مقایسه عملکرد ماتریس‌های پیش‌شرطی در گستره‌ی وسیعی از جریان‌های داخلی و خارجی

جوارشکیان, محمد حسن; مقدس خراسانی, مهدی; محمدی, عدنان

مقایسه عملکرد ماتریس‌های پیش‌شرطی در گستره‌ی وسیعی از جریان‌های داخلی و خارجی

نوع مقاله : مقاله پژوهشی

نویسندگان

¹ استاد، دانشگاه فردوسی مشهد ، مشهد ، ایران

² دانشجوی کارشناسی ارشد،دانشگاه فردوسی مشهد، مشهد، ایران

³ دانشجوی دکترا ،دانشگاه فردوسی مشهد، مشهد، ایران

چکیده

در این پژوهش ابتدا با توجه به ماتریس ژاکوبین پیش‌شرطی برحسب متغیرهای بقایی، بردارهای ویژه، مقادیر ویژه و مقادیر مشخصه به‌صورت یکپارچه برای سه روش ترکل، چوی-مرکل و اریکسون استخراج می‌گردد. بدین‌منظور این روش‌‌های پیش‌شرطی در یک الگوریتم چگالی‌مبنا دو بعدی با روش بالادستی "رو" و یک شبکه بی‌سازمان برای معادلات اولر توسعه داده می‌شود. دقت و نرخ همگرایی این روش‌های پیش‌شرطی برای جریان‌های خارجی حول ایرفویل NACA0012 ، ایرفویل سه المانه 30P-30N و جریان داخل کانال با برامدگی در شرایط مختلف جریان مورد بررسی قرار می‌گیرد. این پژوهش نشان می‌دهد که استفاده از روش‌های پیش‌شرطی نه‌تنها نرخ همگرایی را برای جریان قابل و غیر قابل تراکم افزایش میدهند; بلکه دقت حل را برای جریان تراکم‌ناپذیر را نیز نسبت به روش کلاسیک به‌طور چشمگیر بهبود می‌دهد. همچنین مقایسه روش‌های پیش‌شرطی نشان می‌دهد که هر سه روش از نظر دقت، جواب‌های تقریبا یکسانی را ارائه می‌دهند. اما از نظر نرخ همگرایی، روش پیش‌شرطی ترکل نرخ همگرایی بهتری را ارایه می‌دهد.

کلیدواژه‌ها

20.1001.1.23223278.1400.10.2.3.0

عنوان مقاله [English]

Comparing the performance of preconditioning matrixes in wide range of internal and external flows

نویسندگان [English]

Mohammad Hassan Djavareshkian ¹
Mahdi Moghadas khorasani ²
adnan mohammadi ³

¹ Professor, Ferdowsi University of Mashhad, Mashhad, Iran

² Master's student, Ferdowsi University of Mashhad, Mashhad, Iran

³ PhD student, Ferdowsi University of Mashhad, Mashhad, Iran

چکیده [English]

In this study, by considering jacobian matrix based on conservative variables, eigenvalues, eigenvectors, and wave strength for three types of preconditioners which are introduced by Turkel, Choi&Merkel and Eriksson are drawn in a unified mathematical manner. For this aim, these preconditioning methods are implemented in two-dimensional density-base “Roe” upwind scheme on unstructured meshes for Euler equations. Accuracy and rate of convergence are examined by external computing flow over NACA0012 airfoil, three-element 30P-30N airfoil, and internal flow over a bump for different flow conditions. This study shows that the application of preconditioning schemes not only increases the rate of convergence for compressible and incompressible flows dramatically; but also improves accuracy for incompressible flow in comparison with the classical method. This study also indicates that all preconditioning schemes provide approximately the same accuracy, but in terms of convergence rate, the Turkel preconditioning scheme provides a better rate of convergence among all the aforementioned preconditioned matrixes

کلیدواژه‌ها [English]

