Egbert Zienicke

Numerical study of the instability of the Hartmann layer

(Received 7 March 2003 and in revised form 1 July 2003) Direct numerical simulation is applied to investigate instability and transition to turbulence in the flow of an electrically conducting incompressible fluid between two parallel unbounded insulating walls affected by a wall-normal magnetic field (the Hartmann flow). The linear stability analysis of this flow provided unrealistically high critical Reynolds numbers, about two orders of magnitude higher than those observed in experiments. We propose an explanation based on the streak growth and breakdown mechanism described earlier for other shear flows. The mechanism is investigated using a two-step procedure that includes transient growth of two-dimensional optimal perturbations and the subsequent three-dimensional instability of the modulated streaky flow. In agreement with recent experimental investigations the calculations produce a critical range between 350 and 400 for the Hartmann thickness based Reynolds number, where the transition occurs at realistic amplitudes of two- and three-dimensional perturbations.

Numerical study of turbulent magnetohydrodynamic channel flow

Mean flow properties of turbulent magnetohydrodynamic channel flow with electrically insulating channel walls are studied using high-resolution direct numerical simulations. The Lorentz force due to the homogeneous wall-normal magnetic field is computed in the quasi-static approximation. For strong magnetic fields, the mean velocity profile shows a clear three-layer structure consisting of a viscous region near each wall and a plateau in the middle connected by logarithmic layers. This structure reflects the significance of viscous, turbulent, and electromagnetic stresses in the streamwise momentum balance dominating the viscous, logarithmic, and plateau regions, respectively. The width of the logarithmic layers changes with the ratio of Reynolds- and Hartmann numbers. Turbulent stresses typically decay more rapidly away from the walls than predicted by mixing-length models.

Constrained flow around a magnetic obstacle

Many practical applications exploit an external local magnetic field -- magnetic obstacle -- as an essential part of their constructions. Recently, it has been demonstrated that the flow of an electrically conducting fluid influenced by an external field can show several kinds of recirculation. The present paper reports a 3D numerical study whose some results are compared with an experiment about such a flow in a rectangular duct.

Stability and Numerical Simulation of the Liquid Metal Pinch Using the Shallow Water Approximation

In the reference [E. Zienicke, B.W. Li, A. Thess, A. Kratzschmar, P. Terhoeven, Stability Analysis of the Liquid Metal Pinch Using the Shallow Water Approximation, in Fifth International pamir Conference on Fundamental and Applied MHD, Ramatuelle, France-2002, Vol. 1, p.I-51~I-56], a simple physical model for a cylindrical jet of liquid metal passed through by an axial electrical current is introduced and its corresponding mathematical model based on shallow water approximation is deduced. In this paper, the MHD pinch instability is analyzed and compared with results from reference, and hence to guide the application of MHD pinch in construction of electric current switch. Besides this a pseudo spectral method is used to successfully simulate the time evolution of the pinch process. The numerical calculations showed that the nonlinear phase of the pinch process is very short. It starts approximately with a relative perturbation about 1/10 and needs around 0.2ms for GaInSn with a radius of R0 = 1cm and an overload factor of 10 above the critical current density. The shortness of the nonlinear phase is due to a strong self—acceleration of the pinch. If one wants to use the shortness of the nonlinear pinch phase the geometrical design of the switch has to give an initial narrowing of the current path by at least one tenth of the radius.

NUMERICAL SIMULATION OF TRANSITION TO TURBULENCE IN MHD CHANNEL FLOW

Direct numerical simulation is applied to investigate instability and transition to turbulence in the flow of an electrically conducting incompressible fluid between two parallel unbounded insulating walls affected by a wall-normal magnetic field (the Hartmann flow). An explanation is based on the streak growth and breakdown mechanism described earlier for other shear flows. The mechanism is investigated using a two-step procedure that includes 2D optimal perturbations and the subsequent 3D instability of the modulated streaky flow. The calculations produce a critical range between 350 and 400 for the Hartmann thickness based Reynolds number that agrees with recent experimental investigations.

Voltage-Driven Instability of Electrically Conducting Fluids

This paper consists of two parts dealing with magnetohydrodynamic pinch instabilities in cylindrical and in planar geometry.

Computational MHD Part II — Application to a Nonlinear Dynamo Model

Lagrangian chaos of the underlying flow is the driving force for the fast dynamo based on the strectch-twist-fold mechanism on small scales. In this contribution the hypothesis that the magnetic field by the action of the Lorentz force supresses Lagrangian chaos is checked by direct numerical simulations of the MHD equations. As a measure of the level of chaos the Lyapunov exponent of a set of 128 × 128 trajectories of fluid particles is computed in the growth phase and in the saturated phase of the dynamo when the magnetic field has reached its final strength. The numerical code, based on a pseudospectral algorithm, is developped for parallel computation on a multiprocessor system. Magnetic Reynolds numbers up to 240 and scale separations between the wavelength of the hydrodynamical forcing and the scale of the computational domain up to four are reached. For the runs were the kinetic Reyold number is high enough that the hydrodynamical bifurcation sequence to a more chaotic flow already has taken place, the mean value of the Lyapunov exponent is noticeable diminished in the saturated phase compared to the growth phase of the dynamo.

Parallel Computation of the Saturation Process in a Nonlinear Dynamo Model

In this paper the backreaction of a growing magnetic field in a nonlinear dynamo on the flow is investigated. The hypothesis that the magnetic field by the action of the Lorentz force supresses Lagrangian chaos of the flow is checked by direct numerical simulations of the MHD equations. As a measure of the level of chaos the Lyapunov exponent of a set of 128 × 128 trajectories of fluid particles is computed in the linear growth phase of the dynamo and in the saturated phase of the dynamo when the magnetic field has reached its final strength. The numerical code, based on a pseudospectral algorithm, is developed for parallel computation on a multiprocessor system (Cray-T3E). The trajectories for the computation of the Lyapunov exponent are advanced in a timestep parallel to the timestep of the MHD-solver. Magnetic Reynolds numbers up to 240 and scale separations between the wavelength of the hydrodynamical forcing and the scale of the computational domain up to four are reached. For the runs where the kinetic Reyold number is high enough that the hydrodynamical bifurcation sequence to a more chaotic flow already has taken place, the mean value of the Lyapunov exponent is noticeable diminished in the saturated phase compared to the growth phase of the dynamo.