A model for the turbulent Hartmann layer

Here we study the Hartmann layer, which forms at the boundary of any electr ically-conducting fluid flow under a steady magnetic field at high Hartmann number provided the magnetic field is not parallel to the wall. The Hartma nn layer has a well-known form when laminar. In this paper we develop a mod el for the turbulent Hartmann layer based on Prandtl's mixing-length model without adding arbitrary parameters, other than those already included in t he log-law. We find an exact expression for the displacement thickness of t he turbulent Hartmann layer [also given by Tennekes, Phys. Fluids 9, 1876 ( 1966)], which supports our assertion that a fully-developed turbulent Hartm ann layer of finite extent exists. Leading from this expression, we show th at the interaction parameter is small compared with unity and that therefor e 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 Reynold s number and the friction coefficient for the turbulent Hartmann layer in t he limit of large Reynolds number. By considering the distance over which t he stress decays, we find a condition for the two opposite Hartmann layers in duct flows to be isolated (nonoverlapping). (C) 2000 American Institute of Physics. [S1070-6631(00)00906-5].