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Grasping Extreme Aerodynamics on a Low-Dimensional Manifold

Kai Fukami, Kunihiko Taira

arXiv:2305.08024v2physics.flu-dyncs.LG

TL;DR

Extreme gust–airfoil interactions lack established theory and exhibit nonlinear variation across cases. The study uses a lift-augmented autoencoder to identify a low-dimensional manifold, finding that the flows can be represented with three variables.

Abstract

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Modern air vehicles perform a wide range of operations, including transportation, defense, surveillance, and rescue. These aircraft can fly in calm conditions but avoid operations in gusty environments, encountered in urban canyons, over mountainous terrains, and in ship wakes. With extreme weather becoming ever more frequent due to global warming, it is anticipated that aircraft, especially those that are smaller in size, will encounter sizeable atmospheric disturbances and still be expected to achieve stable flight. However, there exists virtually no theoretical fluid-dynamic foundation to describe the influence of extreme vortical gusts on wings. To compound this difficulty, there is a large parameter space for gust-wing interactions. While such interactions are seemingly complex and different for each combination of gust parameters, we show that the fundamental physics behind extreme aerodynamics is far simpler and lower-rank than traditionally expected. We reveal that the nonlinear vortical flow field over time and parameter space can be compressed to only three variables with a lift-augmented autoencoder while holding the essence of the original high-dimensional physics. Extreme aerodynamic flows can be compressed through machine learning into a low-dimensional manifold, which can enable real-time sparse reconstruction, dynamical modeling, and control of extremely unsteady gusty flows. The present findings offer support for the stable flight of next-generation small air vehicles in atmosphere conditions traditionally considered unflyable.

Introduction

Small aircraft face severe, poorly theorized aerodynamic disturbances in adverse weather, and studying the many possible gust–wing interactions is resource-intensive. The study identifies shared dynamics in these flows and compresses them to three variables.

Introduction

  • Aerodynamics is generally built around steady or quasi-steady flight, while extreme aerodynamic problems lack available theory.Strong gusts with G > 1 are described as extreme aerodynamics.Source: Introduction · pp. 1–2
  • Small aircraft can traverse urban and mountainous environments, but adverse-weather turbulence and vortical disturbances pose major operating challenges.Such environments have traditionally been avoided by conventional aircraft.Source: Introduction · p. 1
  • Gust interactions vary in size, strength, orientation, position, and geometry, making broad study resource-intensive.A naive approach would require extensive experimental and computational campaigns across the large parameter space.Source: Introduction · p. 1
  • Extreme gust–airfoil flows can be compressed to three variables with a nonlinear machine-learning technique, revealing shared underlying dynamics.The study’s model problem is a strong vortex hitting a canonical airfoil.Source: Introduction · p. 2

Results

A lift-augmented nonlinear autoencoder maps extreme gust–airfoil responses onto a three-variable manifold, unlike PCA, which loses distinctions between flow cases. The representation reconstructs flows and lift in tested interpolation, extrapolation, noisy, and multi-vortex conditions.

Results

  • The study simulates two-dimensional incompressible flow around a NACA 0012 airfoil at Re = 100, varying angle of attack and vortex size, strength, and position.The vortex–airfoil setup represents strong disturbances whose size can be comparable to the wing.Source: Extreme Vortex-Airfoil Interactions · p. 2, Identification of Extreme Aerodynamic… · p. 5
  • PCA overlaps baseline cases across angles of attack, producing indistinguishable representations and grossly inaccurate flow reconstructions.The comparison uses the first three PCA components; the nonlinear autoencoder instead reproduces the flow accurately.Source: Identification of Extreme Aerodynamic… · p. 5, Figure 4 · p. 5
  • The convolutional autoencoder compresses vorticity through a latent bottleneck, while a lift-prediction network guides the representation toward force-relevant structures.The latent variables are trained to encode the flow field and estimate lift; the added lift objective helps distinguish cases with different lift responses.Source: Identification of Extreme Aerodynamic… · pp. 5, 7
  • Lift augmentation collapses the extreme gust–airfoil responses onto a three-variable, cone-shaped manifold that captures their dominant dynamics.The manifold geometry is discovered without being specified in advance.Source: Figure 4 · p. 5, Identification of Extreme Aerodynamic… · p. 7
  • For held-out cases within the training parameter range, three latent variables reconstruct nonlinear gust–wing flow states and transient lift.The examples use |G| ≤ 4 and are assessed with structural similarity between decoded and reference flow fields.Source: Identification of Extreme Aerodynamic… · p. 7
  • For |G| > 4, the decoder recovers flow states and estimates large transient lift dynamics beyond the training gust-strength range.The reported extrapolation examples use G = 6.4 and G = −6.0.Source: Identification of Extreme Aerodynamic… · pp. 8–9
  • The model reconstructs flow and estimates lift for 30%-noise data and for vertically or horizontally arranged two-vortex gusts.For the horizontal pair, two latent-space inflections coincide with two valleys in the lift response.Source: Identification of Extreme Aerodynamic… · p. 9
  • The model accurately reconstructs flow variables and lift for a five-vortex disturbance despite training only on single-gust cases.The five vortices simulate severe wake turbulence striking the airfoil.Source: Identification of Extreme Aerodynamic… · p. 9

