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Wind-mixing experiments for solar ponds
Solar Energy Vol. 38, No. 6; pp. 389-403, 1987
0038-092X/87 $3.00 + .00 © 1987 Pergamon Journals Ltd.
Printed in the U.S.A.
WIND-MIXING EXPERIMENTS FOR SOLAR PONDS J. F. ATKINSON Department of Civil Engineering, S.U.N.Y. at Buffalo, Buffalo, NY, 14260 and D. R. F. HARLEMAN Department of Civil Engineering, Massachusetts, Institute of Technology, Cambridge, MA 02139 (Received for publication 18 November 1986)
Abstract--Experiments are described in which wind-induced entrainment was measured in a laboratory tank under strongly stratified double-diffusive conditions in order to calibrate a numerical model for application to solar ponds. Results showed that there was no effect of double-diffusive stratification and that the entrainment followed approximately an inverse Richardson number relationship. Shearinduced mixing was found to play a strong role in the entrainment process, and it is suggested that an effort be made to obtain more data on wind-induced currents in operating solar ponds. An additional series of tests is also reported in which a preliminary investigation of the effectiveness of floating wave suppressors was carried out. These tests showed that the wave suppressors, consisting of either plastic netting or PVC pipes, were able to reduce average turbulence levels in the water, though it appears that there may be some enhanced mixing directly beneath the nets or pipes. 1. I N T R O D U C T I O N
Previous solar pond studies have dealt with many of the physical processes of interest in understanding how to operate and maintain a pond for maximum heat generation. The ability to accurately model the temporal development of the temperature and salinity profiles becomes critically important when examining the technical and economic feasibility of constructing a pond in a given location for a given purpose. The present paper is concerned with one of the mechanisms responsible for determining the depth of the upper convecting zone (UCZ) of a solar pond. In particular, experiments were performed to obtain data for calibrating a model for the rate of layer deepening, or entrainment rate, under the action of mechanical (primarily wind-generated) mixing in the UCZ. Several windmixing mixed layer models have appeared in the solar pond literature[I-3], and the model used here is a modified version of the algorithm suggested by Atkinson and Harleman[2]. The present experiments were designed to provide data for calibrating some of the coefficients of this model. Wind shear stress was used to drive the mixing, and entrainment was measured in a laboratory wind/wave tank with average wind speeds of between 4 and 11 m/s. Two-layer stratification was used, with the density step produced either by salinity differences or by both temperature and salinity differences, such that "diffusive" double-diffusive stratification similar to that found in salt gradient solar ponds was obtained. Although the water layer beneath the entrainment interface is normally stratified in a solar pond, it is felt that the present experiments should provide useful information for entrainment modeling in ponds. This is primarily
because the entrainment model described below predicts that for the strong density step at the base of the U C Z there will be a minor effect of stratification in the lower layer. Data are also reported for stratification levels stronger than those usually found in the literature. It is important not only to be able to predict wind-mixing effects but also to be able to control them for improved collection efficiency. A possible means of reducing wind-mixing consists of installing a floating wave-suppression system consisting of a grid of plastic netting[4], a grid of PVC pipe[5] or plastic rings[6]. Twede et a/.[7] have reported a laboratory study of the effects of floating barriers on wind-induced mixing. Their results provide some preliminary guidelines for the design of wave suppressors, but in general little has been done to examine these effects in any detail. Several tests were conducted to determine the effect these floating systems would have on wind-induced mixing. Because of the geometry of the experimental apparatus the circular rings were not tested, but several results for the nets and pipes are presented.
2. MODEL DEVELOPMENT The entrainment model is derived from a parameterization of the one-dimensional (vertical) turbulent kinetic energy (TKE) budget applied to the UCZ, which is written as[2, 8]: Oq
g--
Ot
w'p'
Po
0 - -2- ( w ' q + w'p'/po) oZ -
389
u'w'--
Oz
-
~,
(1)
J. F. ATKINSONand D. R. F. HARLEMAN
390
at the interface and Cg 2 is a drag coefficient. Assuming ui/-ff ~ 1,
where q = T K E per unit mass, g = gravity, u and w are horizontal and vertical velocities, respectively, p = density, Po = mixed layer (UCZ) dens i t y , p -- pressure, z = vertical coordinate (positive downward), ~ = dissipation rate, primes denote fluctuating quantities and overbars represent mean (time-averaged) values. The left-hand side of eqn (1) represents the time rate of change of T K E and the right-hand side represents gravity work, flux divergence, shear production and viscous dissipation, respectively. The model here follows a similar approach as in [8] except for the treatment of the turbulence shear production term, which is described as follows. If it is assumed that interracial shear stress (due to a mean current in the UCZ) provides the major source of T K E and that the primary sink balancing this source is the conversion of T K E to potential energy by entrainment, the energy equation is expressed as a balance between gravity work and shear production,
u~
Kh
-- ~ CoRio I u gm '
(5)
where Rio is defined here with ~ in place of u . i . Again, an inverse Richardson number relationship is obtained when Km oc ~h. Christodoulou[13] has shown that eqn (5) should be valid for a broad transition range of Rio between conditions in the mixed layer which correspond with supercritical (lower Rio) and subcritical (higher R i o ) flow, with the critical value of Rio being of order 0(1). In the supercritical range the exponent o f Rio in eqn (5) is closer to - 1 / 2 , agreeing with several earlier models[1416], and in the subcritical range the exponent value is closer to - 3 / 2 , which agrees with other resuits[17] and is also the form of the entrainment relationship found for salt-stratified experiments with oscillating grids (a review of these latter experiments is provided in [18]). A possible means of (2) (hallowing the model above (eqn (5)) to reproduce reg w ' p ' ~ - u ' w ' 0_~ . sults supporting an exponent value other than - 1 Po OZ would be to assume that Km is dependent on Rio, Following Buch[9], a vertical momentum transfer at least outside of the transition range. For the purcoefficient Km is defined so that u2i = - K,, aT/ poses of this paper, however, it is assumed that the az, where u , i is the friction velocity associated with inverse R i o relationship holds, so that the parainterracial shear (note that the upper layer is as- meterization of the shear production term in the sumed to be flowing, so the velocity gradient O-ff/az T K E budget is - u ' w ' O-ff/Oz oc "ff3/h" Shear production has been neglected in previous < 0). u,i is also approximated by u2,i ~ u ' w ' . Then, assuming w ' p ' ~ ueAp, where Ap is the density step developments of the entrainment model[2, 8]. The across the entrainment interface and ue = d h / d t = present model was developed by Atkinson et a/.[18] entrainment rate, where h = mixed layer (UCZ) and consists of the same basic budget considered previously, though the sink term is modified and depth, we have the shear production term described above is ing cluded. In the course of developing the entrainment u4,i u,A9 ~ K-'--~" (3) model, a velocity scale for motions in the mixed layer is defined as or, where (r accounts for various This equation can be rewritten in a more convenient inputs of kinetic (mixing) energy into the layer. In other words, the rate at which kinetic energy is inform by introducing a bulk Richardson number, R i . i troduced into the U C Z is proportional to (r 3. F o r = gAph/poU2.i. Then, example, in cases where surface shear stress drives the interracial mixing it is common to assume ~ U---Le ¢c R i . l i u , i h (4) u . , where u . is the friction velocity at the surface. u,i Km Other researchers have assumed that the bulk mean velocity is a better choice (this is consistent with A common entrainment model is the inverse Ri- the use of Rio in eqn (5) above), so cr ~ ~. After chardson number relationship, which is supported parameterizing the various terms in the T K E equaby a number of experiments ([10, 11], for example) tion, using tr as the scaling velocity, the entrainment and also by field measurements[12]. This result is model is[18] m
Ue O"
C F + Csn(-ff/cr) 3 - ~' + C r R i r P e 7 1 / 2 Ct + R i
easily obtained from eqn (4) i f K m oc u,~h, a plausible assumption. A n alternative way of writing this expression in terms of the mean layer velocity is obtained by noting that r = interracial shear stress = pou2,i = Cg2po ('ff - ui) 2, where ui is the velocity
-
C ~ R i ~ e e 7 v2
(6)
where the Cs are parameterization coefficients, R i is defined with (r as the velocity scale, R i r and Ris are the Richardson numbers defined by the individual b u o y a n c y steps associated with the temperature and salinity gradients, respectively, P e r =
Wind-mixing experiments for solar ponds orh/kr and Pes = (rh/k, are the thermal and saline Peclet numbers, and k r and ks are thermal and saline diffusivities, e' is a term describing the leakage of energy away from the entrainment interface by the formation of internal waves radiating into the gradient region. It is expressed as[18]
C e' = ~ [2Rig' + N2h - 2Ri(g '2 +
N20"2)1/2],
391
o: R i . 1"5117] or ue/u, o: R i . 1'2120], implying that CF and Csn are possibly functions of R i , . This sort of relationship is also consistent with the conclusions of Christodoulou[13], mentioned above. The present experiments were performed to obtain data for large interfacial density steps in a doubly diffusive system in order to evaluate the form of the entrainment relationship that would be applicable for modeling solar ponds.
