Hypersonic 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.

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.

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

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.

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.

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.

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.

Yes. This is the fundamental philosophy of hypersonic aerodynamic design. It's also called the Blunt Body Paradox.

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.

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.

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.

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.

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.

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.

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).

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.