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Turbulent convection from isolated sources
Dynamics of Atmospheres and Oceans 30 Ž1999. 125–148 www.elsevier.comrlocaterdynatmoce
Turbulent convection from isolated sources J. Colomer a , B.M. Boubnov b, H.J.S. Fernando
c,)
a
Department of EnÕironmental Sciences, UniÕersity of Girona, 17071 Girona, Spain A.M. ObukhoÕ Institute of Atmospheric Physics, Russian Academy of Sciences, 109017 Moscow, Russia Department of Mechanical and Aerospace Engineering, EnÕironmental Fluid Dynamics Program, Arizona State UniÕersity, P.O. Box 879809, Tempe, AZ 85287-9809, USA
b c
Received 15 September 1998; received in revised form 28 May 1999; accepted 1 June 1999
Abstract Laboratory experiments were conducted to investigate the evolution of a dense turbulent plume, specified by its buoyancy flux Bo and source diameter D, issuing into a homogeneous, nonrotating, environment. This study was motivated by the desire to delineate velocity and buoyancy scaling for convection from isolated buoyancy sources of finite extent. Such flow configurations have relevance to geophysical Že.g., deep convection., environmental Že.g., urban heat island effect. and engineering Že.g., plume stacks. flows. Special attention was given to study the evolution of the plume following its initiation and the flow near the source when the influence of confining boundaries is insignificant. It was found that, for times t - 1.2Ž D 2rBo .1r3 , the descent of the plume front can be treated as one-dimensional with negligible lateral Žentrainment. mean flow, the plume growth mechanism being the encroachment of underlying nonturbulent fluid. At larger times, the flow achieved a quasi-steady state, in which the plume width first decreases up to a distance 0.28 D Žregion I. and then increases Žregion II.. The quasi-steady state velocity and buoyancy measurements in region I showed that they are strongly influenced by the lateral entrainment flow Žand hence, by D ., and thus classical free convection scaling is inapplicable. On the other hand, at large zrD Žin region II., the velocity and buoyancy scaling tend to be independent of D, indicating the fading influence of source diameter effects. The results establish basic scaling for convection from isolated, but distributed, sources and provide a baseline with which future work that incorporates background rotation and stratification can be compared. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Convection; Plume; Velocity and buoyancy scaling
)
Corresponding author. Tel.: q1-480-965-2807; fax: q1-480-965-8746; e-mail: [email protected]
0377-0265r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 0 2 6 5 Ž 9 9 . 0 0 0 2 3 - 8
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1. Introduction Turbulent convection from isolated, but distributed, sources is a widespread phenomenon in environmental and engineering flows. Some examples of consequence in ocean and atmospheric dynamics are the deep-convection, development of cumulous clouds and urban heat island flows. In engineering context, plumes emanating from wide plume stacks are common in power plant cooling. Oceanic and atmospheric deep convection are characterized by the formation of intense convective regions that penetrate into greater depths. Because of the presence of density stratification, for example, the thermocline in the ocean and inversion layers in the atmosphere, the vertical scale of geophysical convection, is limited to shallow depths, but under special conditions, convective motions can penetrate deep into stratified layers. In oceans, convective fluid elements typically penetrate a few hundred meters ŽAnis and Moum, 1994., but in deep convective regions Žknown as ‘‘chimneys’’., these elements descend up to a few kilometers, the width of the chimneys being 10–100 km ŽMaxworthy, 1997.. Cloud convection is characterized by rising of air currents beyond the level of dew point, wherein the condensation of water vapor and associated latent heat release drive convection ŽLudlam, 1980.. Under unstable conditions, clouds can grow to great vertical extents Že.g., violent thunderstorms., on the order of tens of kilometers, with hundreds of kilometers horizontal scale Žatmospheric deep convection.. Under more stable conditions, the extent of clouds is smaller, with several hundred meters in the vertical and several kilometers in the horizontal, characterizing cumulus clouds. Urban heat islands arise due to the presence of elevated temperatures in localized urban areas relative to their rural surroundings, which drives a circulation due to horizontal temperature gradients. The resulting flow patterns can have a significant impact on the microclimate of cities ŽBornstein, 1987.. The above examples represent, though the details vary, convection from isolated sources of large horizontal extent. In most situations, the horizontal scale Ž D . of convection exceeds the vertical scale Ž H ., thus making the flow approach that of horizontally homogeneous convection Ž DrH ™ `., where the free-convection scaling of Deardorff Ž1970. applies ŽKaimal et al., 1976.. If the buoyancy flux per unit area is Bo , then the Deardorff scaling for length and velocity at normalized distances from the source zrH ) 0.15 becomes H and Ž Bo H .1r2 , respectively. For zrH - 0.15 and beyond the molecular-diffusive sub-layer adjoining the source, the flow obeys local scaling, z and Ž Bo z .1r3, respectively, indicating the unimportance of H ŽAdrian et al., 1986.. A question arises, however, on the role of source diameter effects near the source Ž zrD - 1. for cases with finite values of DrH. In such cases, D is expected to play a role because of the possibility of developing an entrainment Žmean. flow through lateral boundaries. It is the influence of finite diameter of the source that we wish to investigate in this study. The questions of particular interest are: ‘Is the free-convection scaling applicable to finite-diameter sources?’; If applicable, ‘Under what conditions is it valid?’ and if not applicable, ‘What should be the new scaling?’ The flow configuration considered consists of a disk-shaped source of buoyancy Žof diameter D . releasing into a fluid of large horizontal extent and depth H. The important dimensional governing variables for the problem, excluding the molecular parameters,
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are Bo , D, H and z. The measurements, however, were confined to regions zrH 0.8–0.9, wherein the depth Ž H . effects are insignificant for propagating plume fronts and also for plumes that have already impacted on the bottom surface ŽNoh et al., 1992.. Although a large number of scaling studies have been reported on asymptotic cases of horizontally homogeneous sources DrH ™ ` ŽAdrian et al., 1986; Castaing et al., 1989; Siggia, 1994. and point and line plumes and thermals DrH ™ 0 ŽPapanicolaou and List, 1988; Ayotte and Fernando, 1994; Helfrich, 1994., the case of finite diameter sources has received only little attention. More so, existing studies with finite-diameter sources have dealt with specific geophysical situations, and only little attention has been given to establish or verify scaling laws in detail Že.g., oceanic convection work of Maxworthy and Narimousa, 1994 and urban heat island studies by Lu et al., 1997a,b.. This paper shows that the finite-diameter effects can indeed lead to new scaling in the vicinity of the source. The present results are strictly valid only for sources in nonrotating fluids or for flows with insignificant Coriolis effects. It is acknowledged that the effects of rotation can be important in geophysical cases, perhaps from the outset of convection, because of the large horizontal extent and small velocities of the flow. For example, during the development of convection in a rotating fluid, small-scale vortices can be developed near the source due to the breakdown of unstable buoyancy gradients, thus making the flow different from its nonrotating counterpart ŽChen et al., 1989; Klinger and Marshall, 1995; Marshall and Schott, 1998.. In geophysical situations, the appearance of such vortices is rare, but laboratory and numerical studies confirm their presence. Although these vortices may not significantly change the integral properties of turbulence near the source ŽFernando et al., 1991., they can lead to anomalous mixing characteristics at some distances away from the source ŽJulien et al., 1999.. The extension of present nonrotating convection results to geophysical situations, therefore, must be done with caution. Geophysical situations are often dominated by stratification effects, which also should be considered in extrapolating laboratory results obtained with homogeneous backgrounds to natural situations. The present results, nevertheless, provide baseline data and scales with which future modeling involving background rotation and stratification can be compared. Section 2 presents the details of the experimental apparatus and procedure. Section 3 describes the experimental results. A summary of the study is given in Section 4 and in Section 5 a discussion on practical implications of results is given.
2. Experimental apparatus and procedure The experiments were conducted in a cylindrical Plexiglas tank of diameter D s 107 cm and height 60 cm ŽFig. 1.. This tank was placed inside a larger square Plexiglas tank of internal dimensions 108 = 108 cm2 to minimize optical distortions arising due to tank curvature when viewing from the sides. The source was made of a short circular cylinder, which was packed with several layers of porous material Ž1-cm thick sponge and two filter paper layers.. These layers were held in place by a bottom mesh lid of mesh size 1 mm. This design ensured uniform distribution of salt water over the entire
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Fig. 1. A schematic diagram of the experimental apparatus. The nature of the plume at large times is illustrated with near ŽI. and far ŽII. field regions.
source cross-section when the salt water is fed from the top via a constant head reservoir. The salt water seeps through the porous layers and drains into the underlying water with a velocity wo . If the buoyancy of the resulting plume fluid is D bo s Ž rp y ro . grro , where rp and ro are the densities of the plume and the background fluid, respectively, and g is the gravitational acceleration, then the buoyancy flux per unit area Bo , the total buoyancy flux Qo , the momentum flux Mo and the volume flux Vo can be written as Bo s D bo wo ; Qo s
p D 2 Bo 4
; Mo s
p D 2 wo2 4
; and Vo s
p D 2 wo 4
.
Ž 2.1 .
To check the ‘‘plume’’ vis-a-vis the ‘‘jet’’ nature of the source, the length scale ` LM s
Mo3r4 Qo1r2
Ž 2.2 .
was used, analogous to the momentum length scale used in point buoyant jetrplume studies ŽList, 1982; Papanicolaou and List, 1988.. At distances z ) L M from the source, the momentum flux can be regarded as unimportant, and the experimental parameters
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were selected to minimize L M while maintaining the buoyancy flux in a reasonable parameter range. The L M values used varied in the range 0.01 - L M - 0.36 cm. Source diameters of D s 3.1, 9.5 and 19.1 cm were used, but most experiments were conducted with D s 19.1 cm. The working depth of the fluid H was held constant at 53 cm. The plume density was varied over the range 1.008 - rp - 1.071 grcm3 , with rp ) ro . The buoyancy of the plume fluid and the buoyancy flux were varied in the ranges 7.8 - D bo - 132 cmrs 2 and 0.4 - Bo - 56.7 cm2 sy3 . The experiments were initiated by releasing salty water from the constant head reservoir into the plume source. The plume fluid was added with a small amount of fluorescein dye, enabling its visualization under the illumination of a sheet of laser light generated by a laser-beam scanner ŽDeSilva et al., 1990.. The vertical thickness of the laser sheet was 8 cm, which imposed a limit on the vertical field of view. In some cases, intense arc lamps were used to produce light sheets that span the entire depth of the water column. The flow was video-recorded and the tapes were later analyzed using standard techniques of laser-induced fluorescence ŽLIF.. A software package, DigImage, was used for this purpose Žfor details, see Dalziel, 1992.. For experiments with large D bo , refractive index variations across plume boundaries were found to have a significant effect on LIF measurements. In such cases, the refractive indices between plume and background fluids were matched by adding alcohol to the background fluid Žfor details, see Hannoun et al., 1988.. To check the accuracy of the LIF imaging technique, limited measurements were also made using two stationary conductivity probes placed at various distances below the source. Both techniques measured mean concentrations within "5%. In a series of experiments, the velocity field measurements were made using particle tracking velocimetry ŽPTV.. Here, the background fluid was seeded with neutrally buoyant particles of approximate diameter 0.3 mm and the entrainment flow into the plume was tracked using DigImage software. The accuracy of the velocity measurements was estimated to be "10%.
