Magnetohydrodynamic Turbulent Flows Research Articles

Large eddy simulation of magnetohydrodynamic turbulent channel flows with local subgrid-scale model based on coherent structures

For turbulent channel flows with a uniform magnetic field perpendicular to insulated walls, the performance of the coherent structure Smagorinsky model (CSM) is investigated in comparison to the Smagorinsky model (SM) and the dynamic Smagorinsky model (DSM). The Lorentz force acts against a streamwise flow. The effect of the Hartmann flattening leads to an increase in the wall shear stress, so that the skin friction coefficient increases. In contrast, the turbulence suppression by the magnetic field results in a decrease of the Reynolds shear stress near the wall, so that the skin friction coefficient decreases. As the magnetic field increases, a turbulent magnetohydrodynamic (MHD) flow transits to a laminar MHD flow at a critical Hartmann number. The CSM predicts a higher transition Hartmann number than the DSM and SM, because the model parameter of the CSM is locally determined based on coherent structures and the fluctuations are reflected in the shear stress. On the other hand, the model parameter of the DSM is averaged in the homogeneous directions, so that the shear stress is somewhat underestimated for the subcritical Hartmann number. The SM with a model constant and a wall damping function of the Van Driest type reproduces the laminar MHD flow at the lowest transition Hartmann number, because the model parameter (which does not change in the magnetic field) provides significant energy dissipation. Moreover, the CSM and DSM can reproduce properly the laminar MHD flow at high Hartmann number, because the model parameters of the CSM and DSM are drastically damped near the wall and the Reynolds shear stresses are suppressed to zero. The skin friction coefficients predicted by the CSM and DSM agree with the “two-dimensional” laminar solution, whereas the SM gives higher values than the laminar solution. The coherent structures become large and align themselves along the magnetic field in the transition to the laminar MHD flow.

Damping rate of magnetohydrodynamic vortices at low magnetic Reynolds number

The damping rate of vortices in an electrically conducting fluid submitted to a uniform magnetic field is analyzed for a large Hartmann number Ha. The fluid is contained in a layer of constant thickness h, bounded by two insulating walls that are perpendicular to the magnetic field. The damping times and the eigenfunctions along the magnetic field are obtained from a linear eigenvalue problem. According to the damping times and these eigenfunctions, vortices are classified into several classes by the range of combinations of the mode number m in the magnetic field direction and the wave number k2D in the plane perpendicular to the magnetic field. It is found that the damping rate of vortices in the range of k2D∼[(m+12)πHa]1∕2h−1 and m=0,1,2 is of the same order as that of large-scale two-dimensional vortices. This fact suggests that actual quasi-two-dimensional magnetohydrodynamic turbulent flows include not only m=0 but also higher-mode (m⩾1) eigenfunctions of this wave-number range, although the eigenfunction of m=0 has a 30% variation and the higher-mode eigenfunctions change their sign along the magnetic field.

A numerical study of the alpha model for two-dimensional magnetohydrodynamic turbulent flows

We explore some consequences of the “alpha model,” also called the “Lagrangian-averaged” model, for two-dimensional incompressible magnetohydrodynamic (MHD) turbulence. This model is an extension of the smoothing procedure in fluid dynamics, which filters velocity fields locally while leaving their associated vorticities unsmoothed, and has proved useful for high Reynolds number turbulence computations. We consider several known effects (selective decay, dynamic alignment, inverse cascades, and the probability distribution functions of fluctuating turbulent quantities) in magnetofluid turbulence and compare the results of numerical solutions of the primitive MHD equations with their alpha-model counterparts’ performance for the same flows, in regimes where available resolution is adequate to explore both. The hope is to justify the use of the alpha model in regimes that lie outside currently available resolution, as will be the case in particular in three-dimensional geometry or for magnetic Prandtl numbers differing significantly from unity. We focus our investigation, using direct numerical simulations with a standard and fully parallelized pseudospectral method and periodic boundary conditions in two space dimensions, on the role that such a modeling of the small scales using the Lagrangian-averaged framework plays in the large-scale dynamics of MHD turbulence. Several flows are examined, and for all of them one can conclude that the statistical properties of the large-scale spectra are recovered, whereas small-scale detailed phase information (such as, e.g., the location of structures) is lost.