(7) 3. EXPERIMENTS
where g ' = gAp/po, N = [-g(Op/Oz)/po] 1/2 is the b u o y a n c y frequency in the region immediately below the interface, and C is a parameterization coefficient. (Note that it can be shown that e' approaches zero as N approaches zero, which is the expected result since internal waves will propagate away from the interface only when N is non-zero.) Finally, the terms in eqn (6) involving the Peclet numbers represent limits placed on ue due to molecular diffusion. These terms are not expected to be significant except for very high stratification levels with Richardson numbers at least of order 104105, and will probably be negligible in most situations. F u r t h e r discussion of these terms may be found in [18]. It should be noted that the present experiments deal with surface shear-stress-driven mixing in a two-layer system so that ~' may be neglected. We also assume that the terms involving the Peclet numbers are negligible. Then if tr = C j u , , where C1 is a constant, we have U~e = CICF "4 CSH(-~/lg*) 3/C2 u. C, + Ri./C~ '
(8)
where R i , = g'h/u2,. As noted above, some investigators have used (r = K, which would imply (from eqn (6)) that ue/(r g: Rio I , as long as Rio was sufficiently greater than C,. Depending on the coefficient values, eqn (8) may also approximate an inverse Richardson number relationship, which will be shown to describe the present results. It should be noted that eqn(8) is written in terms of u , rather than u,,-, the interracial friction velocity used in parameterizing the shear production term. This is because we are now considering mixing driven by surface shear, and the effect of u,i (i.e., mixing caused by a mean velocity step across the density interface) is already included in the term in eqn (8) multiplied by Cs~r. One of the attributes of entrainment relationships such as eqn (8) is the ability to model ue over a wide range o f R i , . F o r example, ue/u, approaches a finite value as R i , approaches zero, and the slope of the curve (ue/u, vs Ri,) increases as R i , increases, which is a desirable result if the model is to be able to reproduce some of the experimental evidence[19]. Several wind-mixing experiments have indicated that over limited R i , ranges, Ue/U,
Two basic experiments were performed. The first set of tests was designed to provide data for calibrating the model described above by measuring the rate of entrainment in a two-layer system as a function of wind speed and stratification (i.e., as a function of Ri,). The influence of fetch length was also considered by taking measurements in test sections of two different lengths. The second set of tests was performed in order to obtain laboratory data for a preliminary evaluation of the performance of several types of mixing suppression systems. These experiments are described below. Entrainment tests The wind-mixing experiments were conducted in a 25 m long wind/wave flume, sketched in Fig. 1. The wind inlet and outlet sections were made of heavy gauge aluminum and the cover consisted of plyron boards clamped along the top. Tape was used to cover all junctions to keep the tunnel as airtight as possible. The inlet section ended in a threefoot (91 cm) long horizontal section which had flow vanes and straighteners to smooth out the air flow. The fan was located at the outlet end and sucked air through the tunnel and wind speed was controlled by the opening of an exhaust damper on the fan. Measurements were made primarily for three discrete wind speeds, W = 4.4, 7.8 and 10.2 m/s (defined as the cross-sectional average wind speed), though some data were also collected with W = 3.4 and 6.1 m/s in order to calculate average friction velocities for each fetch (see below). The full length of the wind-wave flume was used only for nonstratified conditions and consisted of measurements of wind speed and water velocity profiles at different fetches (distances downwind from the inlet). This provided data for evaluating the effect of the floating wave suppressors (see below). Density-stratified tests were conducted in shortened test sections of the flume, 3.5 m and 7 m, created by installing bulkheads in the flume, equal in height to the water depth, with the upwind bulkhead located at the end of the air inlet section. W a t e r velocity data (horizontal component) were obtained using a laser Doppler anemometer (LDA) mounted on a carriage to allow movement in all three coordinate directions. There were some problems in obtaining good data below a density interface due to the absence of strong motion there.
392
J. F. ATKINSONand D. R. F. HARLEMAN
exhaust
~
inlet section
outlet section
plyron boards
/
.26 m .6
m
T I~
4
bl ower
q
25m
I-
I
31m Fig. 1. Wind/wave tunnel; width = .77 m.
It should also be noted that data was not obtained at positions very close to the interface (within about 1-2 cm depending on the waviness of the interface), since the index of refraction changes caused the laser beams to deflect severely. Vertical profiles of mean and r.m.s, velocities were calculated for various fetches and lateral positions within each test section. Details of the experimental setup are given in [18]. In order to scale the entrainment results it is necessary to first define the friction velocity in air, U,a. These values were obtained as a function of fetch by fitting a logarithmic curve to the wind speed pro-
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file data taken at various fetches. These data were measured with a small Pitot-static tube connected to a pressure transducer with a metered output. Unfortunately, it was impossible to obtain wind speed measurements very close to the water surface, particularly at larger fetches. This was due primarily to the waviness of the surface and, at higher wind speeds, to spray blown off of the waves which tended to clog the Pitot tube. There is some uncertainty in the calculations for u,~ because of this, but the log fit to the rest of the data is very good, as seen in Fig. 2. Average values of u.a were calculated for each of the two test sections incorpo-
4i
I
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I
b
5
6
7
8
w
i
o~
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I
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9
10
11
12
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Fig. 2. Velocity profiles used to determine u,, for 5 different wind speeds, measured at a fetch of 5.8 m; d = height above water surface.
Wind-mixing experiments for solar ponds rating these measurements as well as values obtained at different lateral positions. The friction velocity for water u, was calculated from u,a by assuming that the shear stress was constant across the surface, i.e., r = p~u2a = pou2,. This assumption neglects the portion of the shear stress that is used to generate waves (form drag). In the short test section there was not sufficient fetch for waves to develop to any significant degree (wave amplitude usually 1 cm at most) and even in the longer test section waves developed primarily only for the higher wind speeds. The above assumption is then a fair approximation for the present tests. Average values of u,~ and u, are shown in Fig. 3 for the two test sections for each of the wind speeds considered. Two kinds of stratification were considered for the entrainment tests: (1) salt stratification only and (2) 'diffusive' stratification (lower layer warmer and saltier). The stratification was achieved by first filling the lower layer with a solution of desired salinity and temperature. Heating was accomplished by adding hot water initially or by injecting steam into the lower layer. The upper layer, usually about 10 cm deep before mixing was initiated, was then carefully poured on top. In practice an interfacial thickness of about 0 . 5 - 1 cm was evident after Idling. Wind was applied by turning the blower on and slowly (over several minutes) opening the exhaust d a m p e r from an initially closed position to the desired position. This procedure reduced the effects
393
of an impulsive start, manifested primarily in strong wave motion and mixing along the interface. The interfacial position was monitored visually as well as by measuring vertical salinity and temperature profiles using conductivity and thermistor probes. It was found that ue could be determined equally well by either method. Figure 4 indicates measurement points used for the shorter test section. Similar measurement points were used in the longer section. As expected, the interface was observed to tilt when wind stress was 'applied to the surface (see Fig. 4), with the degree o f tilt depending on the density difference across the interface and on wind speed. In addition an intermediate layer developed just underneath the main interface in the upwind portion of the test section, normally in a wedge-like shape similar to the one described by Kit et a/.[17]. Average interfacial positions were plotted with time in order to estimate the rate of deepening. The wind-induced entrainment rate was found to be nearly constant (see Fig. 5), at least over the time period of the experiments. Flow along the interface was easily visible by eye due to the strong refractive index change. Qualitatively, a surface drift current developed under the action of the wind stress and appeared to turn under at the downwind bulkhead and start flowing back along the interface. What appeared to be a predominantly two-dimensional flow started to be established, but very quickly (within several min-
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U
(e n,/s) I
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2
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6
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Fig. 3. Average friction velocities for the two test sections; circles represent u,a and squares represent u, ; solid points are for the 3.5 m section and open points are for the 7 m section.
394
J . F . ATKINSONand D. R. F. HARLEMAN wind
inlayte~r~diate
• main interface f
~
~
ctivi and temperature
Fig. 4. Measurement locations in 3.5 m test section.
utes) broke down into a complicated three-dimensional pattern (see [18]), Specification of this flow constitutes a complicated problem in itself and this is not attempted here. The presence of the end walls creates a situation which may not be very realistic for applying results to field conditions. In particu-
lar, the same return flow patterns would not usually be expected to be established in the field, not only because of the much reduced fetch of the laboratory model, but because it would be very rare to have a steady wind blowing over the entire surface of a large pond for any appreciable length of time. Pre-
/ 38 m
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P0
18
16
14 h
12
10
I
I
50
i00 t (min)
Fig. 5. Examples of growth of UCZ, with time.
