Self-similar interfacial mixing with variable acceleration

Turbulence is often considered as the only remaining problem of classical physics.^1–3 Its complexity and similarity fascinate scientists and mathematicians, demand the attention of engineers and practitioners, and inspire philosophers and poets.^1–3 Isotropy, homogeneity, and locality are the fundamental hypotheses which advanced our understanding of turbulent dynamics.^2,4 Still, the problem sustains the efforts applied, since realistic processes often depart from idealistic scenarios.⁵ Self-similar Rayleigh–Taylor (RT) mixing is a generic problem we inevitably encounter when expanding our knowledge of turbulence beyond canonical considerations.⁵ 

Rayleigh–Taylor mixing is an intensive interfacial mixing that develops when fluids of different densities are accelerated against their density gradients.^6–8 It occurs in nature and technology, at astrophysical and molecular scales, and under conditions of high and low energy densities.⁸ Supernovas, fusion, oil recovery, and nano-fabrication are examples of processes, to which RT mixing is critically relevant.⁵ In many instances, RT dynamics is driven by variable accelerations, whereas the bulk of existing studies considered only constant and impulsive (e.g., shock-driven) accelerations. The former is referred to as the classical RT instability (RTI), while the latter is referred to as the classical Richtmyer–Meshkov instability (RMI).^6–10 Our work analyzes symmetries and invariants of RT mixing driven by accelerations with power-law time dependence. We discover a special class of self-similar solutions and identify their scaling, correlations, and spectra. We find that dynamics of RT mixing can vary from superballistics to sub-diffusion depending on the exponent of the acceleration's power-law and retains memory of deterministic conditions for any exponent's value.

RT flows with variable acceleration are common to occur in fluids, plasmas, and materials.^5,8 These include: RTI quenching ignition in inertial confinement fusion; blast wave driven RT mixing in a supernova enabling conditions for synthesis of heavy mass elements; RT unstable plasma irregularities in the Earth's ionosphere resulting in climate change on regional scales; RT mixing of water and oil limiting efficiency of oil recovery; and RTI governing materials transformation under the impact in nano-fabrication.^5,11–14 In these vastly different physical regimes, RT flows exhibit some similar features of the evolution.⁸ RTI starts to develop when the fluid interface (or the flow fields) is (are) slightly perturbed near the equilibrium state.^6,7 The flow transitions from an initial stage, where the perturbation amplitude grows quickly, to a nonlinear stage, where the growth-rate slows and the interface is transformed to a composition of a large-scale coherent structure and small-scale shear driven vortical structures. The final stage is intensive interfacial mixing, whose dynamics is believed to be self-similar.^6–10,15

RT mixing is a heterogeneous anisotropic process with non-local interactions among the many scales. It is an extreme challenge to study in its direct manifestations.^5,8,15 In experiments, the transient character and sensitivity of large scale dynamics to small scales and deterministic conditions impose tight requirements on the flow implementation, diagnostics, and control.^16–21 In simulations, the needs to accurately track unstable interfaces and capture small-scale processes, the singular nature of numerical solutions and their dependence on unresolved scales demand the use of highly accurate numerical methods and massive computations.^22–24 In theory, we have to elaborate new approaches to study non-equilibrium, multi-scale, nonlinear, and non-local dynamics, identify universal properties of asymptotic solutions, capture symmetries of RT flows, and account for their noisiness.^8,15,25–33

Despite these challenges, significant success has been recently achieved in the understanding of RTI and RT mixing.^8,15–33 Particularly, it has been found that RT mixing with constant acceleration may exhibit order and has stronger correlation, weaker fluctuations, and steeper spectra when compared to canonical turbulence.^{1–4,8,15–19,32} Inspired by this success, we explore here an aspect of RT problem, which is critically important in realistic environments: the effect of variable acceleration on symmetries and invariants of RT mixing, and on its scaling, correlations, and spectra.^5,8,34–36 We apply group theory^{4,8,25,32–36} to keep the analysis from being too formal and too empirical, elaborate the concept of the modified rate of loss of specific momentum, discover the special class of self-similar solutions, and identify properties of RT mixing with variable acceleration that have not been discussed before. We find that in a broad range of acceleration parameters, the dynamics of RT mixing can vary from superballistics to sub-diffusion, and yet, it can retain memory of deterministic conditions. Our results explain existing experiments,^{15–20,32,36,37,49,50} broaden horizons of studies of RT-relevant processes in nature and technology, and outline conditions at which canonical turbulence might be realized in RT mixing with variable acceleration.

