W Werner Kramer

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.

Vorticity dynamics of a dipole colliding with a no-slip wall

The active role of vorticity in the collision of a Lamb-like dipole with a no-slip wall is studied for Re values ranging between 625 and 20000. The initial approach of the dipole does not differ from the stress-free case or from a point-vortex model that incorporates the diffusive growth of the dipole core. When closer to the wall, the detachment and subsequent roll-up of the boundary layer leads to a viscous rebound, as was observed by Orlandi Phys. Fluids A 2, 1429 1990 in numerical simulations with Re up to 3200. The net translation of the vortex core along the wall is strongly reduced due to the cycloid-like trajectory. For Re2500 wall-generated vorticity is wrapped around the separate dipole halves, which hence become partially shielded monopoles. For ReO104, however, a shear instability causes the roll-up of the boundary layer before it is detached from the wall. This leads to the formation of a number of small-scale vortices, between which intensive, narrow eruptions of boundary-generated vorticity occur. Quantitative measures are given for the influx of vorticity at the wall and the consequent increase of boundary layer vorticity and enstrophy.

β-plane turbulence in a basin with no-slip boundaries

On a domain enclosed by no-slip boundaries, two-dimensional, geostrophic flows have been studied by numerical simulations of the Navier-Stokes equation with the -plane approximation at intermediate Reynolds numbers and a range of values for . The effect causes a refinement of the flow structures, and the presence of basin modes has been revealed by means of frequency spectra. The presence and apparent stability of basin modes on a domain enclosed by no-slip boundaries is a rather surprising observation, because these modes are solutions of the inviscid flow equations on a bounded domain with free-slip boundaries . To understand the persistence of these basin modes, the viscous boundary layers near the no-slip walls have been investigated. The mean flow in forced simulations shows a zonal band structure, much unlike the regular Fofonoff-like solution observed when free-slip boundary conditions are used.

Turbulent oscillating channel flow subjected to a free-surface stress.

The channel flow subjected to a wind stress at the free surface and an oscillating pressure gradient is investigated using large-eddy simulations. The orientation of the surface stress is parallel with the oscillating pressure gradient and a purely pulsating mean flow develops. The Reynolds number is typically Re

On the large-scale structure and spectral dynamics of two-dimensional turbulence in a periodic channel

_\omega

Turbulent oscillating channel flow subjected to wind stress

= 10^6 and the Keulegan–Carpenter number—the ratio between the oscillation period and advection time scale—is KC=80. Results compare favorably to the data from direct numerical simulations obtained over a single period. A slowly pulsating mean flow occurs with the turbulent flow essentially being statistically steady. Logarithmic boundary layers are present at both the bottom wall and the free surface. Turbulent streaks are observed in the bottom and free-surface layer. The viscous sublayer below the free surface is, however, much thinner. This observation is verified by simulations we performed for a purely wind-driven channel flow. For the oscillating flow, additional low-speed splats localized regions of upwelling occur at the free surface when the mean velocity and stress are in the same direction.

The wind-driven turbulent oscillating channel flow subjected to a stable stratification

This paper reports on a numerical study of forced two-dimensional turbulence in a periodic channel with flat no-slip walls. Since corners or curved domain boundaries, which are met in the standard rectangular, square, or circular geometries, are absent in this geometry, the (statistical) analysis of the flow is substantially simplified. Moreover, the use of a standard Fourier–Chebyshev pseudospectral algorithm enables high integral-scale Reynolds number simulations. The paper focuses on (i) the influence of the aspect ratio of the channel and (ii) the integral-scale Reynolds number on the large-scale self-organization of the flow. It is shown that for small aspect ratios, a unidirectional flow spontaneously emerges, notably in the absence of a pressure gradient in the longitudinal direction. For larger aspect ratios, the flow tends to organize into an array of counter-rotating vortical structures. The computed energy and enstrophy spectra provide further evidence that the injection of small-scale vorticity at the no-slip walls modify the inertial-range scaling. Additionally, the quasistationary final state of decaying turbulence is interpreted in terms of the Stokes modes of a viscous channel flow. Finally, the transport of a passive tracer material is studied with emphasis on the role of the large-scale flow on the dispersion and the spectral properties of the tracer variance in the presence of no-slip boundaries.

On the Reynolds number scaling of vorticity production at no-slip walls during vortex-wall collisions

The Westerschelde estuary in the Netherlands is characterized by a strong tidal driven flow with typical velocities in the range of 0.2 to 1 m/s. In addition to the tides the wind (5 m/s) exerts a stress at the free surface driving the upper fluid layers. To investigate this flow we performed resolved Large Eddy Simulations for a generic configuration: a periodic 3D channel domain with an oscillating pressure gradient over the channel length, a fixed wind stress at the flat free-surface and no-slip conditions at the bottom boundary. The wind stress drives a unidirectional flow, which in combination with the oscillating (tidal) part generates a strong shear-layer near the free surface. In this shear layer and in the shear layer near the no-slip bottom turbulence is strongly enhanced. Subsequently, the turbulence is spreading over a larger part of the domain, where it has a large influence on the dispersion of particles. The study is extended by including the effects of rotation and stratification.

Structure-function scaling of bounded two-dimensional turbulence

The tide in estuaries drives a turbulent slowly-oscillating flow. To investigate the effects of wind and density stratification on such flows, we have performed Large Eddy Simulations in a cubic fluid column of depth h. The velocity field u is obtained from the Navier-Stokes equation under the Boussinesq approximation,