395
Wind-mixing experiments for solar ponds
same as the energy input at the surface, it was desired to find an alternative scaling that might collapse the data better. One possibility is the use of the r.m.s, velocities measured under nonstratified conditions at the mixed layer depth. This procedure has been used in many oscillating grid experiments (see Turner[21]). The results using this scaling are shown in Fig. 7, along with results from an oscillating grid experiment[18] where there is no mean flow. F r o m Figs. 6 and 7, it appears that there is some effect of increasing fetch, with a definite progression in the results showing the lowest entrainment rates associated with the grid and the highest rates associated with the longer test section. The effects of increasing fetch included greater return flows along the interface (Fig. 8), as well as increased turbulence (r.m.s.) levels (due in part to the larger waves generated). As shown above, scal-
vious investigators have used the friction velocity associated with wind shear stress to scale their results and found that their entrainment rates were lower than the rates for experiments without end walls; this was attributed to the increased complexity of the flows with end walls which resulted in increased dissipation of energy which might otherwise be used for entrainment[17]. A preliminary scaling of the data is shown in Fig. 6, with u , as the scaling velocity. This allows a comparison with some earlier experiments as shown. It is interesting to note that the results for salt stratification and double-diffusive stratification show no noticeable difference--this point is discussed further below. Due to uncertainties regarding the surface stress continuity assumption and also because the energy available at the density interface may not be the
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~ 10-2 ~
/Kato and Phillips[i0]
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Wu [ii~]
~
~/eqn.
(9 a)
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m.e m ~ eqn. (9b)
Kit et al. [17]--___~ 10-4
I
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102
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Ri, Fig. 6. Non-dimensional entrainment rates measured in the 2 test sections, along with results from several previous studies; solid points are for the 3.5 m section, open points for the 7 m section; circles represent salt stratification and squares represent double-diffusive stratification. The curves drawn for previous studies cover the approximate range of Ri, considered in each of those studies.
396
J . F . ATKINSON and D. R. F. HARLEMAN 0 0
O
0
10 -2
O O
0
u e -uI
•
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Ri Fig. 7. Entrainment results using ul = r.m.s, horizontal velocity for scaling; 0 grid-mixing results; • 3.5 m s e c t i o n , • 7 m s e c t i o n .
1°5
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Fig. 8. A v e r a g e return flow v e l o c i t i e s near the interface; bars s h o w range of o b s e r v e d values; • 3.5 m section; [] 7 m section.
ing the entrainment results with the r.m.s, velocities did not result in as good a collapse of the data for the two test sections as desired. A further possibility for a scaling velocity is the mean return flow along the interface, which is responsible for direct
shear-induced mixing. This procedure was also used in [17]. Values for ur (return flow velocity near the interface) were averaged over all measurements taken for a given wind speed for each test section and values in the 7 m section were, on average,
397
Wind-mixing experiments for solar ponds about double the values for the 3.5 m section (Fig. 8). Entrainment results normalized with these values for Ur are shown in Fig. 9, where it can be seen that there is greater similarity in the results for the two test sections than in Fig. 6, though there is still not a perfect collapsing of the data. One last scaling procedure for the entrainment results involves defining a scaling velocity which is a function of more than one of the velocities used above. This procedure in effect recognizes that mixing energy is made available at the interface from more than one source. In particular, energy is available in the form of mean motion (ur) and also in the form of much smaller eddies (characterized by the r.m.s, values ul). Ifty is now defined as a function of the rates at which energy is supplied at the interface, i.e., Gr = f ( u 3, u~), then we should have a more direct scaling for evaluating the process of conversion of this energy to potential energy by entrainment. The simplest form of this equation is a linear combination, suggested by Sherman et a/.[22]. With O"3 = ( U l3 + 2u3), the entrainment results for the two test sections are very similar, as shown in Fig. 10. The constant value of 2 was chosen simply as one which resulted in a reasonably good collapsing of the data, and does not have any other special significance. F r o m the above results it may be concluded that double-diffusive stratification had no effect on the entrainment process in this set of experiments. It appears that mixing at the interface due to motions
in the UCZ eliminates conditions in the diffusive boundary layer which could lead to double-diffusive convection. Double-diffusive effects would probably only be seen for very weak winds and strong (unstable) temperature gradients, as discussed by Atkinson et a/.[23]. Although it is somewhat difficult to interpret the experimental results in terms of field conditions due to the limited fetch of the apparatus and the resulting flow patterns, it is at least possible to identify field conditions that were represented in the experiments in terms of the Richardson number. Average mixed layer depths for the experiments were usually 10-15 cm, compared with UCZ depths in an operating solar pond on the order of 25-30 cm. Depending on how a density step would be defined for the pond situation, considering that there is actually a gradual change rather than a step as in the two-layer stratification used in the tests, it is felt that the reduced gravity term (g') in defining Ri is roughly of the same order of magnitude in both the experiments and in operating ponds. Then, in order for Ri to be the same for both experiment and field conditions, the velocity scale would have to be increased by a factor of approximately 21/2. This implies that the wind speeds used in the tests model field winds which are actually about 40% greater. Thus, we conclude that double-diffusive effects would not be expected to be important for wind speeds greater than about 5.5 m/s and temperature steps less than about 12°C. The limit for wind speed may be less than this value,
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Fig. 9. Entrainment results using ur for scaling; symbols have same meaning as in Fig. 6.
398
J.F. ATKINSONand D. R. F. HARLEMAN
eqn.
10 -2
(6) with ~
= 0.i, CSH = 0.15, C t = 2.55,
and (~/O) = 0.8
u e O
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but it was not possible to accurately control and measure winds much less than the values stated for the present tests. Wave suppression tests A secondary set of experiments was performed to test a method proposed to reduce wind-induced mixing in solar ponds. The basic idea is to have a grid of floating wave breaks which reduces the effective fetch for the development of surface waves. Surface currents are also reduced. Two different materials were examined here. The first consists of 1 1/2 inch PVC pipe. Nielsen[5] has described the use of a floating grid of similar pipes on a solar pond and concluded that surface waves were reduced. These pipes were laid out in a rectangular array with a 10' (3 m) spacing. The second material consists of plastic netting. This netting has been used on several of the Israeli solar ponds and is laid out in a grid of 1 m wide sections with a 5 m spacing[4]. These nets are apparently effective in reducing mixing[14], though there have not been any controlled experiments to test their effectiveness. Both the nets and pipes are anchored at the sides of the pond. The pipes and nets were placed in the wind/wave tank at approximately 3.5 m spacing and attached by string taped to the sides of the tank. This allowed freedom of movement vertically but prevented
drift. Measurements of centedine mean and r.m.s. horizontal velocities in the water were taken with the LDA at several different longitudinal positions including directly beneath the nets (or pipes). The same velocity measurements were also taken without the nets or pipes so that a comparison could be made and the effect of the nets or pipe could be determined. Note that these tests were done without density stratification and the entire length of the wind/wave tank was used. The purpose was to determine the effect of the wave damper on the mechanical energy levels in the water and any reduction should relate to a corresponding reduction in entrainment. It was found that neither the pipe nor net had much of an effect on the flow for wind speeds much less than 4.4 m/s. For intermediate wind speeds there was some reduction in the r.m.s, velocities, with the nets performing slightly better than the pipes. However, at larger wind speeds there was actually an increase in the velocities (r.m.s.) when the nets or pipes were present, especially with the pipes. Mean velocities were also affected considerably. From direct observation it appeared as though the pipes acted in a similar manner as an oscillating grid to add energy to the water column. This was also true with the nets, but to a lesser extent. Some further insight into the effect of the
Wind-mixing experiments for solar ponds nets or pipes is obtained by simple flow visualization, as shown in Fig. 11. For these figures dye was sprinkled on the water surface just upwind of the pipe. The photographs show the strong turning under of the flow and vertical mixing under the pipe. This effect was also observed with the nets, but to a lesser extent, and became more apparent as wind speed was increased. The above results indicate that some stronger
399
mixing might be expected directly beneath the nets or pipes, depending on wind speed. However, the overall effect of placing these devices on a solar pond has been shown to be a reduction in mixedlayer deepening[4, 24]. A series of tests was then performed to measure centerline velocities at various points between two nets and at corresponding positions without the nets present. The results of these tests showed that the energy (measured in
Fig. 11. Flow visualization tests, showing effect of floating wave barrier on surface waves and on subsurface flow. For the results shown here, wind speed was W = 7.8 m/s (left to right) and the pictures were taken at approximately 10 sec. intervals; horizontal lines are at 5 cm spacing.
