线性耗散紧致格式应用于计算气动声学的基础研究
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摘要
在计算气动声学(CAA)领域,声波在空间传播问题的特征明显不同于常规的空气动力学问题,这些特征要求数值格式具有尽可能小的色散和耗散误差。高精度线性耗散紧致格式(DCS)是邓小刚研究员在中心型紧致格式(CCS)基础上提出的一类单参数的线性高精度格式,同中心型紧致格式相比具有抑制高波数的振荡的优点,本文目的是为DCS格式应用于计算气动声学做一些基础性研究。
声波在空间高精度捕捉的同时,在时间方向也需要被精确地追踪,因此时间积分方法也同样需要高精度。目前,在计算气动声学领域,常用的时间推进格式是显式龙格-库塔方法。为了提高龙格-库塔时间积分方法的耗散和色散性能,Hu提出了低耗散和低色散龙格—库塔方法,本文因此发展了三种针对DCS格式的低耗散和低色散龙格—库塔时间积分算法。
在本文的第二章中,参照Hu和Jan Ramboerd对中心型紧致格式(CCS)优化的基本思想,类似地构造了基于耗散型紧致格式(DCS)的一类低耗散低色散龙格—库塔时间积分方法,但由于DCS与CCS存在一个本质的差别,即耗散性,使得这种方法得到的时间推进优化格式并不具有改善计算结果精准度的能力。
在本文的第三章讨论了色散误差限制的增长因子误差优化方法。参照Hu和Jan Ramboerd对中心型紧致格式(CCS)优化的基本思想得到的LDDRK格式表现不理想,可能是该优化方法不具有同时控制DCS的耗散与色散的能力。因此,提出了强化色散误差控制的一种改进的优化方法,它的优化思想是通过Taylor展开,直接控制色散误差,再利用Hu和Jan Ramboerd的思想控制耗散误差,从而达到同时控制耗散与色散误差的目的。典型算例的计算结果表明,这种优化方法得到的LDDRK方法,能够在一定程度上改善计算结果的精度,但并不十分理想,可能是该优化方法仍没能很好反映DCS格式的特性。
在本文的第四章提出了谱区间限制的增长因子误差优化方法。为了得到理想的LDDRK格式,必须考虑DCS格式耗散性这一基本特性。引入了谱区间选择的优化限制条件,这是确保优化算法保持DCS格式的这种特性。通过Fourier分析和典型算例对比计算发现,引入优化谱区间限制条件的优化方法得到的LDDRK格式具有良好的模拟精度。
众所周知,对于诸如声波模拟等一些流动问题的细致模拟中,边界处理将可能对流场内部结构产生影响。基于Jae Wook Kim和Duck Joo Lee的思想,发展了与DCS格式相匹配特征边界条件,推导出了Euler方程的壁面特征边界条件,改进了高精度格式基于对流信息特征传播的对接边界处理技术,同时提出了基于信息传播的粘性壁面边界条件和特征远场边界条件处理方法。
基于上述特征边界条件发展了多块对接网格的高精度线性耗散紧致格式DCS3和DCS5的计算软件,模拟了二维、无粘/粘性、定常/非定常圆柱绕流,并分析了其中非定常流场产生的噪声,得到了很好的计算结果,本文的第六章展示了这些结果,这显示了DCS格式应用于计算气动声学的潜力。
但是要把DCS格式应用于解决工程上的声学问题,还有需要改进的地方,本文的第六章对这些问题进行了总结。
In the field of computational aero acoustics(CAA), the characteristics of the acoustic waves propagate in space are obviously different from that in the ordinary aero dynamic, which need the numerical schemes have low dissipation errors and low dispersion errors as much as possible. Dissipative Compact Scheme(DCS) derived by Deng Xiaogang base on Central Compact Scheme(CCS) is a kind of linear high-order accurate scheme with single parameter, Compare with CCS, DCS has the particular advantage that it can restrain the oscillations in the high wavenumber range, in this thesis basic research of DCS used in CAA has done.
When the acoustic waves tracked accurately in spatial, they also have to be tracked accurately in time, so the time integration schemes also need high-order accurate. By far the most popular schemes in CAA are the Runge-kutta schemes. In order to improve the dissipative and dispersive behavior of Runge-kutta time integration schemes, low-dissipation and low-dispersion Runge-kutta time integration schemes derived by Hu. So in this thesis based on dissipative compact schemes, three optimal low-dissipation and low-dispersion Runge-kutta time integration methods have proposed.
In second chapter, a class of low-dissipation and low-dispersion Runge-kutta time integration schemes is proposed for the DCS schemes using the optimal method of Hu, but because DCS and CCS have a entitative different which is the dissipation, so it have on efficiency.
Amplification Factor Error Optimized Method is discussed in third chapter. Using the optimal method of Hu to propose low-dissipation and low-dispersion Runge-kutta time integration schemes for DCS have no efficiency, may be this optimal method can't control the dissipation errors and the dispersion errors at the same time causes the problem. So in this chapter a new optimal method which control dispersion errors especially has proposed. This optimal method controlls dispersion error by Taylor series expanding and dissipative error by the optimal method of Hu,. But the results indicate that the improvement is limited, this may be caused by the optimal method doesn't adapt to the DCS which is dissipative.
