Rotating flow in a cylinder with a circular barrier on the bottom

The relative flow of a homogeneous, slightly viscous fluid in a rotating cylinder is induced by differential rotation of the bottom disk, on which a thin circular strip of small height is fixed. The axis of symmetry of the strip coincides with the rotation axis of the cylinder.\ud \ud At the strip a Stewartson layer exists which is partially free, partially attached to the strip. The structure of the Stewartson E1/4-layer E being the Ekman number) is not affected by the height of the strip, but the E1/3-layer problem has to be solved in the two separate intervals. The fact that both solutions do not match at the strip edge necessitates the presence of an intermediate region that exhibits some characteristic features of an Ekman layer

Energy spectra for decaying 2D turbulence in a bounded domain

New results are presented for the energy spectra of decaying 2D turbulence in a square container with no-slip walls for integral-scale Reynolds numbers up to 20 000. The one-dimensional energy spectra measured close to the walls reveal a k(-5/3) inertial range, instead of a k(-3) direct enstrophy cascade, due to the production of small-scale vorticity near no-slip boundaries. During the intermediate decay stage a k(-3) spectrum starts to emerge and the change in location of the injection scale of small-scale vorticity is explained in terms of an average boundary-layer thickness

Dissipation of kinetic energy in two-dimensional bounded flows

The role of no-slip boundaries as an enstrophy source in two-dimensional (2D) flows has been investigated for high Reynolds numbers. Numerical simulations of normal and oblique dipole-wall collisions are performed to investigate the dissipation of the kinetic energy E(t), and the evolution of the enstrophy [Omega] (t) and the palinstrophy P(t). It is shown for large Reynolds numbers that dE(t)/dt = ?2 [Omega] (t)/Re [[proportional]] 1/ [sqrt(Re)] instead of the familiar relation dE(t)/dt [[proportional]] 1/Re as found for 2D unbounded flow

A model for vortical plumes in rotating convection

In turbulent rotating convection a typical flow structuring in columnar vortices is observed. In the internal structure of these vortices several symmetries are approximately satisfied. A model for these columnar vortices is derived by prescribing these symmetries. The symmetry constraints are applied to the Navier¿Stokes equations with rotation in the Boussinesq approximation. It is found that the application of the symmetries results in a set of linearized equations. An investigation of the linearized equations leads to a model for the columnar vortices and a prediction for the heat flux (Nusselt number) that is very appropriate compared to the results from direct numerical simulations of the full governing equation

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

Recently, numerical studies revealed two different scaling regimes of the peak enstrophy Z and palinstrophy P during the collision of a dipole with a no-slip wall [Clercx and van Heijst, Phys. Rev. E 65, 066305, 2002]: Z ∝ Re0.8 and P ∝ Re2.25 for 5 × 102 ≤ Re ≤ 2 × 104 and Z ∝ Re0.5 and P ∝ Re1.5 for Re ≥ 2 × 104 (with Re based on the velocity and size of the dipole). A critical Reynolds number Rec(here, Rec ≈ 2 × 104) is identified below which the interaction time of the dipole with the boundary layer depends on the kinematic viscosity ν. The oscillating plate as a boundary-layer problem can then be used to mimick the vortex-wall interaction and the following scaling relations are obtained: Z ∝ Re^3/4, P ∝ Re^9/4, and dP/dt ∝ Re11/4 in agreement with the numerically obtained scaling laws. For Re ≥ Rec the interaction time of the dipole with the boundary layer becomes independent of the kinematic viscosity and, applying flat-plate boundary-layer theory, this yields: Z ∝ Re1/2 and P ∝ Re3/2

Quasi-two-dimensional turbulence in shallow fluid layers:The role of bottom friction and fluid layer depth

The numerical and experimental studies of the role of bottom friction and fluid layer depth on decaying quasi-two-dimensional turbulence enable several interesting observations. The numerical simulations showed that the evolution of vortex statistics of decaying 2D turbulence with bottom friction can be described by bottom-friction independent power laws. In the latter regime, the computed data showed a strong bottom-friction dependence and are strongly dominated by lateral diffusion.</p