J.F. ATKINSONand D. R. F. HARLEMAN
400
wind speed of 4.4 m]s and became progressively less effective (at least in terms of r.m.s, levels) as wind speed increased. Because of the varibility in terms of fetch, depth and wind speed, it is difficult to quantify the effect of the nets. However, an average of a 20% reduction in r.m.s, levels is estimated for W = 4.4 m/s, compared with an average reduction of 5-10% for W = 7.8 m/s. There appears to be no average reduction for the highest wind speed. The results shown in Fig. 12 are due to the fetch control of the nets. With a reduction in wave buildup the friction velocity is also reduced, and the limited fetch prevents the build-up of surface drift current so that there is a reduction in the energy transferred to the water column. Entrainment would then be expected to be reduced accordingly. This was not tested directly in the present experiments and future tests might pursue this idea further. It should also be pointed out that in the field the nets (or pipes) will probably be most effective in attenuating waves with a wavelength related to the spacing of the nets on the pond surface. In turn, the
terms of r.m.s, velocity) between the nets was generally reduced from the corresponding values measured without the nets in place (Fig. 12). This is consistent with the observation that the waves were attenuated by the presence of nets or pipes (see Fig. 11). Averaged over the entire area of a pond, the effect of having nets in place is then to reduce the mixing energy transported downward to the interface, although there may be some increased local mixing directly beneath the nets themselves. For the present tests the average reduction in r.m.s, levels varied with wind speed and also with depth. Figure 12 shows ratios of r.m.s, values measured at various fetches and depths with the nets in place, to equivalent values measured without the nets in place. Values of unity would indicate no effect of the nets. It can be seen that the r.m.s, values are all reduced very near the surface--this is due directly to the reduction of surface waves. However, in some cases r.m.s, values are actually increased at larger depths with the nets in place, probably due to the more complicated flow patterns set up. Based on these results the nets were most effective for a
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401
Wind-mixing experiments for solar ponds
velocity ratio 0.5
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(c) Fig. 12. (continued) wavelength depends on total fetch of the pond which could not be duplicated in the laboratory. Further analysis of the effect of the nets should be done using field data. 4. CONCLUDING COMMENTS
The present results indicate a fetch dependence for mixed-layer entrainment and, at least for the
conditions considered here, double-diffusive effects were found to be negligible. The data in both Figs. 6 and 10 appear to be approximated by an inverse Ri entrainment relationship, and arguments have been presented to show that this type of relationship is reasonable. For the relatively shallow depth of the UCZ in an operating pond a significant mean current may be established due to wind shear and also due to flows associated with surface wash-
J. F. ATKINSONand D. R. F. HARLEMAN
402
ing. Because of various site-specific characteristics, the most direct method of estimating entrainment will be by taking current measurements and calculating ue from results similar to those shown in Fig. 10. This procedure would become particularly important when surface wave suppression systems are in place, since the water velocities can be altered significantly. At this time, however, the authors are not aware of any facilities where these measurements are made. From a practical point of view, calibration of the entrainment model in terms of u . is probably more useful than in terms of tr, and eqn (8) may be used to fit the present results after substituting ur for ~. However, values of Ri, were not sufficiently low to obtain information on C,, and a representative value was chosen from the literature. Values for C~, C r and CsH were then chosen to reproduce the resuits shown in Fig. 6, so U__~e= 0.68 + 0.28(Ur/U,)3 u, 2.55 + Ri,
(9a)
(see [18] for further details). This function is plotted in Fig. 6 and shows reasonably good agreement with data from the longer test section. However, in order to reproduce data from the 3.5 m section it was found that Csn had to be set to zero, so u~ u,
0.68 2.55 + Ri,
(9b)
It appears that shear-induced mixing was more important in the longer test section. Although the present experiments provide no information for evaluating the dissipation term ~', reported values in the literature indicate that the value of C (in eqn (7)) will have a value of order 0(10-2). Better estimates for each of the coefficients in the entrainment relation will be obtained when more detailed field measurements become available. In terms of modeling entrainment with cr as the scaling velocity, eqn (6) may be used, neglecting the terms involving the Peclet numbers as before. Using the same value for Ct as above, values for CF and CsH were chosen to reproduce the data, as shown in Fig. 10. It should be noted that the use of cr in place of u . for scaling is preferable since tr is a better measure of the energy available at the density step, not merely a measure of the energy transferred across the air-water interface. There are several differences between the entrainment model used here (eqn (6) or eqn (8)) and the model used in a previous study[2]. Perhaps the most obvious difference is the way the shear production term is parameterized. It is felt that the current formulation incorporates experimental observations more realistically (whereas the previous formulation essentially ignored shear production). Another point of difference is the parameterization
of the energy leakage term e'. The expression here is similar to that used by Atkinson and Harleman[8] and predicts that the fraction of energy that might be lost by generation of internal waves decreases as the stratification becomes increasingly stable. Thus, in the limit of large Ri, an (approximately) inverse Ri entrainment relationship is predicted (at least until the 'diffusion' limit represented by the Peclet number terms is reached). This is in contrast to the model previously used[2], where Ue would approach zero for Ri, > 900. Thus, some increase in entrainment would be predicted by the current model when Ri, > 900. A preliminary comparison of the two models was performed (see [18]) and there was not a large difference in the results. However, as previously noted, there are still uncertainties in the model coefficients and it is hoped that additional data will refine the estimates reported here. Results from the wind/wave suppression tests have provided some first-order laboratory observations of the effect that the floating system may have on the water motions. As expected, waves and average turbulence levels were generally reduced with the pipes or nets in place, though it appears that there would be some increased mixing directly beneath a pipe or net. These results can be seen in Fig. 12. Both the pipes and nets appeared to be most effective for lower wind speeds. As wind increased both the nets and pipes appeared to act as a kind of oscillating grid and r.m.s, levels were actually increased (at least below the near-surface region). Overall, the nets appear to perform somewhat better than the pipes. Again, more field data is needed to adequately describe the effect of this kind of wave suppression system.
Acknowledgements--This research has been supported by the National Science Foundation, grant CEE-8119384, and the Department of Energy, through the Solar Energy Research Institute, subcontract XX-3-03066-1. Dr. Atkinson was also supported as a Paul Rappaport Fellow of the American Solar Energy Society. REFERENCES
1. K. A. Meyer, Numerical modeling of layer behavior in a salt gradient solar pond. Rep. LA-UR-82-905, Los Alamos Nat. Lab., Los Alamos, NM (1982. 2. J. F. Atkinson and D. R. F. Harleman, A wind-mixed Jayer model for solar ponds. Solar Energy 31,243259 (1983). 3. S. G. Schladow, The upper mixed zone of a salt gradient solar pond: its dynamics, prediction and control. Solar Energy 33, 417-426 (1984). 4. G. Assaf, personal communication. Solmat Systems, Israel (1983). 5. C. Nielsen, Control of gradient zone boundaries. From Sun II, proc. I.S.E.S., K. Boer and B. Glenn (eds.), 1010-1014 (1979). 6. A. Akbarzadeh, R. W. G. MacDonald and Y. F. Wang, Reduction of mixing in solar ponds by floating rings. Solar Energy 31,377-380 (1983). 7. A. T. Twede, J. C. Batty and J. P. Riley, Suppression of wind-induced hydrodynamics in ponds. Solar Energy 35, 21-30 (1985).
Wind-mixing experiments for solar ponds 8. J. F. Atkinson and D. R. F. Harleman, Wind-mixing in solar ponds. Progress in Solar Energy, 6, 393-398 (1983). 9. EricBuch, On entrainment observed in laboratory and field experiments. Tellus 34, 307-311 (1982). 10. H. Kato and O. M. Phillips, On the penetration of a turbulent layer into stratified fluid. J. Fluid Mech. 37, 643-655 (1969). 11. Jin Wu, Wind-induced turbulent entrainment across a stable density interface. J. Fluid Mech. 61, 275-287 (1973). 12. G. Kullenberg, Entrainment velocity in natural stratified vertical shear flow. Estuarine and Coastal Mar. Sci. 5, 329-338 (1977). 13. G. C. Christodoulou, Interracial mixing in stratified flows. J. Hydraulic Res. 24, 77-92 (1986). 14. R.T. Pollard, P. B. Rhines and R. O. R. Y. Thompson, The deepening of the wind-mixed layer, Geophs. Fluid Dyn. 3, 381-404 (1973). 15. Pearn P. Niiler, Deepening of the wind-mixed layer. J. Marine Res. 33, 405-422 (1975). 16. C. Kranenburg, Wind-induced entrainment in a stably stratified fluid. J. Fluid Mech. 145, 253-273 (1984). 17. E. Kit, E. Berent and M. Vajda, Vertical mixing in-
403
duced by wind and a rotating screen in a stratified fluid in a channel. J. Hyd. Res. 18, 35-58 (1980). 18. J. F. Atkinson, E. Eric Adams, W. K. Melville and D. R. F. Harleman, Entrainment in diffusive thermohaline systems: application to salt gradient solar ponds. Tech. Rep. No. 300, Ralph Parsons Laboratory, M.I.T., Cambridge, MA (1984). 19. L. H. Kantha, O. M. Phillips and R. S. Azad, On turbulent entrainment at a stable density interface. J. Fluid Mech. 79, 753-768 (1977). 20. N. K. Shelkovnikov and G. I. Alyavdin, Experimental study of turbulent entrainment in a two-layer fluid. Oceanology 22, 140-144 (1982). ~21. J. S. Turner, Buoyancy Effects in Fluids. Cambridge Univ. Press (1973). 22. F. S. Sherman, J. Imberger and G. M. Corcos, Turbulence and mixing in stably stratified waters. Ann. Rev. Fluid Mech. 10, 267-288 (1978). 23. J. F. Atkinson, E. E. Adams and D. R. F. Harleman, Double-diffusive fluxes in a salt gradient pond. Submitted to J. Solar Energy Engineering (1985). 24. G. Assaf and E. Dagan, Observations of wind-mixing in solar ponds. Unpublished, undated report, Solmat Systems, Jerusalem, Israel.
0038-092X/87 $3.00 + .00 © 1987 Pergamon Journals Ltd.
Printed in the U.S.A.