Because of the problems mentioned in the two optimal methods above, in fourth chapter, Aplification Factor Error Optimized Method is proposed. In order to get the prefect LDDRK of DCS, the dissipation of DCS must be consider. The new limited condition of controlling frequency spectrum range is proposed just for this. The results indicate an important improvement.
As all we know, in the accurate simulation of some aero dynamic problems such as CAA. The boundary condition may be have effect on the flow. So ,in this thesis characteristic boundary conditions have proposed for DCS base on Jae Wook Kimand Duck Joo Lee et al. The characteristic wall boundary condition of Euler, improved characteristic interface conditions, characteristic wall boundary condition of N-S and far-field characteristic boundary conditions are obtained.
Base on the characteristic boundary conditions obtained above, the solver of DCS3 and DCS5 is confirmed. Also the problems of flow past an cylinder which are 2-D,inviscid or viscid, steady or unsteady have been simulated using DCS, and the sound fields of the unsteady flow fields have been analyzed. The result indicate that it is possible to use DCS in CAA.
But using DCS in application fields to solver problems in CAA, Improvement is needed in the future which is given in the sixth chapter, where the summary of current study work is presented and some work to be carried on in future is pointed out as well.
声波在空间高精度捕捉的同时,在时间方向也需要被精确地追踪,因此时间积分方法也同样需要高精度。目前,在计算气动声学领域,常用的时间推进格式是显式龙格-库塔方法。为了提高龙格-库塔时间积分方法的耗散和色散性能,Hu提出了低耗散和低色散龙格—库塔方法,本文因此发展了三种针对DCS格式的低耗散和低色散龙格—库塔时间积分算法。
在本文的第二章中,参照Hu和Jan Ramboerd对中心型紧致格式(CCS)优化的基本思想,类似地构造了基于耗散型紧致格式(DCS)的一类低耗散低色散龙格—库塔时间积分方法,但由于DCS与CCS存在一个本质的差别,即耗散性,使得这种方法得到的时间推进优化格式并不具有改善计算结果精准度的能力。
在本文的第三章讨论了色散误差限制的增长因子误差优化方法。参照Hu和Jan Ramboerd对中心型紧致格式(CCS)优化的基本思想得到的LDDRK格式表现不理想,可能是该优化方法不具有同时控制DCS的耗散与色散的能力。因此,提出了强化色散误差控制的一种改进的优化方法,它的优化思想是通过Taylor展开,直接控制色散误差,再利用Hu和Jan Ramboerd的思想控制耗散误差,从而达到同时控制耗散与色散误差的目的。典型算例的计算结果表明,这种优化方法得到的LDDRK方法,能够在一定程度上改善计算结果的精度,但并不十分理想,可能是该优化方法仍没能很好反映DCS格式的特性。
在本文的第四章提出了谱区间限制的增长因子误差优化方法。为了得到理想的LDDRK格式,必须考虑DCS格式耗散性这一基本特性。引入了谱区间选择的优化限制条件,这是确保优化算法保持DCS格式的这种特性。通过Fourier分析和典型算例对比计算发现,引入优化谱区间限制条件的优化方法得到的LDDRK格式具有良好的模拟精度。
众所周知,对于诸如声波模拟等一些流动问题的细致模拟中,边界处理将可能对流场内部结构产生影响。基于Jae Wook Kim和Duck Joo Lee的思想,发展了与DCS格式相匹配特征边界条件,推导出了Euler方程的壁面特征边界条件,改进了高精度格式基于对流信息特征传播的对接边界处理技术,同时提出了基于信息传播的粘性壁面边界条件和特征远场边界条件处理方法。
基于上述特征边界条件发展了多块对接网格的高精度线性耗散紧致格式DCS3和DCS5的计算软件,模拟了二维、无粘/粘性、定常/非定常圆柱绕流,并分析了其中非定常流场产生的噪声,得到了很好的计算结果,本文的第六章展示了这些结果,这显示了DCS格式应用于计算气动声学的潜力。
但是要把DCS格式应用于解决工程上的声学问题,还有需要改进的地方,本文的第六章对这些问题进行了总结。
In the field of computational aero acoustics(CAA), the characteristics of the acoustic waves propagate in space are obviously different from that in the ordinary aero dynamic, which need the numerical schemes have low dissipation errors and low dispersion errors as much as possible. Dissipative Compact Scheme(DCS) derived by Deng Xiaogang base on Central Compact Scheme(CCS) is a kind of linear high-order accurate scheme with single parameter, Compare with CCS, DCS has the particular advantage that it can restrain the oscillations in the high wavenumber range, in this thesis basic research of DCS used in CAA has done.
When the acoustic waves tracked accurately in spatial, they also have to be tracked accurately in time, so the time integration schemes also need high-order accurate. By far the most popular schemes in CAA are the Runge-kutta schemes. In order to improve the dissipative and dispersive behavior of Runge-kutta time integration schemes, low-dissipation and low-dispersion Runge-kutta time integration schemes derived by Hu. So in this thesis based on dissipative compact schemes, three optimal low-dissipation and low-dispersion Runge-kutta time integration methods have proposed.
