Hjh Herman Clercx

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.

Decaying two-dimensional turbulence in square containers with no-slip or stress-free boundaries

We report results of direct numerical simulations of decaying two-dimensional (2D) turbulence inside a square container with rigid boundaries. It is shown that the type of boundary condition (no-slip or stress-free) determines the flow evolution essentially. During the initial (0⩽t⩽0.2Re) and intermediate (0.2Re⩽t⩽3Re) stages of decaying 2D turbulence (t≅1 is comparable with an eddy turnover time, Re is the Reynolds number of the flow), the decay scenario for simulations with no-slip boundary conditions can be understood from turbulent spectral transfer and selective decay. A third mechanism can be recognized for t⩾3Re: A decay stage where diffusion dominates over nonlinear advection, i.e., spectral transfer is then absent in favor of self-similar decay. The present results show that at presently accessible Reynolds numbers and computation times, laboratory experiments cannot be accurately compared with quasi-stationary states from ideal maximum-entropy theories or with computed solutions of flows in cont...

Two-dimensional Navier-Stokes turbulence in bounded domains

In this review we will discuss recent experimental and numerical results of quasi-two-dimensional decaying and forced Navier–Stokes turbulence in bounded domains. We will give a concise overview of developments in two-dimensional turbulence research, with emphasis on the progress made during the past 10 years. The scope of this review concerns the self-organization of two-dimensional Navier–Stokes turbulence, the quasi-stationary final states in domains with no-slip boundaries, the role of the lateral no-slip walls on two-dimensional turbulence, and their role on the possible destabilization of domain-sized vortices. The overview of the laboratory experiments on quasi-two-dimensional turbulence is restricted to include only those carried out in thin electromagnetically forced shallow fluid layers and in stratified fluids. The effects of the quasi-two-dimensional character of the turbulence in the laboratory experiments will be discussed briefly. As a supplement, the main results from numerical simulations of forced and decaying two-dimensional turbulence in rectangular and circular domains, thus explicitly taking into account the lateral sidewalls, will be summarized and compared with the experimental observations.

Breakdown of large-scale circulation in turbulent rotating convection

Turbulent rotating convection in a cylinder is investigated both numerically and experimentally at Rayleigh number Ra=109 and Prandtl number σ=6.4. In this letter we discuss two topics: the breakdown under rotation of the domain-filling large-scale circulation (LSC) typical for confined convection, and the convective heat transfer through the fluid layer, expressed by the Nusselt number. The presence of the LSC is addressed for several rotation rates. For Rossby numbers Ro1.2 no LSC is found (the Rossby number indicates relative importance of buoyancy over rotation, hence small Ro indicates strong rotation). For larger Rossby numbers a precession of the LSC in anticyclonic direction (counter to the background rotation) is observed. It is shown that the heat transfer has a maximal value close to Ro=0.18 being about 15% larger than in the non-rotating case Ro=∞. Since the LSC is no longer present at this Rossby value we conclude that the peak heat transfer is independent of the LSC.

Intrinsic three-dimensionality in electromagnetically driven shallow flows

The canonical laboratory set-up to study two-dimensional turbulence is the electromagnetically driven shallow one- or two-layer fluid. Stereo-Particle-Image-Velocimetry measurements in such driven shallow flows revealed strong deviations from quasi–two-dimensionality, which are attributed to the inhomogeneity of the magnetic field and, in contrast to what has been believed so far, the impermeability condition at the bottom and top boundaries. These conjectures have been confirmed by numerical simulations of shallow flows without surface deformation, both in one- and two-layer fluids. The flow simulations reveal that the observed three-dimensional structures are in fact intrinsic to flows in shallow fluids because they do not result primarily from shear at a no-slip boundary: they are a direct consequence of the vertical confinement of the flow.

Self-organization of quasi-two-dimensional turbulence in stratified fluids in square and circular containers

Laboratory experiments on decaying quasi-2D (two-dimensional) turbulence have been performed in stratified fluids in both square and circular containers. The turbulence was generated by towing an array of vertical cylinders through the container, which was filled with either a two-layer or a linearly stratified fluid. By varying the grid configuration a different amount of angular momentum could be added to the initial flow. The evolution of the flow was visualized by 2D particle tracking velocimetry. The observed decay scenario has been investigated with emphasis on the evolution of the kinetic energy and the enstrophy of the horizontal flow, vortex statistics and the angular momentum of the flow. In particular it is shown that the experiments in both the square and the circular container support the observations from numerical simulations of decaying 2D turbulence in bounded domains with no-slip walls. Two striking examples are the experimental observation of the spontaneous spin-up phenomenon (in the square-container experiments) and the confirmation that the angular momentum of the flow in the circular-container experiment is better conserved than the total kinetic energy of the flow. The role of the initial nonzero net angular momentum on the decay of quasi-2D turbulence is investigated for both geometries and indications for an acceleration of the self-organization process are presented.

