Cylindrical Taylor-Couette Flow Research Articles

MATRICS: The implicit matrix-free Eulerian hydrodynamics solver

Context. There exists a zoo of different astrophysical fluid dynamics solvers, most of which are based on an explicit formulation and hence stability-limited to small time steps dictated by the Courant number expressing the local speed of sound. With this limitation, the modeling of low-Mach-number flows requires small time steps that introduce significant numerical diffusion, and a large amount of computational resources are needed. On the other hand, implicit methods are often developed to exclusively model the fully incompressible or 1D case since they require the construction and solution of one or more large (non)linear systems per time step, for which direct matrix inversion procedures become unacceptably slow in two or more dimensions. Aims. In this work, we present a globally implicit 3D axisymmetric Eulerian solver for the compressible Navier–Stokes equations including the energy equation using conservative formulation and a fully simultaneous approach. We use the second-order-in-time backward differentiation formula for temporal discretization as well as the κ scheme for spatial discretization. We implement different limiter functions to prohibit the occurrence of spurious oscillations in the vicinity of discontinuities. Our method resembles the well-known monotone upwind scheme for conservation laws (MUSCL). We briefly present efficient solution methods for the arising sparse and nonlinear system of equations. Methods. To deal with the nonlinearity of the Navier–Stokes equations we used a Newton iteration procedure in which the required Jacobian matrix-vector product was reconstructed with a first-order finite difference approximation to machine precision in a matrix-free way. The resulting linear system was solved either completely matrix-free with a combination of a sufficient Krylov solver and an approximate Jacobian preconditioner or semi-matrix-free with the incomplete lower upper factorization technique as a preconditioner. The latter was dependent on a standalone approximation of the Jacobian matrix, which was optionally calculated and needed solely for the purpose of preconditioning. Results. We show our method to be capable of damping sound waves and resolving shocks even at Courant numbers larger than one. Furthermore, we prove the method’s ability to solve boundary value problems like the cylindrical Taylor-Couette flow (TC), including viscosity, and to model transition flows. To show the latter, we recover predicted growth rates for the vertical shear instability, while choosing a time step orders of magnitude larger than the explicit one. Finally, we verify that our method is second order in space by simulating a simplistic, stationary solar wind.

Numerical investigation of the onset of axisymmetric and wavy Taylor-Couette flows between combinations of cylinders and spherocylinders

The Taylor-Couette flow, which is the flow that occurs between two rotating bodies, has become fundamental to the study of instability and nonlinear behaviour. Many modifications can be made to the Taylor-Couette flow resulting in much more complex flow structures, either by geometrical changes, or by combining different geometries. The main goal of this work is to numerically investigate the effect of the cylinders-spheres combinations on the onset of different instabilities of a fluid confined between two concentric bodies. The modelling strategy was developed by studying three types of flow configurations: cylindrical Taylor-Couette flow (with fixed endcaps), flow between two concentric cylinders with hemispheres on the lower end wall, and flow between two coaxial spherocylinders (cylinders with hemispheres on the upper and lower end surfaces). The inner element rotates while the outer one is at rest. The numerical calculations were carried out to determine the transition zone from a laminar Couette flow to the onset of Taylor vortices and wavy vortex flow. The parameter that determines the flow regimes is the Reynolds number based on the angular velocity of the inner element. The fluid dynamic behaviours for different flow configurations are characterized by the wall shear stress and the skin friction coefficient. Laminar, axisymmetric and wavy Taylor-Couette flows are predicted for different configurations. It is established that the different combinations deeply affect the flow behaviour and the appearance of the instabilities. The transition from LCF to TVF and then to WVF in the combined flow systems is substantially retarded compared to the cylindrical Taylor-Couette flow system.