3. Finite-sized plumes in homogeneous fluids This section presents qualitative observations and quantitative measurements of the experiments described above. 3.1. General flow description Detailed observations made during the plume descent revealed the following. Ži. The thin dense fluid layer that develops under the source, immediately following the release of plume fluid, breaks down into small-scale turbulence via Rayleigh–Taylor type overturning instabilities. The visually estimated time at which this breakdown occurs is assigned as t s 0. The resulting plume fluid near the source develops a ‘‘neck’’ Žcontraction of the width. just below the source Žsee Fig. 2Ža. and Žb.. This observation can be attributed to enhanced mixing along the periphery of the source, which entails strongly buoyant fluid near the center of the plume and lighter Žslow
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Fig. 2. Ža–e. The evolution of the plume Žobtained using LIF images. with time Ž t .. The surface region corresponds to half the source diameter Ds19.1 cm. Bo s1.07 cm2 sy3 . The pseudocolor representation used is shown below Že.. The background corresponds to black and the lowest and the highest concentrations are represented by red and purple, respectively. The brown in the background appears as a result of noise and diffused light.
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Fig. 2 Žcontinued..
descending. fluid at the periphery. The relatively lighter fluid produced in the periphery is dragged down by the dense source fluid residing toward the center of the plume, thus producing the ‘‘neck’’. Detailed measurements of fluid emanating from the source at different radii showed that the buoyancy flux is uniform across the source, thus indicating that the appearance of the neck is not an artifact of inhomogeneities of the plume, but a characteristic of finite-diameter plumes. Žii. The turbulent dense fluid layer near the source starts to spread downward, lowering its ‘‘neck’’ and expanding its width below the neck. For a time of about TD f 1.2Ž D 2rBo .1r3 , the depth of the plume front h follows the propagation law h ; Ž Bo t 3 .1r2 , much the same as that in horizontal homogeneous convection ŽFernando et al., 1991.; also, see Žiii. below and Section 3.2. This observation is consistent with the fact that no significant lateral entrainment flow is evident during this period, either from velocity measurements or LIF visualization. Concentration distribution within the plumes based on LIF imaging is shown in Fig. 2Ža. – Že., where the pseudocolor representation varies from red Žthe lowest detectable concentration. to purple Žthe largest concentration.. Note the appearance of high-concentration regions interior to the locally mixed Žand elongated. fluid layer at the edge of the plume at t - TD , indicating negligible entrainment of ambient fluid; this is also corroborated by the PTV data that showed lack of entrainment into the plume. Žiii. At t - TD , the propagation of the plume front appears to be dominated by the engulfment of fluid from below Žand to a certain extent from the sides. by turbulent
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eddies. This is seen in Fig. 2Ža. – Žd., where fluid parcels from the immediate vicinity of the plume Žred. are seen to engulf into the turbulent layer, thus creating strong concentration inhomogeneities within the plume. Entrained fluid parcels, however, are distorted and homogenized within the plume quite rapidly. Živ. After the plume descends for a certain time, approximately 1.8Ž D 2rBo .1r3, the downward movement of the ‘‘neck’’ ceases, settling at a distance approximately 0.28 D from the source Žsee Fig. 2e and also Section 3.2.. The dense fluid continues to drift downwards, inducing an entrainment flow and expanding the plume laterally. The development of the entrainment flow is well evident from the PTV records taken during the experiments. Fig. 3 shows a resultant velocity vector plot, taken at various times, starting from the time where a lateral entrainment velocity could be discerned. The entrainment flow first appeared near the source, but with time, it spreads along the entire plume. Žv. Upon reaching the tank bottom, the plume front deflects and flows as a gravity current, which, being constrained by tank boundaries, accumulates fluid at the bottom of the tank. Flow structures in the plume do not change appreciably until the bottom dense layer rises to a height of about 0.2 H, where H is the fluid depth in the tank. The flow in the bulk of the plume can be considered quasi-stationary until this time. Thereafter,
Fig. 3. A vertical cross-section through the plume, indicating resultant velocity vectors at different times: Ža. t s 20 s, Ž Bo r D 2 .1r 3 t s 3.82; Žb. 40, 7.64 and Žc. 60, 11.46. The external parameters are Ds19.1 cm and Bo s 2.54 cm2 sy3 . The arrow at the top left indicates the scale.
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secondary flow cells become evident in the tank due to the confinement of flow by sidewalls. During the quasi-steady state, two distinct flow regions could be identified in the plume, and they are shown in Fig. 1. In Region I, the plume contracts until a minimum diameter is achieved and then the flow expands in Region II. Fig. 4 shows a streakline photograph taken in the quasi-steady state, using neutrally buoyant particles suspended in the background fluid. Because of the rapid increase of particle velocity upon entrainment into the plume, the particle tracks within the plume are not picked up by the resolution limits of the image processing system. Also note that the entrainment flow in the vicinity of the source converges toward region I, indicating
Fig. 4. A front-view streak photograph of flow field around the plume under quasi-steady conditions. The streaks are produced by pathlines of suspended particles in the ambient fluid. Ds19.1 cm and Bo s 2.54 cm2 sy3 . Regions I and II are defined in Fig. 1.
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high entrainment demand to replenish buoyant fluid leaving the source region. In region II, the flow is dominated by the entrainment of background fluid along almost horizontal streamlines. 3.2. Scales of flow adjustment As discussed in Section 3.1, after some time, the plume assumes a quasi-steady state, as evident from velocity and concentration field measurements. The parameters governing the plume evolution are D, Bo , D bo , n Žthe kinematic viscosity. and k Žthe molecular diffusivity of salt., where the influence of rprro is neglected based on Boussinesq approximation. In the experiments, the outer boundary of the plume was mapped using LIF, and the time scale te required for the ‘‘neck’’ Žor the minimum width. of the plume to achieve its steady depth z e was measured. Based on the above governing parameters, it is possible to expect te 2
Ž D rBo .
1r3
s f 1Ž E . ,
Ž 3.1 .
where f 1 , f 2 , . . . are functions of the dimensionless parameters E s D bo DrŽ Bo D . 2r3. Here, the dependence on molecular parameters Ž n and k . has been neglected by assuming Reynolds and Pecelt number similarities. As discussed in Section 3.1, the measurements were made either during the descending phase of the plume front or away from the bottom walls of the tank after the plume is impacted. No measurements were made after the plume fluid accumulating at the bottom rose beyond 0.2 H. The depth of the tank H, therefore, was not considered as an important governing parameter. Fig. 5 shows a test of Eq. Ž3.1., where the normalized te is plotted against E. The results show that terŽ D 2rBo .1r3 is independent on E in the range 5 - E - 120, with te 2
Ž D rBo .
1r3
s 1.78 " 0.14.
Ž 3.2 .
The scatter of data at E - 5 can be attributed to comparatively high exit velocities of plume fluid employed for those runs. Similarly, the depth of the neck z c ŽFig. 1. was also measured and the results are presented in Fig. 6. Each data point represents the average of 100 frames. The data are scattered, but only a weak dependence of z crD on E could be identified for E ) 5. The data for E ) 5 can be best represented using the mean value zc s 0.28 " 0.13. Ž 3.3 . D The mean Žtime-averaged. width LŽ z . of the plume is plotted as a function of depth in Fig. 7 Žsee Section 3.3.. By fitting a fourth-order polynomial to the width vs. depth data, the minimum mean Žneck. width of the plume Lo was evaluated ŽFig. 1. and the results were found to follow LorD s 0.55 " 0.05.
Ž 3.4 .
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Fig. 5. The normalized time te rŽ D 2 r Bo .1r 3 for the plume boundaries in 0 - z - D to achieve the steady state as a function of the parameter Es D bo DrŽ Bo D . 2r 3 . ^ — Ds 3.1 cm; I — Ds9.9; q — Ds19.1. 7.78- D bo -132 cmrs 2 .
The mean spreading angle of the time-averaged plume width for zrD ) 0.5 was 158 " 28. Another quantity of interest is the rate of propagation of turbulent front under the source, which ought to be dependent on D, Bo and t, given the insensitivity of results to the variations of E. The depth of the front h under the source, thus, can be expressed as h 3 1r2
Ž Bo t .
s f2
žŽ
D Bo t 3 .
1r2
/
.
Ž 3.5 .
A plot of hrŽ Bo t 3 .1r2 vs. DrŽ Bo t 3 .1r2 is shown in Fig. 8, which lends support for the functional form Ž3.5.. Two regimes of layer growth are evident from this plot: for DrŽ Bo t 3 .1r2 ) 0.80 or t - 1.2Ž D 2rBo .1r3, the growth follows f 2 s constantf 0.35 Ž 3.6a . and at larger t, it is found that h
Ž Bo t 3 .
1r2
s C1
žŽ
n
D Bo t 3 .
1r2
/
,
Ž 3.6b .
