Free convection in a slender, laterally-heated cavity: Inclination effects

Math/ Comput.Modelling, Vol.14,pp.810-813,1990 Printed inGreatBritain

0895-7177/90 53.00+0.00 Pergamon Pressplc

FREE CONVECTION IN A SLENDER, LATERALLY-HEATED CAVITY: INCLINATION EFFECTS G.S.H. Lock and J-C Han Department of Mechanical Engineering University of Alberta Edmonton, Alberta, Canada

The paper is concerned with the laminar, natural convection of Abstract. air in a slender, laterally-heated, square-section cavity situated in a uniform gravitational field. With an aspect ratio of 5:l and a Rayleigh number of 104, consideration is given to the flow structure and heat transfer rate obtained when the orientation of the cavity is changed about three mutually-perpendicular axes: specifically as the result of tilting, skewing and rolling. The study is undertaken in numerical form and serves two main purposes: to extend the data and observations of others who have treated three dimensional stable conditions; and to explore the multiplicity of solutions when the heatedI surface is underneath and close to horizontal. INTRODUCTION Natural convection in a laterally-heated, slender cavity has received a great deal of attention, as witnessed by several reviews (Ostrach, 1972; Catton, 1979; Yang, 1987). If the span is reduced until it is comparable flow becomes threewith the width, the dimensional. This introduces two additional variables: the spanwise aspect ratio H/S, and the skew angle 0 measured from the vertical Such a position, as illustrated in Fig. 1. system is more complex than its two dimensional simplification, but has received less attention. For liquid fillings, notably water and silicone oil, attention has been given to the effect of side walls and tilt on stability in the classical Rayleigh problem and motion of a fluid heated from below (Davis, 1967; Stork and Muller, 1972; Koster and Muller, 1982; Hart, 1971; Ozoe et al., 1975). For air fillings, reported work appears to be more limited. Morrison and Tran (1978) describe experiments on a slender (H/W - 5) vertical cavity, detailing the effects of side wall conduction on the flow pattern. More recently, Symons and Peck (1984) studied flow and heat transfer with H/W - 7.5 and H/S - 45. winters and Brown (1985) have undertaken a numerical study of a short cavity (H/S - H/W 5 2) which is rolled about its long axis (see Fig. 1). Perhaps the most extensive numerical coverage to date is given in a series of papers by Yang et al. (1986, 1987, 1988) which discuss the effect of roll and tilt on flow and heat transfer in cavities with selected aspect ratios.

9

t heated surface shown shaded

Fig. 1

Reference configurations

FORMULATION The coordinate system chosen for the cavity is shown in Fig. 2. The X-axis is parallel to the longitudinal axis with the Y-axis and Zaxis forming a right-handed set. As indicated, the extent of the cavity is H, W and S in the X, Y and Z directions, respectively. The heated surface is thus in the X-Z plane at Y-O with the cooled surface being directly opposite. These surfaces were taken to be two isothermals with the temperature varying Fig. 2 810

Coordinate system

Proc. 7th Int. Conf. on Mathematical and Computer Modelling

811

linearly over the four remaining surfaces between Y-O and Y-W. Following the dsscription and definitions of Fig. 1, skew is executed about the Y-axis with the skew angle 0 being measured from configuration A where the X-axis is vertical. Roll is executed about that is, the X-axis when it lies horizontal: the roll angle 1 is measured from configuZ-axis is vertical. where the ration C Finally, tilt is executed about the Z-axis with the tilt angle a being measured from configuration A.

x=3.75

within the cavity is convection Natural the continuity and energy governed by with the equation of equations together rectangular, expressed in here motion, Cartesian coordinates. have been solved following The equations the following boundary normalization for conditions. Velocity:

u-v-w-o

Temperature:

y-0,1: x-0,1: z-0,1:

3

x‘

x=2.50

over all six boundaries. d-1, -1 for all x,z. +1-2y d-l-2y

The results are limited to a Prandtl number of 0.71. trials with different After a series of versions of the Patankar finite difference algorithm (Patankar, 1980) it was decided to use the SIMPLE-C modification developed by Van Although this Doormaal and Raithby (1964). does not eliminate the diagnostic difficulties associated with the pressure correction, it did prove satisfactory; with careful control over relaxation factors, the solutions could be made to converge efficiently. Convergence required successive agreement better than 1% in each of the dependent variables: in fact, agreement was usually much better than this. Validation studies established confidence in the ability of a 13 x 51 x 13 network to reproduce the details of the internal flow system and to predict heat transfer data with an accuracy of about 5%. The Nusselt number used in the results presented below is defined by Nu-

