scholarly journals From three-dimensional to quasi-two-dimensional: transient growth in magnetohydrodynamic duct flows

2018 ◽  
Vol 861 ◽  
pp. 382-406 ◽  
Author(s):  
Oliver G. W. Cassells ◽  
Tony Vo ◽  
Alban Pothérat ◽  
Gregory J. Sheard
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Layer Thickness ◽  
Hartmann Number ◽  
Three Dimensional ◽  
Transient Growth ◽  
Two Dimensional ◽  
Growth Mechanisms ◽  
Electrically Conducting ◽  
Magnetohydrodynamic Models

This study seeks to elucidate the linear transient growth mechanisms in a uniform duct with square cross-section applicable to flows of electrically conducting fluids under the influence of an external magnetic field. A particular focus is given to the question of whether at high magnetic fields purely two-dimensional mechanisms exist, and whether these can be described by a computationally inexpensive quasi-two-dimensional model. Two Reynolds numbers of $5000$ and $15\,000$ and an extensive range of Hartmann numbers $0\leqslant \mathit{Ha}\leqslant 800$ were investigated. Three broad regimes are identified in which optimal mode topology and non-modal growth mechanisms are distinct. These regimes, corresponding to low, moderate and high magnetic field strengths, are found to be governed by the independent parameters; Hartmann number, Reynolds number based on the Hartmann layer thickness $R_{H}$ and Reynolds number built upon the Shercliff layer thickness $R_{S}$, respectively. Transition between regimes respectively occurs at $\mathit{Ha}\approx 2$ and no lower than $R_{H}\approx 33.\dot{3}$. Notably for the high Hartmann number regime, quasi-two-dimensional magnetohydrodynamic models are shown to be excellent predictors of not only transient growth magnitudes, but also the fundamental growth mechanisms of linear disturbances. This paves the way for a precise analysis of transition to quasi-two-dimensional turbulence at much higher Hartmann numbers than is currently achievable.

Hydromagnetic instability of a free shear layer at small magnetic Reynolds numbers

1971 ◽  
Vol 49 (1) ◽  
pp. 21-31 ◽  
Author(s):  
Kanefusa Gotoh
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Shear Layer ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Free Shear Layer ◽  
Electrically Conducting ◽  
The Stability ◽  
The Magnetic Field

The effect of a uniform and parallel magnetic field upon the stability of a free shear layer of an electrically conducting fluid is investigated. The equations of the velocity and the magnetic disturbances are solved numerically and it is shown that the flow is stabilized with increasing magnetic field. When the magnetic field is expressed in terms of the parameter N (= M2/R2), where M is the Hartmann number and R is the Reynolds number, the lowest critical Reynolds number is caused by the two-dimensional disturbances. So long as 0 [les ] N [les ] 0·0092 the flow is unstable at all R. For 0·0092 < N [les ] 0·0233 the flow is unstable at 0 < R < Ruc where Ruc decreases as N increases. For 0·0233 < N < 0·0295 the flow is unstable at Rlc < R < Ruc where Rlc increases with N. Lastly for N > 0·0295 the flow is stable at all R. When the magnetic field is measured by M, the lowest critical Reynolds number is still due to the two-dimensional disturbances provided 0 [les ] M [les ] 0·52, and Rc is given by the corresponding Rlc. For M > 0·52, Rc is expressed as Rc = 5·8M, and the responsible disturbance is the three-dimensional one which propagates at angle cos−1(0·52/M) to the direction of the basic flow.

