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МАТЕМАТИЧНЕ ТА КОМП'ЮТЕРНЕ МОДЕЛЮВАННЯ
МАТЕМАТИЧЕСКОЕ И КОМПЬЮТЕРНОЕ МОДЕЛИРОВАНИЕ
MATHEMATICAL _AND COMPUTER MODELING_
UDC 517.9:532.5
Lamtyugova S. N.1, Sidorov M. V.2, Sytnykova I. V.3
1Ph.D, Associate Professor of the Department of Advanced Mathematics of O. M. Beketov National University of Urban Economy
in Kharkiv, Kharkiv, Ukraine
2Ph.D, Associate Professor, Associate Professor of the Department of Applied Mathematics of Kharkiv National University of Radio
Electronics, Kharkiv, Ukraine
3Ph.D, Associate Professor of the Department of Advanced Mathematics of O. M. Beketov National University of Urban Economy
in Kharkiv, Kharkiv, Ukraine
METHOD OF NUMERICAL ANALYSIS OF THE PROBLEM OF MASS TRANSFER OF A CYLINDRICAL BODY WITH THE UNIFORM _TRANSLATIONAL FLOW_
Context. The problem of mass transfer of a cylindrical body with a uniform translational flow of a viscous incompressible fluid is examined in the paper.
Objective. The purpose of this work is to develop a new method for numerical analysis of the problem of mass transfer of a cylindrical body with a uniform translational flow, which based on the joint application of the ^-functions structural method and the Galerkin projection method.
Method. In general case, the problem of stationary mass transfer of a cylindrical body with a viscous incompressible fluid flow is reduced to the solution of the equation of hydrodynamic flow passing a surface and an equation for concentration with corresponding boundary conditions on the surface of the body and far away from it. The geometry of the area, and also the boundary conditions (including the condition at infinity) may be taken into account precisely by using the constructive apparatus of the ^-functions theory by V. L. Rvachev, the Academician of Ukrainian National Academy of Sciences. In this study, a complete structure of the solution of a linear boundary value problem for the concentration that exactly satisfies the boundary conditions on the boundary and condition at infinity is constructed on the basis of the ^-functions theory methods, and this made it possible to lead the tasks in the infinite domain to tasks in the finite domain. To solve the linear problem for concentration the numerical algorithm on the basis of Galerkin method is developed.
Results. The computational experiment for the problem of the flow past circular and elliptical cylinders at various Reynolds and Peclet numbers was carried out.
Conclusions. The conducted experiments have confirmed the efficiency of the proposed method of numerical analysis of the problem of mass transfer of a cylindrical body with a uniform translational flow, based on the joint application of the ^-functions structural method and Galerkin projection method. The prospects for the further research may be to use the developed method for the implementation of iterative methods for solving the task of nonlinear mass transfer, semi-discrete and projection methods for solving the non-stationary tasks, as well as in solving the tasks of optimal management of relevant technological processes.
Keywords: flow task, stationary flow, viscous fluid, stream function, mass transfer, concentration function, ^-functions method, Galerkin method.
NOMENCLATURE
c = c(r, 9) - a function of concentration; c0 - the constant concentration given on the boundary dQ of a streamlined body;
dQ - the boundary of a streamlined body; l = 1,2 - the number of equation; n - the external normal to dQ ; Pe - Peclet number;
© Lamtyugova S. N., Sidorov M. V., Sytnykova I. V., 2018 DOI 10.15588/1607-3274-2018-2-3
r , 9, z - variables of the cylindrical coordinate system; Re - Reynolds number;
ui - the solutions of auxiliary problems with homogeneous boundary conditions;
Ux - unperturbed fluid velocity at infinity; v = Re-1 - the coefficient of viscosity; {9 k (r, 9)} - a complete system of particular solutions of equation AU = 0 with respect to the exterior of a cylinder of finite radius;
{t j (r, 9)} - a complete system of particular solutions of equation Au = 0 relative to domain {ro(r,9) <M};
{^i (r, 9)} - a sequence of functions that is complete with respect to the whole plane;
®1, ®2 - undefined components of the solution structure;
W = w(r, 9) - the stream function;
ro - a sufficiently smooth function built by using the constructive apparatus of the ^-functions theory and describing the geometry of the domain Q ;
= 0 - a normalized equation of dQ;
Q - the flow domain;
Qn - the bounded domain {0 < ro(r, 9) < Mn }. INTRODUCTION
The problems of calculating viscous flows, which are complicated by mass transfer, are widely used in heat power engineering, chemical and food technologies, geo- and astrophysical studies, and environmental protection. Many processes of chemical technology are associated with fluid movement in process equipment. In the preparation of reagents and in the isolation of reaction products, such operations as leaching, absorption, extraction and distillation play an important role. The laws of hydrodynamics, heat and mass transfer are essential for the entire technological process. Processes of heat and mass transfer are also ones of the most important in the energy sector, as well as in a number of the technological processes in the metallurgical and other industries. In addition, the problems of mass transfer of bodies with a uniform viscous flow underlie the calculation of many technological processes, which are associated with dissolution, extraction, evaporation, precipitation of colloids, etc. Therefore, the development of new, as well as the improvement of existing methods of mathematical modeling and numerical analysis of external stationary problems of hydrodynamics of a viscous incompressible fluid, which take into consideration the mass transfer, is an actual scientific issue.