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Fig. 6. A plot of normalized distance z c r D from the source to the depth where the minimum plume width Žneck. occurs as a function of E. Symbols have the same meaning as in Fig. 5.
where C1 f 0.54 and n f 0.5. Note that Eq. Ž3.6a. implies that at t - TD , the growth law is independent of D, or h s hŽ Bo ,t . ; Ž Bo t 3 .1r2 as observed in horizontally homogeneous turbulent convective flows ŽFernando et al., 1991.. This observation supports the notion that the entrainment flow Žwhich brings forth the dependence of D . is insignificant at t F 1.2Ž D 2rBo .1r3 and the growth is occurring by the engulfment of fluid from below the front by turbulent eddies. When the frontal depth is h, these eddies have a velocity Ž Bo h.1r3 ŽFernando et al., 1991., and hence, a frontal propagation velocity of d hrdt ; Ž Bo h.1r3 is expected. Meanwhile, information on lateral processes are transmitted into the plume at a rate d xrdt ; Ž Bo h.1r3 ; Ž Bo t .1r2 , which is felt over the entire plume Ž x ; D . at a time scale of t ; Ž D 2rBo .1r3. 3.3. Velocity measurements A typical instantaneous velocity distribution map of a plume operating under quasi-steady conditions is shown in Fig. 9. The velocity distribution here has similarities
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Fig. 7. A plot of plume width bw vs. depth z. The experimental parameters are as follows: Ds19.1 cm and v — Bo s 2.71; ` — 2.63; q — 0.75; x — 6.28; B — 4.43; I — 7.80; ' — 2.19 and ^ — 1.35 cm2 sy3 .
to that of an expanding plume, with the maximum mean velocity first increasing and then decreasing. The width of the velocity zone also appears to decrease in region I Ž z - 0.28 D . and then increase, much the same as the plume width based on concentration profiles ŽSection 3.2.. The quasi-steady velocity field established near the source was mapped using PTV. The maximum horizontal velocity Vx in region I, measured by averaging the data taken from 200 frames, was analyzed to delineate velocity scales in that region. The data were found to correlate well with the scale Ž Bo D .1r3 , as shown in Fig. 10. The other scales considered for a possible correlation were Ž Bo z .1r3 and Ž Bo H .1r3 ; here, H is the depth of the fluid layer and z is the location where this maximum horizontal velocity occurs. The results clearly showed Vx s 0.66 Ž Bo D .
1r3
.
Ž 3.7 .
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Fig. 8. The variation of the normalized frontal position h rŽ Bo t 3 .1r 2 with the parameter DrŽ Bo r t 3 .1r 2 .
This velocity scale is consistent with the flow evolution scenarios described in Eq. Ž3.2., wherein the development of region I is completed upon feeling the lateral plume boundary effects over the plume width at a time t ; Ž D 2rBo .1r3. The velocity scale of turbulence at this time becomes Ž Bo h.1r3 ; Ž Bo t .1r3 ; Ž Bo D .1r3, which is consistent with Eq. Ž3.7.. The centerline vertical velocities Wc Žaveraged over 200 frames. measured using PTV as a function of zrD are shown in Fig. 11. The continuity condition implies VxrD ; Wcrz, and hence, the velocity in the vicinity of the source is expected to behave as Wc ;
Bo
1r3
ž / D2
z,
Ž 3.8a .
or in normalized form Wc
Ž Bo D .
1r3
s C2
z
ž / D
,
Ž 3.8b .
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Fig. 9. A whole field velocity vector plot, taken after the flow has become quasi-stationary Žat t s80 s.. Ds19.1 cm, Bo s 0.53 cm2 sy3 and Es 49.5. The arrow at the top left represents the scale. Regions I and II are also marked.
where C2 is a constant. The data show a fair agreement with Eq. Ž3.8b. for zrD - 0.3, except the data point taken very close to the source; based on the eye-fitted line drawn in Fig. 11, C2 f 2.7. For zrD ) 0.3, in region II, the data tend to deviate from Eqs. Ž3.8a. and Ž3.8b. and the vertical velocity evolves as Wc
Ž Bo D .
1r3
s C3
z
ž / D
n1
,
Ž 3.9 .
with a best fit line indicating C3 f 1.2 and n1 f 0.30. It is interesting that Eq. Ž3.9. implies Wc ; Ž Bo z .1r3 type behavior, independent of the source diameter, suggesting that the scales imposed by the source diameter are waning at large distances from the source, at least in 0.3 - zrD - 6. 3.4. Measurements of buoyancy The variation of instantaneous buoyancy within the plume and its evolution in time were depicted in Fig. 2, from which it is evident that the buoyancy in the plume is highly variable in space and time. Further support for this observation can be seen from
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Fig. 10. The maximum time-averaged horizontal velocity Vx observed in the vicinity of the source Ž z - 0.3 D . as a function of Ž Bo D .1r 3 . ` — Ds19.1 cm; $ — 9.5.
the quantitative data shown in Fig. 12. Here, instantaneous buoyancy profiles bŽ r,t . taken at z s 4 cm Ž zrD s 0.21. at two different times of the flow evolution are shown. Shown in Fig. 13 are the time-averaged mean buoyancy data b Ž r , z . taken at various radial Ž r . locations for different zrD values, as a plot between b Ž r , z . rŽ Bo2rD .1r3 and rrLŽ z ., where LŽ z . is the averaged width of the plume. Special attention has been given to small zrD values so that the effects of finite source-diameter can be delineated. Here, the location where the buoyancy falls below 10% of the maximum buoyancy of the profile was defined as an edge of the plume, and the plume width was evaluated by averaging instantaneous widths of 200 profiles. The normalization variables were selected based on the governing parameters z, r, D and Bo ; hence, variables such as LŽ z ., the time-averaged mean buoyancy b Ž r , z . and the radially averaged b Ž r , z . over the plume width bav can be written as LŽ z . D
s f3
ž /
bŽ r , z .
Ž Bo2rD .
z
1r3
Ž 3.10 .
D
s f4
ž
r
,
z
D D
/
Ž 3.11 .
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Fig. 11. The centerline mean vertical velocity Wc , normalized by Ž Bo D .1r 3 , as a function of the normalized distance from the source. I, $, `, ^ — Ds19.1 cm and Bo s1.66, 1.03, 0.80, 0.52 cm2 sy3 , respectively. l — Ds9.5 cm and Bo s 24.68 cm2 sy3 .
and
Ž
bav 1r3 2 Bo rD
s f5
.
z
ž / D
,
Ž 3.12 .
where f 3 , f 4 and f5 are functions. Expressions Ž3.10. and Ž3.11. can be written as bŽ r , z .
Ž Bo2rD .
1r3
s f6
ž
r LŽ z . , , D D
/
Ž 3.13a,b .
which, using self-similarity Žintermediate asymptotic. arguments, becomes bŽ r , z .
Ž Bo2rD .
1r3
s
LŽ z .
p1
r
LŽ z .
L z
D
ž / ž Ž . /ž / D
f7
p2
,
Ž 3.14 .
where f6 and f 7 denote functions and p 1 and p 2 are exponents. Careful inspection of Figs. 12 and 13 reveals the following. Ži. Although the instantaneous distribution of buoyancy is highly irregular with sharp spatial variations ŽFig. 12., the averaged profiles show smooth stable spatial structure ŽFig. 13., thus enabling to evaluate the functional forms Ž3.10. – Ž3.12.. Papantoniou and
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Fig. 12. The variation of instantaneous buoyancy of the plume b with the dimensional radial distance r, at a normalized depth of z r Ds 0.21. Ds19.1 cm and Bo s1.07 cm2 sy3 .
List Ž1989. have also noted the wide disparity between ‘‘instantaneous’’ and ‘‘averaged’’ profiles for the case of point plumes and pointed out the implications of such disparities in turbulent modeling. Žii. The data of Fig. 13 for zrD - 0.15 cluster together, forming a collapsed curve at least for < rrLŽ z .< - 0.2. This, together with Eq. Ž3.14., implies the independence of results on LŽ z .rD or zrD, p 1 f p 2 f 0 and constancy of f5 in the above depth range. Žiii. At zrD ) 0.15, f 4 to f 7 appears to be strong functions of zrD, indicating the depth dependence of the normalized buoyancy. The above results indicate the dependence of functional forms f 3 to f 7 on the spatial regime of the flow characterized by zrD. Different asymptotic forms are possible for different ranges of zrD. To further illustrate the presence of disparate asymptotic regime, a plot of bav rŽ Bo2rD .1r3 vs. zrD, is shown in Fig. 14, concurrent with Eq. Ž3.12.. The data confirm that indeed f5 is a constant for zrD - 0.1 or so and the following relations are valid: f5
z
ž / D
s
Ž
bav 1r3 2 Bo rD
f 10 for zrD - 0.1,
Ž 3.15 .
, for zrD ) 0.4,
Ž 3.16 .
.
and f5
z
ž / D
s 3.7
z
ž / D
yn 2
J. Colomer et al.r Dynamics of Atmospheres and Oceans 30 (1999) 125–148
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Fig. 13. A plot of the nondimensional time-averaged buoyancy brŽ Bo2 r D .1r3 vs. nondimensional horizontal distance rr LŽ z . for different z r D values. Ds19.1 cm and Bo s1.07 cm2 sy3.
with n 2 f 1r3. The above results confirm those revealed by the velocity measurements that the plume diameter is the most important governing length scale close to the source. In addition, as evident from the velocity records in the range 0.3 - zrD - 2, the diameter effects fade away resulting in
bav f 3.7
Bo2
ž / z
1r3
,
Ž 3.17 .
based on Eqs. Ž3.12. and Ž3.16.. Thus, in the asymptotic limit of zrD ™ 0 or more specifically zrD - 0.15, D becomes the governing length scale of plume evolution, whereas when zrD ™ ` Žspecifically zrD ) 0.4., the diameter effects begin to wane, leaving z as the governing variable. This result is consistent with the observation that in Region I Ž zrD - 0.28., the velocity scale of the flow is Ž Bo D .1r3 , which differs from
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Fig. 14. A plot of the spatially averaged Žover the plume width. nondimensional buoyancy Ž b .av rŽ Bo2 r D .1r3 vs. the nondimensional depth z r D covering both regions I and II. Ds19.1 cm for all runs. B — Bo s1.07 cm2 sy3 ; I — 1.45; ` — 0.16; e — 0.87: l —0.27;: ' — 1.52 and ^ — 1.34.
the scaling for horizontally homogeneous sources Ž Bo z .1r3 or Ž Bo H .1r3 ŽDeardorff, 1970. or for point plumes wŽ Bo D 2 .rz x1r3 ŽList, 1982.. 4. Summary and discussion An experimental study on the evolution of a turbulent buoyant plume in a homogeneous fluid was described in foregoing sections. This study differs from many previous investigations carried out on point plumes, in that it considers finite diameter effects and the nature of flow near the source. The results should have applications to a number of oceanic and atmospheric convective flows wherein the typical horizontal extent of forcing D is much larger than the fluid layer depth. Experiments carried out with varying source diameters D and source buoyancy fluxes Bo , using the LIF and PTV techniques for flow diagnostics, revealed the following. Ži. The initial development of the plume during the time t - 1.2Ž D 2rBo .1r3 occurs as if convection is horizontally homogeneous; i.e., the depth of the front grows as h f 0.35Ž Bo t 3 .1r2 , lateral entrainment flow is insignificant and the frontal propagation occurs by the engulfment of ambient fluid by turbulent eddies. The entrainment flow, however, could be detected at larger times t - 1.2Ž D 2rBo .1r3 , especially after the plume underwent morphological changes and achieved a quasi-steady state at a time te f 1.8Ž D 2rBo .1r3.
