改进虚拟边界算法在超声速流动问题求解中的应用
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摘要
提出了一种改进的虚拟单元浸没边界法,并与一种高阶格式的有限差分算法相结合,运用于求解超声速复杂几何绕流问题.该算法的核心思想在于在固体边界的内部和外部分别施加满足边界关系的作用点,使得几何边界离散更加细化,起到了壁面附近网格局部加密的作用.采用源空间内流体点作为反距离插值算法的重构点,有效避免了插值点数目过少而与作用点相重合情况.通过对二维激波反射现象(马赫数为2.81)和三维超声速球体绕流问题(马赫数为1.2)的数值模拟,与实验结果对比表明,本文改进算法相对一般的虚拟边界法来说能显著提高数值精度,减小计算误差.计算结果揭示了球体绕流中剪切层、压缩波系和尾迹的相互作用导致自由剪切层失稳的机理.剪切层厚度和湍流雷诺脉动经历了线性增长、大幅度震荡和小幅度波动三个阶段,导致剪切层表面褶皱因子变化呈指数规律增长.其湍流结构表现出明显的各向异性,具体在流向雷诺正应力在湍流脉动中占主导地位,激波的压缩作用对不同方向雷诺正应力的影响存在空间迟滞效应.
An improved ghost-cell immersed boundary method proposed in this paper, coupled with a high order finite difference solver, is applied to simulate the supersonic compressive flows around the complex obstacles. The main improvement of this algorithm is the treatment of the solid boundary that both ghost points inside the solid domain and forcing points inside the fluid domain due to the extension of the boundary are chosen to reconstruct the flow information considering the effect of solid wall on fluid. This brings refined boundary with discrete points and strengthens the wall conditions, which plays the role of local mesh refinement. The fluid points are limited in a certain source space as the interpolating points of the inverse distance algorithm, which effectively avoids the fact that the interpolating points are too few to possibly lead to coincide with the forcing points. Two problems of two dimensional shock reflection(Ma=2.81)and three dimensional flow around the smooth sphere(Ma=1.2) demonstrate the significant improvement of the numerical accuracy compared to the general ghost cell method. The results reveal the instability mechanism of the free shear layer as a result of the interaction between the shear layer, the compression wave system and the wake. The thickness and Reynolds fluctuation of the shear layer experience three regimes of linear growth, large amplitude oscillation and small amplitude fluctuation, resulting in an exponential growth of wrinkling factor. The turbulent structure near the shear layer shows obvious anisotropy because the streamwise Reynolds normal stress is dominant and a spatial hysteresis exists in the effect of the tail shock on Reynolds normal stresses in different directions.
An improved ghost-cell immersed boundary method proposed in this paper, coupled with a high order finite difference solver, is applied to simulate the supersonic compressive flows around the complex obstacles. The main improvement of this algorithm is the treatment of the solid boundary that both ghost points inside the solid domain and forcing points inside the fluid domain due to the extension of the boundary are chosen to reconstruct the flow information considering the effect of solid wall on fluid. This brings refined boundary with discrete points and strengthens the wall conditions, which plays the role of local mesh refinement. The fluid points are limited in a certain source space as the interpolating points of the inverse distance algorithm, which effectively avoids the fact that the interpolating points are too few to possibly lead to coincide with the forcing points. Two problems of two dimensional shock reflection(Ma=2.81)and three dimensional flow around the smooth sphere(Ma=1.2) demonstrate the significant improvement of the numerical accuracy compared to the general ghost cell method. The results reveal the instability mechanism of the free shear layer as a result of the interaction between the shear layer, the compression wave system and the wake. The thickness and Reynolds fluctuation of the shear layer experience three regimes of linear growth, large amplitude oscillation and small amplitude fluctuation, resulting in an exponential growth of wrinkling factor. The turbulent structure near the shear layer shows obvious anisotropy because the streamwise Reynolds normal stress is dominant and a spatial hysteresis exists in the effect of the tail shock on Reynolds normal stresses in different directions.
