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尺度无关高阶WCNS格式

张子轩 董义道 黄梓全 孔令发 刘伟

张子轩, 董义道, 黄梓全, 孔令发, 刘伟. 尺度无关高阶WCNS格式. 力学学报, 2023, 55(1): 254-271 doi: 10.6052/0459-1879-22-399
引用本文: 张子轩, 董义道, 黄梓全, 孔令发, 刘伟. 尺度无关高阶WCNS格式. 力学学报, 2023, 55(1): 254-271 doi: 10.6052/0459-1879-22-399
Zhang Zixuan, Dong Yidao, Huang Ziquan, Kong Lingfa, Liu Wei. A scale-invariant high-order WCNS scheme. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(1): 254-271 doi: 10.6052/0459-1879-22-399
Citation: Zhang Zixuan, Dong Yidao, Huang Ziquan, Kong Lingfa, Liu Wei. A scale-invariant high-order WCNS scheme. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(1): 254-271 doi: 10.6052/0459-1879-22-399

尺度无关高阶WCNS格式

doi: 10.6052/0459-1879-22-399
基金项目: 湖南省自然科学基金(2022JJ40539), 国家自然科学基金(11972370)和国家重大项目(GJXM92579)资助
详细信息
    通讯作者:

    董义道, 讲师, 主要研究方向为高精度方法及湍流数值模拟. E-mail: tianyatingxiao@163.com

  • 中图分类号: V211.3

A SCALE-INVARIANT HIGH-ORDER WCNS SCHEME

  • 摘要: 高速流场的数值模拟中, 既要保证对小尺度结构的高保真分辨, 又要实现对激波稳定、无振荡地捕捉.当前工程中广泛应用的高精度数值格式虽然都能一定程度地满足上述两种要求, 但仍与理想目标存在较大差距.例如, 模拟雷诺应力模型等小尺度问题时, 高精度格式在间断解附近易产生数值振荡.基于高精度格式所存在的上述问题, 本文引入去尺度函数, 探索了一种更加简单稳定的非线性权重构造方法, 并将其应用于7阶精度加权紧致非线性格式WCNS, 提出了一种尺度无关的7阶WCNS格式.该格式的性能与灵敏度参数和尺度因子的选择无关, 并且在小尺度下仍可以有效捕捉流场激波.同时, 该格式在间断处具有基本无振荡性质, 且在任意尺度函数下保持尺度无关, 并且在极值点处也能保持最优精度.本文还推导了7阶D权函数的形式.最后, 在一维线性对流方程中验证了新格式在流场光滑区能够达到设计精度, 并通过一系列数值实验证明了尺度无关的7阶WCNS格式在激波捕捉能力上具有良好表现, 为WCNS格式改进和解决可压缩湍流等非线性问题提供了一种新途径.

     

  • 图  1  $\varepsilon = 1.0 \times {10^{ - 6}}$时Lax问题的密度分布曲线

    Figure  1.  Density distribution curves of the Lax problem with $\varepsilon $= $1.0 \times {10^{ - 6}} $

    图  2  $\varepsilon = 1.0 \times {10^{ - 40}}$时Lax问题的密度分布曲线

    Figure  2.  Density distribution curves of the Lax problem with $\varepsilon $= $ 1.0 \times {10^{ - 40}} $

    图  3  $\varepsilon = 1.0 \times {10^{ - 6}}$$\varepsilon = 1.0 \times {10^{ - 40}}$时使用WCNS7-Dm格式求解Lax问题的密度分布曲线

    Figure  3.  Density distribution curves of the Lax problem computed by the WCNS7-Dm scheme with $\varepsilon = 1.0 \times {10^{ - 6}}$ and $\varepsilon = 1.0 \times {10^{ - 40}}$

    图  4  3套网格单元数$N$下使用WCNS格式计算Sine函数$ {f_1}(x) $的Si误差${E_{Si}}(\lambda )$, $\lambda \in [1.0 \times{\text{1}}{{\text{0}}^{-19}},1.0 \times {\text{1}}{{\text{0}}^{19}}]$, $\varepsilon = 1.0 \times {10^{ - 6}}$ (上行)和$\varepsilon =1.0 \times {10^{ - 40}}$(下行)

    Figure  4.  The Si-error ${E_{Si}}(\lambda )$ of the Sine function$ {f_1}(x) $ calculated by the WCNS schemes with $\varepsilon = 1.0 \times {10^{ - 6}}$ (top row), $\varepsilon = 1.0 \times {10^{ - 40}}$(bottom row), and different scale factors $\lambda \in [1.0 \times{\text{1}}{{\text{0}}^{ - 19}},1.0 \times {\text{1}}{{\text{0}}^{19}}]$ for three number of cells $N$

