Tomographic particle-image velocimetry and thermography in...

Experiments in Fluids 34 (2oo3) 156-172 DOI lo.loo7/soo348-ooz-o534-4

Tomographic particle-image velocimetry and thermography in Rayleigh-B6nard convection using suspended thermochromic liquid crystals and digital image processing

M. Ciofalo, M. Signorino, M. Simiano

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Abstract Steady-state flow and temperature fields in shallow rectangular enclosures heated from below were visualized and quantitatively characterized by using glycerol as the working fluid and suspended thermochromic liquid crystals as tracers. Couples of photographs taken on 120 transparency film for two orthogonal sets of vertical plane sections were digitized by a 1,200-dpi flatbed scanner and split into HSL (hue-saturation-Lightness) components by using commercial general-purpose image processing software. Two-dimensional velocity fields were obtained from the lightness component by a two-frame cross-correlation technique using a commercial particle- image velocimetry (PIV) package. Temperature fields were obtained from the hue component on the basis of an in situ calibration procedure, conducted under conditions of stable thermal stratification. Finally, 2D flow and temperature distributions were interpolated by a purpose-written Fortran program to give 3D flow and thermal fields in the enclosure. Results are presented here for the case of a 1:2:4 aspect ratio cavity at a Rayleigh number of ,,-14,500, for which a complex 3D flow and temperature distribution was observed.

1 Introduction The main purpose of the present work was the development of imaging techniques for the visualization and the quantitative characterization of flow and temperature fields in fluids. However, the problem that was chosen as the model problem, Rayleigh-B6nard convection, is in it- self one of the most basic and fascinating problems of fluid dynamics.

Received: 27 November 2000/Accepted: 11 March 2002 Published online: 7 January 2003 �9 Springer-Verlag 2003

M. Ciofalo (Y~), M. Signorino, M. Simiano Dipartimento di Ingegneria Nucleare, Universit/l di Palermo, Viale delle Scienze, 90128 Palermo, Italy E-mail: [email protected] Tel.: +39-091-232257 Fax: +39-091-232215

The authors are grateful to Professor Peter Daniels (Department of Mathematics, City University, London) for the interest shown and for providing support in the acquisition of the VISIFLOW software. They would also thank Prof. Jan Stasiek (Technical University of Gdansk, Poland) for sharing his invaluable experience in the use of TLCs.

Rayleigh-B6nard convection in a shallow cavity heated from below is a classic example of a non-linear system exhibiting a sequence of transitions as a control parameter increases. Here, this parameter is the Rayleigh number Ra=(gflATH~)/(vct), H being the cavity height, AT the temperature difference between the walls, g the accelera- tion caused by gravity, and t , v, and ~t respectively the thermal expansion coefficient, kinematic viscosity, and thermal diffusivity of the fluid.

Despite the long time elapsed since the first contribu- tion by Lord Rayleigh (1916) and the many experimental (Stork and Mfiller 1972; Dubais and Berge 1978) and theoretical (Davis 1967; Swift and Hohenberg 1977; Frick and Clever 1982) studies dedicated to the problem, it is still only partially understood and is the subject of continuing research. An updated review is given, for example, by Bodenschatz et al. (2000).

In a layer of infinite horizontal extent, the first insta L biLity (transition from pure conduction to 2D convection) occurs at a first critical Rayleigh number Ra1~1,708; 2D convection rolls become unstable to 3D disturbances at a second critical Rayleigh number Ra2,-~22,600 (Busse 1978). In finite aspect ratio enclosures the situation is compli- cated by the influence of the side walls. The first instability (transition from conduction to 2D convection) still occurs at a Rayleigh number close to the theoretical value for infinite-aspect ratio cavities (Rare1,708). Flow patterns initially take the form of steady transverse rolls, parallel to the shorter side. At Rayleigh numbers of the order of 104, depending on the aspect ratio of the enclosure and on the Prandtl number of the fluid, rolls orthogonal to the above ones appear near the short sides, where they are superimposed on the base transverse-roll pattern. A complex interface ("grain boundaries") develops between the regions dominated by the two alternative flow patterns (Daniels and Weinstein 1992). Further increases of Ra lead to a growth of the regions dominated by longitudinal rolls; eventually, through different and complex mechanisms, steady-state flow becomes unstable and a fully 3D and time-dependent regime is established in the enclosure.

For practical and theoretical reasons, our investigation has focused so far on moderate aspect ratios (height/ width/length=l/2/4 or 1/4/8) and Rayleigh numbers ranging from ~5,000 to ~30,000. The experimental approach was based on the use of thermochromic liquid crystals (TLC) suspended in glycerol as simultaneous temperature and flow tracers.

The use of TLC in heat transfer research is widespread (Akino et al. 1989; Stasiek et al. 1996), but applications of

TLC in the suspended form are rare (Hiller and Kowa- lewski 1987; Richards and Richards 1998) and are usually limited to the qualitative visualization of flow and temperature distributions in a single plane. Only recently, techniques based on scanning different image planes and reconstructing 3D flow and/or temperature fields have been reported in the literature; see for example Fujisawa and Funatani (2000), who used suspended TLC, and Sakakibara and Adrian (1999), who used two-component laser-induced fluorescence (LIF).

The quantitative reconstruction of velocity and temperature fields, based on TLC droplets suspended in glycerol and on a multiple-exposure photographic technique (PIVT, or simultaneous particle-image velocimetry and thermography), was demonstrated for free convection in rectangular enclosures in our previous work (Ciofalo et al. 1994). There, it was limited to a single plane or a few planes at most, which - in the case of inclined or vertical enclosures - was justified by the flow being essentially 2D. The extraction of velocity vectors was performed by a rather crude - essentially manual - technique.

The aim of the present work was to improve the above investigation by:

�9 extending the study to 3D flows, via the acquisition and post-processing of images for two families of orthogonal planes and the subsequent numerical interpolation of the results, yielding 3D flow and temperature fields (tomographic particle-image velocimetry and thermography, or TPIVT);

�9 using a PIV-dedicated software package automatically to extract in-plane velocity distributions from digitized pictures.

A flow chart of the overall method is reported in Fig. 1. Preliminary results for a 1:4:8 aspect ratio enclosure were presented by Palazzolo et al. (2000). In the following, the various steps involved will be described in detail and results will be presented for a 60•215 mm (1:2:4 aspect ratio) cavity subjected to a temperature difference of ,-~1.5~ giving Ra,~14,500.

Hopefully, whole-field 3D results obtained for low- Rayleigh number Rayleigh-B~nard convection in shallow enclosures will help the specialists to elucidate the structure of the flow, especially in the proximity of the side boundaries, and will assist the intense theoretical efforts currently being dedicated to this problem.

