Magnetic and Acoustic Investigations of Turbulent Spherical Couette Flow

Title of dissertation: MAGNETIC AND ACOUSTIC INVESTIGATIONS OF TURBULENT SPHERICAL COUETTE FLOW Matthew M. Adams, Doctor of Philosophy, 2016 Dissertation directed by: Professor Daniel Lathrop Department of Physics This dissertation describes experiments in spherical Couette devices, using both gas and liquid sodium. The experimental geometry is motivated by the Earth's outer core, the seat of the geodynamo, and consists of an outer spherical shell and an inner sphere, both of which can be rotated independently to drive a shear flow in the fluid lying between them. In the case of experiments with liquid sodium, we apply DC axial magnetic fields, with a dominant dipole or quadrupole component, to the system. We measure the magnetic field induced by the flow of liquid sodium using an external array of Hall effect magnetic field probes, as well as two probes inserted into the fluid volume. This gives information about possible velocity patterns present, and we extend previous work categorizing flow states, noting further information that can be extracted from the induced field measurements. The limitations due to a lack of direct velocity measurements prompted us to work on developing the technique of using acoustic modes to measure zonal flows. Using gas as the working fluid in our 60~cm diameter spherical Couette experiment, we identified acoustic modes of the container, and obtained excellent agreement with theoretical predictions. For the case of uniform rotation of the system, we compared the acoustic mode frequency splittings with theoretical predictions for solid body flow, and obtained excellent agreement. This gave us confidence in extending this work to the case of differential rotation, with a turbulent flow state. Using the measured splittings for this case, our colleagues performed an inversion to infer the pattern of zonal velocities within the flow, the first such inversion in a rotating laboratory experiment. This technique holds promise for use in liquid sodium experiments, for which zonal flow measurements have historically been challenging.

Numerical study of bifurcation solutions of spherical Taylor-Couette flow

The steady bifurcation flows in a spherical gap (gap ratio sigma=0.18) with rotating inner and stationary outer spheres are simulated numerically for Re(c1)less than or equal to Re less than or equal to 1 500 by solving steady axisymmetric incompressible Navier-Stokes equations using a finite difference method. The simulation shows that there exist two steady stable flows with 1 or 2 vortices per hemisphere for 775 less than or equal to Re less than or equal to 1 220 and three steady stable flows with 0, 1, or 2 vortices for 1 220flows at the same Reynolds number is related with different initial conditions which on be generated by different accelerations of the inner sphere. Generation of zero-or two-vortex flow depends mainly on the acceleratio n, but that of one-vortex flow also depends on the perturbation breaking the equatorial symmetry. The mechanism of development of a saddle point in the meridional plane at higher Re number and its role in the formation of two-vortex flow are analyzed.

Inverse Spectral Methods in Acoustic Normal Mode Velocimetry of High Reynolds Number Spherical Couette Flows

Experimental geophysical fluid dynamics often examines regimes of fluid flow infeasible for computer simulations. Velocimetry of zonal flows present in these regimes brings many challenges when the fluid is opaque and vigorously rotating; spherical Couette flows with molten metals are one such example. The fine structure of the acoustic spectrum can be related to the fluid’s velocity field, and inverse spectral methods can be used to predict and, with sufficient acoustic data, mathematically reconstruct the velocity field. The methods are to some extent inherited from helioseismology. This work develops a Finite Element Method suitable to matching the geometries of experimental setups, as well as modelling the acoustics based on that geometry and zonal flows therein. As an application, this work uses the 60-cm setup Dynamo 3.5 at the University of Maryland Nonlinear Dynamics Laboratory. Additionally, results obtained using a small acoustic data set from recent experiments in air are provided.

Heat transfer in plane Couette flow of a rarefied gas using Bhatnagar-Gross-Krook model

Using the linearized BGK model and the method of moments of half-range distribution functions the temperature jumps at two plates are determined, and it is found that the results are in fair agreement with those of Gross and Ziering, and Ziering.

Heat transfer in rarefied couette flow on the basis of discrete ordinate method

The method of discrete ordinates, in conjunction with the modified "half-range" quadrature, is applied to the study of heat transfer in rarefied gas flows. Analytic expressions for the reduced distribution function, the macroscopic temperature profile and the heat flux are obtained in the general n-th approximation. The results for temperature profile and heat flux are in sufficiently good accord both with the results of the previous investigators and with the experimental data.

Non-linear couette flow with heat transfer using the B-G-K model

In recent years a large number of investigators have devoted their efforts to the study of flow and heat transfer in rarefied gases, using the BGK [1] model or the Boltzmann kinetic equation. The velocity moment method which is based on an expansion of the distribution function as a series of orthogonal polynomials in velocity space, has been applied to the linearized problem of shear flow and heat transfer by Mott-Smith [2] and Wang Chang and Uhlenbeck [3]. Gross, Jackson and Ziering [4] have improved greatly upon this technique by expressing the distribution function in terms of half-range functions and it is this feature which leads to the rapid convergence of the method. The full-range moments method [4] has been modified by Bhatnagar [5] and then applied to plane Couette flow using the B-G-K model. Bhatnagar and Srivastava [6] have also studied the heat transfer in plane Couette flow using the linearized B-G-K equation. On the other hand, the half-range moments method has been applied by Gross and Ziering [7] to heat transfer between parallel plates using Boltzmann equation for hard sphere molecules and by Ziering [83 to shear and heat flow using Maxwell molecular model. Along different lines, a moment method has been applied by Lees and Liu [9] to heat transfer in Couette flow using Maxwell's transfer equation rather than the Boltzmann equation for distribution function. An iteration method has been developed by Willis [10] to apply it to non-linear heat transfer problems using the B-G-K model, with the zeroth iteration being taken as the solution of the collisionless kinetic equation. Krook [11] has also used the moment method to formulate the equivalent continuum equations and has pointed out that if the effects of molecular collisions are described by the B-G-K model, exact numerical solutions of many rarefied gas-dynamic problems can be obtained. Recently, these numerical solutions have been obtained by Anderson [12] for the non-linear heat transfer in Couette flow,