Numerical Simulation of Magnetohydrodynamic Flows

The development of a versatile computational tool for the solution of turbulent magnetohydrodynamic flows is presented. The flow solver can simulate the full magnetohydrodynamic equations or simplified equations based on the low magnetic Reynolds number approximation. Both laminar and turbulent flows are investigated. The Baldwin‐Lomax turbulence model evaluates the eddy viscosity to represent turbulence. Modifications accounting for the presence of a magnetic field are presented to extend its range of application to magnetohydrodynamic flows. The modification of the turbulence model is performed based on the turbulent Hartmann flow. The numerical solutions are compared with existing analytical solutions and experimental data, for low- and high-speed magnetohydrodynamic flows.

A direct-numerical-simulation-based second-moment closure for turbulent magnetohydrodynamic flows

A magnetic field, imposed on turbulent flow of an electrically conductive fluid, is known to cause preferential damping of the velocity and its fluctuations in the direction of Lorentz force, thus leading to an increase in stress anisotropy. Based on direct numerical simulations (DNS), we have developed a model of magnetohydrodynamic (MHD) interactions within the framework of the second-moment turbulence closure. The MHD effects are accounted for in the transport equations for the turbulent stress tensor and energy dissipation rate—both incorporating also viscous and wall-vicinity nonviscous modifications. The validation of the model in plane channel flows with different orientation of the imposed magnetic field against the available DNS (Re=4600,Ha=6), large eddy simulation (Re=2.9×104,Ha=52.5,125) and experimental data (Re=5.05×104 and Re=9×104, 0⩽Ha⩽400), show good agreement for all considered situations.

Minimum correlation time in turbulent magnetohydrodynamic flows

The correlation time of the velocity in a magnetohydrodynamic flow plays an important role in several models of dynamo theory. In particular δ-correlated flows and asymptotic analyses for very small correlation times have been often considered in the study of kinematic dynamos with random plasma flows. It is rigorously proved here that the correlation time in a real flow is bounded below by a positive quantity depending only on the total enstrophy. As a consequence, two-dimensional flows as well as physically realistic three-dimensional ones, including the classical models of homogeneous turbulence, posses a minimum correlation time weakly dependent on the initial condition. This emphasises the need to be cautious when taking random flows as depicting real physical processes.

Von Kármán-Howarth relationship for helical magnetohydrodynamic flows.

We derive an exact equation for homogeneous isotropic magnetohydrodynamic (MHD) turbulent flows with nonzero helicity; this result is of the same nature as the classical von Kármán-Howarth (VKH-HM) formulation for the kinetic energy of turbulent fluids. Helical MHD is relevant to the astrophysical flows such as in the solar corona, or the interstellar medium, and in the dynamo problem. The derivation involves the new writing of the general form of tensors for that case, for either vectors or (pseudo)axial vectors. It is shown that, for general third-order tensors, four generating functions are needed when taking into account the nonmirror invariance of helical fluids, instead of two as in the fully isotropic case. The new equation obtained, denoted by VKH-HM, links the dissipation of magnetic helicity to the third-order correlations involving combinations of the components of the velocity, the magnetic field, and the magnetic potential. Finally, in the long-time and nonresistive limit, this relationship leads to a linear scaling with separation of the third-order tensor, correlating the two normal components of the electromotive force and of the magnetic potential.

