Gh Geert Keetels

Fourier spectral and wavelet solvers for the incompressible Navier-Stokes equations with volume-penalization: Convergence of a dipole-wall collision

In this study, we use volume-penalization to mimic the presence of obstacles in a flow or a domain with no-slip boundaries. This allows in principle the use of fast Fourier spectral methods and coherent vortex simulation techniques (based on wavelet decomposition of the flow variables) to compute turbulent wall-bounded flow or flows around solid obstacles by simply adding one term in the equation. Convergence checks are reported using a recently revived, and unexpectedly difficult dipole-wall collision as a benchmark computation. Several quantities, like the vorticity isolines, truncation error, kinetic energy and enstrophy are inspected for a collision of a dipole with a no-slip wall and compared with available benchmark data obtained with a standard Chebyshev pseudospectral method. We quantify the possible deteriorating effects of the Gibbs phenomenon present in the Fourier based schemes due to continuity restrictions of the penalized Navier-Stokes equations on the wall. It is found that Gibbs oscillations have a negligible effect on the flow evolution allowing higher-order recovery of the accuracy on a Fourier basis by means of postprocessing. An advantage of coherent vortex simulations, on the other hand, is that the degrees of freedom of the flow computation can strongly be reduced. In this study, we quantify the possible reduction of degrees of freedom while keeping the accuracy. For an optimal convergence scenario the penalization parameter has to scale with the number of Fourier and wavelet modes. In addition, an implicit treatment of the Darcy drag term in the penalized Navier-Stokes equations is beneficial since this allows one to set the time step independent from the penalization parameter without additional computational or memory requirements.

The minimum-enstrophy principle for decaying 2D turbulence in circular domains

In several numerical and experimental studies [1, 2] on freely evolving or decaying two-dimensional (2D) turbulence on a square bounded domain it is observed that a flow, initially containing no net angular momentum (L), spontaneously acquires angular momentum by flow-wall interaction. From earlier work, by Li and Montgomery [3], it could be conjectured that on a circular domain with a no-slip boundary angular momentum production is absent. Decaying turbulence experiments in stratified fluids conducted a few years later by Maassen et al. [4] provided additional evidence supporting this conjecture. These observations have recently been confirmed by Schneider and Farge [5] for decaying 2D turbulence with substantially higher initial integral scale Reynolds numbers.

Fourier Spectral Solver for the Incompressible Navier-Stokes Equations with Volume-Penalization

In this study we use a fast Fourier spectral technique to simulate the Navier-Stokes equations with no-slip boundary conditions. This is enforced by an immersed boundary technique called volume-penalization. The approach has been justified by analytical proofs of the convergence with respect to the penalization parameter. However, the solution of the penalized Navier-Stokes equations is not smooth on the surface of the penalized volume. Therefore, it is not a prioriknown whether it is possible to actually perform accurate fast Fourier spectral computations. Convergence checks are reported using a recently revived, and unexpectedly difficult dipole-wall collision as a test case. It is found that Gibbs oscillations have a negligible effect on the flow evolution. This allows higher-order recovery of the accuracy on a Fourier basis by means of a post-processing procedure.