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Investigation of laminar flow through solution of inverse problem for heat conduction equation
Mechanics
Research Communications,
Pergamon
Vol. 28. No. 6, pp. 623428.2001
Copyright
0 2001 Elsewer
F’rmted in the USA.
All
Science Ltd
rights reserved
0093413/01/&see
front matter
PII: SOO93-6413(02)00214-S
INVESTIGATION OF LAMINAR FLOW THROUGH SOLUTION OF INVERSE PROBLEM FOR HEAT CONDUCTION EQUATION M. Cialkowski’, A. Frackowiak*, J. A. Kolodziej** *Institute of Thermal Engineering, **Institute of Applied Mechanics, Poznan University of Technology, 60-965 Poznan, Poland
Introduction The paper presents a solution of a plane flow of viscous fluid as an inverse problem for parabolic equation (heat conduction equation) with an unknown harmonic source function. This firnction is derived from a known boundary condition. Distribution of stream function Yis determined by its approximation within the whole domain by means of polyharmonic functions. Descrintion of method A plane flow of viscous fluid is governed by the following system of equations - continuity equation
(1) -
motion equation (without mass forces)
av
-+uat
av+v-=av ar
ay
lap ---+vAv, Pav
v=v,
=const
(3)
These equations form a complete system with unknowns u(x,y, t), v(x,y,t) of velocity vector and an unknown of pressure function. The system of equations (l-3) is non-linear. Velocity distribution II and v is assumed at the initial time in region fi and on its boundary X? for t > 0. The system of equations (l-3) can be reduced to the system of equations with two unknowns by eliminating the pressure function. Differentiating equation (2) with respect to unknown y and equation (3) with respect to unknown x and then subtracting the sides we get
623
624
M. CIALKOWSKI, A. FRqCKOWlAK and J. A. KOLODZIEJ
Note, that due to continuity equation (1) the fourth component of the sum in equation (4) vanishes, and all the other parentheses are a function of vorticity. The vorticity function w is defined as i (3=rotv=
k
j
2-d
_c
U
w
(5)
a~* az v
which for a two-dimensional case u = IC(X,Y,I), v = V(X,y,t), w = 0 has the following form:
(6) After we introduce vorticity t%nction (6) into equation (4) we get
ati _+udw+vdw=v*~ at
dx
ay
’
(XtY)ER
(7)
On the other hand after we introduce vorticity function (6) into continuity equation (1) we get: - as a result of differentiating equation (1) with respect to x and introducing mnction w Au+?=0 4,
’
@,Y) E Q
hence -A-‘@+H&,y,l) au as a result of differentiating equation (1) with respect toy and introducing function o u =
-
Av-?%O ax
’
(X,Y) E Q
hence v= -A-'$f+H,(x,y,t) The system of equations (7)-(9) has been solved by expanding functions w,u and v into Taylor series with respect to time (which leads to an explicit differential scheme) and by acting with the help of operator A-’ on equations (8) and (9). Numerical results are presented in the work by authors [I].
INVERSE HEAT CONDUCTION SOLUTIONS
625
Equations (7)~(9) form a system of 3 non-linear equations with unknowns u(x,y,t), v(x,y,t), w(r,y,t) The common characteristic of equations (7)-(9) is the occurrence of Laplace’s operator A as an operator of the highest order of differentiation. Acting with the help of the operator A-’ on the system of equations (7)~(9) will result in a harmonic finction of a structure HW.0
= ~[A,(r).F,(x,~)+B,(t)-G,(x,y)] n=O
(10)
where functions (Fn(x, y),G,(x, y) }form a complete set of harmonic fbnctions. Because differential mesh was introduced, the sum (10) will be finite. Analoev between flow of viscous fluid and heat conduction Replacing functions w,u,v in equation (7) with the relation of these finctions with the stream tbnction AI/=-W
,
u,*
, ay
v=-- av 8x
(11)
we get the following equation (12) or the system of equations (13)
Acting with the help of the operator A-’ [l, 21 on equation (13) we get a parabolic equation with an unknown source function H, av -=vAy-Q+H, at
(14)
where source tknctions @ and H, are defined by formulas
@=A’
(15)
Hv(wv)= ~[C,(~).F,(X,Y)+D,(~).G,(X,Y)I, n=o
Defining an unknown function of the source H, in a parabolic equation (14) is, indeed, an inverse problem. Function H, , which is a harmonic function, will be defined for a given boundary condition for velocities u and v.
626
M. CIALKOWSKI, A. FRA&KOWIAK and J. A. KOLODZIEJ
The system of equations (l)-(3) was reduced to the parabolic equation with a source function, which leads to a simple analogy with the problem of heat conduction. Temperature corresponds with a stream fiurction. A stream function difference equals the stream of fluid volume whereas heat flux corresponds with temperature difference. Aleorithm of solution of eauation (141 Replacing a derivative with respect to time in equation (14) with its corresponding forward differential quotient of a step AIwe obtain an explicit scheme with an unknown stream function H,(x, y, t - At), which is determined on the basis of a boundary condition for functions u and v (which is equivalent to the knowledge of the derivative dY / &I ). Knowing the function V on the boundary and the initial time we can now to solve equation (14). Function Y and non-linear source function Q = f(Y) (15) were determined by polyharmonic fkrctions [ 1.21. Coeffkients of function expansion \Y series according to polyharmonic functions were determined knowing the function ‘9 in the nodes inside the domain and in the boundary nodes of the domain - Fig.2 at the initial time; total number of nodes is 18 1.
