基于有限积分法的输流直管轴向流固耦合有限积分法数值模拟
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摘要
针对弹性输流直管轴向耦合振动四方程模型,应用有限积分法分别配合隐式欧拉法和拉普拉斯数值反演法建立两种不同的数值模式,研究瞬时关阀时输流直管轴向耦合振动响应特性。采用径向基函数和多项式基函数组成的混合函数通过节点构造有限积分法的插值函数,将函数积分结果转换为节点函数值的线性累加,简单易行且稳定性好。将所提出的数值模式计算结果与前人的研究成果进行对比,吻合良好,表明有限积分法在处理间断波问题时具有较高的鲁棒性和稳定性,可精确地捕捉到水锤压力的非线性变化过程,但在时间项处理上,拉普拉斯数值反演法在间断处存在一定的数值振荡,隐式欧拉法并未出现,可见在水击问题上有限积分法配合隐式欧拉法的精度更高、稳定性更好。研究结果表明:在考虑泊松和连接耦合共同作用时,管道压力水头波动与经典水锤相比,相位出现延迟,与只有泊松耦合相比,由于管道的振动叠加了与液体产生的强迫振动,压力脉动明显加剧。
Based on the four-equation model of the elastic fluid-conveying straight pipe,the axial coupling vibration characteristics with water hammer effect are studied in this paper.Two numerical schemes were developed by combining the finite integration method(FIM)with the implicit Euler method(IEM)and with the numerical inversion of Laplace transform(NILT),respectively,to efficiently and accurately analyze the axial coupling vibration of the pipe.A mixed function,consisting of the radial basis function(RBF)and the polynomial basis function(PBF),is adopted to construct the interpolation function of the FIM at each node.Then,the integral value of a function can be discretized into linear combinations of the function values at the node.This feature makes the FIM easy to program,straightforward and efficient when it is applied to a complicated computational region.To evaluate the accuracy and robustness of the proposed scheme for dealing with the discontinuous wave problems,it was compared with other numerical methods and reasonable agreements were achieved.It indicates that FIM can accurately capture the nonlinear process with water hammer effect.Nevertheless,on processing the temporal term,the numerical oscillation occurs at the discontinuity for NILT while IEM does not have.Moreover,when considering the interaction of Poisson and connection coupling,the pipe pressure wave appears to have some delay in the phase compared with the classic water hammer.Due to the superposition of the pipe vibration and the forced vibrations produced by liquids,the pressure fluctuation increases significantly compared with the situation when only considering Poisson coupling.
Based on the four-equation model of the elastic fluid-conveying straight pipe,the axial coupling vibration characteristics with water hammer effect are studied in this paper.Two numerical schemes were developed by combining the finite integration method(FIM)with the implicit Euler method(IEM)and with the numerical inversion of Laplace transform(NILT),respectively,to efficiently and accurately analyze the axial coupling vibration of the pipe.A mixed function,consisting of the radial basis function(RBF)and the polynomial basis function(PBF),is adopted to construct the interpolation function of the FIM at each node.Then,the integral value of a function can be discretized into linear combinations of the function values at the node.This feature makes the FIM easy to program,straightforward and efficient when it is applied to a complicated computational region.To evaluate the accuracy and robustness of the proposed scheme for dealing with the discontinuous wave problems,it was compared with other numerical methods and reasonable agreements were achieved.It indicates that FIM can accurately capture the nonlinear process with water hammer effect.Nevertheless,on processing the temporal term,the numerical oscillation occurs at the discontinuity for NILT while IEM does not have.Moreover,when considering the interaction of Poisson and connection coupling,the pipe pressure wave appears to have some delay in the phase compared with the classic water hammer.Due to the superposition of the pipe vibration and the forced vibrations produced by liquids,the pressure fluctuation increases significantly compared with the situation when only considering Poisson coupling.
引文
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[17]Li M,Chen C S,Hon Y C,et al.Finite integration method for solving multi-dimensional partial differential equations[J].Applied Mathematical Modelling,2015,39(17):4979-4994.
[18]Li M,Hon Y C,Korakianitis T,et al.Finite integration method for nonlocal elastic bar under static and dynamic loads[J].Engineering Analysis with Boundary Elements,2013,37(5):842-849.
[19]Li M,Tian Z,Hon Y,et al.Improved finite integration method for partial differential equations[J].Engineering Analysis with Boundary Elements,2016,64:230-236.
[20]Yu Y,Xu D,Hon Y.Reconstruction of inaccessible boundary value in a sideways parabolic problem with variable coefficients-forward collocation with finite integration method[J].Engineering Analysis with Boundary Elements,2015,61:78-90.
[21]Yun D F,Wen Z H,Hon Y C.Adaptive least squares finite integration method for higher-dimensional singular perturbation problems with multiple boundary layers[J].Applied Mathematics and Computation,2015,271:232-250.
[22]Li Y,Hon Y C.Finite integration method with radial basis function for solving stiff problems[J].Engineering Analysis with Boundary Elements,2017,82:32-42.
[23]Tijsseling A S.Exact solution of linear hyperbolic four-equation system in axial liquid-pipe vibration[J].Journal of Fluids&Structures,2003,18(2):179-196.