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Žii. The plume under this steady state consists of two regions. In region I, 0 - z - z c , the plume width shrinks with the distance z and achieves a minimum diameter of Lo f 0.5D at a depth of z c f 0.28D. In region II, z ) z c , the plume expands, much the same way as a point plume, and at larger z, the source diameter effects fade away leaving Bo and z as governing parameters. Žiii. Under quasi-steady conditions, the averaged maximum horizontal velocity in region I was found to be Vx f 0.66 Ž Bo D .
1r3
,
and the vertical velocity in this region is proposed to have the dependence z 1r3 Wc f 2.7 Ž Bo D . . D
ž /
The instantaneous buoyancy within the plume shows high space-time variability, but stable statistical averages can be discerned. For zrD - 0.1, the time-averaged buoyancy b Ž r , z . spatially averaged across the plume width at a given z, was found to be bav f 10 Ž Bo2rD .
1r3
.
Živ. The mean centerline velocity and the width-averaged buoyancy measurements in region II showed the following for larger zrD values: Wc f 1.2 Ž Bo z .
1r3
for zrD ) 0.3,
and bav f 3.7
Bo2
ž / z
1r3
, for zrD ) 0.4,
indicating the absence of source diameter effects.
5. Geophysical implications The results of this paper will be useful in the studies of numerous geophysical situations with convection from isolated, but distributed, sources. Fig. 2 provides a useful description of the establishment of convection from isolated sources. First, the heavy fluid layer near the source descends downward, and the resulting mean drift causes the plume to develop a ‘neck’ beneath the source. Soon, this drifting heavy fluid becomes unstable and breaks down into turbulence. The resulting turbulent region grows by the engulfment of underlying fluid, as in the case of horizontally homogeneous convection. At this stage, the growth of the turbulent layer is dominated by the engulfment of fluid from below, rather than by the mean drift. With time, the mean drift of the mixed fluid is established, so as the lateral entrainment flow into the plume. Eventually, a plume dominated by lateral entrainment is generated. The above description of flow evolution in finite-diameter plumes is in agreement with the description of cumulus cloud growth advocated by numerous previous studies.
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By assuming a typical buoyancy Žheat. flux of 5 = 10y4 m2rs 3 at a cloud base of 0.5 km in horizontal extent Že.g., Ludlam, 1980., it is possible to infer the time scale of initial cloud growth by the engulfment of eddies as TD f 1.2Ž D 2rBo .1r3 ; 15 min. The time scale for the cloud to achieve a steady profile with well-defined regions I and II is about te ; 25 min. Since this is the time scale where the cloud growth is arrested by the stable stratification, it is expected that growing clouds do not have time to achieve the quasi-steady state described before. Hence, the cloud growth ought to be dominated by the entrainment of air from aloft, with very little lateral flow. These laboratory-based inferences are consistent with field observations and numerical studies, which, in general, show that: Ži. plume models based on lateral entrainment predict the properties of clouds Že.g., water content. poorly ŽWarner, 1970.; Žii. the lateral growth rates of clouds are small, and there is no discernible negative correlation between velocity and lengthscales in clouds that would have been typical in entraining plumes ŽLemone and Zisper, 1980.; Žiii. there are rapid uprdowndrafts in the cloud, with mixing occurring between them ŽHeymsfield and Scotz, 1985; Yuter and Houze, 1995.; and, Živ. undiluted fluid parcels from the base area can be seen near the cloud top ŽPaluch, 1979.. The importance of uprdowndrafts in cumulus convection, especially near the cloud top, has been accentuated in a recent article by Vonnegut Ž1997.. In the context of deep ocean convection, typical surface buoyancy fluxes are of the order 10y7 m2rs 3 with a horizontal extent of the source on the order of 50 km. Accordingly, the time scale TD with which the convective region grows by the engulfment mechanism is much larger than the time to achieve its maximum depth Ž; 2 km.. Also, TD is much larger than the rotational time scale fy1 Žwhere f is twice the Earth’s rate of rotation. over which baroclinic instabilities develop surrounding the chimneys and small-scale vortices develop within the chimneys ŽChen et al., 1989.. Thus, the oceanic convection is expected to develop much the same way as in Fig. 2Ža. – Žc., consisting of updrafts and downdrafts with little lateral entrainment flow or a mean plume flow, until the plume front reaches its maximum extent. This inference is consistent with several field measurements which show that there is no vertical mean flow within deep convective regions when the measurements are averaged over a reasonable time period; e.g., see the observations in central Greenland sea and in the Gulf of Lions ŽMediterranean. by Schott et al. Ž1993. and Schott et al. Ž1996.. Upon becoming baroclinically unstable, the convective region is expected to be governed by a complex pattern of eddies moving out of the region and associated mean flow into the convective region. As stated earlier, slower nocturnal cooling of urban areas compared to their surroundings may generate urban heat islands that act as plume sources. In low latitudes, such flows are not affected by the Earth’s rotation ŽLu et al., 1997a,b.. In the absence of background winds and rough topography, typical parameters for moderate sized heat islands are D ; 10 km and Bo ; 8 = 10y4 m2rs 3, thus yielding Ž Bo D .1r3 ; 2 mrs or maximum horizontal velocities on the order 0.66Ž Bo D .1r3 ; 1.3 mrs. Given that heat islands typically appear at night, however, it is expected that the plume rise is determined by the stratification Žor the buoyancy frequency N . of the nocturnal boundary layer, which was not mimicked in our experiments. As pointed out by Whitehead et al. Ž1996., Lu et al. Ž1997b. and Colomer et al. Ž1998., the height of a
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plume under such conditions is given by z max s b Ž Bo D .1r3rN where different values for b have been proposed: 2.86 ŽLu et al., 1997b. and 3.65 ŽWhitehead et al., 1996.. Based on these values, the height of the plume can be estimated as z max ; 3Ž Bo D .1r3rN ; 600 m, where N ; 0.01 sy1 has been assumed. This calculation shows that the urban heat island can indeed be considered as a low aspect ratio plume with z maxrD ; 0.06. Acknowledgements Primary support for this work was provided by the Army Research Office and the NATO travel grant CRG 960740. Stratified flow research at Arizona State University is also sponsored by the National Science Foundation ŽFluid Mechanics and Hydraulics Program. and the Office of Naval Research. The post-doctoral work of the first author at ASU was supported by a grant from the University of Girona. References Adrian, R.J., Ferreira, R.T.D.S., Boberg, T., 1986. Turbulent thermal convection in wide horizontal fluid layers. Exp. Fluids 4, 121. Anis, A., Moum, J.N., 1994. Convection with rotation in a neutral ocean: a study of open-ocean deep convection. J. Phys. Oceanogr. 24, 2142. Ayotte, B.A., Fernando, H.J.S., 1994. The motion of a turbulent thermal in the presence of background turbulence. J. Atmos. Sci. 51 Ž13., 1989. Bornstein, R., 1987. Mean diurnal circulation and thermodynamic evolution of urban boundary layer. Modeling the Urban Boundary Layer. Am. Met. Soc., p. 53. Castaing, B., Gunaratne, G., Heslot, F., Kadanoff, L., Libchaber, A., Thomae, S., Wu, X., Zaleski, S., Zanetti, G., 1989. Scaling of hard thermal turbulence in Rayleigh–Benard convection. J. Fluid Mech. 204, 1. ´ Chen, R.R., Fernando, H.J.S., Boyer, D.L., 1989. Formation of isolated vortices in a rotating convecting fluid. J. Geophys. Res. 94 ŽDI5., 18445–18453. Colomer, J., Zieren, L.D., Fernando, H.J.S., 1998. Comment on ‘Localized convection in rotating stratified fluid’. In: Whitehead, J.A., et al. ŽEds... J. Geophys. Res. 103 ŽC6. 12891. Dalziel, S., 1992. Decay of rotating turbulence: some particle tracking experiments. Appl. Sci. Res. 49, 217. Deardorff, J.W., 1970. Convective velocity and temperature scales for the unstable planetary boundary layer and for Rayleigh Convection. J. Atmos. Sci. 27, 1211. DeSilva, I.P.D., Montenegro, L., Fernando, H.J.S., 1990. Measurement of interfacial distortions at a stratified entrainment interface. Exp. Fluids 9, 174. Fernando, H.J.S., Chen, R., Boyer, D.L., 1991. Effect of rotation on convective turbulence. J. Fluid Mech. 228, 513. Hannoun, I.A., Fernando, H.J.S., List, E.J., 1988. Turbulence structure near a density interface. J. Fluid Mech. 189, 189. Helfrich, K., 1994. Thermals with background rotation and stratification. J. Fluid Mech. 259, 265. Heymsfield, G.M., Scotz, S., 1985. Structure and evolution of a severe squall line over Oklahoma. Mon. Weather Rev. 113, 1580. Julien, K., Legg, S., McWilliams, J., Werne, J., 1999. Plumes in rotating convection: Part 1. Ensemble statistics and dynamics balances. J. Fluid Mech. 391, 151–187. Kaimal, J.C., Wyngaard, J.C., Haugen, D.A., Cote, ´ O.R., Izumi, Y., Caughey, S.J., Readings, C.J., 1976. Turbulence structure in the convective boundary layer. J. Atmos. Sci. 33, 2152. Klinger, B.A., Marshall, J., 1995. Regimes and scaling laws for rotating deep convention in the ocean. Dyn. Atmos. Ocean 21, 227.
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Lemone, M.A., Zipser, E., 1980. Cumulonimbus vertical velocity events in GATE: Part I. Diameter, intensity and mass flux. J. Atmos. Sci. 37, 2444. List, E.J., 1982. Turbulent jets and plumes. Annu. Rev. Fluid Mech. 14, 189. Lu, J., Arya, S.P., Snyder, W.H., Lawson, R.E. Jr., 1997a. A laboratory study of the urban heat island in a calm and stably stratified environment: Part I. Temperature field. J. Appl. Meteorol. 36, 1377. Lu, J., Arya, S.P., Snyder, W.H., Lawson, R.E. Jr., 1997b. A laboratory study of the urban heat island in a calm and stably stratified environment: Part I. Velocity field. J. Appl. Meteorol. 36, 1392. Ludlam, F.H., 1980. Clouds and Storms. Pennsylvania Univ. Press. Marshall, J., Schott, F., 1998. Open-ocean convection: observations, theory and models. Report No. 52. MIT, Center for Global Change Science. Maxworthy, T., 1997. Convection into domains with open boundaries. Annu. Rev. Fluid Mech. 29, 327. Maxworthy, T., Narimousa, S., 1994. Unsteady, turbulent convection into a homogeneous, rotating fluid, with oceanographic applications. J. Phys. Oceanogr. 24, 865. Noh, Y., Fernando, H.J.S., Ching, C.Y., 1992. Flow induced by the impingement of a thermal on a density interface. J. Phys. Oceanogr. 22 Ž10., 1207. Paluch, I.R., 1979. The entrainment mechanism in Colorado cumuli. J. Atmos. Sci. 36, 2467. Papanicolaou, P.N., List, E.J., 1988. Investigations of round vertical turbulent buoyant jets. J. Fluid Mech. 195, 341. Papantoniou, D., List, E.J., 1989. Large-scale structures in the far field of buoyant jets. J. Fluid Mech. 209, 151–190. Schott, F., Visbeck, M., Fischer, J., 1993. Observations of vertical currents and convection in the central Greenland Sea during the winter of 1988r89. J. Geophys. Res. 98 ŽC8., 14401. Schott, F., Visbeck, M., Send, U., Fischer, J., Stramma, L., Desaubies, Y., 1996. Observations of deep convection in the Gulf of Lions, Northern Mediterranean during the winter of 1991r92. J. Phys. Oceanogr. 26, 505. Siggia, E.D., 1994. High Rayleigh number convection. Annu. Rev. Fluid Mech. 26, 137. Vonnegut, B., 1997. Quaint cumulus convection conviction. EOS 78 Ž23., 281. Warner, J., 1970. On steady-state one-dimensional models of cumulus convection. J. Atmos. Sci. 27, 1035. Whitehead, J.A., Marshall, J., Hufford, G.E., 1996. Localized convection in rotating stratified fluid. J. Geophys. Res. 101 ŽC1., 25705. Yuter, S.E., Houze, R.A. Jr., 1995. Three-dimensional kinematic and microphysical evolution of Florida cumulonimbus: Part I. Spatial distribution of updrafts, downdrafts and precipitation. Mon. Weather Rev. 123, 1921.
Turbulent convection from isolated sources J. Colomer a , B.M. Boubnov b, H.J.S. Fernando
c,)
a
Department of EnÕironmental Sciences, UniÕersity of Girona, 17071 Girona, Spain A.M. ObukhoÕ Institute of Atmospheric Physics, Russian Academy of Sciences, 109017 Moscow, Russia Department of Mechanical and Aerospace Engineering, EnÕironmental Fluid Dynamics Program, Arizona State UniÕersity, P.O. Box 879809, Tempe, AZ 85287-9809, USA
b c
Received 15 September 1998; received in revised form 28 May 1999; accepted 1 June 1999
Abstract Laboratory experiments were conducted to investigate the evolution of a dense turbulent plume, specified by its buoyancy flux Bo and source diameter D, issuing into a homogeneous, nonrotating, environment. This study was motivated by the desire to delineate velocity and buoyancy scaling for convection from isolated buoyancy sources of finite extent. Such flow configurations have relevance to geophysical Že.g., deep convection., environmental Že.g., urban heat island effect. and engineering Že.g., plume stacks. flows. Special attention was given to study the evolution of the plume following its initiation and the flow near the source when the influence of confining boundaries is insignificant. It was found that, for times t - 1.2Ž D 2rBo .1r3 , the descent of the plume front can be treated as one-dimensional with negligible lateral Žentrainment. mean flow, the plume growth mechanism being the encroachment of underlying nonturbulent fluid. At larger times, the flow achieved a quasi-steady state, in which the plume width first decreases up to a distance 0.28 D Žregion I. and then increases Žregion II.. The quasi-steady state velocity and buoyancy measurements in region I showed that they are strongly influenced by the lateral entrainment flow Žand hence, by D ., and thus classical free convection scaling is inapplicable. On the other hand, at large zrD Žin region II., the velocity and buoyancy scaling tend to be independent of D, indicating the fading influence of source diameter effects. The results establish basic scaling for convection from isolated, but distributed, sources and provide a baseline with which future work that incorporates background rotation and stratification can be compared. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Convection; Plume; Velocity and buoyancy scaling
)
Corresponding author. Tel.: q1-480-965-2807; fax: q1-480-965-8746; e-mail: [email protected]
0377-0265r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 0 2 6 5 Ž 9 9 . 0 0 0 2 3 - 8
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1. Introduction Turbulent convection from isolated, but distributed, sources is a widespread phenomenon in environmental and engineering flows. Some examples of consequence in ocean and atmospheric dynamics are the deep-convection, development of cumulous clouds and urban heat island flows. In engineering context, plumes emanating from wide plume stacks are common in power plant cooling. Oceanic and atmospheric deep convection are characterized by the formation of intense convective regions that penetrate into greater depths. Because of the presence of density stratification, for example, the thermocline in the ocean and inversion layers in the atmosphere, the vertical scale of geophysical convection, is limited to shallow depths, but under special conditions, convective motions can penetrate deep into stratified layers. In oceans, convective fluid elements typically penetrate a few hundred meters ŽAnis and Moum, 1994., but in deep convective regions Žknown as ‘‘chimneys’’., these elements descend up to a few kilometers, the width of the chimneys being 10–100 km ŽMaxworthy, 1997.. Cloud convection is characterized by rising of air currents beyond the level of dew point, wherein the condensation of water vapor and associated latent heat release drive convection ŽLudlam, 1980.. Under unstable conditions, clouds can grow to great vertical extents Že.g., violent thunderstorms., on the order of tens of kilometers, with hundreds of kilometers horizontal scale Žatmospheric deep convection.. Under more stable conditions, the extent of clouds is smaller, with several hundred meters in the vertical and several kilometers in the horizontal, characterizing cumulus clouds. Urban heat islands arise due to the presence of elevated temperatures in localized urban areas relative to their rural surroundings, which drives a circulation due to horizontal temperature gradients. The resulting flow patterns can have a significant impact on the microclimate of cities ŽBornstein, 1987.. The above examples represent, though the details vary, convection from isolated sources of large horizontal extent. In most situations, the horizontal scale Ž D . of convection exceeds the vertical scale Ž H ., thus making the flow approach that of horizontally homogeneous convection Ž DrH ™ `., where the free-convection scaling of Deardorff Ž1970. applies ŽKaimal et al., 1976.. If the buoyancy flux per unit area is Bo , then the Deardorff scaling for length and velocity at normalized distances from the source zrH ) 0.15 becomes H and Ž Bo H .1r2 , respectively. For zrH - 0.15 and beyond the molecular-diffusive sub-layer adjoining the source, the flow obeys local scaling, z and Ž Bo z .1r3, respectively, indicating the unimportance of H ŽAdrian et al., 1986.. A question arises, however, on the role of source diameter effects near the source Ž zrD - 1. for cases with finite values of DrH. In such cases, D is expected to play a role because of the possibility of developing an entrainment Žmean. flow through lateral boundaries. It is the influence of finite diameter of the source that we wish to investigate in this study. The questions of particular interest are: ‘Is the free-convection scaling applicable to finite-diameter sources?’; If applicable, ‘Under what conditions is it valid?’ and if not applicable, ‘What should be the new scaling?’ The flow configuration considered consists of a disk-shaped source of buoyancy Žof diameter D . releasing into a fluid of large horizontal extent and depth H. The important dimensional governing variables for the problem, excluding the molecular parameters,
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are Bo , D, H and z. The measurements, however, were confined to regions zrH 0.8–0.9, wherein the depth Ž H . effects are insignificant for propagating plume fronts and also for plumes that have already impacted on the bottom surface ŽNoh et al., 1992.. Although a large number of scaling studies have been reported on asymptotic cases of horizontally homogeneous sources DrH ™ ` ŽAdrian et al., 1986; Castaing et al., 1989; Siggia, 1994. and point and line plumes and thermals DrH ™ 0 ŽPapanicolaou and List, 1988; Ayotte and Fernando, 1994; Helfrich, 1994., the case of finite diameter sources has received only little attention. More so, existing studies with finite-diameter sources have dealt with specific geophysical situations, and only little attention has been given to establish or verify scaling laws in detail Že.g., oceanic convection work of Maxworthy and Narimousa, 1994 and urban heat island studies by Lu et al., 1997a,b.. This paper shows that the finite-diameter effects can indeed lead to new scaling in the vicinity of the source. The present results are strictly valid only for sources in nonrotating fluids or for flows with insignificant Coriolis effects. It is acknowledged that the effects of rotation can be important in geophysical cases, perhaps from the outset of convection, because of the large horizontal extent and small velocities of the flow. For example, during the development of convection in a rotating fluid, small-scale vortices can be developed near the source due to the breakdown of unstable buoyancy gradients, thus making the flow different from its nonrotating counterpart ŽChen et al., 1989; Klinger and Marshall, 1995; Marshall and Schott, 1998.. In geophysical situations, the appearance of such vortices is rare, but laboratory and numerical studies confirm their presence. Although these vortices may not significantly change the integral properties of turbulence near the source ŽFernando et al., 1991., they can lead to anomalous mixing characteristics at some distances away from the source ŽJulien et al., 1999.. The extension of present nonrotating convection results to geophysical situations, therefore, must be done with caution. Geophysical situations are often dominated by stratification effects, which also should be considered in extrapolating laboratory results obtained with homogeneous backgrounds to natural situations. The present results, nevertheless, provide baseline data and scales with which future modeling involving background rotation and stratification can be compared. Section 2 presents the details of the experimental apparatus and procedure. Section 3 describes the experimental results. A summary of the study is given in Section 4 and in Section 5 a discussion on practical implications of results is given.
2. Experimental apparatus and procedure The experiments were conducted in a cylindrical Plexiglas tank of diameter D s 107 cm and height 60 cm ŽFig. 1.. This tank was placed inside a larger square Plexiglas tank of internal dimensions 108 = 108 cm2 to minimize optical distortions arising due to tank curvature when viewing from the sides. The source was made of a short circular cylinder, which was packed with several layers of porous material Ž1-cm thick sponge and two filter paper layers.. These layers were held in place by a bottom mesh lid of mesh size 1 mm. This design ensured uniform distribution of salt water over the entire
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Fig. 1. A schematic diagram of the experimental apparatus. The nature of the plume at large times is illustrated with near ŽI. and far ŽII. field regions.
source cross-section when the salt water is fed from the top via a constant head reservoir. The salt water seeps through the porous layers and drains into the underlying water with a velocity wo . If the buoyancy of the resulting plume fluid is D bo s Ž rp y ro . grro , where rp and ro are the densities of the plume and the background fluid, respectively, and g is the gravitational acceleration, then the buoyancy flux per unit area Bo , the total buoyancy flux Qo , the momentum flux Mo and the volume flux Vo can be written as Bo s D bo wo ; Qo s
p D 2 Bo 4
; Mo s
p D 2 wo2 4
; and Vo s
p D 2 wo 4
.
Ž 2.1 .
To check the ‘‘plume’’ vis-a-vis the ‘‘jet’’ nature of the source, the length scale ` LM s
Mo3r4 Qo1r2
Ž 2.2 .
was used, analogous to the momentum length scale used in point buoyant jetrplume studies ŽList, 1982; Papanicolaou and List, 1988.. At distances z ) L M from the source, the momentum flux can be regarded as unimportant, and the experimental parameters
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were selected to minimize L M while maintaining the buoyancy flux in a reasonable parameter range. The L M values used varied in the range 0.01 - L M - 0.36 cm. Source diameters of D s 3.1, 9.5 and 19.1 cm were used, but most experiments were conducted with D s 19.1 cm. The working depth of the fluid H was held constant at 53 cm. The plume density was varied over the range 1.008 - rp - 1.071 grcm3 , with rp ) ro . The buoyancy of the plume fluid and the buoyancy flux were varied in the ranges 7.8 - D bo - 132 cmrs 2 and 0.4 - Bo - 56.7 cm2 sy3 . The experiments were initiated by releasing salty water from the constant head reservoir into the plume source. The plume fluid was added with a small amount of fluorescein dye, enabling its visualization under the illumination of a sheet of laser light generated by a laser-beam scanner ŽDeSilva et al., 1990.. The vertical thickness of the laser sheet was 8 cm, which imposed a limit on the vertical field of view. In some cases, intense arc lamps were used to produce light sheets that span the entire depth of the water column. The flow was video-recorded and the tapes were later analyzed using standard techniques of laser-induced fluorescence ŽLIF.. A software package, DigImage, was used for this purpose Žfor details, see Dalziel, 1992.. For experiments with large D bo , refractive index variations across plume boundaries were found to have a significant effect on LIF measurements. In such cases, the refractive indices between plume and background fluids were matched by adding alcohol to the background fluid Žfor details, see Hannoun et al., 1988.. To check the accuracy of the LIF imaging technique, limited measurements were also made using two stationary conductivity probes placed at various distances below the source. Both techniques measured mean concentrations within "5%. In a series of experiments, the velocity field measurements were made using particle tracking velocimetry ŽPTV.. Here, the background fluid was seeded with neutrally buoyant particles of approximate diameter 0.3 mm and the entrainment flow into the plume was tracked using DigImage software. The accuracy of the velocity measurements was estimated to be "10%.
3. Finite-sized plumes in homogeneous fluids This section presents qualitative observations and quantitative measurements of the experiments described above. 3.1. General flow description Detailed observations made during the plume descent revealed the following. Ži. The thin dense fluid layer that develops under the source, immediately following the release of plume fluid, breaks down into small-scale turbulence via Rayleigh–Taylor type overturning instabilities. The visually estimated time at which this breakdown occurs is assigned as t s 0. The resulting plume fluid near the source develops a ‘‘neck’’ Žcontraction of the width. just below the source Žsee Fig. 2Ža. and Žb.. This observation can be attributed to enhanced mixing along the periphery of the source, which entails strongly buoyant fluid near the center of the plume and lighter Žslow
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Fig. 2. Ža–e. The evolution of the plume Žobtained using LIF images. with time Ž t .. The surface region corresponds to half the source diameter Ds19.1 cm. Bo s1.07 cm2 sy3 . The pseudocolor representation used is shown below Že.. The background corresponds to black and the lowest and the highest concentrations are represented by red and purple, respectively. The brown in the background appears as a result of noise and diffused light.
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Fig. 2 Žcontinued..
descending. fluid at the periphery. The relatively lighter fluid produced in the periphery is dragged down by the dense source fluid residing toward the center of the plume, thus producing the ‘‘neck’’. Detailed measurements of fluid emanating from the source at different radii showed that the buoyancy flux is uniform across the source, thus indicating that the appearance of the neck is not an artifact of inhomogeneities of the plume, but a characteristic of finite-diameter plumes. Žii. The turbulent dense fluid layer near the source starts to spread downward, lowering its ‘‘neck’’ and expanding its width below the neck. For a time of about TD f 1.2Ž D 2rBo .1r3 , the depth of the plume front h follows the propagation law h ; Ž Bo t 3 .1r2 , much the same as that in horizontal homogeneous convection ŽFernando et al., 1991.; also, see Žiii. below and Section 3.2. This observation is consistent with the fact that no significant lateral entrainment flow is evident during this period, either from velocity measurements or LIF visualization. Concentration distribution within the plumes based on LIF imaging is shown in Fig. 2Ža. – Že., where the pseudocolor representation varies from red Žthe lowest detectable concentration. to purple Žthe largest concentration.. Note the appearance of high-concentration regions interior to the locally mixed Žand elongated. fluid layer at the edge of the plume at t - TD , indicating negligible entrainment of ambient fluid; this is also corroborated by the PTV data that showed lack of entrainment into the plume. Žiii. At t - TD , the propagation of the plume front appears to be dominated by the engulfment of fluid from below Žand to a certain extent from the sides. by turbulent
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eddies. This is seen in Fig. 2Ža. – Žd., where fluid parcels from the immediate vicinity of the plume Žred. are seen to engulf into the turbulent layer, thus creating strong concentration inhomogeneities within the plume. Entrained fluid parcels, however, are distorted and homogenized within the plume quite rapidly. Živ. After the plume descends for a certain time, approximately 1.8Ž D 2rBo .1r3, the downward movement of the ‘‘neck’’ ceases, settling at a distance approximately 0.28 D from the source Žsee Fig. 2e and also Section 3.2.. The dense fluid continues to drift downwards, inducing an entrainment flow and expanding the plume laterally. The development of the entrainment flow is well evident from the PTV records taken during the experiments. Fig. 3 shows a resultant velocity vector plot, taken at various times, starting from the time where a lateral entrainment velocity could be discerned. The entrainment flow first appeared near the source, but with time, it spreads along the entire plume. Žv. Upon reaching the tank bottom, the plume front deflects and flows as a gravity current, which, being constrained by tank boundaries, accumulates fluid at the bottom of the tank. Flow structures in the plume do not change appreciably until the bottom dense layer rises to a height of about 0.2 H, where H is the fluid depth in the tank. The flow in the bulk of the plume can be considered quasi-stationary until this time. Thereafter,
Fig. 3. A vertical cross-section through the plume, indicating resultant velocity vectors at different times: Ža. t s 20 s, Ž Bo r D 2 .1r 3 t s 3.82; Žb. 40, 7.64 and Žc. 60, 11.46. The external parameters are Ds19.1 cm and Bo s 2.54 cm2 sy3 . The arrow at the top left indicates the scale.
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secondary flow cells become evident in the tank due to the confinement of flow by sidewalls. During the quasi-steady state, two distinct flow regions could be identified in the plume, and they are shown in Fig. 1. In Region I, the plume contracts until a minimum diameter is achieved and then the flow expands in Region II. Fig. 4 shows a streakline photograph taken in the quasi-steady state, using neutrally buoyant particles suspended in the background fluid. Because of the rapid increase of particle velocity upon entrainment into the plume, the particle tracks within the plume are not picked up by the resolution limits of the image processing system. Also note that the entrainment flow in the vicinity of the source converges toward region I, indicating
Fig. 4. A front-view streak photograph of flow field around the plume under quasi-steady conditions. The streaks are produced by pathlines of suspended particles in the ambient fluid. Ds19.1 cm and Bo s 2.54 cm2 sy3 . Regions I and II are defined in Fig. 1.
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high entrainment demand to replenish buoyant fluid leaving the source region. In region II, the flow is dominated by the entrainment of background fluid along almost horizontal streamlines. 3.2. Scales of flow adjustment As discussed in Section 3.1, after some time, the plume assumes a quasi-steady state, as evident from velocity and concentration field measurements. The parameters governing the plume evolution are D, Bo , D bo , n Žthe kinematic viscosity. and k Žthe molecular diffusivity of salt., where the influence of rprro is neglected based on Boussinesq approximation. In the experiments, the outer boundary of the plume was mapped using LIF, and the time scale te required for the ‘‘neck’’ Žor the minimum width. of the plume to achieve its steady depth z e was measured. Based on the above governing parameters, it is possible to expect te 2
Ž D rBo .
1r3
s f 1Ž E . ,
Ž 3.1 .
where f 1 , f 2 , . . . are functions of the dimensionless parameters E s D bo DrŽ Bo D . 2r3. Here, the dependence on molecular parameters Ž n and k . has been neglected by assuming Reynolds and Pecelt number similarities. As discussed in Section 3.1, the measurements were made either during the descending phase of the plume front or away from the bottom walls of the tank after the plume is impacted. No measurements were made after the plume fluid accumulating at the bottom rose beyond 0.2 H. The depth of the tank H, therefore, was not considered as an important governing parameter. Fig. 5 shows a test of Eq. Ž3.1., where the normalized te is plotted against E. The results show that terŽ D 2rBo .1r3 is independent on E in the range 5 - E - 120, with te 2
Ž D rBo .
1r3
s 1.78 " 0.14.
Ž 3.2 .
The scatter of data at E - 5 can be attributed to comparatively high exit velocities of plume fluid employed for those runs. Similarly, the depth of the neck z c ŽFig. 1. was also measured and the results are presented in Fig. 6. Each data point represents the average of 100 frames. The data are scattered, but only a weak dependence of z crD on E could be identified for E ) 5. The data for E ) 5 can be best represented using the mean value zc s 0.28 " 0.13. Ž 3.3 . D The mean Žtime-averaged. width LŽ z . of the plume is plotted as a function of depth in Fig. 7 Žsee Section 3.3.. By fitting a fourth-order polynomial to the width vs. depth data, the minimum mean Žneck. width of the plume Lo was evaluated ŽFig. 1. and the results were found to follow LorD s 0.55 " 0.05.
Ž 3.4 .
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Fig. 5. The normalized time te rŽ D 2 r Bo .1r 3 for the plume boundaries in 0 - z - D to achieve the steady state as a function of the parameter Es D bo DrŽ Bo D . 2r 3 . ^ — Ds 3.1 cm; I — Ds9.9; q — Ds19.1. 7.78- D bo -132 cmrs 2 .
The mean spreading angle of the time-averaged plume width for zrD ) 0.5 was 158 " 28. Another quantity of interest is the rate of propagation of turbulent front under the source, which ought to be dependent on D, Bo and t, given the insensitivity of results to the variations of E. The depth of the front h under the source, thus, can be expressed as h 3 1r2
Ž Bo t .
s f2
žŽ
D Bo t 3 .
1r2
/
.
Ž 3.5 .
