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Numerical approximation of statistical solutions of the incompressible Navier-Stokes Equations

by P. Bansal

(Report number 2021-18)

Abstract
Statistical solutions, which are time-parameterized probability measures on spaces of square-integrable functions, have been established as a suitable framework for global solutions of incompressible Navier-Stokes equations (NSE). We compute numerical approximations of statistical solutions of NSE on two-dimensional domains with non-periodic boundary conditions and empirically investigate the convergence of these approximations and their observables. For the numerical solver, we use Monte Carlo sampling with an H(div)-FEM based deterministic solver. Our numerical experiments for high Reynolds number turbulent flows demonstrate that the statistics and observables of the approximations converge. We also develop a novel algorithm to compute structure functions on unstructured meshes.

Keywords: fluid dynamics, turbulence, numerical approximation, computational methods, Monte Carlo, finite element method

@Techreport{B21_960,
  author = {P. Bansal},
  title = {Numerical approximation of statistical solutions of the incompressible
Navier-Stokes Equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2021-18},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2021/2021-18.pdf },
  year = {2021}
}

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