The fully-implicit log-conformation formulation and its application to three-dimensional flows
The fully-implicit log-conformation formulation and its application to three-dimensional flows
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DOI：
10.1016/j.jnnfm.2015.07.004
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发表时间：
2015-09-01
影响因子：
3.1
通讯作者：
Knechtges, Philipp
中科院分区：
文献类型：
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作者：
Knechtges, Philipp
由于高Weissenberg数问题，粘弹性流动的稳定和有效的数值模拟一直是一个持续的斗争。虽然Fattal和Kupferman提出的对数构象方法可以大大提高宏观描述的稳定性，但将有效的Newton-Raphson算法应用于由对数构象方程和Navier-Stokes方程组成的完整整体控制方程系统一直存在问题。特别是，它是制定的本构方程的频谱分解，阻碍了进一步的分析工具的应用。因此，到目前为止，完全单一的方法只能在两个维度上实现，例如，这篇论文的目的是找到先前对三维的考虑的一般化，使得在这种情况下也可以实现基于对数构象公式的整体牛顿-拉夫森求解器。其基本思想类似于二维情况，用一个分析上更“良好”的项来代替本构方程中的本征值分解，并仅依赖于本征值分解进行实际计算。此外，为了证明所提出的方法的实用性，数值计算结果的新推导出的配方的情况下，沉积球和椭球基准的Oldrophil-B和Giesekus模型。结果表明，牛顿法可以达到预期的二次收敛。(C)2015 Elsevier B. V.版权所有。
The stable and efficient numerical simulation of viscoelastic flows has been a constant struggle due to the High Weissenberg Number Problem. While the stability for macroscopic descriptions could be greatly enhanced by the log-conformation method as proposed by Fattal and Kupferman, the application of the efficient Newton-Raphson algorithm to the full monolithic system of governing equations, consisting of the log-conformation equations and the Navier-Stokes equations, has always posed a problem. In particular, it is the formulation of the constitutive equations by means of the spectral decomposition that hinders the application of further analytical tools. Therefore, up to now, a fully monolithic approach could only be achieved in two dimensions, as, e.g., recently shown by Knechtges et al. (2014).The aim of this paper is to find a generalization of the previously made considerations to three dimensions, such that a monolithic Newton-Raphson solver based on the log-conformation formulation can be implemented also in this case. The underlying idea is analogous to the two-dimensional case, to replace the eigenvalue decomposition in the constitutive equation by an analytically more "well-behaved" term and to rely on the eigenvalue decomposition only for the actual computation. Furthermore, in order to demonstrate the practicality of the proposed method, numerical results of the newly derived formulation are presented in the case of the sedimenting sphere and ellipsoid benchmarks for the Oldroyd-B and Giesekus models. It is found that the expected quadratic convergence of Newton's method can be achieved. (C) 2015 Elsevier B.V. All rights reserved.