WIND-MIXING EXPERIMENTS FOR SOLAR PONDS J. F. ATKINSON Department of Civil Engineering, S.U.N.Y. at Buffalo, Buffalo, NY, 14260 and D. R. F. HARLEMAN Department of Civil Engineering, Massachusetts, Institute of Technology, Cambridge, MA 02139 (Received for publication 18 November 1986)
Abstract--Experiments are described in which wind-induced entrainment was measured in a laboratory tank under strongly stratified double-diffusive conditions in order to calibrate a numerical model for application to solar ponds. Results showed that there was no effect of double-diffusive stratification and that the entrainment followed approximately an inverse Richardson number relationship. Shearinduced mixing was found to play a strong role in the entrainment process, and it is suggested that an effort be made to obtain more data on wind-induced currents in operating solar ponds. An additional series of tests is also reported in which a preliminary investigation of the effectiveness of floating wave suppressors was carried out. These tests showed that the wave suppressors, consisting of either plastic netting or PVC pipes, were able to reduce average turbulence levels in the water, though it appears that there may be some enhanced mixing directly beneath the nets or pipes. 1. I N T R O D U C T I O N
Previous solar pond studies have dealt with many of the physical processes of interest in understanding how to operate and maintain a pond for maximum heat generation. The ability to accurately model the temporal development of the temperature and salinity profiles becomes critically important when examining the technical and economic feasibility of constructing a pond in a given location for a given purpose. The present paper is concerned with one of the mechanisms responsible for determining the depth of the upper convecting zone (UCZ) of a solar pond. In particular, experiments were performed to obtain data for calibrating a model for the rate of layer deepening, or entrainment rate, under the action of mechanical (primarily wind-generated) mixing in the UCZ. Several windmixing mixed layer models have appeared in the solar pond literature[I-3], and the model used here is a modified version of the algorithm suggested by Atkinson and Harleman[2]. The present experiments were designed to provide data for calibrating some of the coefficients of this model. Wind shear stress was used to drive the mixing, and entrainment was measured in a laboratory wind/wave tank with average wind speeds of between 4 and 11 m/s. Two-layer stratification was used, with the density step produced either by salinity differences or by both temperature and salinity differences, such that "diffusive" double-diffusive stratification similar to that found in salt gradient solar ponds was obtained. Although the water layer beneath the entrainment interface is normally stratified in a solar pond, it is felt that the present experiments should provide useful information for entrainment modeling in ponds. This is primarily
because the entrainment model described below predicts that for the strong density step at the base of the U C Z there will be a minor effect of stratification in the lower layer. Data are also reported for stratification levels stronger than those usually found in the literature. It is important not only to be able to predict wind-mixing effects but also to be able to control them for improved collection efficiency. A possible means of reducing wind-mixing consists of installing a floating wave-suppression system consisting of a grid of plastic netting[4], a grid of PVC pipe[5] or plastic rings[6]. Twede et a/.[7] have reported a laboratory study of the effects of floating barriers on wind-induced mixing. Their results provide some preliminary guidelines for the design of wave suppressors, but in general little has been done to examine these effects in any detail. Several tests were conducted to determine the effect these floating systems would have on wind-induced mixing. Because of the geometry of the experimental apparatus the circular rings were not tested, but several results for the nets and pipes are presented.
2. MODEL DEVELOPMENT The entrainment model is derived from a parameterization of the one-dimensional (vertical) turbulent kinetic energy (TKE) budget applied to the UCZ, which is written as[2, 8]: Oq
g--
Ot
w'p'
Po
0 - -2- ( w ' q + w'p'/po) oZ -
389
u'w'--
Oz
-
~,
(1)
J. F. ATKINSONand D. R. F. HARLEMAN
390
at the interface and Cg 2 is a drag coefficient. Assuming ui/-ff ~ 1,
where q = T K E per unit mass, g = gravity, u and w are horizontal and vertical velocities, respectively, p = density, Po = mixed layer (UCZ) dens i t y , p -- pressure, z = vertical coordinate (positive downward), ~ = dissipation rate, primes denote fluctuating quantities and overbars represent mean (time-averaged) values. The left-hand side of eqn (1) represents the time rate of change of T K E and the right-hand side represents gravity work, flux divergence, shear production and viscous dissipation, respectively. The model here follows a similar approach as in [8] except for the treatment of the turbulence shear production term, which is described as follows. If it is assumed that interracial shear stress (due to a mean current in the UCZ) provides the major source of T K E and that the primary sink balancing this source is the conversion of T K E to potential energy by entrainment, the energy equation is expressed as a balance between gravity work and shear production,
u~
Kh
-- ~ CoRio I u gm '
(5)
where Rio is defined here with ~ in place of u . i . Again, an inverse Richardson number relationship is obtained when Km oc ~h. Christodoulou[13] has shown that eqn (5) should be valid for a broad transition range of Rio between conditions in the mixed layer which correspond with supercritical (lower Rio) and subcritical (higher R i o ) flow, with the critical value of Rio being of order 0(1). In the supercritical range the exponent o f Rio in eqn (5) is closer to - 1 / 2 , agreeing with several earlier models[1416], and in the subcritical range the exponent value is closer to - 3 / 2 , which agrees with other resuits[17] and is also the form of the entrainment relationship found for salt-stratified experiments with oscillating grids (a review of these latter experiments is provided in [18]). A possible means of (2) (hallowing the model above (eqn (5)) to reproduce reg w ' p ' ~ - u ' w ' 0_~ . sults supporting an exponent value other than - 1 Po OZ would be to assume that Km is dependent on Rio, Following Buch[9], a vertical momentum transfer at least outside of the transition range. For the purcoefficient Km is defined so that u2i = - K,, aT/ poses of this paper, however, it is assumed that the az, where u , i is the friction velocity associated with inverse R i o relationship holds, so that the parainterracial shear (note that the upper layer is as- meterization of the shear production term in the sumed to be flowing, so the velocity gradient O-ff/az T K E budget is - u ' w ' O-ff/Oz oc "ff3/h" Shear production has been neglected in previous < 0). u,i is also approximated by u2,i ~ u ' w ' . Then, assuming w ' p ' ~ ueAp, where Ap is the density step developments of the entrainment model[2, 8]. The across the entrainment interface and ue = d h / d t = present model was developed by Atkinson et a/.[18] entrainment rate, where h = mixed layer (UCZ) and consists of the same basic budget considered previously, though the sink term is modified and depth, we have the shear production term described above is ing cluded. In the course of developing the entrainment u4,i u,A9 ~ K-'--~" (3) model, a velocity scale for motions in the mixed layer is defined as or, where (r accounts for various This equation can be rewritten in a more convenient inputs of kinetic (mixing) energy into the layer. In other words, the rate at which kinetic energy is inform by introducing a bulk Richardson number, R i . i troduced into the U C Z is proportional to (r 3. F o r = gAph/poU2.i. Then, example, in cases where surface shear stress drives the interracial mixing it is common to assume ~ U---Le ¢c R i . l i u , i h (4) u . , where u . is the friction velocity at the surface. u,i Km Other researchers have assumed that the bulk mean velocity is a better choice (this is consistent with A common entrainment model is the inverse Ri- the use of Rio in eqn (5) above), so cr ~ ~. After chardson number relationship, which is supported parameterizing the various terms in the T K E equaby a number of experiments ([10, 11], for example) tion, using tr as the scaling velocity, the entrainment and also by field measurements[12]. This result is model is[18] m
Ue O"
C F + Csn(-ff/cr) 3 - ~' + C r R i r P e 7 1 / 2 Ct + R i
easily obtained from eqn (4) i f K m oc u,~h, a plausible assumption. A n alternative way of writing this expression in terms of the mean layer velocity is obtained by noting that r = interracial shear stress = pou2,i = Cg2po ('ff - ui) 2, where ui is the velocity
-
C ~ R i ~ e e 7 v2
(6)
where the Cs are parameterization coefficients, R i is defined with (r as the velocity scale, R i r and Ris are the Richardson numbers defined by the individual b u o y a n c y steps associated with the temperature and salinity gradients, respectively, P e r =
Wind-mixing experiments for solar ponds orh/kr and Pes = (rh/k, are the thermal and saline Peclet numbers, and k r and ks are thermal and saline diffusivities, e' is a term describing the leakage of energy away from the entrainment interface by the formation of internal waves radiating into the gradient region. It is expressed as[18]
C e' = ~ [2Rig' + N2h - 2Ri(g '2 +
N20"2)1/2],
391
o: R i . 1"5117] or ue/u, o: R i . 1'2120], implying that CF and Csn are possibly functions of R i , . This sort of relationship is also consistent with the conclusions of Christodoulou[13], mentioned above. The present experiments were performed to obtain data for large interfacial density steps in a doubly diffusive system in order to evaluate the form of the entrainment relationship that would be applicable for modeling solar ponds.
