Laminar Free Convection in a Vertical Slot

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J . Fluid Me&. (1965), vol. 23, part I , p p . 77-98 Printed in Great Britain 77 Laminar free convection in a vertical slot By J. W. ELDER Department of Applied Mathematics and Theoretical Physics, Cambridge? (Received 6 November 1964 and in revised form 10 May 1965) This is largely an experimental study of the interaction of buoyancy and shear forces in the free convective flow of a liquid in a rectangular cavity across which there is a uniform temperature difference, AT, produced by maintai
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  J. Fluid Me&. (1965), vol. 23, part I, pp. 77-98 Printed in Great Britain 77 Laminar free convection in a vertical slot By J. W. ELDER Department of Applied Mathematics and Theoretical Physics, Cambridge?(Received 6 November 1964 and in revised form 10 May 1965) This is largely an experimental study of the interaction of buoyancy and shearforces in the free convective flow of a liquid in a rectangular cavity across whichthere is a uniform temperature difference, AT, produced by maintaining the twovertical walls at two different temperatures. The height of the cavity, H, s madelarger than the width of the cavity, L, and the cavity is sufficiently long in thethird dimension for the mean flow to be nearly everywhere two-dimensional.The flow is specified by three dimensionless parameters: CT, the Prandtl number; h = H/L, the aspect ratio; A = yg hTL3/~v, he Rayleigh number. The experi-ments are generally restricted to h = 1-60, (r + lo3 and A < los. For A < lo3 the temperature field closely satisfies Laplace’s equation but aweak stable unicellular circulation is generated. The flow is vertical throughoutthe slot except for regions within a distance of order L from the ends.For lo3 < A < lo5, arge temperature gradients grow near the walls, and in aninterior region a uniform vertical temperature gradient is established. The flow issimilartothat near an isolated, heated, vertical plate except that the verticalgrowth of the wall layers is inhibited in the central part of the slot by the presenceof the other layer which prevents entrainment of fluid.Near A = lo5 he interior region of the flow generates a steady secondary flow. A regular cellular pattern becomes superimposed on the basic flow to produce a ’ cats-eye ’ pattern of streamlines. Near A = lo6 when the secondary cell ampli-tude is large, a further steady cellular motion is generated in the weak shearregions between each cell. 1. Preliminary remarks Many geophysical and astrophysical phenomena are maintained by buoyancyforces, but the role of these forces is generally strongly modified by co-existingshear, rotation of the system as a whole, processes at a free surface, and so on.The free convection of a viscous fluid in a vertical slot, whose walls are held at two different temperatures, provides one of the simplest cases of an interactionbetween buoyancy and shearing forces; a study of this interaction is the centraltheme of this paper.This investigation was started simply as an attempt to obtain an experimentalvisualization of the boundary-layer flows which precede and follow the onset of t Present address : Institute of Geophysics and Planetary Physics, University of California, LE olla, California.  78 J. W. Elder convective turbulence, but it was sooii discovered that the primary laminarcirculatory flow does not at first produce boundary-layer waves but, rather,steady large-scale secondary flows appear in the interior of the slot. Further, itwas found that these secondary flows, when of sufficient amplitude, were able togenerate other steady secondary flows, to be called tertiary flows. Here wediscuss the primary circulatory flow and the steady secondary flows which cangrow in it, leaving for another paper the study of the unsteady disturbances andthe subsequent turbulent flow. 2. Statement of the problem 2.1. Dimensional analysis Consider the arrangement shown in figure 1. A hollow rectangular prism of width L, breadth B and height H has a co-ordinate frame OXI'Z with its srcinlocated in one corner. The face x = L is maintained at temperature To, he face L 0 1Y FIGURE . Diagram of the slot. (Note : n all diagrams the hot wall is on the left-hand side.) x = 0 at temperature (To +AT) and the other walls are insulated. Provided B is sufficiently greater than L, the primary motion of the fluid which fills the prismwill be nearly everywhere two-dimensional and confined to planes y = const.,except near y = 0, B. In the analysis BIL = 00. A possible isotherm and stream-line is shown projected on y = 0 in figure 1. Making the Boussinesq approximation, that density variations are significantonly in their generation of buoyancy forces, and that other fluid parameters areindependent of temperature, the problem is defined by: the kinematic viscosity, v;  Laminar free convection in a vertical slot 79 the thermal diffusivity, K; the acceleration due to buoyancy, yg AT, where y isthe coe%cient of cubical expansion; L nd H. Hence, since these involve onlythe dimensions of length and time, three dimensionless parameters are needed tospecify the system, A convenient set is: S2 = (T = V/K, Prandtl number; (14 A = yg ATL3/~v, Rayleigh number; (1 b) h = H/L, aspect ratio. (14 The field variables can be conveniently made dimensionless by choosing units of length, temperature, pressure, velocity: L, AT, POL% (KV)W. (2) Where confusion may arise, dimensional variables are written with an asterisk (*). The dimensionless temperature is written as 8. We shall write 5 = 1 - . 