A comparison of experimental, theoretical, and numerical simulation Rayleigh–Taylor mixing rates

Abstract

We present a Rayleigh–Taylor mixing rate simulation with an acceleration rate falling within the range of experiments. The simulation uses front tracking to prevent interfacial mass diffusion. We present evidence to support the assertion that the lower acceleration rate found in untracked simulations is caused, at least to a large extent, by a reduced buoyancy force due to numerical interfacial mass diffusion. Quantitative evidence includes results from a time-dependent Atwood number analysis of the diffusive simulation, which yields a renormalized mixing rate coefficient for the diffusive simulation in agreement with experiment.

Since Read and Youngs (1, 2) published the first experimental study of Rayleigh–Taylor instability with a randomly perturbed fluid interface, attention has been drawn to the nondimensional acceleration rate of the bubble envelope. Assume that two fluids are separated by a randomly perturbed interface and that the gravitational field points from the heavy fluid (density ρ_H) to the light fluid (density ρ_L). Read and Youngs confirmed the Sharp–Wheeler theoretical prediction of late stage Rayleigh–Taylor instability (1961, unpublished technical report, Institute of Defense Analyses) that the average bubble front moves with acceleration scaling

1

where h is the height of the bubble envelope, A = (ρ_H − ρ_L)/(ρ_H + ρ_L) is the Atwood number, g is the gravity, and t is time. Read and Youngs show that the acceleration rate α is almost a constant, with α ≈0.063−0.077 in three-dimensional experiments. The experiments have been repeated by various authors with different apparatus, and similar values of α have been obtained; we mention the experiments of Dimonte and Schneider (3–5), giving α = 0.05 ± 0.01. The theoretically determined value, α ≈0.05−0.06, is obtained from a bubble merger renormalization group model (B. Cheng, J.G., and D. H. Sharp, unpublished work). Computation of the center of mass of the mixing zone introduces a coupling between its two edges. Therefore, a characterization of the center of mass (nearly stationary for A ≤ 0.8, for example) determines the total mixing zone size in terms of α alone (6, 7).

The coefficient α is thus important. It characterizes the size of the mixing zone, and thus largely determines the amount of material that is mixed. It has been reported by experimentalists as being approximately universal, in the sense that it is nearly independent of the random initial conditions of an experiment.

Researchers in several laboratories have tried to reproduce the Sharp–Wheeler theoretical scaling law with the experimental value for α through numerical simulation. Most researchers report a time-dependent, decreasing value for α, ranging from 0.015 to 0.03.

These simulations are from computational codes using numerical schemes with interfacial mass diffusion. We have compared numerical simulations by using a high-resolution front tracking code frontier with zero interfacial mass diffusion to our own simulations using an untracked total variation diminishing (TVD)-level set code with interfacial mass diffusion similar to the others. We also introduce an analytic study of the effects of mass diffusion on buoyancy reduction, and we predict the numerically observed reduction in α for untracked simulations. Our main result is that all values of α (theory, experiment, simulation) are consistent if the diffusive calculation of α is renormalized to account for mass diffusion.

Diffusive and Nondiffusive Simulations

An earlier comparison shows that frontier simulations produce values for α close to agreement with experiment whereas untracked TVD simulations produce low values for α (8). These comparisons were limited in the simulation time and in the penetration depth of mixing achieved. Here we extend the comparison to a later time, comparable to most other simulations. Fig. 1 shows the evolution of the fluid interface in the frontier simulation. The color coding displays the height through the mixing zone, and the cut plane near the bubble surface at the top of Fig. 1 Right shows the location of the 5% volume fraction contour for the light fluid. Note that there are a number of light fluid bubbles at the later time. The dynamics is multimode, not dominated by a single large space-filling bubble up to this time. Such a large bubble would indicate the end of any possible self similar flow regime, as the acceleration scaling depends on a continued growth in the transverse scale of the mixing structures. We expand on this idea. The dynamics of continued acceleration of the mixing zone edge, as expressed in the t² scaling in Eq. 1, depends on a continued growth of the large-scale structures (the bubbles). See for example ref. 9. Bubbles grow through a process of bubble competition and merger. Thus the t² scaling and the determination of α requires a simulation that is still in the multimode regime, where bubble competition and merger can occur.

Figure 1.