引文
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3 Kadoch B,Reimann T,Schneider K,et al.Comparison of a spectral method with volume penalization and a finite volume method with body fitted grids for turbulent flows.Computers&Fluids,2016,133:140-150
4 Sotiropoulos F,Yang X.Immersed boundary methods for simulating fluid-structure interaction.Progress in Aerospace Sciences,2014,65:1-21
5 Peskin CS.Flow patterns around heart valves:A numerical method.Journal of Computational Physics,1972,10(2):252-271
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7 Saiki E M,Biringen S.Numerical simulation of a cylinder in uniform flow:Application of a virtual boundary method.Journal of Computational Physics,1996,123(2):450-465
8 Lee C.Stability characteristics of the virtual boundary method in three-dimensional applications.Journal of Computational Physics,2003,184(2):559-591
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13 Fadlun EA,Verzicco R,Orlandi P,et al.Combined immersedboundary finite-difference methods for three-dimensional complex flow simulations.Journal of Computational Physics,2000,161(1):35-60
14 Mittal R,Dong H,Bozkurttas M,et al.A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries.Journal of Computational Physics,2008,227(10):4825-4852
15 Majumdar S,Iaccarino G,Durbin P,RANS solvers with adaptive structured boundary non-conforming grids,Annual Research Briefs,NASA Ames Research Center/Stanford University.Stanford,CA,2001,353-366
16 Tseng YH,Ferziger JH.A ghost-cell immersed boundary method for flow in complex geometry.Journal of Computational Physics,2003,192(2):593-623
17 Qu Y,Shi R,Batra RC.An immersed boundary formulation for simulating high-speed compressible viscous flows with moving solids.Journal of Computational Physics,2018,354:672-691
18 Ghias R,Mittal R,Dong H.A sharp interface immersed boundary method for compressible viscous flows.Journal of Computational Physics,2007,225(1):528-553
19 Chaudhuri A,Hadjadj A,Chinnayya A.On the use of immersed boundary methods for shock/obstacle interactions.Journal of Computational Physics,2011,230(5):1731-1748
20 Bouchon F,Dubois T,James N.A second-order cut-cell method for the numerical simulation of 2D flows past obstacles.Computers&Fluids,2012,65:80-91
21 Ikeno T,Kajishima T.Finite-difference immersed boundary method consistent with wall conditions for incompressible turbulent flow simulations.Journal of Computational Physics,2007,226(2):1485-1508
22 Seo JH,Mittal R.A high-order immersed boundary method for acoustic wave scattering and low-Mach number flow-induced sound in complex geometries.Journal of Computational Physics,2011,230(4):1000-1019
23 Haugen NEL,Kragset S.Particle impaction on a cylinder in a crossflow as function of Stokes and Reynolds numbers.Journal of Fluid Mechanics,2010,661:239-261
24 Luo K,Zhuang Z,Fan J,et al.A ghost-cell immersed boundary method for simulations of heat transfer in compressible flows under different boundary conditions.International Journal of Heat and Mass Transfer,2016,92:708-717
25 Luo K,Mao C,Zhuang Z,et al.A ghost-cell immersed boundary method for the simulations of heat transfer in compressible flows under different boundary conditions Part-II:Complex geometries.International Journal of Heat and Mass Transfer,2017,104:98-111
26 Li XL,Fu DX,Ma YW,et al.Direct numerical simulation of shock/turbulent boundary layer interaction in a supersonic compression ramp.Science China Physics,Mechanics&Astronomy,2010,53(9):1651-1658
27 Li X,Fu D,Ma Y.Direct numerical simulation of hypersonic boundary layer transition over a blunt cone with a small angle of attack.Physics of Fluids,2010,22(2):025105
28 童福林,李新亮,唐志共.激波与转捩边界层干扰非定常特性数值分析.力学学报,2017,49(1):93-104(Tong Fulin,Li Xinliang,Tang Zhigong.Numerical analysis of unsteady motion in shock wave/transitional boundary layer interaction.Chinese Journal of Theoretical and Applied Mechanics,2017,49(1):93-104(in Chinese))