    图  5  3套网格单元数$N$下使用WCNS格式计算$ {\text{ReLU}} $函数$ {f_2}(x) $的Si误差${E_{Si}}(\lambda )$, $\lambda \in [1.0 \times{\text{1}}{{\text{0}}^{-19}},1.0 \times{\text{1}}{{\text{0}}^{19}}]$, $\varepsilon = 1.0 \times{10^{ - 6}}$(上行)和$\varepsilon = 1.0 \times{10^{ - 40}}$(下行)

    Figure  5.  The Si-error ${E_{Si}}(\lambda )$ of the $ {\text{ReLU}} $function$ {f_2}(x) $ calculated by the WCNS schemes with $\varepsilon = 1.0 \times{10^{ - 6}}$ (top row), $\varepsilon =1.0 \times {10^{ - 40}}$(bottom row), and different scale factors $\lambda \in [1.0 \times{\text{1}}{{\text{0}}^{-19}},1.0 \times{\text{1}}{{\text{0}}^{19}}]$ for three number of cells $N$

    图  6  3套网格单元数$N$下使用WCNS格式计算Heaviside函数$ {f_3}(x) $的Si误差${E_{Si}}(\lambda )$, $\lambda \in [1.0 \times{\text{1}}{{\text{0}}^{ - 19}},1.0 \times{\text{1}}{{\text{0}}^{19}}]$, $\varepsilon = 1.0 \times{10^{ - 6}}$(上行)和$\varepsilon = 1.0 \times{10^{ - 40}}$(下行)

    Figure  6.  The Si-error ${E_{Si}}(\lambda )$ of the Heaviside function $ {f_3}(x) $ calculated by the WCNS schemes with $\varepsilon = 1.0 \times{10^{ - 6}}$ (top row), $\varepsilon = 1.0 \times{10^{ - 40}}$(bottom row), and different scale factors $\lambda \in [1.0 \times{\text{1}}{{\text{0}}^{ - 19}},1.0 \times{\text{1}}{{\text{0}}^{19}}]$ for three number of cells $N$

    图  7  在极值点${n_{cp}} = n = 1,2,3,4$下WCNS格式的收敛精度和${L_\infty }$误差, 且$\lambda = 1$, $\varepsilon = 1.0 \times {10^{ - 6}}$ (上行), $\varepsilon = 1.0 \times {10^{ - 40}}$(下行)

    Figure  7.  ${L_\infty }$error of the WCNS schemes with $\varepsilon = 1.0 \times {10^{ - 6}}$ (top row), $\varepsilon = 1.0 \times {10^{ - 40}}$(bottom row), and the scale factor $\lambda = 1$ for ${n_{cp}} = n = 1,2,3,4$

    图  8  $\varepsilon = 1.0 \times {10^{ - 6}}$$\varepsilon = 1.0 \times {10^{ - 40}}$时分别使用WCNS格式计算$\lambda = 1.0 \times {10^{ - 7}},1,{100}$下的Lax问题

    Figure  8.  Density of the Lax problem computed by the WCNS schemes with $\varepsilon = 1.0 \times {10^{ - 6}}$ and $\varepsilon =1.0 \times {10^{ - 40}}$ for $\lambda = 1.0 \times {10^{ - 7}},1,{100}$

    图  9  $\varepsilon =1.0 \times {10^{ - 40}}$时使用WCNS格式计算$ \lambda = 1.0 \times {10^{ - 7}} $的Sod问题

    Figure  9.  Density of the Sod problem computed by the WCNS schemes with $\varepsilon =1.0 \times {10^{ - 40}}$ for $ \lambda =1.0 \times {10^{ - 7}} $

    图  10  $ \varepsilon =1.0 \times {10^{ - 40}} $时使用WCNS格式计算$\lambda =1.0 \times {10^{ - 7}}$激波密度波问题

    Figure  10.  Density of the shock-density wave interaction problem computed by the WCNS schemes with $ \varepsilon =1.0 \times {10^{ - 40}} $ for $\lambda $= $1.0 \times {10^{ - 7}} $

    图  11  $ N = 400 $时使用WCNS格式计算爆炸波相互作用问题

    Figure  11.  Density of the blast-waves interaction problem computed by the WCNS schemes with $ N = 400 $

    图  12  $N = 800$时使用WCNS格式计算爆炸波相互作用问题

    Figure  12.  Density of the blast-waves interaction problem computed by the WCNS schemes with $ N = 800 $

