حل تحلیلی معادله دوبعدی و غیرماندگار انتقال آلودگی برای شرایط اولیه و مرزی دلخواه

نوع مقاله: مقاله پژوهشی

نویسندگان

1 دانشجوی دکتری گروه سازه های آبی ، دانشگاه تربیت مدرس تهران

2 عضو هیات علمی گروه سازه های آبی دانشگاه تربیت مدرس

3 عضو هیئت علمی گروه سازه های آبی دانشگاه تربیت مدرس

چکیده

در این تحقیق حل تحلیلی صورت دوبعدی معادلة جابه‌جایی- ‌پراکندگی- واکنش در دامنة محدود در رودخانه، با استفاده ازروش تبدیل انتگرالی تعمیم یافته استخراج شده است. شرط مرزی بالادست دیریشلت، به همراه تابع غلظت ورودی با الگوی زمانی نامنظم و دلخواه، شرط مرزی پایین دست و سواحل رودخانه نیومن در نظر گرفته شد. همچنین شرط اولیه نیز به‌صورت تابع مکانی کلی در دامنه لحاظ شد. به‌منظور ارزیابی حل استخراج شده، نتایج حاصله از حل پیشنهادی با حل تحلیلی به‌دست‌آمده با استفاده از روش تابع گرین مقایسه شد. به‌این‌ترتیب که در دو مثال فرضی مجزا برای حالتی که آلاینده ورودی از مرز صفر و شرط اولیه به‌صورت تخلیه ناگهانی جرم مشخصی از یک ماده آلاینده در یک نقطه معین در دامنه باشد؛ نیز در حالتی که هم‌زمان شرط اولیه و شرط مرزی با الگوی زمانی نامنظم و دلخواه در دامنه فعال باشند، مقایسه انجام شده و شاخص‌های آماری محاسبه شد. مقدار شاخص‌آماری ضریب هم ‌بستگی برابر با یک و میانگین خطای نسبی حدود 0/1 درصد به دست آمد. مقادیر شاخص‌های محاسبه شده بیانگر انطباق کامل نتایج حاصل از هر دو حل تحلیلی با یکدیگر است. حل تحلیلی پیشنهادی به دلیل انعطاف ‌ پذیری بالادر اتخاذ توابع گوناگون به عنوان شرط مرزی و اولیه، قابلیت بالایی به‌منظور کاربرد در صحت‌سنجی حل‌های عددی پیچیده معادله انتقال آلودگی در حالت‌های دوبعدی را دارد.

کلیدواژه‌ها


عنوان مقاله [English]

An analytical Solution to Bi-dimensional Unsteady Contaminant Transport Equation with Arbitrary Initial and Boundary Conditions

نویسندگان [English]

  • Neda Mashhadgarme 1
  • Mehdi Mazaheri 2
  • jamal mohammad vali samani 3
1 water structure department, Tarbiat modares university, Tehran
2 water structures department , Tarbiat modares university, Tehran
3 Water structure department, Tarbiat modares university, Tehran
چکیده [English]

In this research, the analytical solution to bi-dimensional Advection-Dispersion-Equation was obtained in the finite domain at the open channels using Generalized Integral Transform Technique (GITT). The upstream boundary condition was considered Dirichlet type with arbitrary and irregular time pattern of the entrance concentration. The downstream, right and left bank boundary condition was considered zero gradient. The initial condition function was assumed in the general form. The Evaluation of the derived solution was performed using two hypothetical examples and by comparing the results with the analytical solution resulting from the Green’s Function Method (GFM). In this way, in the first example, the entrance concentration from the upstream boundary was assumed zero and the initial condition function was considered impulsive at the specific point at the domain. At the second example, the irregular time pattern function of the entrance concentration from the upstream boundary and impulsive initial condition function was considered simultaneously. The results of both examples were compared with the results of GFM and the concentration contours at different times were presented. The results show good agreement between the proposed solution and the GFM solution and report the performance of the proposed solutions is satisfactory and accurate. The proposed analytical solution has high flexibility in adopting the various functions as the initial and boundary conditions. So it is very applicable and useful for verification of the two-dimensional complex numerical models.

کلیدواژه‌ها [English]

  • Pollutant transport Equation
  • initial condition
  • boundary condition with irregular time pattern
  • finite domain
  • Generalized Integral Transform Technique

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