一种基于Picard迭代求解不可压缩Navier-Stokes方程的速度修正投影法

doi:10.19596/j.cnki.1001-246x.8810

摘要/Abstract

摘要：

介绍基于速度修正投影技术处理不可压缩Navier-Stokes方程的实施流程, 并引入Picard线性化处理速度控制方程中的对流项, 从而发展出一种基于Picard迭代的速度修正投影法。相较于传统方法, 经过Picard线性化处理后的投影法可以采用更大的时间步长进行求解, 提高了求解方法的稳定性, 并且收敛精度也符合要求, 证实了求解方法的可靠性。

关键词: Navier-Stokes方程, Picard线性化, 速度修正投影法

Abstract:

The implementation flow of processing incompressible Navier-Stokes equations based on velocity-correction schemes is introduced, and velocity-correction schemes based on Picard iteration is developed by introducing Picard linearization to process the convection term in the velocity governing equation. Compared with the traditional method, the projection method after Picard linearization can be solved with a larger time step, which improves the stability of the solution method, and the convergence accuracy meets the requirements, which confirms the reliability of the solution method.

Key words: Navier-Stokes equations, Picard linearization, velocity-correction schemes

尹林茂, 张玉龙, 杨洋, 肖福建, 蒋秉颜. 一种基于Picard迭代求解不可压缩Navier-Stokes方程的速度修正投影法[J]. 计算物理, 2025, 42(1): 10-17.

Linmao YIN, Yulong ZHANG, Yang YANG, Fujian XIAO, Bingyan JIANG. Velocity-correction Schemes for Solving Incompressible Navier-Stokes Equations Based on Picard Iteration[J]. Chinese Journal of Computational Physics, 2025, 42(1): 10-17.

图/表 11

图1 速度修正投影法流程图

Fig.1 Flow chart of velocity-correction schemes a

表1 σ=1 000时两种投影法求解Kovasznay流所得E2和收敛阶

Table 1 Comparison of E2 variation with grid and degree of convergence obtained by two projection methods for Kovasznay flow with σ=1 000

	60	90	120	150	收敛阶
E₂_u-TPM	4.64×10^-4	2.01×10^-4	1.14×10^-4	7.39×10^-5	1.952 3
E₂_u-PPM	4.16×10^-4	1.87×10^-4	1.06×10^-4	6.88×10^-5	1.954 6
E₂_v-TPM	1.28×10^-4	5.65×10^-5	3.19×10^-5	2.05×10^-5	1.979 2
E₂_v-PPM	1.43×10^-4	6.33×10^-5	3.56×10^-5	2.28×10^-5	1.997 9
E₂_p-TPM	9.19×10^-4	4.22×10^-4	2.48×10^-4	1.66×10^-4	1.820 5
E₂_p-PPM	1.64×10^-3	7.34×10^-4	4.18×10^-4	2.72×10^-4	1.940 8

表2 不同网格下两种投影法求解Kovasznay流的σmin

Table 2 Two projection methods for solving σmin of Kovasznay flow under different grids

	x方向网格数	60	90	120	150
σ_min	传统投影法	38	43	45	46
σ_min	Picard投影法	10	12	15	20

图2 方腔顶盖驱动流物理模型

Fig.2 Physical model of flow driven by the square cavity top cover

图3 σ=100时，不同Re数下的方腔流中心线上的速度分量(a) x=0.5；(b) y=0.5

Fig.3 Velocity components on the center line of square cavity flow with different Re with σ=100 (a) x=0.5;(b) y=0.5

图4 σ=500时, 不同Re数下的方腔流中心线上的速度分量(a) x=0.5；(b) y=0.5

Fig.4 Velocity components on the center line of square cavity flow with different Re with σ=500 (a) x=0.5;(b) y=0.5

表3 不同Re下两种投影法求解方腔顶盖驱动流的σmin

Table 3 Two projection methods for solving σmin of the driven flow of the square cavity top cover under different Re

	Re	1	100	400	1 000	5 000
σ_min	传统投影法	0.0	1.5	30.4	71.2	243.5
	Picard投影法	0.0	0.0	5.4	23.9	147.6

表4 不同Re下求解方腔顶盖驱动流的最优σ和收敛所需迭代步数

Table 4 Number of iterations required to solve the optimal σ and convergence of flow driven by the square cavity head under different Re

Re	传统投影法		Picard投影法
Re	最优σ	迭代步数	最优σ	迭代步数
400	30.4	1 706	5.4	525
1 000	71.2	8 131	23.9	3 037

图5 封闭方腔自然对流物理模型

Fig.5 Physical model of natural convection in a closed square cavity

表5 本文和文献平均努塞尔数

Table 5 The average Nusselt number in this paper and the literature

Ra	σ	平均努塞尔数Nu
Ra	σ	传统投影法	Picard投影法	Ref.[25]	Ref.[26]
10³	1 000	1.117 4	1.117 9	1.118 0	1.114 0
10⁴	1 000	2.245 5	2.245 1	2.243 0	2.245 0
10⁵	5 000	4.524 5	4.522 9	4.451 9	4.510 0
10⁶	25 000	8.830 4	8.827 6	8.800 0	8.806 0

表6 不同Ra下两种投影法求解自然对流的σmin

Table 6 σmin of natural convection solved by two projection methods at different Ra

	Ra	10³	3×10³	10⁴	10⁵	10⁶
σ_min	传统投影法	3	50	207	2 152	22 775
σ_min	Picard投影法	0	0	90	1 560	11 339

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