scholarly journals MAGNETOROTATIONAL TURBULENCE TRANSPORTS ANGULAR MOMENTUM IN STRATIFIED DISKS WITH LOW MAGNETIC PRANDTL NUMBER BUT MAGNETIC REYNOLDS NUMBER ABOVE A CRITICAL VALUE

2011 ◽  
Vol 740 (1) ◽  
pp. 18 ◽  
Author(s):  
Jeffrey S. Oishi ◽  
Mordecai-Mark Mac Low
Keyword(s):  
Reynolds Number ◽  
Angular Momentum ◽  
Prandtl Number ◽  
Critical Value ◽  
Magnetic Reynolds Number ◽  
Magnetic Prandtl Number

scholarly journals The onset of turbulent rotating dynamos at the low magnetic Prandtl number limit

2017 ◽  
Vol 822 ◽  
Author(s):  
Kannabiran Seshasayanan ◽  
Vassilios Dallas ◽  
Alexandros Alexakis
Keyword(s):  
Reynolds Number ◽  
Prandtl Number ◽  
Large Scale ◽  
Injection Rate ◽  
Magnetic Reynolds Number ◽  
Strong Decrease ◽  
New Paradigm ◽  
Turbulent Fluctuations ◽  
Rotation Rates ◽  
Magnetic Prandtl Number

We demonstrate that the critical magnetic Reynolds number $Rm_{c}$ for a turbulent non-helical dynamo in the limit of low magnetic Prandtl number $Pm$ (i.e. $Pm=Rm/Re\ll 1$) can be significantly reduced if the flow is subjected to global rotation. Even for moderate rotation rates the required energy injection rate can be reduced by a factor of more than $10^{3}$. This strong decrease in the onset is attributed to the transfer of energy to the large scales, forming a large-scale condensate, and the reduction in the turbulent fluctuations that cause the flow to have a much larger cutoff length scale than in a non-rotating flow of the same Reynolds number. The dynamo thus behaves as if it is driven just by the large scales that act as a laminar flow (i.e. it behaves as a high $Pm$ dynamo) even though the actual Reynolds number is much higher than the magnetic Reynolds number (i.e. low $Pm$). Our finding thus points to a new paradigm for the design of new experiments on liquid metal dynamos.

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 $10^{4}$.

scholarly journals Kazantsev dynamo in turbulent compressible flows

2019 ◽  
Vol 475 (2223) ◽  
pp. 20180591 ◽  
Author(s):  
Marco Martins Afonso ◽  
Dhrubaditya Mitra ◽  
Dario Vincenzi
Keyword(s):  
Growth Rate ◽  
Reynolds Number ◽  
Critical Exponent ◽  
Compressible Flows ◽  
Critical Value ◽  
Magnetic Reynolds Number ◽  
Scaling Exponent ◽  
Scaling Exponents ◽  
Potential Part

We consider the kinematic fluctuation dynamo problem in a flow that is random, white-in-time, with both solenoidal and potential components. This model is a generalization of the well-studied Kazantsev model. If both the solenoidal and potential parts have the same scaling exponent, then, as the compressibility of the flow increases, the growth rate decreases but remains positive. If the scaling exponents for the solenoidal and potential parts differ, in particular if they correspond to typical Kolmogorov and Burgers values, we again find that an increase in compressibility slows down the growth rate but does not turn it off. The slow down is, however, weaker and the critical magnetic Reynolds number is lower than when both the solenoidal and potential components display the Kolmogorov scaling. Intriguingly, we find that there exist cases, when the potential part is smoother than the solenoidal part, for which an increase in compressibility increases the growth rate. We also find that the critical value of the scaling exponent above which a dynamo is seen is unity irrespective of the compressibility. Finally, we realize that the dimension d  = 3 is special, as for all other values of d the critical exponent is higher and depends on the compressibility.

scholarly journals High-speed standard magneto-rotational instability

2019 ◽  
Vol 865 ◽  
pp. 492-522 ◽  
Author(s):  
Kengo Deguchi
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Prandtl Number ◽  
Stability Condition ◽  
Stability Criterion ◽  
High Speed ◽  
Hydrodynamic Instability ◽  
Rotational Instability ◽  
Large Reynolds Number ◽  
Magnetic Prandtl Number