[24]Dubner H,Abate J.Numerical inversion of laplace transforms by relating them to the finite fourier Cosine Transform[J].Journal of the ACM,1968,15(1):115-123.
[2]Tijsseling A S.Fluid-structure interaction in liquidfilled pipe systems:a review[J].Journal of Fluids&Structures,1996,10(2):109-146.
[3]Skalak R.An extension of the theory of water hammer[J].Transactions of the AMSE,1956,8:17-22.
[4]Lin T C,Morgan G W.Wave propagation through fluid contained in a cylindrical,elastic Shell[J].Journal of the Acoustical Society of America,1956,28(6):1165-1176.
[5]Tijsseling A S.Fluid-structure interaction in case of water hammer with cavitation[D].Ph.D.Thesis of Delft University of Technology,The Netherlands,1993.
[6]张挺,吕勇臻,杨志强,等.水锤激励下黏弹性输流直管轴向振动响应特性[J].振动工程学报,2017,30(2):241-248.ZHANG Ting,LYong-zhen,YANG Zhi-qiang,et al.Axial vibration response of the viscoelastic fluidconveying straight pipe induced by water hammer[J].Journal of Vibration Engineering,2017,30(2):241-248.
[7]Janez Gale,Iztok Tiselj.Applicability of the godunov's method for fundamental four equation FSI model[C].Proceeding of the International Conference Nuclear of Energy for New Europe 2005,Slovenia,2005:1-10.
[8]Tijsseling A S,Lavooij C S W.Waterhammer with fluid-structure interaction[J].Applied Scientific Research,1990,47(3):273-285.
[9]Lavooij C S W,Tijsseling A S.Fluid-structure interaction in liquid-filled piping systems[J].Journal of Fluids and Structures,1991,5(5):573-595.
[10]杨超,范士娟.管材参数对输液管流固耦合振动的影响[J].振动与冲击,2011,30(7):210-213.YANG Chao,FAN Shi-juan.Influence of pipe parameters on fluid-structure coupled vibration of a fluid-conveying pipe[J].Journal of Vibration and Shock,2011,30(7):210-213.
[11]孙玉东,王锁泉,刘忠族,等.液-管耦合空间管路系统振动噪声的有限元分析方法[J].振动工程学报,2005,18(2):149-154.SUN Yu-dong,WANG Suo-quan,LIU Zhong-zu,et al.Unified finite element method for analyzing vibration and noise in 3-D piping system with liquid-pipe coupling[J].Journal of Vibration Engineering,2005,18(2):149-154.
[12]Sreejith B,Jayaraj K,Ganesan N,et al.Finite element analysis of fluid-structure interaction in pipeline systems[J].Nuclear Engineering and Design,2004,227(3):313-322.
[13]Wang Z M,Tan S K.Vibration and pressure fluctuation in a flexible hydraulic power systemon an aircraft[J].Computers and Fluids,1998,27(1):1-9.
[14]Ahmadi A,Keramat A.Investigation of fluid-structure interaction with various types of junction coupling[J].Journal of Fluids and Structures,2010,26(7):1123-1141.
[15]Wen P H,Hon Y C,Li M,et al.Finite integration method for partial differential equations[J].Applied Mathematical Modelling,2013,37(24):10092-10106.
[16]李书伟,徐定华,余跃.双调和方程的有限积分方法[J].浙江理工大学学报,2016,35(1):133-139.LI Shuwei,XU Dinghua,YU Yue.Finite Integration Method for Biharmonic Equations[J].Journal of Zhejiang Sci-tech University,2016,35(1):133-139.
[17]Li M,Chen C S,Hon Y C,et al.Finite integration method for solving multi-dimensional partial differential equations[J].Applied Mathematical Modelling,2015,39(17):4979-4994.
[18]Li M,Hon Y C,Korakianitis T,et al.Finite integration method for nonlocal elastic bar under static and dynamic loads[J].Engineering Analysis with Boundary Elements,2013,37(5):842-849.
[19]Li M,Tian Z,Hon Y,et al.Improved finite integration method for partial differential equations[J].Engineering Analysis with Boundary Elements,2016,64:230-236.
[20]Yu Y,Xu D,Hon Y.Reconstruction of inaccessible boundary value in a sideways parabolic problem with variable coefficients-forward collocation with finite integration method[J].Engineering Analysis with Boundary Elements,2015,61:78-90.
[21]Yun D F,Wen Z H,Hon Y C.Adaptive least squares finite integration method for higher-dimensional singular perturbation problems with multiple boundary layers[J].Applied Mathematics and Computation,2015,271:232-250.
[22]Li Y,Hon Y C.Finite integration method with radial basis function for solving stiff problems[J].Engineering Analysis with Boundary Elements,2017,82:32-42.
[23]Tijsseling A S.Exact solution of linear hyperbolic four-equation system in axial liquid-pipe vibration[J].Journal of Fluids&Structures,2003,18(2):179-196.
[24]Dubner H,Abate J.Numerical inversion of laplace transforms by relating them to the finite fourier Cosine Transform[J].Journal of the ACM,1968,15(1):115-123.