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Rayleigh-Taylor Turbulence with Singular Nonuniform Initial Conditions

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Rayleigh-Taylor Turbulence with Singular Nonuniform Initial Conditions Rayleigh-Taylor turbulence with singular non-uniform initial conditions L. Biferale,1 G. Boffetta,2 A.A. Mailybaev,3 and A. Scagliarini4 1Department of Physics and INFN, University of Rome “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Rome, Italy 2Department of Physics and INFN, University of Torino, via P. Giuria 1, 10125 Torino, Italy 3Instituto Nacional de Matemática Pura e Aplicada – IMPA, 22460-320 Rio de Janeiro, Brazil 4Istituto per le Applicazioni del Calcolo ’M. Picone’ – IAC-CNR, Via dei Taurini 19, 00185 Rome, Italy We perform Direct Numerical Simulations of three dimensional Rayleigh-Taylor turbulence with a non-uniform singular initial temperature background. In such conditions, the mixing layer evolves under the driving of a varying effective Atwood number; the long time growth is still self-similar, but not anymore proportional to t2 and depends on the singularity exponent c of the initial profile ∆T / zc. We show that universality is recovered when looking at the efficiency, defined as the ratio of the variation rates of the kinetic energy over the heat flux. A closure model is proposed that is able to reproduce analytically the time evolution of the mean temperature profiles, in excellent agreement with the numerical results. Finally, we reinterpret our findings on the light of spontaneous stochasticity where the growth of the mixing layer is mapped in to the propagation of a wave of turbulent fluctuations on a rough background. Introduction. Turbulent mixing is a mechanism of evolve in a non-homogeneous background. In particular utmost importance in many natural and industrial pro- we investigate analytically and by using direct numerical cesses, often induced by the Rayleigh-Taylor (RT) insta- simulations in three dimensions the generic case when the bility which takes place when a fluid is accelerated against initial unstable vertical temperature distribution is given a less dense one [1–5]. RT turbulence occurs in disciplines by the power law: as diverse as in astrophysics [7–9], atmospheric science [10], confined nuclear fusion [11, 12],plasma physics [13], z c T0(z) = (θ0=2) sgn(z) j j ; (1) laser-matter interactions [14, 15] (see [4–6] for recent re- − L views). One important application of RT instability is the case of convective flow, in which density differences where L is a characteristic length scale and L z L. reflect temperature fluctuations of a single fluid and the The exponent of the singularity belongs to− the≤ interval≤ acceleration is provided by gravity. 1 < c < 1, where the upper limit corresponds to a In the simplest configuration of Boussinesq approxima- smooth− profile and the lower limit ensures that the po- tion for an incompressible flow, RT turbulence considers tential energy density, βgzT0(z), does not diverge near a planar interface which separates a layer of cooler (heav- the interface among the− two miscible fluids at z = 0. The ier, of density ρH ) fluid over a layer of hotter (lighter, of value c = 0 recovers the standard RT configuration. density ρ ) fluid under a constant body force such as L We develop a closure model based on the Prandtl Mix- gravity. The driving force is constant in time and pro- ing Length approach, which is able to reproduce with portional to g , where g is the acceleration due to the good accuracy the evolution of the mean temperature body force andA = (ρ ρ )=(ρ + ρ ) = βθ =2 is the H L H L 0 profile at all scales and for all values of the singularity Atwood number,A expressed− in term of the thermal expan- exponent c. Beside the importance of testing the robust- sion coefficient β and the temperature jump θ between 0 ness with respect to the initial configuration, the above the two layers. However, in some relevant circumstances setup allows us to investigate the idea that the Mixing one has to cope with time varying acceleration (as in in- Layer (ML) growth can be mapped to a traveling wave in ertial confinement fusion or in pulsating stars [16–18]) appropriate renormalized variables. This wave describes or with a varying Atwood number, that emerges when the self-similar evolution of the probability distribution the mixing proceeds over a non-uniform background as function (PDF) of turbulent fluctuations from small to arXiv:1802.05021v2 [physics.flu-dyn] 13 Sep 2018 in thermally stratified atmosphere [19–21]. large scales in a rough background given by the initial