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Mathematical and Numerical Modeling of Natural Convection in an Enclosure Region with Heat-conducting Walls by the R-functions
and Galerkin Method
Artyukh A.
Abstract—This paper is dedicated to the investigation of the natural convection in an enclosed region. The mathematical model has been formulated using the dimensionless variables for the stream function and temperature. The numerical results have been obtained by means of the R-functions and Galerkin methods.
Index Terms—natural convection, stream function,
temperature, R-functions method, Galerkin method.
I. Introduction
THE problem of the natural convection in an enclosed region has vital importance in many technical applications such as microelectronics, radio electronics, energetics etc. Obviously, such problem has a lot of important implications which makes the corresponding investigation actual.
Such problems are mainly resolved using the finite difference and finite element methods. They are easy to program, but they are not universal since a new grid generation is required every time a transition to a new area is made. The R-functions method developed by the academician of the Ukrainian Academy of Sciences V. L. Rvachev allows considering the geometry of the problem accurately [5].
The objective of this work is the mathematical simulation of the natural convection in an enclosed region by means of the R-functions method and Galerkin method.
II. Problem Statement
The mathematical model of the natural convection in an enclosed region with heat-conducting walls in an arbitrary closed region is shown in Fig.1.
Let’s consider the Q = Qs U Of area, where Qf is the gas cavity, Qs - solid walls, dQsf - impermeable and fixed bound between Qf and Qs. It is assumed that the fluid is Newtonian, incompressible, and viscous.
Manuscript received December 13, 2012.
A. Artyukh is with the National University of Radioelectronics, Kharkov, Ukraine (phone: +38-095-917-42-24; e-mail: [email protected]).
5Q2
Fig. 1. Problem Solution Region
The mathematical model using the dimensionless variables takes the following form [3]: in the cavity:
дАу ду дАу дт ду dx
59 ду д0 ду d0 1 дт Dy дх дх Dy л/Ra^Pr in the solid walls:
ду дАу дх ду
/ Pr 2 д9
--А2у + —
Ra дх
(1)
(2)
ае
дт
asf
гА9
v/Ra • Pr
Where х, y are the dimensionless coordinates,
т - dimensionless time,
A - Laplace operator, у - dimensionless stream function,
9 - dimensionless temperature,
Ra=
gpTL3
vaf
- Rayleigh number,
(3)
v
Pr =-------Prandtl number,
af
g - acceleration of gravity,
P - coefficient of volumetric thermal expansion, v - kinematic coefficient of viscosity,
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103
af - temperature diffusivity coefficient of the gas,
asf
asolid
afluid
A
- relative temperature diffusivity coefficient,
(4)
(5)
0la
Ш
ae an
at internal borders
0laQ3 e2 ,
- 0.
5Q2
l5Q.
- 0,
aef
ae
an
dy
an
5Q4
- 0,
=0
(6)
(7)
5Qsf
ae.
^ -Xsf ^ on 5Qsf = dn dn
(9)
The is the most widespread R-function system: 1
x ла y =
1 + a
(x + y -^/x2 + y2 -2axy).
Xsf = solid— relative heat conduction coefficient,
Xfluid
L - length of the gas cavity.
Equation (1) is considered for Qf , and equations (2) -(3) are considered for Qf and Qs respectively.
Initial conditions for the problem (1) - (3) are set as follows:
4t-0 = Уо (x,y),
0U = e0(x,y).
The boundary conditions have the following form: at external borders:
where dQ - dQ1 UdQ2 U dQ3 U dQ4 , 0s - temperature in the solid wall, 0f - temperature in the gas cavity, n is a normal vector to the boundary.
III. The R-functions Method Consider the inverse problem of analytical geometry.
Let’s consider a geometric object Q in space R2 with a piecewise smooth bound dQ. It is required to construct a function o(x,y) that would be positive inside Q , negative outside of Q and equal to zero at dQ. The equation The equation o(x,y) - 0 determines an implicit form of the locus for the points that belong to the boundary dQ of the region Q .