Three-dimensional MHD flows in rectangular ducts with internal obstacles

2000 ◽  
Vol 418 ◽  
pp. 265-295 ◽  
Author(s):  
B. MÜCK ◽  
C. GÜNTHER ◽  
U. MÜLLER ◽  
L. BÜHLER
Keyword(s):  
Magnetic Field ◽  
Numerical Simulation ◽  
Reynolds Number ◽  
Magnetic Fields ◽  
Interaction Parameter ◽  
Hartmann Number ◽  
Three Dimensional ◽  
Side Length ◽  
Two Dimensional ◽  
The Magnetic Field

This paper presents a numerical simulation of the magnetohydrodynamic (MHD) liquid metal flow around a square cylinder placed in a rectangular duct. In the hydrodynamic case, for a certain parameter range the well-known Kármán vortex street with three-dimensional flow patterns is observed, similar to the flow around a circular cylinder. In this study a uniform magnetic field aligned with the cylinder is applied and its influence on the formation and downstream transport of vortices is investigated. The relevant key parameters for the MHD flow are the Hartmann number M, the interaction parameter N and the hydrodynamic Reynolds number, all based on the side length of the cylinder. The Hartmann number M was varied in the range 0 [les ] M [les ] 85 and the interaction parameter N in the range 0 [les ] N [les ] 36. Results are presented for two fixed Reynolds numbers Re = 200 and Re = 250. The magnetic Reynolds number is assumed to be very small. The results of the numerical simulation are compared with known experimental and theoretical results. The hydrodynamic simulation shows characteristic intermittent pulsations of the drag and lift force on the cylinder. At Re = 200 a mix of secondary spanwise three-dimensional instabilities (A and B mode, rib vortices) could be observed. The spanwise wavelength of the rib vortices was found to be about 2–3 cylinder side lengths in the near wake. At Re = 250 the flow appears more organized showing a regular B mode pattern and a spanwise wavelength of about 1 cylinder side length. With an applied magnetic field a quasi-two-dimensional flow can be obtained at low N ≈ 1 due to the strong non-isotropic character of the electromagnetic forces. The remaining vortices have their axes aligned with the magnetic field. With increasing magnetic fields these vortices are further damped due to Hartmann braking. The result that the ‘quasi-two-dimensional’ vortices have a curvature in the direction of the magnetic field can be explained by means of an asymptotic analysis of the governing equations. With very high magnetic fields the time-dependent vortex shedding can be almost completely suppressed. By three-dimensional visualization it was possible to show characteristic paths of the electric current for this kind of flow, explaining the action of the Lorentz forces.

Magnetic damping of jets and vortices

1995 ◽  
Vol 299 ◽  
pp. 153-186 ◽  
Author(s):  
P. A. Davidson
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Angular Momentum ◽  
Kinetic Energy ◽  
Lorentz Force ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Magnetic Damping ◽  
Field Lines ◽  
The Magnetic Field

It is well known that the imposition of a static magnetic field tends to suppress motion in an electrically conducting liquid. Here we look at the magnetic damping of liquid-mental flows where the Reynolds number is large and the magnetic Reynolds number is small. The magnetic field is taken as uniform and the fluid is either infinite in extent or else bounded by an electrically insulating surface S. Under these conditions, we find that three general principles govern the flow. First, the Lorentz force destroys kinetic energy but does not alter the net linear momentum of the fluid, nor does it change the component of angular momentum parallel to B. In certain flows, this implies that momentum, linear or angular, is conserved. Second, the Lorentz force guides the flow in such a way that the global Joule dissipation, D, decreases, and this decline in D is even more rapid than the corresponding fall in global kinetic energy, E. (Note that both D and E are quadratic in u). Third, this decline in relative dissipation, D / E, is essential to conserving momentum, and is achieved by propagating linear or angular momentum out along the magnetic field lines. In fact, this spreading of momentum along the B-lines is a diffusive process, familiar in the context of MHD turbulence. We illustrate these three principles with the aid of a number of specific examples. In increasing order of complexity we look at a spatially uniform jet evolving in time, a three-dimensional jet evolving in space, and an axisymmetric vortex evolving in both space and time. We start with a spatially uniform jet which is dissipated by the sudden application of a transverse magnetic field. This simple (perhaps even trivial) example provides a clear illustration of our three general principles. It also provides a useful stepping-stone to our second example of a steady three-dimensional jet evolving in space. Unlike the two-dimensional jets studied by previous investigators, a three-dimensional jet cannot be annihilated by magnetic braking. Rather, its cross-section deforms in such a way that the momentum flux of the jet is conserved, despite a continual decline in its energy flux. We conclude with a discussion of magnetic damping of axisymmetric vortices. As with the jet flows, the Lorentz force cannot destroy the motion, but rather rearranges the angular momentum of the flow so as to reduce the global kinetic energy. This process ceases, and the flow reaches a steady state, only when the angular momentum is uniform in the direction of the field lines. This is closely related to the tendency of magnetic fields to promote two-dimensional turbulence.

