Supersonic Flow

The most fundamental difference is the direction of information propagation. In subsonic flow, pressure disturbances propagate in all directions, so information about downstream obstacles also reaches upstream. However, in supersonic flow, since the flow exceeds the speed of sound, disturbances only propagate downstream. This property gives rise to shock wave formation, the existence of Mach cones, and a change in the type of governing equations (from elliptic to hyperbolic).

When the Mach number $M = U/a$ exceeds 1, disturbances only propagate within a cone of Mach angle $\mu$.

For $M = 2$, $\mu = 30°$; for $M = 3$, $\mu \approx 19.5°$.

It's directly related. The angle $\beta$ of the oblique shock wave formed at the tip of a wedge-shaped object is determined by the relationship between the wedge half-angle $\theta$ and the Mach number $M$ (the $\theta$-$\beta$-$M$ relation).

For the same $\theta$, there exist two solutions: a weak shock wave solution and a strong shock wave solution; typically, the weak one is realized.

When supersonic flow turns around a convex corner, a continuous expansion wave (Prandtl-Meyer expansion fan) is formed. As the flow passes through the expansion wave, it accelerates and the Mach number increases. The relationship between the turning angle $\Delta\theta$ and the Mach number is described by the Prandtl-Meyer function $\nu(M)$.

The relation $\nu(M_2) - \nu(M_1) = \Delta\theta$ holds.

We create a numerical table or solve it inversely using Newton's method. In CFD, the solver naturally resolves the expansion wave, but this equation is essential for verifying the results. It also plays a central role in the MOC (Method of Characteristics) for supersonic nozzle design.

The diamond airfoil is a classic example of supersonic linear theory. Shock waves and expansion waves form at the leading and trailing edges respectively, generating lift from the pressure difference between the upper and lower surfaces, and wave drag from the flow-direction component of pressure. According to supersonic linear theory (Ackeret theory), the pressure coefficient is:

where $\theta$ is the local wall inclination angle. Under the thin airfoil approximation, the lift coefficient and wave drag coefficient are:

There are three main differences. First, upwind-type schemes are needed for shock wave capturing. Second, chemical reactions (high-temperature gas effects) must be considered at high Mach numbers. Third, boundary condition settings are different. For supersonic inlets, all variables are prescribed; for supersonic outlets, all variables are extrapolated.

Shock fitting treats the shock wave position as an unknown, precisely placing the shock wave on a grid boundary. The advantage is that the shock wave is sharply resolved. On the other hand, shock capturing automatically captures the shock wave within the range of numerical diffusion.

In practice, shock capturing is overwhelmingly mainstream. The reasons are as follows.

MOC is still a standard method for supersonic nozzle contour design. In steady 2D supersonic flow, Riemann invariants are conserved along characteristic lines, so by tracing these lines, the flow field can be constructed.

Specifically, along the $C^+$ characteristic line (Mach line),

holds. Starting from the throat and solving these relations at each grid point yields the nozzle wall shape that achieves the desired Mach number distribution. It's still used in the design of supersonic wind tunnel nozzles like NASA CELV.

MOC provides an exact solution under the assumptions of inviscid, isentropic flow, so it can be used for CFD verification. Also, for nozzle design, its strength over CFD is that it can directly solve the inverse problem (desired Mach number distribution → wall shape).

At around $M > 5$ (hypersonic regime), temperatures behind the shock wave reach several thousand K, making the following effects non-negligible.

• Vibrational Excitation: At $T > 800$ K, molecular vibrational modes are excited, decreasing $\gamma$.
• Dissociation: At $T > 2500$ K, $O_2$ dissociates; at $T > 4000$ K, $N_2$ dissociates.
• Ionization: At $T > 9000$ K, ionization begins, causing plasma effects.

To handle these, the ideal gas equation of state $p = \rho R T$ must be replaced with a chemical equilibrium model or a chemical non-equilibrium model. Fluent's Species Transport model, OpenFOAM's hy2Foam solver, and table lookup methods using the NASA CEA database are used.