A. Thess

Direct numerical simulation of forced MHD turbulence at low magnetic Reynolds number

The transformation of initially isotropic turbulent flow of electricallyn conducting incompressible viscous fluid under the influence of an imposed homogeneousn magnetic field is investigated using direct numerical simulation. Under the assumptionn of large kinetic and small magnetic Reynolds numbers (magnetic Prandtl number P m [Lt ]1) the quasi-static approximation is applied for the computation of the magnetic field fluctuations. The flow is assumed to be homogeneous and containedn in a three-dimensional cubic box with periodic boundary conditions. Large-scalen forcing is applied to maintain a statistically steady level of the flow energy.n It is found that the pathway traversed by the flow transformation depends decisivelyn on the magnetic interaction parameter (Stuart number). If the magnetic interactionn number is small the flow remains three-dimensional and turbulent and no detectablen deviation from isotropy is observed. In the case of a strong magnetic field (largen magnetic interaction parameter) a rapid transformation to a purely two-dimensionaln steady state is obtained in agreement with earlier analytical and numerical resultsn for decaying MHD turbulence. At intermediate values of the magnetic interaction parametern the system exhibits intermittent behaviour, characterized by organizedn quasi-two-dimensional evolution lasting several eddy-turnover times, which is interrupted byn strong three-dimensional turbulent bursts. This result implies that then conventional picture of steady angular energy transfer in MHD turbulence must be refined.n The spatial structure of the steady two-dimensional final flow obtained inn the case of large magnetic interaction parameter is examined. It is found that duen to the type of forcing and boundary conditions applied, this state always occurs in then form of a square periodic lattice of alternating vortices occupying the largest possiblen scale. The stability of this flow to three-dimensional perturbations is analysed usingn the energy stability method.

Square cells in surface-tension-driven Bénard convection: experiment and theory

The convective flow in a thin liquid layer with a free surface heated from below is studied using a combination of accurate experiments with silicone oil ( =0 : 1c m 2 s 1 ) and high-resolution direct numerical simulations of the time-dependent governing equations. It is demonstrated that above a certain value s of the threshold of primary instability, = 0, square convection cells rather than the seemingly all-embracing hexagons are the persistent dominant features of B enard convection. The transition from hexagonal to square cells sets in via a subcritical bifurcation and is accompanied by a sudden rapid increase of the Nusselt number. This implies that square cells are the more ecient mode of heat transport. Their wavenumber exceeds that of hexagonal cells by about 8%. The transition depends on the Prandtl number and it is shifted towards higher s if the Prandtl number is increased. The replacement of hexagonal by square cells is mediated by pentagonal cells. In the transitional regime from hexagonal to square cells, characterized by the presence of all three planforms, the system exhibits complex irregular dynamics on large spatial and temporal scales. The time dependence becomes more vivid with decreasing Prandtl number until nally non-stationary square cells appear. The simulations agree with the experimental observations in the phenomenology of the transition, and in the prediction of both the higher Nusselt number of square B enard cells and the subcritical nature of the transition. Quantitative dierences occur with respect to the values of s and the Prandtl number beyond which the time dependence vanishes. These dierences are the result of a considerably weaker mean flow in the simulation and of residual inhomogeneities in the lateral boundary conditions of the experiment which are below the threshold of control.

Inertial Bénard–Marangoni convection

Two-dimensional surface-tension-driven Benard convection in a layer with a free-slip bottom is investigated in the limit of small Prandtl number using accurate numerical simulations with a pseudospectral method complemented by linear stability analysis and a perturbation method. Upon increasing the Marangoni number Ma the system experiences a transition between two typical convective regimes. The first one is the regime of weak convection characterized by only slight deviations of the isotherms from the linear conductive temperature profile. In contrast, the second regime, called inertial convection, shows significantly deformed isotherms. The transition between the two regimes becomes increasingly sharp as the Prandtl number is reduced. The possibility of experimental verification of inertial Benard-Marangoni convection is briefly discussed.

Natural convection in a liquid metal heated from above and influenced by a magnetic field

Abstract Natural convection in a liquid metal heated locally at its upper surface and affected by a vertical magnetic field is investigated both experimentally and numerically. The experiments are conducted in a cylindrical test cell of large aspect ratio which is typical for application. The cell is filled with the liquid alloy GaInSn in eutectic composition. Temperature and velocity are measured using thermocouples and an electric potential probe, respectively. In the absence of the magnetic field the experimental results indicate a dependence of the Nusselt number on the Rayleigh number according to the law Nu∝Ra0.191. The particular value of the scaling exponent is in excellent agreement with the prediction of a scaling analysis for laminar, boundary layer-type flow in a low-Prandtl number fluid. Furthermore the experiments demonstrate that the Nusselt number and therefore the convective heat losses can be decreased by about 20% when a magnetic field of moderate strength (B=0.1xa0T) is present. The numerical simulations solve the Boussinesq equations in an axisymmetric geometry using a finite element method. The results of the simulations are both quantitatively and qualitatively in good agreement with the experimental observations. Deviations are attributed to the three-dimensional characteristics of the flow.

