Rayleigh-Taylor Driven Mixing in a Multiply Stratified Environment

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1 Rayleigh-Taylor Driven Mixing in a Multiply Stratified Environment Abstract Andrew George Weir Lawrie and Stuart Dalziel Department of Applied Mathematics and Theoretical Physics, University of Cambridge A.G.W.Lawrie@damtp.cam.ac.uk S.Dalziel@damtp.cam.ac.uk Mixing is ubiquitous in both the natural environment and industrial applications, and its consequences are far-reaching. This paper focuses on the details of the small scale mixing processes which are driven by buoyancy in a gravitational field. In particular we explore the evolution of an interleaved heavy-light-heavy miscible liquid system with one Rayleigh- Taylor unstable density interface, and one statically stable interface. Experiments are performed using LIF illumination of a chemically reactive but dynamically passive tracer, and we seek to quantify mixing induced across both interfaces. In a complementary numerical study, adaptive mesh refinement is used to target computational capacity at the small scales in the region surrounding the stable density interface. It is of particular interest whether simulations accurately capture mixing across stable density interfaces in liquid systems, when frequently such codes operate with Schmidt numbers of order unity. 1. Introduction The driving instability in the present study has been a focus for scientific curiosity since Rayleigh (1883) when the problem of dense fluid above less dense fluid in a gravitational field was first considered. Such instability is observed in a variety of fluids with applications ranging from geophysical and astrophysical to industrial fluid systems. Understanding the suppression of turbulent mixing by stable density stratification also has implications for geophysical systems and potential industrial application, and the interleaved heavy-light-heavy problem configuration (Jacobs and Dalziel (2005)) considered here is an appropriate case study, since turbulence generated by the development of the unstable interface induces mixing across the stable interface. The ability of modern numerical techniques to capture the flows with such varied mixing characteristics is not yet confirmed, and this is a focus of current investigation Computational Approach Despite the approximately exponential growth in computing power over time, numerical simulation of Rayleigh-Taylor instability remains a challenging problem. An approach called Implicit Large Eddy Simulation (ILES) is used in this paper, with the code of Almgren et al. (1998). Historically, the approach originated from the surprisingly successful application of modern numerical methods for compressible gas dynamics to problems involving intense mixing. Total Variation Diminishing (TVD) methods (eg. Leveque (1992)) were developed as a (non-linear) means of controlling truncation error terms in numerical algorithms. Dispersive error (odd order truncation error terms) manifests itself as spurious oscillations around discontinuities and high gradients. Such errors could be eliminated by using a numerical scheme which interpolates, using some empirically derived function,