The object of this study is the stationary hydrodynamic process of flow past bodies by a viscous incompressible fluid, complicated by mass transfer and described by a system of equations with respect to the stream function and concentration.
The subject of this study is a mathematical model of the stationary task of flow past body by a viscous incompressible fluid with allowance for mass transfer, and a method for its numerical analysis.
The purpose of the work is to develop a new method for mathematical modeling of the mass transfer of a cylindrical body with a uniform translational flow which based on the ^-functions method and Galerkin method.
1 PROBLEM STATEMENT
At small Peclet numbers, to describe the mass transfer process the Oseen approximation (equation 1) is used [1]:
Ac = PeUx| cos9— -
dc sin 9 dc
99
in Q .
(1)
In the general case, the mass transfer process is described by the equation for the concentration (equation 2) in the form [1, 2]:
Pe f dw dc dw dc Ac = — l —----— I in Q
r ^ 99 dr dr 99 y
(2)
„s w^ . 1 d I dc 1 1 d2c
In equations (1) and (2) Ac =--1 r— I +---.
r dr \ dr ) r2 992
Equations (1) and (2) in a rectangular coordinate system have the following form:
Ac = PeU „
dc dx
Ac = Pe|^£
dy dx dx dy
in Q
(3)
(4)
where Ac =
d 2 c
dx2 dy2
Equations (1) and (2) should be supplemented by the boundary condition on dQ and the condition at infinity [1]:
- + -
5 2 c
2
l5Q
= c0,
lim c = 0.
r
(5)
(6)
The stream function y(r, 9) can be found, for example, as the solution of the following nonlinear task of flow past a cylindrical body by a viscous incompressible fluid [3-6]:
a2 1 5(Ay) 1 dw 5(Ay)
r 99 dr r dr 99
Q
IdQ
= 0,
dn
= 0,
dQ
lim r 1w = U x sin 9
r^+x '
(7)
(8)
(9)
= 1 5 fr dw1+ 1 52w a2
where Aw =--1 r-I +
Aw = A(AW) .
r dr y dr ) r2 992
The task (1), (5), (6) does not depend on the stream function w(r, 9), and the solution of the task (2), (5), (6) consists of two steps:
a) the search of the stream function as a solution of the task (7)-(9);
b) the solution of the task for concentration.
2 REVIEW OF THE LITERATURE
Various tasks which are arising in the study of external viscous fluid flows can be investigated theoretically or by means of physical experiment. At present, mathematical modeling and computational experiment are increasingly
r
being used in the study of hydrodynamic problems. Basically, finite difference method, finite element method, boundary integral element method and others are used for the numerical analysis of such problems. These methods are easy to implement, but do not have the necessary property of universality: when moving to a new area (especially non-classical geometry), it is necessary to generate a new grid, and often to replace complex sections of the boundary with simple ones, composed of, for example, straight line segments. The use of the R-functions structural-variational method [7, 8] by V. L. Rvachev, the Academician of Ukrainian National Academy of Sciences, is an alternative to the existing methods of numerical computation of hydrodynamic problems.
The R-functions method in computational hydrodynamics was applied in [9-14]. The task of viscous fluid external flow around bodies of revolution in a spherical coordinate system, which is complicated by mass transfer, was solved in [15, 16] with using the R-functions method.
In this study, we propose to apply the R-functions and Galerkin method for mathematical modeling the problem of mass transfer of a cylindrical body with a uniform translational flow.
