Daniel Livescu

Variable-density mixing in buoyancy-driven turbulence

The homogenization of a heterogeneous mixture of two pure fluids with different densities by molecular diffusion and stirring induced by buoyancy-generated motions, as occurs in the Rayleigh-Taylor (RT) instability, is studied using direct numerical simulations. The Schmidt number, Sc, is varied by a factor of 20, 0.1 ≤Sc≤ 2.0, and the Atwood number, A, by a factor of 10, 0.05 ≤A≤0.5. Initial-density intensities are as high as 50% of the mean density. As a consequence of differential accelerations experienced by the two fluids, substantial and important differences between the mixing in a variable-density flow, as compared to the Boussinesq approximation, are observed. In short, the pure heavy fluid mixes more slowly than the pure light fluid: an initially symmetric double delta density probability density function (PDF) is rapidly skewed and, only at long times and low density fluctuations, does it relax to a Gaussian-like PDF. The heavy-light fluid mixing process asymmetry is relevant to the nature of molecular mixing on different sides of a high-Atwood-number RT layer. Diverse mix metrics are used to examine the homogenization of the two fluids. The conventional mix parameter, θ, is mathematically related to the variance of the excess reactant of a hypothetical fast chemical reaction. Bounds relating θ and the normalized product, Ξ, are derived. It is shown that θ underpredicts the mixing, as compared to Ξ, in the central regions of an RT layer; in the edge regions, θ is larger than S. The shape of the density PDF cannot be inferred from the usual mix metrics popular in applications. For example, when θ, Ξ ≥ 0.6, characteristic of the interior of a fully developed RT layer, the PDFs can have vastly different shapes. Bounds on the fluid composition using two low-order moments of the density PDF are derived. The bounds can be used as realizability conditions for low-dimensional models. For the measures studied, the tightest bounds are obtained using S and mean density. The structure of the flow is also examined. It is found that, at early times, the buoyancy production term in the spectral kinetic energy equation is important at all wavenumbers and leads to anisotropy at all scales of motion. At later times, the anisotropy is confined to the largest and smallest scales: the intermediate scales are more isotropic than the small scales. In the viscous range, there is a cancellation between the viscous and nonlinear effects, and the buoyancy production leads to a persistent small-scale anisotropy.

Compressibility effects on the Rayleigh–Taylor instability growth between immiscible fluids

The linearized Navier–Stokes equations for a system of superposed immiscible compressible ideal fluids are analyzed. The results of the analysis reconcile the stabilizing and destabilizing effects of compressibility reported in the literature. It is shown that the growth rate n obtained for an inviscid, compressible flow in an infinite domain is bounded by the growth rates obtained for the corresponding incompressible flows with uniform and exponentially varying density. As the equilibrium pressure at the interface p∞ increases (less compressible flow), n increases towards the uniform density result, while as the ratio of specific heats γ increases (less compressible fluid), n decreases towards the exponentially varying density incompressible flow result. This remains valid in the presence of surface tension or for viscous fluids and the validity of the results is also discussed for finite size domains. The critical wavenumber imposed by the presence of surface tension is unaffected by compressibility. Ho...

High-Reynolds number Rayleigh–Taylor turbulence

The turbulence generated in the variable density Rayleigh–Taylor mixing layer is studied using the high-Reynolds number fully resolved 30723 numerical simulation of Cabot and Cook (Nature Phys. 2 (2006), pp. 562–568). The simulation achieves bulk Reynolds number, Re = H Ḣ/ν = 32,000, turbulent Reynolds number, Re t = [ktilde] 2/νϵ = 4600, and Taylor Reynolds number, R λ = 170. The Atwood number, A, is 0.5, and the Schmidt number, Sc, is 1. Typical density fluctuations, while modest, being one quarter the mean density, lead to non-Boussinesq effects. A comprehensive study of the variable density energy budgets for the kinetic energy, mass flux and density specific volume covariance equations is undertaken. Various asymmetries in the mixing layer, not seen in the Boussinesq case, are identified and explained. Hypotheses for the variable density turbulent transport necessary to close the second moment equations are studied. It is found that, even though the layer width becomes temporally self-similar relatively fast, the transient effects in the energy spectrum remain important for the duration of the simulation. Thus, the dissipation does not track the spectral energy cascade rate and the integral lengthscale does not follow the expected Kolmogorov scaling, [ktilde] 3/2/ϵ. As a result, the popular eddy diffusivity expression, ν t ∼[ktilde] 2/ϵ, does not model the temporal variation of the turbulent transport in any of the moment equations. An eddy diffusivity based on a lengthscale related to the layer width is found to work well in a gradient transport hypothesis for the turbulent transport; however, that lengthscale is a global quantity and does not lead to pointwise, local closure. Therefore, although the transient effects may vanish asymptotically, it is suggested that, even long after the onset of the self-similar growth, two separate lengthscale equations (or equivalent) are needed in a moment closure strategy for Rayleigh–Taylor turbulence: one for the turbulent transport and the other for the dissipation. Despite the fact that the intermediate scales are nearly isotropic, the small scales have a persistent anisotropy; this is due to a cancellation between the viscous and nonlinear effects, so that the anisotropic buoyancy production remains important at the smallest scales.

