dT3,h(T)=aT+b2T2+⋯ .c_p(T) = a + bT + cT^2 + dT^3, \quad h(T) = aT + \frac{b}{2}T^2 + \cdots.

10. Stagnation Heating and Aerodynamic Heating

10.1 Stagnation Temperature

T0=T(1+γ−12M2).T_0 = T (1 + \frac{γ-1}{2}M^2).

10.2 Recovery Factor

Due to boundary layer effects: Tr=T∞(1+rγ−12M2),r=Pr1/3.T_r = T_∞ (1 + r\frac{γ-1}{2}M^2), \quad r = Pr^{1/3}.

10.3 Convective Heat Flux (Fay–Riddell Equation)

At stagnation point for laminar boundary layer on blunt body: qw=0.763(ρeμe)1/2(he−hw)(dpedx)1/2.q_w = 0.763 (ρ_e μ_e)^{1/2} (h_e - h_w) (\frac{dp_e}{dx})^{1/2}.

11. Radiative Heat Transfer in Hypersonic Flow

At high enthalpy, electronic excitation produces radiation:

  • Continuum radiation: from free–free and free–bound transitions.
  • Line radiation: from discrete electronic transitions (e.g., N₂*, NO*).

Radiative flux: qrad=∫0∞κλ(Bλ−Iλ)dλ.q_{rad} = \int_0^∞ κ_λ (B_λ - I_λ) dλ.

Coupling with convective heating is crucial for thermal protection system (TPS) design.

12. Entropy and Exergy in Nonequilibrium Flows

Entropy production rate: σs=∑iJi⋅∇μiT+q⋅∇TT2+τ:∇uT.σ_s = \sum_i \frac{J_i·∇μ_i}{T} + \frac{q·∇T}{T^2} + \frac{τ:∇u}{T}.

Chemical nonequilibrium adds new production terms: σchem=∑rArξ˙rT,σ_{chem} = \sum_r \frac{A_r \dot{ξ}_r}{T}, where ArA_r is the chemical affinity.

Exergy destruction rate: E˙D=T0σsV.\dot{E}_D = T_0 σ_s V.

In high-enthalpy flows, major exergy losses arise from vibrational relaxation, dissociation, and radiation.

13. Numerical and Experimental Methods

13.1 Computational Approaches

  • DSMC: molecular-level rarefied modeling.
  • Navier–Stokes + Finite Rate Chemistry: continuum region.
  • Coupled CFD–DSMC Hybrid: multi-regime reentry flow simulation.

13.2 Experimental Techniques

  • Shock tunnels (milliseconds)
  • Plasma wind tunnels (steady-state)
  • Laser diagnostics: LIF, CARS, TDLAS for nonequilibrium temperature fields.

14. Applications in Hypersonic Aerothermodynamics

  • Atmospheric reentry (capsules, shuttles)
  • Scramjets and detonation-based propulsion
  • High-altitude aerodynamics
  • Ablation and TPS design
  • Plasma sheath and radio blackout phenomena

15. Summary of Key Relations

ConceptRelationNotes
Knudsen numberKn=λ/LKn = λ/LRegime indicator
Slip velocityus=(2−σv)/σvλ(∂u/∂n)u_s = (2-σ_v)/σ_v λ (∂u/∂n)Slip flow correction
Temperature jumpΔT=(2−σT)/σT(2γ/(γ+1))λ(∂T/∂n)ΔT = (2-σ_T)/σ_T (2γ/(γ+1))λ(∂T/∂n)Wall thermal nonequilibrium
Mean free pathλ=kT/(√2πd2p)λ = kT/(√2 π d^2 p)Molecular spacing
Vibrational relaxationlog10τv=A+BT−1/3log_{10} τ_v = A + B T^{-1/3}Millikan–White law
Stagnation heat fluxqw=0.763(ρeμe)1/2(he−hw)(dpe/dx)1/2q_w = 0.763 (ρ_e μ_e)^{1/2}(h_e-h_w)(dp_e/dx)^{1/2}Fay–Riddell correlation
  • Thermodynamics/10_NonEquilibrium_Thermodynamics.md — irreversible processes and coupled transport.
  • Fluid_Dynamics/09_Compressible_and_Supersonic_Flow.md — shock waves and compressible dynamics.
  • Heat_Transfer/HighTemperature_Radiation.md — radiative transfer and emission modeling.
  • Aero_Thermodynamics/Reentry_Physics.md — practical reentry and TPS design analysis.