In second chapter, a class of low-dissipation and low-dispersion Runge-kutta time integration schemes is proposed for the DCS schemes using the optimal method of Hu, but because DCS and CCS have a entitative different which is the dissipation, so it have on efficiency.
Amplification Factor Error Optimized Method is discussed in third chapter. Using the optimal method of Hu to propose low-dissipation and low-dispersion Runge-kutta time integration schemes for DCS have no efficiency, may be this optimal method can't control the dissipation errors and the dispersion errors at the same time causes the problem. So in this chapter a new optimal method which control dispersion errors especially has proposed. This optimal method controlls dispersion error by Taylor series expanding and dissipative error by the optimal method of Hu,. But the results indicate that the improvement is limited, this may be caused by the optimal method doesn't adapt to the DCS which is dissipative.
Because of the problems mentioned in the two optimal methods above, in fourth chapter, Aplification Factor Error Optimized Method is proposed. In order to get the prefect LDDRK of DCS, the dissipation of DCS must be consider. The new limited condition of controlling frequency spectrum range is proposed just for this. The results indicate an important improvement.
As all we know, in the accurate simulation of some aero dynamic problems such as CAA. The boundary condition may be have effect on the flow. So ,in this thesis characteristic boundary conditions have proposed for DCS base on Jae Wook Kimand Duck Joo Lee et al. The characteristic wall boundary condition of Euler, improved characteristic interface conditions, characteristic wall boundary condition of N-S and far-field characteristic boundary conditions are obtained.
Base on the characteristic boundary conditions obtained above, the solver of DCS3 and DCS5 is confirmed. Also the problems of flow past an cylinder which are 2-D,inviscid or viscid, steady or unsteady have been simulated using DCS, and the sound fields of the unsteady flow fields have been analyzed. The result indicate that it is possible to use DCS in CAA.
But using DCS in application fields to solver problems in CAA, Improvement is needed in the future which is given in the sixth chapter, where the summary of current study work is presented and some work to be carried on in future is pointed out as well.
引文
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[2]MacCormack R.W.,A perspective on a quarter century of CFD research,AIAA-93-3291
[3]张涵信,求解气动方程的高阶精度格式及其相关的问题,第四届计算空气动力学年会论文集,3-15(1995)
[4]张涵信,呙超,宗文刚,网格与高精度差分计算问题,力学学报,Vol.3,1999
[5]Tam,C.K.W.,Computational Aeroacoustics:Issues and Methods,[J]AIAA,J,V.33,No.10,1995:1788-1796
[6]刘儒勋,舒其望,计算流体力学的若干新方法,科学出版社,2003
[7]Harten A.,High solution schemes for hypersonic conservation laws,Journal.Computational.Physics,Vol.49,357-393(1983)
[8]Clarksean R.,Direct numerical simulation of a plane jet using the spectral-compact finite difference technique,AIAA-91-019
[9]Sanjiva K.Lele,Compact Finite Difference Scheme with Spectral-like Resolution,Journal.Computational.Physics.Vol.103,16-42(1992)
[10]Yu S.T.,Hultgren L.S.and Liu N.S.,Direct calculations of waves in fluid flows using high-order compact difference scheme,AIAA-93-0148
[11]Alain Rigal,High order difference schemes for unsteady one-dimensional diffusion-convection problems,Journal Computational Physics,Vol.114,59-76(1994)
[12]William Frederick Spotz,High-order compact finite difference schemes for computational mechanics,Dissertation for Ph.D in the University of Texas at Austin,December 1995
[13]Boschitsch A.H.and Quackenbush T.R.,Efficient high-order compact scheme for deforming unstructured meshes,AIAA-96-0415
[14]D.Gaitonde,J.S.Shang and J.L.Yong,Practical aspects of high-order accurate finite-volume schemes for electromagnetics,AIAA-97-0363
[15]R.V.Wilson,A.Q.Demuren and M.H.Carpenter,High-order compact schemes for numerical simulations of incompressible flows,ICASE Report 98-13
[16]M.H.Kobayashi,On a class of Pade' finite volume methods,Journal Computational Physics,Vol.156,137-180(1999)
[17]Sergey Smirnov,Chrislacor and Martine Baelmans,A finite volume formulation for compact scheme with applications to LES,AIAA-2001-2546
[18]王强,傅德薰,马延文,平面可压基频涡卷非线性深化行为数值研究,力学学报,Vol.33,No.1,1-10(2001)
[19]沈孟育,蒋莉,满足熵增原则的高精度高分辨率格式,清华大学学报(自然科学版),Vol.16,56-62(1998)
[20]Alain Lerat and Christophe Corre,A residual-based compact scheme for the compressible Navier-Stokes equations,Journal Computational Physics,Vol.170,642-675(2001)
[21]Zigo Haras and Shlomo Ta'asan,Finite difference schemes for long-time integration,Journal Computational Physics,Vol.114,265-279(1994)
[22]J.S.Shang,High-order compact-difference schemes for time-dependent Maxwell equations,Journal Computational Physics,Vol.153,312-333(1999)
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