Rayleigh–Taylor and Richtmyer–Meshkov instabilities and RT/RM interfacial mixing are subject of active research. Examples of recent studies include the effects of viscosity and compressibility on Rayleigh–Taylor instability, the dynamic stabilization of RTI in miscible liquids, RTI in convergent geometry, the simulations of Richtmyer–Meshkov dynamics with re-shocks and in complex geometry, the studies of strong-shock driven RMI by particle methods, numerical modeling of RTI and RT interfacial mixing by using coupled Cahn–Hilliard/Navier–Stokes models and by applying large eddy simulation, Richtmyer–Meshkov instability in convergent geometry with magnetohydrodynamics, and the numerical modeling of RTI at kinetic and continuous scales with the interface tracking.^{22,23,31,32,56–65}

There is a strong need for a systematic study of RTI/RMI and RT/RM mixing in realistic conditions in order to provide benchmarks for the experiments and simulations and to better understand a broad range of processes in nature and technology to which RT/RM dynamics are relevant. The novelties of our work are in systematic rigorous study of self-similar RT/RM dynamics driven by the acceleration varying as power-law with time, in the identification of fundamental properties of RT/RM mixing in a broad parameter regime, and in the elaboration of new extensive benchmarks for future research.

RT dynamics is governed by the conservation of mass, momentum, and energy.^4,6–8 In continuous approximation, for ideal fluids, the governing equations are

where $x=(x1,x2,x3)=(x,y,z)$ are the spatial coordinates, $t$ is the time, $(ρ,v,P,E)$ are the fields of density, velocity, pressure, and energy density $E=ρ(e+v2/2)$⁠, and $e$ is the specific internal energy. Gravity $g=(0,0,−g),g=|g|$ is directed along the $z$ axis from the heavy to the light fluid.⁴ These nonlinear partial differential equations are augmented with the boundary value problem represented by a sub-set of nonlinear partial differential equations at a nonlinear freely evolving unstable interface and by the conditions at the outside boundaries of the domain.^4,8 They are further augmented with the initial value problem, including initial perturbations of the flow fields at the interface and in the bulk. The boundary value problem is influenced by singularities and secondary instabilities; the initial value problem is ill-posed.^{6–8,25–35} In spatially extended systems, the flow can be periodic in the $(x,y)$ plane normal to the $z$ direction of gravity and is free of mass sources.^8,25 For non-ideal fluids, governing equations are further modified; in the presence of kinematic viscosity $ν$⁠, the momentum equation in Eq. (1) is augmented with the term $(−ρν∂2vi/∂xj2)$⁠.⁴ 

Since the time of the first hypotheses on existence of self-similar RT mixing and first attempts to observe it in experiments and simulations, tremendous efforts have been undertaken to understand RT dynamics.^8,15–30 Significant success has been achieved by linear and weakly nonlinear theories in quantifying the growth of RTI with constant and impulsive accelerations for various initial conditions, by interpolation models in the describing observational data of late time RT dynamics with nearly same sets of parameters, and by turbulence models in the estimating turbulence effect on RT mixing with the use of canonical scaling laws.^{6–8,15,25–33,39–45}

It is worth noting that while RT dynamics is a complex process with many coupled scales, it has some features of universality and order at early and at late times.^{6–8,15–25} Hence, it can be studied from the first principles by applying group theory.^8,15,25 For linear and nonlinear RTI with constant and impulsive accelerations, group theory employs theory of space groups; its key results are symmetries of RT flows, multi-scale character of nonlinear dynamics (to which the coherent structure wavelength and amplitude both contribute), and tendency to keep isotropy in the plane normal to gravity.^8,15,25 For RT mixing, group theory is implemented in the momentum model having the same scaling symmetries as the governing equations; its key result—RT mixing with constant acceleration may exhibit order—self-consistently explains a broad set of experiments.^{8,15–20,25,32,33,36}

While the theory of RT dynamics illustrates what a real complexity is, in realistic environment, the problem can be even more challenging, since RTI and RT mixing is often driven by variable acceleration.^5,11–15 Here, we consider accelerations that are power-law functions of time. On the side of fundamentals, power laws can lead to special scaling properties of the dynamics.^4,8,38 On the side of applications, they can be used to adjust the acceleration's time-dependence in observations.^11–15 Group theory analysis of the boundary value problem of linear and nonlinear RTI with variable acceleration is given in Refs. 33 and 34. Here, we focus on the analysis of invariants, scaling laws, correlations, and fluctuations of RT mixing by implementing group theory in the momentum model and by further applying Lie groups and scaling analysis.

