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Numerical Heat Transfer, Part A: Applications: AnInternational Journal of Computation and MethodologyPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/unht20

Natural Convection in Trapezoidal CavitiesMilovan Perić aa Institut für Schiffbau, Lammersieth 90, D-22305 Hamburg, Germany

Version of record first published: 16 Jan 2007.

To cite this article: Milovan Perić (1993): Natural Convection in Trapezoidal Cavities, Numerical Heat Transfer, Part A:Applications: An International Journal of Computation and Methodology, 24:2, 213-219

To link to this article: http://dx.doi.org/10.1080/10407789308902614

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Numerical Heat Tlansrer, Part A, vol. 24, pp. 213-219, 1993

NATURAL CONVECTION I N TRAPEZOIDAL CAVITIES

I Milouan Peric' Ins~itutf i i r Schiffbau, Lamnzcrsieth 90, 0-22305 Hamburg, G e r n ~ o ~ ~ ?

This note was inspired by a poperpublishd recently in this joumol by T. S. Leo I l l . Here, results ofcolcuLyiorts oflho some problem ore presented, which difler both qrlulilolively and qunnritatirdy from those of Lee I l l . By u i n g o series oJsystemruicaUy reJined grids fmm 10 x 10 to 160 X 160 control uolumq s, second-order dircreliialion method, and an eJjieient mulligrid olgorilhm, the convergence of the rerults loword grid independent solulions is demonslmred, and the erron ore prousn to negligibly smoU. Features oJ/lows ore d&cused. and disagreements wilh resub of Lee 111 are onnlyzed.

INTRODUCTION

In the paper by Lee [I], laminar natural convection in a trapezoidal two- dimensional cavity was studied. Figure 1 shows the geometry and the boundary conditions applied. The author performed parameter studies by varying aspect ratio, tilt angle, Rayleigh number, and Prandtl number. However, a closer look at the results reveals that, contrary to expectation, the predicted isotherms approach adiabatic walls at angles as low as ISo, indicating large heat fluxes! Obviously, the results are wrong: either the whole solution method is wrong, or the boundary conditions were improperly implemented, or the solution had not converged. Solutions of the same problem, which are believed to be accurate to within 0.1%, are presented below.

SOLUTION METHOD AND ERROR ANALYSIS

Calculations are performed for flows at Ra = lo5, which was the highest value used in the study by Lee [I]. The Rayleigh number was based on temperature difference between the hot and cold wall, T, - Tc, and a characteristic length L, defined by Lee [I] as L = = d f i :

The following values were chosen in this study: p = 1, d = 1, p = 1, Pr = 0.7, T , = 1, and T, = 0. The viscosity f i was set such that the desired value of Ra is obtained for the chosen Pr. The direction of the gravitation vector was changed by

This work was carried out while the author was a visiting scholar at Stanford University, California, under the sponsorship of the Deulsche Forschungsgemeinschaff. The author thanks J . H. Ferziger for helpful discussions.

Capyrighl 0 1993 Taylor & Francis 1040-7782/93 $10.00 + .00

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NOMENCLATURE

depth of cavity magnitude of ~ravity vector length of cavity (= 3dl characteristic length (= Ji;i) Nusselt number Prandtl number Rayleigh number temperature Cartesian coordinates coefficient of volumetric expansion increment; difference tilt angle, Fig. 1

p dynamic viscosity p density

4 numerical solution on a particular grid

9, exact solution of partial differential equstions

J, stream function

Subscripts

C cold H hat max maximum value

setting its x and y components to * 1 or zero, since only the angles 0 = 0", 90", 180Q, and 270" are considered (cf. Fig. 1 ) .

The finite volume discretization method presented by Demirdiif and Perid [21 is used to solve the conservation equations for mass, momentum, and enthalpy in thcir natural (dimensional) form, using boundaly-fitted nonorthogonal grids. Sec- ond-order central differencing was employed in the discretization of both the convection and diffusion fluxes. The multigrid algorithm described by Hortmann et al. [31 was used to accelerate the convergence of the iterations. Calculations were performed on a series of systematically refined grids from 10 x 10 to 160 X 160 control volumes (CVs).

Calculations were started on the coarsest grid with zero velocity and pressure fields and temperature set to 0.5(TH + Tc) Iterations were stopped when the sum of absolute residuals had fallen six orders of magnitude. The solution on one grid is used to obtain initial fields for the next finer grid [3]. An equivalent of about 50 iterations on the finest grid is typically required to obtain solutions on all grids. With the above convergence criterion, the relative convergence errors were of the order of

Fig. I Geometry and boundary conditions for the test prab- lem.

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NATURAL CONVECTION IN TRAPEZOIDAL CAVITIES 215

Solution was first obtained for the case of pure conduction. The heat flux through isothermal walls is used for calculating the Nusselt number; the latter is defined as the ratio of the wall heat flux with natural convection to the heat flux of pure conduction, i.e., Nu = I for pure conduction.