The correlation between magnetic pressure and density in compressible MHD turbulence

We study, both analytically and numerically, the behavior of magnetic pressure and density fluctuations in turbulent isothermal magnetohydrodynamic (MHD) flows in a slab geometry. We first consider “simple” MHD waves, which are the nonlinear analogue of the slow, fast and Alfvén linear waves, and show that the dependence of magnetic field strength B on density ρ in a simple wave depends on the mode which is considered: for the slow mode, , while for the fast mode, . We also perform a perturbative analysis about a circularly-polarized plane Alfvén wave to investigate Alfvén wave pressure, recovering the results of McKee and Zweibel that , with at large Ma, at moderate Ma and long wavelengths, and at low Ma. This wide variety of behaviors implies that a single polytropic description of magnetic pressure is not possible in general, but instead depends on which mode dominates the density fluctuation production. This in turn depends on the angle θ between the magnetic field and the direction of wave propagation and on the Alfvénic Mach number Ma. Typically, at small Ma, the slow mode dominates, and B is anticorrelated with ρ . At large Ma, both modes contribute to density fluctuation production, and the magnetic pressure decorrelates from density, exhibiting a large scatter, which however decreases towards higher densities. In this case, magnetic “pressure” does not act as a restoring force, but rather as a random forcing. These results have implications for the probability density function (PDF) of mass density. The non-systematic behavior of the magnetic pressure causes the PDF to maintain the log-normal shape corresponding to non-magnetic isothermal turbulence, except in cases where the slow mode dominates, in which the PDF develops an excess at low densities because the magnetic “random forcing” becomes density-dependent. Our results are consistent with the low values and apparent lack of correlation between the magnetic field strength and density in surveys of the lower-density molecular gas, and also with the recorrelation apparently seen at higher densities, if the Alfvénic Mach number is relatively large there.

Application of the “ K– ε” model to open channel flows in a magnetic field

In magnetohydrodynamic (MHD) flows turbulence reduction occurs due to the Joule dissipation. It results in heat transfer degradation. In open channel flows, heat transfer degradation is also caused by the turbulence redistribution near the free surface. Both effects can be significant in fusion applications with low-conductivity fluids such as molten salts. In the present study, the “ K– ε” model equations for turbulent flows and the free surface boundary condition are adjusted with taking into account MHD effects. Different orientations of the magnetic field, perpendicular and parallel to the main flow, have been considered. The model coefficients have been tuned by a computer optimization using available experimental data for the friction factor. The effect of free surface heat transfer degradation due to the turbulence redistribution has been implemented through the variation of the turbulent Prandtl number. As an example, the model is used for the analysis of a turbulent MHD flow down an inclined chute with the heat flux applied to the free surface.

Magnetohydrodynamic turbulent flow in a channel at low magnetic Reynolds number

Effects of the Lorentz force on near-wall turbulence structures are investigated using the direct numerical simulation technique with the assumption of no induced magnetic field at low magnetic Reynolds number. A uniform magnetic field is applied in the streamwise (x), wall-normal (y) or spanwise (z) direction to turbulent flow in an infinitely long channel with non-conducting walls. The Lorentz force induced from the magnetic field suppresses the dynamically significant coherent structures near the wall. The skin friction decreases with increasing streamwise and spanwise magnetic fields, whereas it increases owing to the Hartmann effect when the strength of the wall-normal magnetic field exceeds a certain value. All the turbulence intensities and the Reynolds shear stress decrease with the wall-normal and spanwise magnetic fields, but the streamwise velocity fluctuations increase with the streamwise magnetic field although all other turbulence intensities decrease. It is also shown that the wall-normal magnetic field is much more effective than the streamwise and spanwise magnetic fields in reducing turbulent fluctuations and suppressing the near-wall streamwise vorticity, even though the wall-normal magnetic field interacts directly with the mean flow and results in drag increase at strong magnetic fields. In the channel with a strong streamwise magnetic field, two-dimensional streamwise velocity fluctuations u(y, z) exist, even after other components of the velocity fluctuations nearly vanish. In the cases of strong wall-normal and spanwise magnetic fields, all turbulence intensities, the Reynolds shear stress and vorticity fluctuations decrease rapidly and become zero. The turbulence structures are markedly elongated in the direction of the applied magnetic field when it is strong enough. It is shown that this elongation of the structures is associated with a rapid decrease of the Joule dissipation in time.