29 Steger JL,Warming RF.Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods.Journal of Computational Physics,1981,40(2):263-293
30 Jiang GS,Shu CW.Efficient implementation of weighted ENO schemes.Journal of Computational Physics,1996,126(1):202-228
31 Franke R.Scattered data interpolation:Tests of some methods.Mathematics of Computation,1982,38(157):181-200
32 Yang J,Liu Y,Lomax H.Computation of shock wave reflection by circular cylinders.AIAA Journal,1987,25(5):683-689
33 Giepman RHM,Schrijer FFJ,Van Oudheusden BW.Flow control of an oblique shock wave reflection with micro-ramp vortex generators:Effects of location and size.Physics of Fluids,2014,26(6):066101
34 Igra D,Takayama K,Igra O.Shock wave reflection over roughened wedges//30th International Symposium on Shock Waves,Cham:Springer,2017:621-625
35 Fang J,Yao Y,Zheltovodov AA,et al.Investigation of three dimensional shock wave turbulent boundary layer interaction initiated by a single fin.AIAA Journal,2016,55(2):509-523
36 Bryson AE,Gross RWF.Diffraction of strong shocks by cones,cylinders,and spheres.Journal of Fluid Mechanics,1961,10(1):1-16
37 Kaca J.An interferometric investigation of the diffraction of a planar shock wave over a semicircular cylinder.NASA STI/Recon Technical Report N,1988,89
38 Constantinescu G,Squires K.Numerical investigations of flow over a sphere in the subcritical and supercritical regimes.Physics of Fluids,2004,16(5):1449-1466
39 Rodriguez I,Borell R,Lehmkuhl O,et al.Direct numerical simulation of the flow over a sphere at Re=3700.Journal of Fluid Mechanics,2011,679:263-287
40 胡政敏,肖志坚,陶铸等.基于LES的亚临界和超临界圆球绕流数值研究.水动力学研究与进展,2016,31(4):496-502(Hu Zhengmin,Xiao Zhijian,Tao Zhu,et al.LES investigations of flow over a sphere in the subcritical and supercritical regimes.Chinese Journal of Hydrodynamics,2016,31(4):496-502(in Chinese))
41 Von Terzi DA,Sandberg RD,Fasel HF.Identification of large coherent structures in supersonic axisymmetric wakes.Computers&Fluids,2009,38(8):1638-1650
42 Yun G,Kim D,Choi H.Vortical structures behind a sphere at subcritical Reynolds numbers.Physics of Fluids,2006,18(1):015102
43 林孟达,崔桂香,张兆顺等.飞机尾涡演变及快速预测的大涡模拟研究.力学学报,2017,49(6):1185-1200(Lin Mengda,Cui Guixiang,Zhang Zhaoshun,et al.Large eddy simulation on the evolution and the fast-time prediction of aircraft wake vortices.Chinese Journal of Theoretical and Applied Mechanics,2017,49(6):1185-1200(in Chinese))
44 及春宁,花阳,许栋等.不同剪切率来流作用下柔性圆柱涡激振动数值模拟.力学学报,2018,50(1):21-31(Ji Chunning,Hua Yang,Xu Dong,et al.Numerical simulation of vortex-induced vibration of a flexible cylinder exposed to shear flow at different shear rates.Chinese Journal of Theoretical and Applied Mechanics,2018,50(1):21 -31(in Chinese))
45 任安禄,李广望,邹建锋.中等雷诺数圆球绕流的数值研究.浙江大学学报(工学版),2004,38(5):644-648(Ren Anlu,Li Guangwang,Zou Jianfeng.Numerical study of uniform flow over sphere at intermediate Reynolds numbers.Journal of Zhejiang University(Engineering Science),2004,38(5):644-648(in Chinese))
46 邹建锋,任安禄,邓见.圆球绕流场的尾涡分析和升阻力研究.空气动力学报.2004,22(3):303-308(Zou Jianfeng,Ren Anlu,Deng Jian.Numerical investigations of wake and force for flow past a sphere.Acta Aerodynamica Sinica,2004,22(3):303-308(in Chinese))
47 Achenbach E.Experiments on the flow past spheres at very high Reynolds numbers.Journal of Fluid Mechanics,1972,54(3):565-575
2 Zhao D,Chen J,Zheng Y,et al.Fine-grained parallel algorithm for unstructured surface mesh generation.Computers&Structures,2015,154:177-191
3 Kadoch B,Reimann T,Schneider K,et al.Comparison of a spectral method with volume penalization and a finite volume method with body fitted grids for turbulent flows.Computers&Fluids,2016,133:140-150
4 Sotiropoulos F,Yang X.Immersed boundary methods for simulating fluid-structure interaction.Progress in Aerospace Sciences,2014,65:1-21