Fig. 1. Boundary conditionsfor flow in the cavity
Fig. 2. Nodes of calculation mesh
Fig. 3. Flow in the square cavity. Stream function: a) Re = 100, b) Re = 400
INVERSE HEAT CONDUCTION
527
b)
a) rm
SOLUTIONS
1
Rys. 4 Comparisonof velocity componentsin tbe cross-section x = 0,5: a) Re = 100,b) Re = 400
a) 0401
b) 040,
Fig. 5. Comparisonof velocity componentsin the cross setion y = 0,5: a) Re = 100,b) Re = 400
Conclusions The applied method of inverse operators together with polyharmonic functions gave good results as compared with other works [3,4]. In the work the solution was approximated in the whole domain. Coefficients in finction expansions Y into a series according to polyharmonic functions were determined knowing the function Y at the initial time in 181 nodes. In work [4] the number of nodes was 900 whereas in work [3] the number was 16900. Acknowledgements This work was supported by the Polish Committee of Scientific Research (KBN) through a grant from Poznan University of Technology in Grant DPB 32/337/2000-32/337/2001 related to the first and the third authors. The second author carried out this work under grant KBN 8TlOB06820.
628
M. CIALKOWSKI. A. FR&CKOWIAK and J. A. KOLODZIEJ
References
1. M. Cialkowski, A. Fqckowiak, Heat functions (T-functions) and their application to solution of heat transfer and mechanics problems (in Polish). Wydawnictwo Politechniki Poznatiskiej, Poznan 2000. 2. A. Frackowiak, Applications of hest functions to solution of some problems of heat transfer and mechanics (in Polish). Ph D. Thesis., Poznar’t2000. 3. U. Ghia, K. N. Ghia, C. Shin, Journal of Computational Physics, 48, pp. 387-411, (1982). 4. Z. Kosma, Determination of viscous fluid flow (in Polish). Wydawnictwo Politechniki Radomskiej, Radom 1999, p. 192. 5. W. J. Prosnak, Fluid mechanics (in Polish), vol. 1, PWN, Warszawa 1970.
Research Communications,
Pergamon
Vol. 28. No. 6, pp. 623428.2001
Copyright
0 2001 Elsewer
F’rmted in the USA.
All
Science Ltd
rights reserved
0093413/01/&see
front matter
PII: SOO93-6413(02)00214-S
INVESTIGATION OF LAMINAR FLOW THROUGH SOLUTION OF INVERSE PROBLEM FOR HEAT CONDUCTION EQUATION M. Cialkowski’, A. Frackowiak*, J. A. Kolodziej** *Institute of Thermal Engineering, **Institute of Applied Mechanics, Poznan University of Technology, 60-965 Poznan, Poland
Introduction The paper presents a solution of a plane flow of viscous fluid as an inverse problem for parabolic equation (heat conduction equation) with an unknown harmonic source function. This firnction is derived from a known boundary condition. Distribution of stream function Yis determined by its approximation within the whole domain by means of polyharmonic functions. Descrintion of method A plane flow of viscous fluid is governed by the following system of equations - continuity equation
(1) -
motion equation (without mass forces)
av
-+uat
av+v-=av ar
ay
lap ---+vAv, Pav
v=v,
=const
(3)
These equations form a complete system with unknowns u(x,y, t), v(x,y,t) of velocity vector and an unknown of pressure function. The system of equations (l-3) is non-linear. Velocity distribution II and v is assumed at the initial time in region fi and on its boundary X? for t > 0. The system of equations (l-3) can be reduced to the system of equations with two unknowns by eliminating the pressure function. Differentiating equation (2) with respect to unknown y and equation (3) with respect to unknown x and then subtracting the sides we get
623
624
M. CIALKOWSKI, A. FRqCKOWlAK and J. A. KOLODZIEJ
Note, that due to continuity equation (1) the fourth component of the sum in equation (4) vanishes, and all the other parentheses are a function of vorticity. The vorticity function w is defined as i (3=rotv=
k
j
2-d
_c
U
w
(5)
a~* az v
which for a two-dimensional case u = IC(X,Y,I), v = V(X,y,t), w = 0 has the following form:
(6) After we introduce vorticity t%nction (6) into equation (4) we get
ati _+udw+vdw=v*~ at
dx
ay
’
(XtY)ER
(7)
On the other hand after we introduce vorticity function (6) into continuity equation (1) we get: - as a result of differentiating equation (1) with respect to x and introducing mnction w Au+?=0 4,
’
@,Y) E Q
hence -A-‘@+H&,y,l) au as a result of differentiating equation (1) with respect toy and introducing function o u =
-
Av-?%O ax
’
(X,Y) E Q
hence v= -A-'$f+H,(x,y,t) The system of equations (7)-(9) has been solved by expanding functions w,u and v into Taylor series with respect to time (which leads to an explicit differential scheme) and by acting with the help of operator A-’ on equations (8) and (9). Numerical results are presented in the work by authors [I].