The behaviour of gas bubbles in a turbulent liquid metal magnetohydrodynamic flow: Part I: Dispersion in quasi-two-dimensional magnetohydrodynamic turbulence

We investigate the dispersion of gas bubbles injected from a single orifice into a liquid metal flow subjected to transverse and longitudinal magnetic fields. The local void fraction is detected by means of resistivity probes. The results show a damping of velocity fluctuations corresponding to a decrease in the bubble dispersion with increasing magnetic field. An isotropic distribution of the gas phase is preserved if a longitudinal field is applied, while an anisotropic distribution is observed in case the applied magnetic field is transverse to the flow indicating the existence of quasi-two-dimensional vortices as typical of turbulent magnetohydrodynamic flows.

Parametric investigation of self-similar decay laws in MHD turbulent flows

An investigation of the decay laws of energy and of higher moments of the Elsässer fields z±=v±b in the self-similar regime of magnetohydrodynamic (MHD) turbulence is presented, using phenomenological models as well as two-dimensional numerical simulations with periodic boundary conditions and up to 20482 grid points. The results are compared with the generalization of the parameter-free model derived by Galtier et al. [Phys. Rev. Lett.79, 2807 (1997)], which takes into account the slowing down of the dynamics due to the propagation of Alfvén waves. The new model developed here allows for a study in terms of one parameter governing the wavenumber dependence of the energy spectrum at scales of the order of (and larger than) the integral scale of the flow. The one-dimensional compressible case is also dealt with in two of its simplest configurations. Computations are performed for a standard Laplacian diffusion as well as with a hyperdiffusive algorithm. The results are sensitive to the amount of correlation between the velocity and the magnetic field, but rather insensitive to all other parameters such as the initial ratio of kinetic to magnetic energy or the presence or absence of a uniform component of the magnetic field. In all cases, the decay is significantly slower than for neutral fluids in a way that favours for MHD flows the phenomenology of Iroshnikov [Soviet Astron.7, 566 (1963)] and Kraichnan [Phys. Fluids8, 1385 (1965)] as opposed to that of Kolmogorov [Dokl. Akad. Nauk. SSSR31, 538 (1941)]. The temporal evolution of q-moments of the generalized vorticities 〈[mid ]ω±[mid ]q〉 =〈[mid ]ω±j[mid ]q〉 up to order q=10 is also given, and is compared with the prediction of the model. Less agreement obtains as q grows – a fact probably due to intermittency and the development of coherent structures in the form of eddies, and of vorticity and current sheets.

Experimental Evidence for Differences in the Extended Self-Similarity Scaling Laws between Fluid and Magnetohydrodynamic Turbulent Flows

Experimental Evidence for Differences in the Extended Self-Similarity Scaling Laws between Fluid and Magnetohydrodynamic Turbulent Flows

Evidences for extended self‐similarity in hydromagnetic turbulence

We investigate the occurrence of Extended Self‐Similarity in the Solar Wind Magnetohydrodynamic (MHD) turbulence by using Helios 2 bulk velocity data recorded in the inner heliosphere at five different heliocentric distances. For each analysed time interval, we compute the sets of scaling exponents for the q‐th order velocity structure functions and we show that these sets turn out to be to a good approximation the same whatever the different heliocentric distances. This result shows that a universal non gaussian probability distribution function, which depends on a unique parameter, describes the velocity differences in a MHD turbulent flow also outside the inertial range and in presence of intermittency. Moreover comparing the values of the obtained scaling exponents with those derived from the existing theoretical models of intermittency, we can infer that the most intermittent MHD structures should be two‐dimensional current sheets.

Experimental evidence for differences in the extended self-similarity scaling laws between fluid and magnetohydrodynamic turbulent flows.

It has been recently suggested that the various $q$th order velocity structure functions in turbulent flows are related to each other through well defined scaling laws, which extend outside the usual inertial domain. Even if theoretical models provide different scaling laws for fluid and magnetohydrodynamic turbulent flows, no attempt has been made up to now to furnish experimental evidence for these differences. By using measurements from the solar wind turbulence and from turbulence in ordinary fluid flows, we show that the differences can be observed only by looking at the high-order velocity structure functions.