The dynamics of a parcel of fluid undergoing RT mixing is governed by a balance per unit mass of the rate of momentum gain, $μ$⁠, and the rate of momentum loss, $μ̃$⁠, as

where $h$ is the length scale along the acceleration $g$⁠, $v$ is the corresponding velocity, and $μ̃ (μ)$ is the magnitude of the rate of gain (loss) of specific momentum in the acceleration direction.^{1–4,8,15,29–33,35,36} The rate of gain (loss) of specific momentum $μ̃ (μ)$ is related to the rate of gain (loss) of specific energy $ε̃ (ε)$ as $μ̃=ε̃/v (μ=ε/v)$⁠, similarly to the link between the power and force.^4,8,29,30 The rate of energy gain is $ε̃=Bgv$⁠, and the rate of energy dissipation (loss) is $ε=Cv3/L$⁠, where $B$ is the buoyancy, $C$ is the drag, and $L$ is the length scale for energy dissipation. While the buoyancy and the drag can depend on the density ratio and the other parameters and can also be random, for the purposes of this work we consider constant $B>0$ and $C>0$⁠, rescale $B g→g$ and consider drag as free parameter $C∈(0,∞)$⁠. This leads to $μ̃=g$ and $μ=Cv2/L$⁠. We study the dynamics in the domain $h>0, v>0, and t>t0>0$ with some initial time $t0$⁠.

The momentum model has the same symmetries and scaling transformations as the governing equations (1) and (2). Thus, it captures the asymptotic behaviors, and invariant and scaling properties of the dynamics. For certain acceleration patterns, it can be solved by applying the Lie groups. We consider acceleration that is a power-law function of time, $g=Gta$⁠, with exponent $a∈(−∞,+∞)$ and pre-factors $G>0$ with dimensions $dim G=m/s(2+a)$ and $dim a=1$⁠. We refer to the dynamics as to RT type when it is driven by the acceleration, and as to RM type when it is driven by the initial growth-rate and the acceleration's effect is negligible, and we mark these types sub-scripts RT and RM.^{6–10,33–36}

In RT/RM flows, two macroscopic length scales can contribute to the dynamics: the wavelength $λ$ in the normal plane and the amplitude $h$ in the direction of gravity $g$⁠, or the horizontal and the vertical scales.^{8,15,25,28–30} The former is set by some deterministic conditions, e.g., by the wavelength of the initial perturbation or the mode of fastest growth.^28,30,33 The latter is regarded as an integral scale; it is believed to grow self-similarly in the mixing regime.^8,15,32–34 The scale for energy dissipation can be horizontal $L∼λ$⁠, or vertical $L∼h$⁠, or be a combination of scales, $L∼L(h,λ)$⁠. The case $L∼λ∼const$ corresponds to linear and nonlinear RT/RM dynamics, and the case scale $L∼h$ to RT/RM self-similar mixing.^8,15,29,30 Note that the linear, nonlinear, and mixing RT/RM dynamics may have their own values $C$ due to their distinct symmetries. The acceleration $G$ and initial growth-rate $v0$⁠, $v0=|v(t0)|$⁠, define natural time-scales $τG=(λ/G)1/(a+2)$ and $τ0=(λ/v0)$⁠. The initial time is $t0≫{τG,τ0}$ to preempt the effect of time origin on the dynamics.

For the linear and nonlinear dynamics $L=λ$ with $a>−2$⁠, the smallest timescale is $τG≪τ0$⁠, and the dynamics is RT type. In Eq. (2), in the early time regime, the asymptotic solution is $vRT=(λ/τG)(t0/τG)a((t−t0)/τG)$ with $|v̇|∼|μ̃|, |μ/μ̃|→0$⁠, and in the late-time regime, the asymptotic solution is $vRT=(λ/τG)(t/τG)a/2C−1/2$ with $|μ|∼|μ̃|, |v̇/μ|→0$⁠. For the linear and nonlinear dynamics $L=λ$ with $a<−2$⁠, the smallest timescale is $τ0≪τG$⁠, and the dynamics is RM-type. In Eq. (2), in the early time regime, the asymptotic solution is $vRM=v0−Cv0(t−t0)/τ0$⁠, and in the late-time regime, the asymptotic solution is $vRM=(λ/C)/t$ with $|v̇|∼|μ|,|μ̃/μ|→0$ in either regime. At $a=−2$⁠, the solutions transit from RT-to-RM with the variation of the acceleration strength $G$⁠, in consistency with group theory analysis of the boundary value problem.³³ 

