Natural convection around a horizontal circular cylinder under constanttemperature or constant heat flux conditions in an infinite space has been the subjectof numerous investigations in recent years. However, these studies use the inflowoutflowboundary in cylindrical coordinates that gives a sensible error, especially whenthe Rayleigh number is small. This investigation, that enters within the framework ofgeneral study dealing with natural convection from an array of cylinders, states theproblem in cartesian coordinates system, involves the use of a control-volume methodand resolves various apparent redundancies in boundary conditions. This problemwas investigated numerically for laminar case by solving the full vorticity transportequation together with the stream function and energy equations. Results are obtainedfor 101 local and mean Nusselt number, velocities and temperature distribution are clarifiedin detail. The numerical approach presented here appears to be sufficiently versatileto permit computation of a vertical array of cylinders.
Two-dimensional laminar natural convection from horizontal cylinders under steady-state conditions has been extensively investigated numerically and experimentally. There is a growing demand for a better understanding of this phenomenon in areas like heat exchangers, air heating systems for solar dryers, utilisation of natural circulation for energy storage systems and passive solar heating among others. The investigations reported earlier used different kinds of numerical methodologies. However, the principal results indicate that at small Rayleigh numbers, the cylinder behaves like a line heat source. Asymptotic matching solutions have been obtained at low Rayleigh numbers by Nakai and Okazaki [1] where an inner conduction dominated region is matched to an outer region governed mainly by convection. For moderately large Rayleigh numbers, the flow is laminar and forms a boundary layer around the cylinder. The assumptions usually made are that the curvature effects and the pressure difference across the boundary layer are negligible. Under these assumptions, the resulting
* Author to whom correspondence should be addressed
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78 Technical Note
simplified boundary layer equations have been solved using a variety of techniques like the integral methods, or Blasius or Gortler expansions [2].
In order to obtain solutions over a wide range of Rayleigh numbers from 10°-107, where neither asymptotic matching techniques nor boundary layer assumptions are accurate, Kuehn and Goldstein [3] have solved the complete Navier-Stokes and energy equations using an elliptic numerical procedure. They used a finite difference scheme and they adopted the inflow and outflow boundary conditions at the artificially placed outer boundary. But the assumption that the change of inflow to outflow at the outer boundary is near 150 ° has not been adopted by most investigators because recent numerical tests indicate that the position of this demarcation line between inflow and outflow vary with Rayleigh number [4].
Just after Kuehn and Goldstein's work, Fujii et al. [5] investigated numerically and experimentally the natural convection heat transfer about a horizontal wire by using two computational domains: one is a domain near the cylinder prescribed by a cylindrical coordinate system and the other is one outside this domain prescribed by a rectangular coordinate system. In order to check the validity of the boundary conditions, they used two outer boundary conditions : a solid boundary condition and an open boundary condition. But, they did not present the numerical results for relatively large Rayleigh numbers.
Recently, Saitoh et al. [6] presented a high accuracy bench mark solutions for the same problem. They adopted a high accuracy forth order finite difference method and a logarithmic coordinate transformation. They used a solid boundary condition at 1000 D-20,000 D in order to check the validity of the inflow-outflow condition at the outer boundary condition used by Kuehn and Goldstein. As a consequence, the inflow-outflow condition gives a significant discrepancy compared with the solid boundary condition, especially when the Rayleigh number is small. But the principal problem of the bench mark solution is the large computer running time.
Mainly motivated by the above circumstances, we aim to find the numerical solution for the natural convection around a horizontal isothermal cylinder with a minimum number of computational points over a wide range of Rayleigh numbers from 10~-106, where neither inflow-outflow nor solid boundary conditions are used. For this, the coupled elliptic transport equations are solved numerically using the control-volume scheme. An open boundary condition is adopted.
2. MODELISATION OF THE FLOW
Using the stream function-vorticity approach and assuming constant properties except for the density in the body force term, the coupled elliptic transport equations for qJ, co and T can be written in Cartesian coordinates, in dimensionless form and for steady state 2-D. laminar natural convection, as follows :
Since the flow is symmetric about a vertical plane passing through the axis of the cylinder, only the half-plane need be considered (Fig. 1). Thus, the solution domain is bounded by the half-cylinder surface, the two lines of symmetry, the outer boundary, and the inflow and outflow boundaries. On one hand, the Cartesian coordinate where chosen for the present problem as it is found to be flexible and capable of modelling any pitch-to-diameter ratio for the array of cylinders considered. On the other hand, this choice allows to avoid the use of both inflow-outflow and solid boundary conditions.
