Numerical study of the instability of the Hartmann layer

Abstract

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.

Transition to Turbulence in the Hartmann Boundary Layer

“…The transition consists of a growth of streamwise two-dimensional perturbations which subsequently become unstable with respect to fully three-dimensional perturbations. The numerical simulations [5] yield a range of 350 < R < 400 for the critical Reynolds number which is in good agreement with the experimental findings. Thereby a long-standing open problem of MHD has been solved.…”

“…The goal of this paper is to familiarize the reader with the mechanisms responsible for the loss of stability of this flow and for the ways in which the Hartmann flow becomes turbulent. Our paper is not intended as a review but will rather focus on the results of a recent joint experimental-numerical work whose details are given in [4] and [5].…”

“…This unsatisfactory state of affairs prompted researchers to perform new experiments [4] and direct numerical simulations [5] which finally led to a better understanding of the instability of the Hartmann flow. The main results of these investigations will be summarized next.…”

Patterned Turbulence in Liquid Metal Flow: Computational Reconstruction of the Hartmann Experiment

“…This is not registered in the high-Ha experiments because the total friction is dominated by the friction in the laminar Hartmann layers. Accordingly, the change in friction behavior caused by transition in isolated Hartmann layers at R ∼ 400 [14] is found.…”

“…One should, therefore, speak of a range of flow parameters in which the transition or laminarization occur (e.g., recent experiments [13] in pipe), rather than about a sharp threshold. Previous numerical studies, such as [14,15], examined the transition following the evolution of finite-amplitude perturbations imposed on a laminar flow. Here, we adopt the same approach as in the Hartmann experiments and perform most of the simulations as laminarization tests at constant flow rate, in which B is increased until the initially turbulent flow becomes laminar.…”