The Метод крупных вихрей для моделирования вырождения МГД турбулентности конечно-разностным гибридным методов
DOI:
https://doi.org/10.26577/JMMCS-2018-3-519Ключевые слова:
Магнитогидродинамика, вихревая задача Тейлора-Грина, конечно- разностный гибридный метод, спектральный метод, вырождение турбулентностиАннотация
Данная работа посвящена моделированию вырождении магнитогидродинамической (МГД)
турбулентности конечно-разностным гибридным методом (КРГМ), сочетающейся из двух
различных численных методов: конечно-разностный и спектральный. Разработан численный
алгоритм гибридного метода на основе решения уравнения Навье-Стокса и уравнения для
магнитного поля конечно-разностным методом в сочетании с циклической пятидиагональной
матрицей, которая дает точность четвертого порядка по пространству и точность
третьего порядка по времени. Уравнение Пуассона для давление решается спектральным
методом. Для валидации разработанного алгоритма рассматривается классическая задача
трехмерного вихревого потока Тейлора и Грина без учета магнитного поля, и полученные
турбулентные характеристики при моделировании имеют отличные согласование с
результатами аналитического решения на краткосрочном отрезке времени. Также показано,
что разработанный эффективный численный алгоритм может быть использован для
моделирования вырождения магнитогидродинамической турбулентности при различных
числах Рейнольдса.
Библиографические ссылки
[1] Abdibekova A., Zhakebayev D., Abdigaliyeva A. and Zhubat K., "Modelling of turbulence energy decay based on hybrid methods,"Engineering Computations 35(5),(2018):1965-77.
[2] Batchelor G. K. "On the spontaneous magnetic field in a conducting liquid in turbulent motion,"Proc. Roy. Soc. A201. 16(1950): 405-16.
[3] Batchelor G. K.,The theory of homogeneous turbulence (Cambridge University Press: 1953)
[4] Batista M., "A Method for Solving Cyclic Block Penta-diagonal Systems of Linear Equations,"ArXiv, March 6, 2008.
Accessed March 14, 2008. http://arxiv.org/abs/0803.0874.
[5] Brachet M.E. , Meiron D. I., Orszag S. A., Nickel B. G., Morf R.H. and Frisch U., "Small-scale structure of the
Taylor-Green vortex"Fluid Mechanics 130(1983): 411-52.
[6] Brachet M.E., "Direct simulation of three - dimensional turbulence in the Taylor - Green vortex,"Fluid Dynamics Research 8(1991): 1-8.
[7] Burattini P., Zikanov O. and Knaepen B., "Decay of magnetohydrodynamic turbulence at low magnetic Reynolds number"Fluid Mechanics 657(2010): 502-38.
[8] DeBonis J.R., "Solutions of the Taylor-Green vortex problem using high - resolution explicit finite difference methods" (paper presented at 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Dallas, Texas, January 07-10, 2013).
[9] Hossain M., "Inverse energy cascades in three dimensional turbulence,"Physics of Fluids B: Plasma Physics 3(2)(1991): 511-14.
[10] Frisch U., Turbulence. The legacy of A.N. Kolmogorov (Cambridge University Press: 1995)
[11] Ievlev V. M., The method of fractional steps for solution of problems of mathematical physics (Moscow: Science Nauka: 1975) [12] Kampanis N.A. and Ekaterinaris J.A., "A staggered grid, high-order accurate method for the incompressible Navier- Stokes equations,"Computational Physics 215(2006): 589-613.
[13] Kim J. and Moin P., "Application of a fractional - step method to incompressible Navier- Stokes equations,"Computational Physics 59(1985): 308-23.
[14] Knaepen B., Kassinos S. and Carati D., "Magnetohydrodynamic turbulence at moderate magnetic Reynolds number"Fluid Mechacincs 513(3)(2004): 199-220.
[15] Kolmogorov A. N., "Local structure of turbulence in an incompressible fluid at very high Reynolds numbers"Dokladi Akademii Nauk USSR 30(1941): 299-303.
[16] Moatt H. K., "On the suppression of turbulence by a uniform magnetic field,"Fluid Mechanics 28(1967): 571-592.
[17] Monin A. C., Yaglom A. M., Statistical fluid mechanics( Cambridge: MIT Press: 1975)
[18] Navon M., "Pent: A periodic pentadiagonal systems solver,"Communications in applied numerical methods 3(1987): 63-9.
[19] Schumann U., "Numerical simulation of the transition from three- to two-dimensional turbulence under a uniform magnetic field,"Fluid Mechanics 74(1976):31-58.
[20] Sirovich L., Smith L., Yakhot V., "Energy spectrum of homogeneous and isotropic turbulence in far dissipation range,"Physical Review Letters 72(3)(1994): 344-47.
[21] Taylor G.I. and Green A.E., "Mechanism of production of small eddies from large ones,"Proceedings of the royal society, Mathematics and physical sciences 158(895)(1937): 499-521.