The boundary conditions are as follows : - - O n the symmetry lines :
~T* OV* 09* = ~ * . . . . U * = 0
~ x * ~ x *
- - O n the isothermal cylinder surface :
T * = 1 , U * = V * = W * = 0
A major difficulty of vorticity-stream function formulation is to obtain the vorticity at the cylinder wall. In the present study, we derive the boundary condition for o9" by considering the following steps. As shown in Fig. 2, we obtain wall vorticities 09" and COs* in terms of ~ki. ~a, ¢i,j ~ and An* by expanding the stream function near the wall as a Taylor series and then using the continuity and no- slip condition.
2U,'* -~'*) 09 * -
A / , / . 2
n* denotes the normal direction from the wall. ~b* is the stream function value at the surface cylinder and ~k* is the value at a short distance An* in to the fluid.
Interpolating along the wall, we determine the wall vorticity value 09".,,j in terms of ~bi, j ~ , ~k*+ ~,.j, Ax*, Ay* and 0.
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~i+l,i
!1
Fig. 2. Wall vorticity.
--Inflow-outflow boundaries : It is assumed that if the inflow and outflow boundaries are set sufficiently far away from the
cylinder, then the velocity components in the x-direction are negligible, implying that in the far field, all the flow must be in the direction parallel to the symmetry lines. This is not strictly true for the outflow section, unless it is considered at a very large distance from the cylinder centre. The approach here is to somehow neglect the details of flow further downstream and obtain realistic answers upstream [7]. So, this assumption does not have a strong effect on the flow pattern and heat transfer over the cylinder. * Inflow boundary:
The fluid ambient is assumed to enter the solution domain at ambient temperature. Thus,
~V* ~ , * &o* T* = U* - - 0 ~y* Oy* Oy*
* Outflow boundary : The fluid is assumed to leave the solution domain without any vertical temperature gradient. Thus,
OT* OV* t ~ * &o* = U * - - - 0 ~y* oy* ~y* ey*
---Outer boundary : The commonly used outer boundary condition is to assume that the temperature gradient normal
to this pseudo-boundary is zero. This means that at location (x ~ ~ ) , the convection is the prevailing heat transfer mode. This obviously requires that the velocities are sufficiently large, a condition that is probably satisfied within the scope of the present study, which is restricted to steady-state convec- tion. The boundary condition for ~ is determined by considering the x derivative of the stream function to be set equal to zero. Vorticity is set equal to zero by considering the entrained flow to be irrotational. The truncated values of the infinity location in the x-direction can be increased until the
Technical Note 81
variables become insensitive to further variation. In this study, for the low Rayleigh numbers, the infinity value in the x-direction is equal to about 4D from the lines of symmetry. As the Rayleigh number increases, the outer boundary is moved inward.
~U* dT* ~b* V* . . . . (o* = 0
~x* t3x* ~x*
3. SOLUTION PROCEDURE
The coupled elliptic transport equations for e~, W and T are solved numerically using the control- volume scheme developed by Patankar [9]. The most attractive feature of this approach is that the conservation of energy and vorticity are exactly satisfied, and thus, even a coarse grid formulation can yield physically realistic results. Also, the mathematical formulations involved in this method do not obscure the physical meaning of each term in the governing equations. First, the calculation domain is divided into a number of non-overlapping control-volumes such that there is one control- volume surrounding each grid point. Then, instead of using standard Taylor series expansions, each governing differential equation is discretized by integrating it over each control-volume by assuming piecewise linear profiles for the variation of a dependent variable in space. A hybrid technique of Patankar and Spalding [8] is used to maintain stability at large Rayleigh numbers. As a result, the governing differential equations get transformed into simple algebraic equations. Such discretization equations, one for each control-volume, are then solved by using the simultaneous overrelaxation method (S.O.R). Finally, we use in the present study an unequally spaced grids that provide good results with a minimum number of computational points.
4. NUMERICAL RESULTS
The flow and temperature fields and heat transfer results were obtained for Pr = 0.7 and Rayleigh numbers ranging from Ra = 10L106. Several computed results for isotherms and streamlines are shown in Figs 3 4 . As indicated in the previous studies, the boundary layer becomes thin with increasing Rayleigh numbers. At Ra = 104, the boundary layer thickness is approximately equal to the cylinder radius. The assumption of negligible curvature effect is not valid at this Rayleigh number, so the solution of boundary layer equations does not give valid results here. However, at Ra = 10 6,
the boundary-layer thickness has become much thinner than the cylinder radius, so the boundary- layer model should give fairly accurate results. We note that at high Rayleigh numbers, the majority of the flow comes from the sides instead of bottom, but the trend is reversed for smaller Rayleigh numbers. At Ra = 106, the streamlines take the form of the cylinder. From Fig. 5, we note that the temperature field indicates the presence of a thermal plume above the cylinder where the boundary layer separates. In fact, the isotherms move upward at the top of the cylinder (region of the plume) while continue adhering to it at the bottom. The velocity vectors in the mid-plane have been plotted in Fig. 6 at the representative Rayleigh numbers to display the role of natural convection in modifying the flowfield in the domain considered. Figures 7 and 8 show the vertical (or angular) velocity and temperature distributions at 0 = 90 ° for different Rayleigh numbers. An increase in the Rayleigh number caused the maximum of vertical velocities to shift toward the cylinder surface, i.e. toward Dr* = 0. From Fig. 8, we note that the thermal boundary layer thickness decreases as Ra increases. Velocity and temperature distributions at Ra = 105 are given in Figs 9-10. For 0 = 90 °, the vertical (or angular) velocity distributions are very similar to what boundary layer solutions predict. Near the cylinder, the centre line velocity is the largest velocity in the plume. The horizontal (or radial) velocities are fairly small and negative. This indicates that the flow is moving toward the cylinder. At 0 = 90 °, the horizontal (or radial) temperature profiles are similar to those of boundary layer region, but at 0 = 180 °, the plume alters the temperature distributions.
