Session A9: General Fluid Dynamics: Rotating Flows

Giant Hydrodynamic Fluctuations Due to Coriolis Redistribution of Turbulent Kinetic Energy

The influence of chemical reaction on turbulent mixing of a chemically reactive constituent in a rotating channel flow indicates that the transverse transport of the reactive constituent is mitigated by a coupling between the shear component of the Reynolds stress and the longitudinal component of the mean flux of the reactive constituent. In the region of zero intrinsic vorticity, the dispersion coefficient in the cross flow direction is significantly larger than the dispersion coefficient in the spanwise direction. The dispersion coefficient in the longitudinal direction is relatively small. Koppula, K.S., A. B\'{e}nard, and C. A. Petty, 2011, ``Turbulent Energy Redistribution in Spanwise Rotating Channel Flows'', Ind. Eng. Chem. Res., 50 (15), 8905-8916.   

Suppressing Taylor vortices in a Taylor--Couette flow system with free surface

Taylor--Couette flows have been extensively investigated due to their many industrial applications, such as catalytic reactors, electrochemistry, photochemistry, biochemistry, and polymerization. Mass transfer applications include extraction, tangential filtration, crystallization, and dialysis. A 3D study is carried out to simulate a Taylor--Couette flow with a rotating and pulsating inner cylinder. We utilize FLUENT to simulate the incompressible flow with a free surface. The study reveals that flow structuring is initiated with the development of an Ekman vortex at low Taylor number, Ta $=0.01$. For all encountered flow regimes, the Taylor vortices are systematically inhibited by the pulsatile motion of the inner cylinder. A spectral analysis shows that this pulsatile motion causes a rapid decay of the free surface oscillations, from a periodic wavy movement to a chaotic one, then to a fully turbulent motion. This degenerative free surface behavior is interpreted as the underlying mechanism responsible for the inhibition of the Taylor vortices.   

A slowly rotating impeller in a rapidly rotating fluid.

We characterize the two-dimensionalization process in the turbulent flow produced by an impeller rotating at a rate $\omega $ in a fluid rotating at a rate $\Omega $ around the same axis for Rossby number Ro $=\omega $ / $\Omega $ down to 0.01. The flow can be described as the superposition of a large-scale vertically invariant global rotation and small-scale shear layers detached from the impeller blades. As Ro decreases, the large-scale flow is subjected to azimuthal modulations. In this regime, the shear layers can be described in terms of wakes of inertial waves traveling with the blades, originating from the velocity difference between the non-axisymmetric large-scale flow and the blade rotation. The wakes are well defined and stable at low Rossby number, but they become disordered and interact nonlinearly at Ro of order of 1.   

The generalized Onsager model and DSMC simulations of high-speed rotating flows with product and waste baffles

The generalized Onsager model for the radial boundary layer and of the generalized Carrier-Maslen model for the axial boundary layer in a high-speed rotating cylinder ((S. Pradhan {\&} V. Kumaran, J. Fluid Mech., 2011, vol. 686, pp. 109-159); (V. Kumaran {\&} S. Pradhan, J. Fluid Mech., 2014, vol. 753, pp. 307-359)), are extended to a multiply connected domain, created by the product and waste baffles. For a single component gas, the analytical solutions are obtained for the sixth-order generalized Onsager equations for the master potential, and for the fourth-order generalized Carrier-Maslen equation for the velocity potential. In both cases, the equations are linearized in the perturbation to the base flow, which is a solid-body rotation. An explicit expression for the baffle stream function is obtained using the boundary layer solutions. These solutions are compared with direct simulation Monte Carlo (DSMC) simulations and found excellent agreement between the analysis and simulations, to within 15{\%}, provided the wall-slip in both the flow velocity and temperature are incorporated in the analytical solutions.   

Turbulent strength in ultimate Taylor-Couette turbulence

We provide the local scaling of the Taylor-Reynolds number ($\text{Re}_\lambda$) as a function of driving strength (Ta), in the ultimate regime of Taylor-Couette flow for the inner cylinder rotation case. The calculation is done via local flow measurements using Particle Image Velocimetry (PIV) to reconstruct the velocity fields. We approximate the value of the local dissipation rate $\epsilon(r)$ using the scaling for the second order structure functions in the longitudinal and transversal directions within the inertial regime where Taylor's hypothesis is not invoked. We find an effective local scaling of $\langle \epsilon(r) \rangle_r/(\nu^{3}d^{-4})\sim \text{Ta}^{1.4}$, which is the same as the global dissipation rate obtained from both torque measurements and Direct Numerical Simulations (DNS). Additionally, we calculate the Kolmogorov length scale and find $\langle \eta(r) \rangle_r /d \sim \text{Ta}^{-0.35}$. The turbulence intensity is also calculated and it is found to scale with the driving strength as $i_\theta \sim \text{Ta}^{-0.056}$. Finally, with both the local dissipation rate and the local fluctuations available we find that the Taylor-Reynolds number scales as Re$_\lambda\sim \text{Ta}^{0.18}$.   