Qw

A(TH-TC)k

where A - HS is the heated surface area, Q is the heat flux and k is the fluid thermal conductivity.

x=1.25

x=0.292

Fig. 3

Longitudinal velocity profiles in a vertical cell: H/W - H/S - 5.,Ra - lo4

As the skew angle B increases from zero, the primary, single-loop flow pattern is subject the to effect of horizontal temperature gradients in the presence an increasing component, gsin0, of the gravitational field. The effect of the flow on heat transfer is not monotonic, however, as Fig. 4 reveals. The initial effect of skew near the vertical position is to decrease the heat transfer rate slightly as the fluid spirals along a* increased path length. But the shorter path length associated with the longitudinal roll is soon achieved, thus leading to a higher heat transfer rate at higher skew angles. 2.0

r

RESULTS AND DISCUSSION The effect of skew Configuration A corresponds to a threedimensional version of the classical vertical cavity. With Ra - 104, the velocity field fills the cavity, as illustrated in Fig. 3. As anticipated, a boundary layer does not exist and the flow is anti-symmetric about the mid-y plane and symmetric about the mid-s plane.

1.2 0

I 15

I 30

1 45

I 60

I 7.5

, 90

0, degrees Fig. 4

Effect of skew on Nusselt number: H/S - H/S - 5, Ra - lo4

812

P~OC.

7th Inr. Conf. on Mathematical and Computer Modelling

The effect of roll In rotating the cavity slowly about the roll axis from configuration C to configuration B, it was anticipated that the flow pattern would gradually change from a longitudinal roll to a set of five transverse rolls, in accordance with previous results (Davis, 1967; Stork and Muller, 1972). This result was not obtained. Instead, it was found that the longitudinal at roll was maintained all roll angles, although with decreasing strength.

2.0-

z' 1.6-

1.4 -

The corresponding heat transfer curve for this path CB is shown in Fig. 5, from which it is evident that the initial effect of roll is to increase the heat transfer rate slightly. Each point on the CB path in Fig. 5 was obtained using the previous roll solution as the initial field. Starting at B with an initial field nominally zero produced very different This input generated the familiar results. transverse rolls, as demonstrated in Fig. 6 which shows the velocity field on the mid-z It is interesting to note that this plane. does not reveal the anticipated five rolls; the six nor do rolls generated possess symmetry about the mid-x plane. This threedimensionality may be ascribed to the jet-like character of the updraughts and downdraughts as they impinge and spread on the horizontal surfaces. Fig. 7 illustrates this impingement on the cooled upper surface. When the cavity was rolled back along the path BC, the heat transfer curve travelled the path indicated on Fig. 5. A dramatic drop in heat transfer rate occurred as 7+80 degrees when the original longitudinal roll was restored.

o path

CB

o path

BC

I 15

1.2 0

I 30

I 45

I 80

I 75

I 90

Y, degrees Fig. 5

Effect of

roll

on

Ngsselt

number:

Fig. 6

Longitudinal velocity field mid-z plane: configuration B.

Fig. 7

Updraught impingement on the cooled upper surface: configuration B, y - 0.917

at

the

The effect of tilt Increasing the tilt angle from zero to 90 degrees leads to the heat transfer curve shown in Fig. 8 which suggests that the single loop flow pattern persists up to a - 75 degrees. This was confirmed from the velocity field in the mid-z longitudinal plane. However, for a - 90 degrees, the velocity data in this same plane exhibited a set of 5.5 transverse rolls; configuration B yielded yet another solution In the vicinity of o with three updraughts. 85 degrees, rapidly increasing subdivision of the flow rapidly increases the heat transfer rate. This is in contrast to the monotonic behaviour for 0' < 7 < 90" observed by Yang et al. (1987) with H/W - H/S - 7.5. However, their observations were made with Ra - 3 x 105, as opposed to the present value of 104. It therefore appears that the precise shape of the Nu v’s a curve is strongly dependent upon Rayleigh number and aspect ratio.