In this work we address a question with both funda- singular profile [22, 23]. Such a description would then mental and applied importance: what happens when the naturally explain the universality of the ML evolution initial unstable profile is more general than the usual RT and its spontaneously stochastic behavior in the inertial step-function, as in the case of non-differentiable power- range [26]. We introduce a shell model for the RT evolu- law density profile. As a result, the mixing layer will tion to illustrate and quantify the ML statistical proper- ties. Results for Navier–Stokes Equations. We con- Postprint version of the manuscript Phys. Rev. Fluids 3, sider the Boussinesq approximation for an incompress- 092601(R) (2018). ible velocity field u(r; t) coupled to the temperature field 2 2 1.0 =2 100 γ .6 ) =1 0.5 t γ Σ( .67 =2 z 0 0 2 4 6 γ /L ) t (t t0(c))/t ( 1 − ∗ h 10− 25 . =0 c =0 0.25 c c = − FIG. 1. (Color online). Snapshots of the vertical section of 2 10− the temperature field T for three simulations of RT turbulence 0.5 1 5 10 with power law initial condition (1) with c = −0:25 (left), t/t c = 0 (center) and c = 0:25 (right) at three different times ∗ corresponding to the same mixing length h(t) ' 0:4Lz. High FIG. 2. Temporal evolution of the mixing layer h(t). From left (low) temperature is represented by yellow (blue). to right: c = −0:25 (green triangles), c = 0 (red squares) and c = 0:25 (blue circles). The three lines represent the power law predicted by the formula (5) with γ = 2=(1−c). Inset: Ef- T (r; t) by a buoyancy term: ficiency of kinetic energy production Σ = −(dE=dt)=(dP=dt) as a function of time for the three cases c = −0:25 (green tri- 2 @tu + u ru = rp + ν u βgT; (2) angles), c = 0 (red squares) and c = 0:25 (blue circles). The · − 2 r − time axis is shifted by a time t0 which depends on c defining @tT + u rT = κ T; (3) · r the onset of the self-similar growth. The bars indicate the where g = (0; 0; g) is the gravity acceleration, ν typical amplitude of fluctuations around the mean value in the plateau region. and κ are the kinematic− viscosity and thermal diffusiv- ity respectively. The choice to rely on Navier-Stokes- Boussinesq equations for studies of RT at high Reynolds determined on dimensional grounds [31–33] from (1) and numbers is very widespread, and it is justified by the ob- (2) in the form servation that the turbulent Mach number has an upper 2 c bound [27], thus making RT turbulence an effectively in- u(t) =h(t) βgθ0(h(t)=L) ; (4) compressible (or low compressible) phenomenon. On the ' other hand, it is known that, when detectable, compress- where u(t) is a large-scale velocity. Assuming that u ibility effects amount mainly to break the up-down sym- dh=dt, one ends with ' metry of the mixing layer growth, with ’spikes’ (down- 2 1+c ward falling temperature fluctuations) being on average t 1−c t 1−c faster than ’bubbles’ (upward rising); nevertheless, such h(t) L ; u(t) U ; (5) ' t∗ ' t∗ asymmetry is limited to the prefactor, while the scaling in time of the full mixing layer width, which is our main where U = L=t∗ and t∗ was defined above. Notice that interest here, is preserved [28, 29]. the first expression can be reinterpreted as a standard The initial condition for the velocity at position r = RT diffusion (x; y; z) is u(r; 0) = 0, while for the temperature field 2 T (r; 0) = T (z) we consider a generic power-law distri- h(t) = αc c(t)gt (6) 0 A bution given by (1). The only inviscid parameter that c 1=(1−c) 2 c=(1−c) relates spatial and temporal scales is ξ = βgθ0=L which where c(t) = (βθ0) gt =L is the time de- 1−c 2 A has physical dimensions of length =time . Thus, for pendent Atwood number and the pre-factor αc represents a given length L, the corresponding integral temporal the generalization of the standard RT α coefficient [34]. −1=2 (1−c)=2 p scale is given by t∗ = ξ L = L=(βgθ0). In order to test the above predictions, we performed di- The distribution (1) is unstable and the dimensional rect numerical simulations (DNS) of the system of equa- argument provides the inviscid growth exponent λ tions (2–3) in a periodic domain of size Lx Ly Lz 1=2 (1−c)=2 ' × × ξ k for the modes with wavenumber k, where with Ly = Lx and Lz = 4Lx by means of a fully parallel the dimensionless proportionality coefficient can be de- pseudo-spectral code at resolution 512 512 2048 for × × termined by solving the linear stability problem [30]. initial conditions (1) with different c and L = Lz. For all This dispersion relation predicts that the instability is runs we have βg = 1=2, θ0 = 1 and Pr = ν/κ = 1.
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