Stability of plane parallel flows of electrically conducting fluids

1953 ◽  
Vol 49 (1) ◽  
pp. 166-168 ◽  
Author(s):  
D. H. Michael
Keyword(s):  
Reynolds Number ◽  
Velocity Profile ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Electrically Conducting ◽  
Plane Parallel ◽  
Parallel Flows ◽  
Lower Reynolds Number ◽  
The Stability ◽  
Conducting Fluids

The ordinary theory of stability of plane parallel flows is considerably simplified by a result due to Squire (2) which says that if a velocity profile becomes unstable to a small three-dimensional disturbance at a given Reynolds number, then it will become unstable to a small two-dimensional disturbance at a lower Reynolds number. This result enables us to restrict investigation of the stability to the cases of two-dimensional disturbances.

On the measurement of the turbulent diffusivity of a large-scale magnetic field

2013 ◽  
Vol 717 ◽  
pp. 347-360 ◽  
Author(s):  
S. M. Tobias ◽  
F. Cattaneo
Keyword(s):  
Magnetic Field ◽  
Large Scale ◽  
Laboratory Experiments ◽  
Three Dimensional ◽  
Effective Diffusivity ◽  
Dimensional System ◽  
Two Dimensional ◽  
Electrically Conducting ◽  
State Response ◽  
Large Scale Magnetic Field

AbstractWe argue that a method developed by Ångström (Ann. Phys. Chem., vol. 114, 1861, pp. 513–530) to measure the thermal conductivity of solids can be adapted to determine the effective diffusivity of a large-scale magnetic field in a turbulent electrically conducting fluid. The method consists of applying an oscillatory source and measuring the steady-state response. We illustrate this method in a two-dimensional system. This geometry is chosen because it is possible to compare the results with independent methods that are restricted to two-dimensional flows. We describe two variants of this method: one (the ‘turbulent Ångström method’) that is better suited to laboratory experiments and a second (the ‘method of oscillatory sines’) that is effective for numerical experiments. We show that, if correctly implemented, all methods agree. Based on these results we argue that these methods can be extended to three-dimensional numerical simulations and laboratory experiments.

Numerical simulation of liquid-metal MHD flows in rectangular ducts

1990 ◽  
Vol 216 ◽  
pp. 161-191 ◽  
Author(s):  
A. Sterl
Keyword(s):  
Magnetic Field ◽  
Liquid Metal ◽  
Boundary Layers ◽  
Hartmann Number ◽  
Three Dimensional ◽  
Mhd Flow ◽  
Two Dimensional ◽  
Square Duct ◽  
Mhd Flows ◽  
Pressure Losses

To design self-cooled liquid metal blankets for fusion reactors, one must know about the behaviour of MHD flows at high Hartmann numbers. In this work, finite difference codes are used to investigate the influence of Hartmann number M, interaction parameter N, wall conductance ratio c, and changing magnetic field, respectively, on the flow.As liquid-metal MHD flows are characterized by thin boundary layers, resolution of these layers is the limiting issue. Hartmann numbers up to 103 are reached in the two-dimensional case of fully developed flow, while in three-dimensional flows the limit is 102. However, the calculations reveal the main features of MHD flows at large M. They are governed by electric currents induced in the fluid. Knowing the paths of these currents makes it possible to predict the flow structure.Results are shown for two-dimensional flows in a square duct at different Hartmann numbers and wall conductivities. While the Hartmann number governs the thickness of the boundary layers, the wall conductivities are responsible for the pressure losses and the structure of the flows. The most distinct feature is the side layers where the velocities can exceed those at the centre by orders of magnitude.The three-dimensional results are also for a square duct. The main interest here is to investigate the redistribution of the fluid in a region where the magnetic field changes. Large axial currents are induced leading to the ‘M-shaped’ velocity profiles characteristic of MHD flow. So-called Flow Channel Inserts (FCI), of great interest in blanket design, are investigated. They serve to decouple the load carrying wall from the currents in the fluid. The calculations show that the FCI is indeed a suitable measure to reduce the pressure losses in the blanket.