3 MATERIALS AND METHODS
The method for solving the task for the stream function, based on the application of R-functions, successive approximations and Galerkin method, is described in [17,
18]. Substituting the stream function y(r, ф) so obtained into equation (2), let us solve the problems (1), (5), (6) and (2), (5), (6) by the R-functions method. To do this, using the constructive means of the R-functions theory [7, 8] let us construct the structure of the boundary value task solution, i.e. the functions bundle that exactly satisfies the boundary condition and the condition at infinity.
Let us consider a sufficiently smooth function [19]
fM (x) =
Mx
1 - exp-—, 0 < x < M ;
x-M
(10)
1,
x > M (M = const > 0).
The function (10) fM (x) e Cx [0,+œ) and satisfies the conditions:
a) fM (0) = 0; b) fM (0) = 1; c) fM (x) > 0 Vx > 0; d) fM (x) - 1 Vx > M . Let us denote a m = fM (ro) , then
1) aM > 0 in Q ; 2) aM = 0;
5ra m
3)
dn
= -1; 4) ю m = 1, lf ю M > M.
dQ
In the problems (3), (5), (6), and (4)-(6) we make the replacement
с = Co(1 -юm ) + ui.
The choice of such a substitution is due to the fact that the function C0(1 — ) satisfies the boundary condition (5) and the condition at infinity (6).
Then for ui, l = 1,2, we obtain the tasks
-Диi + Biui = Fi In Q ,
ui
dQ
=0,
lim ui = 0 ,
r
(11) (12) (13)
where
B1 = PeU B2 = Pe[^U2
dx ^ dy dx dx dy
F = Д(с0(1 -юm ))-PeU>°(1:"M ,
dx
F2 = Д(с0(1-юM))-д(с0 (1 - Юм )) dy д(с0 (1 - ®m ))1
- Pe
дУ
dx
dx
dy
Note that Ft = 0, l = 1,2, in the domain (ra(x, y) > M} .
Let us find the generalized solution u of tasks (11)—(13) as the limit, when n ^ x, of solutions un of equations (11), which are considered in a sequence of domains {Q n }, that is a monotonic exhaustion of an infinite area Q.
In domains Q n we will consider the boundary value tasks
-Ди i + Biu i = Fi m Qn
u„j\ = 0, IdQ
(14)
(15)
where functions un,l are continued by zero outside of Qn .
An approximate solution of tasks (14) - (15) for each n = 1,2,... according to the Bubnov-Galerkin method, will be sought in the form:
N
un,i,N = Z cn,i,j Ф j . j=1
(16)
The numbers cn,i,j , j = 1,..., N , are found from the system of linear algebraic equations
N
Z Cn,i, j {[Ф j, Ф/ ] + (Ki Ф j, Фг )} = (Fi, фг ), (17)
j=1
where
= 1, N, i = 1,2,
(F1,Фt ) = -jC0V(1 -юm ) ^ФtdQ +
+ j C0Pe^œ-(1 -Юм)^dQ,
Q dx
Q
n
(F2, * z) = -J Co V(1 -» M) -V* tdQ +
+ J coPe(1 -»m-^W^U.
Q y dy dx dx dy )
"n
To construct the coordinate sequence, the complete system of particular solutions of the Laplace equation [20] and the ^-functions method [7, 8] will be used.
We have proved that for any choice of sufficiently smooth functions ®1, ®2 and at the requirement that ®1 ^ 0 as r ^ +a> a function of the form
u = »M®1 + raM (1 -»M)®2
exactly satisfies the boundary conditions (12) and (13), that is, it is the structure of the solution of the boundary value problem (11)-(13).
Approximations of the functions ®1 h ® 2 in the domain Q n will be seek in the form
®1 = 1a k-9k, ® 2 = 1p
k=1
rj "]'
j=1
-ncosn9
where {9 k (r, 9)} = <{r , n = 1,2,...
sin n9
k cos k9
{t j (r, 9)} = <¡1, rk , k = 1,2,...
sin k9
Then the sequence of functions ,which is complete with relatively to the whole plane, has the form:
(r, 9)} = {»M (r, 9)9k (r, 9), »M (r, 9)(1 - »m (r, 9))t j (r, 9)} .
The values of the coefficients ak (k = 1,2,..., m^ and Pj ( j = 1,2,...,m2) in accordance with the Bubnov-Galerkin method will be found from the condition of residual orthogonality to the first N (N = m1 + m2) elements of the sequence (18), which leads to a system of linear algebraic equations in the form (17).
We have proved the convergence of the Galerkin approximations unj,N of the form (16) at N ^ x, Mn ^ <x> in the energy norm to the unique generalized solutions of tasks (11)-(13).
