Plumes and jets with time-dependent sources in stratified and unstratified environments

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1 Plumes and jets with time-dependent sources in stratified and unstratified environments Abstract Matthew Scase 1, Colm Caulfield 2,1, Stuart Dalziel 1 & Julian Hunt 3 1 DAMTP, Centre for Mathematical Sciences, University of Cambridge UK. 2 BP Institute, University of Cambridge UK. 3 CPOM, University College London, UK. m.scase@damtp.cam.ac.uk The classical bulk model for isolated jets and plumes due to Morton, Taylor & Turner (1956) have been generalized to allow for time-dependence in the various fluxes driving the flow (Scase et al., 2006a). This new system models the spatio-temporal evolution of both Boussinesq and non-boussinesq jets and plumes in unstratified and uniformly stratified fluids. Separable time-dependent solutions for plumes and jets were found. These separable solutions are characterized by having time-independent plume or jet radii, with appreciably smaller spreading angles than either constant source buoyancy flux pure plumes or constant source momentum flux pure jets. In stratified environments, stall times of plumes and jets was predicted. In the present paper, complementary experiments have been conducted to compare with our theoretical predictions, and we find good agreement. 1. Introduction and background When the source buoyancy flux (for a plume) or source momentum flux (for a jet) is decreased generically from an initial to a lower final value, numerical solutions and complementary experiments exhibit three qualitatively different regions of behaviour. There exists an upper region which behaves like a steady plume, based on the initial source conditions, which is unaffected by changes at the source, since information has not yet propagated up the plume far enough. A lower region exists which behaves like a steady plume, based on the final source conditions, and a transitional region exists which connects the upper and lower regions. In our theoretical model we assume top-hat profiles for the plumes with the plume radius denoted by b, the vertical plume velocity by w, and the reduced gravity by g. The reduced gravity in a Boussinesq plume is defined as g = g (ρ ρ) /ρ. Now, defining a mass flux πq, momentum flux πm, and buoyancy flux πf as Q = b 2 wρ, M = b 2 w 2 ρ, F = b 2 w (ρ ρ), (1) then for a Boussinesq plume, or jet, the system of equations describing their evolution is (Scase et al., 2006a) 6th International Conference on Stratified Flows, Perth, Western Australia Page 112

2 t t ( ) Q 2 + Q M z Q t + M ) z ( QF M + F z = 2αρ 1/2 M1/2, = QF M, (2a) (2b) = N 2 Q. (2c) The background ambient fluid has density ρ (z) = ρ 0 exp { N 2 z/g}, where ρ 0 is the density of the ambient fluid at the height of the nozzle, N is the constant buoyancy frequency, and g is the acceleration due to gravity. The system (2) is parabolic, with triply repeated real eigenvalue w, demonstrating that information propagates through the system with the local plume velocity. Numerical solutions to the system (2) indicate that when the driving buoyancy flux is rapidly reduced, three qualitatively different regimes are realized along the plume (Scase et al., 2006a). The upper region remains largely unaffected by the change in buoyancy flux or momentum flux at the source. The lower region is an effectively steady plume or jet based on the final (lower) buoyancy flux or momentum flux. The intermediate transition region, in which the plume or jet adjusts between the states in the lower and upper regions converges closely to (Scase et al., 2006b) ( Q = Nα2 ρ 0 Nt z 3 cot 9 2 F = N3 α 2 ρ 0 z 4 cot 72 ) (, M = N2 α 2 ρ 0 Nt z 4 cot ( ) [ ( ) Nt Nt cot ), (1.3a b) ] 1. (1.3c) The equivalent plume radius, plume velocity and reduced gravity are b = 2αz 3, w = Nz ( ) [ ( ) ] Nt 4 cot, g = N2 z Nt cot 2 1, (1.4) and in the limit N 0, (1.4) becomes b = 2αz/3, w = z/ (2t), g = z/ (2t 2 ). A key result is that when the background fluid is stably stratified, i.e. N > 0, we predict a stall time within the plume at t s = π/n. Similarly, for a jet we predict a stall time at t s = π/ (2N). It is well-known that a turbulent plume rising through a linear stratification with constant buoyancy frequency, N, has a maximum rise height given by (see e.g. Turner 1973; Scase et al. 2006b) ( ) 1/4 F0 z h = (1.5) 4α 2 ρ 0 N 3 Hence, a reduction in the driving source buoyancy flux, F 0 leads eventually to the establishment of a new plume with a lower maximum rise height. As discussed in Scase et al. (2006b), as this new plume is established there exists a region in the plume which stalls corresponding to where (1.4) is realized. 6th International Conference on Stratified Flows, Perth, Western Australia Page 113