On the stability of plane Poiseuille flow with a finite conductivity in an aligned magnetic field

1968 ◽  
Vol 33 (3) ◽  
pp. 433-443 ◽  
Author(s):  
Sung-Hwan Ko
Keyword(s):  
Magnetic Field ◽  
Differential Equation ◽  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Order Differential Equation ◽  
Three Dimensional ◽  
Sixth Order ◽  
Finite Conductivity ◽  
Two Dimensional ◽  
The Stability

A study is made of the stability of a viscous, incompressible fluid with a finite conductivity flowing between parallel planes in a parallel magnetic field. The general form of the magnetohydrodynamic stability equation is a sixth-order differential equation. The complete sixth-order differential equation is solved numerically as an eigenvalue problem. Stability curves are obtained for a range of values of the magnetic Reynolds number Rm and the Alfvé n number A based on two-dimensional disturbances. It is found that the minimum critical Reynolds number is raised as Rm increases for a given A2 and as A2 increases for a given Rm, respectively. The stability curve closes and finally degenerates to a point which gives the critical value for Rm or A2. Results obtained for two-dimensional disturbances are modified to take into account three-dimensional disturbances. Then the minimum critical Reynolds number where three-dimensional disturbances become apparent is obtained, below which two-dimensional disturbances are the most unstable.

scholarly journals Instability and transition to turbulence in a free shear layer affected by a parallel magnetic field

2007 ◽  
Vol 574 ◽  
pp. 131-154 ◽  
Author(s):  
A. VOROBEV ◽  
O. ZIKANOV
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Shear Layer ◽  
Three Dimensional ◽  
Magnetic Reynolds Number ◽  
Transition To Turbulence ◽  
Free Shear Layer ◽  
Parallel Magnetic Field ◽  
Oblique Waves ◽  
Electrically Conducting

Instability and transition to turbulence in a temporally evolving free shear layer of an electrically conducting fluid affected by an imposed parallel magnetic field is investigated numerically. The case of low magnetic Reynolds number is considered. It has long been known that the neutral disturbances of the linear problem are three-dimensional at sufficiently strong magnetic fields. We analyse the details of this instability solving the generalized Orr–Sommerfeld equation to determine the wavenumbers, growth rates and spatial shapes of the eigenmodes. The three-dimensional perturbations are identified as oblique waves and their properties are described. In particular, we find that at high hydrodynamic Reynolds number, the effect of the strength of the magnetic field on the fastest growing perturbations is limited to an increase of their oblique angle. The dimensions and spatial shape of the waves remain unchanged. The transition to turbulence triggered by the growing oblique waves is investigated in direct numerical simulations. It is shown that initial perturbations in the form of superposition of two symmetric waves are particularly effective in inducing three-dimensionality and turbulence in the flow.

Effective transition of steady flow over a square leading-edge plate

2012 ◽  
Vol 698 ◽  
pp. 335-357 ◽  
Author(s):  
Mark C. Thompson
Keyword(s):  
Reynolds Number ◽  
Shear Layer ◽  
Three Dimensional ◽  
Leading Edge ◽  
Maximum Energy ◽  
Transient Growth ◽  
Two Dimensional ◽  
Optimal Perturbation ◽  
Perturbation Mode ◽  
Optimal Mode