A plot of hrŽ Bo t 3 .1r2 vs. DrŽ Bo t 3 .1r2 is shown in Fig. 8, which lends support for the functional form Ž3.5.. Two regimes of layer growth are evident from this plot: for DrŽ Bo t 3 .1r2 ) 0.80 or t - 1.2Ž D 2rBo .1r3, the growth follows f 2 s constantf 0.35 Ž 3.6a . and at larger t, it is found that h
Ž Bo t 3 .
1r2
s C1
žŽ
n
D Bo t 3 .
1r2
/
,
Ž 3.6b .
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Fig. 6. A plot of normalized distance z c r D from the source to the depth where the minimum plume width Žneck. occurs as a function of E. Symbols have the same meaning as in Fig. 5.
where C1 f 0.54 and n f 0.5. Note that Eq. Ž3.6a. implies that at t - TD , the growth law is independent of D, or h s hŽ Bo ,t . ; Ž Bo t 3 .1r2 as observed in horizontally homogeneous turbulent convective flows ŽFernando et al., 1991.. This observation supports the notion that the entrainment flow Žwhich brings forth the dependence of D . is insignificant at t F 1.2Ž D 2rBo .1r3 and the growth is occurring by the engulfment of fluid from below the front by turbulent eddies. When the frontal depth is h, these eddies have a velocity Ž Bo h.1r3 ŽFernando et al., 1991., and hence, a frontal propagation velocity of d hrdt ; Ž Bo h.1r3 is expected. Meanwhile, information on lateral processes are transmitted into the plume at a rate d xrdt ; Ž Bo h.1r3 ; Ž Bo t .1r2 , which is felt over the entire plume Ž x ; D . at a time scale of t ; Ž D 2rBo .1r3. 3.3. Velocity measurements A typical instantaneous velocity distribution map of a plume operating under quasi-steady conditions is shown in Fig. 9. The velocity distribution here has similarities
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Fig. 7. A plot of plume width bw vs. depth z. The experimental parameters are as follows: Ds19.1 cm and v — Bo s 2.71; ` — 2.63; q — 0.75; x — 6.28; B — 4.43; I — 7.80; ' — 2.19 and ^ — 1.35 cm2 sy3 .
to that of an expanding plume, with the maximum mean velocity first increasing and then decreasing. The width of the velocity zone also appears to decrease in region I Ž z - 0.28 D . and then increase, much the same as the plume width based on concentration profiles ŽSection 3.2.. The quasi-steady velocity field established near the source was mapped using PTV. The maximum horizontal velocity Vx in region I, measured by averaging the data taken from 200 frames, was analyzed to delineate velocity scales in that region. The data were found to correlate well with the scale Ž Bo D .1r3 , as shown in Fig. 10. The other scales considered for a possible correlation were Ž Bo z .1r3 and Ž Bo H .1r3 ; here, H is the depth of the fluid layer and z is the location where this maximum horizontal velocity occurs. The results clearly showed Vx s 0.66 Ž Bo D .
1r3
.
Ž 3.7 .
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Fig. 8. The variation of the normalized frontal position h rŽ Bo t 3 .1r 2 with the parameter DrŽ Bo r t 3 .1r 2 .
This velocity scale is consistent with the flow evolution scenarios described in Eq. Ž3.2., wherein the development of region I is completed upon feeling the lateral plume boundary effects over the plume width at a time t ; Ž D 2rBo .1r3. The velocity scale of turbulence at this time becomes Ž Bo h.1r3 ; Ž Bo t .1r3 ; Ž Bo D .1r3, which is consistent with Eq. Ž3.7.. The centerline vertical velocities Wc Žaveraged over 200 frames. measured using PTV as a function of zrD are shown in Fig. 11. The continuity condition implies VxrD ; Wcrz, and hence, the velocity in the vicinity of the source is expected to behave as Wc ;
Bo
1r3
ž / D2
z,
Ž 3.8a .
or in normalized form Wc
Ž Bo D .
1r3
s C2
z
ž / D
,
Ž 3.8b .
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Fig. 9. A whole field velocity vector plot, taken after the flow has become quasi-stationary Žat t s80 s.. Ds19.1 cm, Bo s 0.53 cm2 sy3 and Es 49.5. The arrow at the top left represents the scale. Regions I and II are also marked.
where C2 is a constant. The data show a fair agreement with Eq. Ž3.8b. for zrD - 0.3, except the data point taken very close to the source; based on the eye-fitted line drawn in Fig. 11, C2 f 2.7. For zrD ) 0.3, in region II, the data tend to deviate from Eqs. Ž3.8a. and Ž3.8b. and the vertical velocity evolves as Wc
Ž Bo D .
1r3
s C3
z
ž / D
n1
,
Ž 3.9 .
with a best fit line indicating C3 f 1.2 and n1 f 0.30. It is interesting that Eq. Ž3.9. implies Wc ; Ž Bo z .1r3 type behavior, independent of the source diameter, suggesting that the scales imposed by the source diameter are waning at large distances from the source, at least in 0.3 - zrD - 6. 3.4. Measurements of buoyancy The variation of instantaneous buoyancy within the plume and its evolution in time were depicted in Fig. 2, from which it is evident that the buoyancy in the plume is highly variable in space and time. Further support for this observation can be seen from
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Fig. 10. The maximum time-averaged horizontal velocity Vx observed in the vicinity of the source Ž z - 0.3 D . as a function of Ž Bo D .1r 3 . ` — Ds19.1 cm; $ — 9.5.
the quantitative data shown in Fig. 12. Here, instantaneous buoyancy profiles bŽ r,t . taken at z s 4 cm Ž zrD s 0.21. at two different times of the flow evolution are shown. Shown in Fig. 13 are the time-averaged mean buoyancy data b Ž r , z . taken at various radial Ž r . locations for different zrD values, as a plot between b Ž r , z . rŽ Bo2rD .1r3 and rrLŽ z ., where LŽ z . is the averaged width of the plume. Special attention has been given to small zrD values so that the effects of finite source-diameter can be delineated. Here, the location where the buoyancy falls below 10% of the maximum buoyancy of the profile was defined as an edge of the plume, and the plume width was evaluated by averaging instantaneous widths of 200 profiles. The normalization variables were selected based on the governing parameters z, r, D and Bo ; hence, variables such as LŽ z ., the time-averaged mean buoyancy b Ž r , z . and the radially averaged b Ž r , z . over the plume width bav can be written as LŽ z . D
s f3
ž /
bŽ r , z .
Ž Bo2rD .
z
1r3
Ž 3.10 .
D
s f4
ž
r
,
z
D D
/
Ž 3.11 .
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Fig. 11. The centerline mean vertical velocity Wc , normalized by Ž Bo D .1r 3 , as a function of the normalized distance from the source. I, $, `, ^ — Ds19.1 cm and Bo s1.66, 1.03, 0.80, 0.52 cm2 sy3 , respectively. l — Ds9.5 cm and Bo s 24.68 cm2 sy3 .
and
Ž
bav 1r3 2 Bo rD
s f5
.
z
ž / D
,
Ž 3.12 .
where f 3 , f 4 and f5 are functions. Expressions Ž3.10. and Ž3.11. can be written as bŽ r , z .
Ž Bo2rD .
1r3
s f6
ž
r LŽ z . , , D D
/
Ž 3.13a,b .
which, using self-similarity Žintermediate asymptotic. arguments, becomes bŽ r , z .
Ž Bo2rD .
1r3
s
LŽ z .
p1
r
LŽ z .
L z
D
ž / ž Ž . /ž / D
f7
p2
,
Ž 3.14 .
where f6 and f 7 denote functions and p 1 and p 2 are exponents. Careful inspection of Figs. 12 and 13 reveals the following. Ži. Although the instantaneous distribution of buoyancy is highly irregular with sharp spatial variations ŽFig. 12., the averaged profiles show smooth stable spatial structure ŽFig. 13., thus enabling to evaluate the functional forms Ž3.10. – Ž3.12.. Papantoniou and
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Fig. 12. The variation of instantaneous buoyancy of the plume b with the dimensional radial distance r, at a normalized depth of z r Ds 0.21. Ds19.1 cm and Bo s1.07 cm2 sy3 .
List Ž1989. have also noted the wide disparity between ‘‘instantaneous’’ and ‘‘averaged’’ profiles for the case of point plumes and pointed out the implications of such disparities in turbulent modeling. Žii. The data of Fig. 13 for zrD - 0.15 cluster together, forming a collapsed curve at least for < rrLŽ z .< - 0.2. This, together with Eq. Ž3.14., implies the independence of results on LŽ z .rD or zrD, p 1 f p 2 f 0 and constancy of f5 in the above depth range. Žiii. At zrD ) 0.15, f 4 to f 7 appears to be strong functions of zrD, indicating the depth dependence of the normalized buoyancy. The above results indicate the dependence of functional forms f 3 to f 7 on the spatial regime of the flow characterized by zrD. Different asymptotic forms are possible for different ranges of zrD. To further illustrate the presence of disparate asymptotic regime, a plot of bav rŽ Bo2rD .1r3 vs. zrD, is shown in Fig. 14, concurrent with Eq. Ž3.12.. The data confirm that indeed f5 is a constant for zrD - 0.1 or so and the following relations are valid: f5
z
ž / D
s
Ž
bav 1r3 2 Bo rD
f 10 for zrD - 0.1,
Ž 3.15 .
, for zrD ) 0.4,
Ž 3.16 .
.
and f5
z
ž / D
s 3.7
z
ž / D
yn 2
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Fig. 13. A plot of the nondimensional time-averaged buoyancy brŽ Bo2 r D .1r3 vs. nondimensional horizontal distance rr LŽ z . for different z r D values. Ds19.1 cm and Bo s1.07 cm2 sy3.
with n 2 f 1r3. The above results confirm those revealed by the velocity measurements that the plume diameter is the most important governing length scale close to the source. In addition, as evident from the velocity records in the range 0.3 - zrD - 2, the diameter effects fade away resulting in
bav f 3.7
Bo2
ž / z
1r3
,
Ž 3.17 .
based on Eqs. Ž3.12. and Ž3.16.. Thus, in the asymptotic limit of zrD ™ 0 or more specifically zrD - 0.15, D becomes the governing length scale of plume evolution, whereas when zrD ™ ` Žspecifically zrD ) 0.4., the diameter effects begin to wane, leaving z as the governing variable. This result is consistent with the observation that in Region I Ž zrD - 0.28., the velocity scale of the flow is Ž Bo D .1r3 , which differs from
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Fig. 14. A plot of the spatially averaged Žover the plume width. nondimensional buoyancy Ž b .av rŽ Bo2 r D .1r3 vs. the nondimensional depth z r D covering both regions I and II. Ds19.1 cm for all runs. B — Bo s1.07 cm2 sy3 ; I — 1.45; ` — 0.16; e — 0.87: l —0.27;: ' — 1.52 and ^ — 1.34.
the scaling for horizontally homogeneous sources Ž Bo z .1r3 or Ž Bo H .1r3 ŽDeardorff, 1970. or for point plumes wŽ Bo D 2 .rz x1r3 ŽList, 1982.. 4. Summary and discussion An experimental study on the evolution of a turbulent buoyant plume in a homogeneous fluid was described in foregoing sections. This study differs from many previous investigations carried out on point plumes, in that it considers finite diameter effects and the nature of flow near the source. The results should have applications to a number of oceanic and atmospheric convective flows wherein the typical horizontal extent of forcing D is much larger than the fluid layer depth. Experiments carried out with varying source diameters D and source buoyancy fluxes Bo , using the LIF and PTV techniques for flow diagnostics, revealed the following. Ži. The initial development of the plume during the time t - 1.2Ž D 2rBo .1r3 occurs as if convection is horizontally homogeneous; i.e., the depth of the front grows as h f 0.35Ž Bo t 3 .1r2 , lateral entrainment flow is insignificant and the frontal propagation occurs by the engulfment of ambient fluid by turbulent eddies. The entrainment flow, however, could be detected at larger times t - 1.2Ž D 2rBo .1r3 , especially after the plume underwent morphological changes and achieved a quasi-steady state at a time te f 1.8Ž D 2rBo .1r3.
