A bi-projection method for Bingham type flows

International audienceWe propose and study a new numerical scheme to compute the isothermal and unsteady flow of an incompressible viscoplastic Bingham medium.The main difficulty, for both theoretical and numerical approaches, is due to the non-differentiability of the plastic part of stress tensor in regionswhere the rate-of-strain tensor vanishes. This is handled by reformulating the definition of the plastic stress tensor in terms ofa projection.A new time scheme, based on the classical incremental projection method for the Newtonian Navier-Stokes equations, is proposed. The plastictensor is treated implicitly in the first sub-step of the projection scheme and is computed by using a fixed point procedure. A pseudo-timerelaxation is added into the Bingham projection whose effect is to ensure a geometric convergence of the fixed point algorithm. This is akey feature of the bi-projection scheme which provides a fast and accurate computation of the plastic tensor.Stability and error analyses of the numerical scheme are provided. The error induced by the pseudo-time relaxation term is controlled bya prescribed numerical parameter so that a first-order estimate of the time error is derived for the velocity field.A second-order cell-centred finite volume scheme on staggered grids is applied for the spatial discretisation.The scheme is assessed against previously published benchmark results for both Newtonian and Bingham flows in a two-dimensional lid-drivencavity for Reynolds number equals 1 000.Moreover, the proposed numerical scheme is able to reproduce the fundamental property of cessation in finite time of a viscoplasticmedium in the absence of any energy source term in the equations.For a fixed value (100) of the Bingham number, various numerical simulations for a range of Reynolds numbers up to 200 000 were performedwith the bi-projection scheme on a grid with 1024x1024 mesh points. The effect of this (physical) parameter on the flow behaviour is discussed