2.0 r

z”

1.6LF 1.4 -

CY,degrees Fig. 8

Effect of path AB

tilt

on

Nusselt

number:

Proc. 7th Int. ConJ on Murhemaiical and Computer Mode/kg CONCLUSIONS Consideration has been given to the flow and heat transfer in air occupying a slender, laterally-heated cavity situated in a uniform gravitational field. For a Rayleigh number of 104, and aspect ratios HjW - H/S - 5, a numerical study has been made of the resulting three-dimensional laminar natural convection system when the orientation of the cavity was varied systematically in three ways: tilting, rolling and skewing. For small tilt, roll and exhibited stable, skew angles, the system unique solutions with heat transfer characagreement with the teristics in general nearest related data of other workers. For skew in particular, this behaviour actually extended over the complete range of angles, bearing in mind the repetition created by rotation through all four quadrants. A transition from a single transverse roll, in the vertical position, to a single longitudinal roll in the horizontal position was noted in the vicinity of B - 15 degrees where a local minimum served to separate the low, near-vertical heat transfer performance from the higher heat transfer rates obtained for 0 > 30 degrees. The effect of roll was found to be more complex. In approaching the configuration of a cavity heated from below, the flow system underwent no qualitative change, thus leading to hysteretic behaviour upon reducing the roll angle after a new (transverse) flow system had been generated. In accord with similar findings of Winters and Brown, the system did not then exhibit a unique solution. This multiplicity of solutions evidently reflects the importance of initial conditions. Similarly, the effect of tilt is equally complex when the heated surface lies closer to the horizontal position. Branching of the solution was again found. In this situation, uniqueness only extended for angles up to about 45 degrees, thus giving greater latitude to "anomalous modes": a single longitudinal roll and 1, 1.5, 2, 2.5, 3, 4.5, 5 and 6 transverse rolls were all found to be possible solutions depending upon the tilt.

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REFERENCES (1979). Natural convection in Catton, I. enclosures. Proc. of the 6th Int. Heaf Transf.f, Toronto 6, 13-43. Davis, S.H. (1967). Convection in a box: linear theory. J m 30(3), 465-478. Hart, J.E. (1971). Stability of the flow in a differentially heated inclined box. L fluid 47(3), 547-582 Koster, J.N. and U. Muller (1982). Free Convection in vertical gaps. J. Fluid &&, m, 429-451. Ostrach, S. (1972). Natural convection in enclosures. In Advances in Heat Transfer, Academic Press t?, 161-227. ozoe, H., H. Sayama and S.W. Churchill (1975). Natural convection in an inclined, rectangular channel at various aspect ratios. Int. J. Heat Mass Transf., 18, 1425-1431. Morrison, G.L. and V.Q. Tran (1978). Laminar flow structure vertical free in convective cavities. mI t Transf. 21, 203-213. Patankar, S.V. Numerical Heat (1980). Transfer. Hemisphere Publishing Corporation. Stork, K. and U. Muller, (1972). Convection in boxes: experiments. J. Fluid Mech 54 (4), 599-611. Symons, J.G. and M.K. Peck (1984). Natural convection heat transfer through inclined, longitudinal slots. J. Heat Transf.,j&, 824-829. Van Doormaal, J.P. and G.D. Raithby (1984). Enhancements of the SIMPLE method for predicting incompressible fluid flows. Numer., z, 147Winters, K.H. and T.W. Brown (April, 1985). convection in a threeBernard dimensional cavity: the effects of aspect ratio and tilt. T.P. 1121, Theoretical Physics Division, AERE Harwell, 10 pp. Yang, H.Q., K.T. Yang and J.R. Lloyd (1988). Three-dimensional bimodal buoyant flow transitions in tilted enclosures. Int_. Heat 9(2), 90-97. Yang, H.Q., K.T. Yang and J.R. Lloyd (1987). Laminar natural-convection flow transitions in tilted three-dimensional longitudinal rectangular enclosures. Int. J. Heat Mass Transf., a(8), 1637-1644. Yang, H.Q., K.T. Yang and J.R. Lloyd (1986). Flow transition in laminar buoyant flow in a three-dimensional tilted rectangular channel. rP oc Conf., Hemisphere 4, 1495-1500. (1987). Natural convection in Yang, K.T., enclosures. In S. Kakac, R.K. Shah and W. Aung (eds.) Handbook of Sinele Phase co n, vect ve WileyInterscience. New York.