(7) 3. EXPERIMENTS
where g ' = gAp/po, N = [-g(Op/Oz)/po] 1/2 is the b u o y a n c y frequency in the region immediately below the interface, and C is a parameterization coefficient. (Note that it can be shown that e' approaches zero as N approaches zero, which is the expected result since internal waves will propagate away from the interface only when N is non-zero.) Finally, the terms in eqn (6) involving the Peclet numbers represent limits placed on ue due to molecular diffusion. These terms are not expected to be significant except for very high stratification levels with Richardson numbers at least of order 104105, and will probably be negligible in most situations. F u r t h e r discussion of these terms may be found in [18]. It should be noted that the present experiments deal with surface shear-stress-driven mixing in a two-layer system so that ~' may be neglected. We also assume that the terms involving the Peclet numbers are negligible. Then if tr = C j u , , where C1 is a constant, we have U~e = CICF "4 CSH(-~/lg*) 3/C2 u. C, + Ri./C~ '
(8)
where R i , = g'h/u2,. As noted above, some investigators have used (r = K, which would imply (from eqn (6)) that ue/(r g: Rio I , as long as Rio was sufficiently greater than C,. Depending on the coefficient values, eqn (8) may also approximate an inverse Richardson number relationship, which will be shown to describe the present results. It should be noted that eqn(8) is written in terms of u , rather than u,,-, the interracial friction velocity used in parameterizing the shear production term. This is because we are now considering mixing driven by surface shear, and the effect of u,i (i.e., mixing caused by a mean velocity step across the density interface) is already included in the term in eqn (8) multiplied by Cs~r. One of the attributes of entrainment relationships such as eqn (8) is the ability to model ue over a wide range o f R i , . F o r example, ue/u, approaches a finite value as R i , approaches zero, and the slope of the curve (ue/u, vs Ri,) increases as R i , increases, which is a desirable result if the model is to be able to reproduce some of the experimental evidence[19]. Several wind-mixing experiments have indicated that over limited R i , ranges, Ue/U,
Two basic experiments were performed. The first set of tests was designed to provide data for calibrating the model described above by measuring the rate of entrainment in a two-layer system as a function of wind speed and stratification (i.e., as a function of Ri,). The influence of fetch length was also considered by taking measurements in test sections of two different lengths. The second set of tests was performed in order to obtain laboratory data for a preliminary evaluation of the performance of several types of mixing suppression systems. These experiments are described below. Entrainment tests The wind-mixing experiments were conducted in a 25 m long wind/wave flume, sketched in Fig. 1. The wind inlet and outlet sections were made of heavy gauge aluminum and the cover consisted of plyron boards clamped along the top. Tape was used to cover all junctions to keep the tunnel as airtight as possible. The inlet section ended in a threefoot (91 cm) long horizontal section which had flow vanes and straighteners to smooth out the air flow. The fan was located at the outlet end and sucked air through the tunnel and wind speed was controlled by the opening of an exhaust damper on the fan. Measurements were made primarily for three discrete wind speeds, W = 4.4, 7.8 and 10.2 m/s (defined as the cross-sectional average wind speed), though some data were also collected with W = 3.4 and 6.1 m/s in order to calculate average friction velocities for each fetch (see below). The full length of the wind-wave flume was used only for nonstratified conditions and consisted of measurements of wind speed and water velocity profiles at different fetches (distances downwind from the inlet). This provided data for evaluating the effect of the floating wave suppressors (see below). Density-stratified tests were conducted in shortened test sections of the flume, 3.5 m and 7 m, created by installing bulkheads in the flume, equal in height to the water depth, with the upwind bulkhead located at the end of the air inlet section. W a t e r velocity data (horizontal component) were obtained using a laser Doppler anemometer (LDA) mounted on a carriage to allow movement in all three coordinate directions. There were some problems in obtaining good data below a density interface due to the absence of strong motion there.
392
J. F. ATKINSONand D. R. F. HARLEMAN
exhaust
~
inlet section
outlet section
plyron boards
/
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m
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q
25m
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It should also be noted that data was not obtained at positions very close to the interface (within about 1-2 cm depending on the waviness of the interface), since the index of refraction changes caused the laser beams to deflect severely. Vertical profiles of mean and r.m.s, velocities were calculated for various fetches and lateral positions within each test section. Details of the experimental setup are given in [18]. In order to scale the entrainment results it is necessary to first define the friction velocity in air, U,a. These values were obtained as a function of fetch by fitting a logarithmic curve to the wind speed pro-
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file data taken at various fetches. These data were measured with a small Pitot-static tube connected to a pressure transducer with a metered output. Unfortunately, it was impossible to obtain wind speed measurements very close to the water surface, particularly at larger fetches. This was due primarily to the waviness of the surface and, at higher wind speeds, to spray blown off of the waves which tended to clog the Pitot tube. There is some uncertainty in the calculations for u,~ because of this, but the log fit to the rest of the data is very good, as seen in Fig. 2. Average values of u.a were calculated for each of the two test sections incorpo-
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6
7
8
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i
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I
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10
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Fig. 2. Velocity profiles used to determine u,, for 5 different wind speeds, measured at a fetch of 5.8 m; d = height above water surface.
Wind-mixing experiments for solar ponds rating these measurements as well as values obtained at different lateral positions. The friction velocity for water u, was calculated from u,a by assuming that the shear stress was constant across the surface, i.e., r = p~u2a = pou2,. This assumption neglects the portion of the shear stress that is used to generate waves (form drag). In the short test section there was not sufficient fetch for waves to develop to any significant degree (wave amplitude usually 1 cm at most) and even in the longer test section waves developed primarily only for the higher wind speeds. The above assumption is then a fair approximation for the present tests. Average values of u,~ and u, are shown in Fig. 3 for the two test sections for each of the wind speeds considered. Two kinds of stratification were considered for the entrainment tests: (1) salt stratification only and (2) 'diffusive' stratification (lower layer warmer and saltier). The stratification was achieved by first filling the lower layer with a solution of desired salinity and temperature. Heating was accomplished by adding hot water initially or by injecting steam into the lower layer. The upper layer, usually about 10 cm deep before mixing was initiated, was then carefully poured on top. In practice an interfacial thickness of about 0 . 5 - 1 cm was evident after Idling. Wind was applied by turning the blower on and slowly (over several minutes) opening the exhaust d a m p e r from an initially closed position to the desired position. This procedure reduced the effects
393
of an impulsive start, manifested primarily in strong wave motion and mixing along the interface. The interfacial position was monitored visually as well as by measuring vertical salinity and temperature profiles using conductivity and thermistor probes. It was found that ue could be determined equally well by either method. Figure 4 indicates measurement points used for the shorter test section. Similar measurement points were used in the longer section. As expected, the interface was observed to tilt when wind stress was 'applied to the surface (see Fig. 4), with the degree o f tilt depending on the density difference across the interface and on wind speed. In addition an intermediate layer developed just underneath the main interface in the upwind portion of the test section, normally in a wedge-like shape similar to the one described by Kit et a/.[17]. Average interfacial positions were plotted with time in order to estimate the rate of deepening. The wind-induced entrainment rate was found to be nearly constant (see Fig. 5), at least over the time period of the experiments. Flow along the interface was easily visible by eye due to the strong refractive index change. Qualitatively, a surface drift current developed under the action of the wind stress and appeared to turn under at the downwind bulkhead and start flowing back along the interface. What appeared to be a predominantly two-dimensional flow started to be established, but very quickly (within several min-
Uea (cr /s)
U
(e n,/s) I
--
45
--
1
r-I
0
--
30
[]
0 •
n
15L r
•
0
Q
0
•
I
I
1
I
I
2
4
6
8
10
w
(m/~)
Fig. 3. Average friction velocities for the two test sections; circles represent u,a and squares represent u, ; solid points are for the 3.5 m section and open points are for the 7 m section.
394
J . F . ATKINSONand D. R. F. HARLEMAN wind
inlayte~r~diate
• main interface f
~
~
ctivi and temperature
Fig. 4. Measurement locations in 3.5 m test section.
utes) broke down into a complicated three-dimensional pattern (see [18]), Specification of this flow constitutes a complicated problem in itself and this is not attempted here. The presence of the end walls creates a situation which may not be very realistic for applying results to field conditions. In particu-
lar, the same return flow patterns would not usually be expected to be established in the field, not only because of the much reduced fetch of the laboratory model, but because it would be very rare to have a steady wind blowing over the entire surface of a large pond for any appreciable length of time. Pre-
/ 38 m
A__~0
P0
18
16
14 h
12
10
I
I
50
i00 t (min)
Fig. 5. Examples of growth of UCZ, with time.
395
Wind-mixing experiments for solar ponds
same as the energy input at the surface, it was desired to find an alternative scaling that might collapse the data better. One possibility is the use of the r.m.s, velocities measured under nonstratified conditions at the mixed layer depth. This procedure has been used in many oscillating grid experiments (see Turner[21]). The results using this scaling are shown in Fig. 7, along with results from an oscillating grid experiment[18] where there is no mean flow. F r o m Figs. 6 and 7, it appears that there is some effect of increasing fetch, with a definite progression in the results showing the lowest entrainment rates associated with the grid and the highest rates associated with the longer test section. The effects of increasing fetch included greater return flows along the interface (Fig. 8), as well as increased turbulence (r.m.s.) levels (due in part to the larger waves generated). As shown above, scal-
vious investigators have used the friction velocity associated with wind shear stress to scale their results and found that their entrainment rates were lower than the rates for experiments without end walls; this was attributed to the increased complexity of the flows with end walls which resulted in increased dissipation of energy which might otherwise be used for entrainment[17]. A preliminary scaling of the data is shown in Fig. 6, with u , as the scaling velocity. This allows a comparison with some earlier experiments as shown. It is interesting to note that the results for salt stratification and double-diffusive stratification show no noticeable difference--this point is discussed further below. Due to uncertainties regarding the surface stress continuity assumption and also because the energy available at the density interface may not be the
I0-I Kranenburg[16] _
~ 10-2 ~
/Kato and Phillips[i0]
\-,<
Wu [ii~]
~
~/eqn.