In the mixing regime $L=h$ for $a≥acr$⁠, the dynamics is RT type and the asymptotic solution is $hRT=BRTt2+a$ with $μ̃≠μ, |μ̃|∼|μ|∼|v̇|∼ta$⁠. The solution's exponent is set by the acceleration's exponent as $(2+a)$⁠, and the pre-factor $BRT$ is set by the acceleration parameters and drag.^33,35,36 The solution $hRT$ has the invariant $h/t(a+2)$ due to the scaling symmetry of the governing equations $t→eγt, h→eβγh$ with the parameter $γ$ and constant $β=a+2$ of the transformation. In the mixing regime $L=h$ for $a≤acr$⁠, the dynamics is RM-type and the asymptotic solution is $hRM=BRM t2+acr$ with $|μ̃/μ|→0, |μ|∼|v̇|∼tacr$⁠. The solution's exponent is set by drag, $acr=−2+(1+C)−1$⁠, and the pre-factor $BRM$ is set by the deterministic conditions. The solution $hRM$ has a point symmetry associated with the arbitrariness of choice of the time origin.^33,35,36 At $a∼acr$⁠, RT-to-RM mixing transition occurs with the variation of the acceleration exponent $a$⁠. The critical exponent is in the interval $acr∈(−2,−1)$ with $acr→−1$ for $C→0$ and with $acr→−2$ for $C→∞$⁠. While, in the mixing regime, the RT and RM asymptotic solutions must be coupled as particular and homogeneous solutions of nonlinear equations Eq. (2), they are effectively de-coupled due to their distinct symmetries.^{8,15,33,35,36}

Hence, for $a>acr$⁠, the linear, nonlinear, and mixing dynamics are RT type; for $a<−2$⁠, the linear, nonlinear, and mixing dynamics are RM type; and for $−2<a<acr$⁠, the linear and nonlinear dynamics are RT type and the mixing is RM type.³³ RT mixing may develop for $a>acr$ due to the algebraic imbalance of the rates of momentum $μ̃≠μ$ when the scale for energy dissipation is the vertical scale $L∼h$⁠, and/or when the scale for energy dissipation is the horizontal scale $L∼λ$ increasing as $λ∼Gta+2$⁠. RM mixing may develop for $a<acr$ due to the asymptotic imbalance of the rates of momentum $|μ̃/μ|→0$ when the scale for energy dissipation is the vertical scale $L∼h$⁠, and/or when the scale for energy dissipation is the horizontal scale $L∼λ$ increasing as $λ∼t(acr+2)+(acr−a)$ for $−2<a<acr$ and as $λ∼t(acr+2)$ for $a<−2$⁠. The growth of the horizontal scale is possible, and it is a not necessary condition for self-similar mixing to occur, in agreement with group theory analysis of RT/RM structures and transitions.^{8,15,25,33–36}

In RT mixing with $a>acr$⁠, the length changes with time as $L∼ta+2$⁠, and the velocity as $v∼ta+1$⁠. The length scale increases with time for any $a>acr$⁠, whereas the velocity scale increases with time for $a>−1$⁠, is time-independent at $a=−1$⁠, and decreases with time for $a<−1$⁠. By comparing with scaling laws for diffusion $L∼t1/2$ and ballistics $L∼t2$⁠, and by marking similar/quicker/slower processes with prefixes quasi/super/sub, we find that for RT mixing with $a>acr$⁠, the dynamics $L∼ta+2$ is superballistic for $a>0$⁠, ballistic at $a=0$⁠, superdiffusive for $a>−3/2$⁠, quasi-diffusive at $a=−3/2$⁠, and sub-diffusive for $a∈(acr,−3/2)$⁠.^2,4,33 We call the value $a=−1$ as the “steady flex” point because at $a=−1$⁠, the mixing dynamics is steady, and because $a=−1$ is an inflection point. At the steady flex point $a=−1$⁠, a transition occurs from the dynamics with larger velocities at larger length scales to the dynamics with larger velocities at smaller length scales. In RM mixing with $a<acr$⁠, the length scale increases with time $L∼t(acr+2)$⁠, the velocity scale decreases with time $v∼t(acr+1)$⁠, larger velocities correspond to smaller length scales, and the dynamics is sub-diffusive for $C>1$ (Fig. 1).³³ Canonical turbulence is superdiffusive, $L∼t3/2$⁠, with larger velocities at larger length scales.^1–4,33

RT mixing and canonical turbulence have symmetries in statistical sense. For isotropic and homogeneous turbulence, these are Galilean transformations, translations in time, and translations, rotations, and reflections in space.^2,4 In the governing equations, incompressible turbulent dynamics is invariant with respect to scaling transformation $L→LK$⁠, $v→vKn$⁠, and $t→tK1−n$⁠. In the limit of vanishing viscosity, $ν/vL→0$ with $ν→νK1+n$⁠, the scaling exponent is $n=1/3$⁠, and the invariance quantity of the transformation is the rate of energy dissipation, $ε=ν(∂vi/∂xj) 2$⁠, with $ε∼v3/L$ and $ε→εK3n−1$⁠.^1–4 