2.2. Other investigations Batchelor (1954) has given a theoretical analysis of the problem, the case A < lo3 being treated in some detail, but for large A only a qualitative discussion ispossible. Batchelor’s work, like that of Pillow (1952), suggests that as A -+ 03 theinner region has both constant non-zero vorticity and constant temperature.Eckert & Carlson (1961) have obtained with an interferometer, detailed observa-tions of the primary temperature distribution in air (cr = 0.7) and, in particular,measurements of the local heat transfer at the walls. They substantially confirmBatchelor’s calculations at small A, but at large values of A find no evidence forBatchelor and Pillow’s contention that the inner region will have a constanttemperature; rather, they find a region of constant vertical temperature-gradient.These observations are confirmed by Mordchelles-Regnier & Kaplan (1963) usingcarbon dioxide gas at high pressures. Further evidence that the inner region has a constant vertical temperature-gradient and also zero vorticity is given in acalculation by Weinbaum (1964) and some experiments by Martini & Churchill (1960) for convection in a horizontal cylinder. 2.3. The present investigation The experiments reported here are generally restricted to h = 1-60, u + lo3 and A < 108. Near A = lo7 the wall region becomes unstable; travelling wavesystems grow independently on both the hot and cold walls. At A = lo9, thecentral portion of the slot is turbulent. The flows for A > 108 will be discussed inanother paper.Our interest will centre around: (i) the uniform vertical temperature gradientfound in the interior of the flow for A > 104; and (ii) the mechanics of thesecondary flows. It is found for A > lo5, that ph = const., independent of S and A. In this case the flow in the central portion of the slot is most stronglyinfluenced by m E. ($PA)*. Secondary flows appear for m 2 2n. The experimental observations are discussed in Q 3-5, followed by an analyticaldiscussion in 3 6-8.  80 .I. W. Elder 3. Experimental method The flows were established in a rectangular cavity, two walls of which weremaintained at two constant but different temperatures by pumping water fromtwo thermostatic units through cavities behind the walls. The temperaturedifference AT was held constant to generally better than 2 .05 C, the metalwalls ensuring a negligible vertical temperature-gradient in the walls (measuredvalues, no more than O.Ol/H, Cicm). The lower end of the cavity was packed withan insulator, either a block of porous rubber or Perspex; the upper (free) surfaceof the fluid was left undisturbed in contact with a dead-air space above the filledportion of the cavity. Thus, the boundary conditions on the upper fluid surfacediffer from those on the lower surface, but this was the most convenient arrange-ment for probe access.Velocity measurements were made by direct observation through the glass or Perspex sides (y* = 0, B) of aluminium powder suspended in the fluid; eithervisually, by timing the passage of a single particle between fixed marks in theeyepiece of a travelling microscope, or from time photographs, in which thestreak-length is proportional to the velocity. Where possible, the plane of observa-tion was chosen halfway between the glass sides. Velocities are accurate to betterthan 5 5 %. A very helpful point is that aluminium particles are roughly disk-shaped and tend to lie with the plane of the disk on the stream surface, so that thebroad features of the stream surfaces are immediately apparent.Temperatures were measured with copper-constantin thermocouples and apotentiometer to k .01 C. A probe was entered from above; the lower elid wasofdiameter 1 mm, with 3 ern of the thermocouple wires (46s.w.c.) ticking out ofthe end. Direct observation showed very little disturbance to the flow due to thepresence of the probe, except for a very slow drift of the cellular pattern.Three experimental arrangements were used:Apparatus I: H < 60 cm; B = 5 em; L = 1,2,3,4,5 m.Apparatus 11: H < 80 em; B = 10 cm; L = 4.08 cm.Apparatus 111: H = 55 cm; annular cylinder of radii 3.3 cm, 6.3 em; L = 3 em.The initial study was with apparatus 111, in which the discovery of the verticaltemperature-gradient and the cats-eye mode were first made. This apparatus wassupplied with a uniform heat flux on the inner boundary wall.Apparatus I was designed so that L could be set to an arbitrary value, whileapparatus I1 was specifically designed for the turbulence work.Two fluidswere used: medicinal paraffinand a silicone oil MS 200/ 100 centistoke.They both allow studies over the Rayleigh-number range of theexperiments usingapparatuses of convenient size and temperature differences sufficiently large foraccurate measurement. A particularly convenient feature is the ease with whichaluminium powder can be suspended. Such viscous oils necessarily have highPrandtl numbers so that in effect the 3 parameters (1) are reduced to 2. Both oilshave similar properties, but paraffin has a very much larger variation of viscositywith temperature. The specification (1) requires y, K, v to be independent oftemperature and yA T < 1. Within the present experimental accuracy theserequirements are satisfied except that the variation of viscosity with temperature
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