scholarly journals High-speed shear-driven dynamos. Part 2. Numerical analysis

2019 ◽  
Vol 876 ◽  
pp. 830-858 ◽  
Author(s):  
Kengo Deguchi
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Couette Flow ◽  
Uniform Magnetic Field ◽  
Asymptotic Theory ◽  
Companion Paper ◽  
Mhd Equations ◽  
Alfven Wave ◽  
Large Reynolds Number ◽  
Plane Couette Flow

This paper aims to numerically verify the large Reynolds number asymptotic theory of magneto-hydrodynamic (MHD) flows proposed in the companion paper Deguchi (J. Fluid Mech., vol. 868, 2019, pp. 176–211). To avoid any complexity associated with the chaotic nature of turbulence and flow geometry, nonlinear steady solutions of the viscous resistive MHD equations in plane Couette flow have been utilised. Two classes of nonlinear MHD states, which convert kinematic energy to magnetic energy effectively, have been determined. The first class of nonlinear states can be obtained when a small spanwise uniform magnetic field is applied to the known hydrodynamic solution branch of plane Couette flow. The nonlinear states are characterised by the hydrodynamic/magnetic roll–streak and the resonant layer at which strong vorticity and current sheets are observed. These flow features, and the induced strong streamwise magnetic field, are fully consistent with the vortex/Alfvén wave interaction theory proposed in the companion paper. When the spanwise uniform magnetic field is switched off, the solutions become purely hydrodynamic. However, the second class of ‘self-sustained shear-driven dynamos’ at the zero external magnetic field limit can be found by homotopy via the forced states subject to a spanwise uniform current field. The discovery of the dynamo states has motivated the corresponding large Reynolds number matched asymptotic analysis in the companion paper. Here, the reduced equations derived by the asymptotic theory have been solved numerically. The asymptotic solution provides remarkably good predictions for the finite Reynolds number dynamo solutions.

Tertiary solutions and their stability in rotating plane Couette flow

1998 ◽  
Vol 358 ◽  
pp. 357-378 ◽  
Author(s):  
M. NAGATA
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Stability Boundary ◽  
Streamwise Vortex ◽  
Reynolds Numbers ◽  
Time Dependent ◽  
Plane Couette Flow ◽  
Streamwise Direction ◽  
The Stability ◽  
Oscillatory Instabilities

The stability of nonlinear tertiary solutions in rotating plane Couette flow is examined numerically. It is found that the tertiary flows, which bifurcate from two-dimensional streamwise vortex flows, are stable within a certain range of the rotation rate when the Reynolds number is relatively small. The stability boundary is determined by perturbations which are subharmonic in the streamwise direction. As the Reynolds number is increased, the rotation range for the stable tertiary motions is destroyed gradually by oscillatory instabilities. We expect that the tertiary flow is overtaken by time-dependent motions for large Reynolds numbers. The results are compared with the recent experimental observation by Tillmark & Alfredsson (1996).

scholarly journals The linear instability of the stratified plane Couette flow

2018 ◽  
Vol 853 ◽  
pp. 205-234 ◽  
Author(s):  
Giulio Facchini ◽  
Benjamin Favier ◽  
Patrice Le Gal ◽  
Meng Wang ◽  
Michael Le Bars
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Numerical Simulations ◽  
Couette Flow ◽  
Vertical Direction ◽  
Stability Diagram ◽  
Linear Instability ◽  
Plane Couette Flow ◽  
Horizontal Shear ◽  
The Stability

We present the stability analysis of a plane Couette flow which is stably stratified in the vertical direction orthogonal to the horizontal shear. Interest in such a flow comes from geophysical and astrophysical applications where background shear and vertical stable stratification commonly coexist. We perform the linear stability analysis of the flow in a domain which is periodic in the streamwise and vertical directions and confined in the cross-stream direction. The stability diagram is constructed as a function of the Reynolds number $Re$ and the Froude number $Fr$, which compares the importance of shear and stratification. We find that the flow becomes unstable when shear and stratification are of the same order (i.e. $Fr\sim 1$) and above a moderate value of the Reynolds number $Re\gtrsim 700$. The instability results from a wave resonance mechanism already known in the context of channel flows – for instance, unstratified plane Couette flow in the shallow-water approximation. The result is confirmed by fully nonlinear direct numerical simulations and, to the best of our knowledge, constitutes the first evidence of linear instability in a vertically stratified plane Couette flow. We also report the study of a laboratory flow generated by a transparent belt entrained by two vertical cylinders and immersed in a tank filled with salty water, linearly stratified in density. We observe the emergence of a robust spatio-temporal pattern close to the threshold values of $Fr$ and $Re$ indicated by linear analysis, and explore the accessible part of the stability diagram. With the support of numerical simulations we conclude that the observed pattern is a signature of the same instability predicted by the linear theory, although slightly modified due to streamwise confinement.

scholarly journals Deformation of a magnetic liquid drop in an applied non-stationary uniform magnetic field

2018 ◽  
Vol 185 ◽  
pp. 09006
Author(s):  
Alexander Tyatyushkin
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Steady State ◽  
Uniform Magnetic Field ◽  
Liquid Drop ◽  
Magnetic Liquid ◽  
Stationary Approximation ◽  
The Magnetic Field

Small steady-state deformational oscillations of a drop of magnetic liquid in a nonstationary uniform magnetic field are theoretically investigated. The drop is suspended in another magnetic liquid immiscible with the former. The Reynolds number is so small that the inertia can be neglected. The variation of the magnetic field is so slow that the quasi-stationary approximation for the magnetic field and the quasi-steady approximation for the flow may be used.

scholarly journals High-speed shear-driven dynamos. Part 1. Asymptotic analysis

2019 ◽  
Vol 868 ◽  
pp. 176-211 ◽  
Author(s):  
Kengo Deguchi
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Solar Phys ◽  
Wave Interaction ◽  
Resonant Absorption ◽  
Base Flow ◽  
Transition To Turbulence ◽  
Alfven Wave ◽  
Hydrodynamic Theory ◽  
Large Reynolds Number

Rational large Reynolds number matched asymptotic expansions of three-dimensional nonlinear magneto-hydrodynamic (MHD) states are the concern of this contribution. The nonlinear MHD states, assumed to be predominantly driven by a unidirectional shear, can be sustained without any linear instability of the base flow and hence are responsible for subcritical transition to turbulence. Two classes of nonlinear MHD states are found. The first class of nonlinear states emerged out of a nice combination of the purely hydrodynamic vortex/wave interaction theory by Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666) and the resonant absorption theories on Alfvén waves, developed in the solar physics community (e.g. Sakurai et al. Solar Phys., vol. 133, 1991, pp. 227–245; Goossens et al. Solar Phys., vol. 157, 1995, pp. 75–102). Similar to the hydrodynamic theory, the mechanism of the MHD states can be explained by the successive interaction of the roll, streak and wave fields, which are now defined both for the hydrodynamic and magnetic fields. The derivation of this ‘vortex/Alfvén wave interaction’ state is rather straightforward as the scalings for both of the hydrodynamic and magnetic fields are identical. It turns out that the leading-order magnetic field of the asymptotic states appears only when a small external magnetic field is present. However, it does not mean that purely shear-driven dynamos are not possible. In fact, the second class of ‘self-sustained shear-driven dynamo theory’ shows a magnetic generation that is slightly smaller in size in the absence of any external field. Despite its small size, the magnetic field causes the novel feedback mechanism in the velocity field through resonant absorption, wherein the magnetic wave becomes more strongly amplified than the hydrodynamic counterpart.

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