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Convection in an open-topped slot
Fluid Dynamics
Research
Y (I YY2) 207-2
I8
North-Holland
Convection in an open-topped D.J. Tritton
slot
‘, P.A. Smith and D.S. Jeffcock
Departmcrrt of‘ Phycics, Unic~enity of‘ Nmvcu,stk upon Tyne. NE1 7RU, l/K Received
15 September
Abstract. An exploratory ture fluctuations,
1991
investigation
of free convection
the slot was uniform
horizontally
has been made, by flow visualisation
of water
and decreased
tank. The flow always had downgoing mainly in the downgoing
with mixed inhibited
convection
in vertical
intermittently.
changes are interpreted in heat transfer
and pipes: processes
into the base of a large
in the long horizontal
The rise in mean temperature
is lost. The difference channels
through
of the tempera-
in each of the large walls of
upwards. The top of the slot opened
regions. The intermittent
the buoyancy driving the circulation
and observations
slot. The temperature
and upgoing regions separated
slot, but the position of these changed occurred
in a vertical
direction
of the
of the circulating as a consequence
is in turn interpreted which turbulence
fluid
of this;
by analogy
is enhanced
or
by buoyancy effects in such flows are discussed.
1. Introduction We describe observations on the broad features of free convection in a slot extending downwards from the base of a tank. A vertical temperature gradient was maintained in the walls of the slot, with, in principle, no imposed horizontal temperature gradients. The wall temperature was the same at the top of the slot as the fluid temperature in the tank and increased downwards. Convection thus occurred as a result of the unstable temperature gradient in the slot. Our reason for experimenting with this configuration was possible application to convection in fissures in the ocean floor (Strens and Cann, 1986), but we report the work here as a basic fluid dynamical study. The observations and their interpretation have implications for heat transfer processes in configurations extending beyond our specific system, in mixed as well as in free convection. The aim of the experiment in the first place was simply to discover the broad features of the convection pattern: whether the circulation extended over the full depth of the slot or was broken into a series of vertically stacked rolls; and, if the former, whether the upgoing and downgoing currents were separated in the direction of the long or short dimension of the slot. Anticipating our observations, the circulation did extend over the full depth, with the separation in the long direction; but there were intermittent changes in the pattern of this large-scale circulation. After we had terminated the experiments, we discovered that an “oscillation” which is almost certainly essentially the same phenomenon had been previously observed in a similar, although not identical, system by Siegel and Norris (1957). Our additional observations enable us to give a fuller interpretation of this behaviour. Our essentially exploratory experiments do, however, in turn indicate a need for a more extensive investigation to resolve the more speculative aspects of the interpretation. Unfortu’ Correspondence Grenoble
to: D.J. Tritton,
lnstitut
de Mkcanique
de Grenoble,
Domaine
Universitaire,
Cedex. France.
0169.5983/92/$04.00
0 1902 - The Japan Society of Fluid Mechanics.
All rights reserved
B.P. 53X,
38041
nately, for personal foreseeable future.
2. Apparatus
reasons,
we shall not be in a position
to carry out such experiments
in the
and procedure
The dimensions of the apparatus are shown in fig. 1. The working fluid was water. Coordinate directions x, y, and z as shown in fig. 1 are used to facilitate the description of the apparatus and observations below. Walls A and B of the slot were made of aluminium alloy 6.25 mm thick. The objective was to maintain the same uniform vertical temperature gradient in these two walls, without any horizontal temperature variations either within one wall or between the two. A good approximation to this was achieved in the following way. Outside each wall was a jacket through which air was pumped. The air entered at room temperature just below the top tank and was heated by passing over an array of 20 heating elements, stretched horizontally across the air gap, before venting to the room at the level of the bottom of the slot. On each side the airflow was driven by four fans (hair driers with their heating elements removed) and ducted via diffusers to different regions in the x--direction of the top of the jacket. The temperature distribution so established in each wall was monitored by 20 thermojunctions in a rectangular array. Adjustments were made to the current through the heating elements (four consecutive elements being in series, so that there were five controls on each side) to give the required vertical tcmpcrature gradient: and to the fans to minimise horizontal temperature variations. Departures from required temperatures were of the order of 5% of the total temperature difference between top and bottom. The observed flows showed no bias that could be attributed to unwanted tcmperaturc variations in the s-direction or differences in the y-direction. A cooling-water coil was placed in the top tank to keep the water temperature there steady. The tops of the slot sidewalls were in contact with this water. as a way of achieving the requirement that their temperatures should be the same. This matching could be optimised by varying the tlow rate through the cooling coil.
Free surface 580
2-i 2-5 END VIEW
TOP VIEW Fig. I. Dimensions
of the
apparatus; all quantities
are mm
The bottom wall and end walls of the slot were of perspex. Clearly, having to view the flow through these narrow edges was somewhat restricting in the information that could be obtained from flow visualisation. However, it proved adequate to discover the broad features of the flow described below. Usually, the flow was illuminated through the base and/or one of the end walls and viewed through the other end wall and/or down through the top tank. The principal method of flow visualisation was to dissolve thymol blue indicator in the water and to inject a small amount of sodium hydroxide solution through hypodermic tubing to generate a patch of alkaline and thus dark blue fluid. In the later stages of the experiment it was noticed that small suspended particles originating as rust from walls of the top tank, were also helpful in revealing the flow pattern. (The tank was therefore not given the planned repaint!) Temperature measurements in the slot were made with thermojunctions. These provided the main information on which our interpretation is based. All measurements were made at the centre of the slot in the y-direction, but at various positions in the X- and z-directions as indicated in section 4.
3. Specification We denote the width of the slot by D (= 25.4 mm>, the temperature of the walls by T,(z) and that of the fluid by T(x, y, z, t). Fluid properties are denoted: (Y, coefficient of expansion; V, kinematic viscosity; K, thermal diffusivity. Because the maximum range of T, is about 60°C these quantities vary substantially; for example, a, which is involved in our scaling parameters (see below) varies by a factor of about 3 over this temperature range. Because we observe (section 4) that most of the fluid is at a temperature not much above that of the top tank, values at this temperature are used in calculating quantities quoted below. The Rayleigh number gaD”
dT,/dz
Ra = VK
ranged between 2.3 x 10” and 8.5 x lo’, the upper limit corresponding to the 60°C maximum range of T, achievable. Critical Rayleigh numbers for various modes of linear instability of the rest configuration in a slot of infinite vertical dimension are of the order 10’ to lo4 (Gershuni and Zukhovitskii, 1972), so one expects vigorous, probably turbulent, convection. The Prandtl number, Pr = V/K, was about 6.3 (and would be lower if based on values of v and K at, e.g., an average value of T,). The geometrical non-dimensional parameters specifying the situation are, of course, L,/D and L,/D, where L, and L; are the slot dimensions in the X- and z-directions. As seen from fig. 1, these were respectively 25.5 and 22.7. To non-dimensionalise the quantitative aspects of our observations, we introduce temperature difference, time, and velocity scales as respectively, 0 = D dT,/dz, It should
be noted
that
T = (ga
dT,/dz)
0 is small compared
-I”,
V= D/r.
with the total imposed
temperature
difference,
L,O/D.
