Transonic Buffet Modeling via Invariant Manifolds

TL;DR

The paper tackles predicting nonlinear transonic buffet, a Hopf-type global instability in shock–boundary-layer interactions, by developing a two-dimensional invariant-manifold ROM that predicts the full flow evolution on a graph-style manifold and expresses the reduced dynamics in an extended normal-form. A data-driven approach identifies the manifold with a linear encoder and a polynomial decoder, followed by a least-squares fit of a low-dimensional vector field; an extended normal-form transform yields a Stuart–Landau-type amplitude equation with a physically meaningful modal decomposition into a shift mode, a fundamental mode, and higher harmonics. The framework is demonstrated on 2D buffet over the OAT15A airfoil at $M_{\infty}=0.71$, using a single training trajectory to reconstruct the full flow and accurately predict nonlinear transients and the limit cycle, with full-field reconstruction errors typically below 5–10%. The work advances nonlinear, state-reconstructive reduced-order modeling for compressible flows with shocks, enabling faster prediction and potential control-oriented extensions across operating conditions.

Abstract

In transonic flow over aircraft wings, shock-boundary-layer interactions can give rise to transonic buffet, which degrades maneuverability through unsteady aerodynamic loads. Beyond its practical importance, two-dimensional transonic buffet represents a canonical example of a global instability for which reduced-order modeling remains challenging due to nonlinearity, sharp spatial gradients, and the coexistence of an unstable equilibrium with an attracting limit cycle. Commonly, reduced-order models of such phenomena capture nonlinear dynamics only in aerodynamic observables, while prediction of the full flow state is achieved through linear representations valid only near the unstable equilibrium or on the limit cycle. In this work, we present a reduced-order model that predicts the nonlinear evolution of the full flow field by exploiting the existence of an attracting two-dimensional invariant manifold. We adapt an existing data-driven framework for identifying invariant manifolds and the associated reduced dynamics, making it suitable for scaling to large-scale CFD applications. The invariant manifold is identified as a graph over its tangent space using an iterative encoder-update and the reduced dynamics are obtained via least-squares regression. A subsequent extended normal-form transformation enables physical interpretability of the model through a modal decomposition of the flow. The reduced-order model is identified for transonic buffet over the OAT15A supercritical airfoil, showing that it is possible to achieve this accurately using just a single training trajectory. Validation against independent simulations demonstrates accurate prediction of nonlinear behavior, together with reliable reconstruction of the full flow field, particularly in the late-transient and limit-cycle regimes.

Figures (14)

• Figure 1: Unsteady lift coefficient $C_l'$ signals for different Mach numbers and angles of attack of the OAT15A airfoil, computed with the SA-neg turbulence model in nitzsche2022effect. Red curves correspond to cases exhibiting buffet oscillations, whereas blue curves indicate steady cases without self-sustained unsteadiness.
• Figure 2: Invariant-manifold formulation versus autoencoder formulation. (a) Invariant-manifold setting. An attracting, normally hyperbolic invariant manifold $\mathcal{W} \subset \mathbb R^n$ is globally parameterized by reduced coordinates $\boldsymbol{\eta}\in\mathbb R^r$ through a decoder (embedding) $\mathbf W:\mathbb R^r\to\mathbb R^n$, with $\mathcal{W}=\mathbf W(\mathbb R^r)$. The reduced dynamics $\dot{\boldsymbol{\eta}}=\mathbf r(\boldsymbol{\eta})$ and the embedding $\mathbf W$ satisfy the invariance equation which ensures trajectory pairing $\Phi_t(\mathbf q_0)=\mathbf W(S_t(\boldsymbol{\eta}_0))$. The inverse map $\mathbf W^{-1}$ is defined only on the manifold $\mathcal{W}$. (b) Autoencoder (data-driven) setting. An encoder $\mathbf U:\mathbb R^n\to\mathbb R^r$ is introduced, defined on the entire ambient space $\mathbb R^n$, extending $\mathbf W^{-1}$ away from $\mathcal{W}$. Together with the decoder $\mathbf W$, the pair $(\mathbf U,\mathbf W)$ is characterized by the consistency relations $\mathbf U\circ\mathbf W=\mathrm{Id}$ and, approximately on data, $\mathbf W\circ\mathbf U\approx \mathrm{Id}$, enabling identification of a candidate manifold from sampled trajectories.
• Figure 3: Computational mesh employed in this study, adopted from nitzsche2019fluid: (a) mesh in the vicinity of the airfoil and (b) close-up of the near-wall quasi-structured boundary-layer mesh.
• Figure 4: Surface pressure coefficient $C_p$ distributions over the OAT15A airfoil for several angles of attack at $M_{\infty}=0.71$. The suction-side pressure plateau followed by a sharp gradient indicates the presence of a transonic shock, which moves upstream toward the leading edge with increasing angle of attack.
• Figure 5: Steady streamwise momentum $\rho u$ field at $M_{\infty}=0.71$ for (a) $\alpha = 4.45^\circ$ and (b) $\alpha = 5.25^\circ$. The sharp spatial gradient corresponds to the shock. Regions of reversed streamwise momentum downstream of the shock indicate flow separation. For the higher angle of attack shown, the shock is located further upstream on the suction side and the separated region has a larger spatial extent.