    图  13  $\varepsilon = 1.0 \times {10^{ - 40}}$时使用WCNS格式计算Riemann问题中的构型3

    Figure  13.  Density of the configuration 3 of 2-D Riemann problems computed by the WCNS7-JSm/Zm/Dm schemes with $ \varepsilon = 1.0 \times {10^{ - 40}} $

    13  $\varepsilon =1.0 \times {10^{ - 40}}$时使用WCNS格式计算Riemann问题中的构型3 (续)

    13.  Density of the configuration 3 of 2-D Riemann problems computed by the WCNS7-JSm/Zm/Dm schemes with $ \varepsilon =1.0 \times {10^{ - 40}} $ (continued)

    图  14  $\varepsilon = 1.0 \times {10^{ - 40}}$时使用WCNS7-D/Dm格式计算$\lambda = 1.0 \times {10^{ - 7}}$下的二维 Riemann 问题中的构型6

    Figure  14.  Density of the configuration 6 of 2-D Riemann problems computed by the WCNS7-D/Dm schemes with $\lambda = 1.0 \times {10^{ - 7}}$ and $\varepsilon = 1.0 \times {10^{ - 40}}$

    14  $\varepsilon = 1.0 \times {10^{ - 40}}$时使用WCNS7-D/Dm格式计算$\lambda = 1.0 \times {10^{ - 7}}$下的二维 Riemann 问题中的构型6 (续)

    14.  Density of the configuration 6 of 2-D Riemann problems computed by the WCNS7-D/Dm schemes with $\lambda =1.0 \times {10^{ - 7}}$ and $\varepsilon = 1.0 \times {10^{ - 40}}$ (continued)

    图  15  $\varepsilon =1.0 \times {10^{ - 40}}$时使用WCNS7-Dm格式计算双马赫反射问题

    Figure  15.  Density of the double Mach reflection problem computed by the WCNS7-Dm scheme with $\varepsilon =1.0 \times {10^{ - 40}}$

    图  16  $\varepsilon = 1.0 \times {10^{ - 40}}$时使用WCNS7-Dm格式计算Richtmyer-Meshko不稳定性问题

    Figure  16.  Density of the Richtmyer-Meshkov instability problem computed by the WCNS7-Dm scheme with $\varepsilon = 1.0 \times {10^{ - 40}}$

    表  1  WCNS格式在$\varepsilon = 1.0 \times 1{{\text{0}}^{ - 40}}$,${n_{cp}} = 3$时的${L_\infty }$误差${E_N}$和收敛阶$O(\Delta {x^s})$

    Table  1.   ${L_\infty }$ error ${E_N}$, and order of accuracy $O(\Delta {x^s})$ of WCNS schemes with $\varepsilon = 1.0 \times 1{{\text{0}}^{ - 40}}$and ${n_{cp}} = 3$