4. Observations The central feature of our observations ing currents were separated in the long
was that, at any instant, the upgoing and downgo(x) direction, but the pattern of this circulation
changed intermittently. This result came in the first place from flow visualisation. Subsequent temperature measurements proved a more convenient indicator of what pattern was occurring, and also provided the clue to the mechanism of the changes. Dye produced by alkali injection close to one of the end windows showed that the flow was turbulent. The dye was mixed across most of the slot in the y-direction in a time short compared with the time taken to travel round the slot. The movement of the dye patch showed little variation of velocity with y, implying that such variation occurred mainly in boundary layers thin compared with the slot width. The speed could be estimated; the maximum through the time sequence described below was typically 3V, but varied between approximately 1.5V and 4V. Observations of the above features were possible only close to a window, but either method of flow visualisation could give a good indication of the general circulation pattern throughout the slot. At any instant there were either one or two cells in the slot, i.e. a downgoing flow at one end and upgoing at the other, or downgoing at both ends and upgoing in the middle, or the reverse of the latter. Sometimes, particularly for single cell patterns, there was marked asymmetry, i.e. the downgoing flow would the broad and slow and the upgoing narrow and fast, or vice versa. The speed range 1.5V to 4V quoted above is related to this; the average speed probably varied less. A particular pattern never persisted permanently. Quantitative information on the timescale of the changes was given also by the temperature measurements and will be discussed below. From the flow visualisation it appeared that the circulation was most rapid shortly after a change of pattern. Also at this stage there was strong interchange of fluid between the toi, tank and the slot. (The fluid entering the slot was drawn from close to the base of the tank; fluid leaving the slot penetrated to close to the top of the tank.) The circulation then slowed down, leading to the fluid being almost stationary just before the onset of a new pattern of circulation. During the later stages, there was a tendency for the interchange between slot and tank to be reduced, i.c. there was significant horizontal motion within the upper part of the slot. Figure 2 shows an example of one sequence observed more fully than usual; because of the limitations to the viewing of the flow, only the broad features, not the details, of these diagrams are significant. A few experiments were made with only the slot, and not the top tank, filled with water. In this case, the circulation always consisted of a permanent two-cell pattern, rising at the centre (in the x-direction). No changes of the sort occurring when there was water in the tank were observed. Returning to the experiments with the top tank filled, temperature measurements in the slot close to its top showed alternations between two levels (specified more quantitatively below) with relatively small fluctuations about each level. This behaviour evidently corre-
-1 /
L-------(
i
ID 77
Fig. 2. Example similar
hut
with
approximate
of observed sequence of flow patterns. reversed
sense of circulation.
Only
involving change from an initial pattern the
times for which the flow was predominantly
broad
features
of each
pattern
of the form shown are indicated.
to a final one that is are
significant.
(Ra = 8.3 X IO’)
The
D.J. Tritton et al. / Com,ection in m open-topped slot
21 I
Temoerature
>
4
1007 Fig. 3. Simultaneous temperature 7.57. y/D = 0, z/D = 0.4, (-
-Trme
variations at three points close to top of slot, based on sampling ~x/D=-12.0,(~~~~~~~x/D=0,~----_)x/D=12.0~Ra=X.O~10~~.
each at inten&
of
sponded to alternations between downward and upward motion at the point of observation (an interpretation confirmed by simultaneous flow visualisation and temperature measurement). Figure 3 shows an example of simultaneous temperature measurements at three points separated in the x-direction. Similar observations with an array of five equally spaced thermojunctions gave a good indication of the circulation pattern. We illustrate the changes in the circulation pattern with an example of an observed sequence (at Ra = 3.7 x IO"). In this 1 and 2 indicate the number of cells, C and A indicate clockwise and anticlockwise (as seen from one, arbitrarily chosen, side of the apparatus), and U and D indicate up and down in the middle. The sequence was 2U/ 1C/ 2U/ 1A/ lC/ 2D/ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ lA/ 2D/ 1A. This example is typical of the behaviour at the lower values of the Rayleigh number (roughly 2 < Ra/lO” < 4). For higher values (roughly 4 < Ra/lO” < S>, two cell patterns occurred less frequently, so that the behaviour was predominantly an alternation between clockwise and anticlockwise circulation. Occasionally, as revealed by the flow visualisation, the motion would die away as if a change were about to occur, but then re-start with the same sense of circulation. Such behaviour would probably not have been apparent from the temperature observations. [It should be added that, although the common occurrence of two cell motion has been attributed above to Ra being lower, it is not certain that this is a genuine Rayleigh number effect; it is possible that the small departures from ideal boundary conditions that could be influential in the detailed behaviour (although not the broad features) were different for different values of dT’,/dz.] The time between consecutive changes in pattern was quite variable and showed no obvious periodicity or other regularity. The average time was 41~ and the standard deviation of the distribution (representing actual variability and not experimental uncertainty) was 15~. The longest time for which the flow was observed to remain in the same general pattern was 1307. The shortest measured times between changes were of the order of the time for a fluid particle to circulate through the slot, i.e. about the shortest time for which it is meaningful to say that a particular pattern exists. Any variations of the average time and standard deviation with Rayleigh number were slight. We turn now to other features of the temperature measurements. Figure 4 shows an example of temperature fluctuations at three vertically separated points. These and other similar observations showed the following features. We recall (fig. 3) that the fluctuations in the slot but close to its top were primarily an alternation between two levels; this is shown again by the corresponding trace in fig. 4 (which has more frequent observations but over a shorter total time). We denote these two levels by 7” and T, (down and up). Where the temperature near the top was close to T,,the temperatures at points directly below were also close to T,.Where the temperature near the top was close to Td, the temperatures below fluctuated much more vigorously. These fluctuations had Td as their “baseline”. The size of
Temperature
A
-s
*
100 Fig. 4. Simultaneous of 1.8ST. X/D
temperature
= -7.9.
J/D
-A
T
variations
at three
vertically
) z/D = 0.4,
= 0. (--
separated
points.
Time based on sampling each at intervals
.) z/D = 6.3,C----l
(.
z/D =
10.2 (Ra = 3.65~
IO’).
the fluctuations increased with depth, so that around and below mid-depth they extended up to and occasionally above T,. This behaviour is summarised and specified more quantitatively by fig. 5. In this the temperature is divided into ranges, and the fraction of time spent in each range plotted as histograms. The two histograms relate to close to the top of the slot and mid-depth. (Each combines data from the centre of the slot in the x-direction and towards one end; there were no apparent systematic differences between the two positions.) One sees a bimodal structure in the first histogram corresponding to the alternation between levels. In the second one only the higher temperature peak is present corresponding to the behaviour described above. The observations of some temperatures at mid-depth higher than the highest observed at the top correspond mainly to large fluctuations during downflow, rather than small ones during upflow.
0 3-
r--, ---I ;
0.2--1
.S F L? 1
s 2
Olr--J
r__J 1 I
I , I I
:-
---
1 I :--,
r__J
i__
I
1
0
1__T
I
Temperature -----@I-----,
Fig. 5. Histograms intervals
of
1.85~.
showing Data
fraction from
x/D
of occasions
on which
= - 7.9 and z/D
x/D
the temperature
= 0 combined,
= 10.2 (Ra = 3.65 x 10’).
y/D
was in various
ranges when
sampled
= 0, ( ---J
z,‘D=O.4.
(---_)
at
D.J. Tritton et al. / Correction in an open-topped slot
213
The value of CT, - 7’,‘,)/0, based on the separation of histogram peaks, varied between 1.00 at Ra = 2.3 X 10’ and 0.75 at Ra = 8.5 X 10”. The difference may not be significant; it is smaller than the changes that would be produced by using a different choice of N in the evaluation of 0. The fact that
5. Previous
experiments
Siegel and Norris (1957) carried out experiments on convection in air between two vertical plates. The cases studied included that in which the bottom and ends were closed but the top was open to the room, giving a system similar to that in our experiments. The boundary conditions were, however, different; Siegel and Norris imposed uniform heat flux over the plates. They observed an alternation between clockwise and anticlockwise circulation. It seems probable that this was essentially the same phenomenon as we observed. The difference in boundary conditions makes determination of the time-scale corresponding to our T somewhat problematic. However, a rough evaluation indicates that the time quoted for the average period of flow reversals (about 20 s) corresponded to 15~. In view of differences in geometry and Prandtl number, as well as the boundary conditions, order of magnitude agreement with our results seems consistent with the above interpretation. Siegel and Norris varied the relative dimensions of their slot. Their values of L,/L, were always smaller than ours, so one might expect to get only alternations between single cell patterns and not the appearance of two cell ones, Their observations indicate that the changes in flow pattern do not occur if Lx/L, is too small or D/L, is too large. Their observations of temperature fluctuations were confined to positions level with the top of the slot. They thus did not have information comparable with our fig. 4, which leads to the interpretation to be proposed in section 6.
