Supersonic Flow
Theory and Physics
Fundamental Properties of Supersonic Flow
Professor, what is fundamentally different between supersonic flow and subsonic 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 $M = 3$, $\mu \approx 19.5°$.
Is that concept of Mach angle related to the angle of shock waves around an airfoil?
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.
Prandtl-Meyer Expansion Wave
What happens when supersonic flow expands?
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.
This equation is quite complex. How is it used in practice?
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.
Supersonic Aerodynamic Characteristics of Diamond Airfoils
How do you calculate the lift and drag of a supersonic airfoil?
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:
In subsonic flow, lift is linear with angle of attack, but wave drag is proportional to the square of the angle of attack. I understand why supersonic flight is so costly.
The Day Chuck Yeager Broke the "Sound Barrier"
On October 14, 1947, test pilot Chuck Yeager achieved Mach 1.06 in the Bell X-1, accomplishing humanity's first supersonic flight. Surprisingly, he had broken two ribs in a horse riding accident the day before. It was the moment when engineers and a pilot, whose courage some might call reckless, overcame the shock wave/compressibility barrier that aeronautical engineers of the time believed would cause wings to disintegrate near the speed of sound. When calculating normal shock waves with modern CFD, one can glimpse the challenge of that day beyond the equations.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Imagine turning on a faucet. At first, water comes out in an unstable, spluttering manner, but after a while, the flow becomes steady, 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—these are all unsteady phenomena. So what is steady-state analysis? It's looking only at "after sufficient time has passed and the flow has settled down"—in other words, setting this term to zero. Since this drastically reduces computational cost, starting with a steady-state solution is a basic CFD strategy.
- Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? 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 becomes faster, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar things" → They're 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, right? That's molecular diffusion. Now a question—which flows more easily, honey or water? 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 is dominant. Conversely, in high Re number flow, convection overwhelms 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, it might be due to mixing up absolute/gauge pressure.
- Source Term $S_\phi$: Heated air rises—why? Because it becomes lighter (lower density) than its surroundings, so it's pushed up 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 applied to molten metal by an electromagnetic pump in a factory... These are all actions that "inject energy or force into the fluid from the outside," expressed by source terms. What happens if you forget a source term? In natural convection analysis, forgetting to include buoyancy means the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a room with the heater on in winter.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (molecular 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 numbers above 0.3, compressibility effects must be considered.
- Boussinesq Approximation (Natural Convection): Consider density variation 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: ~1.225 kg/m³ @20°C, Water: ~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 | $Re = \rho u L / \mu$. Criterion for laminar/turbulent transition. |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability. |
Numerical Methods and Implementation
CFD Methods for Supersonic Flow
In CFD analysis of supersonic flow, what special considerations are needed compared to subsonic flow?
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.
I've heard there are schemes that capture shock waves and methods that place shock waves on grid lines (shock fitting)?
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.
| Comparison Item | Shock Fitting | Shock Capturing |
|---|---|---|
| Sharpness of Shock | Accurate (discontinuous) | Spreads over several cells |
| Applicability to Complex Shapes | Difficult | Easy |
| Shock Intersection/Reflection | Requires manual handling | Automatically handled |
| Unsteady Problems | Requires shock tracking | Directly applicable |
| Ease of Implementation | Complex | Relatively easy |
Method of Characteristics (MOC)
Is the Method of Characteristics still used?
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.
Why use MOC when we have CFD?
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).
High-Temperature Gas Effects
I've heard the ideal gas assumption breaks down at high Mach numbers.
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.
Numerical Methods for Supersonic Flow—Godunov-type Schemes and Upwind Difference Choices
In numerical analysis of supersonic flow, the greatest challenge is solving shock waves stably "without numerical oscillations." The approach proposed by Godunov (1959) of "solving the Riemann problem at each interface" became the foundation of modern compressible CFD. For higher accuracy, PPM (Piecewise Parabolic Method), MUSCL, and WENO (Weighted Essentially Non-Oscillatory) schemes are used. WENO suppresses oscillations near shock waves while achieving high-order accuracy (5th order or higher) in smooth regions. Finite Volume Method (FVM) with structured grids is the standard for supersonic CFD, implemented in solvers like Ansys Fluent and OpenFOAM's rhoCentralFoam.
Upwind Differencing (Upwind)
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 (Central Differencing)
2nd-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 using 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 multiphysics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant Number)
Explicit methods: CFL ≤ 1 is the stability condition. Implicit methods: Stable even for CFL > 1, but affects accuracy and iteration count. LES: CFL ≈ 1 recommended. Physical meaning: Information should not travel more than one cell per time step.
Residual Monitoring
Convergence is judged when residuals for continuity, momentum, and energy equations drop by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factors
Pressure: 0.2~0.3, Velocity: 0.5~0.7 are typical initial values. If diverging, lower the relaxation factor. After convergence, increase to accelerate.
Internal Iterations for Unsteady Calculations
Iterate within each time step until a steady solution converges. Internal iteration count: 5~20 iterations is a guideline. If residuals fluctuate between time steps, review the time step size.
Analogy for the SIMPLE Method
The SIMPLE method is an "alternating adjustment" technique. First, velocity is tentatively determined (predictor step), then pressure is corrected so that mass conservation is satisfied with that velocity (corrector step), and then velocity is revised with the corrected pressure—this back-and-forth is repeated to approach the correct solution. It resembles two people leveling a shelf: one adjusts the height, the other balances it, and they repeat this alternately.
Analogy for Upwind Differencing
Upwind differencing is "standing in a river flow and prioritizing information from upstream"
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