Discussion

A lift-informed autoencoder identifies a low-dimensional manifold for strong vortex–airfoil wake dynamics. The representation suggests routes to sparse flow reconstruction, reduced-order modeling, and control, while the study’s scope remains two-dimensional.

Discussion

  • Three variables represent the complex vortical flow, and projection onto the identified manifold can reduce the representation to two.The nonlinear autoencoder incorporates aerodynamic insights in its formulation.Source: Discussion · p. 10
  • A small number of airfoil sensors may support real-time reconstruction of the surrounding flow from the low-dimensional representation.The discussion points to decoder networks that map sparse sensor inputs to high-resolution flow estimates.Source: Discussion · p. 10
  • The manifold may support reduced-order dynamics models and control strategies, including phase-based selection of actuation timing and locations.The discussion describes phase-reduction analysis as a potential way to modify wake dynamics.Source: Discussion · p. 10
  • The study examines two-dimensional vortex–airfoil interactions; other gust types require three-dimensional analysis to assess the approach further.The discussion identifies three-dimensional flow extensions as important for examining the approach’s potential.Source: Discussion · p. 10

Methods

The study simulates extreme vortex–airfoil interactions across gust conditions and trains convolutional autoencoders to compress and reconstruct the resulting flow fields. A lift-augmented model also uses lift reconstruction to guide the learned latent representation.

Simulations of Extreme Vortex-Airfoil Interactions

  • The simulations use a NACA 0012 airfoil at Re = 100 and vary angle of attack and vortex gust ratio, size, and vertical position.Forty randomly sampled disturbed cases per angle of attack are split evenly between training and testing.Source: Simulations of Extreme Vortex-Airfoil… · p. 10
  • The analysis uses 1.26×10^5 frames from disturbed and undisturbed cases, with 1200 vorticity snapshots per case and lift histories from the simulations.Vorticity analysis focuses on a subdomain around the airfoil where vortex-airfoil interactions primarily occur.Source: Simulations of Extreme Vortex-Airfoil… · pp. 10–11

Autoencoder Setup

  • The autoencoder weights are optimized to minimize a loss, and Adam is used to determine those weights.The encoder maps high-dimensional flow input to a low-dimensional latent vector, while the decoder reconstructs the input flow.Source: Autoencoder Setup · p. 11
  • The convolutional autoencoder uses CNN layers to capture global flow features and MLP layers to extract features and further reduce data size.Nonlinear activations allow compression beyond linear PCA; the hyperbolic tangent accommodates positive and negative gust influence in latent space.Source: Autoencoder Setup · p. 11
  • The lift-augmented autoencoder reconstructs vorticity and lift together, encouraging its latent vector to retain information relevant to lift.Its loss balances vorticity-field and lift reconstruction losses with β = 0.05.Source: Autoencoder Setup · p. 11

Appendix

The appendix compares flow-field reconstructions by PCA and autoencoders, including cases with multiple gust vortices. The lift-augmented autoencoder reconstructs unseen paired-vortex cases with over 90% SSIM, while PCA misses key flow structures.

Reconstruction of vorticity fields

Autoencoder Setup

  • The appendix describes the lift-augmented autoencoder’s network structure, with convolutional filter size and count and pooling or upsampling ratios listed by layer.Source: Appendix · p. 11, Table 1 · p. 12

Reconstruction of vorticity fields

  • The nonlinear convolutional autoencoder substantially improves reconstruction over PCA and compresses the flow fields to three variables.It accurately estimates the extreme-vortex position over time, including during vortex impingement and strong nonlinear interactions.Source: Figure 8 · p. 13, Reconstruction of vorticity fields · p. 15
  • Over 90% SSIM is achieved by the lift-augmented autoencoder for both unseen two-vortex arrangements, despite training on single-vortex disturbed flows.PCA fails to capture the vortices and their influence on the wake in these examples.Source: Figure 9 · p. 14, Reconstruction of vorticity fields · p. 15
  • For two single-vortex cases, PCA fails to reconstruct the gust vortex, wake structures, and flow around the wing.The linear reconstructions show incorrect superpositions of wings.Source: Reconstruction of vorticity fields · p. 15

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