A spectral/compact finite-difference method with a third-order Adams-Bashforth-Moulton time-evolution scheme is used to perform a direct numerical simulation (DNS) of Rayleigh-Taylor flow. The initial conditions are modeled by parameterizing the multi-mode velocity and density perturbations measured just off of the splitter plate in water channel experiments. Parameters in the DNS are chosen to match the experiment as closely as possible. The early-time transition from a weakly-nonlinear to a strongly-nonlinear state, as well as the onset of turbulence, is examined by comparing the DNS and experimental results. The mixing layer width, molecular mixing parameter, vertical velocity variance, and density variance spectrum obtained from the DNS are shown to be in good agreement with the corresponding experimental values

Andrews, M

Mueschke, N

Schilling, O

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UCRL-CONF-227192Rayleigh-Taylor instability-inducedmixing: initial conditions modeling,three-dimensional simulations andcomparisons with experimentN. Mueschke, O. Schilling, M. AndrewsJanuary 11, 2007International Workshop on the Physics of CompressibleTurbulent MixingParis, FranceJuly 17, 2006 through July 21, 2006Disclaimer   This document was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor the University of California nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or the University of California, and shall not be used for advertising or product endorsement purposes.  e-mail: schilling1@llnl.gov10th IWPCTM – PARIS (FRANCE) July 2006Rayleigh–Taylor instability-induced mixing: initial conditions modeling,three-dimensional simulations and comparisons with experimentNicholas MUESCHKE1, Oleg SCHILLING2 and Malcolm ANDREWS31 Texas A&M University, College Station, Texas 77843, USA2 University of California, Lawrence Livermore National Laboratory, Livermore, California 94551, USA3 Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USAAbstract: A spectral/compact finite-difference method with a third-order Adams–Bashforth–Moulton time-evolution scheme is used to perform a direct numerical simulation (DNS) of Rayleigh–Taylor flow. The initialconditions are modeled by parameterizing the multi-mode velocity and density perturbations measured just offof the splitter plate in water channel experiments. Parameters in the DNS are chosen to match the experimentas closely as possible. The early-time transition from a weakly-nonlinear to a strongly-nonlinear state, as wellas the onset of turbulence, is examined by comparing the DNS and experimental results. The mixing layerwidth, molecular mixing parameter, vertical velocity variance, and density variance spectrum obtained fromthe DNS are shown to be in good agreement with the corresponding experimental values.1 INTRODUCTIONThe purpose of this research is to elucidate the physics of turbulent transport, and transitional dynam-ics in Rayleigh–Taylor instability-driven mixing layers using coupled experiments and high-resolution three-dimensional direct numerical simulations (DNS). To accomplish this, experimental measurements characterizingthe initial conditions and the evolution of various large-scale and turbulence statistics were performed within asmall Atwood number, incompressible, miscible Rayleigh–Taylor mixing layer [1,2]. A DNS model of the experi-ment was formulated using parameters that closely correspond to the experiment and experimentally-measuredinitial density and velocity perturbations. Data from this DNS was also used to investigate the relative im-portance of terms in the turbulent kinetic energy transport equation and the validity of the gradient-diffusionapproximation in modeling these terms [3].2 EXPERIMENTAL FOUNDATIONMeasurements were taken is an open-loop water channel facility at Texas A&M University. A schematic ofthe water channel and the associated diagnostics can be found elsewhere [1,2]. An adverse density strat-ification was created by heating the bottom water stream. The temperature difference (∆T ≈ 5◦C) cre-ated a density difference through thermal expansion of the bottom stream, resulting in an Atwood numberA = (ρ1 − ρ2)/(ρ1 + ρ2) ≈ 7.5× 10−4. Downstream distance, x, is converted to time using Taylor’s hypothesisand normalized such that τ = x/(Um√gA/H), where Um = 4.5 cm/s is the mean velocity of the waterstreams, g = 981 cm/s2, and H = 32 cm is the height of the channel.3 