Front plot for a frontier simulation of Rayleigh–Taylor mixing, with A = 0.5. (Left) Early time. (Right) Late time. Color coding represents vertical height. The initial mean height of the interface is 4, and the height scale on the color bar applies only to the later time.

The t = 0 interface is constructed out of Fourier modes with random amplitudes and frequencies in the range of 8 to 16 modes per computational domain width. See ref. 8 for further information concerning these simulations. The 2 × 2 × 8 computational domain used here allows computationally efficient late-time, deep-penetration simulations. Within this computational aspect ratio, the Fourier mode numbers represent a balance between the conflicting requirements of spatial resolution, favoring low numbers of modes, and late-time statistical validity, favoring large numbers of modes. Except as noted, the simulations used a 128² × 512 grid. Our simulations, thus balanced, have about 12² = 144 initial bubbles and a grid resolution of about 10 cells in each dimension per initial bubble. The final time considered here has about five bubbles (see Fig. 1).

A comparison of the mixing rates for the two simulations is shown in Fig. 2 Left, plotting bubble height h vs. Agt². frontier has a distinctly higher growth rate than does the interface mass diffusive TVD simulation. The value of h(t) is the difference between the t = 0 bubble height and the time t bubble height. The latter quantity is defined in terms of a 1% volume fraction, i.e., the greatest height at which the fluid is 99% heavy and 1% light according to the front tracking front or the TVD level set. This definition is somewhat unstable statistically, and a few spurious oscillations associated with the definition were removed in the plots of Fig. 2.

Figure 2.

Mixing growth comparison of a frontier (nondiffusive) with a TVD (diffusive) simulation. For the TVD simulation, two grid levels are shown, the coarser being 64² × 128. In all cases, h is the height of the 1% volume fraction contour, and the initial mean height of the interface is 4. (Left) h vs. A(t = 0)gt² for frontier and TVD. (Right) h vs. 2 ∫ ∫ A(s)gdsds₁ for frontier and TVD. The solid line represents the frontier simulation, the dashed line is the finer-grid TVD simulation, and the dotted line is the coarser-grid TVD simulation.

Mass diffusion is a common feature of most untracked simulation codes. Because of the interpolation constraint, numerical schemes (finite difference, finite volume) can have only first-order accuracy in their spatial derivatives near a discontinuity. For a contact discontinuity, the corresponding characteristic is linear for the wave equation of the Riemann invariant

2

and so the truncation error will spread to the interior region. Assuming that a finite difference scheme is second order in time and first order in space at a contact discontinuity, we have the equivalent equation

3

so that the width ΔL of the numerically diffused density profile satisfies ΔL ∼ .

To understand the difference between the two simulations, we compare the cross-sectional density plots in a series of horizontal slices from the bubble (upper) portion of the mixing region. Fig. 3 shows the cross-sectional density plots in these simulations. Observe that there is a substantial smearing-out of the density across the boundary between the two fluids in the untracked TVD simulation, whereas the frontier simulation maintains a sharp boundary with a discontinuous density profile throughout the simulation. As a further difference, we note the fine-scale structure size in the frontier simulation in comparison to the TVD simulation.

Figure 3.

Cross-sectional plots showing density on a common rainbow color scale. The pure light fluid is colored blue and the pure heavy fluid is red. Yellow and green represent various levels of microscopic mixing. The ratio of extreme density values is 3.3:1. (Right) Shown is a higher slice in the z direction. (Upper) frontier. (Lower) TVD. The simulations are shown at comparable penetration distances, but at different times (Agt² = 23 for frontier, Agt² = 66 for TVD). It is evident that the density contrast for the TVD simulation has been reduced by about 50% because of mass diffusion. See also Fig. 4. (The complete 2 × 2 cross section is displayed.)

We compute an effective Atwood number A(t) as a function of time for the TVD simulations. This number is determined from the highest and lowest densities in a horizontal slice, with the resulting time- and space-dependent Atwood number averaged over heights in the upper third of the mixing zone at a fixed time to get an Atwood number dependent on time alone. In Fig. 4, we plot A(t) vs. t for three simulations (fine- and coarse-grid TVD and fine-grid frontier). The time dependence of A(t) in the frontier simulation is caused purely by (small) compressibility effects. For the mass diffusive TVD simulation, the initial density contrast, A(t = 0) = 0.5, is almost completely washed out; the earliest time displayed shows A(t = 2) ≈ 0.15. As new pure (heavy and light) fluid is injected into the mixing region, the effective Atwood number increases, but it is still reduced to about A ≈ 0.3 on a time-averaged basis, or nearly a 50% reduction relative to its initial value.

