On the decay of low-magnetic-Reynolds-number turbulence in an imposed magnetic field

2010 ◽  
Vol 651 ◽  
pp. 295-318 ◽  
Author(s):  
N. OKAMOTO ◽  
P. A. DAVIDSON ◽  
Y. KANEDA
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Isotropic Turbulence ◽  
Magnetic Reynolds Number ◽  
Integral Invariant ◽  
Mhd Turbulence ◽  
Velocity Correlation ◽  
Developed Turbulence ◽  
Integral Scales ◽  
Point Velocity

We examine the integral properties of freely decaying homogeneous magnetohydrodynamic (MHD) turbulence subject to an imposed magnetic field B0 at low-magnetic Reynolds number. We confirm that, like conventional isotropic turbulence, the fully developed state possesses a Loitsyansky-like integral invariant, in this case I// = − ∫ r⊥2 〈 u⊥·u′⊥〉 dr, where 〈u(x) ·u(x + rc)〉 = 〈u·u′〉 is the usual two-point velocity correlation and the subscript ⊥ indicates components perpendicular to the imposed field. The conservation of I// for fully developed turbulence places a fundamental restriction on the way in which the integral scales can develop, i.e. it implies u⊥2 ℓ⊥4 ℓ// ≈ constant where u⊥, ℓ⊥ and ℓ// are integral scales. This constraint can be used to estimate the evolution of u⊥(t; B0), ℓ⊥(t; B0) and ℓ//(t; B0), and these theoretical decay laws are shown to be in good agreement with numerical simulations.

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.

Fully developed MHD turbulence near critical magnetic Reynolds number

1981 ◽  
Vol 104 ◽  
pp. 419-443 ◽  
Author(s):  
J. Léorat ◽  
A. Pouquet ◽  
U. Frisch
Keyword(s):  
Reynolds Number ◽  
Kinetic Energy ◽  
Isotropic Turbulence ◽  
Magnetic Energy ◽  
Reynolds Numbers ◽  
Magnetic Reynolds Number ◽  
Mhd Equations ◽  
Liquid Sodium ◽  
Turbulent Heat ◽  
Mhd Turbulence

Liquid-sodium-cooled breeder reactors may soon be operating at magnetic Reynolds numbers RM where magnetic fields can be self-excited by a dynamo mechanism (as first suggested by Bevir 1973). Such flows have kinetic Reynolds numbers RV of the order of 107 and are therefore highly turbulent.This leads us to investigate the behaviour of MHD turbulence with high RV and low magnetic Prandtl numbers. We use the eddy-damped quasi-normal Markovian closure applied to the MHD equations. For simplicity we restrict ourselves to homogeneous and isotropic turbulence, but we do include helicity.We obtain a critical magnetic Reynolds number RMc of the order of a few tens (non-helical case) above which magnetic energy is present. RMc is practically independent of RV (in the range 40 to 106). RMc can be considerably decreased by the presence of helicity: when the overall size of the flow L is much larger than the integral scale l0, RMc can drop below unity as suggested by an α-effect argument. When L ≈ l0 the drop can still be substantial (factor of 6) when helicity is a maximum. We examine how the turbulence is modified when RM crosses RMc: presence of magnetic energy, decreased kinetic energy, steepening of kinetic-energy spectrum, etc.We make no attempt to obtain quantitative estimates for a breeder reactor, but discuss some of the possible consequences of exceeding RMc, such as decreased turbulent heat transport. More precise information may be obtained from numerical simulations and experiments (including some in the subcritical regime).

scholarly journals The decay of isotropic magnetohydrodynamics turbulence and the effects of cross-helicity

2018 ◽  
Vol 84 (1) ◽  
Author(s):  
Antoine Briard ◽  
Thomas Gomez
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Energy Spectrum ◽  
Kinetic Energy ◽  
Total Energy ◽  
Isotropic Turbulence ◽  
Reynolds Numbers ◽  
Inertial Range ◽  
Mhd Turbulence ◽  
Previous Prediction

Decaying homogeneous and isotropic magnetohydrodynamics (MHD) turbulence is investigated numerically at large Reynolds numbers thanks to the eddy-damped quasi-normal Markovian (EDQNM) approximation. Without any background mean magnetic field, the total energy spectrum $E$ scales as $k^{-3/2}$ in the inertial range as a consequence of the modelling. Moreover, the total energy is shown, both analytically and numerically, to decay at the same rate as kinetic energy in hydrodynamic isotropic turbulence: this differs from a previous prediction, and thus physical arguments are proposed to reconcile both results. Afterwards, the MHD turbulence is made imbalanced by an initial non-zero cross-helicity. A spectral modelling is developed for the velocity–magnetic correlation in a general homogeneous framework, which reveals that cross-helicity can contain subtle anisotropic effects. In the inertial range, as the Reynolds number increases, the slope of the cross-helical spectrum becomes closer to $k^{-5/3}$ than $k^{-2}$. Furthermore, the Elsässer spectra deviate from $k^{-3/2}$ with cross-helicity at large Reynolds numbers. Regarding the pressure spectrum $E_{P}$, its kinetic and magnetic parts are found to scale with $k^{-2}$ in the inertial range, whereas the part due to cross-helicity rather scales in $k^{-7/3}$. Finally, the two $4/3$rd laws for the total energy and cross-helicity are assessed numerically at large Reynolds numbers.