Large eddy simulation of magnetohydrodynamic turbulent channel flows under a uniform magnetic field

Magnetohydrodynamic turbulent channel flows under a uniform magnetic field are studied by using a large eddy simulation technique. The effect of a uniform magnetic field is incorporated into the subgrid-scale model in the form of a local damping factor. Compared with the experiment, this model is shown to work better than the usual Smagorinsky model in both turbulent and laminar states. This model predicts the detailed structures of the present flows, for which, one-point global turbulence models of ensemble mean type are helpless.

Magnetohydrodynamic buoyancy-driven turbulence in thermally stratified conducting flow

Magnetohydrodynamic (MHD) buoyancy-driven instability has much in common with gravity-driven natural convection instability in thermally stratified fluids, in that account must be taken of the temperature dependence of conductivity in the former case and of density in the latter case. Employing the Monin–Obukhov theory of mixed convection in a stratified atmosphere, temperature and velocity profiles in MHD turbulent shear flow are obtained. The problem of the boundary-layer development is solved, and the heat and momentum fluxes are found. It is shown that the buoyancy force effects drastically change the fluxes when the Stuart number exceeds unity. Anode–cathode asymmetry observed experimentally on MHD devices is explained.

Liquid-metal flows in sliding electrical contacts with arbitrary magnetic-field orientations

In certain situations, liquid-metal sliding electrical contacts for high-current and low-voltage electrical machines may prove a viable alternative to solid metal brushes. Before it can be ascertained whether such an option is feasible, the problems inherent in a liquid-metal flow through a narrow gap between a fixed and a moving surface with free surfaces beyond each gap end must be explored. The flow occurs in the presence of an arbitrarily oriented magnetic field. By assuming that the secondary flow is negligible, the problem reduces to a fully developed magnetohydrodynamic (MHD) duct flow problem. In the parameter range presented here, the liquid-metal flow can be laminar or turbulent, requiring that both regimes be analyzed. The numerical results from the mathematical model presented herein for laminar flow with arbitrary Hartmann number M and with arbitrary magnetic-field orientation indicate that, even with an O(1) Hartmann number, the flow is already beginning to evolve into the distinct regions predicted by the asymptotic solution for M≫1 derived from singular perturbation theory. However, the actual velocity and electric potential distributions only agree approximately with those predicted by the large M asymptotic solutions. The numerical results presented in this work for the turbulent MHD flow are profoundly different from their corresponding laminar counterparts. Velocity magnitudes are significantly reduced in the turbulent MHD flows, and as a consequence, the magnitude of the current density increases. Thus, according to the mathematical model presented in this paper, liquid-metal sliding electrical contacts in external magnetic fields appear to transport current more efficiently in the turbulent regime than the laminar regime under the conditions of the calculations.

Statistical Analysis of Magnetohydrodynamic Turbulent Shear Flows at Low Magnetic Reynolds Number

Magnetohydrodynamic turbulent shear flows at low magnetic Reynolds number are studied by using a two-scale direct-interaction approximation combined with the double expansion with regard to the magnetic Reynolds number and the interaction parameter. Using these statistical results, a two-equation turbulence model which incorporates the effects of magnetic fluctuations is constructed. This model can quantitatively explain existing experimental results for the skin friction in MHD channel flows.

Statistical Analysis of Magnetohydrodynamic Turbulent Shear Flows at Low Magnetic Reynolds Number

Magnetohydrodynamic turbulent shear flows at low magnetic Reynolds number are studied by using a two-scale direct-interaction approximation combined with the double expansion with regard to the magnetic Reynolds number and the interaction parameter. Using these statistical results, a two-equation turbulence model which incorporates the effects of magnetic fluctuations is constructed. This model can quantitatively explain existing experimental results for the skin friction in MHD channel flows.