Shercliff layers in strongly magnetic cylindrical Taylor–Couette flow

We numerically compute axisymmetric Taylor-Couette flow in the presence of axially periodic magnetic fields, with Hartmann numbers up to $Ha^2=10^7$. The geometry of the field singles out special field lines on which Shercliff layers form. These are simple shear layers for insulating boundaries, versus super-rotating or counter-rotating layers for conducting boundaries. Some field configurations have previously studied spherical analogs, but fundamentally new configurations also exist, having no spherical analogs. Finally, we explore the influence of azimuthal fields $B_\phi\sim r^{-1}{\bf\hat e}_\phi$ on these layers, and show that the flow is suppressed for conducting boundaries but enhanced for insulating boundaries.

Metamorphosis of helical magnetorotational instability in the presence of axial electric current.

This paper presents numerical linear stability analysis of a cylindrical Taylor-Couette flow of liquid metal carrying axial electric current in a generally helical external magnetic field. Axially symmetric disturbances are considered in the inductionless approximation corresponding to zero magnetic Prandtl number. Axial symmetry allows us to reveal an entirely new electromagnetic instability. First, we show that the electric current passing through the liquid can extend the range of helical magnetorotational instability (HMRI) indefinitely by transforming it into a purely electromagnetic instability. Two different electromagnetic instability mechanisms are identified. The first is an internal pinch-type instability, which is due to the interaction of the electric current with its own magnetic field. Axisymmetric mode of this instability requires a free-space component of the azimuthal magnetic field. When the azimuthal component of the magnetic field is purely rotational and the axial component is nonzero, a new kind of electromagnetic instability emerges. The latter, driven by the interaction of electric current with a weak collinear magnetic field in a quiescent fluid, gives rise to a steady meridional circulation coupled with azimuthal rotation.

The role of boundaries in the magnetorotational instability

In this paper, we investigate numerically the flow of an electrically conducting fluid in a cylindrical Taylor-Couette flow when an axial magnetic field is applied. To minimize Ekman recirculation due to vertical no-slip boundaries, two independently rotating rings are used at the vertical endcaps. This configuration reproduces setup used in laboratory experiments aiming to observe the magnetorotational instability (MRI). Our 3D global simulations show that the nature of the bifurcation, the non-linear saturation, and the structure of axisymmetric MRI modes are significantly affected by the presence of boundaries. In addition, large scale non-axisymmetric modes are obtained when the applied field is sufficiently strong. We show that these modes are related to Kelvin-Helmholtz destabilization of a free Shercliff shear layer created by the combined action of the applied field and the rotating rings at the endcaps. Finally, we compare our numerical simulations to recent experimental results obtained in the Princeton MRI experiment.

Inviscid helical magnetorotational instability in cylindrical Taylor-Couette flow

This paper presents the analysis of axisymmetric helical magnetorotational instability (HMRI) in the inviscid limit, which is relevant for astrophysical conditions. The inductionless approximation defined by zero magnetic Prandtl number is adopted to distinguish the HMRI from the standard MRI in the cylindrical Taylor-Couette flow subject to a helical magnetic field. Using a Chebyshev collocation method convective and absolute instability thresholds are computed in terms of the Elsasser number for a fixed ratio of inner and outer radii λ = 2 and various ratios of rotation rates and helicities of the magnetic field. It is found that the extension of self-sustained HMRI modes beyond the Rayleigh limit does not reach the astrophysically relevant Keplerian rotation profile not only in the narrow- but also in the finite-gap approximation. The Keppler limit can be attained only by the convective HMRI mode provided that the boundaries are perfectly conducting. However, this mode requires not only a permanent external excitation to be observable but also has a long axial wavelength, which is not compatible with limited thickness of astrophysical accretion disks.

Nonaxisymmetric Magnetorotational Instabilities in Cylindrical Taylor-Couette Flow

We study the stability of cylindrical Taylor-Couette flow in the presence of azimuthal magnetic fields, and show that one obtains nonaxisymmetric magnetorotational instabilities, having azimuthal wave number m=1. For Omega{o}/Omega{i} only slightly greater than the Rayleigh value (r{i}/r{o}){2}, the critical Reynolds and Hartmann numbers are Re{c} approximately 10{3} and Ha{c} approximately 10{2}, independent of the magnetic Prandtl number Pm. These values are sufficiently small that it should be possible to obtain these instabilities in the Potsdam Rossendorf Magnetic Instability Experiment.