RT mixing is non-inertial and is invariant with respect to translations, rotations, and reflections in the plane normal to the acceleration $g$⁠. It is invariant with respect to scaling transformation, $L→LK$⁠, $v→vKn$⁠, and $t→tK1−n$⁠. For constant acceleration, $a=0$ and $g=Gt0=g0$ with $g0→g0K2n−1$⁠, in the limit of vanishing viscosity, $ν/vL→0$ with $ν→νK1+n$⁠, the scaling exponent is $n=1/2$ and the invariance quantity of the transformation is the rate of loss of specific momentum in the direction of gravity, $μ$⁠, or $μ∼v2/L$⁠, with $μ→μK2n−1$⁠, where $μi=ν(∂2vi/∂xj2)$ are the components of the rate of momentum loss vector. The scaling exponent $n=1/2$ leads to the invariance of $μ̃, μ$⁠, whereas the rates of energy gain and energy dissipation are time-dependent, $ε̃, ε∼g03t$⁠.^8,15,29,33

For variable acceleration, $g=Gta$⁠, in RT mixing with $a>acr$⁠, the dynamics is invariant with respect to scaling transformation $L→LK$⁠, $v→vKn$⁠, and $t→tK1−n$⁠, conditional that $ν→νK1+n$ and $G→GKn(a+2)−(a+1)$⁠. In this case in the limit of vanishing viscosity, $ν/vL→0$⁠, the rates of loss and gain of energy $ε (ε̃)$ and momentum $μ (μ̃)$ are scale-dependent. The exponent $n=(a+1)/(a+2)$ leads to invariance of $M=μ/ta$⁠, which we refer to as to the modified rate of momentum loss, $M∼v(a+2)/L(a+1)$⁠, and the invariance of the modified rate of momentum gain $M̃=μ̃/ta$⁠, with $M̃=G$⁠. The value $M (M̃)$ can also be viewed as the fractional $a$ th order time-derivative $∂a/∂ta$ of the rate of loss (gain) of specific momentum in the direction of gravity.^8,15,33,46 In RM mixing with $a<acr$⁠, the invariance quantity is $Mcr=μ/tacr$⁠, with $Mcr∼v(acr+2)/L(acr+1)$ and with $acr$ set by drag. The quantity $M̃cr=μ̃/tacr$ is zero asymptotically, with $M̃cr∼ta−acr$ and $M̃cr→0$⁠.³³ 

In RT (RM) mixing with $a>acr (a>acr)$⁠, the invariance of $M (Mcr)$ implies that the modified rate of momentum loss is constant, and there is the momentum transport between the scales.^8,15 In canonical turbulence, the invariance of energy dissipation rate $ε∼v3/L$ is compatible with existence of inertial interval and non-dissipative energy transfer between the scales.^1–4 Enstrophy and helicity are other invariant quantities of canonical turbulence.^1–4 In RT mixing with variable acceleration, these values are scale-dependent. In RT mixing with $a>acr$⁠, the invariance $M∼v(a+2)/L(a+1)$ leads to $ε∼L(2a+1)/(a+2)$ and $ε∼t2a+1$⁠. The rate of energy dissipation increases (decreases) with time for $a>−1/2 (a>−1/2)$ and is constant at $a=−1/2$⁠. The rate of momentum loss, $μ∼La/(a+2)$ and $μ∼ta$⁠, increases (decreases) with time for $a>0 (a<0)$ and is constant at $a=0$⁠. In RM mixing with $a<acr$⁠, rates of energy dissipation and momentum loss decay with time for any drag $C$⁠, as $ε∼t2acr+1, μ∼tacr$⁠.

In RT mixing with $a>acr$⁠, the invariance $M∼v(a+2)/L(a+1)$ leads to $v(a+2)/L(a+1)∼vl(a+2)/l(a+1)$⁠, where $vl$ is a velocity at a length scale $l$⁠. This results in the scaling for velocity $vl/v∼(l/L)(a+1)/(a+2)$ and the $N$ th order structure function $∼(lM)N(a+1)/(a+2)$⁠. In canonical turbulence, the scaling for velocity is $vl/v∼(l/L)1/3$⁠, and for the $N$ th order structure function, it is $∼(lε)N/3$ due to the invariance $ε∼v3/L$⁠. By comparing these scaling exponents, we find that in RT mixing, the velocity correlations are stronger (weaker) for $a>−1/2 (a>−1/2)$ when compared to turbulence and have scaling laws of canonical turbulence at $a=−1/2$⁠. In RM mixing with $a<acr$⁠, the invariance $Mcr∼v(acr+2)/L(acr+1)$ leads to the scaling for velocity $vl/v∼(l/L)(acr+1)/(acr+2)$ and the $N$th order structure function $∼(lMcr)N(acr+1)/(acr+2)$⁠, with larger velocities at smaller length scales (Fig. 1).

