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.

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.

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.

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.

Investigation of MHD RANS Modeling Base on DNS Database Under the Advanced Blanket Design Conditions Utilized Molten Salt

2014 ◽  
Author(s):  
Naoki Osawa ◽  
Yoshinobu Yamamoto ◽  
Tomoaki Kunugi
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Turbulence Model ◽  
Turbulent Channel Flow ◽  
Turbulent Heat Transfer ◽  
Navier Stokes ◽  
Turbulent Heat ◽  
Mhd Turbulence ◽  
Rans Modeling ◽  
Heat Transfer Degradation

In this study, validations of Reynolds Averaged Navier-Stokes Simulation (RANS) based on Kenjeres & Hanjalic MHD turbulence model (Int. J. Heat & Fluid Flow, 21, 2000) coupled with the low-Reynolds number k-epsilon model have been conducted with the usage of Direct Numerical Simulation (DNS) database. DNS database of turbulent channel flow imposed wall-normal magnetic field on, are established in condition of bulk Reynolds number 40000, Hartmann number 24, and Prandtl number 5. As the results, the Nagano & Shimada model (Trans. JSME series B. 59, 1993) coupled with Kenjeres & Hanjalic MHD turbulence model has the better availability compared with Myong & Kasagi model (Int. Fluid Eng, 109, 1990) in estimation of the heat transfer degradation in MHD turbulent heat transfer.

scholarly journals Reynolds-number dependence of turbulence enhancement on collision growth

2016 ◽  
Vol 16 (19) ◽  
pp. 12441-12455 ◽  
Author(s):  
Ryo Onishi ◽  
Axel Seifert
Keyword(s):  
Reynolds Number ◽  
Isotropic Turbulence ◽  
Reynolds Numbers ◽  
Stokes Number ◽  
Homogeneous Isotropic Turbulence ◽  
Collision Efficiency ◽  
Cloud Droplets ◽  
Clustering Effect ◽  
Reynolds Number Dependence ◽  
Coagulation Kernel

Abstract. This study investigates the Reynolds-number dependence of turbulence enhancement on the collision growth of cloud droplets. The Onishi turbulent coagulation kernel proposed in Onishi et al. (2015) is updated by using the direct numerical simulation (DNS) results for the Taylor-microscale-based Reynolds number (Reλ) up to 1140. The DNS results for particles with a small Stokes number (St) show a consistent Reynolds-number dependence of the so-called clustering effect with the locality theory proposed by Onishi et al. (2015). It is confirmed that the present Onishi kernel is more robust for a wider St range and has better agreement with the Reynolds-number dependence shown by the DNS results. The present Onishi kernel is then compared with the Ayala–Wang kernel (Ayala et al., 2008a; Wang et al., 2008). At low and moderate Reynolds numbers, both kernels show similar values except for r2 ∼ r1, for which the Ayala–Wang kernel shows much larger values due to its large turbulence enhancement on collision efficiency. A large difference is observed for the Reynolds-number dependences between the two kernels. The Ayala–Wang kernel increases for the autoconversion region (r1, r2 < 40 µm) and for the accretion region (r1 < 40 and r2 > 40 µm; r1 > 40 and r2 < 40 µm) as Reλ increases. In contrast, the Onishi kernel decreases for the autoconversion region and increases for the rain–rain self-collection region (r1, r2 > 40 µm). Stochastic collision–coalescence equation (SCE) simulations are also conducted to investigate the turbulence enhancement on particle size evolutions. The SCE with the Ayala–Wang kernel (SCE-Ayala) and that with the present Onishi kernel (SCE-Onishi) are compared with results from the Lagrangian Cloud Simulator (LCS; Onishi et al., 2015), which tracks individual particle motions and size evolutions in homogeneous isotropic turbulence. The SCE-Ayala and SCE-Onishi kernels show consistent results with the LCS results for small Reλ. The two SCE simulations, however, show different Reynolds-number dependences, indicating possible large differences in atmospheric turbulent clouds with large Reλ.