5 Peskin CS.Flow patterns around heart valves:A numerical method.Journal of Computational Physics,1972,10(2):252-271
6 Goldstein D,Handler R,Sirovich L.Modeling a no-slip flow boundary with an external force field.Journal of Computational Physics,1993,105(2):354-366
7 Saiki E M,Biringen S.Numerical simulation of a cylinder in uniform flow:Application of a virtual boundary method.Journal of Computational Physics,1996,123(2):450-465
8 Lee C.Stability characteristics of the virtual boundary method in three-dimensional applications.Journal of Computational Physics,2003,184(2):559-591
9 Kolomenskiy D,Schneider K.A Fourier spectral method for the Navier-Stokes equations with volume penalization for moving solid obstacles.Journal of Computational Physics,2009,228(16):5687-5709
10 Kadoch B,Kolomenskiy D,Angot P,et al.A volume penalization method for incompressible flows and scalar advection-diffusion with moving obstacles.Journal of Computational Physics,2012,231(12):4365-4383
11 Iaccarino G,Verzicco R.Immersed boundary technique for turbulent flow simulations.Applied Mechanics Reviews,2003,56(3):331-347
12 Mohd-Yosuf J.Combined immersed boundary/B-spline methods for simulations of flow in complex geometries,Annual Research Briefs,Center for Turbulence Research,NASA Ames Research Center/Stanford University 1997,317-328
13 Fadlun EA,Verzicco R,Orlandi P,et al.Combined immersedboundary finite-difference methods for three-dimensional complex flow simulations.Journal of Computational Physics,2000,161(1):35-60
14 Mittal R,Dong H,Bozkurttas M,et al.A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries.Journal of Computational Physics,2008,227(10):4825-4852
15 Majumdar S,Iaccarino G,Durbin P,RANS solvers with adaptive structured boundary non-conforming grids,Annual Research Briefs,NASA Ames Research Center/Stanford University.Stanford,CA,2001,353-366
16 Tseng YH,Ferziger JH.A ghost-cell immersed boundary method for flow in complex geometry.Journal of Computational Physics,2003,192(2):593-623
17 Qu Y,Shi R,Batra RC.An immersed boundary formulation for simulating high-speed compressible viscous flows with moving solids.Journal of Computational Physics,2018,354:672-691
18 Ghias R,Mittal R,Dong H.A sharp interface immersed boundary method for compressible viscous flows.Journal of Computational Physics,2007,225(1):528-553
19 Chaudhuri A,Hadjadj A,Chinnayya A.On the use of immersed boundary methods for shock/obstacle interactions.Journal of Computational Physics,2011,230(5):1731-1748
20 Bouchon F,Dubois T,James N.A second-order cut-cell method for the numerical simulation of 2D flows past obstacles.Computers&Fluids,2012,65:80-91
21 Ikeno T,Kajishima T.Finite-difference immersed boundary method consistent with wall conditions for incompressible turbulent flow simulations.Journal of Computational Physics,2007,226(2):1485-1508
22 Seo JH,Mittal R.A high-order immersed boundary method for acoustic wave scattering and low-Mach number flow-induced sound in complex geometries.Journal of Computational Physics,2011,230(4):1000-1019
23 Haugen NEL,Kragset S.Particle impaction on a cylinder in a crossflow as function of Stokes and Reynolds numbers.Journal of Fluid Mechanics,2010,661:239-261
24 Luo K,Zhuang Z,Fan J,et al.A ghost-cell immersed boundary method for simulations of heat transfer in compressible flows under different boundary conditions.International Journal of Heat and Mass Transfer,2016,92:708-717
25 Luo K,Mao C,Zhuang Z,et al.A ghost-cell immersed boundary method for the simulations of heat transfer in compressible flows under different boundary conditions Part-II:Complex geometries.International Journal of Heat and Mass Transfer,2017,104:98-111
26 Li XL,Fu DX,Ma YW,et al.Direct numerical simulation of shock/turbulent boundary layer interaction in a supersonic compression ramp.Science China Physics,Mechanics&Astronomy,2010,53(9):1651-1658
