Through　performing　numerical　simulations,　the　three-dimensional　thermally　driven　flows,　namely　the　Rayleigh-Benard　convection　(RBC　hereafter)　flows,　have　been　studied　under　the　influence　of　magnetic　field　in　the　confined　rectangular　enclosure　full　of　liquid　metal.　To　validate　our　numerical　methodologies,　some　of　our　numerical　solutions　are　compared　with　the　available　experimental　results　that　good　agreements　are　found　between　them.　Besides,　the　correlations　between　the　Nusselt　number　(Nu　hereafter)　and　the　Rayleigh　number　(Ra　hereafter)　are　also　in　accordance　with　the　experimental　results,　whereas　the　Hartmann　number　(Ha　hereafter)　keeps　constant.　Nevertheless,　in　the　present　study,　a　new　correlation　law　between　Nu　and　varied　Ha　is　established　when　Ra　is　fixed,　this　formula　is　more　helpful　when　particularly　considering　the　influence　of　the　magnetic　field.　Regarding　the　horizontal　magnetic　field,　it　is　found　that　the　evolution　of　Nu　versus　magnetic　intensity　can　be　divided　into　three　stages:　when　Ha　is　small,　Nu　increases　with　Ha　because　the　three-dimensional　RBC　flow　transits　to　transient-two　dimensional　pattern;　when　Ha　is　moderate,　Nu　increases　with　Ha　because　the　transient-two　dimensional　flow　is　more　stable　that　the　dynamic　motion　of　vortex　cells　is　disappeared　gradually;　when　Ha　continues　to　increase,　Nu　still　increases　with　Ha　due　to　the　growth　in　number　of　the　convective　cells.　It　should　be　noted　that　such　transient　process　of　the　flow　field　can　be　hardly　observed　in　the　experiments,　because　the　liquid　metal　is　opaque,　however,　we　are　able　to　present　the　evolutions　of　the　temperature　fields　as　well　as　the　flow　structures　with　the　help　of　numerical　techniques.　(C)　2017　Elsevier　Ltd.　All　rights　reserved.