6. Interpretation Although our experiments were exploratory and so the observations of limited extent (particularly with regard to the absence to temperature measurements for y/D Z 01, they are sufficient to indicate the principal aspects of the interpretation. In this section we consider this in a largely qualitative manner. One does, of course, need to check that the orders of magnitude of the quantities involved are consistent with the interpretation; this is done in appendix 1.
The principal observations are the intermittent changes in the large-scale motion and, as an inference from the temperature measurements, the larger heat transfer in downflowing regions than in upflowing. The former may be seen as a consequence of the latter: the buoyancy driving the circulation is lost as a result of rapid warming of the downflow. (We consider this a bit more fully towards the end of this section.) Hence, an explanation of the difference in heat transfer provides an explanation for the overall behaviour. An analogous difference has been observed in mixed convective flows (i.e. forced flows with non-negligible buoyancy forces) through vertical channels and pipes (Carr et al., 1973; Axcell and Hall, 1978; Easby, 1978; Jackson and Hall, 1978; Nakajima et al., 1980; Petukhov et al., 1982; Cotton and Jackson, 1987; Tanaka et al., 1987; Polyakov and Shindin, 1988). Provided that the flow is turbulent, heat transfer coefficients are increased relative to forced convection values for downtlow with heated walls or upflow with cooled ones and decreased for upflow with heated walls or downflow with cooled ones. (The latter trend is reversed as the free convection regime is approached). Equivalent processes may act locally in the present flow. Ultimately, of course, the motion is entirely due to buoyancy forces. One may, however, distinguish between the generation of the large-scale circulation by temperature differences in the x-direction and local processes within this circulation due to variations of the buoyancy force with y and/or time, and apply the analogy to the latter. The mixed convection heat transfer observations have been interpreted in terms of buoyancy induced changes to the intensity and structure of the turbulence. The principal mechanism usually considered (e.g., Jackson and Hall, 1978; Cotton and Jackson, 1987) is modification of the mean velocity profile by the mean buoyancy in a way that is either “destabilising” or “stabilising”, loosely analogous to the effect of an adverse or favourable pressure gradient. It may be sufficient explanation of our observations to suppose that this mechanism is operative locally in the present flow. (See appendix 1 for a more quantitative consideration of this.) However, some further comments may be made. The Reynolds number based on the slot width D and the observed speed of the large-scale motion is of the order 103, lower than typical for the mixed convection experiments mentioned above and rather low for turbulent shear flow. On the other hand, we have noted in section 3 that the rest configuration is unstable with respect to many different convective modes. This suggests that the turbulence might be primarily or partially thermal turbulence rather than shear turbulence, i.e. that buoyancy is directly involved in the turbulent energy generation, through the temperaturevertical velocity correlation. It is worth noting that there is potentially a second mechanism, involving direct effects of buoyancy on the turbulent energy balance, for the changes in mixed convection heat transfer. The mechanism is considered more fully in appendix 2. It is usually, but not universally, taken to be the less important mechanism. Speculatively, we suggest that it is of more importance in the present context. The second stage of interpretation concerns the relationship between the above considerations and the observed changes in the main pattern of circulation. Basically, the point is that the downgoing fluid becomes too light for its density contrast with the upgoing to continue to drive the circulation. This must happen in a time shorter than a typical circulation time; otherwise, the heated fluid would travel round into the upgoing flow and preserve the motion. Quantitative consideration of this (appendix 1) suggests that flow resistance (skin friction) is important in the deceleration process; i.e. that T,, - Td does not become sufficiently negative over a sufficiently large region to provide the deceleration. Buoyancy is “lost” but not sufficiently “reversed”. The variation in the time between changes is presumably a consequence of the turbulent nature of the flow. Just when buoyancy is lost may depend on what temperature fluctuations happen to occur over an extended region. On the other hand, it is possible in view of the
D.J. Tritton et al. / Conmction
in an open-topped
slot
known chaotic behaviour of convection in loops and thermosyphons Hart, 1984; Gorman et al., 1986; Yorke et al., 1987) that the variations intrinsic chaotic behaviour.
Appendix
215
(Creveling et al., 1975; are a manifestation of
1. Orders of magnitude
We consider the orders of magnitudes of various quantities involved in the convection in order to check that the ideas discussed in the main body of the paper are quantitatively consistent. Some relevant definitions are given in section 3. The principal quantitative observations used in the considerations below are that, if U is a typical speed of the large scale motion, U-aV
with a-3
and the temperature
difference
T,-T,-60
with b-l,
The dimensions of the apparatus no need to distinguish between denotes a length-scale of either. L/D
come into the considerations, them in order of magnitude Thus
but because L_, = L,, there is analysis; in the following L
- 25.
Some of the discussion treats the instantaneous large-scale flow patterns as if they were unchanging. This requires that the typical time for the fluid to circulate through the slot should be short compared with the average time between changes. In order of magnitude the former is #’ 2L/U
- 2Lr/aD
- 1%
and the latter has been given in section 4 as 417. It has been supposed that the main circulation is driven by the temperature difference between the downgoing and upgoing limbs. The buoyancy force must be sufficient to provide the accelerations of the fluid moving round the slot. In order of magnitude, the ratio of the relevant inertia to buoyancy forces is
u 2/L - a’D/bL
- i.
ga( T, - Td) Detailed consideration of the ideas in section 6 implies that there must be differences in the temperature profiles in the upgoing and downgoing regions, such that the average (over y) temperature difference will be larger than T, - Td. Thus probably the inertia/buoyancy ratio is sufficiently less than 1 to indicate that skin friction is significantly opposing the buoyancy force. This is more definitely the case when we consider below the dynamics of changes in the flow pattern. The temperature rise as the fluid circulates must be provided by conduction from the walls (primarily in the downgoing region). The discussion has assumed that the associated thermal boundary layer is thin compared with the slot width. The thermal balance requires 2K(aT/ay),
- UD aT,‘dz.
#’ At various places in this analysis, factors magnitude considerations, but are included they were cumulative.
of 2 are included. They are not of much significance in order of partly because they may make the reasoning clearer and partly in case
Defining
a thermal
boundary
layer thickness
6r by
- AT/a,..
(V,‘ay),
where AT is the temperature so is - LO/2D, gives KL@/L)&,.
-
difference
between
the wall and the main body of the fluid and
abVDH/L.
Thus 6,/D
- L2/abD2
Ra”’
Pr’/‘-
l/10.
The difference in the heat transfer in downgoing and upgoing regions has been tated by analogy with mixed convection. In most of the work on mixed convection, reduced by use of a parameter of the form I’= Gr/Re”
interpredata are
Pr”
(where Gr is a local Grashof number based on AT and D) but with considerable variation the values of tn and II chosen. Order of magnitude analysis as above can be used to give I‘_ L,2Da”’
Rat?‘/‘-
1 P,.“+ 1-Ill/2
in
(1)
for the present flow, and this used to see how big an effect the analogy indicates - but we omit details. For any of the suggested values of rn and n. the effect would be large enough for a substantial difference between downgoing and upgoing regions. There is, however, a complication that, in the “stabilised” case, the mixed convection trend is reversed at large enough Gr. If r given by eq. (1) is above the reversal, then the analogy is problematic. Unfortunately, the matter cannot be resolved because different comparisons give different conclusions: using the trend for mixed convection given by Jackson and Hall (1978) or that given by Polyakov and Shindin (1988) does indicate that I’ is above its reversal value, but using that given by Cotton and Jackson (1987) does not. We turn attention to the dynamics of the pattern changes. As indicated by the temperature fluctuations (e.g., fig. 3 or fig. 4), the change in where the fluid is rising and falling usually occurs in time of order 5~. Flow visualisation (e.g., fig. 2) suggests, however, that the deceleration is more extended than this. The interpretation requires that the process occurs in a time less than the circulation time (estimated as 15~ above). We thus take the time scale of a change as t, -CT
with c - 10.
