Linear instability and nonlinear flow states in a horizontal pipe flow under bottom heating and transverse magnetic field

Linear stabilities of the liquid metal mixed convection in a horizontal pipe under bottomheating and transverse magnetic field are studied through linear global stability analyses.Three branches of the linear stability boundary curves are determined by the eigenvaluecomputation ofthe most unstable modes. One branch is located in the region of largeHartmann number and determined by the linear unstable mode which was first revealed bythe numerical simulations of Zikanov, Listratov & Sviridov (J. Fluid Mech., vol. 720, 2013,pp. 486-516). This branch curve shows that the global unstable mode exists abovea threshold of Hartmann number, which agrees with the experiment of Geninet al.(Temperature fluctuations in a heated horizontal tube affected by transverse magneticfield. InProc. 8th PAMIR Conf. Fund. Appl. MHD, Borgo, Corsica, France, pp. 37-41,2011). The other two branch curves determined by two differentlong-wave unstable modes intersect with each other in the region of small Hartmann number. The critical Grash of number on these two curves increases exponentially with the increase of the Hartmann number. Through energy budget analyses at the criticalthresholds of these unstable modes,it is found that, for the unstable mode at large Hartmann numbers, buoyancy is the dominant destabilizing term which demonstrates the hypothetical explanation of Zikanovet al.(2013) who regard natural convection as adestabilization mechanism. It is further revealed that, with respect to the unstable modes on the critical stability curves of small Hartmann numbers,the domin ant destabilization comes from the streamwise shear ofthe basic flow. Finally, within the linear unstable region, fully developed nonlinear flow states of the mixed convection are investigated by direct numerical simulations (DNS)with several sets of selected dimensionless parameters. Thespatio-temporal structures ofthese nonlinear flow states are discussed in detail with comparisonwith the linear unstableglobal modes.