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: $\text{Kn} = \frac{\lambda}{L_c},$ where $\lambda$ is the molecular mean free path and $L_c$ is a characteristic length scale (e.g. boundary layer thickness or body size).

Flow Regime	Knudsen number	Description
Continuum (Navier–Stokes)	Kn < 10⁻³	Standard fluid mechanics valid
Slip Flow	10⁻³ < Kn < 10⁻¹	Velocity slip, temperature jump
Transition	10⁻¹ < Kn < 10	Continuum breakdown, kinetic corrections
Free Molecular	Kn > 10	No intermolecular collisions; ballistic regime

2. Molecular Kinetic Theory of Gases

2.1 Boltzmann Equation

The molecular distribution function $f(\mathbf{x},\mathbf{v},t)$ defines the probability density of finding a molecule with velocity $\mathbf{v}$ : $\frac{Df}{Dt} = \frac{\partial f}{\partial t} + \mathbf{v}\cdot\nabla f + \frac{\mathbf{F}}{m}\cdot\frac{\partial f}{\partial\mathbf{v}} = \left(\frac{\partial f}{\partial t}\right)_{coll}.$

Moments of $f$ yield macroscopic quantities: $\rho = m \int f d^3v, \quad \rho\mathbf{u} = m \int \mathbf{v} f d^3v, \quad p = \frac{1}{3}m\int c^2 f d^3v.$

2.2 Maxwell–Boltzmann Distribution

At equilibrium: $f(v) = \left( \frac{m}{2\pi kT} \right)^{3/2} \exp\left(-\frac{m(v-u)^2}{2kT}\right).$

Mean free path: $\lambda = \frac{kT}{\sqrt{2}\pi d^2p}.$

2.3 Molecular Transport Properties

From Chapman–Enskog expansion: $\mu = \frac{5}{16} \frac{(mkT/\pi)^{1/2}}{\pi d^2\Omega_\mu(T)},$ $k = \frac{15}{4} \frac{R}{M} \mu, \quad Pr = \frac{c_p\mu}{k} \approx 2/3.$

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)

$u_{wall} - u_{gas} = \frac{2 - \sigma_v}{\sigma_v} \lambda \left.\frac{\partial u}{\partial n}\right|_{wall},$ where $\sigma_v$ is the tangential momentum accommodation coefficient (0 < σ_v ≤ 1).

3.2 Temperature Jump

$T_{wall} - T_{gas} = \frac{2 - \sigma_T}{\sigma_T} \frac{2\gamma}{\gamma+1} \lambda \left.\frac{\partial T}{\partial n}\right|_{wall}.$

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: $\mathbf{\tau} = -\mu(\nabla u + \nabla u^T - \frac{2}{3}(\nabla\cdot u)I) - \beta_1 \lambda^2\nabla^2(\nabla u) + \cdots.$

5. Direct Simulation Monte Carlo (DSMC)

Numerical method for solving Boltzmann equation statistically:

Divide domain into cells smaller than mean free path.
Track representative molecules.
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: $e = e_{trans} + e_{rot} + e_{vib} + e_{elec} + e_{chem}.$

Mode	Typical activation (K)	Description
Translational	—	Random molecular motion
Rotational	50–300	Excited in diatomic gases
Vibrational	1000–5000	Quantum vibrational states
Electronic	>10000	Ionization, 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: $\text{N}_2 + \text{M} \right leftharpoons 2\text{N} + \text{M}, \quad \text{O}_2 + \text{M} \right leftharpoons 2\text{O} + \text{M},$ $\text{N} + \text{O} \right leftharpoons \text{NO}, \quad \text{N} + e^- \right leftharpoons \text{N}^+ + 2e^-.$

Equilibrium constant: $K_p = \exp\left(-\frac{\Delta G^°}{RT}\right).$

7.2 Rate Equations (Arrhenius Form)

$\dot{\omega} = k_f \prod_i C_i^{\nu_i} - k_b \prod_j C_j^{\nu_j}, \quad k_f = A T^n e^{-E_a/RT}.$

Relaxation time for vibrational energy (Millikan–White correlation): $\log_{10} \tau_v = A + B T^{-1/3}.$

8. Shock Layer Structure

8.1 Translational–Rotational–Vibrational Nonequilibrium

Temperature separation behind hypersonic shock: $T_{trans} > T_{rot} > T_{vib}.$

8.2 Energy Conservation Across Shock Layer

$h_{total} = h_{trans} + h_{rot} + h_{vib} + h_{chem} + \frac{u^2}{2}.$

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

8.3 Electron Energy Equation

$\rho_e c_{ve} \frac{DT_e}{Dt} = -\nabla\cdot q_e + J\cdot E + Q_{ionization}.$

9. High-Temperature Gas Models

Model	Validity	Notes
Calorically Perfect Gas	< 800 K	c_p constant
Thermally Perfect Gas	< 2000 K	c_p(T), single T
Chemically Reacting Gas	2000–6000 K	dissociation, recombination
Ionized Gas	> 6000 K	plasma regime

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

10. Stagnation Heating and Aerodynamic Heating

10.1 Stagnation Temperature

$T_0 = T (1 + \frac{\gamma-1}{2}M^2).$

10.2 Recovery Factor

Due to boundary layer effects: $T_r = T_\infty (1 + r\frac{\gamma-1}{2}M^2), \quad r = Pr^{1/3}.$

10.3 Convective Heat Flux (Fay–Riddell Equation)

At stagnation point for laminar boundary layer on blunt body: $q_w = 0.763 (\rho_e \mu_e)^{1/2} (h_e - h_w) (\frac{dp_e}{dx})^{1/2}.$

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: $q_{rad} = \int_0^\infty \kappa_\lambda (B_\lambda - I_\lambda) d\lambda.$

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

12. Entropy and Exergy in Nonequilibrium Flows

Entropy production rate: $\sigma_s = \sum_i \frac{J_i\cdot\nabla\mu_i}{T} + \frac{q\cdot\nabla T}{T^2} + \frac{\tau:\nabla u}{T}.$

Chemical nonequilibrium adds new production terms: $\sigma_{chem} = \sum_r \frac{A_r \dot{\xi}_r}{T},$ where $A_r$ is the chemical affinity.

Exergy destruction rate: $\dot{E}_D = T_0 \sigma_s V.$

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

Concept	Relation	Notes
Knudsen number	$Kn = \lambda/L$	Regime indicator
Slip velocity	$u_s = (2-\sigma_v)/\sigma_v \lambda (\partial u/\partial n)$	Slip flow correction
Temperature jump	$\Delta T = (2-\sigma_T)/\sigma_T (2\gamma/(\gamma+1))\lambda(\partial T/\partial n)$	Wall thermal nonequilibrium
Mean free path	$\lambda = kT/(\sqrt{}2 \pi d^2 p)$	Molecular spacing
Vibrational relaxation	$log_{10} \tau_v = A + B T^{-1/3}$	Millikan–White law
Stagnation heat flux	$q_w = 0.763 (\rho_e \mu_e)^{1/2}(h_e-h_w)(dp_e/dx)^{1/2}$	Fay–Riddell correlation

16. Cross-Links

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.