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There exists a kind of quadratic nonlinear system with specific type of turbulent fluctuation excitation in nature, which belongs to a special non-Gaussian input signal system. Its characteristic is that the input signal spectrum is generated by turbulent fluctuations, and the power spectrum distribution of this turbulence fluctuation signal is close to Gaussian distribution. Starting with the work of Choi et al. (
1985 J. Sound Vib. 99 309 ) and Kim et al. [1987 IEEE J. Ocean. Eng. OE-12 568 ), we extend the simulation of a specific turbulent fluctuation excited response-type quadratic nonlinear system represented by the wave excited mooring ship response, and fully develop the internal development of turbulence based on bispectral analysis technology. We also extend the simulation system and conduct systematic modeling analysis. The complete iterative method [2020 Phys. Scr. 95 055202 ] is used to solve the model, and calculate the linear transfer function and quadratic nonlinear transfer function. The comparison of simulation and modeling results with the real systems and their models confirms the correctness of the results from system simulation, system modeling, and model solving. The results obtained are all in line with expectations. The coherence analysis shows that the quadratic coherence of the random wave-ship swaying system is much greater than the linear coherence, but the linear coherence of the fully developed turbulence is greater for the near Gaussian input type. The reverse computation verification or comparison with real systems indicates that the turbulence simulation and system modeling method in this work have good accuracy and high efficiency in solving algorithms, and the research results can be effectively applied to the model description and system analysis of the quadratic nonlinear systems related to specific turbulent fluctuation excitation response.-
Keywords:
- turbulence /
- quadratic nonlinear system /
- bispectral analysis /
- simulation /
- modeling
[1] Choi D, Miksad R W, Powers E J 1985 J. Sound Vib. 99 309
Google Scholar
[2] Kim K I, Powers E J, Ritz Ch P, Miksad R W, Fischer F J 1987 IEEE J. Ocean. Eng. OE-12 568
Google Scholar
[3] Cherneva Z, Soares C G 2008 Appl. Ocean Res. 30 215
Google Scholar
[4] Howard R S, Finneran J J, Ridgway S H 2006 Anesth. Analg. 103 626
Google Scholar
[5] Zhang J, Benoit M, Kimmoun O, Chabchoub A, Hsu H C 2019 Fluids 4 99
Google Scholar
[6] Zhang S G, Lian J J, Li J X, Liu F, Ma B 2022 Ocean Eng. 264 112473
Google Scholar
[7] Smith D E, Powers E J 1973 Phys. Fluids 16 1373
Google Scholar
[8] Hasegawa A, Maclennan C G 1979 Phys. Fluids 22 2122
Google Scholar
[9] Schmidt O T 2020 Nonlinear Dynam. 102 2479
Google Scholar
[10] Cui G, Jacobi I 2021 Phys. Rev. Fluids 6 014604
Google Scholar
[11] O’Brien M J, Burkhart B, Shelley M J 2022 Astrophys. J. 930 149
Google Scholar
[12] Unnikrishnan S, Gaitonde D V 2020 J. Fluid Mech. 905 A25
Google Scholar
[13] Enugonda R, Anandan V K, Ghosh B 2023 J. Electromagnet. Wave. 37 69
Google Scholar