The large Reynolds number asymptotic approximations of the neutral curve of Taylor–Couette flow subject to an axial uniform magnetic field are analysed. The flow has been extensively studied since the early 1990s as the magneto-rotational instability (MRI) occurring in the flow may explain the origin of the instability observed in some astrophysical objects. Elsewhere, the ideal approximation has been used to study high-speed flows, which sometimes produces paradoxical results. For example, ideal flows must be completely stabilised for a sufficiently strong applied magnetic field. On the other hand, the vanishing magnetic Prandtl number limit of the stability should be purely hydrodynamic, so instability must occur when Rayleigh’s stability condition is violated. Our first discovery is that this apparent contradiction can be resolved by showing the abrupt appearance of the hydrodynamic instability at a certain critical value of the magnetic Prandtl number. This is found using the asymptotically large Reynolds number limit but with a sufficiently long wavelength to retain some diffusive effects. Our second finding concerns the so-called Velikhov–Chandrasekhar paradox, namely the mismatch of the zero external magnetic field limit of the Velikhov–Chandrasekhar stability criterion and Rayleigh’s stability criterion. We show for fully wide-gap cases that the high Reynolds number asymptotic analysis of the MRI naturally yields the simple stability condition that describes smooth transition from Rayleigh to Velikhov–Chandrasekhar stability criteria with increasing Lundquist number.

scholarly journals Small-scale dynamo action in rotating compressible convection

2011 ◽  
Vol 690 ◽  
pp. 262-287 ◽  
Author(s):  
B. Favier ◽  
P. J. Bushby
Keyword(s):  
Reynolds Number ◽  
Energy Density ◽  
Thermal Stratification ◽  
Magnetic Energy ◽  
Critical Value ◽  
Magnetic Reynolds Number ◽  
Small Scale ◽  
Dynamo Action ◽  
Lower Bounding ◽  
The Mean

AbstractWe study dynamo action in a convective layer of electrically conducting, compressible fluid, rotating about the vertical axis. At the upper and lower bounding surfaces, perfectly conducting boundary conditions are adopted for the magnetic field. Two different levels of thermal stratification are considered. If the magnetic diffusivity is sufficiently small, the convection acts as a small-scale dynamo. Using a definition for the magnetic Reynolds number ${R}_{M} $ that is based upon the horizontal integral scale and the horizontally averaged velocity at the mid-layer of the domain, we find that rotation tends to reduce the critical value of ${R}_{M} $ above which dynamo action is observed. Increasing the level of thermal stratification within the layer does not significantly alter the critical value of ${R}_{M} $ in the rotating calculations, but it does lead to a reduction in this critical value in the non-rotating cases. At the highest computationally accessible values of the magnetic Reynolds number, the saturation levels of the dynamo are similar in all cases, with the mean magnetic energy density somewhere between 4 and 9 % of the mean kinetic energy density. To gain further insights into the differences between rotating and non-rotating convection, we quantify the stretching properties of each flow by measuring Lyapunov exponents. Away from the boundaries, the rate of stretching due to the flow is much less dependent upon depth in the rotating cases than it is in the corresponding non-rotating calculations. It is also shown that the effects of rotation significantly reduce the magnetic energy dissipation in the lower part of the layer. We also investigate certain aspects of the saturation mechanism of the dynamo.

Fluid flow generated by switching a magnetic field on or off

1973 ◽  
Vol 61 (2) ◽  
pp. 209-217 ◽  
Author(s):  
Alfred Sneyd
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Uniform Magnetic Field ◽  
Magnetic Reynolds Number ◽  
Simple Expression ◽  
Conducting Fluid ◽  
Uniform Field ◽  
Vorticity Distribution ◽  
Magnetic Prandtl Number ◽  
Initial Vorticity

A uniform magnetic field is switched on at time t = 0 outside a body of conducting fluid. It is assumed that the field strength increases in time in proportion to 1 -e−αt, where α is a constant of the circuit generating the field. Under the assumption of small magnetic Reynolds number and small magnetic Prandtl number the equations governing the diffusion of the field into the fluid are derived and a simple expression is given for the initial vorticity distribution produced in the fluid. The situation in which an initially uniform field is switched off is also considered. It is shown that, for sufficiently symmetrically shaped bodies of fluid, the vorticity generated by the switching-on of the field is the same as that generated by the switching-off. The particular case of an infinitely long circular cylinder of conducting fluid is considered in detail and an explicit expression is derived for the vorticity distribution.