Definition 1. The function with the sign entirely determined by the signs of its arguments is called the R-function corresponding to the partition of the numerical axis within the (-<»,0) and [0, +<») intervals, i.e. the function z - f(x,y) is called the R-function if the Boolean function F exists and S[z(x,y)] - F[S(x), S(y)], where
10, x < 0,
S(x) is a double-valued predicate S(x) - <j 1 > 0
1 I 2 2
x Va y = ----(x + y + V x + y - 2axy),
1+a
x = -x,
where
-1 < a(x,y) < 1, a(x,y) = a(y,x) = a(-x,y) = a(x, -y) . Let’s consider the Q region that can be created based on simpler regions Q1 - (o1(x,y) > 0},..., Qm -
- {»m(x,y) > 0}, by means of the of set-theoretic operations such as union, intersection and complement. Therefore, let’s assume that the predicate
Q- F(Qb Q2, k, Qm) (10)
corresponding to the region Q is equal to 1 if (x,y) e Q and is equal to 0 if (x,y) g Q .
The transition from the predicate-based form of the region defining (10) to an ordinary analytical geometry equation is made using the formal substitution of Q with o(x,y), Qi with c»i(x,y) (i -1, 2, ..., m), and the
{I, U, —} are substituted with the R-operations symbols (8) {Aa, va, -} respectively. As a result, an analytic
expression for o(x,y) is derived. This expression defines the required equation o(x,y) - 0 of the bound dQ for the elementary functions. Note that o(x,y) > 0 for the interior points and o(x,y) < 0 for the exterior points of Q .
Definition 2. The equation o(x,y) - 0 for the bound dQ of Q c R2 is normalized to the order n if the
°Ш 0;
function o(x,y) satisfies these conditions:
- -1, - 0 (k - 2, 3, ..., n), where n is an
do
an
dQ
anK
dQ
outer normal vector to dQ, that is defined for all regular points of Q .
The equation o(x,y) - 0 normalized to the first order can be obtained from the equation o(x,y) - 0 as described below.
Theorem 1. If o(x,y) e Cm(R2) satisfies the conditions
> 0 , then the function
dQ
raLQ - 0 and —
dQ an
ut - I------------==
•yo2 +|V«
: c™-1(r2) , M.|f )2+[|
satisfies the conditions i» regular points of the bound dQ
1 d»i
-„ - 0 and —-
dQ an
- -1 for all
dQ
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We can use this simplified formula: ю _ equation normalized to the first order if
raj
К | |V®1
for the Ф 0 in
Q = Q U dQ .
Let’s consider the R-function application scheme for the boundary problems solving. The problem of the physical field calculation can be reduced to finding the solution u of the equation Au = f within the region Q under the following conditions on the bound dQ of Q : Liu = ф;, i = 1, ..., m, where A and L; are known differential operators; f and ф; - functions defined inside Q and in the areas of its boundary dQ. The areas dO; are not necessarily all different, and may coincide with the whole bound dQ. The functions u, f , ф; and operators A and Li mentioned in the boundary problem statement are called analytic components of the boundary problem, the area Q , its boundary dQ, border areas dO; are called geometric
components.
The existence of two different types of information (analytical and geometrical) is a major obstacle for the solution finding. Not only the look of the formulas included into the problem statement should be considered, but the geometrical information should be transferred to the analytical look to so that it can be involved into the solution algorithm. The R-functions method allows this procedure implementation.
The sheaves of functions can be built by means of the normalized equations. The normal derivatives of such functions or an arbitrary linear combination of the normal derivative and the function itself take the given values on the region bounds.
In order to achieve this, let’s consider the following operator
D
дю d dx dx
дю d dy dy ’
where ю (x, y) is a normalized equation of the region bound.
Moreover, for any sufficiently smooth function f on the bound dQ this statement will be valid:
D1f | = -—
1 ldQ dn
dQ
where n is an outer normal vector to dQ. Let
D(i) _ dc»i d dc»i d
dx dx dy dy
denote the analog of D1 corresponding to the areas dO; of the bound dQ, where ca;(x,y) are normalized equations of for the areas dQi.
