Transonic Buffet
Theory and Physics
What is Transonic Buffet?
Professor, what exactly is happening in the phenomenon of transonic buffet?
Transonic buffet is an unsteady phenomenon where the shock wave on the wing surface oscillates spontaneously back and forth. In the transonic regime around Mach 0.7-0.85, as the angle of attack or flight speed increases, the shock wave strengthens, and beyond a certain point, it begins to move periodically forward and backward. This oscillation imposes unsteady aerodynamic loads on the wing, causing aircraft vibration (buffeting).
You mean the shock wave oscillates by itself? Without any external excitation?
Exactly. The most important characteristic of transonic buffet is that it is a self-sustained oscillation. Even without periodic disturbances from outside, the shock wave continues to oscillate due to an internal feedback mechanism within the flow field. Its frequency is typically $St = fL/U_{\infty} \approx 0.06-0.08$, scaled by the wing chord length and freestream velocity.
Buffet Onset Mechanism
What is the feedback mechanism for the self-sustained oscillation?
The Lee model (1990) is the most frequently cited. The feedback loop consists of the following four stages.
1. When the shock wave moves downstream, the shock wave/boundary layer interaction intensifies, causing the boundary layer to separate.
2. Pressure waves (acoustic waves) propagate upstream from the separation region.
3. The pressure waves reach the leading edge, generating new disturbances near the leading edge.
4. These disturbances are convected downstream and push the shock wave back upstream.
The period of this cycle can be estimated as $T = L_{ss}/a_{down} + L_{ss}/U_{conv}$. $L_{ss}$ is the distance between the shock wave and the trailing edge, $a_{down}$ is the speed of sound downstream, and $U_{conv}$ is the convection velocity of the disturbances.
Can we predict the conditions for buffet onset?
The buffet onset boundary is an important parameter in flight envelope design. The divergence Mach number $M_{div}$ can be detected in CFD as the condition where the RMS value of wall pressure exceeds a certain threshold. Practically, the drag divergence Mach number ($dC_D/dM = 0.1$) serves as a good indicator for buffet onset.
In aircraft design, the cruise Mach number is set with a sufficient margin (typically 0.03-0.05 Mach) from the buffet boundary.
Airbus A320 Buffet Problem
Are there any real-world cases where buffet became a problem?
Transonic buffet is one of the factors limiting the flight envelope of all commercial airliners. When flying at high altitudes and high Mach numbers, the buffet boundary of the upper surface shock wave defines the operational limit. When encountering turbulence, if the angle of attack increases, there is a possibility of exceeding the buffet boundary and causing airframe vibration. Therefore, FAR/CS 25.251 requires a margin of 1.3g or more.
The True Nature of That "Shuddering Sensation" on Airliners
Have you ever been on a plane flying around Mach 0.85 and felt the airframe shudder? That's transonic buffet. The shock wave formed on the upper wing surface moves periodically back and forth, interacting with boundary layer separation to cause unsteady lift fluctuations. In the 1960s, predicting this phenomenon was difficult, and there were cases where it was discovered during the first flight, leading to sudden restrictions on the flight envelope. Nowadays, it can be detected before flight using CFD URANS calculations, but it remains a challenging problem due to discrepancies with wind tunnel results.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Think of the moment you turn on a faucet. At first, water comes out in an unstable, spluttering manner, but after a while, the flow becomes steady, right? This "during the 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, solving first with a steady-state approach is a basic CFD strategy.
- Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: If you drop a leaf into a river, what happens? It's 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 because the "carrier," air, transports 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" → Not at all! Convection is transport by flow, conduction is transmission 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, next 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 flows (slow, viscous), diffusion is dominant. Conversely, in high Re number flows, 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, and 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. When switching to compressible analysis, if results become strange, it might be due to confusion between absolute/gauge pressure.
- Source Term $S_\phi$: Warmed 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 outside," expressed by the source term. What happens if you forget the source term? In natural convection analysis, if you forget to include buoyancy, the fluid doesn't move at all—a physically impossible result, like turning on a heater in a winter room but the warm air doesn't rise.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (mean free path of molecules ≪ characteristic length)
- Newtonian Fluid Assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
- Incompressibility Assumption (for Ma < 0.3): Density is treated as constant. For Mach numbers above 0.3, compressibility effects must be considered.
