Modelling of the diffusion-convection process of a pollutant with a periodic source
UDC
517.958+556.5.072DOI:
https://doi.org/10.31429/vestnik-19-1-25-34Abstract
The Krasnodar region with its unique natural resources requires a thorough environmental assessment, an adequate evaluation of the harmful effects the technical and economic facilities have on natural processes, as well as means to forecast the consequences of their influence on the region's ecosystem.
The use of various approaches to modeling the process of the pollutant spread allows implementing methods applicable for the assessment of the state of the regional ecological system. The work is devoted to the development of numerical-analytical methods for solving problems of the impurities migration in the atmosphere and the aquatic environment, based on the use of integral transformations and new efficient algorithms for the construction of the Green's symbols functions for a multilayer medium.
The paper considers a periodic convection-diffusion problem that describes the propagation and decay of substances in a multilayer medium. Emission sources are modeled by functions that allow for selection of a periodic function that can be represented as a Fourier series, as a time-dependent multiplier. We present a method for constructing a solution to a boundary value problem, which is based on the construction of an integral representation for the solution in Fourier images. Calculations of the substance concentration function for the problem of convection--diffusion--decay with a periodic localized source, implemented according to the described algorithm, are presented. In the numerical model, the calculation of the Fourier integral is based on the higher-order Gauss--Kronrod formulas.
The presented model allows accounting for the daily, weekly and seasonal technological and natural work cycles of polluting enterprises. The results can find practical application in identification of contaminated zones and territories resulting from planned and emergency emissions into the atmosphere or water areas.
Keywords:
turbulent diffusion, convection, impurity degradation, time-periodic source, Green's function, Fourier transformReferences
- Марчук Г.И. Математическое моделирование в проблеме окружающей среды. Наука, Москва, 1982. [Marchuk G.I. Matematicheskoe modelirovanie v probleme okruzhayushchey sredy = Mathematical modeling in the problem of the environment. Moscow, Nauka, 1982. (in Russian)]
- Марчук Г.И. Численное решение задач динамики атмосферы и океана. Наука, Москва, 1973. [Marchuk G.I. Chislennoe reshenie zadach dinamiki atmosfery i okeana = Numerical solution of problems of atmospheric and ocean dynamics. Nauka, Moscow, 1973. (in Russian)]
- Алоян А.Е. Моделирование динамики и кинетики газовых примесей и аэрозолей в атмосфере. Наука, Москва, 2008. [Aloyan A.E. Modelirovanie dinamiki i kinetiki gazovykh primesey i aerozoley v atmosfere = Modeling of dynamics and kinetics of gas impurities and aerosols in the atmosphere. Nauka, Moscow, 2008. (in Russian)]
- Пененко В.В., Алоян А.Е. Модели и методы для задач охраны окружающей среды. Наука, Новосибирск, 1985. [Penenko V.V., Aloyan A.E. Modeli i metody dlya zadach okhrany okruzhayushchey sredy = Models and methods for environmental protection tasks. Nauka, Novosibirsk, 1985. (in Russian)]
- Агошков В.И., Асеев Н.А., Новиков И.С. Методы исследования и решения задач о локальных источниках при локальных или интегральных наблюдениях. ИВМ РАН, Москва, 2015. [Agoshkov V.I., Aseev N.A., Novikov I.S. Metody issledovaniya i resheniya zadach o lokal'nykh istochnikakh pri lokal'nykh ili integral'nykh nablyudeniyakh = Methods of research and solving problems about local sources with local or integral observations. IVM RAN, Moscow, 2015. (in Russian)]
- Самарский А.А., Вабищевич П.Н. Численные методы решения задач конвекции-диффузии. Книжный дом "Либроком", Москва, 2015. [Samarsky A.A., Vabishevich P.N. Chislennye metody resheniya zadach konvektsii-diffuzii = Numerical methods for solving convection-diffusion problems. Book House "Librocom", Moscow, 2015 (in Russian)]
- Chawla M.M., Al-Zanaidi M.A., Al-Sahhar M.S. Stabilized fourth order extended methods for the numerical solution of ODEs. Intern. J.Computer Math., 1994, vol. 52, pp. 99–107.