2 Test section, photographic equipment and working fluid Figure 2 shows the enclosure assembly. The rectangular, fluid-filled cavity (test section proper) was delimited by two isothermal aluminium walls and by a acrylic glass frame, having the appropriate height H and side dimen- sions L, W, which was tightly compressed between them and provided the four (transparent) side wails. A separate frame was used for each geometrical configuration investigated; each frame was made of 11-mm thick acrylic glass and was provided with a 2-ram thick rubber gasket to ensure fluid tightness.

j . . . . . h . . . . . . (camera + 80 rnm f:2.8 lens + flash) l

6"9 cm colour , ~ transparency

~[F [ (OE~s~176176 . . . . . ) [

(PaintShop pro) J

/ \ 1 I

J,I, ,I,1 ~176 .... I

FORTRAN program

Fig. 1. Flow chart of the TPIVT technique

b y

c '

Fig. 2. Exploded view of the working section, a Acrylic glass side walls of the enclosure; b,b': aluminium plates with milled grooves (isothermal walls); c,c': perspex back plates

The isothermal walls were 20-mm thick, black anod- yzed aluminium slabs, bearing milled grooves in which water was forced to circulate. Hot and cold water flows were provided by thermostatic laboratory baths equipped with water circulators, which kept the temperature within +0.1~ with good precision and repeatability. Readings of the heating and cooling water temperatures were also provided by 0.05~ mercury thermometers located at the inlet and outlet sections of each milled groove. For the experimental configurations tested, convection heat

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transfer through the cavity was so small (<50 W) that a water flow of ~0.1 l/s through each plate was amply sufficient to keep the inlet-outlet temperature difference within 0.1~

The assembly composed by the two aluminium plates (isothermal walls) and the interposed acrylic glass frame (side walls) was tightly compressed into the final configuration by a robust steel vice, mounted on a swinging table.

Light sheets were generated by using a linear-bulb flash gun capable of 300 I light emission at full power, followed by a simple slit collimator device. The flash recharging time was ,--2.7 s. By using purely geometric considerations, i.e. neglecting diffraction, the thickness of the light sheet (plane beam) thus obtained was estimated to be ,--3 mm at the exit slit and to increase linearly by ,--10 mm/m. For the case of a 120x240 mm planform enclosure, with the beam parallel to the long sides (worst case), this thickness increased from ,-,3.5 to ,--6 mm going from the front wall to the back wall. Such a light sheet, although relatively thick and slightly diverging, was sufficient to select a practically 2D "slice" of the flow and temperature field. It should be observed that in complex 3D flows, some finite thickness of the light sheet is necessary for PIV, or the velocity components orthogonal to the plane investigated would prevent individual particles from leaving two images at two consecutive instants. A greater uniformity of the plane beam could be obtained by using a more sophisticated optical collimator, e.g. including a cylindrical lens.

Pictures were taken by using a medium format professional camera equipped with a 6x9-cm back for 120- type films and an 80 mm f.2.8 lens. The flashgun and camera could slide along rigid aluminium rails with respect to the cavity, and were kept in a fixed position relative to each other by a steel bracket, so that the geometrical distance between the cavity plane being pho- tographed and the focal plane of the camera did not change as the cavity was traversed. The rails were con- nected to a steel frame that could be rapidly mounted in two orthogonal positions on the swinging table holding the cavity, so as to scan the two series of planes parallel to either the long or the short sides of the cavity.

The whole traversing device, and the cross-sections for which images were obtained for the case of a 60x120x240 mm (1:2:4 aspect ratio) enclosure, are sche- matically shown in Fig. 3. Throughout the remainder of this paper, reference will be made to this latter geometry. The figure shows also the nomenclature adopted for the co-ordinate axes (x, y, z) and for the associated velocity components (u, v, w).

Glycerol was selected as the working fluid for the following reasons:

I. when glycerol is used, Rayleigh numbers of ~3,000- 20,000, i.e. in the range of highest interest as regards flow regime transitions, can easily be obtained by letting H vary between 4 and 6 cm and AT between 1 and 2~ With water, the Rayleigh number is about 400 times higher for any given H and AT, so that

a

~ e ra

b sh

Ca 11

4 5

if9 7

C . ." ( u )

O

z (w)

Fig. 3a-c. Arrangement of enclosure, flash and camera showing the plane cross-sections for which images were obtained in the case of a 1:2:4 aspect ratio cavity, a Longitudinal planes; b transverse planes; c nomenclature of axes and velocity components

2.

steady-state regimes can be attained only by using very shallow cavities (H<I cm). pure TLCs can be used only in fluids with which they are completely immiscible, such as glycerol and silicone oil, but not in water, where the micro-encapsulated form of the TLCs, which exhibits much less brilliant and saturated colours, must be used instead. Moreover, TLCs are roughly neutrally buoyant in glycerol, where they float slowly to the surface of shallow layers (few cm) only after several hours, thus allowing ample times for the conduction of the experiments.

Commercial grade glycerol is available at a moderate cost. The relevant physical properties of pure glycerol at 20~ are summarized in Table 1 (Perry and Green 1984). It should be observed that the pure substance has a melting point of 17.8~ but even small traces of impurities such as water or ethylene glycol inhibit solidification, so that commercial or even laboratory grade glycerol exist as a liquid well below 0~

In the present work, both the density and the viscosity of the glycerol used in the experiments were independently measured (Palazzolo 2000).

Density measurements , conducted by simply weighing calibrated volumes of fluid, gave a value of p at 25~ within 0.1% of the theoretical one in Table 1. Viscosity measurements were conducted by using a Brookfield ro- tational viscosity meter in the range 8-50~ Above the theoretical melting point of ~17.8~ only negligible differences were obtained with respect to the pure substance data, while some discrepancy was obtained - as expected - at lower temperatures. All the tests described here were conducted within the range 18-22~ for which the properties of the pure substance can be used with confidence.

Other physical propert ies that influence the flow and temperature fields are the thermal conductivity, the thermal expansion coefficient and the specific heat of the fluid. However, unlike viscosity, these properties are not significantly affected by the presence of small fractions of impurities, so that the pure substance data were simply used to evaluate the Rayleigh and Prandtl numbers.

3 Thermochromic liquid crystals Cholesteric or chiral-nematic TLCs are transparent (clear) at low temperatures, turn red at a characteristic temperature known as "red start", cross the entire visible spectrum f rom red to deep blue, and become transparent again at a temperature known as "clearing point". The deep blue colour remains basically unchanged in a broad T range, so that, for practical purposes, the "blue start" - rather than the clearing point - is used as a second characteristic temperature. The range between red start and blue start is called "colour play". TLC mixtures are available with nominal red start temperatures ranging from -30 to 120~ and colour plays from 0.5 to 20~

TLCs can be used either as pure liquids or in the micro- encapsulated (m.e.) form. In the latter case, small TLC

droplets (typically ~20-50 pm in diameter) are coated with an organic jelly that makes them resistant to water, moisture and - to some extent - contamination by sol- vents, oil or grease. Only m.e. TLCs can be used as suspended tracers if the working fluid is water; they are also used as pigments in the preparat ion of inks or paints. The pure liquid form exhibits more brilliant and saturated colours; therefore, using pure TLCs is preferable whenever possible, i.e. with fluids like glycerol and silicone oils in which they are immiscible and remain suspended as droplets of various size (depending on the mixing technique used).