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Žii. The plume under this steady state consists of two regions. In region I, 0 - z - z c , the plume width shrinks with the distance z and achieves a minimum diameter of Lo f 0.5D at a depth of z c f 0.28D. In region II, z ) z c , the plume expands, much the same way as a point plume, and at larger z, the source diameter effects fade away leaving Bo and z as governing parameters. Žiii. Under quasi-steady conditions, the averaged maximum horizontal velocity in region I was found to be Vx f 0.66 Ž Bo D .
1r3
,
and the vertical velocity in this region is proposed to have the dependence z 1r3 Wc f 2.7 Ž Bo D . . D
ž /
The instantaneous buoyancy within the plume shows high space-time variability, but stable statistical averages can be discerned. For zrD - 0.1, the time-averaged buoyancy b Ž r , z . spatially averaged across the plume width at a given z, was found to be bav f 10 Ž Bo2rD .
1r3
.
Živ. The mean centerline velocity and the width-averaged buoyancy measurements in region II showed the following for larger zrD values: Wc f 1.2 Ž Bo z .
1r3
for zrD ) 0.3,
and bav f 3.7
Bo2
ž / z
1r3
, for zrD ) 0.4,
indicating the absence of source diameter effects.
5. Geophysical implications The results of this paper will be useful in the studies of numerous geophysical situations with convection from isolated, but distributed, sources. Fig. 2 provides a useful description of the establishment of convection from isolated sources. First, the heavy fluid layer near the source descends downward, and the resulting mean drift causes the plume to develop a ‘neck’ beneath the source. Soon, this drifting heavy fluid becomes unstable and breaks down into turbulence. The resulting turbulent region grows by the engulfment of underlying fluid, as in the case of horizontally homogeneous convection. At this stage, the growth of the turbulent layer is dominated by the engulfment of fluid from below, rather than by the mean drift. With time, the mean drift of the mixed fluid is established, so as the lateral entrainment flow into the plume. Eventually, a plume dominated by lateral entrainment is generated. The above description of flow evolution in finite-diameter plumes is in agreement with the description of cumulus cloud growth advocated by numerous previous studies.
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By assuming a typical buoyancy Žheat. flux of 5 = 10y4 m2rs 3 at a cloud base of 0.5 km in horizontal extent Že.g., Ludlam, 1980., it is possible to infer the time scale of initial cloud growth by the engulfment of eddies as TD f 1.2Ž D 2rBo .1r3 ; 15 min. The time scale for the cloud to achieve a steady profile with well-defined regions I and II is about te ; 25 min. Since this is the time scale where the cloud growth is arrested by the stable stratification, it is expected that growing clouds do not have time to achieve the quasi-steady state described before. Hence, the cloud growth ought to be dominated by the entrainment of air from aloft, with very little lateral flow. These laboratory-based inferences are consistent with field observations and numerical studies, which, in general, show that: Ži. plume models based on lateral entrainment predict the properties of clouds Že.g., water content. poorly ŽWarner, 1970.; Žii. the lateral growth rates of clouds are small, and there is no discernible negative correlation between velocity and lengthscales in clouds that would have been typical in entraining plumes ŽLemone and Zisper, 1980.; Žiii. there are rapid uprdowndrafts in the cloud, with mixing occurring between them ŽHeymsfield and Scotz, 1985; Yuter and Houze, 1995.; and, Živ. undiluted fluid parcels from the base area can be seen near the cloud top ŽPaluch, 1979.. The importance of uprdowndrafts in cumulus convection, especially near the cloud top, has been accentuated in a recent article by Vonnegut Ž1997.. In the context of deep ocean convection, typical surface buoyancy fluxes are of the order 10y7 m2rs 3 with a horizontal extent of the source on the order of 50 km. Accordingly, the time scale TD with which the convective region grows by the engulfment mechanism is much larger than the time to achieve its maximum depth Ž; 2 km.. Also, TD is much larger than the rotational time scale fy1 Žwhere f is twice the Earth’s rate of rotation. over which baroclinic instabilities develop surrounding the chimneys and small-scale vortices develop within the chimneys ŽChen et al., 1989.. Thus, the oceanic convection is expected to develop much the same way as in Fig. 2Ža. – Žc., consisting of updrafts and downdrafts with little lateral entrainment flow or a mean plume flow, until the plume front reaches its maximum extent. This inference is consistent with several field measurements which show that there is no vertical mean flow within deep convective regions when the measurements are averaged over a reasonable time period; e.g., see the observations in central Greenland sea and in the Gulf of Lions ŽMediterranean. by Schott et al. Ž1993. and Schott et al. Ž1996.. Upon becoming baroclinically unstable, the convective region is expected to be governed by a complex pattern of eddies moving out of the region and associated mean flow into the convective region. As stated earlier, slower nocturnal cooling of urban areas compared to their surroundings may generate urban heat islands that act as plume sources. In low latitudes, such flows are not affected by the Earth’s rotation ŽLu et al., 1997a,b.. In the absence of background winds and rough topography, typical parameters for moderate sized heat islands are D ; 10 km and Bo ; 8 = 10y4 m2rs 3, thus yielding Ž Bo D .1r3 ; 2 mrs or maximum horizontal velocities on the order 0.66Ž Bo D .1r3 ; 1.3 mrs. Given that heat islands typically appear at night, however, it is expected that the plume rise is determined by the stratification Žor the buoyancy frequency N . of the nocturnal boundary layer, which was not mimicked in our experiments. As pointed out by Whitehead et al. Ž1996., Lu et al. Ž1997b. and Colomer et al. Ž1998., the height of a
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plume under such conditions is given by z max s b Ž Bo D .1r3rN where different values for b have been proposed: 2.86 ŽLu et al., 1997b. and 3.65 ŽWhitehead et al., 1996.. Based on these values, the height of the plume can be estimated as z max ; 3Ž Bo D .1r3rN ; 600 m, where N ; 0.01 sy1 has been assumed. This calculation shows that the urban heat island can indeed be considered as a low aspect ratio plume with z maxrD ; 0.06. Acknowledgements Primary support for this work was provided by the Army Research Office and the NATO travel grant CRG 960740. Stratified flow research at Arizona State University is also sponsored by the National Science Foundation ŽFluid Mechanics and Hydraulics Program. and the Office of Naval Research. The post-doctoral work of the first author at ASU was supported by a grant from the University of Girona. References Adrian, R.J., Ferreira, R.T.D.S., Boberg, T., 1986. Turbulent thermal convection in wide horizontal fluid layers. Exp. Fluids 4, 121. Anis, A., Moum, J.N., 1994. Convection with rotation in a neutral ocean: a study of open-ocean deep convection. J. Phys. Oceanogr. 24, 2142. Ayotte, B.A., Fernando, H.J.S., 1994. The motion of a turbulent thermal in the presence of background turbulence. J. Atmos. Sci. 51 Ž13., 1989. Bornstein, R., 1987. Mean diurnal circulation and thermodynamic evolution of urban boundary layer. Modeling the Urban Boundary Layer. Am. Met. Soc., p. 53. Castaing, B., Gunaratne, G., Heslot, F., Kadanoff, L., Libchaber, A., Thomae, S., Wu, X., Zaleski, S., Zanetti, G., 1989. Scaling of hard thermal turbulence in Rayleigh–Benard convection. J. Fluid Mech. 204, 1. ´ Chen, R.R., Fernando, H.J.S., Boyer, D.L., 1989. Formation of isolated vortices in a rotating convecting fluid. J. Geophys. Res. 94 ŽDI5., 18445–18453. Colomer, J., Zieren, L.D., Fernando, H.J.S., 1998. Comment on ‘Localized convection in rotating stratified fluid’. In: Whitehead, J.A., et al. ŽEds... J. Geophys. Res. 103 ŽC6. 12891. Dalziel, S., 1992. Decay of rotating turbulence: some particle tracking experiments. Appl. Sci. Res. 49, 217. Deardorff, J.W., 1970. Convective velocity and temperature scales for the unstable planetary boundary layer and for Rayleigh Convection. J. Atmos. Sci. 27, 1211. DeSilva, I.P.D., Montenegro, L., Fernando, H.J.S., 1990. Measurement of interfacial distortions at a stratified entrainment interface. Exp. Fluids 9, 174. Fernando, H.J.S., Chen, R., Boyer, D.L., 1991. Effect of rotation on convective turbulence. J. Fluid Mech. 228, 513. Hannoun, I.A., Fernando, H.J.S., List, E.J., 1988. Turbulence structure near a density interface. J. Fluid Mech. 189, 189. Helfrich, K., 1994. Thermals with background rotation and stratification. J. Fluid Mech. 259, 265. Heymsfield, G.M., Scotz, S., 1985. Structure and evolution of a severe squall line over Oklahoma. Mon. Weather Rev. 113, 1580. Julien, K., Legg, S., McWilliams, J., Werne, J., 1999. Plumes in rotating convection: Part 1. Ensemble statistics and dynamics balances. J. Fluid Mech. 391, 151–187. Kaimal, J.C., Wyngaard, J.C., Haugen, D.A., Cote, ´ O.R., Izumi, Y., Caughey, S.J., Readings, C.J., 1976. Turbulence structure in the convective boundary layer. J. Atmos. Sci. 33, 2152. Klinger, B.A., Marshall, J., 1995. Regimes and scaling laws for rotating deep convention in the ocean. Dyn. Atmos. Ocean 21, 227.
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