Elsevier

Computers & Fluids

Volume 105, 10 December 2014, Pages 16-27
Computers & Fluids

Numerical study of MHD Taylor vortex flow with low magnetic Reynolds number in finite-length annulus under uniform magnetic field

https://doi.org/10.1016/j.compfluid.2014.09.001Get rights and content

Highlights

  • Spectral-projection method for flow patterns at various strength of magnetic field in finite-length annulus.

  • Three endwall boundary conditions are considered.

  • Effects of uniform magnetic field on Taylor vortex structures.

Abstract

The effects of axial uniform magnetic field on Taylor vortex flow with low magnetic Reynolds number in a concentric finite-length annulus are investigated numerically. The inner cylinder is rotating and the outer one is held at rest. As the endwall effects play a significant role in determining the vortical structure, three endwall conditions are taken into account in present study: all are fixed, all are rotating with the inner cylinder, one is fixed but the other is rotating with the inner cylinder. All the boundaries are assumed to be insulating, and the chosen aspect ratio, the height of the cylinder to the gap width, is moderate Γ = 10, hence the endwall effects and the Taylor vortex can be both considered. The governing equations are solved using collocation spectral method and projection method which allows pressure–velocity decoupling. The flow patterns and developments of vortical structure with different endwall conditions are presented at various strength of axial magnetic field. The effects of axial magnetic field on magnetohydrodynamics flow are discussed.

Introduction

The motion of an incompressible viscous fluid between concentric rotating cylinders, called Taylor–Couette flow, is the classical problem in hydrodynamics.

In infinite long cylinders, where the outer cylinder is fixed and the inner is rotating, as the speed of the inner cylinder is reaching critical value, the initial laminar flow (named circular Couette flow) becomes unstable to perturbation. If the perturbation is imposed, steady axisymmetric toroidal vortices will develop. This secondary flow is Taylor vortex flow, which is observed experimentally and calculated in Taylor’s pioneering work [1].

In order to ignore the endwall effects, in many experimental studies [1], computational works [2], [3], [4], [5] and theoretical works [1], the periodic boundary in axial direction is assumed, or the cylinders are assumed to be very long, or more precisely the ratio (defined as Γ) of the height of the cylinder to the gap width is set to be large enough.

In fact, since the cylinders in any apparatus are of necessity of finite length, especially for a short annulus, the endwall effects play a significant role in determining the vortical structure and introduce a series of complex problems [6]. Although we could ignore the endwall effects for long cylinders, there will be a slow circulation near the endwall indeed. For fixed endwall, the endwall boundary layer flow tends to be radially inward [6], [7]. The centrifugal force that caused by the rotation of the inner cylinder tends to push the fluid outwards. Due to the braking effect of the endwall, the trend near the endwall is weaker than that in other region. So fluid must flow inwards near the endwall. In the case of the Reynolds number is less than the critical one, weak Taylor-like vortex, known as Ekman vortex [8], [9], will appear closing to the endwall. Ekman vortex may drive adjacent vortices in turn. With the increasing of Reynolds number, more counter-rotating vortices will appear, this is not a result of Taylor instability. Upon reaching the critical valve, due to centrifugal instability, there is classical Taylor vortex flow over the most of the length of the cylinders [10], [11], [12].

Coles [13] chose a shorter cylinder than that of Taylor’s experiment, and obtained 26 distinct stable flow states. Snyder [14] found that, the finite length of the cylinders and specific endwall boundary conditions are important in determining the flow patterns. The similar and further work was done in Andereck’s experiments (Γ = 30) [15]. The toroidal vortex motion started in the Ekman layers adjacent to the motionless boundaries much earlier than Taylor vortices. Alziary and Grillaud [11] employed a finite difference method with implicit fractional steps to numerically calculate the time-dependent equations. Beginning with Ekman vortex at each endwall for small subcritical Reynolds numbers and ending with toroidal vortices filling the entire annulus, smooth transition to Taylor vortex flow was investigated. Burkhalter and Koschmieder [12] experimentally studied endwall effects on the size and wavelength of Taylor vortex in finite-length annulus. By using various cylinder lengths and three different endwall boundaries in their experiments, variations of interior-cell wavelength were observed. The endwall boundaries affected the size of end ring strongly, but second ring little. Neitzel [16] numerically calculated time-dependent Taylor vortex flow and compared his results with Burkhalter’s experiments. The motion was initiated by an impulsive start of the inner cylinder from a state of rest. He discussed the development of Taylor vortex from the standpoint of onset time and vortex-front propagation.

If the fluid is electrically conducting and an external magnetic field is applied, Taylor–Couette problem become magnetohydrodynamic (MHD) Taylor–Couette problem. Then, we will be curious about what will happen, when Taylor vortex flow is subjected to an externally uniform axial magnetic field.