(9 a)
%/ e u.
m
l0-3 --
Kullenberg[12] Alya~in [ 2 0 ] / ~
"
m.e m ~ eqn. (9b)
Kit et al. [17]--___~ 10-4
I
I
!
I
102
I
103
1
I
I
...
104
Ri, Fig. 6. Non-dimensional entrainment rates measured in the 2 test sections, along with results from several previous studies; solid points are for the 3.5 m section, open points for the 7 m section; circles represent salt stratification and squares represent double-diffusive stratification. The curves drawn for previous studies cover the approximate range of Ri, considered in each of those studies.
396
J . F . ATKINSON and D. R. F. HARLEMAN 0 0
O
0
10 -2
O O
0
u e -uI
•
Y• •
IO •
•
o %'800% .•• O
o%•I. 10-3
O
-
•
•
• O0 • oo •
'%
oo~o", o.
v •
O0 ©
10-4
[
I
I
1
10 2
I
I
1
10 3
I 10 4
Ri Fig. 7. Entrainment results using ul = r.m.s, horizontal velocity for scaling; 0 grid-mixing results; • 3.5 m s e c t i o n , • 7 m s e c t i o n .
1°5
P
1
-
U r
.5
--
I
I
I
I
I
2
4
6
8
10
w
(~/~)
Fig. 8. A v e r a g e return flow v e l o c i t i e s near the interface; bars s h o w range of o b s e r v e d values; • 3.5 m section; [] 7 m section.
ing the entrainment results with the r.m.s, velocities did not result in as good a collapse of the data for the two test sections as desired. A further possibility for a scaling velocity is the mean return flow along the interface, which is responsible for direct
shear-induced mixing. This procedure was also used in [17]. Values for ur (return flow velocity near the interface) were averaged over all measurements taken for a given wind speed for each test section and values in the 7 m section were, on average,
397
Wind-mixing experiments for solar ponds about double the values for the 3.5 m section (Fig. 8). Entrainment results normalized with these values for Ur are shown in Fig. 9, where it can be seen that there is greater similarity in the results for the two test sections than in Fig. 6, though there is still not a perfect collapsing of the data. One last scaling procedure for the entrainment results involves defining a scaling velocity which is a function of more than one of the velocities used above. This procedure in effect recognizes that mixing energy is made available at the interface from more than one source. In particular, energy is available in the form of mean motion (ur) and also in the form of much smaller eddies (characterized by the r.m.s, values ul). Ifty is now defined as a function of the rates at which energy is supplied at the interface, i.e., Gr = f ( u 3, u~), then we should have a more direct scaling for evaluating the process of conversion of this energy to potential energy by entrainment. The simplest form of this equation is a linear combination, suggested by Sherman et a/.[22]. With O"3 = ( U l3 + 2u3), the entrainment results for the two test sections are very similar, as shown in Fig. 10. The constant value of 2 was chosen simply as one which resulted in a reasonably good collapsing of the data, and does not have any other special significance. F r o m the above results it may be concluded that double-diffusive stratification had no effect on the entrainment process in this set of experiments. It appears that mixing at the interface due to motions
in the UCZ eliminates conditions in the diffusive boundary layer which could lead to double-diffusive convection. Double-diffusive effects would probably only be seen for very weak winds and strong (unstable) temperature gradients, as discussed by Atkinson et a/.[23]. Although it is somewhat difficult to interpret the experimental results in terms of field conditions due to the limited fetch of the apparatus and the resulting flow patterns, it is at least possible to identify field conditions that were represented in the experiments in terms of the Richardson number. Average mixed layer depths for the experiments were usually 10-15 cm, compared with UCZ depths in an operating solar pond on the order of 25-30 cm. Depending on how a density step would be defined for the pond situation, considering that there is actually a gradual change rather than a step as in the two-layer stratification used in the tests, it is felt that the reduced gravity term (g') in defining Ri is roughly of the same order of magnitude in both the experiments and in operating ponds. Then, in order for Ri to be the same for both experiment and field conditions, the velocity scale would have to be increased by a factor of approximately 21/2. This implies that the wind speeds used in the tests model field winds which are actually about 40% greater. Thus, we conclude that double-diffusive effects would not be expected to be important for wind speeds greater than about 5.5 m/s and temperature steps less than about 12°C. The limit for wind speed may be less than this value,
l0 -2
5
i= • •
1 slope
U
e U-
•
0
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r
i..
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o
\
~
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"" Kit et al. [17] 10 -4
J 5
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l
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2
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Fig. 9. Entrainment results using ur for scaling; symbols have same meaning as in Fig. 6.
398
J.F. ATKINSONand D. R. F. HARLEMAN
eqn.
10 -2
(6) with ~
= 0.i, CSH = 0.15, C t = 2.55,
and (~/O) = 0.8
u e O
10-3 --
5
_
2
--
• l
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but it was not possible to accurately control and measure winds much less than the values stated for the present tests. Wave suppression tests A secondary set of experiments was performed to test a method proposed to reduce wind-induced mixing in solar ponds. The basic idea is to have a grid of floating wave breaks which reduces the effective fetch for the development of surface waves. Surface currents are also reduced. Two different materials were examined here. The first consists of 1 1/2 inch PVC pipe. Nielsen[5] has described the use of a floating grid of similar pipes on a solar pond and concluded that surface waves were reduced. These pipes were laid out in a rectangular array with a 10' (3 m) spacing. The second material consists of plastic netting. This netting has been used on several of the Israeli solar ponds and is laid out in a grid of 1 m wide sections with a 5 m spacing[4]. These nets are apparently effective in reducing mixing[14], though there have not been any controlled experiments to test their effectiveness. Both the nets and pipes are anchored at the sides of the pond. The pipes and nets were placed in the wind/wave tank at approximately 3.5 m spacing and attached by string taped to the sides of the tank. This allowed freedom of movement vertically but prevented
drift. Measurements of centedine mean and r.m.s. horizontal velocities in the water were taken with the LDA at several different longitudinal positions including directly beneath the nets (or pipes). The same velocity measurements were also taken without the nets or pipes so that a comparison could be made and the effect of the nets or pipe could be determined. Note that these tests were done without density stratification and the entire length of the wind/wave tank was used. The purpose was to determine the effect of the wave damper on the mechanical energy levels in the water and any reduction should relate to a corresponding reduction in entrainment. It was found that neither the pipe nor net had much of an effect on the flow for wind speeds much less than 4.4 m/s. For intermediate wind speeds there was some reduction in the r.m.s, velocities, with the nets performing slightly better than the pipes. However, at larger wind speeds there was actually an increase in the velocities (r.m.s.) when the nets or pipes were present, especially with the pipes. Mean velocities were also affected considerably. From direct observation it appeared as though the pipes acted in a similar manner as an oscillating grid to add energy to the water column. This was also true with the nets, but to a lesser extent. Some further insight into the effect of the
Wind-mixing experiments for solar ponds nets or pipes is obtained by simple flow visualization, as shown in Fig. 11. For these figures dye was sprinkled on the water surface just upwind of the pipe. The photographs show the strong turning under of the flow and vertical mixing under the pipe. This effect was also observed with the nets, but to a lesser extent, and became more apparent as wind speed was increased. The above results indicate that some stronger
399
mixing might be expected directly beneath the nets or pipes, depending on wind speed. However, the overall effect of placing these devices on a solar pond has been shown to be a reduction in mixedlayer deepening[4, 24]. A series of tests was then performed to measure centerline velocities at various points between two nets and at corresponding positions without the nets present. The results of these tests showed that the energy (measured in
Fig. 11. Flow visualization tests, showing effect of floating wave barrier on surface waves and on subsurface flow. For the results shown here, wind speed was W = 7.8 m/s (left to right) and the pictures were taken at approximately 10 sec. intervals; horizontal lines are at 5 cm spacing.
J.F. ATKINSONand D. R. F. HARLEMAN
400
wind speed of 4.4 m]s and became progressively less effective (at least in terms of r.m.s, levels) as wind speed increased. Because of the varibility in terms of fetch, depth and wind speed, it is difficult to quantify the effect of the nets. However, an average of a 20% reduction in r.m.s, levels is estimated for W = 4.4 m/s, compared with an average reduction of 5-10% for W = 7.8 m/s. There appears to be no average reduction for the highest wind speed. The results shown in Fig. 12 are due to the fetch control of the nets. With a reduction in wave buildup the friction velocity is also reduced, and the limited fetch prevents the build-up of surface drift current so that there is a reduction in the energy transferred to the water column. Entrainment would then be expected to be reduced accordingly. This was not tested directly in the present experiments and future tests might pursue this idea further. It should also be pointed out that in the field the nets (or pipes) will probably be most effective in attenuating waves with a wavelength related to the spacing of the nets on the pond surface. In turn, the
terms of r.m.s, velocity) between the nets was generally reduced from the corresponding values measured without the nets in place (Fig. 12). This is consistent with the observation that the waves were attenuated by the presence of nets or pipes (see Fig. 11). Averaged over the entire area of a pond, the effect of having nets in place is then to reduce the mixing energy transported downward to the interface, although there may be some increased local mixing directly beneath the nets themselves. For the present tests the average reduction in r.m.s, levels varied with wind speed and also with depth. Figure 12 shows ratios of r.m.s, values measured at various fetches and depths with the nets in place, to equivalent values measured without the nets in place. Values of unity would indicate no effect of the nets. It can be seen that the r.m.s, values are all reduced very near the surface--this is due directly to the reduction of surface waves. However, in some cases r.m.s, values are actually increased at larger depths with the nets in place, probably due to the more complicated flow patterns set up. Based on these results the nets were most effective for a
fetch (m) 5.4 [] 6.3 •
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(a) Fig. 12. Ratio of r.m.s, velocities measured with nets in place to measurements without nets; data for fetches 5.4 m and 8.8 m are directly beneath 2 adjacent nets in the flume; a) W = 10.2 m/s, b) W = 7.8 m/s, and c) W = 4.4 m/s.
