Temporal evolution and scaling of mixing in two-dimensional Rayleigh-Taylor turbulence

Abstract

We report a high-resolution numerical study of two-dimensional (2D) miscible Rayleigh-Taylor (RT) incompressible turbulence with the Boussinesq approximation. An ensemble of 100 independent realizations were performed at small Atwood number and unit Prandtl number with a spatial resolution of 2048 × 8193 grid points. Our main focus is on the temporal evolution and the scaling behavior of global quantities and of small-scale turbulence properties. Our results show that the buoyancy force balances the inertial force at all scales below the integral length scale and thus validate the basic force-balance assumption of the Bolgiano-Obukhov scenario in 2D RT turbulence. It is further found that the Kolmogorov dissipation scale η(t) ∼ t 1/8 , the kinetic-energy dissipation rate ε u (t) ∼ t −1/2 , and the thermal dissipation rate ε θ (t) ∼ t −1 . All of these scaling properties are in excellent agreement with the theoretical predictions of the Chertkov model [Phys. Rev. Lett. 91, 115001 (2003)]. We further discuss the emergence of intermittency and anomalous scaling for high order moments of velocity and temperature differences. The scaling exponents ξ r p of the pth-order temperature structure functions are shown to saturate to ξ r ∞ ≃ 0.78 ± 0.15 for the highest orders, p ∼ 10. The value of ξ r ∞ and the order at which saturation occurs are compatible with those of turbulent Rayleigh-Bénard (RB) convection [Phys. Rev. Lett. 88, 054503 (2002)], supporting the scenario of universality of buoyancy-driven turbulence with respect to the different boundary conditions characterizing the RT and RB systems.

Local dissipation scales in two-dimensional Rayleigh-Taylor turbulence

“…It is seen clearly that in the range 1.6 t/τ 4 two types of lines collapse roughly on top of each other, indicating a quadratic growth of h(t). Further studies show that the power spectra (not shown here) of both the velocity and temperature files obtained within the self-similar range cover a broad range of scales [40], indicating that a turbulent state has been well developed. Therefore, we shall later analyze the temporal evolution of Q(η) within this time range.…”

“…The direct numerical simulations are based on a compact fourth-order finite-difference scheme, proposed by Liu et al [41], and the accuracy, stability, and efficiency of the scheme have been examined in great detail [41,42]. Recently, we have applied the same numerical code to study, respectively, small-scale properties in the 2D RT system [40] and turbulent heat transport in 2D Rayleigh-Bénard convection [43]. In the present study, the number of grid points is set to 4096 × 8193 in all the runs to achieve a sub-Kolmogorov-scale resolution.…”

“…Cascade processes in such a system have been extensively investigated in the past decade [26][27][28][29][30][31][32][33][34]. Here, two considerations prompted us to restrict ourselves to the 2D simulations of RT turbulence: (i) The numerical effort required for 2D simulations is much smaller so that a superfine resolution for the sub-Kolmogorovscale statistics becomes feasible for high Reynolds numbers; (ii) temperature becomes a fully active scalar in the 2D RT system, and thus a new type of phenomenology, i.e., a Bolgiano-Obukhov-like (BO59) scaling [35], was theoretically predicted [36,37] and numerically observed [38][39][40]. Indeed, results obtained by Zhou [40] revealed that buoyancy overwhelms nonlinear energy transfer at all inertial scales.…”

“…Here, two considerations prompted us to restrict ourselves to the 2D simulations of RT turbulence: (i) The numerical effort required for 2D simulations is much smaller so that a superfine resolution for the sub-Kolmogorovscale statistics becomes feasible for high Reynolds numbers; (ii) temperature becomes a fully active scalar in the 2D RT system, and thus a new type of phenomenology, i.e., a Bolgiano-Obukhov-like (BO59) scaling [35], was theoretically predicted [36,37] and numerically observed [38][39][40]. Indeed, results obtained by Zhou [40] revealed that buoyancy overwhelms nonlinear energy transfer at all inertial scales. This is in clear contrast to 3D cases, where temperature behaves as a passive scalar, leading to the traditional Kolmogorov-like (K41) phenomenology as those observed in homogeneous isotropic turbulence [30].…”

Translation simulation of a cylinder in a horizontal pipe

“…It is straightforward to deduce that V t has an obvious linear relationship with U. Additionally, the physical parameters, including κ, ϕ and ε, have an impact on the V t -U relationship. Furthermore, for a given cylinder, there exists a U i to make V t /(U − U i ) a constant, and this means that the cylinder translation system is selfsimilar (Zhou, 2013). Therefore, the terminal velocity can be estimated by extrapolation of the linear relationship obtained in this simulation.…”