Validation Test Cases for Multi-physic Problems: Application to Magneto-hydrodynamic Numerical Simulations

The present paper is concerned with the numerical simulation of magneto-hydrodynamic (MHD) problems with industrial tools. MHD has received attention some thirty to twenty years ago as a possible alternative in propulsion applications; MHD propelled ships have even been designed to that purpose. However such propulsion systems have been proved of low efficiency and fundamental researches in the area have progressively received much less attention over the past decades. Numerical simulation of MHD problem could however provide interesting solutions in the field of turbulent flow control. The development of recent efficient numerical techniques for multi-physic applications provide promising tool for the engineer for that purpose. In the present paper, some elementary test cases in laminar flow with magnetic forcing terms are analyzed; equations of the coupled problem are exposed and analytical solutions are derived in each case, highlighting the relevant non-dimensional number which drives the physics of the problem. Several analytical calculations are then proposed and discussed. The present work will serve as basis for validation of numerical tools (based on the finite element method) for academic as well as industrial application purposes. INTRODUCTION Magneto-Hydro-Dynamic effects have received attention some thirty to twenty years ago as a possible alternative in propulsion applications; MHD propelled ships have even been designed to that purpose: the most famous example is the Japanese ship Yamamoto 1, which has been designed and build as prototype of MHD-propelled ship (see Fig. 1 featuring a photo of the MHD thrusters from Yamamoto 1). However such propulsion systems have been proved of low efficiency and fundamental researches in the area have progressively received much less attention over the past decades. Fig. 1. A MHD thruster from the experimental Japanese ship Yamato 1 at the Ship Science Museum in Odaiba, Tokyo. Numerical simulation of MHD problem could however provide interesting solutions in the field of turbulent flow control. The development of recent efficient numerical techniques for multi-physic applications provide promising tool for the engineer for that purpose. In the present paper, analytical test-cases in laminar flow with magnetic forcing terms are analyzed, namely the Hartman problem (section 1), the Couette problem (section 2) and the Rayleigh problem (section 3). An analytical solution is derived in each case 1 and the physic of the problem is discussed through the influence of a relevant non-dimensional number highlighted by the analytical expressions. As an illustration on MHD-based propulsion system, application of the Hartmann problem solution to an elementary propulsion nozzle is exposed. The present work will serve as basis for validation of numerical tools for multi-physic applications. 1 To the authors’ knowledge, some of the presented analytical solutions have never been reported previously in the literature. ha l-0 04 57 15 1, v er si on 1 16 F eb 2 01 0 Author manuscript, published in "ASME 2008 Pressure Vessels and Piping Conference (PVP2008), Chicago : United States (2008)" DOI : 10.1115/PVP2008-61736