401
Wind-mixing experiments for solar ponds
velocity ratio 0.5
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(c) Fig. 12. (continued) wavelength depends on total fetch of the pond which could not be duplicated in the laboratory. Further analysis of the effect of the nets should be done using field data. 4. CONCLUDING COMMENTS
The present results indicate a fetch dependence for mixed-layer entrainment and, at least for the
conditions considered here, double-diffusive effects were found to be negligible. The data in both Figs. 6 and 10 appear to be approximated by an inverse Ri entrainment relationship, and arguments have been presented to show that this type of relationship is reasonable. For the relatively shallow depth of the UCZ in an operating pond a significant mean current may be established due to wind shear and also due to flows associated with surface wash-
J. F. ATKINSONand D. R. F. HARLEMAN
402
ing. Because of various site-specific characteristics, the most direct method of estimating entrainment will be by taking current measurements and calculating ue from results similar to those shown in Fig. 10. This procedure would become particularly important when surface wave suppression systems are in place, since the water velocities can be altered significantly. At this time, however, the authors are not aware of any facilities where these measurements are made. From a practical point of view, calibration of the entrainment model in terms of u . is probably more useful than in terms of tr, and eqn (8) may be used to fit the present results after substituting ur for ~. However, values of Ri, were not sufficiently low to obtain information on C,, and a representative value was chosen from the literature. Values for C~, C r and CsH were then chosen to reproduce the resuits shown in Fig. 6, so U__~e= 0.68 + 0.28(Ur/U,)3 u, 2.55 + Ri,
(9a)
(see [18] for further details). This function is plotted in Fig. 6 and shows reasonably good agreement with data from the longer test section. However, in order to reproduce data from the 3.5 m section it was found that Csn had to be set to zero, so u~ u,
0.68 2.55 + Ri,
(9b)
It appears that shear-induced mixing was more important in the longer test section. Although the present experiments provide no information for evaluating the dissipation term ~', reported values in the literature indicate that the value of C (in eqn (7)) will have a value of order 0(10-2). Better estimates for each of the coefficients in the entrainment relation will be obtained when more detailed field measurements become available. In terms of modeling entrainment with cr as the scaling velocity, eqn (6) may be used, neglecting the terms involving the Peclet numbers as before. Using the same value for Ct as above, values for CF and CsH were chosen to reproduce the data, as shown in Fig. 10. It should be noted that the use of cr in place of u . for scaling is preferable since tr is a better measure of the energy available at the density step, not merely a measure of the energy transferred across the air-water interface. There are several differences between the entrainment model used here (eqn (6) or eqn (8)) and the model used in a previous study[2]. Perhaps the most obvious difference is the way the shear production term is parameterized. It is felt that the current formulation incorporates experimental observations more realistically (whereas the previous formulation essentially ignored shear production). Another point of difference is the parameterization
of the energy leakage term e'. The expression here is similar to that used by Atkinson and Harleman[8] and predicts that the fraction of energy that might be lost by generation of internal waves decreases as the stratification becomes increasingly stable. Thus, in the limit of large Ri, an (approximately) inverse Ri entrainment relationship is predicted (at least until the 'diffusion' limit represented by the Peclet number terms is reached). This is in contrast to the model previously used[2], where Ue would approach zero for Ri, > 900. Thus, some increase in entrainment would be predicted by the current model when Ri, > 900. A preliminary comparison of the two models was performed (see [18]) and there was not a large difference in the results. However, as previously noted, there are still uncertainties in the model coefficients and it is hoped that additional data will refine the estimates reported here. Results from the wind/wave suppression tests have provided some first-order laboratory observations of the effect that the floating system may have on the water motions. As expected, waves and average turbulence levels were generally reduced with the pipes or nets in place, though it appears that there would be some increased mixing directly beneath a pipe or net. These results can be seen in Fig. 12. Both the pipes and nets appeared to be most effective for lower wind speeds. As wind increased both the nets and pipes appeared to act as a kind of oscillating grid and r.m.s, levels were actually increased (at least below the near-surface region). Overall, the nets appear to perform somewhat better than the pipes. Again, more field data is needed to adequately describe the effect of this kind of wave suppression system.
Acknowledgements--This research has been supported by the National Science Foundation, grant CEE-8119384, and the Department of Energy, through the Solar Energy Research Institute, subcontract XX-3-03066-1. Dr. Atkinson was also supported as a Paul Rappaport Fellow of the American Solar Energy Society. REFERENCES
1. K. A. Meyer, Numerical modeling of layer behavior in a salt gradient solar pond. Rep. LA-UR-82-905, Los Alamos Nat. Lab., Los Alamos, NM (1982. 2. J. F. Atkinson and D. R. F. Harleman, A wind-mixed Jayer model for solar ponds. Solar Energy 31,243259 (1983). 3. S. G. Schladow, The upper mixed zone of a salt gradient solar pond: its dynamics, prediction and control. Solar Energy 33, 417-426 (1984). 4. G. Assaf, personal communication. Solmat Systems, Israel (1983). 5. C. Nielsen, Control of gradient zone boundaries. From Sun II, proc. I.S.E.S., K. Boer and B. Glenn (eds.), 1010-1014 (1979). 6. A. Akbarzadeh, R. W. G. MacDonald and Y. F. Wang, Reduction of mixing in solar ponds by floating rings. Solar Energy 31,377-380 (1983). 7. A. T. Twede, J. C. Batty and J. P. Riley, Suppression of wind-induced hydrodynamics in ponds. Solar Energy 35, 21-30 (1985).
Wind-mixing experiments for solar ponds 8. J. F. Atkinson and D. R. F. Harleman, Wind-mixing in solar ponds. Progress in Solar Energy, 6, 393-398 (1983). 9. EricBuch, On entrainment observed in laboratory and field experiments. Tellus 34, 307-311 (1982). 10. H. Kato and O. M. Phillips, On the penetration of a turbulent layer into stratified fluid. J. Fluid Mech. 37, 643-655 (1969). 11. Jin Wu, Wind-induced turbulent entrainment across a stable density interface. J. Fluid Mech. 61, 275-287 (1973). 12. G. Kullenberg, Entrainment velocity in natural stratified vertical shear flow. Estuarine and Coastal Mar. Sci. 5, 329-338 (1977). 13. G. C. Christodoulou, Interracial mixing in stratified flows. J. Hydraulic Res. 24, 77-92 (1986). 14. R.T. Pollard, P. B. Rhines and R. O. R. Y. Thompson, The deepening of the wind-mixed layer, Geophs. Fluid Dyn. 3, 381-404 (1973). 15. Pearn P. Niiler, Deepening of the wind-mixed layer. J. Marine Res. 33, 405-422 (1975). 16. C. Kranenburg, Wind-induced entrainment in a stably stratified fluid. J. Fluid Mech. 145, 253-273 (1984). 17. E. Kit, E. Berent and M. Vajda, Vertical mixing in-
403
duced by wind and a rotating screen in a stratified fluid in a channel. J. Hyd. Res. 18, 35-58 (1980). 18. J. F. Atkinson, E. Eric Adams, W. K. Melville and D. R. F. Harleman, Entrainment in diffusive thermohaline systems: application to salt gradient solar ponds. Tech. Rep. No. 300, Ralph Parsons Laboratory, M.I.T., Cambridge, MA (1984). 19. L. H. Kantha, O. M. Phillips and R. S. Azad, On turbulent entrainment at a stable density interface. J. Fluid Mech. 79, 753-768 (1977). 20. N. K. Shelkovnikov and G. I. Alyavdin, Experimental study of turbulent entrainment in a two-layer fluid. Oceanology 22, 140-144 (1982). ~21. J. S. Turner, Buoyancy Effects in Fluids. Cambridge Univ. Press (1973). 22. F. S. Sherman, J. Imberger and G. M. Corcos, Turbulence and mixing in stably stratified waters. Ann. Rev. Fluid Mech. 10, 267-288 (1978). 23. J. F. Atkinson, E. E. Adams and D. R. F. Harleman, Double-diffusive fluxes in a salt gradient pond. Submitted to J. Solar Energy Engineering (1985). 24. G. Assaf and E. Dagan, Observations of wind-mixing in solar ponds. Unpublished, undated report, Solmat Systems, Jerusalem, Israel.