2 between low and high order. Increased diffusive error (even order truncation error terms), although undesirable in regions of high gradient, is accepted as a consequence, since it has a physical analogue. The physical relevance of diffusion associated with high gradients motivated the change in application from wave-dominated to highly turbulent flows. It is well known that high gradients are smoothed by viscous effects in a real flow, and the observation that TVD methods also smoothed gradients led to the birth of ILES. Without explicitly applying viscous terms, real flows could be simulated with plausible results (eg. Youngs (1994); Dalziel et al. (1999); Ramaprabhu et al. (2005)). Effectively, the numerical error replaces a conventional eddy-viscosity sub-grid scale model, and performs a similar function. However, a better understanding of the numerical analysis of ILES for mixing problems is only now being developed (Margolin et al. (2006)) Experimental Method While in a broad range of fields much experimental work has focussed on the physics of mixing, obtaining direct visualisation of the mixing process is much less common. Light/Laser-Induced Fluorescence (LIF) techniques have been used in the past for this purpose in miscible liquid shear flows, eg. Koochesfahani and Dimotakis (1985), and more recently also a hybrid fluorescence/phosphorescence technique in gaseous shear flows (Hu and Koochesfahani (2002)), but in Rayleigh-Taylor driven mixing there seem not to be comparable experiments. Experiments using ph indicators in the Rayleigh-Taylor context (Linden et al. (1994)) to mark mixed fluid have been performed before; the present work extends this concept by using a fluorescent ph indicator, hence permitting detailed LIF visualisation of the actual mixing within the mixing region and furthering understanding of its structure. The experiments in the current study were conducted using developments of apparatus and techniques which have been applied previously by Dalziel (1993),Dalziel et al. (1999) and Jacobs and Dalziel (2005). The principal feature of the experimental rig is a horizontal barrier to provide initial separation of fluids across an unstable interface. Using a technique devised by Lane-Serff (1989), the barrier can be removed from the tank without applying the resulting shear to the fluid. The imaging technique used in this study is Light-Induced Fluorescence, using xenon arc lamps to provide incident light. The fluorescent dye used in these experiments is 2,3,5,6-dibenzo pyridine, which absorbs light in the ultraviolet wavelength range, but its emission wavelength is ph dependent. Above ph 5 (Weast (1971)), the emission is violet blue, but below ph 5 the emission is cyan-green. By appropriately filtering the emitted light before reaching the CCD camera, a signal is only received in areas of mixed fluid. The light sheet is thin relative to the dynamics of the flow, so the internal structure of the mixing in a two-dimensional crossection can be examined. The ratio of advection to reaction timescales is very large, so the reaction plays no dynamic role in the experiment. Acid is added to one (salt solution) fluid layer, the fluorescent dye to another. Refractive indices are matched using alcohol. The density is set by the combination of acid and salt. 2. Observations on Rayleigh-Taylor mixing As a starting point for validation of the numerical simulation, detailed experimental work was focussed on the Rayleigh-Taylor component of the flow, since ILES has been found to

3 Figure 1: Time series charting the flow evolution in experiment and simulation, using a comparable diagnostic. The white bar indicates the progress of nondimensional simulation time from 0 to 5. The experimental time origin has been shifted to account for the initialisation differences between experiment and simulation. work well for this problem. Direct comparison of experiment and simulation is not strictly appropriate because there are significant unsimulated effects in the experiments, such as asymmetries and large scale motions introduced in the experiment by the barrier removal, and discrepancies due to the random perturbation which initialises the simulation. However, a virtual time origin can be derived from the comparison of growth profiles. The experimental results have been spatially filtered down to the same resolution as the simulation, and an equivelant diagnostic used to present the simulations. A dual time-sequence of images in figure 1 compares the evolution of the flow. As is evident from the figure, the simulation does not explicitly capture the smallest scales of the flow that are visible even in the filtered experimental time series. However, the mesh size (160x80x200) in the simulation is chosen such that the energy dissipation by numerical error (the implicit sub-grid scale model) reasonably approximates the energy dissipation by turbulent diffusion (nature s sub-grid scale model). Thus the salient mixing statistics could be expected to compare well between experiment and simulation. A good indication of the rate of mixing can be obtained by considering the surface area over which mixing takes place, and the enclosed volume of mixed fluid which grows in time as the mixing front advances. The planar analogue of the surface area (the length by pixel count of the enclosing contour) is shown in 2 and is defined as the boundary of a region where pixel intensity is greater than a threshold. The corresponding volume is shown in figure 3. As before, the processing is performed on the filtered experimental data, for valid comparison with the simulation. The threshold is chosen for convenience, to avoid contaminating the image analysis with digital noise. However, signal intensity (in the absence of other effects) is linear in the local dye concentration, so as further mixing takes place towards later times, the signal

4 Figure 2: Evolution of mixing surface area in time. The length of the 2-D crossection through the surface is measured by pixel count. Close agreement between experiment and simulation is achieved. Figure 3: Evolution of mixed volume in time. The independent variable is measured as a fraction of total area. Agreement is good until the dilution of fluid in the experiment is such that recovered signal intensity is of the order of the noise threshold.