- Boussinesq Approximation (Natural Convection): Density variation is considered only in the buoyancy term; constant density is used 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 coefficient $\nu = \mu/\rho$ [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
Analysis Method Selection
Which method is suitable for CFD analysis of transonic buffet?
Since buffet is an unsteady phenomenon, it naturally cannot be predicted by steady RANS analysis. Let's compare the options for unsteady methods.
| Method | Buffet Frequency Prediction | Amplitude Prediction | 3D Effects | Computational Cost |
|---|---|---|---|---|
| 2D URANS | Good (±10%) | Tends to overestimate | No | Low |
| 3D URANS | Good | Moderate accuracy | Yes | Medium |
| DDES/IDDES | Good | Good | Yes | High |
| Wall-Resolved LES | High accuracy | High accuracy | Yes | Very High |
| ZDES (Zonal DES) | Good | Good | Yes | High |
Can buffet frequency be predicted even with 2D URANS?
For the buffet problem of the ONERA OAT15A airfoil, 2D URANS prediction accuracy for frequency is not bad. However, 2D cannot reproduce the three-dimensional structure of separation (cell structure, spanwise wavenumber), so it tends to overestimate buffet load amplitude. 2D URANS is sufficient for initial screening in design, but 3D analysis is needed for quantitative evaluation of structural loads.
Influence of Turbulence Model
Does the turbulence model in URANS change buffet prediction?
It changes significantly. The influence on buffet onset conditions is particularly notable.
- SA (Spalart-Allmaras): Predicts buffet onset later (by about 0.5-1 degree higher in angle of attack). Buffet frequency is slightly underestimated.
- SST $k$-$\omega$: Buffet onset is close to experiment. Frequency also good. However, amplitude tends to be slightly overestimated.
- EARSM (Explicit Algebraic RSM): Improves separation prediction due to nonlinear eddy viscosity terms.
- $k$-$\omega$ DDES: Can capture the Kelvin-Helmholtz instability of the separated shear layer, making amplitude prediction the most accurate.
So, SST $k$-$\omega$ based DDES is the practical best balance, right?
Exactly. ONERA's ZDES (Zonal DES) is a method that explicitly zones the RANS region near the wall and the LES region in the separated area, and is being vigorously validated by the French aerospace agency. It yields the most reliable results for buffet analysis.
Time Resolution Settings
How small should the time step be set?
The typical buffet frequency is $f \approx 60-80$ Hz (for a chord length of 1m, $M = 0.73$). A time step that divides this period into at least 50-100 parts is necessary.
For DES/LES, an even smaller time step is needed to match the time scale of turbulent structures. Set it to satisfy $\Delta t \cdot U_{\infty} / \Delta x < 1$ (the CFL condition).
To remove the initial transient in unsteady calculations, it is standard to compute at least 10-20 buffet cycles and then take statistical averages over the subsequent 10-20 cycles. That means the total number of calculation steps is on the order of 20-40 cycles × 50-100 steps/cycle = 1000-4000 steps (URANS) to tens of thousands of steps (DES).
URANS vs DES—"What Do You Want to Know?" Rather Than "Which is Correct?"
Engineers who have tried both URANS and DES for transonic buffet analysis hit the wall: "DES matches the shock oscillation frequency better, but takes 10 times longer to compute." In practice, it's common to use fast, coarse URANS for sensitivity analysis in the early design stages, and use DES only for final verification. If the goal is to find the critical Mach number for buffet onset, even just searching for the "kink" in the lift curve with steady RANS can sometimes yield practical accuracy. Method selection is not about tool superiority, but is determined by the answer to the question, "What accuracy is needed in which phase of the design?"
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
2nd-order accuracy, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number, diffusion-dominated flows.
TVD Scheme (MUSCL, QUICK, etc.)
Maintains high accuracy while suppressing numerical oscillations 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.
CFL Condition (Courant Number)
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