Pressure Field Reconstruction from Experimental Measurements on a Transonic Wing

  • des Boscs, Pierre-Emmanuel (ONERA)
  • Fabbiane, Nicolo (ONERA )
  • Nicolas, François (ONERA)
  • Marquet, Olivier (ONERA)

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Following the design and production of a flexible composite wing by ONERA and DLR [1], an extensive measurement campaign in a transonic regime has been conducted. Among many measurements were global forces and momenta measured from a scale, pressure distributions using Pressure Sensitive Paint (PSP) on the suction side, and three-dimensional displacement at 27 points obtained with Modal Deformation Measurement (MDM) on the pressure side. In this presentation, we are interested in reconstructing the pressure and the displacements on both sides of the wing. In a first step, we implemented a finite-element model that reproduces the wing behavior, and couples the pressure $p$ with the displacements $\vec u$ in a relation in the form $\mathcal K \vec u = \mathcal P p$ where $\mathcal K$ is a rigidity operator, and $\mathcal P$ is an operator applying the pressure forces at the surface of the wing. Then, adapting the methodology developed in [2] to solve this inverse problem, we use a weighted cost functional, which evaluates the mismatch between (i) the pressure field and the pressure from the PSP, (ii) the displacements field and the displacements from the MDM, and (iii) the global forces and momenta and those from the scale. We further enforce the model with a Lagrangian multiplier approach, and add a $H_1$ regularisation, penalising the difference of the pressure from a prior estimate. Solving for the Karush–Kuhn–Tucker conditions yields a solution minimising the cost function. We first validate this method on synthetic data. Although the accurately recovered displacement is robust to the choice of weights, they strongly affect pressure distribution. Therefore, weights have to be optimised. Then, applying the method on the experimental data, we observe that the displacement mismatch remains very small. However, a good prior seems to be needed. Using results from flow simulations around a rigid wing from the design phase, we obtained a reasonable output. This simple approach, without fluid model if not for the prior estimate, seems to be sufficient to recover a reasonable estimation of the pressure. [1] N. Fabbiane, F.-X. Irisarri, J. Dillinger and A. Lepage. Aeroelastic-tailoring of a Wind-tunnel Model for Passive Alleviation of Static and Dynamic Loads. \emph{{CEAS} Aero. J.}, 2022. [2] A. Tessler, J.L. Spangler, A Variational Principle For Reconstruction Of Elastic Deformations In Shear Deformable Plates And Shells. \emph{NASA Tech. Memo.}, 2003.