preconditioning
conservative variables
incompressible
“Roe” upwind scheme

مراجع

Chorin, J. “A Numerical Method for Solving Incompressible Viscous Flow Problems”, Journal of Computational Physics, Vol. 2, No. 1, pp. 12-26, 1967.
Turkel, E. “Preconditioned Methods for Solving the Incompressible and Low Speed Compressible Equations”, Journal of Computational Physics, Vol. 72, No. 2, pp. 277-298, 1987.
Choi, Y.H. and Merkle, C.L. “The Application of Preconditioning in Viscous Flows”, Journal of Computational Physics, Vol. 105, No. 2, pp. 207-223, 1993.
Eriksson, L-E. “A preconditioned Navier-Stokes solver for low Mach number flows.”;Proc. Int, Conf. Eccomas computational fluid dynamics, 1996.
Vanleer,B.,Lee, W and Roe, P., “Characteristic time-stepping or local preconditioning of the Euler equations.”;Proc. Int, Conf. Computational Fluid Dynamics ,1991.
Hejranfar, K. and Parseh, K.,. “Application of a preconditioned high-order Accurate artificial compressibility-based incompressible flow solver in wide range of Reynolds numbers”. Journal of Numerical Methods in Fluids, 86, pp.46-77,2018.
Yildirim, B. and Cinnella, P., “On the validation of a global Preconditioner for the Euler Equations” ;Proc. Int, Conf. AIAA Aerospace Sciences Meeting and Exhibit ,2004.
Bas, O., Tuncer, I.H., and Kaynak, U.“A Mach-Uniform Preconditioner for incompressible and subsonic flows”. Journal of Numerical Methods in Fluids, 74, No. 2, pp.100-112,2014.
Bas,, Cakmakcioglu, S. C., and Kaynak, U. “A Novel Intermittency Distribution Based Transition Model for Low-Re Number Airfoils”, ;Proc. Int, Conf. AIAA Applied Aerodynamics, 2013,
Darmofal, D.L. and Schmid, P.J. “The Importance of Eigenvectors for Local Preconditioners of the Euler Equations”, J. Comput. Phys., Vol. 127, No. 2, pp. 346-362, 1996.
Turkel, E., Vatsa, V.N., and Radespiel, R. “Preconditioning Methods for Low-Speed Flows”, Int, Conf. In AIAA. Appl. Aerodynamic. New Orleans ,USA, 1996.
Li, X.-s., Gu, C.-w., and Xu, J.-z. “Development of Roe-type Scheme for All-Speed Flows Based on Preconditioning Method”, Computers & Fluids., Vol. 38, No. 4, pp. 810-817, 2009.
Li, X.-s., Ren, X.-d., and Gu, C.-w. “An Improved Roe Scheme for All Mach-Number Flows Simultaneously Curing Known Problems”, Journal for arXiv:1711.09272, 2017.
Li, X.-s. and Gu, C.-w. “The Momentum Interpolation Method Based on the Time-Marching Algorithm for All-Speed Flows”, Journal of Computational Physics. Vol. 229, No. 20, pp. 7806-7818, 2010.
Li, X.-s. and Gu, C.-w. “On the Mechanism of Roe-type Schemes for All-Speed Flows”, Journal for open-source arXiv. Vol. 209, No. 20, pp. 7806-7818, 2011.
Li, X.-s. “On the All-Speed Roe-type Scheme for Large Eddy Simulation of Homogeneous Decaying Turbulence”, Journal for open-source arXiv. Vol. 308, No. 20, pp. 7505-77517, 2015.
Ren, X.d., Gu, C.w., and Li, X.s. “Role of the Momentum Interpolation Mechanism of the Roe Scheme in Shock Instability”,Journal of Numerical Methods in Fluids, Vol. 84, No. 6, pp. 335-351, 2017.
Maia, A.A.G., Kapat, J.S.J., Tomita, T., Silva, J. F., Bringhenti, C., and Cavalca, D.F. “Preconditioning Methods for Compressible Flow CFD Codes:, Journal of Mechanical Sciences. Vol. 186, 2020.
Maia, A., Ferreira da Silva, J., Tomita, J., and Bringhenti, C. “Implementing a Preconditioning Technique in A RANS Compressible Code to Accelerate the Convergence Rate for Low-Speed Flows”; Proc. Int. Conf. Mechanical, Chemical, and Material Engineering. Canada, 2019.
Maia, A.A., Silva, J.F., Tomita, J.T., and Bringhenti, C. “Applying a Preconditioning Technique to the Euler Equations to Accelerate the Convergence Rate for Low-Speed Flows”, Proc. Int. Conf. Mechanical, Chemical, and Material Engineering. Lisbon, Portugal, 2019.
Darmofal, D.L. and Siu, K. “A Robust Multigrid Algorithm for the Euler Equations with Local Preconditioning and Semi-coarsening”, J. Comput. Phys., Vol. 151, No. 2, pp. 728-756, 1999.
Akbarzadeh, P., Askari Lehdarboni, A., and Derazgisoo, S.M. “A New Smoothing Approach for Accelerating the Convergence of Power-Law Preconditioning Method in Steady and Unsteady Flows Simulation”, Int. J. Mech. Sci. Vol. 141, pp. 316-329, 2018.
Bhatt, M. and Mahesh, K. “A Numerical Approach to Address the Acoustic Stiffness in Cavitating Flows”, Journal of Multiphase Flow, Vol.141,No.19,pp.10-35, 2021.
Guillard, H. and Viozat, C. “On the Behaviour of Upwind Schemes in the Low Mach Number Limit”, Computers & Fluids. Vol. 28, No. 1, pp. 63-86, 1999.
Djavareshkian, M.H. “The Role of Flow Limiter Based on Characteristic Variables Via TVD and Pressure_Based Scheme”, Proc. Int, Conf. ISME. Iran,Tehran, 2000 (in Persian).
Turkel, E., Fiterman, A., and Van Leer, B. “Preconditioning and the Limit to the Incompressible Flow Equations”, in NASA Contractor Report 191500, pp. 215-234, 1993.
Turkel, E. “Preconditioning Methods for Low-Speed Flows”, In 14th Appl. Aerodyn. Conference. New Orleans, LA, U.S.A., 1996.
Jameson, A., Schmidt, W., and Turkel, E. “Numerical Solution of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time Stepping Schemes”, AIAA Paper. Vol. 81, 1981.
Moukalled, F., Mangani, L., and Darwish, M. “The Characteristic Boundary Condition in Pressure-Based Methods”, Numer. Heat Transf. B: Fundam, Vol. 76, pp. 1-17, 2019.
“Advisory, Group, for, Aerospace, Research, and Development, Test Cases for Inviscid Flow Field Methods”, North Atlantic,USA, 1985.
Murayama, M., Nakakita, K., Yamamoto, K., Ura, H., Ito, Y., and Choudhari, M.M. “Experimental Study on Slat Noise from 30P30N Three-Element High-Lift Airfoil at JAXA Hard-Wall Lowspeed Wind Tunnel”, Proc. Int. Conf. AIAA Aeroacoustics. Atlanta, USA, 2014.

Djavareshkian, M.H. and Reza-zadeh, S. “Application of Normalized Flux in Pressure-Based Algorithm”, Computers & Fluids. Vol. 36, No. 7, pp. 1224-1234, 2007