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.

Fluid Flow Analysis in a Pebble Bed Modular Reactor Using RANS Turbulence Models

2008 ◽  
Author(s):  
Margaret Mkhosi ◽  
Richard Denning ◽  
Audeen Fentiman
Keyword(s):  
Reynolds Number ◽  
Fluid Flow ◽  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Turbulence Intensity ◽  
Turbulence Models ◽  
Reynolds Numbers ◽  
Pebble Bed ◽  
Rans Turbulence Models ◽  
Flow Domain

The computational fluid dynamics code FLUENT has been used to analyze turbulent fluid flow over pebbles in a pebble bed modular reactor. The objective of the analysis is to evaluate the capability of the various RANS turbulence models to predict mean velocities, turbulent kinetic energy, and turbulence intensity inside the bed. The code was run using three RANS turbulence models: standard k-ε, standard k-ω and the Reynolds stress turbulence models at turbulent Reynolds numbers, corresponding to normal operation of the reactor. For the k-ε turbulence model, the analyses were performed at a range of Reynolds numbers between 1300 and 22 000 based on the approach velocity and the sphere diameter of 6 cm. Predictions of the mean velocities, turbulent kinetic energy, and turbulence intensity for the three models are compared at the Reynolds number of 5500 for all the RANS models analyzed. A unit-cell approach is used and the fluid flow domain consists of three unit cells. The packing of the pebbles is an orthorhombic arrangement consisting of seven layers of pebbles with the mean flow parallel to the z-axis. For each Reynolds number analyzed, the velocity is observed to accelerate to twice the inlet velocity within the pebble bed. From the velocity contours, it can be seen that the flow appears to have reached an asymptotic behavior by the end of the first unit cell. The velocity vectors for the standard k-ε and the Reynolds stress model show similar patterns for the Reynolds number analyzed. For the standard k-ω, the vectors are different from the other two. Secondary flow structures are observed for the standard k-ω after the flow passes through the gap between spheres. This feature is not observable in the case of both the standard k-ε and the RSM. Analysis of the turbulent kinetic energy contours shows that there is higher turbulence kinetic energy near the inlet than inside the bed. As the Reynolds number increases, kinetic energy inside the bed increases. The turbulent kinetic energy values obtained for the standard k-ε and the RSM are similar, showing maximum turbulence kinetic energy of 7.5 m2·s−2, whereas the standard k-ω shows a maximum of about 20 m2·s−2. Another observation is that the turbulence intensity is spread throughout the flow domain for the k-ε and RSM whereas for the k-ω, the intensity is concentrated at the front of the second sphere. Preliminary analysis performed for the pressure drop using the standard k-ε model for various velocities show that the dependence of pressure on velocity varies as V1.76.

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.

The Drag on Spheres and Cylinders in a Stream of Dust-Laden Air

1959 ◽  
Vol 26 (4) ◽  
pp. 584-586
Author(s):  
Thomas Gillespie ◽  
A. W. Gunter
Keyword(s):  
Particle Size ◽  
Reynolds Number ◽  
Kinetic Energy ◽  
Drag Coefficient ◽  
Flow Pattern ◽  
Reynolds Numbers ◽  
Dust Concentration ◽  
Phase System ◽  
Two Phase ◽  
Coefficient Versus

Abstract A system has been developed for measuring the drag on small spheres and cylinders in a stream of dust-laden air. The drag was found to be proportional to the kinetic energy of the air plus the kinetic energy of the dust, and to be independent of particle size for particles having diameters in the range of 50 to 400μ. The well-known drag-coefficient versus Reynolds-number plots are the same for dust-free and dust-laden air provided the drag coefficient is calculated using the density of the two-phase system and the Reynolds numbers are calculated using the density of air alone. This suggests that the dust has little effect on the flow pattern. The results indicate that an instrument utilizing the drag principle to measure dust concentration could be developed.

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