A bi-projection method for Bingham type flows
Résumé
We 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 regions
where the rate-of-strain tensor vanishes. This is handled by reformulating the definition of the plastic stress tensor in terms of
a projection.
A new time scheme, based on the classical incremental projection method for the Newtonian Navier-Stokes equations, is proposed. The plastic
tensor is treated implicitly in the first sub-step of the projection scheme and is computed by using a fixed point procedure. A pseudo-time
relaxation is added into the Bingham projection whose effect is to ensure a geometric convergence of the fixed point algorithm. This is a
key 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 by
a 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-driven
cavity 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 viscoplastic
medium 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 performed
with the bi-projection scheme on a grid with 1024x1024 mesh points. The effect of this (physical) parameter on the flow behaviour is discussed.
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