Fluid accelerations involved in a change are of the order aV/c~. by the buoyancy force would require a temperature difference (Td - T,)<. - aV/gacT
- aO/c
Production
of these directly
- O/3.
Observed “reversed” temperature differences are generally smaller than this (e.g. figs. 4 and 5), and again it should be remembered that the observations are confined to y/D = 0. Hence, as stated in section 6. flow resistance is important. Defining the thickness 6, of the associated dynamical boundary layer similarly to the definition of a.,. above, 2vaV/6,,
- aVD/c?-.
giving ii,,/D - 2c Pr’/2/Ra’/2 This is consistent
-, A.
with the dye observations.
D.J. Tritton et al. / Com>ection in an open-topped
Appendix
slot
217
2. The role of buoyancy
In section 6 reference was made to the possibility of a direct contribution by buoyancy effects to the turbulent energy balance in mixed convection near vertical walls. Some explanation of this is needed, particularly since it is a process that, by over-simple application of gradient transfer ideas, may be supposed nonexistent (e.g., Tanaka et al., 1987). To be specific, we consider downward flow next to a heated vertical wall as in fig. 6. The coordinates (chosen to correspond to those in fig. 1) and mean velocity and temperature are shown in fig. 6; the velocity fluctuations are denoted by (u, ~1,w) and the temperature fluctuation by 0. The relevant term in the turbulent energy balance is -gcuw0. One anticipates that this will be non-zero, because both 1-w (related to the Reynolds stress) and ~‘0 (related to the heat transfer away from the wall) are non-zero. It is probable that similar eddy structures in the turbulence are responsible for these two processes; it is thus unlikely that I’ and w are correlated and L’and 6’ are correlated without w and 0 being correlated. (The signs are such that, for the particular case in fig. 6, z is negative and a positive, suggesting that w0 will be negative and thus turbulent energy generation positive, i.e. the process is destabilising relative to forced convection.) In fact, experimental evidence suggests that the effect is stronger than is required by the above argument. Experiments on forced convection (Johnson, 1959; Bremhorst and Bullock, 1970) show, in the present notation, that j w0 I is larger than 1~10 1, typically by a factor of 2. If one now carries out the thought experiment of increasing the Richardson number, one of the ways in which buoyancy effects will become non-negligible is through turbulent energy production (or removal in a stabilising case>. This may be expressed by saying that Monin-Obukhov theory (e.g., Tritton, 1988, sect. 21.7) is relevant to flow near vertical surfaces as well as to its usual context of flow near horizontal ones. However, only for horizontal surfaces is this process the only one. As considered in section 6, for a vertical (or inclined) surface, there is also an “indirect” process due to the buoyancy associated with the mean temperature. The relative importance of the two processes for
I--
Y
/ / Fig. 6. Definition
sketch.
differences between mixed and forced convection is not agreed. Petukhov et al. (1982) conclude that the direct effect is dominant. On the other hand, it is often assumed that it is small compared with the indirect effect, and support for this assumption has been given by numerical modelling by Cotton and Jackson (1987) and by Tsai et al. (1987). Measurements of WB have been made in two experiments in which buoyancy effects produced very walls (Carr et change in the production (-non-negligible.
large differences from forced convection; both were for upflow with heated al., 1973; Poiyakov and Shindin, 1988). In these cases, the most substantial turbulent energy balance from that for forced convection was in the shear zw %V/ay) but the direct contribution by buoyancy to this balance was
References Axcell, D.P. and W.B. Hall (197X) Mixed convection to air in a vertical pipe. Proc. 6th. Inr. Heur Trunsfer Conf. Toronto I. pp. 3742. Bremhorst, K. and K.J. Bullock (1’1170) Spectral measurements of temperature and longitudinal velocity fluctuations in fully developed pipe flow. I?tt. J: i%ur ‘4fa.s~ Trctrrs@r f.?, 1313-1329. Carr, A.D., M.A. Connor and H.O. Buhr (1973) Velocity. temperat~lre, and turbulence measurements in air for pipe flow with combined free and forced convection, J. Heat Transfer Y.5. 445-452. Cotton, M.A. and J.D. Jackson (19X7) Calculation of turbulent mixed convection in a vertical tube using a low Reynolds number k -E turbulence model. Preprint, 6tll Symp. Turbulent shear frow, Tou/ouse, 9.6.1-h. Creveling. H.F., J.F. De Paz. J.Y. Baladi and R.J. Schoenhals (1975) Stability characteristics of a single-phase free convection loop, J. Flml Mech. 67. 65-84. Eashy, J.P. (lY7Xf The effect of buoyancy on flow and heat transfer for a gas passing down a vertical pipe at low turbulent Reynolds numbers. I12t. J. Heat Mas.s Trmsfer ZI, 791-X01. Gershuni, G.Z. and E.M. Zukhovitskii (1972) Conr,rctii,e Stability qf Incoqmwible ,fluid.s (Israel Program for Scientific Translations), sec. 12. 1.5. Gorman. M., P.J. Widmann and K.A. Robhins (lYX6) Nonlinear dynamics of a convection loop: a quantitative comparison of experimental witb theory, Ph~~.svsicu D 1Y. 2.55267. Hart, J.E. (19X4) A new analysis of the closed loop thermosyphon, lift. J. Ncclt Muss Tmnsfer 27. 125-136. Jackson, J.D. and W.B. Hall 11978) Influences of buoyancy on heat transfer to fluid flowing in vertical tubes under turbulent conditions, !v4 T(f itSI, Tlirbu~f,~zfForced C(~nl,e~fi~}fz in C~zff~~~c~.~ and Rod Bubb~e.s~~stu~7bul.pp. 1-2X. Johnson, D.S. (1959) Velocity and temperature fluctuation measurements in a turbulent boundary layer downstream of a stepwise discontinuity in wall temperature. J. A&. Mech. 26. 325-336. Nakajima, M., K. Fukui, H. Ueda and T. Mizushina (10X0) Buoyancy effects on turbulent transport in combined free and forced convection between vertical parallel plates. Int. J. Hear Mus.c Transfer 23. 1325-1336. Petukhov. B.S., A.F. Polyakov and O.G. Martynenko (19%) Buoyancy effect on heat transfer in forced channel flows. Froc. 7s!r Inr. Nelzt Tmn.s$v Cbrtj:, ~i~tl~~h. pp. 34% 362. Polyakov, A.F. and S.A. Shindin (I%%) Development of turbulent heat transfer over the length of vertical tubes in the presence of mixed air convection, in/. J. Heat Muss Trunsfer 31. 987-992. Siegel, R. and R.H. Norris (1957) Tests of free convection in a partially enclosed space between two heated vertical plates, Am. Sot. Mech. Eng. Trans. 79. 663-673. Strens, M.R. and J.R. Cann (19X6) A fracture-loop thermal balance model of black smoker circulation. Tecrwroplzys. 122, 307-324. Tanaka, H., S. Maru~ama and S. Hatano f1987) Combined forced and natural convection heat transfer for upward flow in a uniformly heated vertical pipe. Ilzt. .!. Heut Muss Transfer 317, 165-174. Tritton, D.J. (19X8) Physrcul Fluid Dynamics. 2nd Ed, (Clarendon Press. Oxford) sect. 21.7. Tsai, H.M., P.R. Voke and D.C. Leslie (1987) Numerical investigation of a combined free and forced convection turbulent channel flow, Preprint, 6r11 S?;mp. Turbulent shear flow, Toulouse, 5.3.1-6. Yorke, J.A.. ED. Yorke and J. Mallet-Paret (1987) Lorenz-like chaos in a partial differential equation for a heated fluid loop. Ptr+ca D 24, 27Y-291.