The effect of boundaries on the plane Couette flow of granular materials: a bifurcation analysis

The tendency of granular materials in rapid shear flow to form non-uniform structures is well documented in the literature. Through a linear stability analysis of the solution of continuum equations for rapid shear flow of a uniform granular material, performed by Savage (1992) and others subsequently, it has been shown that an infinite plane shearing motion may be unstable in the Lyapunov sense, provided the mean volume fraction of particles is above a critical value. This instability leads to the formation of alternating layers of high and low particle concentrations oriented parallel to the plane of shear. Computer simulations, on the other hand, reveal that non-uniform structures are possible even when the mean volume fraction of particles is small. In the present study, we have examined the structure of fully developed layered solutions, by making use of numerical continuation techniques and bifurcation theory. It is shown that the continuum equations do predict the existence of layered solutions of high amplitude even when the uniform state is linearly stable. An analysis of the effect of bounding walls on the bifurcation structure reveals that the nature of the wall boundary conditions plays a pivotal role in selecting that branch of non-uniform solutions which emerges as the primary branch. This demonstrates unequivocally that the results on the stability of bounded shear how of granular materials presented previously by Wang et al. (1996) are, in general, based on erroneous base states.

The effect of boundaries on the plane Couette flow of granular materials: a bifurcation analysis

The tendency of granular materials in rapid shear ow to form non-uniform structures is well documented in the literature. Through a linear stability analysis of the solution of continuum equations for rapid shear flow of a uniform granular material, performed by Savage (1992) and others subsequently, it has been shown that an infinite plane shearing motion may be unstable in the Lyapunov sense, provided the mean volume fraction of particles is above a critical value. This instability leads to the formation of alternating layers of high and low particle concentrations oriented parallel to the plane of shear. Computer simulations, on the other hand, reveal that non-uniform structures are possible even when the mean volume fraction of particles is small. In the present study, we have examined the structure of fully developed layered solutions, by making use of numerical continuation techniques and bifurcation theory. It is shown that the continuum equations do predict the existence of layered solutions of high amplitude even when the uniform state is linearly stable. An analysis of the effect of bounding walls on the bifurcation structure reveals that the nature of the wall boundary conditions plays a pivotal role in selecting that branch of non-uniform solutions which emerges as the primary branch. This demonstrates unequivocally that the results on the stability of bounded shear flow of granular materials presented previously by Wang et al. (1996) are, in general, based on erroneous base states.

Linear stability, transient growth and the role of viscosity stratification in compressible Couette flow

Linear stability and the nonmodal transient energy growth in compressible plane Couette flow are investigated for two prototype mean flows: (a) the uniform shear flow with constant viscosity, and (b) the nonuniform shear flow with stratified viscosity. Both mean flows are linearly unstable for a range of supersonic Mach numbers (M). For a given M, the critical Reynolds number (Re) is significantly smaller for the uniform shear flow than its nonuniform shear counterpart; for a given Re, the dominant instability (over all streamwise wave numbers, α) of each mean flow belongs to different modes for a range of supersonic M. An analysis of perturbation energy reveals that the instability is primarily caused by an excess transfer of energy from mean flow to perturbations. It is shown that the energy transfer from mean flow occurs close to the moving top wall for “mode I” instability, whereas it occurs in the bulk of the flow domain for “mode II.” For the nonmodal transient growth analysis, it is shown that the maximum temporal amplification of perturbation energy, Gmax, and the corresponding time scale are significantly larger for the uniform shear case compared to those for its nonuniform counterpart. For α=0, the linear stability operator can be partitioned into L∼L̅ +Re2 Lp, and the Re-dependent operator Lp is shown to have a negligibly small contribution to perturbation energy which is responsible for the validity of the well-known quadratic-scaling law in uniform shear flow: G(t∕Re)∼Re2. In contrast, the dominance of Lp is responsible for the invalidity of this scaling law in nonuniform shear flow. An inviscid reduced model, based on Ellingsen-Palm-type solution, has been shown to capture all salient features of transient energy growth of full viscous problem. For both modal and nonmodal instability, it is shown that the viscosity stratification of the underlying mean flow would lead to a delayed transition in compressible Couette flow.

Numerical simulation of axisymmetric unsteady incompressible flow by a vorticity-velocity method

A new numerical method for solving the axisymmetric unsteady incompressible Navier-Stokes equations using vorticity-velocity variables and a staggered grid is presented. The solution is advanced in time with an explicit two-stage Runge-Kutta method. At each stage a vector Poisson equation for velocity is solved. Some important aspects of staggering of the variable location, divergence-free correction to the velocity held by means of a suitably chosen scalar potential and numerical treatment of the vorticity boundary condition are examined. The axisymmetric spherical Couette flow between two concentric differentially rotating spheres is computed as an initial value problem. Comparison of the computational results using a staggered grid with those using a non-staggered grid shows that the staggered grid is superior to the non-staggered grid. The computed scenario of the transition from zero-vortex to two-vortex flow at moderate Reynolds number agrees with that simulated using a pseudospectral method, thus validating the temporal accuracy of our method.