In order to verify the accuracy of the method of solution and the numerical computations, the average values of N u m w e r e compared with results of Kuehn-Goldstein and Saitoh et al., as can be seen in Table 1. This comparison gives strong evidence that the present numerical solutions are close to those of precedent authors.
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(b)
(c)
Fig. 3. Isotherms: (a) Ra = 102; (b) Ra = 104 ; c) Ra = 106.
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Fig. 4. Streamlines: (a) Ra = 102; (b) Ra = 104; (c) Ra = ]06.
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|
Fig. 5. Temperature field for Ra = 10 3.
I 0.90+ 0.83 to 0.90 0.75 to 0.83 0.68 to 0.75 0.61 to 0.68 0.54 to 0.61 0.46 to 0.54 0.39 to 0.46 0.32 to 0.39 0.26 to 0.32
i 0.17 to 0.25 0.10 to 0.17
5. CONCLUSION
Numerical solutions to the Navie~Stokes and energy equations for laminar natural convection around a horizontal isothermal cylinder have been obtained over a wide range of Rayleigh numbers and using an elliptic procedure and a control-volume scheme. The agreement between our results and those of the studies reported earlier is found to be good. So, we can apply this model to an array of vertical cylinders.
D Dr* g Nu Pr x, y X* y* Ra T Ta Tw
NOMENCLATURE
cylinder diameter Dimensionless radial distance from cylinder surface gravitational acceleration Nusselt number Prandtl number Cartesian coordinates dimensionless x coordinate (x/D Ra °~) dimensionless y coordinate (y/D Ra °'~) Rayleigh number (based on D) local temperature ambient temperature temperature of cylinder surface
. . . . Ra,., ,102
tt~xxxxxx~,"'"" YSN'M-~-~-"qX-',, i x x r ~
!!IIIi . . . . / ~ / t ; / / r r r t
, , . / t t t t t t r r r r ~
t t t t t t l f l l W t I t r
r I ~ I ~ t I l l ~ I I I r r I I I I r r I r I [ I l | Z =
dimensionless temperature ( T - Tw)/( T - T~) horizontal velocity dimensionless velocity ( UD/ 2 Ra °25) vertical velocity dimensionless velocity (VD/2 Ra °25)
Greek symbols 2 0
4,
60
(D*
thermal diffusivity angular coordinate (0 < 0 < n) coefficient of thermal expansion stream function dimensionless stream function (if/2 Ra °75) vorticity dimensionless vorticity (~D 2 Ra°25)/)~
Subscripts i, j mesh point
REFERENCES
1. Nakai, S. and Okazaki, T., Heat transfer from a horizontal circular wire at small Reynolds and Grashof numbers. International Journal of Heat and Mass Transfer, 1975, 18, 387-396.
2. Wilks, G., External natural convection about two-dimensional bodies with constant heat flux. International Journal of Heat and Mass Transfer, 1972, 15, 351-354.
3. Kuehn, T. H. and Goldstein, R. J., Numerical solution for laminar natural convection about a horizontal circular cylinder. International Journal of Heat and Mass Transfer, 1980, 23, 971-979.
4. Wang, P. et al., Numerical computation of the natural convection flow about a horizontal cylinder using splines. Numerical Heat Transfer, Part A, 1990, 17, 191-215.
5. Fujii, T. et al., Theoretical and experimental study on free convection around a horizontal wire. J. Soc. Mech. Engrs, 1982, 48(431), 1312-1320.
6. Saitoh, T. et al., Bench mark solutions to natural convection. International Journal of Heat and Mass Transfer, 1993, 36, 1251-1259.
7. Roache, P. J., Computational Fluid Dynamics, Hermosa, Albuquerque, 19??, pp. 152- 161.
8. Patankar, S. V. and Spalding, D. B., A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. International Journal of Heat and Mass Transfer, 1972, 15, 1787-1806.
9. Patankar, S. V., Numerical Heat Transfer and Fluid Flow, Pergamon Press, 1980.