In RT mixing $a>acr$⁠, the Reynolds number $Re=vL/ν$ is scale-dependent, $Re∼G2t(2a+3)/ν$⁠. It increases (decreases) with time for $a>−3/2 (a<−3/2)$⁠. By defining the local Reynolds number as $Rel=vll/ν$⁠, we find the scaling for the Reynolds number $Rel/Re∼(l/L)(2a+3)/(a+2)$⁠, the viscous scale $lν∼(ν(a+2)/M)1/(2a+3)$⁠, and the span of scales, $L/lν∼ta+2(M2/ν)(a+2)/(2a+3)$⁠. In RT mixing, the span of scales increases with time, whereas the viscous scale is finite, is set by the viscosity and acceleration, and is comparable with the fastest growing wavelength $∼(ν(a+2)/G)1/(2a+3)$⁠. In RM mixing $a<acr$⁠, the Reynolds number scaling is $Rel/Re=(l/L)(2acr+3)/(acr+2)$⁠, the viscous scale is $lν∼(ν(acr+2)/Mcr)1/(2acr+3)$⁠, and the span of scales is $L/lν∼tacr+2/lν$⁠. While, in RM mixing, the viscous scale is finite, it is distinct from that at the early times $∼(ν/v0)$⁠. Recall that in canonical turbulence $Rel/Re∼(l/L)4/3$⁠, $lν∼(ν3/ε)1/4$⁠, and $L/lν∼L(ε/ν3)1/4$ (Fig. 1).^{1–4,8,15,33}

Fluctuations are essential for complex and turbulent systems. In turbulence, their strength is expected to exceed the noise from deterministic conditions. In canonical turbulence, the invariance of energy dissipation rate $ε∼v3/L$ leads to the diffusion scaling law for velocity fluctuations $v∼t1/2$⁠.^4,33 In RT mixing with $a>acr$⁠, the invariance of the modified rate of momentum $M$ leads to the velocity fluctuations scaling $v∼t(a+1)$⁠. These fluctuations are superdiffusive for $a>−1/2$⁠, quasi-diffusive at $a=−1/2$⁠, and sub-diffusive for $a∈(acr,−1/2)$⁠. For RM mixing with $a<acr$⁠, the invariance $Mcr$ leads to the velocity fluctuations scaling $v∼t(acr+1)$⁠.

To compare the strengths of velocity fluctuations produced by the process and by deterministic (e.g., initial) conditions, consider two parcels of fluids entrained in the dynamics with a time-delay $τ̃$⁠. Canonical turbulence is a stochastic process with self-generated fluctuations, and fluctuations produced by the turbulence $∼(εvτ̃)1/3$ are substantially stronger than the parcels' relative velocity $∼(ετ̃)1/2$ caused by deterministic (initial) conditions.⁴ In RT (RM) type mixing with $a>acr (a<acr)$⁠, the fluctuations produced by the process $∼Mτ̃(a+1) (Mcrτ̃(acr+1))$ are comparable to the parcels' relative velocity caused by the deterministic initial conditions $∼Mτ̃(a+1) (Mcrτ̃(acr+1))$⁠. Moreover, the ratio of the fluctuating and the mean velocities is $∼(τ̃/t)(a+1) ((τ̃/t)(acr+1))$⁠. Hence, in RT and RM mixing, fluctuations are sensitive to deterministic conditions.^8,15,32,33 In RT mixing with $a>acr$⁠, their strength decays with time for $a>−1$⁠, is time-independent at $a=−1$⁠, and increases with time for $a∈(acr,−1)$⁠. In RM mixing with $a<acr$⁠, the strength of deterministic fluctuations increases with time.

In statistically steady canonical turbulence, the invariance of energy dissipation rate leads to a kinetic energy spectrum $E(k)∼ε2/3k−5/3 (E(ω)∼εω−2)$⁠, where $E(k) (E(ω))$ is the spectral density and $k (ω)$ is the wavevector (frequency).^2,4 In RT/RM mixing, an accurate determination of spectra is a challenge because the dynamics is statistically unsteady.^8,15,29,39 In RT mixing with $a>acr$⁠, the spectra employing the invariance of the modified rate of momentum $M$ suggests the spectral density in the form $E(k)∼M2/(a+2)k−(3a+4)/(a+2) (E(ω)∼M2ω−(2a+3))$⁠. The spectrum is steeper (more gradual) than the Kolmogorov's spectrum for $a>−1/2 (acr<a<−1/2)$ and has the exponent $−5/3 (−2)$ at $a=−1/2$ (Fig. 1). For $acr<a<−1$⁠, scaling exponent in $E(k) (E(ω))$ may change sign because for $a<−1$⁠, larger velocities correspond to smaller scales, in contrast to superdiffusive turbulence (Fig. 1). In RM mixing with $a<acr$ and $E(k)∼Mcr2/(acr+2)k−(3acr+4)/(acr+2) (E(ω)∼Mcr2ω−(2acr+3))$⁠, the scaling exponent sign is consistent with sub-diffusive character of RM mixing (Fig. 1).