The discretization errors in the present solutions were estimated by compar- ing the wall heat fluxes (these were identical on the hot and cold walls to six most significant digits, which is the specified convergence accuracy) and maximum stream function values (the mass flux in the primary eddy) calculated on consecu- tive grids. Figure 2 shows the dependence of Nu and maximum stream function value on the number of grid points for the case where the strongest grid depen- dence was observed ( 0 = 90'). Monotonic convergence toward grid independent values was found for both quantities in all four cases. An estimate of the grid independent values can be made by applying the Richardson extrapolation: assum- ing second-order behavior, the "exact" value can be calculated from

Since the convergence errors were kept below W 5 , errors in the above extrapola- tion are believed to be an order of magnitude lower than the errors in the finest grid solution. This enables estimation of discretization errors on each grid by subtracting the solution of the particular grid from the "exact" one. The errors for Nu and when 0 = 90' are presented in Fig. 3 as a function of the normalized mesh spacing ( A x = I o n the coarsest grid). This figure shows that by reducing the mesh spacing by an order of magnitude, the error reduces two orders of magnitude. This is expected from the second-order scheme used in the calculations. The absolute level of errors o n the finest grid is about 0.1%. Other cases were, as already noted, less grid dependent, and solutions o n the 80 X 80 CV grid were already accurate to within 0.2%.

Fig. 1 Nu (tell) and II.,. (right) as function of grid fineness (Ra - lo5, Pr - 0.7, and 0 - 9V )

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Fig. 3 Errors in Nu and as func- lion of grid spacing for the conditions of Fig. 2.

RESULTS O F CALCULATIONS

Isotherms and streamlines for the flows at Ra = 10' and Pr = 0.7 are presented in Figs. 4-7. The isotherms are always normal to the adiabatic walls, as required. In all cases the fluid rises along the hot wall and falls along the cold wall, as expected. The solutions are symmetrical when the gravitation acts in the direction perpendicular to the symmetry plane. This is expected at Rayleigh numbers for which the solutions are steady. When the direction of gravity is from hot to cold wall ( 0 = 270" ). the flow is very weak: the heat flux exceeds the pure conduction value by only 3.75% (i.e.,Nu = 1.0375). This is to be expected, since in a square cavity there would be no flow at all and Nu would be 1.0. For the opposite direction of gravity (from cold to hot, 0 = 90"; unstable stratification), a signifi- cantly stronger flow results. Nu is now 2.423. The cases in which gravity is perpendicular to the adiabatic walls result in the strongest flow, and the solutions are "a mirror copy" for 0 = 0" and 0 = 180•‹, cf. Figs. 4 and 6. The heat fluxes for thcse two cases are equal: Nu = 4.073.

Lee [I] performed calculations for a variety of aspect ratios and Prandtl numbers. However, all of his solutions exhibit the anomalous behavior mentioned earlier (isotherms not orthogonal to adiabatic walls). Except in one case, both the absolute and relative values of Nu are not in agreement with the present results. Whereas i t might be possible that a different scaling causes disagreement in absolute values, the differences in relative values and trends are remarkable. In Lee's study [I], the highest Nu results for 0 = 180" (Nu = 5.75) and is about 50% higher than for 0 = O" (Nu = 3.79, whereas here, the two values are identical (which seems reasonable). Lee [I] found that for 0 = 90", Nu is about 20% higher than for 0 = 0" and about 25% lower than for 0 = 180" (Nu = 4.4); in the present study, it is 40% lower than for 0 = 0". For 0 = 270", I n e [I] finds Nu only 30% lower than for 0 = 0" (Nu = 2.75), whereas in the present study it is 4 times lower and just 3.75% above the pure conduction value.

All these discrepancies cannot be due to different scaling of Nusselt numbers. While the present results are believed to be accurate to within 0.1%, the observed

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Fig. 6 Same as Fig. 5, but B = 180".

Isotherms A 0.Mo B 0 . w c 0.7w D 0 . m n 0.850 ? 0.- 0 D 3 5 0 B 0.wo I 0 . m J 0 . 0 m \

Fig. 7 Same as Fig. 5, but B - 27liP.

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NATURAL CONVECTION IN TRAPEZOIDAL CAVITIES 219

anomaly in the isotherms presented by Lee Ill leads t o the suspicion that his results are wrong.

REFERENCES

1. T. S. Lee, Numerical Experiments with Fluid Convection in Tilted Nonrectangular Enclosures, Numer. Heat Transfer, Part A , vol. 19, pp. 487-499, 1991.

2. 1. DemirdiiE and M. Peric', Finite Volume Method for Prediction of Fluid Row in Arbitrarily Shaped Domains with Moving Boundaries, Inr. I . Numer. Meth. Fluidr, vol. 10, pp. 771-790. 1990.

3. M. Hortmann, M. Perif, and G. Scheuerer, Finite Volume Multigrid Prediction of Natural Convection: Bench-Mark Solutions, Inr. I . Numer. Merh. Fluids, vol. 11, pp. 189-207, 1990.

Recciud 13 January 1992 Accepred 15 J w u 1992

Address correspondence t o Milovan PeriC.

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