Conduction Heat Transfer Calculations Using Structured and
Unstructured Grids
Zhang Min 1，John C. Chai2，Xu Bin 1，Liu Shaopei 1，Zhang Junbo 1
2
School of Power Eng.，Nanjing University of Science & Technology，Nanjing (210094) School of Mechanical and Aero spacing Eng.，Nanyang Technological University，Singapore
(639798)
E-mail：mz2455@163.com
1
Abstract
Conduction heat transfer problems were solved using a cell-based finite volume method. Solutions were obtained using structured and unstructured grids. When available, results were compared with exact solutions. Although both structured and unstructured grids produce accurate solutions, structured grid solutions are more accurate. The secondary diffusion term is important if accurate solutions were to be obtained. Keywords：Structured grid，unstructured grid，heat conduction
1. Introduction
Over the last three decades, there have been significant advances in numerical procedures for complex diffusion problems. Both finite-volume and finite-element formulations have been presented by various researchers. Initially, procedures are formulated for structured Cartesian and cylindrical computational grids [1]. These procedures were extended to account for complex geometries using body-fitted computational grids [2]. Recently, procedures for unstructured grids were used [3-4]. The objective of this article is to present the results for an unstructured grid procedure applied to a scalar diffusion equation and examine the importance of the secondary diffusion term.
The remainder of this article is divided into four sections. The governing equation and its boundary conditions are presented in the next section. The discretization of the governing equation is then presented in the next section. This is followed by the presentation of two example problems and, finally, some concluding remarks.
2. Governing Equation and Boundary Conditions
The steady-state form of the differential equation representing the conservation of a scalar variable φ can be written as
∂∂xi
variable φ.
⎛∂φ⎜⎜Γ∂x
i⎝⎞⎟⎟+Sφ=0 ⎠
(1)
where Sφ is the net generation rate per unit volume and Γ is the diffusion coefficient appropriate for the
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•PP• (a) quadrilateral (b) triangular Figure 1. Two possible two-dimensional control volumes
In diffusion problems, three kinds of boundary conditions are commonly encountered. These are (1) the given φ, (2) the given normal flux and (3) the given normal flux and boundary φ relation conditions. These conditions can be written as
φB=φgiven
qB=qgiven qB=qc+qφφB
(2a) (2b) (2c)
3. Formulation of the Discretisation Equation
Since the discretization equations for Cartesian, cylindrical and body-fitted coordinates are available in Refs. [1] and [2], only the discretisation equation for unstructured grids is presented here. The governing equation (Eq. 1) can be written as
d+S=0
Integrating Eq. (3) over a typical control volume shown in Fig. 1 gives
nb
(3)
∑Di+Sp∆Vp=0
i=1
(4)
In Eq. (4), nb represents the number of faces associated with cell P. The total diffusion term Di is
Di=Γi∇φ•A
the sum of the primary and the secondary diffusion terms as
→→
(5)
where Γi is an appropriate diffusion coefficient at face i. The diffusion term Di can also be expressed as
Di=Dp,i+Ds,i
(6)
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bDi
E•Dp,ibiaˆt,ieˆs,ieE•ˆin•PDs,i(a)
•Pai (b)
Figure 2. (a) Diffusion terms, and (b) unit vectors
The subscripts p and s represent primary and secondary, respectively. When the computational grids are orthogonal, the secondary diffusion term becomes zero. For the control volume arrangement shown in Fig. 2a, the primary and secondary diffusion terms can be written as
Dp,i=Γi
(φE−φP)Ai•Ai
→dsi
ˆs,iAi•e
→
→
→→
(7a)
Ds,i=Γi
(φb−φa)Ai•Ai
ˆs,i•eˆt,i e→Ai
ˆs,iAi•e
(7b)
ˆs,i and eˆt,i are shown in Fig. 2b. The term dsi is the distance between point P and point E. where e
With the exception of Γi, all the terms in the primary diffusion term (Eq. 7a) can be calculated. The area
→
ˆs,i and distance dsi are geometrical quantities. The neighboring cell value φE is Ai, unit vector e
obtained from the previous iteration or from the initial guess. The term (φE
ˆs,i −φP)dsi is the e
component of the derivative. For the face i shown in Fig. 2b, the following can be written
(φE−φP)→
ˆs,i =(∇φ)ave,i•e
dsi
→
(8)
where (∇φ)ave,i is the average of the derivatives at the two adjacent cells, namely cell P and cell E. Using Eq. (8), Eq. (7a) can be written as
→→
→
ˆs,iDp,i=Γi(∇φ)ave,i•e
→
Ai•Ai
(9)
ˆs,iAi•e
The secondary diffusion is given by Eq. (7b). The values of the dependent variable at the vertices are
needed in the evaluation of the secondary diffusion. These values are interpolated from the most current cell values. Figure 3 shows two possible cell-node arrangements. The following can be observed (a) the values at the vertices depend on the shapes of the surrounding cells, and (b) the vertices can lie on a point of discontinuity as seen in Fig. 3a. In three-dimensional computations, in addition to the above
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ˆt,i. Therefore, an alternate approach is used in the complications, there is no unique tangent vector e
evaluation of the secondary diffusion. From Eq. (6), the secondary diffusion can be written as the
difference between the total diffusion and the primary diffusion as
Ds,i=Di−Dp,i
(10)
Using the expressions for the total diffusion (Eq. 5) and the primary diffusion (Eq. 9), Eq. (10) can be written as
Ds,i
→→⎤⎡
→→→Γ⎢A•Ai⎥
ˆs,ii=i⎢(∇φ)ave,i•Ai−(∇φ)ave,i•e⎥dsi →dsi
ˆs,i⎥⎢Ai•e⎦⎣
(11)
Note that the secondary diffusion (Eq. 11) is identically zero when the grid lines are orthogonal. In this
article, the primary diffusion is treated implicitly. Substituting Eq. (7a) and Eq. (6) into Eq. (4), the discretization equation for the control volume P is given as
∑Bi(φi−φP)+∑Ds,i+SP∆VP=0
i=1
i=1nb
nbnb
(12)
Equation (12) can be rewritten as
aPφP=
where
∑aiφi+b
i=1
(13)
ai=Bi=
nb
ΓiAi•Ai
→dsi
ˆs,iAi•e
→→
(14a)
b=
∑Ds,i+SP∆VP
i=1
(14b)
aP=
∑Bi
i=1
nb
(14c)
T1bXT2XXXT1•••aX••••a••T2X••b•••a•bPhysical Boundary
Internal node
(a) (b)
Figure 3. Two possible cell-node arrangements
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The discretization equation for a control volume is presented in this section. The evaluations of Γiand (∇φ)ave,i are given in [5] and are not repeated here.