[14] Ge Z, Liu P C 2007 Ann. Geophys. 25 1253
Google Scholar
[15] Kim Y C, Powers E J 1979 IEEE Trans. Plasma Sci. PS-7 120
Google Scholar
[16] Smith D E, Powers E J, Caldwell G S 1974 IEEE Trans. Plasma Sci. PS-2 263
Google Scholar
[17] Manz P, Ramisch M, Stroth U, Naulin V, Scott B D 2008 Plasma Phys. Contr. Fusion 50 035008
Google Scholar
[18] Manz P, Ramisch M, Stroth U 2009 Phys. Rev. Lett. 103 165004
Google Scholar
[19] Shen Y, Shen Y H, Dong J Q, Zhao K J, Shi Z B, Li J Q 2022 Chin. Phys. B 31 065206
Google Scholar
[20] Shen Y, Dong J Q, Shi Z B, Nagayama Y, Hirano Y, Yambe K, Yamaguchi S, Zhao K J, Li J Q 2019 Nucl. Fusion 59 044001
Google Scholar
[21] Kim Y C, Wong W F, Powers E J, Roth J R 1979 Proc. IEEE 67 428
Google Scholar
[22] Hong J Y, Kim Y C, Powers E J 1980 Proc. IEEE 68 1026
Google Scholar
[23] Ritz Ch P, Powers E J 1986 Physica D 20 320
Google Scholar
[24] Ritz C P, Powers E J, Miksad R W, Solis R S 1988 Phys. Fluids 31 3577
Google Scholar
[25] Ritz Ch P, Powers E J, Bengtson R D 1989 Phys. Fluids B 1 153
Google Scholar
[26] Kim J S, Durst R D, Fonck R J, Fernandez E, Ware A, Terry P W 1996 Phys. Plasmas 3 3998
Google Scholar
[27] Shen Y H, Li J Y, Li T, Li J 2020 Phys. Scr. 95 055202
Google Scholar
[28] Shen Y H, Li J, Li T 2020 J. Phys. Soc. Jpn. 89 044501
Google Scholar
[29] Hasegawa A, Mimma K 1978 Phys. Fluids 21 87
Google Scholar
[30] Proakis J G, Manolakis D G 2006 Digital Signal Processing-Principles, Algorithem, and Applications (4th Ed.) (Beijing: Electronic Industry Press
[31] Dolgikh G I, Gromasheva O S, Dolgikh S G, Plotnikov A A 2021 J. Mar. Sci. Eng. 9 861
Google Scholar
[32] Xu M, Tynan G R, Holland C, Yan Z, Muller S H, Yu J H 2009 Phys. Plasmas 16 042312
Google Scholar
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图 1 系泊船舶在单向不规则随机海洋中的摇摆运动 (a) 随机海浪波高记录; (b) 入射海浪的自功率谱; (c) 系泊配置; (d) 系泊船舶摇摆响应的时间记录; (e) 系泊船舶摇摆响应的自功率谱(参考自文献[1], 类似的图也可见于文献[2])
Figure 1. Sway motion of a moored vessel in response to a unidirectional irregular beam sea: (a) Incident irregular sea record; (b) auto-power spectrum of incident sea-wave; (c) mooring configuration; (d) time record of moored barge sway response; (e) auto-power spectrum of barge sway response (Referenced by Ref. [1], as well as Ref. [2]).
图 2 充分发展湍流结构图 (a) 湍流信号自功率谱; (b) $ t=16\Delta t, 32\Delta t, 64\Delta t $三个时刻测量的湍流信号, 其中$ \Delta t $表示测试两个相邻信号的时间间隔
Figure 2. Fully developed turbulence: (a) Turbulence signal auto-power spectrum; (b) simulated turbulence signals measured at $ t=16\Delta t, 32\Delta t $ and $ 64\Delta t $.
图 4 随机海波激励船舶摇摆系统试验数据 (a) 输入信号谱密度; (b) 近似输入(随机海波波高仿真)信号; (c) 输出信号谱密度; (d) 近似输出(摇摆位形仿真)信号
Figure 4. Test data of moored vessel sway system excited by random sea waves: (a) Input signal spectra; (b) approximate input (simulated random sea wave height) signals; (c) output signal spectral density; (d) approximate output (simulated sway configuration) signal.
图 5 系统输入/输出信号互双谱幅度$ |\langle{{Y}_{i+j}{X}_{i}^{*}{X}_{j}^{*}}\rangle| $ (a) 立体视图; (b) 等高线视图
Figure 5. Amplitude of the cross bi-spectrum of the system $ |\langle{{Y}_{i+j}{X}_{i}^{*}{X}_{j}^{*}}\rangle| $: (a) Perspective view; (b) contour plot.