AbstractPrevious experimental studies have shown that the steady recirculation bubble that forms as the flow separates at the leading-edge corner of a long plate, becomes unsteady at relatively low Reynolds numbers of only a few hundreds. The reattaching shear layer irregularly releases two-dimensional vortices, which quickly undergo three-dimensional transition. Similar to the flow over a backward-facing step, this flow is globally stable at such Reynolds numbers, with transition to a steady three-dimensional flow as the first global instability to occur as the Reynolds number is increased to 393. Hence, it appears that the observed flow behaviour is governed by transient growth of optimal two-dimensional transiently growing perturbations (constructed from damped global modes) rather than a single three-dimensional unstable global mode. This paper quantifies the details of the transient growth of two- and three-dimensional optimal perturbations, and compares the predictions to other related cases examined recently. The optimal perturbation modes are shown to be highly concentrated in amplitude in the vicinity of the leading-edge corners and evolve to take the local shape of a Kelvin–Helmholtz shear-layer instability further downstream. However, the dominant mode reaches a maximum amplitude downstream of the position of the reattachment point of the shear layer. The maximum energy growth increases at 2.5 decades for each increment in Reynolds number of 100. Maximum energy growth of the optimal perturbation mode at a Reynolds number of 350 is greater than $1{0}^{4} $, which is typically an upper limit of the Reynolds number range over which it is possible to observe steady flow experimentally. While transient growth analysis concentrates on the evolution of wavepackets rather than continuous forcing, this appears consistent with longitudinal turbulence levels of up to 1 % for some water tunnels, and the fact that the optimal mode is highly concentrated close to the leading-edge corner so that an instantaneous projection of a perturbation field from a noisy inflow onto the optimal mode can be significant. Indeed, direct simulations with inflow noise reveal that a root-mean-square noise level of just 0.1 % is sufficient to trigger some unsteadiness at $\mathit{Re}= 350$, while a 0.5 % level results in sustained shedding. Three-dimensional optimal perturbation mode analysis was also performed showing that at $\mathit{Re}= 350$, the optimal mode has a spanwise wavelength of 11.7 plate thicknesses and is amplified 20 % more than the two-dimensional optimal disturbance. The evolved three-dimensional mode shows strong streamwise vortical structures aligned at a shallow angle to the plate top surface.

Bifurcation Characteristics of Flows in Rectangular Sudden Expansion Channels

2005 ◽  
Author(s):  
Francine Battaglia ◽  
George Papadopoulos
Keyword(s):  
Reynolds Number ◽  
Laminar Flow ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Expansion Ratio ◽  
Sudden Expansion ◽  
Two Dimensional ◽  
Effective Expansion ◽  
Three Dimensional Simulations

The effect of three-dimensionality on low Reynolds number flows past a symmetric sudden expansion in a channel was investigated. The geometric expansion ratio of in the current study was 2:1 and the aspect ratio was 6:1. Both experimental velocity measurements and two- and three-dimensional simulations for the flow along the centerplane of the rectangular duct are presented for Reynolds numbers in the range of 150 to 600. Comparison of the two-dimensional simulations with the experiments revealed that the simulations fail to capture completely the total expansion effect on the flow, which couples both geometric and hydrodynamic effects. To properly do so requires the definition of an effective expansion ratio, which is the ratio of the downstream and upstream hydraulic diameters and is therefore a function of both the expansion and aspect ratios. When the two-dimensional geometry was consistent with the effective expansion ratio, the new results agreed well with the three-dimensional simulations and the experiments. Furthermore, in the range of Reynolds numbers investigated, the laminar flow through the expansion underwent a symmetry-breaking bifurcation. The critical Reynolds number evaluated from the experiments and the simulations was compared to other values reported in the literature. Overall, side-wall proximity was found to enhance flow stability, helping to sustain laminar flow symmetry to higher Reynolds numbers in comparison to nominally two-dimensional double-expansion geometries. Lastly, and most importantly, when the logarithm of the critical Reynolds number from all these studies was plotted against the reciprocal of the effective expansion ratio, a linear trend emerged that uniquely captured the bifurcation dynamics of all symmetric double-sided planar expansions.

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