Transition to chaos in a confined two-dimensional fluid flow

For a two-dimensional fluid in a square domain with no-slip walls, new direct numerical simulations reveal that the transition from steady to chaotic flow occurs through a sequence of various periodic and quasiperiodic flows, similar to the well-known Ruelle-Takens-Newhouse scenario. For all solutions beyond the ground state, the phenomenology is dominated by a domain-filling circulation cell, whereas the associated symmetry is reduced from the full symmetry group of the square to rotational symmetry over an angle . The results complement both laboratory experiments in containers with rigid walls and numerical simulations on double-periodic domains

Ekman decay of a dipolar vortex in a rotating fluid

The evolution of quasi-two-dimensional (2D) dipolar vortices over a flat bottom in a rotating fluid system is studied numerically, and the main results are experimentally verified. Our aim is to examine the dipole decay due to bottom friction effects. The numerical simulations are based on the 2D physical model derived by Zavala Sansón and van Heijst [J. Fluid Mech. 412, 75 (2000)], which contains nonlinear Ekman terms, associated with bottom friction, in the vorticity equation. In contrast, the conventional 2D model with bottom friction only retains a linear stretching term in the same equation. It is shown that the dipole trajectory is deflected towards the right (i.e., in the anticyclonic direction) when nonlinear Ekman terms are included. This effect is not observed in simulations based on the conventional model, where the dipole trajectory is a straight line. The basic reason for this behavior is the slower decay of the anticyclonic part of the dipole, with respect to the cyclonic one, due to nonlinear Ekman effects. Another important result is the exchange of fluid between the cyclonic part and the ambient, leaving a tail behind the dipole. By means of laboratory experiments in a rotating tank, these results are qualitatively verified

Some remarks on the role of boundary layers in decaying 2D turbulence in containers with no-slip walls

Direct numerical simulations of decaying two-dimensional (2D) turbulence inside a square container with no-slip boundaries have been carried out for Reynolds numbers up to 2000. The role of the boundary layers during the decay process has been illustrated with ensemble-averaged results for the power law behaviour of several characteristic properties of the coherent vortices which emerge during the decay of 2D turbulence. The evolution of the vortex density, the average vortex radius, the enstrophy and the vorticity extrema have been computed. An algebraic decay regime has been observed during the initial turbulent decay stage. The computed decay exponents disagree, however, with the exponents from the classical scaling theory for 2D decaying turbulence on an unbounded domain. This is attributed to the presence of no-slip boundaries. Additionally, the temporal evolution of the average boundary-layer thickness has been studied by computing the ensemble-averaged viscous stress and normal vorticity gradient near the no-slip boundaries. These computations reveal that d(t)Ct 0.4 and that the average boundary-layer thickness is proportional with Re20.5

Ekman decay of a dipolar vortex in a rotating fluid

The evolution of quasi-two-dimensional (2D) dipolar vortices over a flat bottom in a rotating fluid system is studied numerically, and the main results are experimentally verified. Our aim is to examine the dipole decay due to bottom friction effects. The numerical simulations are based on the 2D physical model derived by Zavala Sansón and van Heijst [J. Fluid Mech. 412, 75 (2000)], which contains nonlinear Ekman terms, associated with bottom friction, in the vorticity equation. In contrast, the conventional 2D model with bottom friction only retains a linear stretching term in the same equation. It is shown that the dipole trajectory is deflected towards the right (i.e., in the anticyclonic direction) when nonlinear Ekman terms are included. This effect is not observed in simulations based on the conventional model, where the dipole trajectory is a straight line. The basic reason for this behavior is the slower decay of the anticyclonic part of the dipole, with respect to the cyclonic one, due to nonlinear Ekman effects. Another important result is the exchange of fluid between the cyclonic part and the ambient, leaving a tail behind the dipole. By means of laboratory experiments in a rotating tank, these results are qualitatively verified