A simple scenario of the laminar breakdown in liquid metal flows

2021 ◽  
Vol 57 (2) ◽  
pp. 191-210
Keyword(s):  
Reynolds Number ◽  
Electrical Potential ◽  
Finite Amplitude ◽  
Magnetic Reynolds Number ◽  
Small Length ◽  
Turbulent Transition ◽  
Local Perturbations ◽  
Magnetohydrodynamic Flows ◽  
The Stability ◽  
Velocity Pressure

In the article, authors present a numerical method for modelling a laminar-turbulent transition in magnetohydrodynamic flows. The small magnetic Reynolds number approach is considered. Velocity, pressure and electrical potential are decomposed to the sum of state values and finite amplitude perturbations. A solver based on the Nektar++ framework is described. The authors suggest using small-length local perturbations as a transition trigger. They can be imposed by blowing or by electrical enforcing. The stability of the Hartmann flow and the flow in the bend are considered as examples. Tables 4, Figs 19, Refs 28.

On the relation between viscoelastic and magnetohydrodynamic flows and their instabilities

2003 ◽  
Vol 476 ◽  
pp. 389-409 ◽  
Author(s):  
GORDON I. OGILVIE ◽  
MICHAEL R. E. PROCTOR
Keyword(s):  
Viscoelastic Medium ◽  
Magnetic Reynolds Number ◽  
Deborah Number ◽  
Elastic Instability ◽  
Magnetorotational Instability ◽  
Electrically Conducting ◽  
Electrically Conducting Fluid ◽  
Magnetohydrodynamic Flows ◽  
Elastic Instabilities ◽  
The Stability

We demonstrate a close analogy between a viscoelastic medium and an electrically conducting fluid containing a magnetic field. Specifically, the dynamics of the Oldroyd-B fluid in the limit of large Deborah number corresponds to that of a magnetohydrodynamic (MHD) fluid in the limit of large magnetic Reynolds number. As a definite example of this analogy, we compare the stability properties of differentially rotating viscoelastic and MHD flows. We show that there is an instability of the Oldroyd-B fluid that is physically distinct from both the inertial and elastic instabilities described previously in the literature, but is directly equivalent to the magnetorotational instability in MHD. It occurs even when the specific angular momentum increases outwards, provided that the angular velocity decreases outwards; it derives from the kinetic energy of the shear flow and does not depend on the curvature of the streamlines. However, we argue that the elastic instability of viscoelastic Couette flow has no direct equivalent in MHD.

scholarly journals Laminar–turbulent transition in pipe flow for Newtonian and non-Newtonian fluids

1998 ◽  
Vol 377 ◽  
pp. 267-312 ◽  
Author(s):  
A. A. DRAAD ◽  
G. D. C. KUIKEN ◽  
F. T. M. NIEUWSTADT
Keyword(s):  
Reynolds Number ◽  
Pipe Flow ◽  
Polymer Solutions ◽  
Reynolds Numbers ◽  
Cylindrical Pipe ◽  
Turbulent Transition ◽  
Wide Range ◽  
Flow Changes ◽  
Transition Points ◽  
The Stability

A cylindrical pipe facility with a length of 32 m and a diameter of 40 mm has been designed. The natural transition Reynolds number, i.e. the Reynolds number at which transition occurs as a result of non-forced, natural disturbances, is approximately 60 000. In this facility we have studied the stability of cylindrical pipe flow to imposed disturbances. The disturbance consists of periodic suction and injection of fluid from a slit over the whole circumference in the pipe wall. The injection and suction are equal in magnitude and each distributed over half the circumference so that the disturbance is divergence free. The amplitude and frequency can be varied over a wide range.First, we consider a Newtonian fluid, water in our case. From the observations we compute the critical disturbance velocity, which is the smallest disturbance at a given Reynolds number for which transition occurs. For large wavenumbers, i.e. large frequencies, the dimensionless critical disturbance velocity scales according to Re−1, while for small wavenumbers, i.e. small frequencies, it scales as Re−2/3. The latter is in agreement with weak nonlinear stability theory. For Reynolds numbers above 30 000 multiple transition points are found which means that increasing the disturbance velocity at constant dimensionless wavenumber leads to the following course of events. First, the flow changes from laminar to turbulent at the critical disturbance velocity; subsequently at a higher value of the disturbance it returns back to laminar and at still larger disturbance velocities the flow again becomes turbulent.Secondly, we have carried out stability measurements for (non-Newtonian) dilute polymer solutions. The results show that the polymers reduce in general the natural transition Reynolds number. The cause of this reduction remains unclear, but a possible explanation may be related to a destabilizing effect of the elasticity on the developing boundary layers in the entry region of the flow. At the same time the polymers have a stabilizing effect with respect to the forced disturbances, namely the critical disturbance velocity for the polymer solutions is larger than for water. The stabilization is stronger for fresh polymer solutions and it is also larger when the polymers adopt a more extended conformation. A delay in transition has been only found for extended fresh polymers where delay means an increase of the critical Reynolds number, i.e. the number below which the flow remains laminar at any imposed disturbance.

