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.

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.

scholarly journals Linear stability of confined flow around a 180-degree sharp bend

2017 ◽  
Vol 822 ◽  
pp. 813-847 ◽  
Author(s):  
Azan M. Sapardi ◽  
Wisam K. Hussam ◽  
Alban Pothérat ◽  
Gregory J. Sheard
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Base Flow ◽  
Two Dimensional ◽  
Two Dimensional Flow ◽  
The Stability

This study seeks to characterise the breakdown of the steady two-dimensional solution in the flow around a 180-degree sharp bend to infinitesimal three-dimensional disturbances using a linear stability analysis. The stability analysis predicts that three-dimensional transition is via a synchronous instability of the steady flows. A highly accurate global linear stability analysis of the flow was conducted with Reynolds number $\mathit{Re}<1150$ and bend opening ratio (ratio of bend width to inlet height) $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 5$. This range of $\mathit{Re}$ and $\unicode[STIX]{x1D6FD}$ captures both steady-state two-dimensional flow solutions and the inception of unsteady two-dimensional flow. For $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 1$, the two-dimensional base flow transitions from steady to unsteady at higher Reynolds number as $\unicode[STIX]{x1D6FD}$ increases. The stability analysis shows that at the onset of instability, the base flow becomes three-dimensionally unstable in two different modes, namely a spanwise oscillating mode for $\unicode[STIX]{x1D6FD}=0.2$ and a spanwise synchronous mode for $\unicode[STIX]{x1D6FD}\geqslant 0.3$. The critical Reynolds number and the spanwise wavelength of perturbations increase as $\unicode[STIX]{x1D6FD}$ increases. For $1<\unicode[STIX]{x1D6FD}\leqslant 2$ both the critical Reynolds number for onset of unsteadiness and the spanwise wavelength decrease as $\unicode[STIX]{x1D6FD}$ increases. Finally, for $2<\unicode[STIX]{x1D6FD}\leqslant 5$, the critical Reynolds number and spanwise wavelength remain almost constant. The linear stability analysis also shows that the base flow becomes unstable to different three-dimensional modes depending on the opening ratio. The modes are found to be localised near the reattachment point of the first recirculation bubble.

The development of three-dimensional wave-packets in unbounded parallel flows

1968 ◽  
Vol 32 (4) ◽  
pp. 801-808 ◽  
Author(s):  
M. Gaster ◽  
A. Davey
Keyword(s):  
Wave Packet ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Mean Flow ◽  
Physical Plane ◽  
Parallel Flows ◽  
Flow Velocity Profile ◽  
The Stability ◽  
Special Case ◽  
Dimensional Wave

In this paper we examine the stability of a two-dimensional wake profile of the form u(y) = U∞(1 – r e-sy2) with respect to a pulsed disturbance at a point in the fluid. The disturbed flow forms an expanding wave packet which is convected downstream. Far downstream, where asymptotic expansions are valid, the motion at any point in the wave packet is described by a particular three-dimensional wave having complex wave-numbers. In the special case of very unstable flows, where viscosity does not have a significant influence, it is possible to evaluate the three-dimensional eigenvalues in terms of two-dimensional ones using the inviscid form of Squire's transformation. In this way each point in the physical plane can be linked to a particular two-dimensional wave growing in both space and time by simple algebraic expressions which are independent of the mean flow velocity profile. Computed eigenvalues for the wake profile are used in these relations to find the behaviour of the wave packet in the physical plane.

Linear Spatial Stability of Developing Flow in a Parallel Plate Channel

1981 ◽  
Vol 48 (1) ◽  
pp. 192-194 ◽  
Author(s):  
S. C. Gupta ◽  
V. K. Garg
Keyword(s):  
Reynolds Number ◽  
Velocity Profile ◽  
Parallel Plate ◽  
Critical Reynolds Number ◽  
Two Dimensional ◽  
Spatial Stability ◽  
Percent Change ◽  
Developing Flow ◽  
The Stability ◽  
Plate Channel

It is found that even a 5 percent change in the velocity profile produces a 100 percent change in the critical Reynolds number for the stability of developing flow very close to the entrance of a two-dimensional channel.

