Simulation of Three-Dimensional Bénard-Marangoni Flows Including Deformed Surfaces (Communications in Computational Physics, 2018)
We present a coupled thermal-fluid model for Bénard-Marangoni convection in a three-dimensional fluid layer. The governing equations are derived in detailfor two reasons: first, we do not assume a flat free surface as commonly done; andsecond, we prepare for the use of flexible discretizations. The governing equationsare discretized using spectral elements in space and an operator splitting approach intime. Since we are here primarily interested in steady state solutions, the focus is onthe spatial discretization. The overall computational approach is very attractive to usefor several reasons: (i) the solution can be expected to have a high degree of regularity,and rapid convergence can be expected; (ii) the spectral element decomposition automatically gives a convenient parameterization of the free surface that allows powerfulresults from differential geometry to easily be exploited; (iii) free surface deformation can readily be included; (iv) both normal and tangential stresses are convenientlyaccounted for through a single surface integral; (v) no differentiation of the surfacetension is necessary in order to include thermocapillary effects (due to integration-by-parts twice); (vi) the geometry representation of the free surface need only be C0 acrosselement boundaries even though curvature effects are included. Three-dimensionalsimulation results are presented, including the free surface deflection due to buoyancyand thermocapillary effects.