4 EXPERIMENTS
A computational experiment was carried out for the tasks of flow past a circular cylinder x 2 + y 2 = 1 and an elliptical
= 1, M = 5, 10, m1 = 10,
2 2 ,-,xy
cylinder--1--= 1 at
J 4 1
m2 = 11 and at various Reynolds and Peclet numbers. The
double integrals in the systems for determination cnj, j were
taken approximately by the Gauss formula with 50 nodes for
each variable.
5 RESULTS
The concentration lines for a circular cylinder are shown in Fig. 1-3. Fig. 4-6 show the concentration lines for elliptical cylinder.
(18)
Figure 1 - The concentration lines for a circular cylinder
at M = 5 and Pe = 0,01
Q
n
b
Figure 2 - The concentration lines for a circular cylinder at Pe = 10 : a - Re = 0, b - Re = 5, c - Re = 15
a
Figure 3 - The concentration lines for a circular cylinder at Pe = 20 : a - Re = 0, b - Re = 5, c - Re = 15
6 DISCUSSION
Figure 4 - The concentration lines for elliptical cylinder at M = 5 and Pe = 0,01
At small Reynolds and Peclet numbers the substance is transported uniformly, dissolving into the liquid. As the Reynolds and Peclet numbers increase, the particles of substance begin to move with the flow. These results are consistent with the physics of the process. The efficiency of the proposed method for a spherical coordinate system was verified on the problem of flow past a sphere [13], for which an exact solution is known [20].
CONCLUSIONS
The numerical method for calculating the mass transfer of a cylindrical body with a uniform translational flow, which based on the joint application of the ^-functions method and Galerkin method, is proposed for the first time in this
Figure 5 - The concentration lines for elliptical cylinder at Pe = 10 :
a - Re = 5 , b - Re = 15, c - Re = 30
c
Figure 6 - The concentration lines for elliptical cylinder at Pe = 20 : a - Re = 5, b - Re = 15, c - Re = 30
study. By using the ^-functions method the structure of the solution of the problems of flow past bodies with allowance for mass transfer, which precisely satisfying the boundary condition on the boundary and the condition at infinity, was constructed, and this made it possible to lead tasks in the infinite domain to tasks in the finite domain. To approximate the uncertain components of the solution structure, the Galerkin method was applied. The stationary problem of flow past a cylindrical body in a cylindrical coordinate system for a circular and elliptical cylinders has been solved numerically for various Reynolds and Peclet numbers. ACKNOWLEDGMENTS
The work was carried out at the department of Applied Mathematics at Kharkiv National University of Radio Electronics and department of Advanced Mathematics at
0. M. Beketov National University of Urban Economy in Kharkiv within the framework of collaborative scientific research conducted by the departments.
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Article was submitted 04.12.2017.
After revision 15.01.2018.
Ламтюгова С. М.1, Сидоров М. В.2, Ситникова Ю. В.3
'Канд. фiз.-мат. наук, доцент кафедри вищо! математики Харгавського нащонального ушверситету мюького господарства iменi О. М. Бекетова, Харгав, Укра!на
2Канд. фiз.-мат. наук, доцент, доцент кафедри прикладно! математики Харгавського нащонального ушверситету радюелектрошки, Харгав, Украша
3Канд. пед. наук, доцент кафедри вищо! математики Харювського нащонального ушверситету мюького господарства iменi О. М. Бекетова, Харгав, Украша
МЕТОД ЧИСЕЛЬНОГО АНАЛ1ЗУ ЗАДАЧ1 МАСООБМ1НУ ЦИЛ1НДРИЧНОГО Т1ЛА З Р1ВНОМ1РНИМ ПОСТУПАЛЬ-НИМ ПОТОКОМ
Актуальтсть. У статп розглядаеться задача масообмшу цилшдричного тша з рiвномiрним поступальним потоком в'язко! нестис-ливо! рщини.
Мета роботи - розробка нового методу чисельного аналiзу задачi масообмшу цилшдричного тша з рiвномiрним поступальним потоком, заснованого на сумюному застосуванш структурного методу К-функцш i проекцшного методу Гальоркша.