3 Constant head tanks Flow meters Mixing valve Figure 1: Schematic of experimental set-up. 2. Experiments In the following section we describe experiments to test the main theoretical predictions described in Scase et al. (2006a, b) and 1. Firstly, that a turbulent plume undergoing a dramatic reduction in its driving source buoyancy flux in a unstratified ambient fluid remains a single plume and does not break up into separate puffs. Secondly, that the effective top-hat profile of the plume necks in from b = 6αz/5 to a narrower b = 2αz/3 profile. Thirdly, we examine whether, in a stratified ambient fluid, a plume which undergoes a reduction in its driving source strength has a region in which the velocity stalls at critical time t s Experimental set-up The experimental set-up is shown schematically in figure 1. A large tank measuring 0.7 m 0.7m 1.3 m is filled with either homogeneous brine or stratified brine to provide the ambient fluid. Two, constant head, header tanks are located above the large tank and contain plume fluids of different densities. The two header tanks are connected, via digital flow meters, to a mixing valve which is in turn connected to a plume nozzle at the bottom of the large tank. At the start of an experiment the mixing valve is put into position 1, allowing low density, high buoyancy fluid to enter the experimental tank and establish a strong plume. When this plume is established, the mixing valve is quickly rotated into position 2 allowing the higher density, lower buoyancy fluid to enter the experimental tank through the plume nozzle, forming a weaker plume. The plume fluid is marked with fluorescein, indicating both the plume s extent and the dilution of the initial plume fluid with unmarked ambient fluid. The experiments are digitally recorded using a Cosmicar f1.8 zoom lens mounted on a 6th International Conference on Stratified Flows, Perth, Western Australia Page 114

4 z (m) (a) (b) (c) x (m) x (m) x (m) Figure 2: (a) Contours of a single frame showing the turbulent initial established plume. The nozzle diameter is m, the initial mass flux is Q 1 = kg s 1 and the initial reduced gravity is g 1 = 1.83 m s 2. (b) An ensemble over 85 experiments of the initial established plume. (c) The outer dashed line is the standard steady top-hat plume radius, b = 6αz/5, the inner dashed line is the predicted minimum plume width b = 2αz/3. The bold solid line is the initial top-hat edge of the experimental plume, calculated from (b). The bold dot-dashed line is the plume radius at time t = 2.00 s after the reduced gravity at the source has been reduced to g 2 = 0.49 m s 2. The plume radius necks in to approximately b = 2αz/3. 0 JAI CVM4+CL 1.3 MPixel digital camera. The digital frames are captured and processed using DigiFlow (Dalziel, 2006). Due to the turbulent nature of the flow an ensemble of experiments are required to match with the theory. Two LEDs are mounted on the front of the experimental tank to indicate which position the mixing valve is in. Each movie synchronized about the time the LEDs, and hence the mixing valve, switch over Experiments with homogeneous ambient fluid The experimental tank is filled with brine with a typical density of ρ = 1186 kg m 3. Header tank 1 contains fresh water with density ρ 1 = 1000 kg m 3 and header tank 2 contains brine with density ρ 2 = 1129 kg m 3. This yields an initial reduced gravity g 1 = 1.83 ms 2 and final reduced gravity g 2 = 0.49 ms 2 giving the ratio g 1/g 2 = Hence the required mass fluxes at the nozzle to ensure pure plume balance (c.f. Caulfield 1991; Hunt and Kaye 2001) are given by Q 1 = kg s 1 and Q 2 = kg s 1. Figure 2(a) shows contours from an instantaneous image of the initial turbulent plume. The experiment was repeated 85 times with nominally identical initial conditions. Figure 2(b) shows contours of light intensity for the ensemble of initial images. Gaussian profiles are fitted to the light intensity at each height z, and an effective top-hat radius can be calculated for the plume. Figure 2(c) shows the standard steady top-hat plume profile b = 6αz/5 and the predicted minimum necking radius b = 2αz/3 in thin dashed lines. The bold solid line shows the initial effective top-hat radius of the plume, calculated from figure 2(b). The bold dot-dashed line shows the effective top-hat plume radius at a time t = 2.00 s after the buoyancy at the source has been reduced. There is good agreement between the observed and predicted minimum necking width. 6th International Conference on Stratified Flows, Perth, Western Australia Page 115