INVERSE HEAT CONDUCTION SOLUTIONS
625
Equations (7)~(9) form a system of 3 non-linear equations with unknowns u(x,y,t), v(x,y,t), w(r,y,t) The common characteristic of equations (7)-(9) is the occurrence of Laplace’s operator A as an operator of the highest order of differentiation. Acting with the help of the operator A-’ on the system of equations (7)~(9) will result in a harmonic finction of a structure HW.0
= ~[A,(r).F,(x,~)+B,(t)-G,(x,y)] n=O
(10)
where functions (Fn(x, y),G,(x, y) }form a complete set of harmonic fbnctions. Because differential mesh was introduced, the sum (10) will be finite. Analoev between flow of viscous fluid and heat conduction Replacing functions w,u,v in equation (7) with the relation of these finctions with the stream tbnction AI/=-W
,
u,*
, ay
v=-- av 8x
(11)
we get the following equation (12) or the system of equations (13)
Acting with the help of the operator A-’ [l, 21 on equation (13) we get a parabolic equation with an unknown source function H, av -=vAy-Q+H, at
(14)
where source tknctions @ and H, are defined by formulas
@=A’
(15)
Hv(wv)= ~[C,(~).F,(X,Y)+D,(~).G,(X,Y)I, n=o
Defining an unknown function of the source H, in a parabolic equation (14) is, indeed, an inverse problem. Function H, , which is a harmonic function, will be defined for a given boundary condition for velocities u and v.
626
M. CIALKOWSKI, A. FRA&KOWIAK and J. A. KOLODZIEJ
The system of equations (l)-(3) was reduced to the parabolic equation with a source function, which leads to a simple analogy with the problem of heat conduction. Temperature corresponds with a stream fiurction. A stream function difference equals the stream of fluid volume whereas heat flux corresponds with temperature difference. Aleorithm of solution of eauation (141 Replacing a derivative with respect to time in equation (14) with its corresponding forward differential quotient of a step AIwe obtain an explicit scheme with an unknown stream function H,(x, y, t - At), which is determined on the basis of a boundary condition for functions u and v (which is equivalent to the knowledge of the derivative dY / &I ). Knowing the function V on the boundary and the initial time we can now to solve equation (14). Function Y and non-linear source function Q = f(Y) (15) were determined by polyharmonic fkrctions [ 1.21. Coeffkients of function expansion \Y series according to polyharmonic functions were determined knowing the function ‘9 in the nodes inside the domain and in the boundary nodes of the domain - Fig.2 at the initial time; total number of nodes is 18 1.
Fig. 1. Boundary conditionsfor flow in the cavity
Fig. 2. Nodes of calculation mesh
Fig. 3. Flow in the square cavity. Stream function: a) Re = 100, b) Re = 400
INVERSE HEAT CONDUCTION
527
b)
a) rm
SOLUTIONS
1
Rys. 4 Comparisonof velocity componentsin tbe cross-section x = 0,5: a) Re = 100,b) Re = 400
a) 0401
b) 040,
Fig. 5. Comparisonof velocity componentsin the cross setion y = 0,5: a) Re = 100,b) Re = 400
Conclusions The applied method of inverse operators together with polyharmonic functions gave good results as compared with other works [3,4]. In the work the solution was approximated in the whole domain. Coefficients in finction expansions Y into a series according to polyharmonic functions were determined knowing the function Y at the initial time in 181 nodes. In work [4] the number of nodes was 900 whereas in work [3] the number was 16900. Acknowledgements This work was supported by the Polish Committee of Scientific Research (KBN) through a grant from Poznan University of Technology in Grant DPB 32/337/2000-32/337/2001 related to the first and the third authors. The second author carried out this work under grant KBN 8TlOB06820.
628
M. CIALKOWSKI. A. FR&CKOWIAK and J. A. KOLODZIEJ
References
1. M. Cialkowski, A. Fqckowiak, Heat functions (T-functions) and their application to solution of heat transfer and mechanics problems (in Polish). Wydawnictwo Politechniki Poznatiskiej, Poznan 2000. 2. A. Frackowiak, Applications of hest functions to solution of some problems of heat transfer and mechanics (in Polish). Ph D. Thesis., Poznar’t2000. 3. U. Ghia, K. N. Ghia, C. Shin, Journal of Computational Physics, 48, pp. 387-411, (1982). 4. Z. Kosma, Determination of viscous fluid flow (in Polish). Wydawnictwo Politechniki Radomskiej, Radom 1999, p. 192. 5. W. J. Prosnak, Fluid mechanics (in Polish), vol. 1, PWN, Warszawa 1970.