A bi-projection method for Bingham type flows

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• 作者：Laurent Chupin ^{Laurent.Chupin@math.univ-bpclermont.fr" class="auth_mail" title="E-mail the corresponding author} ; Thierry Dubois ; ^{Thierry.Dubois@math.univ-bpclermont.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Viscoplastic medium ; Bingham flows ; Lid-driven cavity ; Projection scheme ; Numerical simulation ; Stability and error estimates
• 刊名：Computers & Mathematics with Applications
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：72
• 期：5
• 页码：1263-1286
• 全文大小：1390 K

文摘

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 the 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 1000. 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 1024²${1024}^2$ mesh points. The effect of this (physical) parameter on the flow behaviour is discussed.