5 10 2 log t s log N Figure 3: A plot of experimentally observed stall times (crosses) for two different stratifications, N = 0.99 s 1, N = 0.20 s 1 against the predicted stall time π/n (solid black line) Experiments with density stratified ambient fluid The stratification of the ambient fluid was achieved using the well-known double-bucket technique. The source conditions at the nozzle were identical to those in 2.2, and the individual experiments are conducted in an identical manner. However, the stratification does impose extra experimental difficulties. The overall density of the ambient fluid must remain high, in order that the plumes be buoyant enough to maintain high enough Reynolds numbers at the nozzle. Since the maximum rise height of the plume, in (1.5), scales with (F 0 /N 3 ) 1/4, weak stratifications must be made whereby the density difference between the salty fluid and the fresh fluid in the double-bucket is less than 0.5%, in order to visualize the desired effects. The resulting very weak stratifications are extremely difficult to repeat and no ensemble of stratified plume experiments have been attempted. Furthermore, unlike the unstratified experiments where plume fluid can rise to the top of the tank and contamination of the ambient fluid is minimal, the plumes in the stratification rise to a finite maximum rise height and contaminate the ambient fluid significantly at that height, so repeat experiments in a given stratification are impossible. Figure 3 is a plot of the experimentally observed stall time, t s, along with the predicted stall time t s = π/n (Scase et al., 2006b) against N. The graph shows reasonable agreement between the theoretical prediction and these early experimental results. 3. Conclusions Our experiments have demonstrated the robust nature of turbulent plumes in unstratified ambient fluids. Our predicted minimum necking radius of b = 2αz/3 agrees well with these initial experiments. A key result predicted by our theoretical work was that plumes rising through an unstratified ambient fluid would not break up into individual puffs as a result of a reduction in the driving source conditions. This prediction is supported by our experiments, over the range of parameters used. When the background ambient fluid is linearly density stratified with constant buoyancy frequency, N, we find that a reduction in source buoyancy leads to the establishment of a new lower buoyancy plume, with lower rise height than the initial plume. We also find 6th International Conference on Stratified Flows, Perth, Western Australia Page 116

6 that as this new plume is established there exists a region where the plume fluid stalls, and the time at which this occurs is well predicted by t s = π/n. Acknowledgements The authors would like to acknowledge funding under NERC award NER/A/S/2002/ References Caulfield, C. P. (1991). Stratification and buoyancy in geophysical flows. PhD thesis, University of Cambridge. Dalziel, S. B. (2006). DLResearchPartners, Hunt, G. R. and Kaye, N. B. (2001). Virtual origin correction for lazy turbulent plumes. J. Fluid Mech., 435: Morton, B. R., Taylor, G. I., and Turner, J. S. (1956). Turbulent gravitational convection from maintained and instantaneous sources. Proc. Roy. Soc. Lon. A, 234:1 32. Scase, M. M., Caulfield, C. P., Dalziel, S. B., and Hunt, J. C. R. (2006a). Time-dependent plumes and jets with decreasing source strengths. J. Fluid Mech., 563: Scase, M. M., Caulfield, C. P., and Dalziel, S. B. (2006b). Boussinesq plumes with decreasing source strengths in stratified environments. J. Fluid Mech., 563: Turner, J. S. (1973). Buoyancy Effects in Fluids. Cambridge University Press. 6th International Conference on Stratified Flows, Perth, Western Australia Page 117