دانلود رایگان مقاله محاسبات عددی استرس بازده جریان سیال با برنامه نویسی مخروط مرتبه دوم

Abstract

This work addresses the numerical computation of the two-dimensional flow of yield stress fluids (with Bingham and Herschel–Bulkley models) based on a variational approach and a finite element discretization. The main goal of this paper is to propose an alternative optimization method to existing procedures such as penalization and augmented Lagrangian techniques. It is shown that the minimum principle for Bingham and Herschel–Bulkley yield stress fluid steady flows can, indeed, be formulated as a second-order cone programming (SOCP) problem, for which very efficient primal–dual interior point solvers are available. In particular, the formulation does not require any regularization of the visco-plastic model as is usually the case for existing techniques, avoiding therefore the difficult choice of the regularization parameter. Besides, it is also unnecessary to adopt a mixed stress–velocity approach or discretize explicitly auxiliary variables as frequently proposed in existing methods. Finally, the performance of dedicated SOCP solvers, like the Mosek software package, enables us to solve large-scale problems on a personal computer within seconds only. The proposed method will be validated on classical benchmark examples and used to simulate the flow generated around a plate during its withdrawal from a bath of yield stress fluid.

6. Conclusions

A new optimization technique to solve minimization problems arising in the simulation of yield stress fluid flows has been presented. The method relies on a standard finite element discretization, on the formulation of the discretized minimization problem as a second-order cone program and on the use of dedicated efficient interior-point solvers, like the industrial software package MOSEK. The advantage of using such techniques over traditional approaches, like the Augmented Lagrangian for instance, has been underlined and computing times as functions of the size of the problem have been reported on a benchmark example. An excellent time complexity (almost linear) has been observed for the Bingham as well as for the Herschel–Bulkley model and for a wide range of Bingham numbers. The method has been further validated on the simulation of the flow generated by the plate displacement in a bath of yield stress fluid by comparing the obtained velocity fields to experimental measurements. It is important to note that the proposed approach can be easily generalized to other types of problems. Indeed, for the sake of simplicity, the present work has been restricted to a 2D plane strain steady-state setting. The generalization to axisymmetric or 3D problems will affect only the finite element discretization process whereas the problem formulation as a second-order cone program will have exactly the same structure. Similarly, the simulation of transient flows can be also modeled using the proposed approach as it requires the resolution of similar problems at each time step. To the author’s opinion, the simulation of transient 3D flows in complex geometry would greatly benefit from the computational efficiency of the proposed method.