A SIMPLE based discontinuous Galerkin solver for steady incompressible flows
A SIMPLE based discontinuous Galerkin solver for steady incompressible flows
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DOI：
10.1016/j.jcp.2012.11.051
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发表时间：
2013-03-15
影响因子：
4.1
通讯作者：
Oberlack, Martin
中科院分区：
文献类型：
--
作者：
Klein, Benedikt;Kummer, Florian;Oberlack, Martin
本文介绍了如何将著名的SIMPLE算法推广到用间断Galerkin方法离散的定常不可压N-S方程。对流部分用局部Lax-Friedrichs通量离散，粘性部分用对称内罚方法离散。在SIMPLE算法中，方程在迭代过程中求解。对离散化后的方程进行线性化，在离散水平上推导出压力方程。对于每个速度分量和压力得到的方程是解耦的，因此可以顺序求解，从而导致高效的求解过程。将该格式推广到非定常情况，可采用全隐式时间格式，并对Poiseuille流、恒定蒸腾槽流、Kovasznay流、拐角流和后向台阶流进行了计算。该格式采用k阶速度和k-1阶压力的混阶格式，在所有试验情况下都是数值稳定的。对于速度和压力，分别观察到了k+1和k在L-2范数下的收敛速度。对SIMPLE算法的收敛行为的研究表明，SIMPLE算法不需要压力的欠松弛，这与SIMPLE算法在有限体积法或连续有限元方法中的应用形成了强烈的对比。我们的结论是，在包含hp精度的间断Galerkin方法的背景下，所提出的格式对于求解定常不可压缩的Navier-Stokes方程是有效的。(C)2012 Elsevier Inc.保留所有权利。
In this paper we present how the well-known SIMPLE algorithm can be extended to solve the steady incompressible Navier-Stokes equations discretized by the discontinuous Galerkin method. The convective part is discretized by the local Lax-Friedrichs fluxes and the viscous part by the symmetric interior penalty method. Within the SIMPLE algorithm, the equations are solved in an iterative process. The discretized equations are linearized and an equation for the pressure is derived on the discrete level. The equations obtained for each velocity component and the pressure are decoupled and therefore can be solved sequentially, leading to an efficient solution procedure. The extension of the proposed scheme to the unsteady case is straightforward, where fully implicit time schemes can be used.Various test cases are carried out: the Poiseuille flow, the channel flow with constant transpiration, the Kovasznay flow, the flow into a corner and the backward-facing step flow. Using a mixed-order formulation, i.e. order k for the velocity and order k - 1 for the pressure, the scheme is numerically stable for all test cases. Convergence rates of k + 1 and k in the L-2-norm are observed for velocity and pressure, respectively. A study of the convergence behavior of the SIMPLE algorithm shows that no under-relaxation for the pressure is needed, which is in strong contrast to the application of the SIMPLE algorithm in the context of the finite volume method or the continuous finite element method. We conclude that the proposed scheme is efficient to solve the steady incompressible Navier-Stokes equations in the context of the discontinuous Galerkin method comprising hp-accuracy. (c) 2012 Elsevier Inc. All rights reserved.