Thermocapillary flow in a Hele-Shaw cell

We formulate a simple theoretical model that permits one to investigate surfacetension-driven flows with complex interface geometry. The model consists of a HeleShaw cell lled with two dierent fluids and subjected to a unidirectional temperature gradient. The shape of the interface that separates the fluids can be arbitrarily complex. If the contact line is pinned, i.e. unable to move, the problem of calculating the flow in both fluids is governed by a linear set of equations containing the characteristic aspect ratio and the viscosity ratio as the only input parameters. Analytical solutions, derived for a linear interface and for a circular drop, demonstrate that for large aspect ratio the flow eld splits into a potential core flow and a thermocapillary boundary layer which acts as a source for the core. An asymptotic theory is developed for this limit which reduces the mathematical problem to a Laplace equation with Dirichlet boundary conditions. This problem can be eciently solved utilizing a boundary element method. It is found that the thermocapillary flow in non-circular drops has a highly non-trivial streamline topology. After releasing the assumption of a pinned interface, a linear stability analysis is carried out for the interface under both transverse and longitudinal temperature gradients. For a semi-innite fluid bounded by a freely movable surface long-wavelength instability due to the temperature gradient across the surface is predicted. The mechanism of this instability is closely related to the long-wave instability in surface-tension-driven B enard convection. A linear interface heated from the side is found to be linearly stable. The possibility of experimental verication of the predictions is briefly discussed.

ON BENARD-MARANGONI INSTABILITY IN THE PRESENCE OF A MAGNETIC FIELD

Explicit asymptotic expressions are derived for the first unstable mode of surface tension driven instability in an electrically conducting fluid subjected to a strong magnetic field. The spatial structure of the velocity, temperature, and electric current density is characterized in terms of Hartmann boundary layers—a concept that permits a physical explanation of the role of the magnetic field and an understanding of scaling laws derived in previous work.

On the linear growth rates of the long-wave modes in Bénard–Marangoni convection

Explicit analytical expressions for the linear growth (and decay) rates of long-wave modes in Benard–Marangoni convection are derived and discussed. These analytical predictions are shown to be in good agreement with experimental observations and are used to estimate the minimum experimental time necessary in order to observe the long-wave instability under microgravity conditions.

Direct numerical simulation as a tool for understanding MHD liquid metal turbulence

Abstract The method of direct numerical simulation (DNS) is applied to investigate the most general properties of turbulent flows of liquid metals in the presence of a constant magnetic field. Various aspects of the flow transformation into an anisotropic state are thoroughly examined. The flow is assumed to be homogeneous and the problem is reduced to the classical case of a turbulent flow in a 3D box with periodic boundary conditions. In the framework of this formulation, three specific types of the flow are considered, which are the forced flow, thermal convection, and freely decaying flow. To investigate the long-time evolution of an initially isotropic flow a large-scale forcing is applied to maintain the flow energy at a statistically steady level. The evolution is found to depend strongly on the magnetic interaction parameter (Stuart number). In the case of small Stuart number, the flow remains three-dimensional, turbulent, and approximately isotropic. At large Stuart number (strong magnetic field) the turbulence is suppressed rapidly and the flow becomes two-dimensional and laminar. Very interesting is the intermittent flow evolution detected at moderate Stuart number. Long periods of almost two-dimensional, laminar behaviour are interrupted by strong turbulent three-dimensional bursts. The influence of a constant magnetic field on scalar transport properties of liquid metal turbulence is investigated using the simplified formulation of a homogeneous flow driven by an imposed mean temperature gradient. The flow structure is dominated by two turbulent antiparallel jets providing an effective mechanism of heat transfer. It is shown that the magnetic field parallel to the mean temperature gradient stabilizes the jets and, thus, enhances heat transfer considerably. In the third part, freely decaying MHD turbulence is considered. Numerical simulations are applied to verify the theoretical model proposed in [J. Fluid Mech. 336 (1997) 123]. In particular, it is confirmed that the structure of viscous dissipation and evolution of perpendicular length scale are affected only slightly by the magnetic field. A simple approximation for the mean Joule dissipation is proposed.

On the symmetry of self‐organized structures in two‐dimensional turbulence

We report a systematic investigation of equilibrium states predicted by the statistical theory of vortex patches in a square box, in a channel, and in a square with doubly periodic boundary conditions. The study is limited to initial conditions containing negative and positive vortex patches with equal strength and area. It is demonstrated that the symmetry between positive and negative vorticity is, in general, broken by the self‐organized states. Direct numerical simulations support the predictions of the vortex patch statistics, but agree with point vortex statistics only in the limiting case of small area vortex patches.

A two-dimensional model for slow convection at infinite Marangoni number

The free surface of a viscous fluid is a source of convective flow (Marangoni convection) if its surface tension is distributed non-uniformly. Such non-uniformity arises from the dependence of the surface tension on a scalar quantity, either surfactant concentration or temperature. The surface-tension-induced velocity redistributes the scalar forming a closed-loop interaction. Under the assumptions of (i) small Reynolds number and (ii) vanishing diffusivity this nonlinear process is described by a single self-consistent two-dimensional evolution equation for the scalar field at the free surface that can be derived from the three-dimensional basic equations without approximation. The formulation of this equation for a particular system requires only the knowledge of the closure law, which expresses the surface velocity as a linear functional of the active scalar at the free surface. We explicitly derive these closure laws for various systems with a planar non-deflecting surface and infinite horizontal extent, including an infinitely deep fluid, a fluid with finite depth, a rotating fluid, and an electrically conducting fluid under the influence of a magnetic field.