One can prove that
D^ = 1 + О(ю),
Dl(юФ) = = ^1ю)Ф + roD^ = Ф + О(ю), where ю (x,y) is the normalized equation of the region bound. Definition 3. The expression
u = В(Ф, ю, {ю^, {Фj}j=1)
is called the general boundary problem solution structure if that expression exactly satisfies all boundary conditions of the problem for any undetermined component Ф chosen. В is the operator dependent on the geometry of the region and parts of its border, as well as on the operators of the boundary conditions, but is not dependent on the type of operator A and function f .
Let’s consider the expression u = В;(Ф, ю, ю;, фj) as a
partial solution structure that exactly satisfies the boundary condition only on dQi for any undetermined component.
Thus, the solution structure provides extension of the boundary conditions into the region.
The task of the equation creation for the complex geometric object is a specific case of a more general problem where the unknown function ф takes the given
values on different parts of the bound dQi, i.e.
Ф = ф; on dQi, i = 1, ..., m. (11)
For simplicity, let’s assume that ф; are elementary functions defined everywhere in the region Q U dQ . After the methodology described above is applied, the functions ю0 equal to zero everywhere, except for the area dQ; are constructed. Thus, the function
T m V m j 1
ф= Хф;Ю° (12)
vi=1 Aj=1 у
satisfies (11) and is defined everywhere in the region, with the exception of the points that are common to the different sections. Instead of (12) we can also apply the formula
T m V m V1
ф= Хф* “Г1 X»-1 , (13)
vi=1 yVj=1 у
where ю* = 0 are equations of dQ* of the bound dQ , and ю* > 0 outside dQ*. The function ю* ^ 0 when approaching the area dQi and the limit values of the function ф match the values of the corresponding function
ф;.
Let’s denote the bonding operator for the boundary values defined by any of the above formulas (12) and (13) as EC (ЕСф; = ф).
Practically all of the approximate methods for the boundary problems solving for the partial differential equations are based on the infinite-dimensional problem to a finite-dimensional one reducing. The method of R-
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functions provides the corresponding result achieving by means of the undetermined component of the solution structure representation as the sum:
n
Ф(x, y) * Фп (x, y) = £ ck9k (x, y), k=1
where 9k(x,y) are known elements of the complete functional sequence, and ck (k = 1,2,...,n) are unknown expansion coefficients.
The undefined functions included into the structural formulas should be chosen so that the basic differential equation of the problem is satisfied with the best results. The methods of the undefined function approximations search can be very different. For example, one can use the variational (Ritz, least squares, etc.), projection (Galerkin, collocation, etc.), grid and other methods.
IV. Solution Method
The R-functions and Galerkin methods are used for the initial-boundary problem (1) - (9) solving.
Let’s consider the boundaries 5Q and 5Qsf that are are
piecewise smooth and that can be described by means of the elementary functions ro(x,y) and rosf(x,y). According to
the R-functions method, ro(x,y) and rosf(x,y) satisfy the
below conditions:
1) ro(x,y) > 0 in Q ;
2) ro(x,y) = 0 on dQ ; дю
3) — = -1 on dQ, n is an outer normal vector to dQ ,
an
and
1) rasf (x,y) > 0 in Qf ;
2) rosf(x,y) = 0 on dQsf ;
drasf(x,y)
3) ----------=-1 on dQsf, n is a normal vector
an
pointing into Qf .
The investigation in [5] shows that the boundary conditions (7) - (8) are satisfied by the sheaf of functions
V = ros2f Ф,
where Ф = Ф(x, y, t) is an undefined component.
The solution structure of (2) - (3), i.e. the sheaf of functions which satisfies the boundary conditions (5), (6), (9), was built by means of the region-structure Rvachev-Slesarenko method [6]. Hence
BCD in Qs,
BCD - (1 -ksf)«sfD1B(Y)in Qf,
(14)
where Y = Y(x,y,t) is an undefined component, B(Y) satisfies the boundary conditions on external borders.