On the stability of parallel flows with parallel magnetic fields

1966 ◽  
Vol 293 (1434) ◽  
pp. 342-358 ◽  
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Parallel Magnetic Field ◽  
Parallel Flows ◽  
The Mean ◽  
The Stability ◽  
The Magnetic Field ◽  
Parallel Magnetic Fields

The first part of the paper is a physical discussion of the way in which a magnetic field affects the stability of a fluid in motion. Particular emphasis is given to how the magnetic field affects the interaction of the disturbance with the mean motion. The second part is an analysis of the stability of plane parallel flows of fluids with finite viscosity and conductivity under the action of uniform parallel magnetic fields. We show that, in general, three-dimensional disturbances are the most unstable, thus disagreeing with the conclusion of Michael (1953) and Stuart (1954). We show how results obtained for two-dimensional disturbances can be used to calculate the most unstable three-dimensional disturbances and thence we prove that a parallel magnetic field can never completely stabilize a parallel flow.

Wake behaviour and instability of flow through a partially blocked channel

2007 ◽  
Vol 582 ◽  
pp. 319-340 ◽  
Author(s):  
M. D. GRIFFITH ◽  
M. C. THOMPSON ◽  
T. LEWEKE ◽  
K. HOURIGAN ◽  
W. P. ANDERSON
Keyword(s):  
Reynolds Number ◽  
Three Dimensional ◽  
Underlying Mechanism ◽  
Two Dimensional ◽  
Recirculation Bubble ◽  
Dimensional Flow ◽  
Starting Point ◽  
Two Dimensional Flow ◽  
Flow Through ◽  
The Stability

The two-dimensional flow through a constricted channel is studied. A semi-circular bump is located on one side of the channel and the extent of blockage is varied by adjusting the radius of the bump. The blockage is varied between 0.05 and 0.9 of the channel width and the upstream Reynolds number between 25 and 3000. The geometry presents a simplified blockage specified by a single parameter, serving as a starting point for investigations of other more complex blockage geometries. For blockage ratios in excess of 0.4, the variation of reattachment length with Reynolds number collapses to within approximately 15%, while at lower ratios the behaviour differs. For the constrained two-dimensional flow, various phenomena are identified, such as multiple mini-recirculations contained within the main recirculation bubble and vortex shedding at higher Reynolds numbers. The stability of the flow to three-dimensional perturbations is analysed, revealing a transition to a three-dimensional state at a critical Reynolds number which decreases with higher blockage ratios. Separation lengths and the onset and structure of three-dimensional instability observed from the geometry of blockage ratio 0.5 resemble results taken from backward-facing step investigations. The question of the underlying mechanism behind the instability being either centrifugal or elliptic in nature and operating within the initial recirculation zone is analytically tested.

The stability of a two-dimensional stagnation flow to three-dimensional disturbances

1978 ◽  
Vol 84 (3) ◽  
pp. 517-527 ◽  
Author(s):  
S. D. R. Wilson ◽  
I. Gladwell
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Boundary Layer Thickness ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Stagnation Flow ◽  
Stagnation Line ◽  
Two Dimensional Flow ◽  
The Stability ◽  
Secondary Vortices

Experiments have shown that the two-dimensional flow near a forward stagnation line may be unstable to three-dimensional disturbances. The growing disturbance takes the form of secondary vortices, i.e. vortices more or less parallel to the original streamlines. The instability is usually confined to the boundary layer and the spacing of the secondary vortices is of the order of the boundary-layer thickness. This situation is analysed theoretically for the case of infinitesimal disturbances of the type first studied by Görtler and Hämmerlin. These are disturbances periodic in the direction perpendicular to the plane of the flow, in the limit of infinite Reynolds number. It is shown that the flow is always stable to these disturbances.

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.

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.

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