Figure 4.

Time-dependent A (Atwood number) for fine-grid frontier, fine-grid TVD, and coarse-grid (64² × 128) TVD. At time t = 0, all three simulations have A(t = 0) = 0.5. This plot displays the reduced buoyancy of the diffusive TVD simulations as a function of time.

To compensate for the time-dependent Atwood number A(t), we define an effective alpha, α_eff ≈ h/2∫∫A(s)gdsdt (see Fig. 2 Right). Specifically, α or α_eff is defined here as the slope of the straight line joining the beginning and end of the h(t) curve in Fig. 2. This definition, although somewhat arbitrary, is conventional, and thus allows comparison to the results of others. We observe an improved comparison between frontier and TVD and between TVD and experiment. Note that α_eff lies within the range of experimental values (see Table 1). On this basis, we can state that the diffusive buoyancy renormalization of α is capable of resolving existing discrepancies among simulations, between diffusive simulations and nondiffusive experiments, and with theory.

Table 1.

Values of α determined from experiment, theory, and simulation

Method	α	Reference
Experiment	0.05–0.077	1–5
Theory	0.05–0.06	B., Cheng, J. G., and D. H. Sharp, unpublished work
frontier simulation (unrenormalized)	0.07	This paper (Fig. 2Left); ref. 8
frontier simulation (renormalized)	0.07	This paper (Fig. 2Right)
TVD simulation (unrenormalized)	0.035	This paper (Fig. 2Left)
TVD simulation (renormalized)	0.06	This paper (Fig. 2Right)

All values are consistent except the unrenormalized TVD value (with α determined from a time independent t = 0 Atwood number). 

Diffusion-Induced Buoyancy Reduction

The reduced mixing rate caused by unphysical numerical diffusion can be understood from Fig. 5. Fig. 5a represents an immiscible bubble of radius r. Fig. 5 b and c assumes that this bubble is smeared out numerically to a radius R whereas the total mass inside the sphere of radius R is conserved. The buoyancy forces

4

for the bubbles in Fig. 5 a and c are the same. However, because of the difference between the mass in the nondiffused bubble (Fig. 5a) and the diffused bubble (Fig. 5c), the two acceleration rates

5

are different.

Figure 5.

(a) Unmixed bubble of light fluid. (b) Unmixed bubble and heavy fluid mass that will be mixed with it. (c) Mixed bubble.

As a result of the mass diffusion, the buoyancy force is distributed to a larger amount of mass, thus reducing the acceleration of the bubble. This schematic figure and simple calculation can be illustrated by a TVD simulation of mixed and unmixed light bubbles rising in a heavy fluid (see Fig. 6).

Figure 6.

TVD simulations of mixed and unmixed light fluid bubbles rising in a heavy fluid. (Left) Unmixed bubbles. (Right) Mixed bubbles. (Upper) Shown are the bubbles at the initial time and at the same height. (Lower) Shown are the bubbles at a later common time. The difference in the acceleration of the mixed and unmixed bubbles can be inferred from their height differences in Lower.

Conclusion

We present a frontier simulation run to late time and deep penetration. The simulation is terminated while still in a multimode regime. It has no interfacial mass diffusion, and the overall bubble mixing rate lies within the experimental range. We recalibrate the buoyancy force for mass diffusive TVD simulations and obtain a renormalized α_eff, which is also in agreement with experiment. On this basis, the nondiffusive simulation and the theory of mass diffused buoyancy reduction presented here are capable of resolving the principal differences between simulation and experiment for Rayleigh–Taylor mixing. Our results confirm the earlier agreement between theory and experiment (B. Cheng, J.G., and D. H. Sharp, unpublished work). Finally, we observe that our results open a door to further research and do not close inquiry related to the determination of the mixing rate, as the uncertainties in the experimental, theoretical, and simulation determination of α deserve further investigation. Concerning simulation, which is the main thrust of this paper, we mention the importance of improved resolution. The needs for resolution are numerical accuracy, governed by mesh cells per bubble, statistical accuracy, governed by the number of bubbles, especially at the end of the simulation, and convergence to self similar flow, governed by the length of time of the simulation.

Abbreviation

TVD

total variation diminishing