    $\lambda $$N$WCNS7-JSmWCNS7-ZmWCNS7-DWCNS7-Dm
    ${E_N}$$s$${E_N}$$s$${E_N}$$s$${E_N}$$s$
    $1.0 \times {10^{ - 7}}$$ 80 $$2.51 \times {10^{ - 12}}$$ 2.28 \times {10^{ - 12}} $$ 2.84 \times {10^{ - 18}} $$ 1.70 \times {10^{ - 15}} $
    $ 160 $$ 3.09 \times {10^{ - 13}} $$ 3.02 $$ 2.74 \times {10^{ - 13}} $$ 3.05 $$ 2.24 \times {10^{ - 20}} $$ 6.99 $$ 2.00 \times {10^{ - 17}} $$ 6.41 $
    $ 320 $$ 3.83 \times {10^{ - 14}} $$ 3.01 $$ 3.38 \times {10^{ - 14}} $$ 3.02 $$ 1.75 \times {10^{ - 22}} $$ 6.99 $$ 1.83 \times {10^{ - 19}} $$ 6.77 $
    $ 640 $$ 4.76 \times {10^{ - 15}} $$ 3.01 $$ 4.19 \times {10^{ - 15}} $$ 3.01 $$ 1.37 \times {10^{ - 24}} $$ 7.00 $$ 1.52 \times {10^{ - 21}} $$ 6.91 $
    $ 1280 $$ 5.93 \times {10^{ - 16}} $$ 3.00 $$ 5.21 \times {10^{ - 16}} $$ 3.01 $$ 1.07 \times {10^{ - 26}} $$ 7.00 $$ 1.21 \times {10^{ - 23}} $$ 6.97 $
    $ 2560 $$ 7.40 \times {10^{ - 17}} $$ 3.00 $$ 6.50 \times {10^{ - 17}} $$ 3.00 $$ 8.39 \times {10^{ - 29}} $$ 7.00 $$ 9.57 \times {10^{ - 26}} $$ 6.99 $
    $ 5120 $$ 9.24 \times {10^{ - 18}} $$ 3.00 $$ 8.12 \times {10^{ - 18}} $$ 3.00 $$ 6.56 \times {10^{ - 31}} $$ 7.51 \times {10^{ - 28}} $$ 6.99 $
    $10\;240$$ 1.15 \times {10^{ - 18}} $$ 3.00 $$ 1.01 \times {10^{ - 18}} $$ 3.00 $$ 5.13 \times {10^{ - 33}} $$ 7.00 $$ 5.88 \times {10^{ - 30}} $$ 7.00 $
    ${100}$$ 80 $$ 2.51 \times {10^{ - 3}} $$ 2.28 \times {10^{ - 3}} $$ 7.54 \times {10^{ - 5}} $$ 1.70 \times {10^{ - 6}} $
    $ 160 $$ 3.09 \times {10^{ - 4}} $$ 3.02 $$ 2.74 \times {10^{ - 4}} $$ 3.05 $$ 7.15 \times {10^{ - 7}} $$ 6.72 $$ 2.00 \times {10^{ - 8}} $$ 6.41 $
    $ 320 $$ 3.83 \times {10^{ - 5}} $$ 3.01 $$ 3.38 \times {10^{ - 5}} $$ 3.02 $$ 5.42 \times {10^{ - 9}} $$ 7.04 $$ 1.83 \times {10^{ - 10}} $$ 6.77 $
    $ 640 $$ 4.76 \times {10^{ - 6}} $$ 3.01 $$ 4.19 \times {10^{ - 6}} $$ 3.01 $$ 4.06 \times {10^{ - 11}} $$ 7.06 $$ 1.52 \times {10^{ - 12}} $$ 6.91 $
    $ 1280 $$ 5.93 \times {10^{ - 7}} $$ 3.00 $$ 5.21 \times {10^{ - 7}} $$ 3.01 $$ 3.09 \times {10^{ - 13}} $$ 7.04 $$ 1.21 \times {10^{ - 14}} $$ 6.97 $
    $ 2560 $$ 7.40 \times {10^{ - 8}} $$ 3.00 $$ 6.50 \times {10^{ - 8}} $$ 3.00 $$ 2.37 \times {10^{ - 15}} $$ 7.02 $$ 9.57 \times {10^{ - 17}} $$ 6.99 $
    $ 5120 $$ 9.24 \times {10^{ - 9}} $$ 3.00 $$ 8.12 \times {10^{ - 9}} $$ 3.00 $$ 1.84 \times {10^{ - 17}} $$ 7.01 $$ 7.51 \times {10^{ - 19}} $$ 6.99 $
    $10\;240$$ 1.15 \times {10^{ - 9}} $$ 3.00 $$ 1.01 \times {10^{ - 9}} $$ 3.00 $$ 1.43 \times {10^{ - 19}} $$ 7.01 $$ 5.88 \times {10^{ - 21}} $$ 7.00 $
    下载: 导出CSV

    表  2  WCNS7-D/Dm格式的误差值和收敛阶 ($\varepsilon $ = 1.0 × 10−40)

    Table  2.   Error and accuracy statistics for WCNS7-D/Dm schemes ($\varepsilon $ = 1.0 × 10−40)

