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A priori reduced order model of steady Navier‐Stokes equations for low Reynolds number

This work focuses on a priori reduced order model of the parameterised steady Navier‐Stokes equations for a Reynolds number lower than 100. The approach is based on the Proper Generalized Decomposition [1, 2] where the velocity and the pressure are expressed as series of separated function variables. Each term of the off‐line basis is computed progressively contrary to classical reduced order model techniques which necessitate snapshots of the problem before the reduction [3]. The curse of dimensionality could be overcome if the number of parameters increases. The approximation of the solution can be computed in real‐time once the terms of the reduced basis are known. Here, the Reynolds number µ and the position coordinate vector X are taken as parameters in the classical 2D lid‐driven cavity problem. The key points of the approach, which involve a succession of nonlinear problems expressed in a weak form, are briefly illustrated. Preliminary results, obtained with the use of the open source FEniCS software are presented. The velocity approximation is finally compared to a high fidelity solution. The results show that a small number of modes (fewer than 8) are necessary to obtain a velocity approximation with a global error lower than 1%.