27 Li X,Fu D,Ma Y.Direct numerical simulation of hypersonic boundary layer transition over a blunt cone with a small angle of attack.Physics of Fluids,2010,22(2):025105
28 童福林,李新亮,唐志共.激波与转捩边界层干扰非定常特性数值分析.力学学报,2017,49(1):93-104(Tong Fulin,Li Xinliang,Tang Zhigong.Numerical analysis of unsteady motion in shock wave/transitional boundary layer interaction.Chinese Journal of Theoretical and Applied Mechanics,2017,49(1):93-104(in Chinese))
29 Steger JL,Warming RF.Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods.Journal of Computational Physics,1981,40(2):263-293
30 Jiang GS,Shu CW.Efficient implementation of weighted ENO schemes.Journal of Computational Physics,1996,126(1):202-228
31 Franke R.Scattered data interpolation:Tests of some methods.Mathematics of Computation,1982,38(157):181-200
32 Yang J,Liu Y,Lomax H.Computation of shock wave reflection by circular cylinders.AIAA Journal,1987,25(5):683-689
33 Giepman RHM,Schrijer FFJ,Van Oudheusden BW.Flow control of an oblique shock wave reflection with micro-ramp vortex generators:Effects of location and size.Physics of Fluids,2014,26(6):066101
34 Igra D,Takayama K,Igra O.Shock wave reflection over roughened wedges//30th International Symposium on Shock Waves,Cham:Springer,2017:621-625
35 Fang J,Yao Y,Zheltovodov AA,et al.Investigation of three dimensional shock wave turbulent boundary layer interaction initiated by a single fin.AIAA Journal,2016,55(2):509-523
36 Bryson AE,Gross RWF.Diffraction of strong shocks by cones,cylinders,and spheres.Journal of Fluid Mechanics,1961,10(1):1-16
37 Kaca J.An interferometric investigation of the diffraction of a planar shock wave over a semicircular cylinder.NASA STI/Recon Technical Report N,1988,89
38 Constantinescu G,Squires K.Numerical investigations of flow over a sphere in the subcritical and supercritical regimes.Physics of Fluids,2004,16(5):1449-1466
39 Rodriguez I,Borell R,Lehmkuhl O,et al.Direct numerical simulation of the flow over a sphere at Re=3700.Journal of Fluid Mechanics,2011,679:263-287
40 胡政敏,肖志坚,陶铸等.基于LES的亚临界和超临界圆球绕流数值研究.水动力学研究与进展,2016,31(4):496-502(Hu Zhengmin,Xiao Zhijian,Tao Zhu,et al.LES investigations of flow over a sphere in the subcritical and supercritical regimes.Chinese Journal of Hydrodynamics,2016,31(4):496-502(in Chinese))
41 Von Terzi DA,Sandberg RD,Fasel HF.Identification of large coherent structures in supersonic axisymmetric wakes.Computers&Fluids,2009,38(8):1638-1650
42 Yun G,Kim D,Choi H.Vortical structures behind a sphere at subcritical Reynolds numbers.Physics of Fluids,2006,18(1):015102
43 林孟达,崔桂香,张兆顺等.飞机尾涡演变及快速预测的大涡模拟研究.力学学报,2017,49(6):1185-1200(Lin Mengda,Cui Guixiang,Zhang Zhaoshun,et al.Large eddy simulation on the evolution and the fast-time prediction of aircraft wake vortices.Chinese Journal of Theoretical and Applied Mechanics,2017,49(6):1185-1200(in Chinese))
44 及春宁,花阳,许栋等.不同剪切率来流作用下柔性圆柱涡激振动数值模拟.力学学报,2018,50(1):21-31(Ji Chunning,Hua Yang,Xu Dong,et al.Numerical simulation of vortex-induced vibration of a flexible cylinder exposed to shear flow at different shear rates.Chinese Journal of Theoretical and Applied Mechanics,2018,50(1):21 -31(in Chinese))
45 任安禄,李广望,邹建锋.中等雷诺数圆球绕流的数值研究.浙江大学学报(工学版),2004,38(5):644-648(Ren Anlu,Li Guangwang,Zou Jianfeng.Numerical study of uniform flow over sphere at intermediate Reynolds numbers.Journal of Zhejiang University(Engineering Science),2004,38(5):644-648(in Chinese))
46 邹建锋,任安禄,邓见.圆球绕流场的尾涡分析和升阻力研究.空气动力学报.2004,22(3):303-308(Zou Jianfeng,Ren Anlu,Deng Jian.Numerical investigations of wake and force for flow past a sphere.Acta Aerodynamica Sinica,2004,22(3):303-308(in Chinese))
47 Achenbach E.Experiments on the flow past spheres at very high Reynolds numbers.Journal of Fluid Mechanics,1972,54(3):565-575