Hypersonic Flow
Theory and Physics
Overview
Professor, hypersonic flow is the world above Mach 5, right? What's the difference from regular supersonic flow?
The decisive difference is that the temperature behind the shock wave reaches thousands of Kelvin, changing the chemical properties of the gas. Because air undergoes dissociation and ionization, the ideal gas assumption completely breaks down. It's an unavoidable domain in the design of re-entry capsules and scramjets.
You mean the air molecules break apart because the temperature is too high? That's scary...
Exactly. N₂ and O₂ dissociate into atoms, and at even higher temperatures, they ionize. Therefore, governing equations that account for thermochemical non-equilibrium become necessary.
Governing Equations
Specifically, what kind of equations appear?
First, the fundamental Rankine-Hugoniot relations for a normal shock wave. For an ideal gas, the pressure ratio across the shock is
and the temperature ratio is
What would the temperature ratio be at around M=20?
Calculated with an ideal gas, T₂/T₁ would be several hundred times, but in reality, dissociation reactions absorb energy, suppressing the temperature rise. This is precisely the real gas effect. Newton's modified pressure coefficient is also important in hypersonics, and
is a good approximation for the stagnation point pressure on a blunt body. Furthermore, the stagnation enthalpy
is the starting point for TPS (Thermal Protection System) design. At orbital velocity of Mach 25, $h_0 \approx 30$ MJ/kg.
30 MJ/kg is an enormous amount of energy...
Thermochemical Non-Equilibrium Model
When air dissociates, what kind of model is used?
Additional transport equations for 5 species (N₂, O₂, NO, N, O) or 11 species (including ionic species and electrons) are added. The transport equation for each chemical species $Y_s$ is
where $\dot{\omega}_s$ is the chemical reaction source term, calculated using the Arrhenius reaction rate constant
In Park's (1990) two-temperature model, translational-rotational temperature $T_{tr}$ and vibrational-electronic temperature $T_{ve}$ are treated separately.
You use two temperatures! So behind the shock wave, the vibrational temperature lags behind as it rises.
Exactly. The vibrational relaxation time is estimated using the Millikan-White correlation. This non-equilibrium region significantly affects shock wave stand-off distance and wall heat flux, so proper modeling in CFD is essential.
Wall Heating and Thermal Protection Design
How much does the surface of a re-entry vehicle heat up?
The stagnation point heat flux can be roughly estimated using the Fay-Riddell formula.
A larger nose radius reduces the heat flux, which is why re-entry vehicles have blunt shapes. The nose heat flux for an Apollo-type capsule reaches several MW/m² at its peak.
So that's why re-entry vehicles are round. A sharp tip would concentrate the heating.
Yes. This is the fundamental philosophy of hypersonic aerodynamic design. It's also called the Blunt Body Paradox.
Space Shuttle Re-entry — Air at Mach 25 Becomes Plasma
The biggest challenge in hypersonic flow (above Mach 5) is that "air is no longer an ideal gas." When the Space Shuttle re-entered the atmosphere, the air ahead of the vehicle was instantly heated to over 6000K by a shock wave equivalent to Mach 25. At this temperature, nitrogen molecules (N₂) dissociate into N atoms, and further ionization generates plasma. To calculate this accurately in CFD, a "reacting gas model" is required, which couples chemical reaction equations with the usual flow equations. This chemical reaction CFD also played a crucial role in investigating the cause of the Columbia accident's thermal protection failure.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Imagine the moment you turn on a faucet. At first, water comes out in an unstable, spluttering manner, but after a while, it becomes a steady flow, right? This "period of change" is described by the temporal term. The pulsation of blood flow from a heartbeat, or the flow fluctuation each time an engine valve opens and closes—all are unsteady phenomena. So what is steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled down"—meaning this term is set to zero. Since computational cost drops significantly, starting with a steady-state solution is a basic CFD strategy.
- Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: If you drop a leaf into a river, what happens? It gets carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. Warm air from a heater reaching the far corner of a room is also due to air, the "carrier," transporting heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as the flow speed increases, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There's an order of magnitude difference in efficiency.
- Diffusion Term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever put milk in coffee and left it? Even without stirring, after a while, it naturally mixes. That's molecular diffusion. Now a question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. Higher viscosity strengthens the diffusion term, making the fluid move in a "thick" manner. In low Reynolds number flow (slow, viscous), diffusion dominates. Conversely, in high Re number flow, convection overwhelmingly dominates, and diffusion plays a supporting role.
- Pressure Term $-\nabla p$: When you push the plunger of a syringe, liquid shoots out forcefully from the needle tip, right? Why? Because the piston side is high pressure, the needle tip is low pressure—this pressure difference provides the force that pushes the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Flow is generated where there is a pressure difference"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results go wrong immediately after switching to compressible analysis, mixing up absolute/gauge pressure might be the cause.