5 Figure 4: Numerical simulation of fluorescent dye behaviour when visualising mixing across the stable density interface. The orthogonal lines show a crossection of the grid patches used in the simulation. intensity of this mixed fluid reduces, until eventually falls below the noise threshold. Since this effect is not include in the simulation diagnostic, the experimental and simulation curves diverge at late time. 3. Mixing across the stable interface Adaptive Mesh Refinement is a numerical technique which enables arbitrarily selected regions of a flow to be investigated in considerably more detail, without the associated computational cost of resolving to this level in all regions. The mixed fluid around the stable interface is used to mark the region over which more detail is required, and the resulting region is shown in figure 4. As identified by George et al. (2002) the numerical diffusion inherent by cell averaging conserved quantities in finite volume codes applies not only to momentum, but also to density and tracer advection. This implies that ILES models have a numerical Schmidt number ( µ κ) of O (1). While this is close to the physical Schmidt number in gasses, the values for liquids are frequently two orders of magnitude higher. Quantification of the error is achieved by comparing a simulation which locally well resolves the dynamical scales and in which the Schmidt number is explicitly set by computing the diffusion terms, with a standard ILES simulation. 4. Conclusions An experimental technique has been presented for directly visualising regions of molecularly mixed flow in miscible liquid flows. Particular attention has been paid to the early time evolution of the Rayleigh-Taylor instability, obtaining quantitative measurements from a novel perspective on a widely studied problem. The present study develops previous work on chemical indicators of molecular mixing by gaining a first insight into the detailed structure of the mixing region. ILES numerical simulations have been performed which show comparable mixing behaviour to the experiments, when considering fractal dimension, mixing surface and mixed volume. Adaptive Mesh Refinement techniques are used to simulate in more detail the mixing across the stable interface in the heavy-lightheavy configuration.

6 Acknowledgements The authors would like to thank the UK Natural Environment Research Council (NERC) as primary funding source, and the Co-operative Awards in Science and Engineering (CASE) collaborator, AWE Ltd. Useful discussions with John Bell, Ann Almgren and Nikos Nikiforakis are gratefully acknowledged. References Almgren, A. S., Bell, J. B., Colella, P., Howell, L. H., and Welcome, M. L. (1998). A conservative adaptive projection method for the variable density incompressible Navier- Stokes equations. J. Comp. Phys., 142:1 46. Dalziel, S. B. (1993). Rayleigh-Taylor instability: experiments with image analysis. Dyn. Atmos. Oceans, 20: Dalziel, S. B., Linden, P. F., and Youngs, D. L. (1999). Self-similarity and internal structure of turbulence induced by Rayleigh-Taylor instability. J. Fluid Mech., 399:1 48. George, E., Glimm, J., Li, X.-L., Marchese, A., and Xu, Z.-L. (2002). A comparison of experimental, theoretical and numerical simulation Rayleigh-Taylor mixing rates. Proc. Natl. Acad. Sci., 99: Hu, H. and Koochesfahani, M. M. (2002). A novel method for instantaneous, quantitative measurement of molecular mixing in gaseous flows. Exp. Fluids, 31: Jacobs, J. W. and Dalziel, S. B. (2005). Rayleigh-Taylor instability in complex stratifications. J. Fluid Mech., 542: Koochesfahani, M. M. and Dimotakis, P. E. (1985). Mixing and chemical reactions in a turbulent liquid mixing layer. J. Fluid Mech., 170: Lane-Serff, G. F. (1989). Heat Flow and air movement in buildings. PhD thesis, DAMTP, University of Cambridge, UK. Leveque, R., editor (1992). Numerical Methods for Conservation Laws. Birkhauser. Linden, P. F., Redondo, J. M., and Youngs, D. L. (1994). Molecular mixing in Rayleigh- Taylor instability. J. Fluid Mech., 265: Margolin, L. G., Rider, W. J., and Grinstein, F. F. (2006). Modeling turbulent flow with implicit LES. J. Turbulence, 7:1 27. Ramaprabhu, P., Dimonte, G., and Andrews, M. J. (2005). A numerical study of the influence of initial perturbations on the turbulent Rayleigh-Taylor instability. J. Fluid Mech., 536: Rayleigh, L. (1883). Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc., XIV: Weast, R. C., editor (1971). Handbook of Chemistry and Physics. CRC Press, 52 edition. Youngs, D. L. (1994). Numerical simulation of mixing by Rayleigh-Taylor and Richtmyer- Meshkov instabilities. Laser and Particle Beams, 12: