A model for the turbulent Hartmann layer

Abstract

Here we study the Hartmann layer, which forms at the boundary of any electrically-conducting fluid flow under a steady magnetic field at high Hartmann number provided the magnetic field is not parallel to the wall. The Hartmann layer has a well-known form when laminar. In this paper we develop a model for the turbulent Hartmann layer based on Prandtl's mixing-length model without adding arbitrary parameters, other than those already included in the log-law. We find an exact expression for the displacement thickness of the turbulent Hartmann layer ͓also given by Tennekes, Phys. Fluids 9, 1876 ͑1966͔͒, which supports our assertion that a fully-developed turbulent Hartmann layer of finite extent exists. Leading from this expression, we show that the interaction parameter is small compared with unity and that therefore the Lorentz force is negligible compared with inertia. Hence, we suggest that the turbulence present in the Hartmann layer is of classical type and not affected by the imposed magnetic field, so justifying use of a Prandtl model. A major result is a simple implicit relationship between the Reynolds number and the friction coefficient for the turbulent Hartmann layer in the limit of large Reynolds number. By considering the distance over which the stress decays, we find a condition for the two opposite Hartmann layers in duct flows to be isolated ͑nonoverlapping͒.

Bounds on the attractor dimension for low-Rm wall-bound magnetohydrodynamic turbulence

“…This two-layer structure strongly resembles that of the theoretical prediction of Ref. 19 for the turbulent Hartmann layer, which also involves such a double deck structure, with a viscous sublayer.…”

Modeling and discretization errors in large eddy simulations of hydrodynamic and magnetohydrodynamic channel flows

“…This parameter is very important in the MHD channel flow, since the transition from laminar to turbulent Hartmann layers occurs at R % 350 À 400 [51,52].…”

Secondary optimal energy growth and magnetic damping of turbulence in Hartmann channel flow

“…based on the thickness of the Hartmann layer) being small when R is in the transitional range. They have, therefore, suggested that a turbulence model for these layers does not need to take magnetic damping into account [24]. Two of the present authors have used DNS of turbulent Hartmann flow to evaluate the performance of different simple mixing-length turbulence models [25].…”

“…Therefore, the role of magnetic damping of turbulent fluctuations in the Hartmann flow is also likely to be small. Alboussière and Lingwood [24] have argued that this is because of the local interaction parameter (i.e. based on the thickness of the Hartmann layer) being small when R is in the transitional range.…”

“…In the evaluation of the transfer term it turns out that contributions from the spanwise and wallnormal velocity components V and W of the base flow are negligible. For this reason we consider only the terms (20,24) for two cases. The first one corresponds to a finite Ha > 0 and the most amplified perturbation for these parameters.…”