Forcing for statistically stationary compressible isotropic turbulence

Linear forcing has been proposed as a useful method for forced isotropic turbulence simulations because it is a physically realistic forcing method with a straightforward implementation in physical-space numerical codes [T. S. Lundgren, “Linearly forced isotropic turbulence,” Annual Research Briefs (Center for Turbulence Research, Stanford, CA, 2003), p. 461; C. Rosales and C. Meneveau, “Linear forcing in numerical simulations of isotropic turbulence: Physical space implementations and convergence properties,” Phys. Fluids 17, 095106 (2005)]. Here, extensions to the compressible case are discussed. It is shown that, unlike the incompressible case, separate solenoidal and dilatational parts for the forcing term are necessary for controlling the stationary state of the compressible case. In addition, the forcing coefficients can be cast in a form that allows the control of the stationary state values of the total dissipation (and thus the Kolmogorov microscale) and the ratio of dilatational to solenoidal di...

Rayleigh–Taylor instability in cylindrical geometry with compressible fluids

A linear stability analysis of the Rayleigh–Taylor instability (RTI) between two ideal inviscid immiscible compressible fluids in cylindrical geometry is performed. Three-dimensional (3D) cylindrical as well as two-dimensional (2D) axisymmetric and circular unperturbed interfaces are considered and compared to the Cartesian cases with planar interface. Focuses are on the effects of compressibility, geometry, and differences between the convergent (gravity acting inward) and divergent (gravity acting outward) cases on the early instability growth. Compressibility can be characterized by two independent parameters—a static Mach number based on the isothermal sound speed and the ratio of specific heats. For a steady initial unperturbed state, these have opposite influence, stabilization and destabilization, on the instability growth, similar to the Cartesian case [D. Livescu, Phys. Fluids 16, 118 (2004)]. The instability is found to grow faster in the 3D cylindrical than in the Cartesian case in the converge...

Application of a second-moment closure model to mixing processes involving multicomponent miscible fluids

A second-moment closure model is proposed for describing turbulence quantities in flows where large density fluctuations can arise due to mixing between different density fluids, in addition to compressibility or temperature effects. The turbulence closures used in this study are an extension of those proposed by Besnard et al., which include closures for the turbulence mass flux and density-specific-volume covariance. Current engineering models developed to capture these extended effects due to density variations are scarce and/or greatly simplified. In the present model, the density effects are included and the results are compared to direct numerical simulations (DNS) and experimental data for flow instabilities with low to moderate density differences. The quantities compared include Reynolds stresses, turbulent mass flux, mixture density, density-specific-volume covariance, turbulent length scale, turbulence and material mix time scales, turbulence dissipation, and mix widths and/or growth rates. These comparisons are made within the framework of three very different classes of flows: shear-driven, Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Overall, reasonable agreement is seen between experiments, DNS, and averaging models.