Research
Y (I YY2) 207-2
I8
North-Holland
Convection in an open-topped D.J. Tritton
slot
‘, P.A. Smith and D.S. Jeffcock
Departmcrrt of‘ Phycics, Unic~enity of‘ Nmvcu,stk upon Tyne. NE1 7RU, l/K Received
15 September
Abstract. An exploratory ture fluctuations,
1991
investigation
of free convection
the slot was uniform
horizontally
has been made, by flow visualisation
of water
and decreased
tank. The flow always had downgoing mainly in the downgoing
with mixed inhibited
convection
in vertical
intermittently.
changes are interpreted in heat transfer
and pipes: processes
into the base of a large
in the long horizontal
The rise in mean temperature
is lost. The difference channels
through
of the tempera-
in each of the large walls of
upwards. The top of the slot opened
regions. The intermittent
the buoyancy driving the circulation
and observations
slot. The temperature
and upgoing regions separated
slot, but the position of these changed occurred
in a vertical
direction
of the
of the circulating as a consequence
is in turn interpreted which turbulence
fluid
of this;
by analogy
is enhanced
or
by buoyancy effects in such flows are discussed.
1. Introduction We describe observations on the broad features of free convection in a slot extending downwards from the base of a tank. A vertical temperature gradient was maintained in the walls of the slot, with, in principle, no imposed horizontal temperature gradients. The wall temperature was the same at the top of the slot as the fluid temperature in the tank and increased downwards. Convection thus occurred as a result of the unstable temperature gradient in the slot. Our reason for experimenting with this configuration was possible application to convection in fissures in the ocean floor (Strens and Cann, 1986), but we report the work here as a basic fluid dynamical study. The observations and their interpretation have implications for heat transfer processes in configurations extending beyond our specific system, in mixed as well as in free convection. The aim of the experiment in the first place was simply to discover the broad features of the convection pattern: whether the circulation extended over the full depth of the slot or was broken into a series of vertically stacked rolls; and, if the former, whether the upgoing and downgoing currents were separated in the direction of the long or short dimension of the slot. Anticipating our observations, the circulation did extend over the full depth, with the separation in the long direction; but there were intermittent changes in the pattern of this large-scale circulation. After we had terminated the experiments, we discovered that an “oscillation” which is almost certainly essentially the same phenomenon had been previously observed in a similar, although not identical, system by Siegel and Norris (1957). Our additional observations enable us to give a fuller interpretation of this behaviour. Our essentially exploratory experiments do, however, in turn indicate a need for a more extensive investigation to resolve the more speculative aspects of the interpretation. Unfortu’ Correspondence Grenoble
to: D.J. Tritton,
lnstitut
de Mkcanique
de Grenoble,
Domaine
Universitaire,
Cedex. France.
0169.5983/92/$04.00
0 1902 - The Japan Society of Fluid Mechanics.
All rights reserved
B.P. 53X,
38041
nately, for personal foreseeable future.
2. Apparatus
reasons,
we shall not be in a position
to carry out such experiments
in the
and procedure
The dimensions of the apparatus are shown in fig. 1. The working fluid was water. Coordinate directions x, y, and z as shown in fig. 1 are used to facilitate the description of the apparatus and observations below. Walls A and B of the slot were made of aluminium alloy 6.25 mm thick. The objective was to maintain the same uniform vertical temperature gradient in these two walls, without any horizontal temperature variations either within one wall or between the two. A good approximation to this was achieved in the following way. Outside each wall was a jacket through which air was pumped. The air entered at room temperature just below the top tank and was heated by passing over an array of 20 heating elements, stretched horizontally across the air gap, before venting to the room at the level of the bottom of the slot. On each side the airflow was driven by four fans (hair driers with their heating elements removed) and ducted via diffusers to different regions in the x--direction of the top of the jacket. The temperature distribution so established in each wall was monitored by 20 thermojunctions in a rectangular array. Adjustments were made to the current through the heating elements (four consecutive elements being in series, so that there were five controls on each side) to give the required vertical tcmpcrature gradient: and to the fans to minimise horizontal temperature variations. Departures from required temperatures were of the order of 5% of the total temperature difference between top and bottom. The observed flows showed no bias that could be attributed to unwanted tcmperaturc variations in the s-direction or differences in the y-direction. A cooling-water coil was placed in the top tank to keep the water temperature there steady. The tops of the slot sidewalls were in contact with this water. as a way of achieving the requirement that their temperatures should be the same. This matching could be optimised by varying the tlow rate through the cooling coil.
Free surface 580
2-i 2-5 END VIEW
TOP VIEW Fig. I. Dimensions
of the
apparatus; all quantities
are mm
The bottom wall and end walls of the slot were of perspex. Clearly, having to view the flow through these narrow edges was somewhat restricting in the information that could be obtained from flow visualisation. However, it proved adequate to discover the broad features of the flow described below. Usually, the flow was illuminated through the base and/or one of the end walls and viewed through the other end wall and/or down through the top tank. The principal method of flow visualisation was to dissolve thymol blue indicator in the water and to inject a small amount of sodium hydroxide solution through hypodermic tubing to generate a patch of alkaline and thus dark blue fluid. In the later stages of the experiment it was noticed that small suspended particles originating as rust from walls of the top tank, were also helpful in revealing the flow pattern. (The tank was therefore not given the planned repaint!) Temperature measurements in the slot were made with thermojunctions. These provided the main information on which our interpretation is based. All measurements were made at the centre of the slot in the y-direction, but at various positions in the X- and z-directions as indicated in section 4.
3. Specification We denote the width of the slot by D (= 25.4 mm>, the temperature of the walls by T,(z) and that of the fluid by T(x, y, z, t). Fluid properties are denoted: (Y, coefficient of expansion; V, kinematic viscosity; K, thermal diffusivity. Because the maximum range of T, is about 60°C these quantities vary substantially; for example, a, which is involved in our scaling parameters (see below) varies by a factor of about 3 over this temperature range. Because we observe (section 4) that most of the fluid is at a temperature not much above that of the top tank, values at this temperature are used in calculating quantities quoted below. The Rayleigh number gaD”
dT,/dz
Ra = VK
ranged between 2.3 x 10” and 8.5 x lo’, the upper limit corresponding to the 60°C maximum range of T, achievable. Critical Rayleigh numbers for various modes of linear instability of the rest configuration in a slot of infinite vertical dimension are of the order 10’ to lo4 (Gershuni and Zukhovitskii, 1972), so one expects vigorous, probably turbulent, convection. The Prandtl number, Pr = V/K, was about 6.3 (and would be lower if based on values of v and K at, e.g., an average value of T,). The geometrical non-dimensional parameters specifying the situation are, of course, L,/D and L,/D, where L, and L; are the slot dimensions in the X- and z-directions. As seen from fig. 1, these were respectively 25.5 and 22.7. To non-dimensionalise the quantitative aspects of our observations, we introduce temperature difference, time, and velocity scales as respectively, 0 = D dT,/dz, It should
be noted
that
T = (ga
dT,/dz)
0 is small compared
-I”,
V= D/r.
with the total imposed
temperature
difference,
L,O/D.