Stability of thermocapillary flows in non-cylindrical liquid bridges

2002 ◽  
Vol 458 ◽  
pp. 35-73 ◽  
Author(s):  
CH. NIENHÜSER ◽  
H. C. KUHLMANN
Keyword(s):  
Reynolds Number ◽  
Prandtl Number ◽  
Critical Reynolds Number ◽  
Volume Fraction ◽  
Thermocapillary Flow ◽  
Surface Shape ◽  
Liquid Bridges ◽  
Gravity Level ◽  
Neutral Curves ◽  
Prandtl Numbers

The thermocapillary flow in liquid bridges is investigated numerically. In the limit of large mean surface tension the free-surface shape is independent of the flow and temperature fields and depends only on the volume of liquid and the hydrostatic pressure difference. When gravity acts parallel to the axis of the liquid bridge the shape is axisymmetric. A differential heating of the bounding circular disks then causes a steady two-dimensional thermocapillary flow which is calculated by a finite-difference method on body-fitted coordinates. The linear-stability problem for the basic flow is solved using azimuthal normal modes computed with the same discretization method. The dependence of the critical Reynolds number on the volume fraction, gravity level, Prandtl number, and aspect ratio is explained by analysing the energy budgets of the neutral modes. For small Prandtl numbers (Pr = 0.02) the critical Reynolds number exhibits a smooth minimum near volume fractions which approximately correspond to the volume of a cylindrical bridge. When the Prandtl number is large (Pr = 4) the intersection of two neutral curves results in a sharp peak of the critical Reynolds number. Since the instabilities for low and high Prandtl numbers are markedly different, the influence of gravity leads to a distinctly different behaviour. While the hydrostatic shape of the bridge is the most important effect of gravity on the critical point for low-Prandtl-number flows, buoyancy is the dominating factor for the stability of the flow in a gravity field when the Prandtl number is high.

Numerical Study of Convective Heat Transfer From Expanding Annular Pipe Flows

2016 ◽  
Author(s):  
Khaled J. Hammad
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Prandtl Number ◽  
Convective Heat Transfer ◽  
Diameter Ratio ◽  
Wall Heat Transfer ◽  
Reattachment Point ◽  
Transfer Rates ◽  
Pipe Flows ◽  
Annular Pipe

Convective heat transfer from suddenly expanding annular pipe flows are numerically investigated within the steady laminar flow regime. A parametric study is performed to reveal the influence of the annular diameter ratio, k, the Prandtl number, Pr, and the Reynolds number, Re, over the following range of parameters: k = {0, 0.5, 0.7}, Pr = {0.7, 1, 7, 100}, and Re = {25, 50, 100}. Heat transfer enhancement downstream of the expansion plane is only observed for Pr > 1. Peak wall-heat-transfer-rates always appear downstream of the flow reattachment point, in the case of suddenly expanding round pipe flows, i.e. k = 0. However, for suddenly expanding annular pipe flows, i.e., k = 0.5 and 0.7, peak wall-heat-transfer-rates always appear upstream of the flow reattachment point. The observed heat transfer augmentation is more dramatic for suddenly expanding annular flows, in comparison with the one observed for suddenly expanding pipe flows. For a given annular diameter ratio and Reynolds number, increasing the Prandtl number, always results in higher wall-heat-transfer-rates downstream the expansion plane.

Turbulent dynamo action at low magnetic Reynolds number

1970 ◽  
Vol 41 (2) ◽  
pp. 435-452 ◽  
Author(s):  
H. K. Moffatt
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Large Scale ◽  
Isotropic Turbulence ◽  
Magnetic Energy ◽  
Magnetic Reynolds Number ◽  
Dynamo Action ◽  
Ohmic Dissipation ◽  
Magnetic Energy Density ◽  
Magnetic Diffusivity

The effect of turbulence on a magnetic field whose length-scale L is initially large compared with the scale l of the turbulence is considered. There are no external sources for the field, and in the absence of turbulence it decays by ohmic dissipation. It is assumed that the magnetic Reynolds number Rm = u0l/λ (where u0 is the root-mean-square velocity and λ the magnetic diffusivity) is small. It is shown that to lowest order in the small quantities l/L and Rm, isotropic turbulence has no effect on the large-scale field; but that turbulence that lacks reflexional symmetry is capable of amplifying Fourier components of the field on length scales of order Rm−2l and greater. In the case of turbulence whose statistical properties are invariant under rotation of the axes of reference, but not under reflexions in a point, it is shown that the magnetic energy density of a magnetic field which is initially a homogeneous random function of position with a particularly simple spectrum ultimately increases as t−½exp (α2t/2λ3) where α(= O(u02l)) is a certain linear functional of the spectrum tensor of the turbulence. An analogous result is obtained for an initially localized field.

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