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Comparative Analysis of Obstacle Approximation Strategies for the Steady Incompressible Navier-Stokes Equations

Piotr Krzyżanowski, Sadokat Malikova, Piotr B. Mucha, Tomasz Piasecki

Abstract

This paper aims to compare and evaluate various obstacle approximation techniques employed in the context of the steady incompressible Navier-Stokes equations. Specifically, we investigate the effectiveness of a standard volume penalization approximation and an approximation method utilizing high viscosity inside the obstacle region, as well as their composition. Analytical results concerning the convergence rate of these approaches are provided, and extensive numerical experiments are conducted to validate their performance.

Comparative Analysis of Obstacle Approximation Strategies for the Steady Incompressible Navier-Stokes Equations

Abstract

This paper aims to compare and evaluate various obstacle approximation techniques employed in the context of the steady incompressible Navier-Stokes equations. Specifically, we investigate the effectiveness of a standard volume penalization approximation and an approximation method utilizing high viscosity inside the obstacle region, as well as their composition. Analytical results concerning the convergence rate of these approaches are provided, and extensive numerical experiments are conducted to validate their performance.
Paper Structure (14 sections, 3 theorems, 58 equations, 7 figures)

This paper contains 14 sections, 3 theorems, 58 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Volume penalization
  3. Viscosity penalization
  4. Mixed penalization
  5. Weak formulation
  6. Theoretical bounds on the convergence rate
  7. Volume penalization
  8. Viscosity penalization
  9. Mixed penalization
  10. Numerical simulations
  11. A box-shaped obstacle touching the boundary
  12. An obstacle with sharp corners
  13. Two obstacles, one surrounded by the fluid
  14. Conclusions

Key Result

THEOREM 2.1

Assume $f \in H^{-1}(\Omega)$ and $\Omega$ is a Lipschitz domain. Let ${u}_{n}$ denote a weak solution to equation NSvol. Then Moreover, for a subsequence $u_{n_k}$, denoted in what follows ${u}_{n}$, where $u$ is a weak solution to equation NS. Assuming additionally that $\partial \Omega \in C^{2}$, $f \in L^2(\Omega)$ and $\|f\|_{H^{-1}(\Omega)}$ is small enough with respect to $\nu$ we have

Figures (7)

  • Figure 1: Decomposition of $\Omega$ into fluid domain $\Omega_F$ and solid obstacle $\Omega_S$. The interface between the fluid and the solid is denoted by $\Sigma$ and marked with a solid black line.
  • Figure 2: Streamlines plots, with color corresponding to velocity magnitude, for viscosity penalization (top row) and for volume penalization (bottom row) and varying penalty parameter. Panels on the left correspond to penalty parameter $m=n=10^3$; those on the right, to $m=n=10^5$. (For details regarding the experimental setting, see Section \ref{['sec:experim:taipei']})
  • Figure 3: From left to right: ,,real obstacle" domains $\Omega_F$ considered in Sections \ref{['sec:experim:box']}, \ref{['sec:experim:taipei']} and \ref{['sec:two obstacles']}, respectively, together with corresponding flow streamlines.
  • Figure 4: Penalized solution streamlines in $\Omega$, approximating the flow depicted in Figure \ref{['stream-real']}. The first two correspond to viscosity penalization with $m=10^5$; the last one comes from mixed penalization with $m=n=10^5$. Note how well the streamlines in the fluid domain $\Omega_F$ approximate the actual flow around the obstacles, cf. Figure \ref{['stream-real']}.
  • Figure 5: Approximation errors measured in $H^1$ seminorm (top) or $L^2$ norm (bottom) on $\Omega$ (left) and $\Omega_S$ (right), for the experiment from Section \ref{['sec:experim:box']}. The plots correspond to ${u}_{n}$ (blue), ${u}_{m}$ (red) and ${u}_{m \lor n}$ (green). For ${u}_{m \lor n}$, we set $n=10^2\cdot m$ and treat only $m$ as the independent penalization parameter (the $x$-axis corresponds to values of $m$ in this case).
  • ...and 2 more figures

Theorems & Definitions (5)

  • THEOREM 2.1
  • THEOREM 2.2
  • proof
  • THEOREM 2.3
  • proof