4. Observations The central feature of our observations ing currents were separated in the long
was that, at any instant, the upgoing and downgo(x) direction, but the pattern of this circulation
changed intermittently. This result came in the first place from flow visualisation. Subsequent temperature measurements proved a more convenient indicator of what pattern was occurring, and also provided the clue to the mechanism of the changes. Dye produced by alkali injection close to one of the end windows showed that the flow was turbulent. The dye was mixed across most of the slot in the y-direction in a time short compared with the time taken to travel round the slot. The movement of the dye patch showed little variation of velocity with y, implying that such variation occurred mainly in boundary layers thin compared with the slot width. The speed could be estimated; the maximum through the time sequence described below was typically 3V, but varied between approximately 1.5V and 4V. Observations of the above features were possible only close to a window, but either method of flow visualisation could give a good indication of the general circulation pattern throughout the slot. At any instant there were either one or two cells in the slot, i.e. a downgoing flow at one end and upgoing at the other, or downgoing at both ends and upgoing in the middle, or the reverse of the latter. Sometimes, particularly for single cell patterns, there was marked asymmetry, i.e. the downgoing flow would the broad and slow and the upgoing narrow and fast, or vice versa. The speed range 1.5V to 4V quoted above is related to this; the average speed probably varied less. A particular pattern never persisted permanently. Quantitative information on the timescale of the changes was given also by the temperature measurements and will be discussed below. From the flow visualisation it appeared that the circulation was most rapid shortly after a change of pattern. Also at this stage there was strong interchange of fluid between the toi, tank and the slot. (The fluid entering the slot was drawn from close to the base of the tank; fluid leaving the slot penetrated to close to the top of the tank.) The circulation then slowed down, leading to the fluid being almost stationary just before the onset of a new pattern of circulation. During the later stages, there was a tendency for the interchange between slot and tank to be reduced, i.c. there was significant horizontal motion within the upper part of the slot. Figure 2 shows an example of one sequence observed more fully than usual; because of the limitations to the viewing of the flow, only the broad features, not the details, of these diagrams are significant. A few experiments were made with only the slot, and not the top tank, filled with water. In this case, the circulation always consisted of a permanent two-cell pattern, rising at the centre (in the x-direction). No changes of the sort occurring when there was water in the tank were observed. Returning to the experiments with the top tank filled, temperature measurements in the slot close to its top showed alternations between two levels (specified more quantitatively below) with relatively small fluctuations about each level. This behaviour evidently corre-
-1 /
L-------(
i
ID 77
Fig. 2. Example similar
hut
with
approximate
of observed sequence of flow patterns. reversed
sense of circulation.
Only
involving change from an initial pattern the
times for which the flow was predominantly
broad
features
of each
pattern
of the form shown are indicated.
to a final one that is are
significant.
(Ra = 8.3 X IO’)
The
D.J. Tritton et al. / Com,ection in m open-topped slot
21 I
Temoerature
>
4
1007 Fig. 3. Simultaneous temperature 7.57. y/D = 0, z/D = 0.4, (-
-Trme
variations at three points close to top of slot, based on sampling ~x/D=-12.0,(~~~~~~~x/D=0,~----_)x/D=12.0~Ra=X.O~10~~.
each at inten&
of
sponded to alternations between downward and upward motion at the point of observation (an interpretation confirmed by simultaneous flow visualisation and temperature measurement). Figure 3 shows an example of simultaneous temperature measurements at three points separated in the x-direction. Similar observations with an array of five equally spaced thermojunctions gave a good indication of the circulation pattern. We illustrate the changes in the circulation pattern with an example of an observed sequence (at Ra = 3.7 x IO"). In this 1 and 2 indicate the number of cells, C and A indicate clockwise and anticlockwise (as seen from one, arbitrarily chosen, side of the apparatus), and U and D indicate up and down in the middle. The sequence was 2U/ 1C/ 2U/ 1A/ lC/ 2D/ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ lA/ 2D/ 1A. This example is typical of the behaviour at the lower values of the Rayleigh number (roughly 2 < Ra/lO” < 4). For higher values (roughly 4 < Ra/lO” < S>, two cell patterns occurred less frequently, so that the behaviour was predominantly an alternation between clockwise and anticlockwise circulation. Occasionally, as revealed by the flow visualisation, the motion would die away as if a change were about to occur, but then re-start with the same sense of circulation. Such behaviour would probably not have been apparent from the temperature observations. [It should be added that, although the common occurrence of two cell motion has been attributed above to Ra being lower, it is not certain that this is a genuine Rayleigh number effect; it is possible that the small departures from ideal boundary conditions that could be influential in the detailed behaviour (although not the broad features) were different for different values of dT’,/dz.] The time between consecutive changes in pattern was quite variable and showed no obvious periodicity or other regularity. The average time was 41~ and the standard deviation of the distribution (representing actual variability and not experimental uncertainty) was 15~. The longest time for which the flow was observed to remain in the same general pattern was 1307. The shortest measured times between changes were of the order of the time for a fluid particle to circulate through the slot, i.e. about the shortest time for which it is meaningful to say that a particular pattern exists. Any variations of the average time and standard deviation with Rayleigh number were slight. We turn now to other features of the temperature measurements. Figure 4 shows an example of temperature fluctuations at three vertically separated points. These and other similar observations showed the following features. We recall (fig. 3) that the fluctuations in the slot but close to its top were primarily an alternation between two levels; this is shown again by the corresponding trace in fig. 4 (which has more frequent observations but over a shorter total time). We denote these two levels by 7” and T, (down and up). Where the temperature near the top was close to T,,the temperatures at points directly below were also close to T,.Where the temperature near the top was close to Td, the temperatures below fluctuated much more vigorously. These fluctuations had Td as their “baseline”. The size of
Temperature
A
-s
*
100 Fig. 4. Simultaneous of 1.8ST. X/D
temperature
= -7.9.
J/D
-A
T
variations
at three
vertically
) z/D = 0.4,
= 0. (--
separated
points.
Time based on sampling each at intervals
.) z/D = 6.3,C----l
(.
z/D =
10.2 (Ra = 3.65~
IO’).
the fluctuations increased with depth, so that around and below mid-depth they extended up to and occasionally above T,. This behaviour is summarised and specified more quantitatively by fig. 5. In this the temperature is divided into ranges, and the fraction of time spent in each range plotted as histograms. The two histograms relate to close to the top of the slot and mid-depth. (Each combines data from the centre of the slot in the x-direction and towards one end; there were no apparent systematic differences between the two positions.) One sees a bimodal structure in the first histogram corresponding to the alternation between levels. In the second one only the higher temperature peak is present corresponding to the behaviour described above. The observations of some temperatures at mid-depth higher than the highest observed at the top correspond mainly to large fluctuations during downflow, rather than small ones during upflow.
0 3-
r--, ---I ;
0.2--1
.S F L? 1
s 2
Olr--J
r__J 1 I
I , I I
:-
---
1 I :--,
r__J
i__
I
1
0
1__T
I
Temperature -----@I-----,
Fig. 5. Histograms intervals
of
1.85~.
showing Data
fraction from
x/D
of occasions
on which
= - 7.9 and z/D
x/D
the temperature
= 0 combined,
= 10.2 (Ra = 3.65 x 10’).
y/D
was in various
ranges when
sampled
= 0, ( ---J
z,‘D=O.4.
(---_)
at
D.J. Tritton et al. / Correction in an open-topped slot
213
The value of CT, - 7’,‘,)/0, based on the separation of histogram peaks, varied between 1.00 at Ra = 2.3 X 10’ and 0.75 at Ra = 8.5 X 10”. The difference may not be significant; it is smaller than the changes that would be produced by using a different choice of N in the evaluation of 0. The fact that
5. Previous
experiments
Siegel and Norris (1957) carried out experiments on convection in air between two vertical plates. The cases studied included that in which the bottom and ends were closed but the top was open to the room, giving a system similar to that in our experiments. The boundary conditions were, however, different; Siegel and Norris imposed uniform heat flux over the plates. They observed an alternation between clockwise and anticlockwise circulation. It seems probable that this was essentially the same phenomenon as we observed. The difference in boundary conditions makes determination of the time-scale corresponding to our T somewhat problematic. However, a rough evaluation indicates that the time quoted for the average period of flow reversals (about 20 s) corresponded to 15~. In view of differences in geometry and Prandtl number, as well as the boundary conditions, order of magnitude agreement with our results seems consistent with the above interpretation. Siegel and Norris varied the relative dimensions of their slot. Their values of L,/L, were always smaller than ours, so one might expect to get only alternations between single cell patterns and not the appearance of two cell ones, Their observations indicate that the changes in flow pattern do not occur if Lx/L, is too small or D/L, is too large. Their observations of temperature fluctuations were confined to positions level with the top of the slot. They thus did not have information comparable with our fig. 4, which leads to the interpretation to be proposed in section 6.