Two-dimensional turbulence in square and circular domains with no-slip walls

Several fascinating phenomena observed for 2D turbulence in bounded domains are discussed. The first part of this paper concerns a short overview of the non-trivial behaviour of freely evolving 2D turbulence in square domains with no-slip boundaries. In particular, the Reynolds number dependence of, and the influence of the initial conditions on spontaneous spin-up of the flow, which is characterised by a sudden increase of the absolute value of the angular momentum of the flow, is investigated in more detail. In a second set-up we have investigated forced 2D turbulence in circular containers with no-slip walls. A comparison with the double periodic case reveals that domain-filling structures, always observed in the double periodic cases, are being prevented from emerging. Wall-generated, small-scale structures are continuously injected into the interior of the domain, destroying larger structures and maintaining the turbulent flow field.

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.

Three-dimensional structure and decay properties of vortices in shallow fluid layers

Recently, several laboratory experiments on vortex dynamics and quasi-two-dimensional turbulence have been performed in thin (stratified) fluid layers. Commonly, it is tacitly assumed that vertical motions, giving rise to a three-dimensional character of the flow, are inhibited by the limited vertical dimension. However, shallow water flows, which are vertically bounded by a no-slip bottom and a free surface, necessarily possess a three-dimensional structure due to the shear in the vertical direction. This shear may lead to significant secondary circulations. In this paper, the three-dimensional (3D) structure and the decay properties of vortices in shallow layers of fluid, both homogeneous and stratified, have been studied in detail by 3D direct numerical simulations. The quasi-two-dimensionality of these flows is an important issue if one is interested in a comparison of experiments of this type with purely two-dimensional theoretical models. The influence of several flow parameters, like the depth of the fluid and the Reynolds number, has been investigated. In general, it can be concluded that the flow loses its two-dimensional character for larger fluid depth and larger Reynolds number. Furthermore, it is possible to construct a regime diagram that allows the assessment of the parameter regime, where the flow can be considered as quasi-two-dimensional. It is found that the presence of secondary circulations within a planar vortex flow results in a deformation of the radial profile of axial vorticity. In the limiting case of quasi-two-dimensional flow, the vorticity profiles can be scaled according to a simple diffusion model. In a two-layer stratified system, the decay is reduced and three-dimensional motions are significantly inhibited compared to the corresponding flows in a homogeneous layer.

Dynamics of pancake-like vortices in a stratified fluid : experiments, model and numerical simulations

The dynamics and the three-dimensional structure of vortices in a linearly stratified, non-rotating fluid are investigated by means of laboratory experiments, an analytical model and through numerical simulations. The laboratory experiments show that such vortices have a thin pancake-like appearance. Due to vertical diffusion of momentum the strength of these vortices decreases rapidly and their thickness increases in time. Also it is found that inside a vortex the linear ambient density profile becomes perturbed, resulting in a local steepening of the density gradient. Based on the assumption of a quasi-two-dimensional axisymmetric flow (i.e. with zero vertical velocity) a model is derived from the Boussinesq equations that illustrates that the velocity field of the vortex decays due to diffusion and that the vortex is in so-called cyclostrophic balance. This means that the centrifugal force inside the vortex is balanced by a pressure gradient force that is provided by a perturbation of the density profile in a way that is observed in the experiments. Numerical simulations are performed, using a finite difference method in a cylindrical coordinate system. As an initial condition the three-dimensional vorticity and density structure of the vortex, found with the diffusion model, are used. The influence of the Froude number, Schmidt number and Reynolds number, as well as the initial thickness of the vortex, on the evolution of the flow are investigated. For a specific combination of flow parameters it is found that during the decay of the vortex the relaxation of the isopycnals back to their undisturbed positions can result in a stretching of the vortex. Potential energy of the perturbed isopycnals is then converted into kinetic energy of the vortex. However, when the stratification is strong enough (i.e. for small Froude numbers), the evolution of the vortex can be described almost perfectly by the diffusion model alone.