For $a>0$⁠, RT mixing is superballistic and has strong correlations. For $a→∞$⁠, the invariance quantity is $M∼v/L$⁠, the velocity and the Reynolds number scale as $vl/v∼(l/L)$ and $Rel/Re∼(l/L)2$⁠, and the spectral density is $E(k)∼k−3$⁠. At $a=0$⁠, the invariance is $M=μ∼v2/L$⁠, the velocity and Reynolds number scale as $vl/v∼(l/L)1/2$ and $Rel/Re∼(l/L)3/2$⁠, respectively, and the spectral density is $E(k)∼k−2$⁠. At $a=0$⁠, dynamics of RT mixing is ballistic, with strong correlations and weak fluctuations whose strength is set by deterministic conditions and decays with time $∼t−1$⁠. For $a<0$⁠, RT mixing is sub-ballistic; for $−1<a<0$⁠, it has larger velocities at larger length scales. For $a>−1/2$⁠, RT mixing is super-Kolmogorov. Its correlations are stronger, and spectra are steeper than those in canonical turbulence, suggesting that strongly accelerated RT mixing may laminarize (Fig. 1).^{8,15,29,47,48}

At $a=−1/2$⁠, RT mixing is quasi-Kolmogorov because it has the same scaling properties as canonical turbulence, including the invariance of energy dissipation rate $ε∼v3/L$ and the velocity and Reynolds number scaling laws $vl/v∼(l/L)1/3$ and $Rel/Re∼(l/L)4/3$⁠, respectively, and the spectral density $E(k)∼k−5/3$⁠.⁴ Yet, in RT mixing with $a=−1/2$⁠, fluctuations are sensitive to deterministic conditions, and their strength decays with time $∼t−1/2$⁠, whereas in canonical turbulence, the fluctuations are independent of the deterministic conditions. For $a∈(−1,−1/2)$⁠, correlations are weaker and spectra are more gradual in RT mixing than in canonical turbulence (Fig. 1).

At $a=−1$⁠, RT mixing is steady, $L∼t$⁠, its velocities are scale-independent $vl∼v$⁠, and the Reynolds number scales as $Rel/Re=(l/L)$⁠. At this steady flex point, velocity fluctuations are set by deterministic conditions, their strength is constant, and spectral density is $E(k)∼k−1$⁠. For $a<−1$⁠, RT mixing has larger velocities at smaller scales; for $a∈(−3/2,−1)$⁠, it is superdiffusive (Fig. 1).

At $a=−3/2$⁠, RT mixing is quasi-diffusive, with a diffusion scaling law for the length $L∼t1/2$ and velocity $vl/v∼(l/L)−1$ and with the scale-invariant Reynolds number $Rel∼Re$⁠. At $a=−3/2$⁠, the fluctuations are set by deterministic conditions, and their strength increases with time $∼t1/2$⁠, leading to spectral density $E(k)∼k$⁠. The sign of spectral exponent is consistent with the quasi-diffusive character of the dynamics having larger velocities at smaller length scales. For $a∈(acr,−3/2)$⁠, RT mixing is sub-diffusive. The effect of deterministic conditions is stronger for smaller values of the acceleration exponents (Fig. 1).

At $a∼acr$ and $acr=−2+(1+C)−1$⁠, the mixing becomes RM mixing. For $a∈(−∞, acr)$⁠, RM mixing is defined by the drag and the deterministic conditions. As a sub-diffusive process, RM mixing has intense motions at small scales with larger velocities at smaller length scales. The velocity and the Reynolds number scale as $vl/v∼(l/L)−C$ and $Rel/Re∼(l/L)1−C$⁠, respectively, the strength of deterministic fluctuations increases with time $∼t(C+1)/C$⁠, and the spectral density is $E(k)∼k2C−1$ (Fig. 1).

Our results explain and agree with existing observations and serve to better understand RT relevant phenomena in nature and technology.

