حل تحلیلی معادله دینامیکی جریان متغیر تدریجی با استفاده از تابع فوق‌هندسی گوس

نویسندگان

1 استادیار، گروه عمران، واحد رامهرمز، دانشگاه آزاد اسلامی، رامهرمز

2 استادیار، دانشکده فنی-مهندسی، گروه عمران، دانشگاه مراغه، مراغه

چکیده

وجود سازه‌های هیدرولیکی در مسیر جریان باعث ایجاد جریان متغیر تدریجی (GVF) می‌شود که محاسبه‌ی تغییرات سنجه‌های هیدرولیکی بر مبنای حل معادله‌ی دینامیکی جریان با دقت بسیار زیاد از اهمیت ویژه‌ای بین پژوهشگران برخوردار است. در این پژوهش، ضمن بی‌بعدسازی معادله‌ی دینامیکی GVF در قالب دو روش yn-مبنا و yc-مبنا، از تابع فوق هندسی گوس برای حل تحلیلی این معادله برای پنج شیب ملایم (M)، تند (S)، بحرانی (C)، افقی (H) و معکوس (A) استفاده شده است. همچنین با استفاده از داده‌های آزمایشگاهی جمع‌آوری شده از یک کانال مستطیلی، مقایسه‌ای بین دقت محاسبه‌ی روش حل تحلیلی تابع فوق‌هندسی گوس و روش عددی رونگ‌کوتا مرتبه‌ی چهار بر مبنای شاخص‌های مجذور میانگین مربعات خطا (RMSE)، ضریب تبیین (2R) و متوسط درصد خطا (E) برای پروفیل‌های نوع M1، S2 و C3 انجام گرفته است. مقدار شاخص‌های RMSE و 2R برای پروفیل‌های M1، S2 و C3 در حل‌گر تحلیلی فوق‌هندسی گوس به ترتیب (0173/0، 9986/0)، (0167/0، 9984/0) و (0204/0، 9988/0) و در حل‌گر عددی رونگ‌کوتای مرتبه‌ی چهار به ترتیب (0458/0، 9864/0)، (0259/0، 991/0) و (0327/0، 9869/0) به دست آمدند. نتایج پژوهش نشان داد استفاده از حل‌گر تحلیلی تابع فوق‌هندسی گوس برای حل معادله‌ی دیفرانسیلی جریان GVF ، از دقت بسیار زیادی برخوردار است.

کلیدواژه‌ها


عنوان مقاله [English]

Analytical solution, Gaussian hyper-geometric function, Gradually varied flow, Numerical solution, Water surface profile,

نویسندگان [English]

  • mehdi fuladipanah 1
  • Mahdi Majedi Asl 2
1 Assis. Prof., Dept. of Civil Engineering, Ramhormoz Branch, Islamic Azad University, Ramhormoz, Iran
2 Assis. Prof., Dept. of Civil Engineering, Maragheh University, Maragheh, Iran
چکیده [English]

The presence of the in-flow hydraulic structures create gradually varied flow (GVF) profiles which the computation of the hydraulic parameters based on the dynamic equation of flow with high accuracy has significant aspect among researchers. In this study, while doing dimensionless process of flow dynamic equation using yn and yc, Gaussian Hyper-Geometric Function (GHF) has been implemented to solve the equation analytically for five main channel slopes namely Mild (M), Steep (S), Critical (C), Horizontal (H) and Adverse (A). also, a comparison has been done using laboratory data between the accuracy of numerical Rung-Kutta 4th order method and GHF analytical solver based on root mean square error (RMSE), determination coefficient (R2) and mean percent error € for M1, S2 and C3 profiles. While The values of RMSE and R2 indices for M1, S2 and C3 profiles for GHF solver obtained (0.0173,0.9986), (0.0167,0.9984) and (0.0204,0.9988) respectively, corresponds values for Rong-Kutta method were (0.0458,0.9864), (0.0259,0.991) and (0.0327,0.9869). The results showed that using GHF analytical solver to solve the differential equation of GVF is more accurate.

کلیدواژه‌ها [English]

  • Analytical solution
  • Gaussian hyper-geometric function
  • Gradually varied flow
  • Numerical solution
  • Water surface profile
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