On the stability of pipe-Poiseuille flow to finite-amplitude axisymmetric and non-axisymmetric disturbances

1985 ◽  
Vol 158 ◽  
pp. 289-316 ◽  
Author(s):  
P. K. Sen ◽  
D. Venkateswarlu ◽  
S. Maji
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Finite Amplitude ◽  
Higher Order ◽  
Equilibrium States ◽  
Minimum Threshold ◽  
Threshold Amplitude ◽  
The Stability ◽  
The One ◽  
Axisymmetric Disturbances

The stability of fully developed pipe-Poiseuille flow to finite-amplitude axisymmetric and non-axisymmetric disturbances has been studied using the equilibrium-amplitude method of Reynolds & Potter (1967). In both the cases the least-stable centre-modes were investigated. Also, for the non-axisymmetric case the mode investigated was the one with azimuthal wavenumber equal to one. Many higher-order Landau coefficients were calculated, and the Stuart-Landau series was analysed by the Shanks (1955) method and by using Padé approximants to look for the existence of possible equilibrium states. The results show in both cases that, for each value of the Reynolds number R, there is a preferred band of spatial wavenumbers α in which equilibrium states are likely to exist. Moreover, in both cases it was found that the magnitude of the minimum threshold amplitude for a given R decreases with increasing R. The scales of the various quantities obtained agree very well with those deduced by Davey & Nguyen (1971).

scholarly journals Bifurcations and instabilities in sliding Couette flow

2011 ◽  
Vol 678 ◽  
pp. 156-178 ◽  
Author(s):  
K. DEGUCHI ◽  
M. NAGATA
Keyword(s):  
Reynolds Number ◽  
Stability Result ◽  
Axial Direction ◽  
Travelling Wave ◽  
Finite Amplitude ◽  
Axisymmetric Case ◽  
Sliding Motion ◽  
Axisymmetric Solutions ◽  
The Stability ◽  
Axisymmetric Perturbations

We carry out linear and nonlinear analyses on a flow between two infinitely long concentric cylinders with the radii a and b subject to a sliding motion of the inner cylinder in the axial direction. We confirm the linear stability result of Gittler (Acta Mechanica, vol. 101, 1993, p. 1) for the axisymmetric case, namely the flow is linearly stable against axisymmetric perturbations when the radius ratio η = a/b is greater than 0.1415. We extend his analysis to the non-axisymmetric case and find that the stability of the flow is still determined by axisymmetric perturbations. Our nonlinear analysis exhibits that (i) finite-amplitude axisymmetric solutions exist far below the linear critical Reynolds number for η < 0.1415 and (ii) non-axisymmetric travelling wave solutions appear abruptly at a finite Reynolds number even for η > 0.1415 where the linear critical state is absent.

The effect of external forcing on the stability of plane Poiseuille flow

1978 ◽  
Vol 359 (1699) ◽  
pp. 453-478 ◽  
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Stable Solution ◽  
Reynolds Numbers ◽  
Finite Amplitude ◽  
Plane Poiseuille Flow ◽  
Critical Reynolds Numbers ◽  
Threshold Amplitude ◽  
The Stability

The stability of plane Poiseuille flow in a channel forced by a wavelike motion on one of the channel walls is investigated. The amplitude Є of this forcing is taken to be small. The most dangerous modes of forcing are identified and it is found in general the critical Reynolds number is changed by O (Є) 2 . However, we identify two particular modes of forcing which give rise to decrements of order Є 2/3 and Є in the critical Reynolds number. Some types of forcing are found to generate sub critical stable finite amplitude perturbations to plane Poiseuille flow. This contrasts with the unforced case where the only stable solution is the zero amplitude solution. The forcing also deforms the unstable subcritical limit cycle solution from its usual circular shape into a more complicated shape. This has an effect on the threshold amplitude ideas suggested by, for example, Meksyn & Stuart (1951). It is found that the phase of disturbances must also be considered when finding the amplitude dependent critical Reynolds numbers.

