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不可压Navier-Stokes方程是流体力学的基本控制方程, 其高精度数值模拟具有重要的科学意义. 本综述性文章回顾了求解Navier-Stokes方程的投影方法, 重点介绍了时空一致四阶精度的GePUP方法. 该方法用一个广义投影算子对不可压Navier-Stokes方程进行了重新表述, 使得投影流速的散度由一个热方程控制, 保持了UPPE方法的优点. 与UPPE方法不同的是, GePUP方法的推导不依赖于Leray-Helmholtz投影算子的各种性质, 并且GePUP表述中的演化变量无需满足散度为零的条件, 因此数值近似Leray-Helmholtz投影算子的误差对精度和稳定性的影响非常透明. 在GePUP方法中, 时间积分和空间离散是完全解耦的, 因此对这两个模块都能以“黑匣子”的方式自由替换. 时间积分模块的灵活性实现了时间上的高阶精度, 并使得GePUP算法能同时适用于低雷诺数流体和高雷诺数流体. 空间离散模块的灵活性使得GePUP算法能很好地适应不规则边界. 理论分析和数值测试结果都显示, 相对于二阶投影方法, GePUP方法无论在精度上还是效率上都具有巨大优势.
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关键词:
- 不可压Navier-Stokes方程/
- 投影方法/
- 时空一致四阶精度/
- 无滑移边界条件/
- 广义投影算子/
- 压力Poisson方程
The incompressible Navier-Stokes equations (INSE) are the basic governing equations of fluid dynamics, and their numerical solutions are of great significance. In this review paper, we first recollect some classical projection methods and their relatives in the past 50 years and then fully explain the recent fourth-order projection method called GePUP [Zhang Q2016 J. Sci. Comput. 671134 ]. Based on a generic projection operator and the UPPE formulation of the INSE [Liu J G, Liu J, Pego R L2007 Comm. Pure Appl. Math. 601443 ], we derive the GePUP equations, which retain the advantage of UPPE that the velocity divergence is governed by a heat equation and is thus well under control. In comparison with UPPE, the GePUP formulation is advantageous in three aspects: (1) its derivation depends on none of the properties of the Leray-Helmholtz projection; (2) the evolutionary velocity can be divergent, thus it is directly applicable to numerical calculations with nonzero velocity divergence; (3) the Leray-Helmholtz projection does appear on the right-hand sides of the governing equations, thus making it transparent to analyze the accuracy and stability issues raised by numerically approximating the Leray-Helmholtz projection. As the most appealing feature of GePUP, temporal integration and spatial discretization are completely decoupled and can be treated as black boxes, so that the user can choose his favorite methods for the two parts to form his own GePUP method. In particular, high-order accuracy in time can be easily obtained since no internal details of the ODE solver are needed. The flexibility in time makes the GePUP method applicable to both low-Reynolds-number flow and high-Reynolds-number flow. The flexibility in space makes the GePUP method applicable to both rectangular boxes and irregular domains. The numerical results and elementary analysis show that the fourth-order GePUP method may be much more accurate and efficient than classical second-order projection methods by many orders of magnitude.-
Keywords:
- incompressible Navier-Stokes equations/
- projection methods/
- fourth-order accuracy both in time and in space/
- no-slip boundary conditions/
- generic projection/
- pressure Poisson equation
通信作者:张庆海,qinghai@zju.edu.cn
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h ${1}/{32}$ Rate ${1}/{64}$ Rate $1/{128}$ Rate $1/{256}$ ${u}$ $ L_{\infty} $ 2.63 × 10–4 3.94 1.71 × 10–5 3.98 1.08 × 10–6 4.00 6.78 × 10–8 $ {u} $ $ L_1 $ 1.63 × 10–4 3.87 1.12 × 10–5 3.97 7.14 × 10–7 3.99 4.48 × 10–8 $ {u} $ $ L_2 $ 1.49 × 10–4 3.89 1.01 × 10–5 3.98 6.38 × 10–7 4.00 4.00 × 10–8 p$ L_{\infty} $ 1.63 × 10–3 3.99 1.02 × 10–4 4.00 6.39 × 10–6 4.01 3.97 × 10–7 p$ L_1 $ 3.26 × 10–4 4.00 2.04 × 10–5 4.03 1.25 × 10–6 3.97 7.97 × 10–8 p$ L_2 $ 4.60 × 10–4 3.99 2.89 × 10–5 4.01 1.80 × 10–6 3.99 1.13 × 10–7 
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$ S_{4 |2} $ $ \epsilon = 10^{-4} $ $ \epsilon = 10^{-6} $ $ \epsilon = 10^{-8} $ $ \epsilon = 10^{-10} $ 单涡量测试,Re= $ 2\times 10^4 $ 1.73 5.50 × 101 1.73 × 103 5.47 × 104 二维粘盒测试,Re= $ 10^4 $ 5.03 1.59 × 102 5.03 × 103 1.59 × 105 二维粘盒测试,Re= 100 1.26 × 101 4.00 × 102 1.26 × 104 3.98 × 105 三维粘盒测试,Re= $ 10^4 $ 8.65 × 101 8.65 × 103 8.65 × 105 8.65 × 107 三维粘盒测试,Re= 100 4.24 × 101 4.24 × 103 4.24 × 105 4.24 × 107 下载:
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$ S_{3 |1} $ $ \epsilon = 10^{-2} $ $ \epsilon = 10^{-3} $ $ \epsilon = 10^{-4} $ $ \epsilon = 10^{-5} $ 单涡量测试,Re= $ 2\times 10^4 $ 1.89 × 104 1.89 × 106 1.89 × 108 1.89 × 1010 二维粘盒测试,Re= $ 10^4 $ 2.30 × 105 2.30 × 107 2.30 × 109 2.30 × 1011 三维粘盒测试,Re= $ 10^4 $ 3.66 × 107 1.70 × 1010 7.88 × 1012 3.66 × 1015 下载:
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