Nonlinear simulations of magnetic Taylor-Couette flow with currentfree helical magnetic fields

The magnetorotational instability (MRI) in cylindrical Taylor-Couette flow with external helical magnetic field is simulated for infinite and finite aspect ratios. We solve the MHD equations in their small Prandtl number limit and confirm with timedependent nonlinear simulations that the additional toroidal component of the magnetic field reduces the critical Reynolds number from O(10 6 ) (axial field only) to O(10 3 ) for liquid metals with their small magnetic Prandtl number. Computing the saturated state we obtain velocity amplitudes which help designing proper experimental setups. Experiments with liquid gallium require axial field ∼ 50 Gauss and axial current ∼ 4 kA for the toroidal field. It is sufficient that the vertical velocity uz of the flow can be measured with a precision of 0.1mm/s. We also show that the endplates enclosing the cylinders do not destroy the traveling wave instability which can be observed as presented in earlier studies. For TC containers without and with endplates the angular momentum transport of the MRI instability is shown as to be outwards.

The Traveling-Wave MRI in Cylindrical Taylor-Couette Flow: Comparing Wavelengths and Speeds in Theory and Experiment

We study experimentally the flow of a liquid metal confined between differentially rotating cylinders, in the presence of externally imposed axial and azimuthal magnetic fields. For increasingly large azimuthal fields a wave-like disturbance arises, traveling along the axis of the cylinders. The wavelengths and speeds of these structures, as well as the field strengths and rotation rates at which they arise, are broadly consistent with theoretical predictions of such a traveling wave magnetorotational instability.

New Type of Magnetorotational Instability in Cylindrical Taylor-Couette Flow

We study the stability of cylindrical Taylor-Couette flow in the presence of combined axial and azimuthal magnetic fields, and show that adding an azimuthal field profoundly alters the previous results for purely axial fields. For small magnetic Prandtl numbers Pm, the critical Reynolds number Re(c) for the onset of the magnetorotational instability becomes independent of Pm, whereas for purely axial fields it scales as Pm-1. For typical liquid metals, Re(c) is then reduced by several orders of magnitude, enough that this new design should succeed in realizing this instability in the laboratory.

Second-order accurate particle image velocimetry

An adaptive, second-order accurate particle image velocimetry (PIV) technique is presented. The technique uses two singly exposed images that are interrogated using a modified cross-correlation algorithm. Consequently, any of the equipment commonly available for conventional PIV (such as dual head Nd: YAG lasers, interline transfer CCD cameras, etc.) can be used with this more accurate algorithm. At the heart of the algorithm is a central difference approximation to the flow velocity (accurate to order Δt 2) versus the forward difference approximation (accurate to order Δt) common in PIV. An adaptive interrogation region-shifting algorithm is used to implement the central difference approximation. Adaptive shifting algorithms have been gaining popularity in recent years because they allow the spatial resolution of the PIV technique to be maximized. Adaptive shifting algorithms also have the virtue of helping to eliminate velocity bias errors. The second- order accuracy resulting from the central difference approximation can be obtained with relatively little additional computational effort compared to that required for a standard first-order accurate forward difference approximation. The adaptive central difference interrogation (CDI) algorithm has two main advantages over adaptive forward difference interrogation (FDI) algorithms: it is more accurate, especially at large time delays between camera exposures; and it provides a temporally symmetric view of the flow. By comparing measurements of flow around a single red blood cell made using both algorithms, the CDI technique is shown to perform better than conventional FDI-PIV interrogation algorithms near flow boundaries. Cylindrical Taylor–Couette flow images, both experimental and simulated, are used to demonstrate that the CDI algorithm is significantly more accurate than conventional PIV algorithms, especially as the time delay between exposures is increased. The results of the interrogations are shown to agree quite well with analytical predictions and confirm that the CDI algorithm is indeed second-order accurate while the conventional FDI algorithm is only first-order accurate.