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).

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.

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.

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.

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.

Since buffet is an unsteady phenomenon, it naturally cannot be predicted by steady RANS analysis. Let's compare the options for unsteady methods.

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.

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.

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.

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).