Session E26: Rotating Flow II

Spatial variation of angular momentum transport in turbulent spherical Couette flow.

Turbulent shear flow between two concentric rotating spheres of radius ratio $\eta=0.35$ is studied up to 56 million Reynolds number. The dimensionless differential rotation as expressed by the Rossby number $Ro=(\Omega_i-\Omega_o)/\Omega_o$ is an important parameter in setting the observed large scale flow state and the angular momentum flux through the system at a given Reynolds number $Re=(\Omega_i-\Omega_o)L^2/\nu$. Here we consider the case where the inner sphere super-rotates in the range $0 [Preview Abstract]

Experimental Quantification of Inverse Energy Cascade in Deep Rotating Fluid

We present quantitative experimental measurements of an inverse energy cascade in deep rotating turbulence. Experiments were performed in a rotating cylindrical water tank configuration with energy injection at a large wavenumber, k$_{i}$. For k$<$k$_{i}$ the steady state energy spectrum is very well described by $E_k =C\varepsilon ^{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}k^{{-5} \mathord{\left/ {\vphantom {{-5} 3}} \right. \kern-\nulldelimiterspace} 3},$ the 2D energy spectrum. The temporal evolution of energy spectra towards steady states was measured. We identify a ``front'' which propagates from large to small wave numbers, behind which the spectrum attains its steady state value. The propagation of these fronts was measured for different energy injection rates and rotation frequencies. These measurements are in good quantitative agreement with the predictions for 2D turbulence.   

Taylor-Columns in Deep Water: Testing the Limits of the Taylor-Proudman Theorem

The formation of Taylor-Columns is one of the familiar phenomena observed in flows where strong background rotation is present. The current investigation considers the Taylor-Column generated by a differentially rotating disk in a rotating fluid, a geometry similar to the classic Stewartson-layer problem. Experimental work was performed on the large turntable at the University of Warwick (overall height 5.7m and 1.4m diameter). It offers a water depth of 2m below the disk, allowing for a much longer Taylor-Column to be observed than in previous experimental studies. An automated, traversing Stereo-PIV system was developed to visualize the flow, allowing data-acquisition at different heights in the tank. The experimental results show a z-dependence of the angular velocity in the interior of the Taylor-Column for positive differential rotation, but not for negative differential rotation. These results contradict both the Taylor-Proudman theorem and numerical predictions.   

Experimental modelling of bistability in mid-latitude atmospheric jets

Bistability, a striking property of some climate features, can be obtained in purely hydrodynamic systems in turbulent regimes and can be explained in terms of statistical mechanics. Here, we report laboratory experiments reproducing mid-latitude atmospheric jets in an annular rotating tank. The jet is influenced by a non-axisymmetric bottom topography interfering with the natural wavy pattern of the jet, which is related to a barotropic instability. Two states emerge from this interaction: a wave `blocked' with respect to the topography, and a freely propagating wave. These two states are obtained with the same control parameters, the chosen state depending on the history of the system. No spontaneous transition between the two states is observed, in spite of the strong turbulent fluctuations. Finally, the response of the system to external noise will be presented.   

Vortex formation through inertial wave focusing

We present a novel experimental and numerical study on the formation of columnar vortical structures by inertial waves in a rotating fluid. Two inertial-wave cones are generated by a vertically oscillating torus in a fluid in solid body rotation At the tip of the cones, there is a singular point towards which the energy of the waves gets focused. The particularity of this configuration, as compared to those of previous experiments (e.g. oscillating sphere or disc), is that the singular point's position within the fluid leads to complex non-linear wave interaction, which may lead to the formation of a localized vortex that expands in the vertical in the form of a Taylor column. Using detailed PIV measurements we consider the flow evolution from the localized wave overturning motion to the Taylor column formation as well as the inertial wave dynamics during this process, The results are discussed in the context of turbulence in rotating fluids.   

