Subgrid Models and DNS Studies of Fluid Mixing

New simulation studies, modeling, and theoretical results of the authors and collaborators are presented here. Three−dimensional simulations of the Rayleigh−Taylor instability, carried to late time, give a growth rate h(t) ∼ αb Agt2 with αb ≈ 0.08. This value of αb is high but probably consistent with experimental data due to finite compressibility effects. Our modeling has focused on the internal structure of the mixing zone, expressed in terms of the edge velocities. A Chunk Mix model with few adjustable parameters has been proposed, which moreover is mathematically stable, does not require overly smoothed velocities to suppress instabilities, and which avoids thermodynamic complications of atomically mixed or pressure equilibrated equations of state. Our theoretical analysis of the edge velocities has two components: A bubble merger model to obtain a physics based zero parameter determination of αb in agreement with experiment and an analysis based on the center of mass (COM) to predict the ratio αs/αb, again in agreement with experimental data. 1. Direct Numerical Simulation The three−dimensional Rayleigh−Taylor (RT) instability was simulated with the Front Tracking code FronTier [1,2]. The simulation used the compressible Euler equations with a dimensionless compressibility M2 = λg/ch2 of 0.05 and an Atwood number A = (ρ2 − ρ1)/(ρ2 + ρ1) of 1/3. Here λ is a mean bubble wave length measured at t = 0 and ch is the sound speed in the heavy fluid. ρ1 and ρ2 are the densities of the light and heavy fluids. The computational grid was 100x100x200. The initial resolution is about 10 zones per mode. The initial amplitude of the perturbation is approximately 0.015 as a multiple of the wave length. About one full generation of bubble merger has occurred in the simulation. The initial interface between the two fluids was given by a random sequence to select Fourier modes over a wave number range [κmin, κmax] in both x and y directions, with κmax/κmin = 2. This restriction on κmin is important as we wish to observe the emergence of low wave number modes due to bubble merger, without introducing such modes into the initial data directly. The growth rate of the bubble mixing zone edge Ζb(t) was determined as the z− coordinate of the most advanced bubble, i.e. the 0% contour or volume fraction. For this problem, we observed αb ≈ 0.08, at the upper end of agreement with experimental data. Based on 2D simulations, we predict that the finite compressibility M2 = 0.05 > 0.0 contributes 5% − 10% to this value of αb. In other words, we can surmise an incompressible αb ≈0.07 on the basis of our simulation. See Fig. 1. Use of a 5% contour or level surface to define αb would decrease the observed value, probably by a similar amount. Thus, in comparisons to experiment or to other simulations, it is important to compare α’s defined by the same cutoff contours. Effects of mesh refinement and of increase of spatial domain (allowing more generations of bubble merger before the growth rate is dominated by a single bubble terminal velocity or has so few bubbles that small sample size statistics become important) will be explored in future work. 2. Multiphase Flow Models We have previously reported on a two pressure, two fluid multiphase flow model based on an ensemble average of the two fluid Euler equations [3, 4, 5]. This model is hyperbolic, and thus mathematically stable, without velocity smoothing. It has few adjustable parameters. It imposes no special thermodynamic assumptions or constraints. It agrees with experimental data to the extent that this data is available. Closed form solutions in the incompressible limit have been obtained. This model describes chunk, or large scale, mix. We have recently investigated a one pressure, two temperature model [6]. To achieve a stable, well posed one pressure model, we also impose a single fluid velocity. In this case, the mixing is driven by a diffusion process. We used the known solution of the chunk mix model (above) to determine the effective diffusion coefficient. This computation depends on a determination of the entrainment time, which is the time (within Fig. 1 Computation of the RT growth rate αb ≈ 0.08, consistent with laboratory experiments, as is discussed in the text. the chunk mix model) for a parcel of fluid k, located at position z, volume fraction βk (z, t), and time t, to enter the mixing zone. To obtain analytic expressions for the solution, we assume self similar edge 2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7