    $\lambda $$N$WCNS7-DWCNS7-Dm
    ${L_1}$errororder${L_\infty }$errororder${L_1}$errororder${L_\infty }$errororder
    $1.0 \times {10^{ - 7}}$$ 5.73 \times {10^{ - 9}} $$ 3.25 \times {10^{ - 8}} $$ 5.89 \times {10^{ - 9}} $$ 5.04 \times {10^{ - 8}} $
    $ 40 $$ 1.27 \times {10^{ - 9}} $$ 2.17 $$ 7.92 \times {10^{ - 9}} $$ 2.04 $$ 1.09 \times {10^{ - 9}} $$ 2.43 $$ 1.10 \times {10^{ - 8}} $$ 2.19 $
    $ 80 $$ 4.12 \times {10^{ - 11}} $$ 4.95 $$ 3.70 \times {10^{ - 10}} $$ 4.42 $$ 4.17 \times {10^{ - 11}} $$ 4.71 $$ 3.72 \times {10^{ - 10}} $$ 4.89 $
    $ 160 $$ 3.97 \times {10^{ - 13}} $$ 6.70 $$ 4.17 \times {10^{ - 12}} $$ 6.47 $$ 3.97 \times {10^{ - 13}} $$ 6.71 $$ 4.17 \times {10^{ - 12}} $$ 6.48 $
    $ 320 $$ 3.26 \times {10^{ - 15}} $$ 6.93 $$ 3.47 \times {10^{ - 14}} $$ 6.91 $$ 3.26 \times {10^{ - 15}} $$ 6.93 $$ 3.47\times {10^{ - 14}} $$ 6.93 $
    $ 640 $$ 2.62 \times {10^{ - 17}} $$ 6.96 $$ 2.85 \times {10^{ - 16}} $$ 6.93 $$ 2.62 \times {10^{ - 17}} $$ 6.96 $$ 2.85 \times {10^{ - 16}} $$ 6.93 $
    1$ 20 $$ 5.51 \times {10^{ - 2}} $$ 4.85 \times {10^{ - 1}} $$ 5.89 \times {10^{ - 2}} $$ 5.04 \times {10^{ - 1}} $
    $ 40 $$ 8.75 \times {10^{ - 3}} $$ 2.65 $$ 9.15 \times {10^{ - 2}} $$ 2.40 $$ 1.09 \times {10^{ - 2}} $$ 2.43 $$ 1.10 \times {10^{ - 1}} $$ 2.19 $
    $ 80 $$ 4.07 \times {10^{ - 4}} $$ 4.43 $$ 3.70 \times {10^{ - 3}} $$ 4.63 $$ 4.16 \times {10^{ - 4}} $$ 4.17 $$ 3.72 \times {10^{ - 3}} $$ 4.89 $
    $ 160 $$ 3.97 \times {10^{ - 6}} $$ 6.68 $$ 4.17 \times {10^{ - 5}} $$ 6.47 $$ 3.97 \times {10^{ - 6}} $$ 6.71 $$ 4.17 \times {10^{ - 5}} $$ 6.48 $
    $ 320 $$ 3.26 \times {10^{ - 8}} $$ 6.93 $$ 3.47 \times {10^{ - 7}} $$ 6.91 $$ 3.26 \times {10^{ - 8}} $$ 6.93 $$ 3.47 \times {10^{ - 7}} $$ 6.91 $
    $ 640 $$ 2.62 \times {10^{ - 10}} $$ 6.96 $$ 2.85 \times {10^{ - 9}} $$ 6.93 $$ 2.62 \times {10^{ - 10}} $$ 6.96 $$ 2.85 \times {10^{ - 9}} $$ 6.93 $
    ${10^2}$$ 20 $$ 5.89 \times {10^0} $$ 5.04 \times {10^{ - 1}} $$ 5.89 \times {10^0} $$ 5.04 \times {10^{ - 1}} $
    $ 40 $$ 1.10 \times {10^0} $$ 2.42 $$ 1.11 \times {10^{ - 1}} $$ 2.18 $$ 1.09 \times {10^0} $$ 2.43 $$ 1.10 \times {10^{ - 1}} $$ 2.19 $
    $ 80 $$ 4.43 \times {10^{ - 2}} $$4.63$$ 3.71 \times {10^{ - 1}} $$ 4.90 $$ 4.16 \times {10^{ - 2}} $$ 4.71 $$ 3.72 \times {10^{ - 1}} $$ 4.89 $
    $ 160 $$ 3.97 \times {10^{ - 4}} $$ 6.80 $$ 4.17 \times {10^{ - 3}} $$ 6.47 $$ 3.97 \times {10^{ - 4}} $$ 6.71 $$ 4.17 \times {10^{ - 3}} $$ 6.48 $
    $ 320 $$ 3.26 \times {10^{ - 6}} $$ 6.93 $$ 3.47 \times {10^{ - 5}} $$ 6.91 $$ 3.26 \times {10^{ - 6}} $$ 6.93 $$ 3.47 \times {10^{ - 5}} $$ 6.91 $
    $ 640 $$ 2.62 \times {10^{ - 8}} $$ 6.96 $$ 2.85 \times {10^{ - 7}} $$ 6.93 $$ 2.62 \times {10^{ - 8}} $$ 6.96 $$ 2.85 \times {10^{ - 7}} $$ 6.93 $
    下载: 导出CSV
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出版历程
  • 收稿日期:  2022-08-29
  • 录用日期:  2022-10-20
  • 网络出版日期:  2022-10-21
  • 刊出日期:  2023-01-18

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