The undefined components Ф and Y were found by
means of the Galerkin method. Therefore, we will obtain an approximate solution of the problem (1) - (9).
V. Numerical Results
Let’s consider the mathematical model of natural convection (1) - (3) in a closed region (fig. 2) [3]. it is assumed that the fluid is Newtonian, incompressible and viscous.
x
Fig. 2. Problem Solution Region
The initial conditions for the problem (1) - (3) have the below form:
4=0 =eL =°- <l5>
The boundary conditions are set as follows: on external borders:
0|x=0 = 01 , 0|x=L = 02 , where 0 < у < Ly ^
ae
an
= 0,
y=0
on internal borders
ae
an
= 0, where 0 < x < Lx
lx=h
lx=LY -h
ly=h
ly=Ly -h
= 0.
(16)
(17)
(18)
a^
an
a^ a^ a^
x=h an x=Lx-h an y=h an
= 0, (19)
У=Ly -h
0s = f
a0f = a0s ' = Asf '
(20)
an an
where 0s is the temperature in the solid wall, 0f -temperature in the gas cavity, n - normal vector to the boundary, Lx and Ly are normalized by the length of the
gas cavity L .
The functions ro(x,y) and rosf(x,y) have the following form:
ro(x,y) = t—x(Lx - x) Aq —- y(Ly - y),
Lx Ly
®sf(x,y) =
1 -(x-h)(Lx -x) a0 Т (y -h)(Ly -y).
Lx -2h' A ' " Ly -2h'J y
x У
After (14) is applied, B(Y) satisfies the boundary
L
У
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conditions on external borders (16) - (17), i.e.
91(LX - x) + 92x
B( T)
—(Lx - x)=0 Lx
Lx
dB(T)
Ж
-L(l -y)=0 L y
y
= 0 .
The basic functions used are power polynomials, trigonometric polynomials and Legendre polynomials. The Gauss formula with 16 knots was used for evaluation of integrals in the Galerkin method.
The stream lines, temperature field and vorticity field for
Lx = Ly = L = 1, h = 0.05, Ra = 103, Pr = 0.7, ^sf = 10 , 9j = 0.5 , 92 = -0.5, asf = 1, T = 5 are given in figures 1, 2; 3, 4, and 5, 6 for different time respectively.
The results of numerical experiment well correspond to those obtained by the other authors [3].
0.2 0.4 0.6 0.8
Fig. 3. Stream Lines t = 0.02
1.0
0.8
0.6
0.4.1
0.2
0.0
0.2 0.4 0.6 0.8
Fig. 4. Stream Lines t = 3
0.0 0.2 0.4 0.6 0.8 1.0
Fig. 5. Temperature Field t = 0.02
Fig. 6. Temperature Field t = 3
Fig. 7. Vorticity Field t = 0.02
0.2 0.4 0.6 0.8
Fig. 8. Vorticity Field t = 3
VI. Conclusion
The natural convection in an enclosed region with the presence of local heat is investigated. The solution structures of unknown function were built by means of the R-functions method, and the Galerkin method was used for the approximate undefined components. Thus, the stream function and the temperature were represented in analytical
way.
The algorithm for solving the problem of mathematical modeling and numerical analysis of non-stationary natural convection in an enclosed region based on the R-functions method and the Galerkin method is used. The advantage of the suggested algorithm is that it does not have to be modified for different geometries of the regions being reviewed which illustrates the scientific innovation of the results obtained. As a result, the approximate solution for
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such streams investigation problems is obtained in the nonclassic geometry field.
The methods developed for analysis of natural convection in an enclosed region are easy to use for the program algorithms and are more versatile than those used at the present time, as one only needs to change the boundary equation in order to make the transition from one region to another. The obtained results allow us to carry out computational experiments in mathematical modeling of various physical, mechanical, and biological streams.
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