An Adaptive Wavelet Collocation Method for Fluid-Structure Interaction at High Reynolds Numbers

Abstract

Two mathematical approaches are combined to calculate high Reynolds number incompressible fluid-structure interaction: a wavelet method to dynamically adapt the computational grid to flow intermittency and obstacle motion, and Brinkman penalization to enforce solid boundaries of arbitrary complexity. We also implement a wavelet based multilevel solver for the Poisson problem for the pressure at each time step. The method is applied to two-dimensional flow around fixed and moving cylinders for Reynolds numbers in the range 3 × 10 1 ≤ Re ≤ 10 5. The compression ratios up to 1 000 are achieved. For the first time it is demonstrated in actual dynamic simulations that the compression scales like Re 1/2 over five orders of magnitude, while computational complexity scales like Re. This represents a significant improvement over the classical complexity estimate of Re 9/4 for two-dimensional turbulence.

A high-resolution penalization method for large Mach number flows in the presence of obstacles

“…This technique corresponds to a Brinkman-type porous media model with variable permeability, where the fluid domain has a very large permeability in front of that of the obstacle, and it has been applied successfully in many configurations (fixed and moving obstacles) [2,16,18,31,17,34]. The analysis carried out in [2] demonstrates that the solution of the penalized system converges to the solution of the incompressible Navier-Stokes system in the fluid domain, and that the velocity converges to zero in the solid domain at a theoretical rate of O(η 3/4 ).…”

“…Since the penalized system is solved in an obstacle-free domain, fast and effective methods for Cartesian grids can be used. Different numerical simulations of viscous flows using adaptive wavelet methods [31,17,34], pseudospectral methods [16], or finite difference/volume methods [2,18,26], have put in evidence the efficiency of this technique for incompressible flow simulations. The underlying numerical scheme provides the computational framework for the interplay between the real and the fictitious fluids.…”

On interpolatory divergence-free wavelets

“…The adaptation and usage of second-generation wavelets for the numerical solution of partial differential equations by a collocation method is introduced in [22,23,36,37].…”

“…This is one reason for considering collocation methods; see e.g. [22,23,35,37]. In this case, interpolatory wavelets are required, which can easily be adapted to complicated domains.…”

“…However, to our knowledge, there are no divergence-free interpolatory wavelets available, which is the reason why, e.g. in [22], an additional projection method has to be performed.…”