Late-time growth rate, mixing, and anisotropy in the multimode narrowband Richtmyer–Meshkov instability: The θ-group collaboration

Abstract

Turbulent Richtmyer-Meshkov instability (RMI) is investigated through a series of high resolution three dimensional smulations of two initial conditions with eight independent codes. The simulations are initialised with a narrowband perturbation such that instability growth is due to non-linear coupling/backscatter from the energetic modes, thus generating the lowest expected growth rate from a pure RMI. By independently assessing the results from each algorithm, and computing ensemble averages of multiple algorithms, the results allow a quantification of key flow properties as well as the uncertainty due to differing numerical approaches. A new analytical model predicting the initial layer growth for a multimode narrowband perturbation is presented, along with two models for the linear and non-linear regime combined. Overall, the growth rate exponent is determined as θ = 0.292 ± 0.009, in good agreement with prior studies; however, the exponent is decaying slowly in time. Also, θ is shown to be relatively insensitive to the choice of mixing layer width measurement. The asymptotic integral molecular mixing measures Θ = 0.792 ± 0.014, Ξ = 0.800±0.014 and Ψ = 0.782±0.013 which are lower than some experimental measurements but within the range of prior numerical studies. The flow field is shown to be persistently anisotropic for all algorithms, at the latest time having between 49% and 66% higher kinetic energy in the shock parallel direction compared to perpendicular and does not show any return to isotropy. The plane averaged volume fraction profiles at different time instants collapse reasonably well when scaled by the integral width, implying that the layer can be described by a single length scale and thus a single θ. Quantitative data given for both ensemble averages and individual algorithms provide useful benchmark results for future research.

Turbulent transport and mixing in the multimode narrowband Richtmyer-Meshkov instability

“…An analysis of turbulent transport in an inhomogeneous compressible turbulent mixing layer induced by Richtmyer-Meshkov instability has been presented. The analysis used data from four independent ILES codes which were applied to the quarter-scale problem presented in the θ -group collaboration 16 . All terms in the high-Reynolds number (inviscid) limit of the Reynolds-averaged transport equations for momentum, species mass fraction, turbulent kinetic energy, and mass fraction variance were computed in the inhomogeneous direction.…”

“…With a sufficiently strong impulse, time to develop, and high Reynolds number, linear growth will be followed by a transition to turbulence, where velocity fluctuations decay in time, and the layer grows at a relatively slow rate ∝ t θ with θ ≈ 0.2 to 1 (possible if dominated by linear modes, although a fully turbulent layer is limited to 8 θ ≤ 2/3) with a strong dependence on initial conditions [9][10][11][12][13][14][15] . As an example θ ≈ 0.29 for the configuration studied in this paper 16 , compared to KHI (∝ t) and RTI (∝ t 2 ). Despite the relatively small temporal exponent, the growth rate of RMI is sufficiently high that it significantly impacts the physics of the aforementioned applications, and so there is substantial interest in the accurate modelling of turbulent transport and mixing in RMI.…”

“…However, the statistical noise does not mask the overall trends for the majority of the results presented. Based on the previous study 16 , it is expected that results from Turmoil and Triclade would converge at a similar grid resolution to Flamenco. Flash, however, has historically demonstrated slower convergence compared to the other algorithms, but given the good agreement with the other algorithms presented here, and the fact that Flamenco is reasonably converged at the intermediate grid level (a factor of eight times fewer points than the highest resolution), the Flash results are also expected to be reasonably converged.…”

“…A. Summary of the numerical data employed This paper utilises the 'quarter scale' data from four algorithms presented in the θ -group study of Thornber et al 16 . In that paper the full details of the initial conditions, results, and grid convergence studies for each individual algorithm is presented.…”

“…Recently, a benchmark multimode, three-dimensional RMI case was computed with multiple independent computational fluid dynamics codes, establishing a well understood 'θgroup' dataset 16 . A principal goal of that study was to understand and quantify numerical uncertainty in the computation of such flows, and to provide a foundation for basic quantities of interest.…”