A numerical and experimental investigation of the stability of spiral Poiseuille flow

1981 ◽  
Vol 102 ◽  
pp. 101-126 ◽  
Author(s):  
Donald I. Takeuchi ◽  
Daniel F. Jankowski
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Concentric Cylinders ◽  
Axial Pressure Gradient ◽  
Secondary Motion ◽  
Wave Forms ◽  
Predicted Values ◽  
The Stability

The linear stability of the spiral motion induced between concentric cylinders by an axial pressure gradient and independent cylinder rotation is studied numerically and experimentally for a wide-gap geometry. A three-dimensional disturbance is considered. Linear stability limits in the form of Taylor numbers TaL are computed for the rotation ratios μ, = 0, 0·2, and -0·5 and for values of the axial Reynolds number Re up to 100. Depending on the values of μ and Re, the disturbance which corresponds to TaL can have a toroidal vortex structure or a spiral form. Aluminium-flake flow visualization is used to determine conditions for the onset of a secondary motion and its structure at finite amplitude. The experimental results agree with the predicted values of TaL for μ [ges ] 0, and low Reynolds number. For other cases in which agreement is only fair, apparatus length is shown to be a contributing influence. The comparison between experimental and predicted wave forms shows good agreement in overall trends.

On the stability of plane flow of a conducting fluid in the presence of a coplanar magnetic field

1970 ◽  
Vol 43 (3) ◽  
pp. 591-596 ◽  
Author(s):  
C. Sozou
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Reynolds Numbers ◽  
Magnetic Reynolds Number ◽  
Conducting Fluid ◽  
Unstable Flow ◽  
Plane Flow ◽  
The Stability ◽  
Two Parameters ◽  
The Relationship

The equations governing the propagation of small perturbations to plane flow of a viscous incompressible conducting fluid are re-examined with special reference to the case when the constant unperturbed magnetic field and flow velocity are parallel. We use the relationship between two parameters in one equation and, without computations, show the following: If for a non-zero value of the Alfvén number the flow is unstable when the Reynolds and magnetic Reynolds numbers take particular finite values, then, for that value of the Alfvén number, the flow cannot be completely stabilized for all finite Reynolds numbers, when the magnetic Reynolds number is finite. Since for a finite Alfvén number one expects that unstable flow cannot be stabilized for all finite Reynolds numbers, unless the magnetic Reynolds number exceeds some value, we deduce the following: An unstable parallel flow of a finitely conducting fluid cannot be completely stabilized for all finite Reynolds numbers by a constant magnetic field, which is coplanar with the flow.

The Effect of Finite Amplitude Disturbance Magnitude on Departures From Laminar Conditions in Impulsively Started and Steady Pipe Entrance Flows

2001 ◽  
Vol 124 (1) ◽  
pp. 235-240 ◽  
Author(s):  
E. A. Moss ◽  
A. H. Abbot
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Shape Parameter ◽  
Critical Reynolds Number ◽  
Flow Instability ◽  
Finite Amplitude ◽  
Pipe Flows ◽  
Displacement Thickness ◽  
Layer Flow ◽  
The Stability

The aim of this study was to investigate first departures from laminar conditions in both impulsively started and steady pipe entrance flows. Wall shear stress measurements were conducted of transition in impulsively started pipe flows with large disturbances. These results were reconciled in a framework of displacement thickness Reynolds number and a velocity profile shape parameter, with existing measurements of pipe entrance flow instability, pipe-Poiseuille and boundary layer flow responses to large disturbances, and linear stability predictions. Limiting critical Reynolds number variations for each type of flow were thus inferred, corresponding to the small and gross disturbance limits respectively. Consequently, insights have been provided regarding the effect of disturbance levels on the stability of both steady and unsteady pipe flows.

scholarly journals Azimuthal magnetorotational instability with super-rotation

2018 ◽  
Vol 84 (1) ◽  
Author(s):  
G. Rüdiger ◽  
M. Schultz ◽  
M. Gellert ◽  
F. Stefani
Keyword(s):  
Reynolds Number ◽  
Prandtl Number ◽  
Numerical Solutions ◽  
Hartmann Number ◽  
Outer Cylinder ◽  
Magnetic Reynolds Number ◽  
Neutral Stability ◽  
Magnetorotational Instability ◽  
Magnetic Prandtl Number ◽  
The Stability

It is demonstrated that the azimuthal magnetorotational instability (AMRI) also works with radially increasing rotation rates contrary to the standard magnetorotational instability for axial fields which requires negative shear. The stability against non-axisymmetric perturbations of a conducting Taylor–Couette flow with positive shear under the influence of a toroidal magnetic field is considered if the background field between the cylinders is current free. For small magnetic Prandtl number $Pm\rightarrow 0$ the curves of neutral stability converge in the (Hartmann number,Reynolds number) plane approximating the stability curve obtained in the inductionless limit $Pm=0$. The numerical solutions for $Pm=0$ indicate the existence of a lower limit of the shear rate. For large $Pm$ the curves scale with the magnetic Reynolds number of the outer cylinder but the flow is always stable for magnetic Prandtl number unity as is typical for double-diffusive instabilities. We are particularly interested to know the minimum Hartmann number for neutral stability. For models with resting or almost resting inner cylinder and with perfectly conducting cylinder material the minimum Hartmann number occurs for a radius ratio of $r_{\text{in}}=0.9$. The corresponding critical Reynolds numbers are smaller than $10^{4}$.

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