One potential source of concern with pure TLCs is their sensitivity to shear and pressure, which can change their colour even in the absence of temperature variations. However, in the present application both shear rates and pressure variations across the enclosure are extremely low and it is not credible that they can affect the TLC colour to any significant extent. For example, as will be discussed in detail in Sect. 4.2, peak velocities in an enclosure of height H=60 m m are of the order of 0.2 mm/s for AT=I.5~ ( R a ~ 1 4 , 5 0 0 ) , yielding shear rates of the order of 10 -2 (S q ) at most. Another potential concern is that the chromatic response of the TLCs may vary with time because of contaminat ion by the ambient fluid (glycerol) and other causes. In order to assess these effects, calibration images (obtained for stably stratified fluid as discussed in Sect. 6) were repeated after 24 h and the corresponding hue- temperature curves were compared. Only marginal discrepancies, corresponding to differences of a few percent in normalized temperatures, were observed. In any case, long-term effects were cut out by obtaining specific cali- brat ion curves for each configuration studied immediately before the actual convection tests, i.e. usually a few hours earlier.

The TLC mixture used in the present work (BN/ R20C12 W) was produced by Hallcrest Ltd (Poole, Dorset, UK). The number following the "R" indicates the red start point and that following the "C" is the nominal colour play, both expressed in ~

Preliminary tests have shown that the best results, as far as the quantitative extraction of flow fields is concerned, are obtained with a TLC volume concentrat ion in glycerol of ~300 ppm. At higher concentrations, the image becomes too noisy, although the resulting higher numerical density of particles in the working fluid may be preferable

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Table 1. Properties of glycerol (Perry and Green 1984) Formula: CH2OH CHOH CH2OH

Melting point (pure): 17.8~

Physical properties of pure substance at 20~ Density p Viscosity I-t Specific heat Cp Thermal conductivity z Thermal expansion coefficient fi Kinematic viscosity, ~tl p v Thermal diffusivity, .;./(#Cp) i~ Prandtl number, v/re Pr Refractive index r Surface tension a

Molecular weight: 92.09 Boiling point: 290~

(kg m -3) 1,261.1 (N s m -2) 1.499 (J kg-lK -1) 2,411.4 (W mqK -l) 0.2836 (K -1) 5.05X10 -4 (m2s -1) 1.189x10 -3 (m2s -1 ) 9.326X10 -s (-) 12,740 (-) 1.474 (N m -I) 0.0639

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for qualitative flow visualization and for the characterization of the temperature distribution.

Mixtures of TLCs and glycerol are prepared by pouring the required amount of glycerol into a shallow open con- tainer, dissolving the appropriate amount of TLC in di- ethyl ether (about 1:1 volume ratio) and then spraying the solution on the free surface of the glycerol, using a nee- dleless clinical syringe. After a few minutes the ether evaporates and a thin layer of TLCs (which revert to their liquid crystal structure) is left on the free surface. At this point, the glycerol is gently stirred (manually) with a flat spoon or shovel - avoiding the inclusion of air bubbles - until the TLCs are evenly distributed throughout the fluid. The process may take several minutes and requires some manual dexterity.

The result is a uniform distribution of droplets having an average diameter of,-~0.1 mm, with a limited number of larger (,--0.2-0.3 ram) aggregates. The specific gravity of TLCs at T..~20~ is about 1,200 kg/m 3 and thus is very close to that of glycerol (~1,260 kg/m 3, see Sect.2); therefore, TLC droplets are almost neutrally buoyant in this fluid. Equating the Stokes drag force on a sphere of radius a (6rc~tau) with the buoyant force gV(p-PTLC ) gives a rise velocity

2ga2 (1 PpLC) (1) u = 9v

which, for v~1.2• -3 (kinematic viscosity of glycerol at 20~ see Table 1) and PxLc/p=l,200/1,260, yields u~ l ~tm/s even for "large" particles 0.2 mm in diameter. Thus, in the reference configuration and under isothermal conditions, TLC particles only drift by significant fractions of the cavity height H (60 mm) after several hours. During the tests, convection prevents particle floating and aggregation, but if the test section is turned off, the TLC particles (starting from the largest) adhere to the top wall and the mixture cannot be used a second time.

4 Photographic technique

4 . 1

F o c u s i n g

As mentioned in Sect. 2, during the tests the geometrical distance C between the focal plane of the camera (image) and the light sheet (object) was kept fixed by the traversing device and was ~592 ram. However, the light beams from object to image cross for each plane section of the cavity at different depths of glycerol (which has a refraction index r much larger than unity: r~1.47 at 20~ Therefore, the optical distance and the reproduction ratio varied, and a focusing correction was required for each plane with respect to focusing in air.

The reproduction ratio and focusing correction can be derived from simple geometrical optics considerations as follows. With reference to Fig. 4, let o be the depth of glycerol traversed for the generic plane of the cavity. Snell's law of refraction:

O b j e c [

I I I I I I

"------/~ "---~ ~'~ glycerol / , / ~ i ] ~ \ ' , (refractfon indexr) / /

t + ~ x ~ ~r / w a l l

+

image

Fig. 4. Focusing through media with different refraction indexes

sin ~P2 - - r ( 2 )

sin ~01

links the directions of the beams in glycerol and air. For moderate viewing angles, the following relation can be derived from Eq. (2) between the geometrical depth o and the equivalent optical depth i:

o tan ~02 sin q~z - - - ,-~ - - r ( 3 )

i tan ~o 1 sin ~ol

so that the optical distance between the lens and the object is D=l+i~l+o/r and not l+o, as it would be in air. Now, D and d (distance between lens and focal plane) are related by the fundamental focusing law:

1 1 1 ~ + ~ = ] (4)

in which f is the focal length of the lens, which coincides with d when D--->oo. Although the nominal focal length of the lens was 80 ram, an actual value of ~82 mm was measured and used in the calculations. Finally, the distances o, l and d are related by:

o + I + d = C (constant) (5)

From Eqs. (3), (4) and (5) the following expression can be derived for the lens-to-focal plane distance d:

d - C - Z o , /1 4]" (6) 2 V c - Zo

in which z=l-1/r. The influence of the acrylic glass wall interposed be-

tween object and image, which has a thickness t~11 mm and a refraction index (~1.4) close to that of glycerol, can be approximately taken into account by including it into the plane depth o, so that o=t+b if b is the true depth of glycerol. For o=0, Eq. (6) gives:

dair=~- 1 - (7)

which is the focusing distance in air for the same value of C. Thus, the required focusing correction is Ad=d-dair.

For C=592 ram, f=-82 mm and r=-1.47, the correction Ad is reported in the fourth column of Table 2 as a function of the depth in glycerol b. Tabulated values correspond to the five planes (marked A, B ..... G) parallel to the long sides and to the 11 planes (marked 1, 2 ..... 11) parallel to the short sides, in which images were obtained for the reference 60x120• mm (1:2:4 aspect ratio) enclosure as shown in Fig. 3. The correction is almost linear with b in the range considered and is quite considerable (almost 4 mm) for the farthest section (plane 11), showing that omitting this correction would lead to poor focusing un- less a very small lens opening were used to increase the depth of field.

The above corrections were empirically tested by placing an optical target in the glycerol-filled enclosure at different distances from the front acrylic glass wall and focusing it on the ground glass back of the camera. Test pictures were also taken, and the results basically confirmed the theoretical predictions, with small discrepancies probably caused by minor geometrical imperfections, misalignments, etc. Of course, this focusing calibration needs to be performed only once, at least as far as the object-focal planes distance C remains unchanged.