Critical-band vortex in a precessing sphere

We consider the motion of an incompressible viscous fluid in a rotating sphere with strong precession, where the spin and precession axes are assumed to be orthogonal to each other. By an asymptotic analysis we determine the structure of the steady flow in the entire sphere in the leading order of the asymptotic expansion. It is found that the boundary layer is developed on the whole spherical surface, outside of which the flow is stationary in the leading order. The boundary-layer approximation breaks down on a great circle perpendicular to the precession axis. A partial differential equation which describes the velocity field in the vicinity of this great circle, called the critical band, is derived theoretically and solved numerically to find a pair of vortices localized in the critical band. In the meeting we present the three-dimensional structure of these vortices with streamlines as well as streamsurfaces.   

Rotational Flow of Nonlinear Drilling Mud

To analyze the drilling process, the pseudoplastic flow between coaxial cylinders is investigated. Here, the inner cylinder is assumed to rotate and, at the same time, slide along its axis. A numerical scheme based on the spectral method is used to derive a low-order dynamical system from the conservation of mass and momentum equations under mixed boundary conditions. It is found that the Azimuthal stress develops far greater than other stress components. All stress components increase as pseudoplasticity is decreased. The flow loses its stability to the vortex structure at a critical Taylor number. The emergence of the vortices corresponds to the onset of a supercritical bifurcation. The Taylor vortices, in turn, lose their stability as the Taylor number reaches a second critical number corresponding to the onset of a Hopf bifurcation. The rotational and axial velocities corresponding to the optimum drilling conditions are evaluated. Furthermore, complete stress and viscosity maps are presented for different scenarios in the flow regime.   

Oscillatory Convection in Rotating Liquid Metals

We have performed laboratory experiments in a aspect ratio $\Gamma \simeq 2$ cylinder using liquid gallium ($Pr \simeq 0.023$) as the working fluid. The Ekman number varies from $E = 4 \times 10^{-5}$ to $4 \times 10^{-6}$ and the Rayleigh number varies from $Ra =3 \times 10^5$ to $2 \times 10^7$. Using heat transfer and temperature measurements within the fluid, we characterize the different styles of low $Pr$ rotating convective flow. The convection threshold is first overcome in the form of a container scale inertial oscillatory mode. At stronger forcing, wall-localized modes develop, coexisting with the inertial oscillatory modes in the bulk. When the strength of the buoyancy increases further, the bulk flow becomes turbulent while the wall modes remain. Our results imply that rotating convective flows in liquid metals do not develop in the form of quasi-steady columns, as in $Pr \simeq 1$ planetary and stellar dynamo models, but in the form of oscillatory motions. Therefore, convection driven dynamo action in low $Pr$ fluids can differ substantively than that occurring in typical $Pr \simeq 1$ numerical models. Our results also suggest that low wavenumber, wall modes may be dynamically and observationally important in liquid metal dynamo systems.   

Linear stability analysis of natural convection in an inclined rotating cavity.

The linear stability analysis of the natural convection in an inclined rectangular cavity with rotation is presented. The critical parameters for the onset of longitudinal rolls are obtained by solving the stability equations with the Tau-Chebyshev spectral method. We report under what conditions of the inclination angle, Rayleigh and Taylor numbers, the onset of longitudinal rolls appears. The rectangular cavity with a small aspect ratio (depth/length) is heated from below, cooled from above and thermally isolated at the rest of the boundaries. The rotation axis is orthogonal to the hot and cold surfaces and passes through the center of these surfaces, while the inclination angle varies from 0 to $90\,^{\circ}$ . Based on the results of the linear stability analysis, it was possible to perform non linear, three dimensional numerical simulations based on a spectral element method for a Boussinesq fluid. Our preliminary results show the effect of the rotation rate and the tilted angle on the convective patterns, temperature distribution and heat transfer rate.