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Experimental approach to the problem of the Navier-Stokes singularities

Abstract : This thesis is devoted to the experimental search for prints of the singularities that might occur in the solutions of the 3D incompressible Euler or Navier-Stokes equations. Indeed, the existence of solutions to these partial differential equations has been proven but it is still unknown whether these solutions are regular, i.e. whether they blow up in finite time or not. In this thesis, we postulate the existence of such singularities and look for prints of them in 3D velocity fields acquired experimentally in a turbulent swirling flow. The distribution, 3D structure and time evolution of these prints are detailed. Our detection of prints of possible singularities is based on the work of the mathematicists Duchon and Robert. We look for extreme values of the Duchon-Robert term at small scales, i.e. in the dissipative range. That is what we call prints of singularities. We compute the Duchon-Robert term on velocity fields which are acquired experimentally at the center of a von Kármán turbulent swirling flow. The velocity field is measured by tomographic particle image velocimetry (TPIV), either time-resolved or not. In a first part we perform a scale-by-scale analysis of the statistics of the Duchon-Robert term and compare them to the statistics of the viscous dissipation and of the inter-scale energy transfer terms involved in the LES equations. In a second part, we analyze the topology of the velocity field around the extreme events of the Duchon-Robert term. We first use a method based on the invariants of the velocity gradient tensor (VGT) and then observe directly the velocity fields. A third part presents preliminary results of an Eulerian study of the time-evolution of the extreme events of the Duchon-Robert term.
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Submitted on : Thursday, December 19, 2019 - 8:24:08 PM
Last modification on : Monday, February 10, 2020 - 6:12:22 PM
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  • HAL Id : tel-02420454, version 1


Paul Debue. Experimental approach to the problem of the Navier-Stokes singularities. Fluid Dynamics [physics.flu-dyn]. Université Paris-Saclay, 2019. English. ⟨NNT : 2019SACLS305⟩. ⟨tel-02420454⟩



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