An effective two-dimensional model for MHD flows with transverse magnetic field

Abstract

This paper presents a model for quasi two-dimensional MHD flows between two planes with small magnetic Reynolds number and constant transverse magnetic field orthogonal to the planes. A method is presented that allows to take 3D effects into account in a 2D equation of motion thanks to a model for the transverse velocity profile. The latter is obtained by using a double perturbation asymptotic development both in the core flow and in the Hartmann layers arising along the planes. A new model is thus built that describes inertial effects in these two regions. Two separate classes of phenomena are thus pointed out : the one related to inertial effects in the Hartmann layer gives a model for recirculating flows and the other introduces the possibility of having a transverse dependence of the velocity profile in the core flow. The "recirculating" velocity profile is then introduced in the transversally averaged equation of motion in order to provide an effective 2D equation of motion. Analytical solutions of this model are obtained for two experimental configurations : isolated vortices aroused by a point electrode and axisymmetric parallel layers occurring in the MATUR (MAgneticTURbulence) experiment. The theory is found to give a satisfactory agreement with the experiment so that it can be concluded that recirculating flows are actually responsible for both vortices core spreading and excessive dissipative behavior of the axisymmetric side wall layers. arXiv:2006.15468v1 [physics.flu-dyn]

Spatial evolution of a quasi-two-dimensional Kármán vortex street subjected to a strong uniform magnetic field

“…Using the same approach as per the development for expressions for a and x 0 for pure hydrodynamic flow, parameters b and c are derived from the peak vorticity time history of magnetohydrodynamic cases across 0.1 ≤ β ≤ 0.4, 500 ≤ H ≤ 5000, and 1500 ≤ Re L ≤ 8250, where a laminar periodic shedding regime is captured throughout this parameter range. The valid upper range of Re L is determined by both the assumptions of the SM82 model, i.e., the flow has sufficiently large perpendicular scales, in such a way that the condition of N ≫ (a/l ⊥ ) 3 and H ≫ (a/l ⊥ ) 2 is satisfied, 29,36 and the Hartmann layers must be laminar, i.e., the Reynolds number based on the Hartmann layer thickness Re/H < 250. 1 The former criterion is stricter than the latter, and by taking N > 10 as an indicative threshold for the applicable range of interaction parameters and H = 500, the model will break down at a Reynolds number of order Re L < H 2 /N = 500 2 /10 = 25 000, which is well above the maximum Re L studied, i.e., Re L = 8250.…”

“…29,57 As a result, the SM82 model breaks down locally when the effect of viscosity is relevant, i.e., when Ha ∼ l ∥ /l ⊥ , or when the transverse length scale l ⊥ is of the order of the Shercliff layers thickness, δ S = aHa −1/2 . Despite the inherent limitations of the SM82 model, it has nevertheless been shown to predict the Shercliff layers thickness and an isolated vortex profile to high accuracy when compared to 3D solutions, 36 where the reported errors are less than 10%. 29,38 The model has also been tested for flows in a duct with a cylinder obstacle, where the critical Reynolds number at the onset of vortex shedding in Refs.…”

MHD flow in an insulating rectangular duct under a non-uniform magnetic field

“…The derivation of the complete form of the vorticity transport equation would, however, require the introduction of terms proportional to z 4 in the basic expression (10) for the electric potential. Such a model, would be a generalization of the approach presented in [34], allowing for high ε values. Notice that in the particular case B = B(x), equation (35) can also be interpreted as an expression for the axial part of the current density component j 0x (x, y, z = 0) = -σ (∂ x 0 -Bv 0 ), parallel to ∇ ⊥ B, which is expressed through other unknowns (velocity and vorticity).…”

Quasi-two-dimensional convection in a three-dimensional laterally heated box in a strong magnetic field normal to main circulation

“…26 Models of Q2D flows have been developed for the flow between parallel planes in a uniform magnetic field, 27,28 a nonuniform magnetic field, 29,30 or buoyant convection in a uniform field. 7,8,11,12 More elaborate Q2D models have been developed by Pothérat et al, 31,32 which take into account small but finite Ekman pumping in the Hartmann layers.…”