图 6 计算结果 (a) 线性传递函数$ {L}_{f} $实部; (b) $ {L}_{f} $虚部; (c) $ {L}_{f} $幅度, 其中, 绿线和黑色点线分别表示完备程序(包含非高斯(non-Gaussian)效应)和高斯(Gaussian)程序的计算结果; (d)完备程序计算的二次传递函数幅度$ |{Q}_{f}^{{f}_{1}, {f}_{2}}| $三维分布图; (e)高斯程序计算的二次传递函数幅度$ |{Q}_{f}^{{f}_{1}, {f}_{2}}| $三维分布图
Figure 6. Resultant calculations: (a) The real of linear transfer function $ {L}_{f} $; (b) the imaginary of linear transfer function $ {L}_{f}; $(c) the magnitude of linear transfer function $ {L}_{f} $, where the green and black dotted lines are the results calculated using complete program (including non-Gaussian effects) and Gaussian program, respectively; (d) the perspective view (triple distribution) of the magnitude of quadratic transfer function $ |{Q}_{f}^{{f}_{1}, {f}_{2}}| $ calculated by complete programs; (e) the perspective view (triple distribution) of the magnitude of quadratic transfer function $ |{Q}_{f}^{{f}_{1}, {f}_{2}}| $ calculated by Gaussian programs.
图 7 收敛性分析 (a) $ \left|{L}_{f}\right| $相对误差; (b) $ \left|{Q}_{f}\right| $相对误差
Figure 7. Convergence analysis: (a) Relative error of $ \left|{L}_{f}\right| $; (b) relative error of $ \left|{Q}_{f}\right| $.
图 9 相干系数 (a) 应用完备程序计算结果; (b) 应用高斯输入型程序计算结果
Figure 9. Coherence coefficients: (a) Results from the complete program; (b) results from specific program for Gaussian input signals.
图 8 二次传递函数幅度$ |{Q}_{f}^{{f}_{1}, {f}_{2}}| $等位线图
Figure 8. Contour plot of the magnitude of quadratic transfer function $ |{Q}_{f}^{{f}_{1}, {f}_{2}}| $.
图 10 验算结果 (a)原初始输入信号谱与计算的输出信号谱; (b)复验的摇摆位形; (c)应用高斯输入程序结果复验; (d)在复验处理中移除汉宁窗得到的摇摆位形, 与原输出位形一致
Figure 10. Verification results: (a) The original initial input signal spectrum and the calculated output signal spectrum; (b) sway configure of the retest; (c) sway configure computed after applying Gaussian-input program; (d) sway configure computed after removing the Hanning window.
图 11 充分发展湍流的模拟输入与输出信号谱
Figure 11. Simulated input and output spectra of fully developed turbulence.
图 12 全发展湍流输入/输出信号互双谱幅度$ |\langle{{Y}_{i+j}{X}_{i}^{{\mathrm{*}}}{X}_{j}^{{\mathrm{*}}}}\rangle| $ (a) 立体视图; (b) 等高线视图
Figure 12. Amplitude of the cross bi-spectrum of the full-developed turbulence $ |\langle{{Y}_{i+j}{X}_{i}^{{\mathrm{*}}}{X}_{j}^{{\mathrm{*}}}}\rangle| $: (a) Perspective view; (b) contour plot.
图 13 充分发展湍流系统传递函数计算结果, 其中$ k={k}_{1}^{N}+{k}_{2}^{N},\; {k}_{1}^{N}={f}_{1}/{f}_{{\mathrm{N}}{\mathrm{y}}{\mathrm{q}}} $, $ {k}_{2}^{N}={f}_{2}/{f}_{{\mathrm{N}}{\mathrm{y}}{\mathrm{q}}} $
Figure 13. Calculation results of transfer function for simulated full-developed turbulent systems, where $ k={k}_{1}^{N}+{k}_{2}^{N} $, with $ {k}_{1}^{N}={f}_{1}/{f}_{{\mathrm{N}}{\mathrm{y}}{\mathrm{q}}} $ and $ {k}_{2}^{N}={f}_{2}/{f}_{{\mathrm{N}}{\mathrm{y}}{\mathrm{q}}} $.
图 14 相干性分析
Figure 14. Coherence analysis.
图 15 扩充计算 (a)相位差; (b)平均色散; (c) 线性增长率
Figure 15. Extended calculation: (a) Phase difference; (b) average dispersion; (c) linear growth rate.
图 16 根据输出湍流信号谱反算的信号幅度随测量时间点的变化(部分)
Figure 16. Output signals calculated based on the output turbulent signal spectra (a part).
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[1] Choi D, Miksad R W, Powers E J 1985 J. Sound Vib. 99 309
Google Scholar
[2] Kim K I, Powers E J, Ritz Ch P, Miksad R W, Fischer F J 1987 IEEE J. Ocean. Eng. OE-12 568
Google Scholar
[3] Cherneva Z, Soares C G 2008 Appl. Ocean Res. 30 215
Google Scholar
[4] Howard R S, Finneran J J, Ridgway S H 2006 Anesth. Analg. 103 626
Google Scholar
[5] Zhang J, Benoit M, Kimmoun O, Chabchoub A, Hsu H C 2019 Fluids 4 99
Google Scholar
[6] Zhang S G, Lian J J, Li J X, Liu F, Ma B 2022 Ocean Eng. 264 112473
Google Scholar
[7] Smith D E, Powers E J 1973 Phys. Fluids 16 1373
Google Scholar
[8] Hasegawa A, Maclennan C G 1979 Phys. Fluids 22 2122
Google Scholar
[9] Schmidt O T 2020 Nonlinear Dynam. 102 2479
Google Scholar
[10] Cui G, Jacobi I 2021 Phys. Rev. Fluids 6 014604
Google Scholar
[11] O’Brien M J, Burkhart B, Shelley M J 2022 Astrophys. J. 930 149
Google Scholar
[12] Unnikrishnan S, Gaitonde D V 2020 J. Fluid Mech. 905 A25
Google Scholar
[13] Enugonda R, Anandan V K, Ghosh B 2023 J. Electromagnet. Wave. 37 69
Google Scholar
[14] Ge Z, Liu P C 2007 Ann. Geophys. 25 1253
Google Scholar
[15] Kim Y C, Powers E J 1979 IEEE Trans. Plasma Sci. PS-7 120
Google Scholar
[16] Smith D E, Powers E J, Caldwell G S 1974 IEEE Trans. Plasma Sci. PS-2 263
Google Scholar
[17] Manz P, Ramisch M, Stroth U, Naulin V, Scott B D 2008 Plasma Phys. Contr. Fusion 50 035008
Google Scholar
[18] Manz P, Ramisch M, Stroth U 2009 Phys. Rev. Lett. 103 165004
Google Scholar
[19] Shen Y, Shen Y H, Dong J Q, Zhao K J, Shi Z B, Li J Q 2022 Chin. Phys. B 31 065206
Google Scholar
[20] Shen Y, Dong J Q, Shi Z B, Nagayama Y, Hirano Y, Yambe K, Yamaguchi S, Zhao K J, Li J Q 2019 Nucl. Fusion 59 044001
Google Scholar
[21] Kim Y C, Wong W F, Powers E J, Roth J R 1979 Proc. IEEE 67 428
Google Scholar
[22] Hong J Y, Kim Y C, Powers E J 1980 Proc. IEEE 68 1026
Google Scholar
[23] Ritz Ch P, Powers E J 1986 Physica D 20 320
Google Scholar
[24] Ritz C P, Powers E J, Miksad R W, Solis R S 1988 Phys. Fluids 31 3577
Google Scholar
[25] Ritz Ch P, Powers E J, Bengtson R D 1989 Phys. Fluids B 1 153
Google Scholar
[26] Kim J S, Durst R D, Fonck R J, Fernandez E, Ware A, Terry P W 1996 Phys. Plasmas 3 3998
Google Scholar
[27] Shen Y H, Li J Y, Li T, Li J 2020 Phys. Scr. 95 055202
Google Scholar
[28] Shen Y H, Li J, Li T 2020 J. Phys. Soc. Jpn. 89 044501
Google Scholar
[29] Hasegawa A, Mimma K 1978 Phys. Fluids 21 87
Google Scholar
[30] Proakis J G, Manolakis D G 2006 Digital Signal Processing-Principles, Algorithem, and Applications (4th Ed.) (Beijing: Electronic Industry Press
[31] Dolgikh G I, Gromasheva O S, Dolgikh S G, Plotnikov A A 2021 J. Mar. Sci. Eng. 9 861
Google Scholar
[32] Xu M, Tynan G R, Holland C, Yan Z, Muller S H, Yu J H 2009 Phys. Plasmas 16 042312
Google Scholar
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