Research on Couette flow in the presence of an applied magnetic field can be dated from the pioneering work of Chandrasekhar [17]. Donnelly and Ozima [18] performed experiments on Taylor–Couette flow of mercury between two long cylinders (Γ = 100), with the inner one rotating and the outer one stationary. Tabeling [19] numerically investigated the magnetohydrodynamic Taylor vortex in infinite long cylinders. Tagawa [20] studied a 4-cell flow in the presence of magnetic field for a short annulus (Γ = 4) with fixed endwall via finite difference method. His results showed that, the axial magnetic field can damp out the secondary flow and velocity gradient in the direction of the magnetic field, efficiently.

Balbus and Hawley [21] pointed out the importance of the magnetorotational instability as the origin of the MHD turbulence in accretion disk. Consequently, most investigations have been directly focused on the magnetorotational instability of the MHD Taylor–Couette flow [22]. Also the dynamo action for the generation of the magnetic field in astrophysics is investigated [23], which suggests that the Taylor vortex flow is good candidate for dynamo action in a simple geometry.

In industry, the Taylor vortex reactor (or Taylor–Couette reactor) is widely employed to enhance the performance of reactors, such as particle flocculation [24] and water purification [25], for the unique flow characteristics of mixing and fine control property. The electromagnetic control has enormous potential and good prospect in the application of the Taylor-vortex reactor, as the working medium is electric conducting in many cases.

However, unlike the situations in the astrophysical area, the magnetic Reynolds number is usually low, for example, the liquid metal in industry and laboratory, and the quasi-static approximation [26], [27], [28] is applied, which assumed that the induced magnetic field is negligible in comparison to the external magnetic field. With this assumption the magnetic field is a known quantity that does not depend on the flow.

Only slight attentions were paid to the development of vortical structure and flow patterns in the annulus. Nevertheless, an understanding of the velocity field and vortical structure is crucial to engineering applications of the flow. The purpose of the present paper is to numerically investigate the effects of magnetic field on Taylor vortex flow with low magnetic Reynolds number in finite-length concentric cylinders. We focus on the development and flow patterns of Taylor vortex flows in various conditions.

As the endwall boundary layers can strongly affect the vortical structure, three endwall conditions are used, they are: fixed endwalls, endwalls rotating with inner cylinder, one stationary endwall and the other attached to the rotating inner cylinder (asymmetric endwalls), when the inner cylinder is rotating and outer one is held at rest.

In the present study, we consider a narrow gap with radio ratio (defined later) η = 10/11 and the aspect ratio Γ = 10, which is moderate. End vortex and Taylor vortex are numerically calculated at various strength of magnetic field.

This paper is organized as following. The governing equations of the problem with the boundary conditions and the numerical methods are presented in Section 2. The results and discussions are given in Section 3. Finally, the conclusions are given in Section 4.

Section snippets

Mathematical model and numerical method

We consider the axisymmetric flow of an incompressible viscous electrically conducting fluid with constant kinematic viscosity ν, electrical conductivity σ, magnetic diffusivity λ, magnetic permeability μ0, and density ρ, contained in the annulus between two finite-length concentric cylinders of radii r1 and r2, and height h. We assume that the inner cylinder rotates and the outer cylinder is held at rest, while the endwalls can be fixed or rotate with the inner cylinder. The vortex flow is

Validation

In order to verify our numerical method, we consider the Taylor–Couette flow with axial magnetic field for Γ = 4, η = 1/2, Re = 100 and Ha = 20. The present results are compared with those in Ref. [20], the comparisons are shown in Fig. 1, Fig. 2. Almost the same results are obtained and this indicates that our method is reliable.

Grid numbers of 13 × 61, 17 × 81, 21 × 101, 25 × 121 and 29 × 141 in radial and axial directions, are tested for the grid sensitivity at Ha = 20. In Fig. 3, the distribution curves of

Conclusions

Steady and axisymmetric Taylor vortex flows with low magnetic Reynolds number are numerically investigated for three endwall boundary conditions in the presence of an axial uniform magnetic field. At various strength of magnetic field, a series of flow patterns and vortical structures are observed in this study. The numerical results show that, an axial magnetic field can stabilize the Taylor–Couette flow and reduce the secondary flow, effectively. Whatever endwall boundary condition is chosen,

Acknowledgement

The authors gratefully acknowledge Prof. Andre Thess (TU-Ilmenau) for his constructively discussion and suggestions. The authors wish to acknowledge the financial support of National Nature Science Foundation of China, NSFC (Grant Nos. 10772044 & 51176026).

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