حل عددی معادلات ناویر استوکس در حالت پایای تراکم‌ناپذیر آشفته با استفاده از روش تابع پایه شعاعی چند ربعی

نوع مقاله : مقاله پژوهشی

نویسندگان

1 دانشگاه قم

2 عضو هیات علمی دانشگاه قم

چکیده

در روش‌های عددی هزینه و انرژی قابل توجهی صرف ایجاد و در مراحل بعدی اعمال تغییرات لازم در شبکه می‌شود. به همین دلیل روش‌های بدون شبکه به سرعت در حال توسعه و به کارگیری در مسائل فیزیکی و مهندسی هستند. یکی از انواع این روش‌ها، روشهای تابع پایه شعاعی هستند که روش چندربعی یکی از توانمندترین آنهاست. در این پژوهش، روش تابع پایه شعاعی چندربعی برای حل معادلات تراکم‌ناپدیر جریان پایا شامل معادلات پیوستگی، ناویراستوکس و مدل آشفتگی استاندارد، در یک میدان دوبعدی مورد ارزیابی قرار گرفته‌است. این میدان شامل یک هندسهِ حفره با درپوش متحرک مربعی، به ابعاد m5/0×m5/0 می‌باشد که در پنج عدد رینولدز 105×5/2، 105×5، 106×1، 106×2 و 106×5/5 تحلیل گردیده است. دامنه مذکور دو بار با تعداد نقاط داخلی 36 و 121مورد حل قرار گرفته و متغیر‌های مختلف محاسبه شده‌اند. طی این فرآیند، دو کمیت مهم متغیر شکل c بهینه و مجموعه ضرایب λ بهینه برای هر میدان جریان، مورد بحث و بررسی قرار گرفته است. نتایج نشان می‌دهد، با اتخاذ فرض استقلال مقادیر پارامتر c، برای می‌توان به یک الگوی قابل پیش‌بینی برای مجموعه λ دست‌یافت. الگوهای مذکور به‌همراه توابع پیش‌بین میدان‌های جریان مورد نظر، با نتایج روش حجم محدود (نرم افرار انسیس فلوئنت)، مورد مقایسه قرار گرفتند. ضرایب نش-ساتکلیف 93 الی 99 درصدِ و بیشینه خطای جذر میانگین مربعات نسبی در حد یک درصد، بدست آمده از این مقایسه برای پنج متغیر مستقل نشان‌دهنده قابل اعتماد بودن ترکیب فرض مذکور، الگوهای تکرارپذیر مجموعه λ و نیز توابع پیش‌بینی‌کننده آنها، می‌باشند.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Numerical Solution of Steady Incompressible Turbulent Navier–Stokes Equations using Multiquadric Radial Basis Function (MQ-RBF) Method

نویسندگان [English]

  • Mohammad Hossein Mirabi 1
  • Ehsan Jabbari 2
  • Taher Rajaee 1
1 University of Qom
2 University of Qom
چکیده [English]

The inconveniences of introducing and modifying the mesh grids in mesh-based numerical methods lead the researchers to meshfree methods, among which the RBF methods are probably the most interesting and powerful ones. In this research, the numerical solution of the steady-state incompressible continuity and Navier–Stokes equations, and the standard k-Ɛ turbulence model was investigated in a 2D domain. The computational domain consisting of a 0.5 m×0.5 m square lid-driven cavity was analyzed for five Reynolds numbers of 2.5×105, 5×105, 10×105, 2×106, and 5.5×106. The Multiquadric Radial Basis Function (MQ-RBF), as the most successful RBF, was employed with 36 and 121 domain computational nodes to solve the PDEs. The velocity fields in two directions, the static pressure, the turbulent kinetic energy and the turbulent energy dissipation, were computed. A try–and–error algorithm was used for solving a set of non-linear equations, and the optimal values of the shape parameter c and the λ set coefficients were evaluated and discussed for each flow field. According to the results, assuming the independence of the values of the shape parameter c for each flow field at different Reynolds numbers, a predictable pattern can be obtained for the λ set for different Reynolds’ numbers in the studied range. These patterns with the predictor functions of the flow fields were compared to existing benchmark results of the finite volume method (ANSYS Fluent). The Nash-Sutcliffe coefficients of 93-99% and RRSME of about %1 obtained from this comparison indicated the reasonable accuracy of the assumption concerning the independence of the shape parameter c of the Reynolds’ numbers, the repeatable patterns of the normalized λ set, and polynomial predictor functions in the MQRBF method for each flow field.

کلیدواژه‌ها [English]

  • Multiquadric Method
  • Navier-Stokes
  • Turbulent flow
  • Lid Driven Cavity
  • CFD
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