A formulation for soluble surfactant transport in multiphase flows recently presented by Muradoglu & Tryggvason (JCP 274 (2014) 737–757) is adapted to the context of the Level Contour Reconstruction Method, LCRM, (Shin et al. IJNMF 60 (2009) 753–778) which is a hybrid method that combines the advantages of the Front-tracking and Level Set methods. Particularly close attention is paid to the formulation and numerical implementation of the surface gradients of surfactant concentration and surface tension. Various benchmark tests are performed to demonstrate the accuracy of different elements of the algorithm. To verify surfactant mass conservation, values for surfactant diffusion along the interface are compared with the exact solution for the problem of uniform expansion of a sphere. The numerical implementation of the discontinuous boundary condition for the source term in the bulk concentration is compared with the approximate solution. Surface tension forces are tested for Marangoni drop translation. Our numerical results for drop deformation in simple shear are compared with experiments and results from previous simulations. All benchmarking tests compare well with existing data thus providing confidence that the adapted LCRM formulation for surfactant advection and diffusion is accurate and effective in three-dimensional multiphase flows with a structured mesh. We also demonstrate that this approach applies easily to massively parallel simulations

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Abstract

The flow driven by a rotating disk at the bottom of an open fixed cylindrical cavity is studied numerically and experimentally. The steady axisymmetric Navier-Stokes equations projected onto a curvilinear coordinate system are solved by a Newton-Raphson algorithm. The free surface shape is computed by an iterative process in order to satisfy a zero normal stress balance at the interface. In previous studies, regarding the free surface deflection, there is a significant disagreement between a first-order approximation [M. Piva and E. Meiburg, "Steady axisymmetric flow in an open cylindrical container with a partially rotating bottom wall," Phys. Fluids 17, 063603 (2005)] and a full numerical simulation [R. Bouffanais and D. Lo Jacono, "Unsteady transitional swirling flow in the presence of a moving free surface," Phys. Fluids 21, 064107 (2009)]. For a small deflection, the first-order approximation matches with our numerical simulation and for a large deflection a good agreement is found with experimental measurements.