EQUATIONS SOLVED AND THE NUMERICAL METHODThe DNS uses a hybrid spectral and tenth-order compact differencing spatial discretization scheme to solve theconservation of mass, momentum, and mass fraction evolution equations [4]:∂ρ∂t+∂∂xj(ρ uj) = 0 , (3.1)∂∂t(ρ ui) +∂∂xj(ρ ui uj) = ρ gi − ∂p∂xi+∂σij∂xj, (3.2)10th IWPCTM – PARIS (FRANCE) July 2006∂∂t(ρmr) +∂∂xj(ρmr uj) =∂∂xj(ρD∂mr∂xj), (3.3)where ρ is the density, ui is the velocity, gi is gravity, p is the pressure, σij = µ (∂ui/∂xj + ∂uj/∂xi) −(2/3)µδij∂uk/∂xk is the viscous stress tensor [with µ = ρν the dynamic viscosity and ν = (µ1 + µ2)/(ρ1 + ρ2)the constant kinematic viscosity],mr is the mass fraction of fluid r = 1, 2, andD is the constant mass diffusivity.Time integration is performed using a third-order Adams–Bashforth–Moulton predictor-corrector scheme. Thephysical and numerical parameters are given in Table 3.1.Parameter Valueρ1 0.9986 g/cm3ρ2 0.9970 g/cm3A 7.5× 10−4gz −981 cm/s2µ1 0.009 g/(cm s)µ2 0.011 g/(cm s)Pr (ν/D in DNS) 7Lx × Ly × Lz 16 cm ×10 cm ×16 cmNx ×Ny ×Nz 384× 240× 512Table 3.1. Parameters used in the DNS of the water channel experiment.4 THE MODEL OF THE INITIAL CONDITIONSThe initial interface between the two fluids in the DNS was parameterized by orthogonal perturbations in thex- and y-directions:ζ(x, y) =∑kxζˆx(kx) eikxx +∑kyζˆy(ky) eikyy , (4.1)where kx and ky are the perturbation wavenumbers. The initial velocity was constructed from a potentialformulation, where the potential field is perturbed to give the measured initial vertical velocity fluctuations,φr(x, z) =∑kxwˆ(kx)kxsin [kx x+ ϕ(kx)] e−kx|z| , (4.2)where ϕ(kx) are the phases. The initial velocity field is the gradient of the potential field. In Eqs. (4.1)and (4.2), the perturbation amplitudes ζˆx(kx), ζˆy(ky), and wˆ(kx) are taken directly from the correspondingexperimentally-measured one-dimensional spectra [2]. The measured initial interfacial and velocity spectrafrom the water channel [1,2] are shown in Fig. 4.1 to illustrate the high degree of anisotropy at the onset ofthe instability.5 COMPARISONS OF DNS RESULTS WITH EXPERIMENTAL DATA5.1 Mixing Layer GrowthThe evolution of the f1 = 0.5 volume fraction isosurface is shown in Fig. 5.2. Qualitatively, the mixing layergrowth is dominated by the initial velocity perturbations in the x-direction. The early (τ < 0.5) structure ofthe mixing layer is predominantly two-dimensional, as observed in the experiments. For τ > 0.5, perturbationsto the initial density interface become important as the mixing layer dynamics become highly-nonlinear andmore three-dimensional structure develops.The mixing layer width is measured by the 5–95% thresholds of the volume fraction profiles. The evolutionof the bubble and spike front penetrations is shown in Fig. 5.3. The DNS agrees fairly well with the mixinglayer growth measurements of Snider [5] and Wilson [6]; however, the DNS grows slightly faster than theexperiment, which may be due to the lack of statistical convergence in longer wavelengths. As the mixing layerdevelops, the dominant structures determining its growth become larger, and thus fewer exist within the finitecomputational domain.10th IWPCTM – PARIS (FRANCE) July 2006Fig. 4.1. The initial one-dimensional spectra for the DNS. Left: interfacial perturbations in the x-direction. Middle:interfacial perturbations in the y-direction. Right: vertical velocity fluctuations at the centerplane of mixing layer inthe x-direction.Fig. 5.2. The volume fraction f1 = 0.5 isosurface at τ = 0.25, 0.50, 0.75, 1.00.Figure 5.3 also shows the evolution of the integral-scale Reynolds number of the DNS, where Re = UL/ν andU and L are characteristic velocity and lengthscales, respectively. The mixing layer width h(t) = hb(t) + hs(t)is chosen as the lengthscale for all of the Reynolds numbers shown in Fig. 5.3. Various velocity scales havebeen used, including the total penetration rate of the mixing layer dh/dt, the terminal velocity of the dominantbubble diameter v∞, and the rms centerplane vertical velocity fluctuations w′2. Depending upon the velocityscale used, the final Reynolds number achieved in the DNS ranges from 1200–2000.5.2 Turbulence and Mixing StatisticsIn addition to comparisons of integral-scale quantities, turbulence and mixing statistics are also comparedbetween the DNS and experiment. First, the degree of molecular mixing between the two fluids at the cen-terplane of the mixing layer, θ = 1 − f ′21 /(f1 f2), is compared, where fr is the volume fraction of fluid r (allaverages are taken at the centerplane). A value θ = 1 indicates that the fluids are completely mixed, and θ = 0Fig. 5.3. The evolution of the bubble and spike fronts, hb and hs, and integral-scale Reynolds number, Re.10th IWPCTM – PARIS (FRANCE) July 2006Fig. 5.4. Comparison of DNS results with the experimental measurements of θ, the normalized vertical velocity variance,and the normalized density variance spectrum. Error bars are shown on the measurements of θ and the normalizedvertical velocity variance.indicates complete segregation. Values from the DNS and experiment are shown in Fig. 5.4. The DNS followsthe experimentally measured trends closely, but underpredicts the total amount of mixed fluid. This may beassociated with the method of implementing the initial velocity conditions and how energy from the initialspectrum is distributed among the discrete modes that can be supported on the computational grid.The vertical velocity fluctuations at the centerplane of the mixing layer w′2/(AgH) and the one-dimensionalenergy spectrum of the centerplane vertical velocity fluctuations Ew(kx)/(AgH2)is shown at τ = 0.92 from theDNS and experiment in Fig. 5.4. Both quantities compare very well. In particular, the DNS and experimentalspectra agree quite well over the entire range of scales that overlap.6 CONCLUSIONSA DNS of a small Atwood number, multi-mode Rayleigh–Taylor mixing layer was initialized using interfacial andvelocity perturbations parameterized from experimental measurements. Results from the DNS and experimentshowed favorable agreement. Differences are attributed to details in the method of parameterizing the initialconditions, and the lack of statistical convergence at late time due to a limited computational domain size.Both of these are strongly related to the inclusion of longer wavelengths, which affect the integral-scale andturbulence statistics compared. A transition to highly-nonlinear dynamics was observed at τ ≈ 0.4, which wasseen in the rapid increase of vertical velocity fluctuations and the increase in the degree of molecular mixing.This transition point and the magnitudes of w′2 and θ at this point were observed to be a function of the exactimplementation of the initial conditions. Accordingly, higher resolution DNS investigating the effects of longerwavelengths on the quantities considered here are in progress.This research was supported by the National Nuclear Security Administration under the Stewardship Sci-ence Academic Alliances program through DOE Research Grant #DE-FG03-02NA00060. This work was alsoperformed under the auspices of the U.S. Department of Energy by the University of California LawrenceLivermore National Laboratory under contract No. W-7405-Eng-48.REFERENCES[1] Mueschke, N. J., 2004. An Investigation of the Influence of Initial Conditions on Rayleigh–Taylor Mixing. M.S. thesis, TexasA&M University.[2] Mueschke, N. J., Andrews, M. J. and Schilling O., 2006. Experimental characterization of initial conditions and spatio-temporalevolution of a small Atwood Rayleigh–Taylor mixing layer. J. Fluid Mech. 567, pp. 27–63.[3] Schilling, O., Mueschke, N. J. and Latini, M., 2007. Assessment of gradient-diffusion closures for modeling Rayleigh–Taylorand Richtmyer–Meshkov instability-induced mixing, in Proceedings of the Tenth International Workshop on the Physics ofCompressible Turbulent Mixing, 17–21 July 2006, Paris, France, edited by M. Legrand and M. Vandenboomgaerde.[4] Cook, A. W. and Dimotakis, P. E., 2001. Transition stages of Rayleigh–Taylor instability between miscible fluids. J. FluidMech. 443, pp. 69–99. Corrigendum. J. Fluid Mech. 457, pp. 410–411 (2002).[5] Snider, D. M. and Andrews, M. J., 1994. Rayleigh–Taylor and shear driven mixing with an unstable thermal stratification.Phys. Fluids A 6, pp. 3324–3334.[6] Wilson, P. N., 2002. An Investigation into the Spectral Evolution of Turbulent Mixing by Rayleigh–Taylor Instability. Ph.Dthesis, Texas A&M University.

Rayleigh-Taylor instability-induced mixing: initial conditions modeling, three-dimensional simulations and comparisons with experiment

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Rayleigh-Taylor instability-induced mixing: initial conditions modeling, three-dimensional simulations and comparisons with experiment

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