Equation (4) also shows that the reproduction ratio R=DId (size of object/size of image) is simply f/(d-.t). Therefore, R can be computed for each plane from the values of d given by Eq. (6). Results are listed in the last column of Table 2 for the above reference enclosure geometry, with f=-80 mm and C=590 mm; it can be observed that R is ,~5 in air but varies from 4.89 (planes A and 1) to 4.07 (plane 11) through the various cross-sections, caused by the optical influence of glycerol. Thus, the images of plane sections that are identical in size, but are located at different depths in glycerol, differ considerably. This had to be taken into account in the following digitization and processing of the pictures.

4.2 Exposure and film development In previous work (Palazzolo et al. 2000) both the temperature distribution and the flow field relative to a

Table 2. Focusing correction Ad and reproduction ratio R for the 60x120x240 mm cavity, f=82 mm and C=592 mm

Plane b (ram) d (mm) Ad (ram) R

(in air) 0 98.33 0.00 5.02 A, 1 20 98.77 0.44 4.89 B, 2 40 99.05 0.72 4.81 C, 3 60 99.35 1.02 4.73 D, 4 80 99.65 1.32 4.65 E, 5 100 99.97 1.64 4.56 6 120 100.30 1.97 4.48 7 140 100.64 2.31 4.40 8 160 100.99 2.66 4.32 9 180 101.36 3.03 4.23

10 200 101.75 3.42 4.15 11 220 102.15 3.82 4.07

cross-section of the enclosure were extracted from just one multiple-exposure frame, recording a number of flash pulses, which varied from 2 to 8. However, this introduced an inherent directional ambiguity in the extraction of velocity distributions and led to a poor quality of the reconstructed velocity fields when, as discussed below, the AEA package VISIFLOW (AEA Technology 1997) was used (either in particle-tracking or in single-frame correlation mode). Therefore, a different approach was chosen here: for each cross-sectional plane, two consecutive single-exposure frames were taken for the purpose of quantitative velocity and temperature reconstruction. In some planes, multiple-exposure frames were also obtained using eight flash pulses for the sole purpose of qualitative flow visualization.

The interval At to be used between consecutive frames can be estimated a priori on the basis of the prevailing velocities Umin, Um~ in low- and high-velocity regions, respectively. In fact, At should be such that the distance between consecutive images of a particle is significantly larger than a particle diameter 2a even in low-speed regions, but significantly smaller than the size of convection rolls (~,H) even in high-speed regions:

2a H - - << At << - - ( s ) Umin Um~,

(in which, of course, Umin is a conventional value, repre- sentative of the low-speed regions). As shown by preliminary tests, for the reference geometry and AT=I.5~ (Ra~14,500), one has Umin~-~0.02 mrn/s and U m ~ 0 . 2 mm/s; thus, for 2a=0.1 m m and H=60 mm, Eq. (8) becomes

5 s << At << 300 s (9)

The value chosen in the present work was At=30 s. A slightly shorter interval between consecutive pulses (20 s) was adopted in the eight-exposure frames used for flow visualization.

Lens opening and film sensitivity determine the overall image exposure and thus are limited by the power of the available flash gun-collimator arrangement. The background luminosity of the image (mostly formed by ran- domly scattered light) must be sufficient to have well recognizable colours.

Preliminary tests showed that the combination of an ISO-400 daylight transparency film and f.5.6 lens opening was the best compromise. Reducing the film sensitivity would be beneficial to the resolution and graininess of the images, but would require a larger lens opening, making focusing more critical.

In all tests, the film was developed by a commercial laboratory. Preliminary comparisons with standard colour charts showed that completely accurate and repeatable results in terms of the film's chromatic response could not be reliably achieved, because of fluctuations in the development conditions and/or in the characteristics of different film batches. The in situ calibration procedure described in Sect. 6 does not rely on the absolute accuracy or repeatability of the film response, thus making this issue of limited relevance.

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5 Image digitization and processing As shown in Table 2, the reproduction ratio R (object/ image size) ranges between ,~4 and ,~5 according to the plane considered. For example, section C (mid-plane parallel to the long sides, 60x240 mm in real size) is re- produced with R=4.73 and thus gives an image of about 12.7x50.1 mm on the film. The overall resolution of lens plus film can be estimated to be ,,,50 dots per nun, or 1,270 dots per inch (dpi); therefore, the image virtually contains ,-~635x2,540=1,612,900 dots, and its digitization with no significant loss of information requires a scanner resolution of at least 1,200 dpi.

Transparencies were actually digitized by using a scanner with an optical resolution of 1,200 dpi, just meeting the above criteria. Each image was first clipped via software to exclude regions extraneous to the fluid cross section proper (240x60 mm for the long planes and 120x60 mm for the short planes in real size); the appropriate reproduction ratio R in Table 2 was kept into account for each plane section to facilitate the identification of the real edges (solid walls).

At this stage, a crucial problem was the correct relative positioning of the two consecutive images of each couple. The problem arises because a film camera, and not a CCD camera, was used. Mechanical uncertainties (which in- dude small random movements of the camera during film advancement and errors in positioning the film on the scanner bed) may cause uncertainties in the superposition of the two images. These, in turn, are interpreted by the PIV software as spurious velocities, uniform throughout each plane and superimposed on the "true" velocity field. For At=-30 s, an error of just one pixel (,~90 grn of real size) results in a spurious velocity of ,-,3 gm/s, which is not negligible compared with the prevailing velocities in the enclosure (20-200 p.m/s). In order to avoid such errors, fixed points (usually corresponding to small irregu- larities in the images of the wall surfaces) were carefully identified by eye in each couple of frames and used as boundary markers. It was estimated that this procedure reduced the positional uncertainty to zero pixels - the only problem was, of course, the considerable time and effort required.

In order to facilitate the subsequent processing, all images were then resized to a uniform scale. Since VISI- FLOW yields velocity vectors on grids in multiples of 16 pixds, the final size was chosen to be 2,560• pixels (long sections) or 1,280• pixels (short sections). Thus, 32 image pixels corresponded to 3 mm of real size, which was also the pitch chosen for the grid on which 3D fields were interpolated. Due to slight defocussing, each partide (,-,0.1 mm in diameter) was resolved by 1-3 pixels in each direction, just sufficient to identify it against the complex background.

The above considerations make it clear why a high scanner resolution is mandatory and why the method based on conventional photography followed by scanning, though cumbersome, is strongly competitive with the direct acquisition of the images by a digital camera. In fact, assuming a 3x2 frame aspect ratio, and taking

margins into account, a digital device with at least ,~3,000• pixels is necessary to obtain a comparable resolution. Also, since images are formed by flash pulses, the device must be of the "single-shot" type, cheaper scanning backs being ruled out. A few professional (SLR) digital cameras with the above characteristics are currently produced, but only at prices far beyond the budget available for the present research.

A further processing step consisted of the compensation of the attenuation of the flash planar beam, because of light absorption and scattering in the medium (gtyc- erol-TLC suspension). This was performed by simple purpose-written Fortran programs operating on the lightness (L) component of the digitized images. First, pictures obtained for a long cross-section of the enclosure under stable stratification conditions were averaged along the vertical direction to obtain the mean lightness along the direction x of the flash beam; the best fit of the resulting L(x) function by a simple exponential of the form exp(-xlXR) provided a value of about 750 mm for the relaxation length XR (at the present TLC concentra- tion of ,~300 ppm). The inverse correction factor exp(xl XR) was then applied to the lightness component of all images taken either under stabte stratification or natural convection conditions. It should be observed that because of the build-up of scattered light, the true lightness-distance function is not strictly linear, and actually exhibits a slight maximum shortly following the light entrance side of the enclosure. This effect, however, is negligible and has only a little influence on the subsequent image processing, so that a simple exponential compensation was preferred.

Figure 5 shows a 3D assembly of the flow-visualization, multiple-exposure images. Each image was obtained by eight flash pulses at 20 s intervals and f.8 lens opening. The longitudinal cross-sections, Fig. 5a, dearly show a convective structure composed of four transverse rolls; blue (hot) fluid is associated with upward velocities and red (cold) fluid with downward velocities. The fluid moves upwards along the short side-walls of the enclosure and along the mid-plane orthogonal to the sections shown. The images, relative to the short cross-sections (Fig.5b), are somewhat puzzling, as they also exhibit vortex structures that suggest the simultaneous existence of longitudinal rolls. A full discussion of the flow patterns in the cavity is postponed to Sect. 9 after the 3D reconstruction of the flow and temperature field will be introduced.

6 Colour-temperature calibration The TLC mixture used in the present tests was the one commercialized by Hallcrest Ltd as BN/R20C12 W; its name indicates that it exhibits a red start point of 20~ and a colour play of 12~ between red start and blue start. Of course, the manufacturer's indications are valid under certain standard conditions (thin TLC layers observed in reflected light on a black background), and preliminary tests conducted under these conditions (Signorino and Simiano 2001) actually confirmed this nominal response to within fractions of a ~

Fig. 5a, b. Three-dimensional assembly of individual in-plane images obtained by multiple exposure (eight flash pulses at 20 s interval). a Longitudinal sections A-E; b transverse sections 1-11

However, it is known from previous experience (Ciofalo et al. 1994) that the behaviour of the mixture may be quite different under the working conditions adopted in the present work (droplets suspended in a fluid). Apart from the substantial geometrical and optical differences, also the mixing procedure described in Sect. 3 (in particular, the step involving dissolution in ether) may possibly affect the TLC properties; moreover, these may drift with time. Therefore, an independent calibration procedure was necessary in order to characterize the relation between temperature and hue under conditions identical to those prevailing in the tests proper.

The test was performed by filling the test cavity with the glycerol-TLC suspension and imposing conditions of stable thermal stratification, with the hot waU (T=TH) at the top and the cold wall (T=Tc) at the bottom. The temperatures TH=21.0~ Tc=18.65~ were chosen by preliminary tests so that they encompassed, with some margin, the actual colour play exhibited by the TLCs. After a suitable settling time for the attainment of a complete steady state (2-3 h), photographs were taken following the same procedure as in the convection tests, i.e. using a single flash pulse at a lens opening off.5.6 and, of course, the same film, the same photographic processing and the same scanner-acquisition options.

The images thus obtained were then analyzed as summarized in Fig. 6. All the processing was performed by

using standard, commercially available, image processing packages.

Figure 6a is a digitized image of the stable thermal stratification. Saturation was enhanced for the sole purpose of a better visualization, but played no role in the digital image processing. The red start and colour play regions can be clearly identified. A residual horizontal gradient is also visible, associated with a slight residual temperature difference of the order of 0.1~ between the left and right ends of the cavity. Slight disuniformities of this order were found to be very difficult to eliminate completely during the tests.

Calibration images were carefully analyzed for possible, symmetric horizontal variations of the vertically averaged hue between centre and edges, implying a dependence of the chromatic response upon the viewing angle. No such effects were found, which suggests that the angle dependence of hue is negligible in the present range (semi-angle of view ~13 ~ for a cavity length of 240 mm and a lens-to-plane beam distance of ~0.5 m).

Figure 6b shows the hue map (represented by 256 grey levels) corresponding to the image in Fig. 6a and obtained by HSL channel splitting. The black region at the bottom of Fig. 6a, associated with temperatures below the red start threshold, translates into a region of unresolved hue, i.e. a random dispersion of hue values around a mean value of about 150. However, some of the lightest spots near the red start boundary correspond to pixels of the original image having a deep purple colour. Since hue varies rood(256) and is conventionally set to 0=256 at the red-purple boundary in all standard processing methods employing eight bits per channel, these pixels are associated with a high value of hue (200-255). In order to avoid the averaging of nearby pixels having a similar colour (as represented, for example, by RGB splitting), but a totally different hue, the hue scale was shifted via software from (0,255) to (-85, 170) by subtracting 255 from all hue values above 170.

Figure 6c reports the hue distribution interpolated on a 3x3 mm grid, a step that was necessary in order to smooth out the sharp local hue gradients associated with individual particles. As mentioned above, the same 3x3 mm 2D interpolation grid was also used in the convection tests both for temperature and for velocity in each plane, and a corresponding 3x3x3 mm 3D grid was used to reconstruct the 3D fields.

Figure 6d reports the vertical profile of hue, obtained by averaging the hue map in Fig. 6c over the horizontal direction, together with the corresponding vertical profile of T, obtained by assuming a uniform gradient between TH and To Finally, Fig. 6e reports the functional dependence of T upon hue. Solid symbols correspond to the experimental results in Fig. 6d, cleared of the spurious results corresponding to the black region of the original image. The continuous line is a best fit obtained by choosing a suitable functional form and imposing the red start (hues10) at T~19.2~ the blue saturation (hues156) at high T and the best agreement, in the least-squares sense, with the intermediate data. The following expression was obtained:

163

164

C

21.5

Calibration Hue map - cutoff=170

140

120

100

80

60

40

20

0

A

o

I -

21

20.5

20

19.5

19

18.5

�9 texp l i i ~ T = 1 9 . 2 + 0,75 *(in (147/(157-H)))"0,5 I ...... .& l . . . . . . i

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

0 25 50 75 100 125 150 175

e Hue

( 147_ , 05 T = 1 9 . 2 + 0 . 7 5 x l n 1 5 7 _ h u e j (10)

valid for 10 < hue < 156, corresponding to 19.2~ _< T < 20.9~ No reliable temperature information can be obtained outside this limited range of 1.7~ Note that since the red start (hue~10) corresponds to T~19.2~ and the blue start (hue~130) to T~20.2~ the actual colour play is even smaller, ~I~ As anticipated, both the red start and the colour-play values are quite different from the manufacturer's indication relative to standard conditions (20~ and 12~ respectively), which were independently confirmed by preliminary tests. In particular, the colour play is far smaller than that obtained for thin layers on a black background. This large discrepancy confirms the necessity of a specific colour-temperature calibration for each TLC mixture to be used.

Hue profile 6.E-02 �9

5.E-02 �9 �9 B

_ t _ 4.1=_02 . . . . . . . . ~ -

~, 3.E-02 ,aAA I

2.E-02 ' ~

A 0.E+00 �9 !

0 100 200 Hue

6,E-02

5,E-02

4.E-02

3.E-02

2. E-02

1 .E-02

0.E+00 18

T profile

i

&,

19 20 21 T (~

22

Fig. 6a-e. TLC calibration, a Digitized image of stable stratification; b hue (H) map obtained by HSL splitting; c H distribution on a 3x3 mm grid; d vertical profiles of H, obtained by x-averaging the map in graph c, and of T, obtained by assuming a linear profile b e t w e e n T H and To; e T vs. H. Symbols experimental results from graphs d; line fit

7 Extraction of in-plane temperature distributions The sequence of processing steps followed for the extraction of quantitative in-plane temperature information from the digitized images is illustrated in Fig. 7 for plane B (long cross-section of the enclosure).

Figure 7a is a raw digitized image obtained by a single flash pulse at fi5.6. As will be discussed in greater detail in Sect. 9, this image shows a flow pattern mainly characterized by four transverse rolls; blue (hot) fluid travels upward in the central region and near the two sides of the enclosure, while cold (red) fluid moves downward at two intermediate locations. T h e differences observed between the left and right halves of this cross-section are presumably due to a small, residual asymmetry in the thermal boundary conditions at the side-walls.

b

a

C

165

d Fig. 7a-d. Image processing for a typical cross-section of the enclosure (plane B). a "Raw" image obtained by a single-pulse exposure at ~.5.6; b image after edge clipping, resizing, lightness compensation for beam attenuation and saturation enhance- ment; c hue (/-/) component; d lightness (L) component

The image was now clipped, resized to 2,560x640 pixels and compensated in its lightness component for light attenuation along the flash beam path as discussed in Sect. 5. The result is shown in Fig. 7b, in which the saturation component was enhanced for the sole purpose of a better visualization (saturation is not actually used either for temperature or velocity extraction). Finally, the image was split into its HSL (hue-saturation-lightness) components. Corresponding hue and lightness maps, represented as BW images by 256 grey levels, are shown in Fig. 7c and d, respectively.

The subsequent steps for temperature extraction are Rlustrated in Fig. 8. Figure 8a is a map of the hue component, interpolated on the above-mentioned 3• mm grid after setting the cutoff at 170 as discussed in Sect. 6 on calibration. The conversion of hue into temperature by the calibration curve (Eq. 10) yields the T distribution in Fig. 8b. Minimum and maximum temperatures in this map are ~19.4 and ~-,20.8, respectively, which are very close to the known temperature boundary conditions of the test (Tc=19.45~ T~=20.95~

8 Extraction of in-plane velocity distributions There are two main approaches to PIV analysis, both implemented as alternative options in the software package VISIFLOW. The first (correlation analysis) analyzes one region of the flow at a time, and determines the dis- placement of the particle field that provides the highest correlation between the original and the displaced images of that region. The second (particle-tracking analysis) analyzes the position of each individual particle and at- tempts to match it with its displaced image(s). In both cases, velocities are then trivially derived from displace- ments provided the interval between subsequent images (light pulses) is known. Both techniques can be applied either to single-frame or multiple-frame PIV data, with corresponding differences in the details, and an option for more than two exposures in the same frame explicitly provided. Each technique has advantages and disadvan- tages, and each is more relevant to particular kinds of flows. In the present work, as preliminary comparison tests confirmed, the correlation technique gave the best results, since:

166

a Plane B - Hue map * 0.06-~ O,04q /2"~

~ l/

0.00-~ 1 r 0.00 0.02 0.~34 0,06 0.08 0.10 0.~12 0.14 0.16 0.18 0.20 0.22 0.24

b P l a n e B - T m a p * 0 ~ = ' J i

0 .0 "06- ' - ; ' '

G %J%!~ "~,"

0 0 - : : - - ~ : ~ " . 0 -

0.00 0,02 0.04 0.06 0.08 0.10 0.12 0.14 0. '~6 0.18 0.20 0.22 0.24

1 3 o

5 O

3 0

1 0

. . . . q~ 2 0 _ 9 ~ 2 0 , 7

~ 2 0 . 5

2 0 . 3

2 0 . 1

1 9 . 9

1 9 . 7

1 9 . 5

1 9 . 3

Fig. 8a, b. Reconstruction of the 2D temperature field for plane B (Tc=19.45~ TH=20.95~ a Hue map; b temperature map obtained by using Eq. (t0)

| The seeding density is quite high. For a TLC concen- tration of 300 ppm in volume and an average diameter of the TLC particles of,~0.1 mm (volume ,--.5x10-3mm3), the resulting numerical density of particles is ~60/cm 3. Since the thickness of the light sheet is ~0.5 cm on average (see Sect. 2), 1 cm 2 of any cross-section corresponds to an illuminated volume of ~0.5 cm 3 and thus contains, on average, about 30 particles, The 60•215 mm cavity (volume=l,728 cm 3) contains "--0.5 cm 3 of TLC, scattered between ~105 particles. These estimates are confirmed by the analysis of the actual images as reported, for example, in Fig. 7. In multiple-exposure images like those in Fig. 5, obtained by eight flash pulses, each particle leaves eight distinct images so that some 240 particle images can be observed per square centimeter.

Q Also the background noise is high. Unlike common PtV (in which the background is usually very dark, and the images are basically in black and white), the present technique is based on images in full colour, and the background is bright because of scattered light; these characteristics greatly reduce the luminance contrast between particle images and background. Another source of noise is the graininess of the photographic emulsion, owing to the necessarily high sensitivity of the film used.

A second, crucial option is whether consecutive exposures have to be recorded on a single frame or on distinct, consecutive frames.

The multiple-exposure technique, leading to single- frame correlation analysis for PIV, is simpler and was actually adopted in previous work (Palazzolo et al. 2000) with acceptable results. However, dramatic improvements in the results were obtained when a multiple-frame technique was tested, so that this second alternative was chosen in the present work. The main advantage of the multiple-frame method is that it eliminates the directional ambiguity inherent in the single-frame correlation technique, which makes it particularly effective in the presence of recirculafion. Of course, the extremely slow nature of

the flow, which calls for an optimum interval between flash pulses of the order of 30 s (see Sect. 4.2 on exposure), makes reloading and film advancement problems completely irrelevant in the present application.

An example of the results obtained by the two-frame cross-correlation technique is shown in Fig. 9 for plane B parallel to the long sides of the enclosure, i.e. the same plane for which temperature reconstruction was illustrated in Sect. 7 and Figs. 7 and 8. Figure 9a is a velocity vector plot as produced by VISIFLOW, with superimposed shaded contours of vorticity. The velocity and vorticity scales are also reported. The basic flow pattern is clearly recognized and consists of four transverse roils, with fluid moving upwards in the central region of the enclosure and near the end walls (up-welling plumes), and downwards at two intermediate locations (down-welling plumes). Velocities are very" low, attaining maximum values of ,--30 ~tm/s in the up-we]ling and down-welling plumes; corresponding values of vorticity are of the order of 10 -3 s -1. Figure 9b reports the corresponding flow-visualization, multiple-exposure image obtained by recording eight consecutive flash pulses at 20 s intervals on the same frame. The image was slightly corrected for light attenuation and its saturation was enhanced for a better readability. The correspondence of flow and thermal structures in Fig. 9b with the velocity vector plot in Fig. 9a is easily recognized; up-welling plumes are associated with hot (blue) fluid, whereas down-welling plumes are cold (red). The two rolls on the right are quite regular in shape and are associated with well distinct velocity and vorticity patterns, whereas the two rolls on the left are less well defined because of 3D effects. The second roll from the left is, in fact, partly split into two recirculation regions. This feature is clearly reflected in the more irregular behaviour of vorticity levels and velocity vectors in Fig. 9a.

9 Three-dimensional interpolation As the last stage in the tomographic reconstruction of 3D flow and thermal fields, a simple Fortran program was

a

b

t

N

6E-a

0 Vorticity (s -~) - .

-6E-3

u=O.1 mm/s

Fig. 9a, b. Reconstruction of in-plane velocity fields by using VISIFLOW in two-frame cross- correlation mode (plane B parallel to the long sides of the enclosure), a velocity field plus shaded contours of vorticity (blue: positive, or anti-clockwise; red: negative, or clockwise). The velocity and vorticity scales are reported, b Flow-visualization (eight-pulse) multiple-exposure photograph of the same plane

specifically written to interpolate the experimental velocity linearly interpolated from the experimental data at points and temperature distributions (obtained on planes A, B ..... A and B; and values of the vertical velocity component v or parallel to the long sides and planes 1, 2 ..... parallel to the of the temperature T were obtained by bflinear interpo- short sides) on a 3D grid.

The nomenclature adopted for the reference axes and for the associated velocity components was reported in Fig. 3c. Note that only the (u, v) velocity components are available in the long planes and only the (w, v) components on the short planes. Thus, experimental information on the vertical velocity v is available at more locations than that on the horizontal components u and w. On each plane, temperature and velocity components are defined at the centres of a 3x3 mm grid, thus including 80x20 values of u, v, T in sections A, B ..... and 40x20 values of w, v, T in sections 1, 2 .... (Fig. 10).

Prior to any interpolation, the four side boundaries were added to the experimental cross-sections proper; here, all velocity components were set to zero (no slip conditions) while temperatures were quadratically, extrapolated to the walls from the two nearest experimental planes by assuming zero heat flux conditions on the side walls. In formulae:

T~ = 4 T ' - I T " (11)

being T" and 7" the experimental temperatures on the nearest and second nearest plane (e.g. planes A and B for the front boundary, planes 1 and 2 for the rear boundary). Since the thermal conductivity of the acrylic glass, ~0.12 Wm -~ K -~, is only about half that of glycerol, this adiabaticity assumption is not strictly valid - however, it is preferable to a simple linear extrapolation from inner values.

Now, for any point P located at the centre of a cell of the interpolation grid: values of the u velocity components were linearly interpolated from the experimental data at

lation from those at A, B, C, D as:

~ = (~A(1 - r + ~Br + (OC(1 -- r + ~Dr

in which O=v or 7 and ~=(xp-xA)lA, ~=(z~-zc)lA.

(12)

x

1 2 3 4 5 ...... 6 7 8 9 10 11

@@@i iiiiiiiiiNNiiiiiiNN

~ k k

-F

C i i i

. . . . . . . .

1

1 i t 1

15 A +1

B A

Fig. 10. Planform of the enclosure, showing the experimental planes A, B ..... and 1, 2 ..... together with the 3x3 mm interpolation grid. See also Fig. 3c. The inset shows how values of velocity

points C and D; values of the w velocity components were or temperature are interpolated from experimental data

167

Examples of the 3D fields are illustrated in Fig. 11, 12 and 13. These were all obtained from the 3D arrays containing the interpolated Variables u, v, w, T by using the post-processing computer program CFX-View, which is normally used as part of the CFX-4 CFD code (AEA Technology 2000).

Figure 11 reports an axonometric view of the enclosure with contours of low speed ([v]<12• s -i) on the

horizontal mid-plane y=D/2. Superimposed on these are velocity vector plots on two longitudinal cross-sections coinciding with the experimental planes B and D. Planar views of these latter plots are also reported, for greater readability, in the bottom part of the figure. The colour of the vectors is associated with speed values in the range 0 to ~63x10-6m s -x, as shown by the enclosed key. Low-speed contours visualize the axes of the four transverse rolls

168

D

B

6 . 5 3 0 9 E - 0 5

5 . 4 4 2 5 E - 0 5 ..... 4 . 3 5 4 0 E - 0 5

3 . 2 6 5 5 E - 0 5 2 . 1 7 7 0 1 E - 0 5 I . 0 8 8 5 / ~ - 0 5 O.O000E+O0

B

D

Fig. II. Reconstructed 3D fields: axonometric view of the enclosure showing isolines of low speed (roll axes) on the horizontal mid-plane and velocity vector plots on selected longitudinal planes. The colour of vectors indicates speed (see key in m/s)

~ 6 .5309E-05 5 .4425E-05 4 .3540E-05

~ 3 .2655E-05 2. 1770E-05 1 .0885E-05 O . O O O O E + O O

5

169

i ~' ~ ~ " V '�84 , " ' i

Ot

...................................................................................... 7 ~ " 7 7"7"7 ? ..................

Fig. 12. Reconstructed 3D fields: axonometric view of the enclosure showing isolines of low speed (roll axes) on the horizontal mid-plane and velocity vector plots on selected transverse planes. The colour of vectors indicates speed (see key in m/s)

which clearly stand out as the main flow structures exist- ing in the enclosure. However, these transverse rolls are quite irregular in shape and exhibit meandering, rather than straight, axes. A residual overall asymmetry between the left and right half of the enclosure can also be appreciated. The comparison of the velocity vector plot for section B in Fig. 11 with that in Fig. 9a, which refers to the same plane but before interpolation, shows that the above- described interpolation procedure leads to slight changes in the results also in-plane sections of the 3D grid that

happen to coincide with experimental planes. In fact, in force of Eq. (10), at any point of the interpolation grid, also the two adjacent orthogonal planes always contribute for one half to the reconstructed values of the vertical velocity v (the same holds for temperature T). Interpolated fields are always more regular than "raw" fields.

Figure 12 reports the same axonometric view of the enclosure with mid-plane low-speed contours as Fig. 11, but superimposes on these the velocity vector plot in four transverse planes c~-fi, chosen so as to show recirculation

17o

a

b

~N- 6 . 0 0 0 0 E - 0 5 4 . 4 7 6 2 E - 0 5

:~ 2 . 8 5 7 1 i f - 0 6 - I . 8 0 9 5 E - 0 5 - 3 . 9 0 4 8 E - 0 5 - 6 . 0 0 0 0 E - 0 5

~,%~ 2 . 0 9 5 0 E ' r ' 0 I 2 . 0 6 8 8 E + 0 1 -.2.0426 E + 0 I

, 2 2 . 0 1 6 4 E + 0 1 i 1 . 9 9 0 2 E + 0 1

1.96.-q-0 E + 0 l < 1 . 9 4 5 0 E + 0 1

regions, particularly evident in plane 6. These may appear at first glance as cross-sections of longitudinal roils, so that, as mentioned in Sect. 5, the flow in the cavity may appear to consist of simultaneously present transverse and longitudinal rolls. However, a careful examination of the whole flow field reveals that recirculation regions in transverse cross-sections arise simply from the fact that the transverse rolls are not strictly aligned with the short sides of the enclosure, but exhibit a meandering shape that is not caused by the presence of longitudinal rolls proper. Note that the two recirculations that can be observed in each of the short planes shown are counter-rotating (the left one counter-clockwise, the right one clockwise) in planes cz and 6, but co-rotating (both clockwise) in planes fl and ~. Such behaviour is incompatible with the existence of longitudinal rolls, but is perfectly explained by considering the angles at which the section planes cut the transverse rolls.

