THE RESPONSE OF A PLUME TO A SUDDEN REDUCTION IN BUOYANCY FLUX

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1 THE RESPONSE OF A PLUME TO A SUDDEN REDUCTION IN BUOYANCY FLUX MATTHEW M. SCASE Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 1485, USA COLM P. CAULFIELD * BP Institute, University of Cambridge, Madingley Rise, Cambridge CB 0EZ, UK STUART B. DALZIEL Department of Applied Mathematics, University of Cambridge, Wilberforce Road, Cambridge CB 0WA, UK Abstract. The classical similarity model for the behavior of steady jets and plumes is extended to a time-dependent framework to allow new insights in situations where the driving fluxes are not constant. When the strength of the source is decreased the solution is shown to converge on a new class of similarity solutions. Like the steady plume, this new similarity solution has straight sides for a Boussinesq plume, but with an angle of spread significantly less than that of the corresponding steady plume. The relevance of these similarity solutions is demonstrated both through numerical solution of the timedependent equations, and supported by a large ensemble of laboratory experiments. 1. Introduction The similarity model of Morton, Taylor and Turner, for the behavior of steady jets and plumes, is arguably one of the most successful models in environmental fluid mechanics. The model is simple, robust and gives excellent agreement with experiments and observations ranging from the laboratory scale to flows penetrating a significant fraction of the depth of the atmosphere. However, in almost all naturally occurring circumstances, the strength of the source varies over time scales that can lead to qualitatively different behaviors. Recently, we have extended the steady model to a time-dependent framework. We identify a class of intermediate similarity solutions for plumes and jets from sources with decreasing strength in uniform 4 and stratified environments. * Also Department of Applied Mathematics and Theoretical Physics, University of Cambridge. 1

2 These solutions are closely related to the steady plume solutions discussed by Batchelor 1 in the non-physical context of a statically unstable ambient. In this paper we begin by reviewing the formulation of the time-dependent model and its similarity solution before verifying their relevance through numerical simulation of the time-dependent equations and a large ensemble of laboratory experiments.. Time-dependent model For brevity, we restrict ourselves to models of axisymmetric Boussinesq plumes from a point source of buoyancy. We also restrict our attention to top-hat models of the plume structure; Scase et al. 5 discuss some of the issues associated with other assumed profiles. For a plume of density ρ(,t) and radius b(,t) rising with velocity w(,t) through an ambient stratification described by ρ 0 () in gravity g, we can write down equations for conservation of volume, mass and momentum as ( πb ) + ( πb w) = π bue = απ bw, (1) t ( πρb ) + ( πρb w) = πρ 0bue = απρ 0bw, () t t b w b w b g, () ( πρ ) + ( πρ ) = π ( ρ0 ρ ) where the entrainment velocity u e is given by Batchelor s entrainment hypothesis as α w with α the entrainment coefficient. Defining mass, momentum and buoyancy fluxes as Q = ρb w, M = ρb w and F = (ρ 0 ρ)gb w, respectively, and the buoyancy frequency, N = ( g/ρ 0 dρ 0 /d) 1/, we can rewrite this system as Q Q + = αρ t M M, (4) Q M QF + =, (5) t M QF F + t M = N Q. (6) In the case N = 0, time-dependent power law similarity solutions may be found, specifically

3 Q =, 9 t 4 M =, 9 t 4 F =. (7) 9 t In terms of the basic variables, the structure of this similarity plume is b = α, w =, g =, (8) t t where g = g(ρ 0 ρ)/ρ 0 is the reduced gravity. For comparison with the steady plume solution, it is convenient to rewrite these in terms of the buoyancy flux: 1 1 F w =, F g =. (9) This should be compared with the classical solution for a steady plume where 6α b =, F 1 ρ0 5 9 w = 6 10α, F g = 6 9α ρ 4 0 5/. (10) In particular, although both solutions are of straight sided conical form, the timedependent solution is significantly narrower. While the vertical structure of the velocity and reduced gravity are different between the steady and timedependent plumes, the structure relative to the local buoyancy flux is identical to within a multiplicative constant, as is required by dimensional arguments UPPER REGION: Information about change in source has not yet propagated this far Key t = 0.0 t = 0.1 t = 0. t = 0. t = 0. t = 0.4 t = 0.5 t = 0.6 Steady plume ( 5 6 α) Similarity solution ( α) b/6α TRANSITION REGION: Narrower region, tending towards similarity solution, connecting the upper and lower regions LOWER REGION: New, lower buoyancy plume establishing itself Figure 1: Numerical solution of the time-dependent plume equations for a step decrease in the buoyancy flux. The region descriptions on the right-hand side refer to the plume at t = 0.6.