The role of angular momentum in the magnetic damping of turbulence

1997 ◽  
Vol 336 ◽  
pp. 123-150 ◽  
Author(s):  
P. A. DAVIDSON
Keyword(s):  
Magnetic Field ◽  
Angular Momentum ◽  
Long Range ◽  
Time Scale ◽  
Isotropic Turbulence ◽  
Mhd Turbulence ◽  
Velocity Correlation ◽  
Long Range Correlations ◽  
Rate Of Decay ◽  
The Magnetic Field

Landau & Lifshitz showed that Kolmogorov's E∼t−10/7 law for the decay of isotropic turbulence rests on just two physical ideas: (a) the conservation of angular momentum, as expressed by Loitsyansky's integral; and (b) the removal of energy from the large scales via the energy cascade. Both Kolmogorov's original analysis and Landau & Lifshitz's reinterpretation in terms of angular momentum are now known to be flawed. The existence of long-range velocity correlations means that Loitsyansky's integral is not an exact representation of angular momentum, nor is it strictly conserved. However, in practice the long-range velocity correlations are weak and Loitsyansky's integral is almost constant, so that the Kolmogorov/Landau model provides a surprisingly simple and robust description of the decay. In this paper we redevelop these ideas in the context of MHD turbulence. We take advantage of the fact that the angular momentum of a fluid moving in a uniform magnetic field has particularly simple properties. Specifically, the component parallel to the magnetic field is conserved while the normal components decay exponentially on a time scale of τ=ρ/σB2. We show that the counterpart of Loitsyansky's integral for MHD turbulence is ∫x2⊥Q⊥dx, where Qij is the velocity correlation. When the long-range correlations are weak this integral is conserved. This provides an estimate of the rate of decay of energy. At low values of magnetic field we recover Kolmogorov's law. At high values we find E∼t−1/2, which is a result derived earlier by Moffatt. We also show that ∫x2⊥Q∥dx decays exponentially on a time scale of τ. We interpret these results in terms of the behaviour of isolated vortices orientated normal and parallel to the magnetic field.

scholarly journals Nonlinear amplification in hydrodynamic turbulence

2021 ◽  
Vol 930 ◽  
Author(s):  
Kartik P. Iyer ◽  
Katepalli R. Sreenivasan ◽  
P.K. Yeung
Keyword(s):  
Reynolds Number ◽  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Developed Turbulence ◽  
Advection Term ◽  
Nonlinear Amplification

Using direct numerical simulations performed on periodic cubes of various sizes, the largest being $8192^3$ , we examine the nonlinear advection term in the Navier–Stokes equations generating fully developed turbulence. We find significant dissipation even in flow regions where nonlinearity is locally absent. With increasing Reynolds number, the Navier–Stokes dynamics amplifies the nonlinearity in a global sense. This nonlinear amplification with increasing Reynolds number renders the vortex stretching mechanism more intermittent, with the global suppression of nonlinearity, reported previously, restricted to low Reynolds numbers. In regions where vortex stretching is absent, the angle and the ratio between the convective vorticity and solenoidal advection in three-dimensional isotropic turbulence are statistically similar to those in the two-dimensional case, despite the fundamental differences between them.

scholarly journals Inverse Energy Cascade in Advanced MHD Turbulence (the RNG Method)

1993 ◽  
Vol 157 ◽  
pp. 255-261
Author(s):  
N. Kleeorin ◽  
I. Rogachevskii
Keyword(s):  
Magnetic Field ◽  
Large Scale ◽  
Magnetic Field Effect ◽  
Magnetic Reynolds Number ◽  
Energy Cascade ◽  
Small Scale ◽  
Magnetic Fluctuations ◽  
Mhd Turbulence ◽  
Inverse Energy Cascade ◽  
Large Scale Magnetic Field

The nonlinear (in terms of the large-scale magnetic field) effect of the modification of the magnetic force by an advanced small-scale magnetohydrodynamic (MHD) turbulence is considered. The phenomenon is due to the generation of magnetic fluctuations at the expense of hydrodynamic pulsations. It results in a decrease of the elasticity of the large-scale magnetic field.The renormalization group (RNG) method was employed for the investigation of the MHD turbulence at the large magnetic Reynolds number. It was found that the level of the magnetic fluctuations can exceed that obtained from the equipartition assumption due to the inverse energy cascade in advanced MHD turbulence.This effect can excite an instability of the large-scale magnetic field due to the energy transfer from the small-scale turbulent pulsations. This instability is an example of the inverse energy cascade in advanced MHD turbulence. It may act as a mechanism for the large-scale magnetic ropes formation in the solar convective zone and spiral galaxies.