[22] Vorobev A., Zikanov O., Davidson P. and Knaepen B., "Anisotropy of magnetohydrodynamic turbulence at low magnetic Reynolds number,"Physics of Fluids 17( 2005): 125105-1-125105-12.
[23] Zhakebayev D., Zhumagulov B. and Abdibekova A., "The decay of MHD turbulence depending on the conductive properties of the environment,"Magnetohydrosynamics 50(2)(2014): 121-38.
[24] Zikanov O. and Thess A., "Direct numerical simulation of forced MHD turbulence at low magnetic Reynolds
number,"Fluid Mechanics 358(1998): 299333.
[2] Batchelor G. K. "On the spontaneous magnetic field in a conducting liquid in turbulent motion,"Proc. Roy. Soc. A201. 16(1950): 405-16.
[3] Batchelor G. K.,The theory of homogeneous turbulence (Cambridge University Press: 1953)
[4] Batista M., "A Method for Solving Cyclic Block Penta-diagonal Systems of Linear Equations,"ArXiv, March 6, 2008.
Accessed March 14, 2008. http://arxiv.org/abs/0803.0874.
[5] Brachet M.E. , Meiron D. I., Orszag S. A., Nickel B. G., Morf R.H. and Frisch U., "Small-scale structure of the
Taylor-Green vortex"Fluid Mechanics 130(1983): 411-52.
[6] Brachet M.E., "Direct simulation of three - dimensional turbulence in the Taylor - Green vortex,"Fluid Dynamics Research 8(1991): 1-8.
[7] Burattini P., Zikanov O. and Knaepen B., "Decay of magnetohydrodynamic turbulence at low magnetic Reynolds number"Fluid Mechanics 657(2010): 502-38.
[8] DeBonis J.R., "Solutions of the Taylor-Green vortex problem using high - resolution explicit finite difference methods" (paper presented at 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Dallas, Texas, January 07-10, 2013).
[9] Hossain M., "Inverse energy cascades in three dimensional turbulence,"Physics of Fluids B: Plasma Physics 3(2)(1991): 511-14.
[10] Frisch U., Turbulence. The legacy of A.N. Kolmogorov (Cambridge University Press: 1995)
[11] Ievlev V. M., The method of fractional steps for solution of problems of mathematical physics (Moscow: Science Nauka: 1975) [12] Kampanis N.A. and Ekaterinaris J.A., "A staggered grid, high-order accurate method for the incompressible Navier- Stokes equations,"Computational Physics 215(2006): 589-613.
[13] Kim J. and Moin P., "Application of a fractional - step method to incompressible Navier- Stokes equations,"Computational Physics 59(1985): 308-23.
[14] Knaepen B., Kassinos S. and Carati D., "Magnetohydrodynamic turbulence at moderate magnetic Reynolds number"Fluid Mechacincs 513(3)(2004): 199-220.
[15] Kolmogorov A. N., "Local structure of turbulence in an incompressible fluid at very high Reynolds numbers"Dokladi Akademii Nauk USSR 30(1941): 299-303.
[16] Moatt H. K., "On the suppression of turbulence by a uniform magnetic field,"Fluid Mechanics 28(1967): 571-592.
[17] Monin A. C., Yaglom A. M., Statistical fluid mechanics( Cambridge: MIT Press: 1975)
[18] Navon M., "Pent: A periodic pentadiagonal systems solver,"Communications in applied numerical methods 3(1987): 63-9.
[19] Schumann U., "Numerical simulation of the transition from three- to two-dimensional turbulence under a uniform magnetic field,"Fluid Mechanics 74(1976):31-58.
[20] Sirovich L., Smith L., Yakhot V., "Energy spectrum of homogeneous and isotropic turbulence in far dissipation range,"Physical Review Letters 72(3)(1994): 344-47.
[21] Taylor G.I. and Green A.E., "Mechanism of production of small eddies from large ones,"Proceedings of the royal society, Mathematics and physical sciences 158(895)(1937): 499-521.
[22] Vorobev A., Zikanov O., Davidson P. and Knaepen B., "Anisotropy of magnetohydrodynamic turbulence at low magnetic Reynolds number,"Physics of Fluids 17( 2005): 125105-1-125105-12.
[23] Zhakebayev D., Zhumagulov B. and Abdibekova A., "The decay of MHD turbulence depending on the conductive properties of the environment,"Magnetohydrosynamics 50(2)(2014): 121-38.
[24] Zikanov O. and Thess A., "Direct numerical simulation of forced MHD turbulence at low magnetic Reynolds
number,"Fluid Mechanics 358(1998): 299333.
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Как цитировать
Abdibekova, A., & Zhakebayev, D. B. (2018). The Метод крупных вихрей для моделирования вырождения МГД турбулентности конечно-разностным гибридным методов. Вестник КазНУ. Серия математика, механика, информатика, 99(3), 53–77. https://doi.org/10.26577/JMMCS-2018-3-519
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Прикладная математика