4 4. Numerical solution The relevance of the time-dependent similarity solution is readily demonstrated by computing the numerical solution to the time-dependent plume equations for the case of a step change in the source buoyancy flux between an initial value F = F 0 for t < 0, and a new lower value F = F 1 for t 0. As can be seen in figure 1, the width of the transition region dividing the initial and final steady plume solutions increases with time, and within the transition region the plume radius approaches the time-dependent similarity solution. 4. Experimental confirmation Whereas experiments for steady plumes are relatively straight forward, requiring little more than time averaging to recover close to the idealied conditions of the classical similarity solution, the case for time-dependent plumes is much more difficult. Not only do we require phase-locked averaging over large ensembles of experiments, but the height required for unequivocal identification of the similarity solution is difficult at the laboratory scale whilst maintaining an adequate Reynolds number for the plume to remain turbulent Key Initial plume Transient plume Steady plume ( 5 α) 6 Similarity solution ( α) b Figure : Top-hat plume widths determined from an ensemble of 100 experiments in which the plume lainess was instantaneously increased by a factor of 4. As discussed by Scase et al. 6, we are better able to achieve this compromise at the laboratory scale by considering an increase in the plume lainess (changing the balance between mass, momentum and buoyancy fluxes) rather than simply reducing the buoyancy flux. Numerical solution shows that the similarity solution again acts as an attractor. Figure summaries the result of an ensemble of 100 such experiments. Here we have taken the actual cross-plume density

5 5 profiles, which are approximately Gaussian and re-interpreted them as top-hat distributions. As we can see, the plume width in the intermediate transition region approaches the similarity solution. 5. Discussion and conclusions It is interesting that the classical steady solution has been so widely observed in a vast variety of contexts despite the existence of a time-dependent solution with a superficially different structure. However, the similarity in the structure relative to the local buoyancy flux, as seen by comparing (9) and (10), suggests the structure is not so different. Analysis of the plume equations yields three eigenvalues, all with characteristic speed equal to that of the local plume velocity, but only two distinct left-hand eigenvectors. Thus the equations are parabolic and the solution depends only on the local properties of the plume. In a steady plume in a homogeneous environment, the buoyancy flux is conserved. In a statically unstable stratification, as examined by Batchelor 1, the buoyancy flux increases with height, but the form of the solution relative to the buoyancy flux remains the same. In our similarity solution for a decreasing buoyancy source, the instantaneous buoyancy flux also increases with height, and the solution takes the same functional form for its vertical structure as Batchelor s plume in an unstable stratification, and the same structure local to the local buoyancy flux as the steady plume solution. In this paper we have seen that the similarity solution is approached in the numerical solutions for a step change in buoyancy flux, and the experiments where the lainess of the plume undergoes a step change. Consider now the structure of a steady plume in a statically stable stratification. This problem has an additional time scale, N 1, that gives rise to an additional intrinsic length scale, h = (F 0 /ρ 0 ) 1/4 N /4 = ( F 0 ρ 0 /(g dρ 0 /d )) 1/8, which scales the maximum rise height for the plume, although of course entrainment means that the plume never actually reaches h. Here F 0 is the buoyancy flux at the source. On dimensional grounds the departure from straight sides to the plume is possible only due to this additional length scale. For a plume in a statically unstable stratification, N 1 is complex. Although we can still create a length scale by taking (F 0 ρ 0 /(g dρ 0 /d )) 1/8 in place of h, this scale does not play a direct role in determining the flow, hence Batchelor s 1 finding of a similarity solution. In the case of a time-varying buoyancy flux, the equivalent new time scale is F0 F ɺ 0. As noted above, for a decreasing buoyancy flux, we have a similarity / 4 1/ 4 solution and the corresponding length scale F0 Fɺ 0 ρ0 does not play a physical role. (Note the absolute value around F ɺ 0 is required to yield a real re-

6 6 sult in the same way as the opposite sign for the gradient was required to create a real time scale in the steady unstably stratified case.) For an increasing buoyancy flux, F0 F ɺ 0 is positive and we may anticipate an additional length scale in the problem with behavior qualitatively similar to that found in stable stratifications with strong singularities developing (although here we expect such singularities to propagate upwards with the plume). As we have seen, the classical steady plume model of Morton, Taylor & Turner is readily extended to plumes from sources with decreasing buoyancy flux. The existence of a new class of intermediate similarity solutions sheds light on the behavior of more general reductions in buoyancy flux and acts as an attractor in such cases, the existence of which is verified experimentally. The similarity solution also elucidates the role of the local buoyancy flux in setting the form of the solution. Acknowledgments MMS was funded by NERC award NER/A/S/00/0089 and by the US-UK Fulbright Commission. The authors also acknowledge valuable discussions with Prof. P.F. Linden and Prof. Lord J.C.R. Hunt. References 1. G.K. Batchelor 1954 Heat convection and buoyancy effects in fluids. Q. J. R. Met. Soc. 80, B.R. Morton, G.I. Taylor & J.S. Turner 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. Roy. Soc. Lon. A 4, 1-.. M.M. Scase, C.P. Caulfield & S.B. Daliel 006a Boussinesq plumes with decreasing source strengths in stratified environments. J. Fluid Mech. 56, M.M. Scase, C.P. Caulfield, S.B. Daliel & J.C.R. Hunt. 006b Timedependent plumes and jets with decreasing source strengths. J. Fluid Mech. 56, M.M. Scase, C.P. Caulfield, P.F. Linden & S.B. Daliel 007a Local implications for self-similar turbulent plume models. J. Fluid Mech. 575, M.M. Scase, C.P. Caulfield & S.B. Daliel 007b Temporal variation of non-ideal plumes with sudden reductions in buoyancy flux. Submitted to J. Fluid Mech.

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

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,