Secondary bifurcation of mixed-cross-spirals (MCS) connecting different travelling wave solutions

We investigated numerically in the Taylor-Couette system secondarily bifurcating structures connecting different helical travelling wave solutions with different azimuthal wave numbers, realized by so-called mixed-cross-spirals (MCS), by solving the full Navier-Stokes equations with a combination of a finite difference and a Galerkin method for fixed axial periodicity lengths. Dynamics, stability, and bifurcation behaviour for counter rotating as well as co-rotating cylinders are presented. These secondarily forward bifurcating, stationary flow states - MCS - solutions can be seen as nonlinear superpositions of the involved pure spiral solutions. E.g for a L3R5-MCS. Thereby, the contribution of the respective spiral component (here a L3-SPI and a R5-SPI to the entire MCS varies continuously with the control parameters. In that notation, the well-studied cross-spirals (CSPI) represent a special case of MCS as they consist of two {\em mirror-symmetric} spiral components. Moreover the ribbon (RIB) solution (axially standing waves) can be seen as an CSPI with equal mode amplitudes (here 3-RIB). Thereby these MCS can appear in two different ways. They can appear as a bypass solution connecting the {\em same} SPI branch or they can appear as an interim solution connecting {\em different} SPI branches.   

Computational simulations of rotating-disk induced flows in circular cylinders

Numerical solutions of the axisymmetric Navier-Stokes equations are presented that illustrate the flow regime where vortex breakdown occurs. The parameters of the problem are the aspect ratio of the cylindrical container, $A = H/R$, where $H$ is the height of the cylinder and $R$ is the radius of the cylinder, the radius ratio of the radius of the disk, $R_d/R$, where $R_d$ is the radius of the disk, and the dimensionless rotation speed, viz., the Reynolds number, $Re = R^2 \Omega/\nu$, where $\Omega$ is the angular velocity of the disk and $\nu$ is the kinematic viscosity of the Newtonian fluid within the container. Comparisons with the now classical lid-driven flows are made. Flow visualizations of vortex breakdown in a rotating disk driven flow observed in a relatively crude experimental apparatus is presented to illustrate the robustness of this phenomenon. Comparisons with numerical predictions illustrate that this phenomenon is predominantly an axisymmetric phenomeon. The flow visualizations illustrate interesting three- dimensional features that need further investigation.   

Optimal Taylor-Couette turbulence

Strongly turbulent Taylor-Couette flow with independently rotating inner and outer cylinder with a radius ratio of $\eta = 0.716$ is experimentally analysed. From global torque measurements, the maximum in the angular velocity transport from the inner to the outer cylinder is found at slight counter-rotation, namely at an angular velocity ratio of $a_{opt} = - (\omega_o/\omega_i)_{opt} \approx 0.37$. This value is theoretically interpreted and predictions for general $\eta$ are made. With the help of laser Doppler anemometry, we in addition provide angular velocity profiles. The ratio $a_{opt} \approx 0.37$ is distinguished by zero angular velocity gradient $\partial \left \langle \omega \right \rangle_t /\partial r = 0$ in the bulk. For stronger counter- rotation $|\omega_o| > 0.37 \omega_i$ the probability distribution function of the bulk angular velocity becomes bi-modal, reflecting intermittent bursts of turbulent structures beyond the neutral line into the outer flow domain, which otherwise is stabilized by the counter-rotating outer cylinder.   

Non-Oberbeck-Boussinesq effects in rotating Rayleigh-B\'enard convection

We present results from Direct Numerical Simulations of rotating Rayleigh-B\'enard convection in a cylindrical cell with unity aspect ratio for water with a Prandtl number of $Pr = 4.38$ and glycerol with $Pr = 2547.9$ with a special focus on temperature dependent material properties and thus not within the framework of the conventionally used Oberbeck-Boussinesq (OB) approximation. The generated turbulent flow fields are analysed with respect to the deviations from the OB cases. For the non-Oberbeck-Boussinesq cases we obtain a breakdown of the top-bottom symmetry, that is different boundary layer thicknesses, asymmetric velocity flow patterns and modified mean temperature profiles including an increase of the centre temperature and in the case of rotation a diminished temperature gradient within the bulk. Furthermore we find a slightly different scaling of the Nusselt number with the Rayleigh as well as with the Rossby number.   

Heat transport by liquid metal convection with and without rotation

Convection in liquid metals can be quite different than in other fluids such as air or water, due to their small Prandtl numbers. Studying such systems can therefore give us a different perspective on convective turbulence. Furthermore, rotating convection in liquid metal is responsible for the generation magnetic fields on Earth and other planets. We present results from Rayleigh-B\'enard convection experiments in liquid gallium with and without rotation. The key measurement of interest is heat transport, and heat transport by liquid metal is contrasted with that by water in a similar system.