→
dsi
4. Results
Heat conduction in a rectangular block with constant properties
Problem Specification. Figure 4 shows the schematic of the problem considered. The governing equation with constant properties and no heat generation is
∂⎛∂T⎜k∂x⎝∂x
⎞∂⎛∂T⎟+⎜k⎠∂y⎝∂y⎞⎟=0 ⎠
(15)
The temperatures at all boundaries are given by
T=x+y+xy
(16)
Discussion of Results. This problem is chosen because the exact solution of the problem is the same as the boundary conditions. Figure 4 shows the three different computational grids used in this study. The temperature contours obtained from the exact solutions (Eq. 16) and the numerical solutions also shown in Fig. 4. Two hundred control volumes are used in all computations. The maximum errors for the Cartesian, triangular and hybrid grids are 4.8x10-5 %, 1.67% and 1.19%, respectively. These errors are defined as
Error=E=
Tnum−TexactTmax−Tmin
×100%
(17)
In Eq. (17), Tmax and Tmin are the maximum and minimum temperatures calculated from Eq. (16). It can be seen from the results that the procedure is capable of modeling steady state conduction with constant properties and without heat generation. It is also interesting to note that the Cartesian grid produces the exact solution (machine zero error).
The effect of the secondary diffusion Ds,i is also examined. The maximum errors obtained using triangular grids with and without the secondary diffusion term are 1.67% and 2.42%, respectively. Although these errors are small, the size of error increases by 40% when the secondary diffusion term is not included.
Exact solutions
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Numerical solutions
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24.524.524.51.751.51.25151.32.51.751.51.25151.1.751.51.25151.4443.53.53.5352.32.5yyy2220.7510.750.75110.50.2500.50.2500.50.50.25050.50.00.5100.5100.51xxx
(a)
(b)
(c)
Figure 4 Computational grids and temperature contours: (a) Cartesian, (b) triangular and (c) hybrid
Heat conduction in Irregular Geometry
Problem Specification. Figure 5 shows a solid with some rectangular cut-outs. The governing equation is same as the equation (15). The bottom boundary exchanges heat with the surroundings by convection and radiation. The heat flux at the bottom boundary is given by
44
qB=50(T∞−TB)+2×10−8(T∞−TB)
(18)
where T∞ and TB stand for the temperatures of the surroundings and the boundary, respectively. The numerical values are
Tw1=400,Tw2=500,T∞=300,k=12
(19)
Because of the radiation term in Eq. (19), all temperatures will be taken to be absolute temperatures.
adiabaticadiabaticTw10.6Tw21.6Tw20.50.9convection and radiation2.0Tw21.01.00.41.6 Figure 5. A complex geometry
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33
2.52.5
22
1.51.5
Y1
Y0
0.5
1
1.5
2
2.5
3
1
0.50.5
00
X
00.511.5
X
22.53
(a)
3450
(b)
24001.5Y1390400.5370350000.511.5X
22.549002.5470430Figure 6. Computational grids and temperature distributions: (a) structured, (b) unstructured, and (c) temperature
distributions
Discussion of Results. This problem was solved using both structured (Fig. 6a) and unstructured (Fig. 6b) grids. The total numbers of control volumes used in the structured and unstructured grids are 656 and 644, respectively. Figure 6c shows the temperature distributions predicted by the structured and the unstructured grids. It can be seen that both solutions are identical for all practical purposes.
4103
(c)
5. Concluding Remarks
An unstructured grid procedure is outlined in this article. Two two-dimensional problems are used to study the accuracy of the procedure. For the problem tested, it can be concluded that the secondary diffusion term is important if accurate solutions were to be obtained. The results also indicated that the both quadrilateral and triangular grids could be used and reasonable solutions can be obtained.
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References
1. Patankar, S. V., Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing, New York, 1981.
2. Karki, K. C., and Patankar, S. V., Pressure Based Calculation Procedure for Viscous Flows at All Speeds in Arbitrary
Configurations, AIAA J., Vol. 27, pp. 1167 – 1174, 1989.
3. Barth, T. J., Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations,
Special Course on Unstructured Grid Methods for Advection Dominated Flows, AGARD Report 787, 1992
4. Mathur S. R., and Murthy J. Y., A Pressure-Based Method for Unstructured Meshes, Numerical Heat Transfer, Part
B., Vol. 31, pp. 195 – 215, 1997.
5. Zhang, M., Modeling of Radiative Heat Transfer and Diffusion Processes using Unstructured Grid, Ph.D.
Dissertation, Tennessee Technological University, USA, 2000.
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