- Source Term $S_\phi$: Heated air rises—why? Because it becomes lighter (lower density) than its surroundings, so it's pushed upward by buoyancy. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat generated by a gas stove flame, Lorentz force acting on molten metal in a factory's electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside," expressed by source terms. What happens if you forget the source term? In natural convection analysis, forgetting buoyancy means the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a heated room in winter.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (mean free path ≪ characteristic length)
- Newtonian Fluid Assumption: Linear relationship between shear stress and strain rate (non-Newtonian fluids require viscosity models)
- Incompressibility Assumption (for Ma < 0.3): Treat density as constant. For Mach number above 0.3, compressibility effects must be considered
- Boussinesq Approximation (Natural Convection): Density variation considered only in the buoyancy term, using constant density in other terms
- Non-applicable Cases: Rarefied gas (Kn > 0.1), Supersonic/Hypersonic flow (shock capturing required), Free surface flow (requires VOF/Level Set, etc.)
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Velocity $u$ | m/s | When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units |
| Pressure $p$ | Pa | Distinguish between gauge and absolute pressure. Use absolute pressure for compressible analysis |
| Density $\rho$ | kg/m³ | Air: approx. 1.225 kg/m³ @20°C, Water: approx. 998 kg/m³ @20°C |
| Viscosity Coefficient $\mu$ | Pa·s | Be careful not to confuse with kinematic viscosity $\nu = \mu/\rho$ [m²/s] |
| Reynolds Number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Indicator for laminar/turbulent transition |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability |
Numerical Methods and Implementation
Numerical Scheme Selection
Can hypersonic flow CFD be calculated with a regular compressible solver?
The basic Finite Volume Method (FVM) framework is the same, but the choice of flux calculation scheme becomes extremely important. To stably capture strong shock waves, combining schemes like Roe or AUSM+ family with limiters (Van Leer, Barth-Jespersen, etc.) is standard.
What does AUSM+ stand for?
Advection Upstream Splitting Method. It's a technique that separately evaluates mass flux and pressure flux, reducing carbuncle phenomena in the hypersonic regime. Particularly, the AUSM+-up scheme has high stability across all speed ranges.
Spatial Discretization and Shock Capturing
How shock waves are handled on the mesh is crucial, right?
Exactly. In shock capturing methods, numerical viscosity smooths the discontinuity over a few cell widths. To increase accuracy, use MUSCL reconstruction for second-order accuracy while suppressing oscillations with a TVD (Total Variation Diminishing) limiter.
Here $\phi(r)$ is the limiter function. The MinMod limiter is the most diffusive but stable, while Superbee is sharp but risks overshoot.
Is limiter selection very critical in hypersonics?
Extremely critical. For strong bow shocks at M>10, the Roe method alone tends to cause carbuncle instability. Methods considering multidimensional effects, like the H-CUSP or Rotated Roe schemes, are effective.
Time Integration and Explicit Method Constraints
For time integration, do you use explicit or implicit methods?
For steady-state solutions, implicit methods (LU-SGS, DPLR-type point-implicit, etc.) are efficient. Because CFL numbers can be set above 100, convergence is fast. On the other hand, for tracking unsteady phenomena (like capsule oscillations), second-order implicit methods (BDF2) or explicit two-stage Runge-Kutta are used. However, the explicit method's CFL restriction is
and in hypersonics, the speed of sound $a$ increases due to high temperature, making the time step extremely small.
Chemical Reaction Source Stiffness Problem
I've heard calculations including chemical reactions are difficult to converge.
Because the chemical reaction time constant is many orders of magnitude smaller than the fluid time constant, the source term becomes "stiff." To solve this, either treat the chemical species source term Jacobian implicitly using a point-implicit method, or solve fluid and chemistry separately using an operator splitting method. NASA's DPLR code and the US3D code standardly use point-implicit methods.
What is DPLR?
Data Parallel Line Relaxation. A hypersonic CFD code developed by NASA Ames, incorporating Park's thermochemical model. It's one of the industry standards for re-entry vehicle analysis. Others include LAURA (NASA Langley) and US3D (University of Minnesota).
Don't commercial solvers support this?
Ansys Fluent also supports real gas and finite-rate chemical reactions, but it doesn't have as much verification history as dedicated hypersonic codes. STAR-CCM+ also has a Reacting Flow model, but implementation of 5-species/11-species air models may require user-side customization.
The Real Reason "Mesh is Life" in Hypersonic CFD
The most critical aspect in hypersonic flow CFD is mesh design. Shock wave thickness is on the scale of the mean free path (molecular distance), near the limit of the continuum assumption. Furthermore, boundary layer thickness is orders of magnitude thinner compared to supersonic flow. These multiple thin layers must be resolved simultaneously. In practice, using a technique called "shock fitting"—explicitly incorporating the shock wave position into the mesh—can significantly reduce numerical diffusion behind the shock. A hybrid approach combining structured and unstructured grids is becoming the standard in the field.
Upwind Scheme
1st-order Upwind: Large numerical diffusion but stable. 2nd-order Upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing
Second-order accurate, but numerical oscillations occur for Pe > 2. Suitable for low Reynolds number diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Suppress numerical oscillations while maintaining high accuracy via limiter functions. Effective for capturing shock waves and steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex shapes and multi-physics. Mesh-free methods like SPH are also developing.
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