Метод. У загальному випадку задача про стащонарний масообмш цилшдричного тша з потоком в'язко! нестисливо! рщини зводиться до розв'язання рiвняння гiдродинамiчного обтжання поверхш i рiвняння для концентрацп з вщповщними крайовими умовами на поверхш тша i далеко вщ нього. Точно врахувати геометрто обласп, а також крайовi умови (в т.ч. i умову на нескшченносп), можна, скориставшись конструктивним апаратом теорп К функцiй акад. НАН Укра!ни В. Л. Рвачова. В робот на основi методiв теорп К-функцш побудована повна структура розв'язку лшшно! крайово! задачi для концентрацп, яка точно задовольняе крайовим умовам на межi тша i умовi на нескiнченностi, що дозволило звести задачi в нескiнченнiй областi до задач в скшченнш областi. Для розв'язання лшшно! задачi для концентрацп розроблено чисельний алгоритм на основi методу Гальоркша.
Результати. Обчислювальний експеримент проведений для задачi обтiкання кругового i елiптичного цилiндрiв для рiзних чисел Рейнольдса i Пекле.
Висновки. Проведенi експерименти пiдтвердили працездатнiсть запропонованого методу чисельного аналiзу задачi масообмшу цилшдричного тша з рiвномiрним поступальним потоком, заснованого на сумюному використанш структурного методу К-функцш i проекцшного методу Гальоркша. Перспективи подальших дослщжень можуть полягати у використаннi розробленого методу при реалiзацn iтерацiйних метсдав розв'язання задачi нелiнiйного масообмiну, полудюкретних i проекцiйних методiв розв'язання нестащ-онарних задач, а також при розв'язанш задач оптимального управлiння вщповщними технолопчними процесами.
Ключовi слова: задача обтжання, стацiонарна течiя, в'язка рщина, функцiя течii', масообмiн, функцiя концентрацп, метод Л-функцш, метод Гальоркiна.
Ламтюгова С. Н.1, Сидоров М. В.2, Ситникова Ю. В.3
'Канд. физ.-мат. наук, доцент кафедры высшей математики Харьковского национального университета городского хозяйства имени А. Н. Бекетова, Харьков, Украина
2Канд. физ.-мат. наук, доцент, доцент кафедры прикладной математики Харьковского национального университета радиоэлектроники, Харьков, Украина
3Канд. пед. наук, доцент кафедры высшей математики Харьковского национального университета городского хозяйства имени А. Н. Бекетова, Харьков, Украина
МЕТОД ЧИСЛЕННОГО АНАЛИЗА ЗАДАЧИ МАССООБМЕНА ЦИЛИНДРИЧЕСКОГО ТЕЛА С РАВНОМЕРНЫМ ПОСТУПАТЕЛЬНЫМ ПОТОКОМ
Актуальность. В статье рассматривается задача массообмена цилиндрического тела с равномерным поступательным потоком вязкой несжимаемой жидкости.
Цель работы - разработка нового метода численного анализа задачи массообмена цилиндрического тела с равномерным поступательным потоком, основанного на совместном применении структурного метода К-функций и проекционного метода Галеркина.
Метод. В общем случае задача о стационарном массообмене цилиндрического тела с потоком вязкой несжимаемой жидкости сводится к решению уравнения гидродинамического обтекания поверхности и уравнения для концентрации с соответствующими краевыми условиями на поверхности тела и вдали от него. Точно учесть геометрию области, а также краевые условия (в т.ч. и условие на бесконечности), можно, воспользовавшись конструктивным аппаратом теории К-функций акад. НАН Украины В. Л. Рвачева. В работе на основании методов теории К-функций построена полная структура решения линейной краевой задачи для концентрации, точно удовлетворяющая краевым условиям на границе и условию на бесконечности, что позволило свести задачи в бесконечной области к задачам в конечной области. Для решения линейной задачи для концентрации разработан численный алгоритм на основании метода Галеркина.
Результаты. Вычислительный эксперимент проведен для задачи обтекания кругового и эллиптического цилиндров при различных числах Рейнольдса и Пекле.
Выводы. Проведенные эксперименты подтвердили работоспособность предложенного метода численного анализа задачи массо-обмена цилиндрического тела с равномерным поступательным потоком, основанного на совместном применении структурного метода ^-функций и проекционного метода Галеркина. Перспективы дальнейших исследований могут заключаться в использовании разработанного метода при реализации итерационных методов решения задачи нелинейного массообмена, полудискретных и проекционных методов решения нестационарных задач, а также при решении задач оптимального управления соответствующими технологическими процессами.
Ключевые слова: задача обтекания, стационарное течение, вязкая жидкость, функция тока, массообмен, функция концентрации, метод ^-функций, метод Галеркина.
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