6. Interpretation Although our experiments were exploratory and so the observations of limited extent (particularly with regard to the absence to temperature measurements for y/D Z 01, they are sufficient to indicate the principal aspects of the interpretation. In this section we consider this in a largely qualitative manner. One does, of course, need to check that the orders of magnitude of the quantities involved are consistent with the interpretation; this is done in appendix 1.
The principal observations are the intermittent changes in the large-scale motion and, as an inference from the temperature measurements, the larger heat transfer in downflowing regions than in upflowing. The former may be seen as a consequence of the latter: the buoyancy driving the circulation is lost as a result of rapid warming of the downflow. (We consider this a bit more fully towards the end of this section.) Hence, an explanation of the difference in heat transfer provides an explanation for the overall behaviour. An analogous difference has been observed in mixed convective flows (i.e. forced flows with non-negligible buoyancy forces) through vertical channels and pipes (Carr et al., 1973; Axcell and Hall, 1978; Easby, 1978; Jackson and Hall, 1978; Nakajima et al., 1980; Petukhov et al., 1982; Cotton and Jackson, 1987; Tanaka et al., 1987; Polyakov and Shindin, 1988). Provided that the flow is turbulent, heat transfer coefficients are increased relative to forced convection values for downtlow with heated walls or upflow with cooled ones and decreased for upflow with heated walls or downflow with cooled ones. (The latter trend is reversed as the free convection regime is approached). Equivalent processes may act locally in the present flow. Ultimately, of course, the motion is entirely due to buoyancy forces. One may, however, distinguish between the generation of the large-scale circulation by temperature differences in the x-direction and local processes within this circulation due to variations of the buoyancy force with y and/or time, and apply the analogy to the latter. The mixed convection heat transfer observations have been interpreted in terms of buoyancy induced changes to the intensity and structure of the turbulence. The principal mechanism usually considered (e.g., Jackson and Hall, 1978; Cotton and Jackson, 1987) is modification of the mean velocity profile by the mean buoyancy in a way that is either “destabilising” or “stabilising”, loosely analogous to the effect of an adverse or favourable pressure gradient. It may be sufficient explanation of our observations to suppose that this mechanism is operative locally in the present flow. (See appendix 1 for a more quantitative consideration of this.) However, some further comments may be made. The Reynolds number based on the slot width D and the observed speed of the large-scale motion is of the order 103, lower than typical for the mixed convection experiments mentioned above and rather low for turbulent shear flow. On the other hand, we have noted in section 3 that the rest configuration is unstable with respect to many different convective modes. This suggests that the turbulence might be primarily or partially thermal turbulence rather than shear turbulence, i.e. that buoyancy is directly involved in the turbulent energy generation, through the temperaturevertical velocity correlation. It is worth noting that there is potentially a second mechanism, involving direct effects of buoyancy on the turbulent energy balance, for the changes in mixed convection heat transfer. The mechanism is considered more fully in appendix 2. It is usually, but not universally, taken to be the less important mechanism. Speculatively, we suggest that it is of more importance in the present context. The second stage of interpretation concerns the relationship between the above considerations and the observed changes in the main pattern of circulation. Basically, the point is that the downgoing fluid becomes too light for its density contrast with the upgoing to continue to drive the circulation. This must happen in a time shorter than a typical circulation time; otherwise, the heated fluid would travel round into the upgoing flow and preserve the motion. Quantitative consideration of this (appendix 1) suggests that flow resistance (skin friction) is important in the deceleration process; i.e. that T,, - Td does not become sufficiently negative over a sufficiently large region to provide the deceleration. Buoyancy is “lost” but not sufficiently “reversed”. The variation in the time between changes is presumably a consequence of the turbulent nature of the flow. Just when buoyancy is lost may depend on what temperature fluctuations happen to occur over an extended region. On the other hand, it is possible in view of the
D.J. Tritton et al. / Conmction
in an open-topped
slot
known chaotic behaviour of convection in loops and thermosyphons Hart, 1984; Gorman et al., 1986; Yorke et al., 1987) that the variations intrinsic chaotic behaviour.
Appendix
215
(Creveling et al., 1975; are a manifestation of
1. Orders of magnitude
We consider the orders of magnitudes of various quantities involved in the convection in order to check that the ideas discussed in the main body of the paper are quantitatively consistent. Some relevant definitions are given in section 3. The principal quantitative observations used in the considerations below are that, if U is a typical speed of the large scale motion, U-aV
with a-3
and the temperature
difference
T,-T,-60
with b-l,
The dimensions of the apparatus no need to distinguish between denotes a length-scale of either. L/D
come into the considerations, them in order of magnitude Thus
but because L_, = L,, there is analysis; in the following L
- 25.
Some of the discussion treats the instantaneous large-scale flow patterns as if they were unchanging. This requires that the typical time for the fluid to circulate through the slot should be short compared with the average time between changes. In order of magnitude the former is #’ 2L/U
- 2Lr/aD
- 1%
and the latter has been given in section 4 as 417. It has been supposed that the main circulation is driven by the temperature difference between the downgoing and upgoing limbs. The buoyancy force must be sufficient to provide the accelerations of the fluid moving round the slot. In order of magnitude, the ratio of the relevant inertia to buoyancy forces is
u 2/L - a’D/bL
- i.
ga( T, - Td) Detailed consideration of the ideas in section 6 implies that there must be differences in the temperature profiles in the upgoing and downgoing regions, such that the average (over y) temperature difference will be larger than T, - Td. Thus probably the inertia/buoyancy ratio is sufficiently less than 1 to indicate that skin friction is significantly opposing the buoyancy force. This is more definitely the case when we consider below the dynamics of changes in the flow pattern. The temperature rise as the fluid circulates must be provided by conduction from the walls (primarily in the downgoing region). The discussion has assumed that the associated thermal boundary layer is thin compared with the slot width. The thermal balance requires 2K(aT/ay),
- UD aT,‘dz.
#’ At various places in this analysis, factors magnitude considerations, but are included they were cumulative.
of 2 are included. They are not of much significance in order of partly because they may make the reasoning clearer and partly in case
Defining
a thermal
boundary
layer thickness
6r by
- AT/a,..
(V,‘ay),
where AT is the temperature so is - LO/2D, gives KL@/L)&,.
-
difference
between
the wall and the main body of the fluid and
abVDH/L.
Thus 6,/D
- L2/abD2
Ra”’
Pr’/‘-
l/10.
The difference in the heat transfer in downgoing and upgoing regions has been tated by analogy with mixed convection. In most of the work on mixed convection, reduced by use of a parameter of the form I’= Gr/Re”
interpredata are
Pr”
(where Gr is a local Grashof number based on AT and D) but with considerable variation the values of tn and II chosen. Order of magnitude analysis as above can be used to give I‘_ L,2Da”’
Rat?‘/‘-
1 P,.“+ 1-Ill/2
in
(1)
for the present flow, and this used to see how big an effect the analogy indicates - but we omit details. For any of the suggested values of rn and n. the effect would be large enough for a substantial difference between downgoing and upgoing regions. There is, however, a complication that, in the “stabilised” case, the mixed convection trend is reversed at large enough Gr. If r given by eq. (1) is above the reversal, then the analogy is problematic. Unfortunately, the matter cannot be resolved because different comparisons give different conclusions: using the trend for mixed convection given by Jackson and Hall (1978) or that given by Polyakov and Shindin (1988) does indicate that I’ is above its reversal value, but using that given by Cotton and Jackson (1987) does not. We turn attention to the dynamics of the pattern changes. As indicated by the temperature fluctuations (e.g., fig. 3 or fig. 4), the change in where the fluid is rising and falling usually occurs in time of order 5~. Flow visualisation (e.g., fig. 2) suggests, however, that the deceleration is more extended than this. The interpretation requires that the process occurs in a time less than the circulation time (estimated as 15~ above). We thus take the time scale of a change as t, -CT
with c - 10.
