Buoyancy-driven variable-density turbulence

Abstract

Buoyancy-generated motions in an unstably stratified medium composed of two incompressible miscible fluids with different densities, as occurs in the variable-density Rayleigh–Taylor instability, are examined using direct numerical simulations. The non-equilibrium homogeneous buoyantly driven problem is proposed as a unit problem for variable density turbulence to study: (i) the nature of variable density turbulence, (ii) the transition to turbulence and the generation of turbulence by the conversion of potential to kinetic energy; (iii) the role of non-Boussinesq effects; and (iv) a parameterization of the initial conditions by a static Reynolds number. Simulations are performed for Atwood numbers up to 0.5 with root mean square density up to 50% of the mean density and Schmidt numbers, 0.1 ≤ Sc ≤ 2. The benchmark problem has been designed to have the largest mass flux possible and is, in this configuration, the maximally unstable non-equilibrium flow possible. It is found that the mass flux, owing to its central role in the conversion of potential to kinetic energy, is probably the single most important dynamical quantity to predict in lower-dimensional models. Other primary findings include the evolution of the mean pressure gradient: during the non-Boussinesq portions of the flow, the evolution of the mean pressure gradient is non-hydrostatic (as opposed to a Boussinesq fluid) and is set by the evolution of the specific volume pressure gradient correlation. To obtain the numerical solution, a new pressure projection algorithm which treats the pressure step exactly, useful for simulations of non-solenoidal velocity flows, has been constructed.

Numerical investigation of initial condition effects on Rayleigh-Taylor instability with acceleration reversals

“…Less-than-complete mixing is a hallmark of rapid growth in the future since horizontal density gradients generate the destabilizing baroclinic torque that perpetuates the RT instability, Another facet of this observation [48,49], as developed in the next section, is that net mass flux across the plane of the initial interface is a good proxy for growth of the mixing layer; the mass flux is zero in the case of completely mixed θ =1 fluid, and increases with reducing θ . The value ofθ tends [47] to an asymptotic value of θ ~ 0.8 for the constant acceleration case.…”

“…The normalized mass flux parameter <u 3 c>/ 0.5 tot h is plotted in figure 9. It has been reported in various studies [48,49] that the mass flux plays a crucial role in the conversion of potential energy to kinetic energy of buoyancy driven flows. During deceleration we have statically-stable flow in a wave-like regime.…”

Direct numerical simulation and large-eddy simulation of stationary buoyancy-driven turbulence

“…The advantage to their approach is that no iteration is required. Another approach to this issue is proposed by Livescu & Ristorcelli (2007), who derive an exact nonlinear equation for p ((A 15) in that paper) that requires an iterative solution method but eliminates temporal discretization errors. However, it remains that (2.7) cannot be discretely satisfied owing to the inevitable finite spatial resolution, even if infinite iterations were possible.…”

“…In a triply periodic domain where such far-field boundary conditions cannot be directly imposed, we model Γ i by requiring that ∂u i /∂t m = 0, where m denotes the xy-plane average taken at the mid-plane z = L z /2. As noted by Livescu & Ristorcelli (2007), ∂u i /∂t m ≈ 0 in the Rayleigh-Taylor turbulent mixing zone; they considered a similar model by choosing [∂u i /∂t] x = 0. For definiteness, we choose u i m = 0.…”

“…This is readily achieved by setting ω s = 0 in (2.7) and (2.8), choosing Γ i (t) so that [u i ] x = 0 (see § 2.4) and initializing the flow as random blobs of pure fluids in a cube of size 2π , a procedure documented by Livescu & Ristorcelli (2007) x /ρ 0 ≈ 0.22. As the fluid mixes, the density variance decreases while the kinetic energy increases to peak at t(Ag/ ) 1/2 ≈ 2 before the mixture eventually decays to a homogenized and quiescent state at large times.…”