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Branching Paths Statistics for confined Flows : Adressing Navier-Stokes Nonlinear Transport

Daniel Yaacoub, Gaëtan Brunetto, Stéphane Blanco, Richard Fournier, Gerjan Hagelaar, Jean-François Cornet, Jérémi Dauchet, Thomas Vourc'h

Abstract

Recent advances have allowed to tackle exact path-space probabilistic representations of macroscopic advection-diffusion models involving advection nonlinearities by step forward approaches in terms of continuous branching stochastic processes. Yet, the need of such paradigm shift is huge for the broad flied of fluid flows. In deed, wherever for climate dynamics, engeenering, geophysical and planetary formations, or biomedical applications, complex transport phenomena involving diffusion and advection in confined domains set the physics. In this work, we advance this framework by casting such branching representations within the class of Navier-Stokes strongly nonlinear transport. This yields novel propagator representations for fluid dynamics and opens new routes for efficient simulations of fluids in confined domains by use of new Backward Monte Carlo algorithms.

Branching Paths Statistics for confined Flows : Adressing Navier-Stokes Nonlinear Transport

Abstract

Recent advances have allowed to tackle exact path-space probabilistic representations of macroscopic advection-diffusion models involving advection nonlinearities by step forward approaches in terms of continuous branching stochastic processes. Yet, the need of such paradigm shift is huge for the broad flied of fluid flows. In deed, wherever for climate dynamics, engeenering, geophysical and planetary formations, or biomedical applications, complex transport phenomena involving diffusion and advection in confined domains set the physics. In this work, we advance this framework by casting such branching representations within the class of Navier-Stokes strongly nonlinear transport. This yields novel propagator representations for fluid dynamics and opens new routes for efficient simulations of fluids in confined domains by use of new Backward Monte Carlo algorithms.

Paper Structure

This paper contains 12 sections, 14 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Context.
  3. Probabilistic representations of Navier-Stokes non-linear PDEs.
  4. The idea of CBSP for Navier-Stokes nonlinear transport and outline.
  5. Branching Path-space probabilistic representation
  6. Monte Carlo method and statistical estimators
  7. Monte Carlo algorithm.
  8. Velocity path sampling.
  9. Results and discussions
  10. Free-space unsteady Lamb-Oseen vortex.
  11. Confined unsteady damped Taylor-Couette flow.
  12. conclusions and perspectives

Figures (3)

  • Figure 1: a) Ray tracing intersection with $\partial\Omega$ by linear interpolation between the latest sampled position in $\mathring{\Omega}$ and the first sampled position in $\mathbb{R}^3\backslash\overline{\Omega}$. b) First passage percolation position infered by the nearest orthogonal projection.
  • Figure 2: Temporal and spatial profiles of the velocity field. Each Branching Backward Monte Carlo estimation is computed for $N=7\times 10^3$ samples, $\nu=2,2\times 10^2$ [m$^2$s$^{-1}$], $\Gamma=1\times 10^1$ [m$^2$s$^{-1}$] and $\rho=1\times 10^3$ [kg.m$^{-3}$]. Branching paths are sampled by $\delta s=1,7\times 10^{-5}$ [s] for $t=1\times 10^{-4}$ [s], by $\delta s=6\times 10^{-5}$ [s] for $t=5\times 10^{-4}$ [s] and by $\delta s=1,2\times 10^{-4}$ [s] for $t=1\times 10^{-3}$ [s].
  • Figure 3: Temporal profile of the angular frequencies and spatial profiles of the velocity field. Each Branching Backward Monte Carlo estimation is computed for $N=1\times 10^4$, $\nu=1\times 10^1$ [m$^2$s$^{-1}$], $R1=1$ [m], $\Omega_{2,\text{o}}=1$ [Hz], $\rho=1.10^3$ [kg.m$^{-3}$], $\lambda=1,5\times 10^2$ [Hz] and $\delta s=6\times 10^{-4}$ [s].