Direct numerical simulation of deformable bubbles in wall-bounded shear flows

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We present a method for fully-resolved simulations of bubbly flows using a front-tracking/front-capturing technique. The method is a modification of a marker-and-cell method developed previously for free-surface flows. The basic approach is somehow similar to the front-tracking method of Tryggvason: the continuity and Navier-Stokes equations are solved in a single Eulerian grid, and the interface is represented by an unstructured Lagrangian grid moving through the Eulerian grid, with the velocities at the Lagrangian grid obtained by interpolation from the Eulerian grid. However, contrary to Tryggvason's method, our method uses a sharp interface between the gas and the liquid, since it "captures" the interface within one cell of the Eulerian grid. The surface tension is expressed as a body-force in the Eulerian grid, and is computed using a least-squares fit, together with a mass-conserving filter to remove sub-grid oscillations. The Navier-Stokes equations are solved using a finite difference scheme on the Eulerian grid and the continuity is enforced using a standard projection technique, with the resulting Poisson equation being solved by a conjugate gradient method. We present some results of the simulations of single bubbles of different sizes in laminar wall-bounded uniform-shear flows at moderate Reynolds numbers (bubble-Reynolds-number in the range of 20-50). In wall-bounded shear flows, it has been observed experimentally that depending on the bubble size the lift force can push the bubble either towards or away from the wall. Our simulations show a good agreement with the experiments, both qualitatively and quantitatively, and an explanation for the lift inversion mechanism is provided based on the analysis of the forces acting on the bubble.