New phenomena in variable-density Rayleigh?Taylor turbulence

This paper presents several issues related to mixing and turbulence structure in buoyancy-driven turbulence at low to moderate Atwood numbers, A, found from direct numerical simulations in two configurations: classical Rayleigh–Taylor instability and an idealized triply periodic Rayleigh–Taylor flow. Simulations at A up to 0.5 are used to examine the turbulence characteristics and contrast them with those obtained close to the Boussinesq approximation. The data sets used represent the largest simulations to date in each configuration. One of the more remarkable issues explored, first reported in (Livescu and Ristorcelli 2008 J. Fluid Mech. 605 145–80), is the marked difference in mixing between different density fluids as opposed to the mixing that occurs between fluids of commensurate densities, corresponding to the Boussinesq approximation. Thus, in the triply periodic configuration and the non-Boussinesq case, an initially symmetric density probability density function becomes skewed, showing that the mixing is asymmetric, with pure heavy fluid mixing more slowly than pure light fluid. A mechanism producing the mixing asymmetry is proposed and the consequences for the classical Rayleigh–Taylor configuration are discussed. In addition, it is shown that anomalous small-scale anisotropy found in the homogeneous configuration (Livescu and Ristorcelli 2008 J. Fluid Mech. 605 145–80) and Rayleigh–Taylor turbulence at A=0.5 (Livescu et al 2008 J. Turbul. 10 1–32) also occurs near the Boussinesq limit. Results pertaining to the moment closure modelling of Rayleigh–Taylor turbulence are also presented. Although the Rayleigh–Taylor mixing layer width reaches self-similar growth relatively fast, the lower-order terms in the self-similar expressions for turbulence moments have long-lasting effects and derived quantities, such as the turbulent Reynolds number, are slow to follow the self-similar predictions. Since eddy diffusivity in the popular gradient transport hypothesis is proportional to the turbulent Reynolds number, the dissipation rate and turbulent transport have different length scales long after the onset of the self-similar growth for the layer growth. To highlight the importance of turbulent transport, variable density energy budgets for the kinetic energy, mass flux and density-specific volume covariance equations, necessary for a moment closure of the flow, are provided.

Small scale structure of homogeneous turbulent shear flow

The structure of homogeneous turbulent shear flow is studied using data generated by direct numerical simulations (DNS) and a linear analysis for both compressible and incompressible cases. At large values of the mean shear rate, the rapid distortion theory (RDT) limit is approached. Analytical solutions are found for the inviscid compressible RDT equations at long times. The RDT equations are also solved numerically for both inviscid and viscous cases. The RDT solutions, confirmed by the DNS results, show that the even order transverse derivative moments of the dilatational and solenoidal velocity fields are anisotropic, with the dilatational motions more anisotropic than their solenoidal counterparts. The results obtained for the incompressible case are similar to those obtained for the solenoidal motions in the compressible case. The DNS results also indicate an increase in the anisotropy of the even order transverse derivative moments with the order of the moment, in agreement with the RDT predictions...

Adaptive wavelet collocation method simulations of Rayleigh–Taylor instability

Numerical simulations of single-mode, compressible Rayleigh–Taylor instability are performed using the adaptive wavelet collocation method (AWCM), which utilizes wavelets for dynamic grid adaptation. Due to the physics-based adaptivity and direct error control of the method, AWCM is ideal for resolving the wide range of scales present in the development of the instability. The problem is initialized consistent with the solutions from linear stability theory. Non-reflecting boundary conditions are applied to prevent the contamination of the instability growth by pressure waves created at the interface. AWCM is used to perform direct numerical simulations that match the early-time linear growth, the terminal bubble velocity and a reacceleration region.

Decay of isotropic turbulence: Fixed points and solutions for nonconstant G∼Rλ palinstrophy

A Rλ scaling for the palinstrophy coefficient, G, has been observed in grid turbulence by Antonia et al. [J. Turbulence3, 1 (2002) ]. As a consequence it appears that there exist decay laws other than the self-preserving Taylor-microscale decay of Speziale and Bernard [J. Fluid Mech.241, 645 (1992) ], for which G is constant. Analytic solutions of Ristorcelli [Phys. Fluids15, 3248 (2003) ], for the self-similar Taylor microscale decay, are modified to apply to the G∼Rλ decay. The solution indicates an asymptotic k(t)∼t−1 and a Rt→Rt∞ behavior, as seen in the self-preserving Taylor microscale decay. There are two important differences between the two decays: (1) the fixed point Rt∞ and (2) its rate of approach are substantially smaller for the G∼Rλ decay. It appears that the observation of an asymptotic k(t)→t−1 decay with Rt→const is not a demonstration of Taylor microscale self-similarity, and is achieved, at a much later time, in a wider class of flows. This appears to explain the absence of observation...