Fluid accelerations involved in a change are of the order aV/c~. by the buoyancy force would require a temperature difference (Td - T,)<. - aV/gacT
- aO/c
Production
of these directly
- O/3.
Observed “reversed” temperature differences are generally smaller than this (e.g. figs. 4 and 5), and again it should be remembered that the observations are confined to y/D = 0. Hence, as stated in section 6. flow resistance is important. Defining the thickness 6, of the associated dynamical boundary layer similarly to the definition of a.,. above, 2vaV/6,,
- aVD/c?-.
giving ii,,/D - 2c Pr’/2/Ra’/2 This is consistent
-, A.
with the dye observations.
D.J. Tritton et al. / Com>ection in an open-topped
Appendix
slot
217
2. The role of buoyancy
In section 6 reference was made to the possibility of a direct contribution by buoyancy effects to the turbulent energy balance in mixed convection near vertical walls. Some explanation of this is needed, particularly since it is a process that, by over-simple application of gradient transfer ideas, may be supposed nonexistent (e.g., Tanaka et al., 1987). To be specific, we consider downward flow next to a heated vertical wall as in fig. 6. The coordinates (chosen to correspond to those in fig. 1) and mean velocity and temperature are shown in fig. 6; the velocity fluctuations are denoted by (u, ~1,w) and the temperature fluctuation by 0. The relevant term in the turbulent energy balance is -gcuw0. One anticipates that this will be non-zero, because both 1-w (related to the Reynolds stress) and ~‘0 (related to the heat transfer away from the wall) are non-zero. It is probable that similar eddy structures in the turbulence are responsible for these two processes; it is thus unlikely that I’ and w are correlated and L’and 6’ are correlated without w and 0 being correlated. (The signs are such that, for the particular case in fig. 6, z is negative and a positive, suggesting that w0 will be negative and thus turbulent energy generation positive, i.e. the process is destabilising relative to forced convection.) In fact, experimental evidence suggests that the effect is stronger than is required by the above argument. Experiments on forced convection (Johnson, 1959; Bremhorst and Bullock, 1970) show, in the present notation, that j w0 I is larger than 1~10 1, typically by a factor of 2. If one now carries out the thought experiment of increasing the Richardson number, one of the ways in which buoyancy effects will become non-negligible is through turbulent energy production (or removal in a stabilising case>. This may be expressed by saying that Monin-Obukhov theory (e.g., Tritton, 1988, sect. 21.7) is relevant to flow near vertical surfaces as well as to its usual context of flow near horizontal ones. However, only for horizontal surfaces is this process the only one. As considered in section 6, for a vertical (or inclined) surface, there is also an “indirect” process due to the buoyancy associated with the mean temperature. The relative importance of the two processes for
I--
Y
/ / Fig. 6. Definition
sketch.
differences between mixed and forced convection is not agreed. Petukhov et al. (1982) conclude that the direct effect is dominant. On the other hand, it is often assumed that it is small compared with the indirect effect, and support for this assumption has been given by numerical modelling by Cotton and Jackson (1987) and by Tsai et al. (1987). Measurements of WB have been made in two experiments in which buoyancy effects produced very walls (Carr et change in the production (-non-negligible.
large differences from forced convection; both were for upflow with heated al., 1973; Poiyakov and Shindin, 1988). In these cases, the most substantial turbulent energy balance from that for forced convection was in the shear zw %V/ay) but the direct contribution by buoyancy to this balance was
References Axcell, D.P. and W.B. Hall (197X) Mixed convection to air in a vertical pipe. Proc. 6th. Inr. Heur Trunsfer Conf. Toronto I. pp. 3742. Bremhorst, K. and K.J. Bullock (1’1170) Spectral measurements of temperature and longitudinal velocity fluctuations in fully developed pipe flow. I?tt. J: i%ur ‘4fa.s~ Trctrrs@r f.?, 1313-1329. Carr, A.D., M.A. Connor and H.O. Buhr (1973) Velocity. temperat~lre, and turbulence measurements in air for pipe flow with combined free and forced convection, J. Heat Transfer Y.5. 445-452. Cotton, M.A. and J.D. Jackson (19X7) Calculation of turbulent mixed convection in a vertical tube using a low Reynolds number k -E turbulence model. Preprint, 6tll Symp. Turbulent shear frow, Tou/ouse, 9.6.1-h. Creveling. H.F., J.F. De Paz. J.Y. Baladi and R.J. Schoenhals (1975) Stability characteristics of a single-phase free convection loop, J. Flml Mech. 67. 65-84. Eashy, J.P. (lY7Xf The effect of buoyancy on flow and heat transfer for a gas passing down a vertical pipe at low turbulent Reynolds numbers. I12t. J. Heat Mas.s Trmsfer ZI, 791-X01. Gershuni, G.Z. and E.M. Zukhovitskii (1972) Conr,rctii,e Stability qf Incoqmwible ,fluid.s (Israel Program for Scientific Translations), sec. 12. 1.5. Gorman. M., P.J. Widmann and K.A. Robhins (lYX6) Nonlinear dynamics of a convection loop: a quantitative comparison of experimental witb theory, Ph~~.svsicu D 1Y. 2.55267. Hart, J.E. (19X4) A new analysis of the closed loop thermosyphon, lift. J. Ncclt Muss Tmnsfer 27. 125-136. Jackson, J.D. and W.B. Hall 11978) Influences of buoyancy on heat transfer to fluid flowing in vertical tubes under turbulent conditions, !v4 T(f itSI, Tlirbu~f,~zfForced C(~nl,e~fi~}fz in C~zff~~~c~.~ and Rod Bubb~e.s~~stu~7bul.pp. 1-2X. Johnson, D.S. (1959) Velocity and temperature fluctuation measurements in a turbulent boundary layer downstream of a stepwise discontinuity in wall temperature. J. A&. Mech. 26. 325-336. Nakajima, M., K. Fukui, H. Ueda and T. Mizushina (10X0) Buoyancy effects on turbulent transport in combined free and forced convection between vertical parallel plates. Int. J. Hear Mus.c Transfer 23. 1325-1336. Petukhov. B.S., A.F. Polyakov and O.G. Martynenko (19%) Buoyancy effect on heat transfer in forced channel flows. Froc. 7s!r Inr. Nelzt Tmn.s$v Cbrtj:, ~i~tl~~h. pp. 34% 362. Polyakov, A.F. and S.A. Shindin (I%%) Development of turbulent heat transfer over the length of vertical tubes in the presence of mixed air convection, in/. J. Heat Muss Trunsfer 31. 987-992. Siegel, R. and R.H. Norris (1957) Tests of free convection in a partially enclosed space between two heated vertical plates, Am. Sot. Mech. Eng. Trans. 79. 663-673. Strens, M.R. and J.R. Cann (19X6) A fracture-loop thermal balance model of black smoker circulation. Tecrwroplzys. 122, 307-324. Tanaka, H., S. Maru~ama and S. Hatano f1987) Combined forced and natural convection heat transfer for upward flow in a uniformly heated vertical pipe. Ilzt. .!. Heut Muss Transfer 317, 165-174. Tritton, D.J. (19X8) Physrcul Fluid Dynamics. 2nd Ed, (Clarendon Press. Oxford) sect. 21.7. Tsai, H.M., P.R. Voke and D.C. Leslie (1987) Numerical investigation of a combined free and forced convection turbulent channel flow, Preprint, 6r11 S?;mp. Turbulent shear flow, Toulouse, 5.3.1-6. Yorke, J.A.. ED. Yorke and J. Mallet-Paret (1987) Lorenz-like chaos in a partial differential equation for a heated fluid loop. Ptr+ca D 24, 27Y-291.