Turbulent energy density in scale space for inhomogeneous turbulence

2018 ◽  
Vol 842 ◽  
pp. 532-553 ◽  
Author(s):  
Fujihiro Hamba
Keyword(s):  
Energy Spectrum ◽  
Energy Density ◽  
Isotropic Turbulence ◽  
Scale Space ◽  
Scale Dependence ◽  
Energy Cascade ◽  
Homogeneous Isotropic Turbulence ◽  
Velocity Correlation ◽  
Inhomogeneous Turbulence ◽  
Point Velocity

The energy spectrum is commonly used to describe the scale dependence of the turbulent fluctuations in homogeneous isotropic turbulence. In contrast, one-point statistical quantities, such as the turbulent kinetic energy, are employed for inhomogeneous turbulence modelling. To obtain a better understanding of inhomogeneous turbulence, some attempts have been made to describe its scale dependence by using the second-order structure function and the two-point velocity correlation. However, previous expressions for the energy density in the scale space do not satisfy the requirement that it should be non-negative. In this work, a new expression for the energy density in the scale space is proposed on the basis of the two-point velocity correlation; the integral with a filter function is introduced to satisfy the non-negativity of the energy density. Direct numerical simulation (DNS) data of homogeneous isotropic turbulence were first used to assess the role of the energy density by comparing it with the energy spectrum. DNS data of a turbulent channel flow were then used to investigate the energy density and its transport equation in inhomogeneous turbulence. It was shown that the new energy density is positive in the scale space of the homogeneous direction. The energy transfer was successfully examined in the scale space both in the homogeneous and inhomogeneous directions. The energy cascade from large to small scales was clearly observed. Moreover, the inverse energy cascade from large to very large scales was observed in the scale space of the spanwise direction.

A comparison of heisenberg's spectrum of turbulence with experiment

1951 ◽  
Vol 47 (1) ◽  
pp. 158-176 ◽  
Author(s):  
Ian Proudman ◽  
G. K. Batchelor
Keyword(s):  
Reynolds Number ◽  
Correlation Functions ◽  
Isotropic Turbulence ◽  
Reynolds Numbers ◽  
Velocity Correlation ◽  
Approximate Form ◽  
Velocity Correlation Functions ◽  
Limiting Case

AbstractIn this paper, the theoretical double and triple velocity correlation functions, f(r), g(r) and h(r), which correspond to Heisenberg's spectrum of isotropic turbulence, are obtained numerically for two Reynolds numbers. One set of these correlations is for the limiting case of infinite Reynolds number. In addition, a method is developed for deriving the approximate form of the double correlations for any Reynolds number, which is not too small, from the corresponding correlations for infinite Reynolds number. These theoretical correlations are then compared with the results of experiment.

Minimum-drag shapes in magnetohydrodynamics

2011 ◽  
Vol 467 (2136) ◽  
pp. 3371-3392 ◽  
Author(s):  
Michael Zabarankin
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Optimality Condition ◽  
Boundary Integral Equations ◽  
Integral Formula ◽  
Magnetic Reynolds Number ◽  
Boundary Integral ◽  
The Body ◽  
Mhd Equations ◽  
Minimum Drag

A necessary optimality condition for the minimum-drag shape for a non-magnetic solid body immersed in the uniform flow of an electrically conducting viscous incompressible fluid under the presence of a magnetic field is obtained. It is assumed that the flow and magnetic field are uniform and parallel at infinity, and that the body and fluid have the same magnetic permeability. The condition is derived based on the linearized magnetohydrodynamic (MHD) equations subject to a constraint on the body’s volume, and generalizes the existing optimality conditions for the minimum-drag shapes for the body in the Stokes and Oseen flows of a non-conducting fluid. It is shown that for any Hartmann number M , Reynolds number Re and magnetic Reynolds number Re m , the minimum-drag shapes are fore-and-aft symmetric and have conic vertices with an angle of 2 π /3. The minimum-drag shapes are represented in a function-series form, and the series coefficients are found iteratively with the derived optimality condition. At each iteration, the MHD problem is solved via the boundary integral equations obtained based on the Cauchy integral formula for generalized analytic functions. With respect to the equal-volume sphere, drag reduction as a function of the Cowling number S= M 2 /( Re m   Re ) is smallest at S=1. Also, in the considered examples, the drag values for the minimum-drag shapes and equal-volume minimum-drag spheroids are sufficiently close.

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