False-colour shaded contours of the vertical velocity component v in the horizontal mid-plane of the enclosure (range -60 to +60 ~m/s) are shown in Fig. 13a. The v distribution evidences the alternate transverse up-welling

and down-welling regions, already observed in longitudinal cross-sections, and makes evident the irregularity of the right-hand side down-welting flow structure, which is practically split into two sinking plumes of cold fluid. The same features can also be appreciated by considering the 3D velocity vector plot in the endosure mid-plane

Fig. 13a, b. Reconstructed 3D fields, a distribution of the vertical velocity component v in the horizontal mid-plane of the enclosure (key is in m/s); b 3D velocity vectors in the same mid- plane (the colour of vectors indicates temperature, see key in ~

reported in Fig. 13b. Here, the vector colour is associated with temperature as indicated in the enclosed key. The temperature distribution evidences a residual left-right asymmetry, associated with a slight horizontal temperature gradient, which is visible also in Fig. 5, 7, 8 and 9b. Note that only with the use of 3D tomography is information on the flow discovered that is not immediately contained in the experimental data.

10 Experimental uncertainty A rigorous assessment of the experimental uncertainty associated with the experimental technique presented in this paper is not simple. The whole approach should be regarded as a visualization method augmented by techniques for the quantitative extraction of temperature and velocities; thus, its main advantage resides in the simultaneous characterization of whole flow and thermal fields, rather than in the accuracy by which local quantities can be measured. However, some estimates of the experimental uncertainty will be given here for the sake of completeness.

Temperature ~ be considered first. Most of the final uncertainty in any local value will eventually arise from (a) the in situ calibration and (b) the uncertainty in the observed hue. Other sources of error (such as random errors in the measurement of boundary temperatures and position errors related to the identification of the physical

boundaries of the enclosure) are expected to play a minor rote. Systematic errors in the boundary temperature measurements cancel out if the final temperature distribution is normalized as O=(T-Tc)/(TH-Tc).

As regards calibration, a hue-temperature calibration curve was obtained only in the limited range T=19.2- 20.9~ Within this range, the dispersion of experimental data with respect to the curve is of the order of +0.1~ for any given hue (see Fig. 6). This corresponds to an uncertainty of a few percent in the normalized temperature 0 (_+6% if the hot and cold wall temperatures established under convection conditions coincide with the above upper and lower calibration limits). If the imposed boundary temperatures T~, Tc exceed the calibration limits, there will be upper and lower boundary regions characterized by a large temperature uncertainty.

As regards hue, that observed for individual pixels is heavily affected by noise sources such as film graininess, light reflections and, above all, the discontinuous nature of the medium (TLC particle suspension). However, the associated random fluctuations average out when finite regions of the image are considered. This holds, in particular, for the 3x3 mm square cells considered for interpolation purposes, each containing 1,024 pixels and (as discussed in Sect. 8) three or more TLC particles. An order of magnitude for the residual uncertainty is pro~.dedby the fact that, in stable stratification images, 3x3 mm cells belonging to the same horizontal row (which are expected, on physical grounds, to be at the same temperature) exhibited a standard deviation of 2-3 units in their mean hue, corresponding to a temperature uncertainty of 0.05~ at most on the basis of the calibration curve.

Coming now to the velocity issue, the two-frame cross- correlation technique adopted in the present study has an intrinsic limit in the spatial accuracy by which maxima in the cross-correlation function of consecutive images (reduced to their lightness components) can be located. By assuming this accuracy to be of the order of one pixel (90 ,am of real size), the corresponding uncertainty in the velocity module is of the order of ~3 ,am/s since the interval between frames is 30 s. This amounts to a typical relative uncertainty of the order of -+10% since the prevailing speed in the flow field is of the order of 30 pm/s as discussed in Sect. 4. Larger relative uncertainties in speed, as well as large directional uncertainties, may arise in the flow regions close to recirculation centres, where, however, the issue is of little relevance.

As discussed in Sect. 5, further errors in the velocities (in this case of a systematic nature) might arise from position errors in the superposition of consecutive frames; these errors were kept to a minimum (hopefully zero) by manually checking the correct superposition of the images.

Of course, all the above remarks hold only for "true" experimental data, obtained for the longitudinal and transverse planes A ..... E and 1 ..... 11. At intermediate locations, the 3D interpolation discussed in Sect. 9 gives rise to further errors that are difficult to quantify as they depend on some theoretical model of the flow and temperature behaviour between experimental points.

11 Conclusions and future work The feasibility of simultaneous 3D particle-image velocimetry and thermography was demonstrated for Rayleigh-B~nard convection in shallow rectangular enclosures, using glycerol as the working fluid and suspended TLC particles as flow and temperature tracers. Attention was focussed here on the case of a cavity having an aspect ratio of 1:2:4 at a Rayleigh number of ,~14,500. A careful calibration of the TLC allowed the quantitative reconstruction of temperature maps in different longitudinal and transverse cross-sections of the flow domain. The VISIFLOW package (used in two- frame, cross-correlation mode) provided a satisfactory reconstruction of in-plane velocity fields from digitized couples of images of each cross-section. Purpose-written software was used to interpolate the above in-plane distributions of velocity and temperature onto a 3D fine grid, thus yielding 3D flow and thermal fields.

On average, the experimental uncertainty associated with this experimental technique was estimated to be of the order of +10% both for the fluid velocities and for the (relative) temperatures. Of course, such an accuracy is lower than that allowed by more specialized techniques such as laser-Doppler velocimetry, conventional particle- image velocimetry with laser-generated light beams and dedicated tracers, and local temperature measurements. Nevertheless, it is amply sufficient to resolve all the relevant flow and thermal structures without ambiguities, while the whole-field nature of the method, its overall simplicity and low cost, and the possibility of simultaneously recording the flow and temperature fields, make it particularly suitable for broad-range parametrical studies, e.g. to investigate the influence of aspect ratios, inclination angle and thermal boundary conditions on free convection.

The main limitations of the technique, at least in its present form, lie in the restriction to steady-state flows (imposed by the inevitable time lag by which different portions of the flow are investigated), and in the small temperature range (imposed by the thermochromic response of suspended liquid crystals). Further limitations, such as those related to the minimum interval between frames and to the optical resolution, are not specific to the present technique but rather to PIV in general, and could be significantly relaxed by improvements in the light source and in the photographic hardware.

For the specific case analyzed here, the main flow pattern was found to consist of four transverse rolls, which, however, were not exactly orthogonal to the long sides of the enclosure but exhibited irregular and twisted shapes. This feature caused vortical structures to appear also in transverse cross-sections of the enclosure, but a careful analysis of the reconstructed 3D flow field revealed that no longitudinal rolls proper were involved.

In the immediate future, further work will include the study of free convection in a similar enclosure at different Rayleigh numbers, for which a simpler circulation pattern is expected. This will be followed by the extension of the study to cavities with different aspect ratios and thermal

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b o u n d a r y cond i t ions at the side-walls. Hopeful ly , results will usefully c o m p l e m e n t theore t ica l work on pa t te rn se- lec t ion in Ray le igh-B~nard convec t ion and will p rov ide the basis for the appl ica t ion o f the same t echn ique to o ther f low problems .

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