- Бабешко О.М., Евдокимова О.В., Евдокимов С.М. Об учете типов источников и зон оседания загрязняющих веществ. Доклады Академии наук, 2000, т. 371-1, pp. 32–34. [Babeshko O.M., Evdokimova O.V., Evdokimov S.M. On taking into account the types of sources and settling zones of pollutants. Doklady Akademii nauk = Proc. of the Russian Academy of Sciences, 2000, vol. 371-1, pp. 32–34 (in Russian)]
- Бабешко В.А., Павлова A.B., Бабешко О.М., Евдокимова О.В. Математическое моделирование экологических процессов распространения загрязняющих веществ. Кубанский гос. ун-т, Краснодар, 2009. [Babeshko V.A., Pavlova A.B., Babeshko O.M., Evdokimova O.V. Mathematical modeling of environmental processes of pollutants distribution. Kuban State University, Krasnodar, 2009. (in Russian)]
- Сыромятников П.В. Матричный метод построения символа функции Грина для стационарных задач турбулентной диффузии в многослойных средах. Экологический вестник научных центров Черноморского экономического сотрудничества, 2018, т. 15, №3, с. 62–71. [Syromyatnikov P.V. Matrix method for constructing the symbol of the Green's function for stationary problems of turbulent diffusion in multilayer media. Ecological Bulletin of the Scientific Centers of the Black Sea Economic Cooperation, 2018, vol. 15, no. 3, pp. 62–71. (in Russian)] DOI 10.31429/vestnik-15-3-62-71
- Сыромятников П.В. Матричный метод решения нестационарных задач конвекции-диффузии в полуограниченных многослойных и градиентных средах. Наука Юга России, 2018, т. 14, № 4, с. 3–13. [Syromyatnikov P.V. Matrix method for solving nonstationary convection-diffusion problems in semi-bounded multilayer and gradient media. Nauka Yuga Rossii = Science in the South of Russia, 2018, vol. 14, no. 4, pp. 3–13. (in Russian)] DOI 10.7868/S25000640180401
- Кривошеева М.А., Лапина О.Н., Нестеренко А.Г., Никитин Ю.Г., Сыромятников П.В. Аналитическое и численное моделирование стационарной краевой задачи диффузии-конвекции-распада для однородного слоя на основе уравнений турбулентной диффузии. Экологический вестник научных центров Черноморского экономического сотрудничества, 2020, т. 17, № 3, с. 37–47. [Krivosheeva M.A., Lapina O.N., Nesterenko A.G., Nikitin Yu.G., Syromyatnikov P.V. Analytical and numerical modeling of a stationary boundary value problem of diffusion-convection-decay for a homogeneous layer based on the equations of turbulent diffusion. Ekologicheskiy vestnik nauchnykh tsentrov Chernomorskogo ekonomicheskogo sotrudnichestva = Ecological Bulletin of the Scientific Centers of the Black Sea Economic Cooperation, 2020, vol. 17, no. 3, pp. 37–47. (in Russian)] DOI 10.31429/vestnik-17-3-37-47
- Сыромятников П.В., Кривошеева М.А., Лапина О.Н., Нестеренко А.Г., Никитин Ю.Г. Решение методом факторизации смешанной краевой задачи диффузии-конвекции-распада для однородного слоя на основе уравнений турбулентной диффузии. Экологический вестник научных центров Черноморского экономического сотрудничества, 2021, т. 18, № 1, с. 36–45. [Syromyatnikov P.V., Krivosheeva M.A., Lapina O.N., Nikitin Yu.G. Solution by the factorization method of a mixed boundary value problem of diffusion-convection-decay for a homogeneous layer based on the equations of turbulent diffusion. Ekologicheskiy vestnik nauchnykh tsentrov Chernomorskogo ekonomicheskogo sotrudnichestva = Ecological Bulletin of the Scientific Centers of the Black Sea Economic Cooperation, 2021, vol. 18, no. 1, pp. 36–45. (in Russian)] DOI 10.31429/vestnik-18-1-36-45
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Copyright (c) 2022 Lapina O.N., Nesterenko A.G., Nikitin Yu.G., Pavlova A.V.
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