Our theory finds that in RT-type mixing with acceleration exponents $a>−1/2$⁠, the correlations are stronger and spectra are steeper than in canonical turbulence. This suggests that strongly accelerated RT mixing can laminarize. This result is consistent with classical experiments in accelerated flows, including flows in curved pipes and accelerated boundary layers.^48,49 Experiments^48,49 find that turbulent flows can, indeed, laminarize when they are sharply accelerated. In our analysis, sharp accelerations correspond to large acceleration exponents, thus being consistent with the observations.^48,49

Our theory finds that RT mixing dynamics can vary from superballistic to sub-diffusive depending on the acceleration and, yet, can be sensitive to deterministic conditions for any acceleration. This result is in excellent agreement with experiments.^16,17,29 The experiments are conducted in gas–gas and gas–liquid system and apply strong and weak shocks, and the compressed air and gas mixture detonation to strongly accelerate the fluids and achieve self-similar regimes with Reynolds numbers $Re∼3.2×106$ in a broad range of setups and conditions.^15–17,29 To control initial perturbations in the gas–liquid system, experiments apply the jelly technique, by using jellies of small concentration of aqueous gelatin solutions with zero plasticity and weak strength behaving as incompressible fluids under strong pressure.^15–17,29 The experiments unambiguously find that in a broad parameter regime (including periodic and localized perturbations of the interface and the flow fields, convergent and spatially extended geometries, and steady and variable accelerations), RT mixing is interfacial, heterogeneous, and anisotropic; it retains memory of deterministic conditions and can keep order.^15–17,29 Our analysis achieves excellent agreement and explain these experiments.

Our theory self-consistently explains experiments and simulations of RT mixing with constant acceleration, which found spectra steeper than Kolmogorov.^20,49,50 It also explains the flattening of the initially steep spectra in simulations of RM mixing.²⁴ Our results can be further applied to study the interplay of turbulence and acceleration. According to our theory, RT mixing is quasi-Kolmogorov at the acceleration exponent $a=−1/2$ since it has the same scaling laws, correlations, and spectra as canonical turbulence.⁴ May this imply that variable acceleration $g∼t−1/2$ may produce and maintain turbulence? Our theory suggests the following: Yes, it may, conditional there is also external source supplying turbulent energy to the system at a constant rate,^1–4 and the flow is being homogenized.²⁹ While the acceleration may quickly set proper scaling laws, high Reynolds numbers and large spans of scales, the source of turbulent energy, and the absence of interfaces are needed to keep fluctuations strong and stochastic and the flow isotropic and homogenous.^1–4 

Our analysis finds significant departures of the invariant, scaling, spectral, and correlation properties of self-similar RT mixing from those of canonical turbulence. In realistic turbulent processes, departures from Kolmogorov's scaling laws are found in isotropic homogeneous turbulence, turbulent boundary layers, passive scalar mixing, buoyancy-driven turbulence, turbulent convection, and compressible turbulence.^3,50–55 These departures are referred to as to “anomalous scaling.” Our analysis identifies the special class of self-similar solutions whose dynamics can vary from superballistics to sub-diffusion depending on a single dimensionless parameter. It suggests that the anomalous scaling can serve to systematically incorporate conditions of realistic environments and to better understand complex and turbulent processes in nature and technology.^3,50–55

Supernovas are a central problem in astrophysics due to their role in stellar evolution and nuclear synthesis.¹² A supernova's explosion is driven by a blast-wave causing intensive RT/RM mixing of materials of a progenitor star and enabling conditions for synthesis of heavy mass elements.^12,33 Blast-wave induced acceleration is a power-law function of time, since blast waves are described by self-similar solutions, whose invariant $F, [F]=m/sθ, 0<θ<1$⁠, is a function on energy release, density, and pressure.³⁸ For the power-law solution, the length scale is $L∼tθ$ and acceleration is $∼tθ−2$⁠.^33,38 By comparing exponents $(θ−2)$ and $acr$⁠, we find that blast-wave-driven mixing can be RT mixing with $a∼acr$ for $C<(θ−1−1)$ and be RM mixing with $a<acr$ for $C>(θ−1−1)$⁠. This mixing is sensitive to deterministic conditions and has larger velocities at smaller length scales, thus explaining anisotropy and richness of structures in supernova remnants. Our theory further suggests that in such dynamics a superdiffusive turbulence is a challenge to implement. Mechanisms other than turbulence can enable transport and accumulation of energy at small scales needed for nuclear synthesis. These can include energy localizations and trapping, which are typical for sub-diffusive processes, in excellent agreement with observations of small-scale non-uniform structures in RM flows.³³ 

The conditions of high energy density in supernovas can be achieved at high power laser facilities.^18,19,36,37 Our analysis is in excellent agreement with experiments designed to model blast-wave-driven RT mixing in laboratory plasmas.^18,19,36,37 These experiments find that RT mixing has high degree of coherence. Its late-time dynamics is sensitive to the deterministic (initial) conditions, is dominated by the vertical scale, and also exhibits some small-scale structures. In the experiments, the acceleration exponent and the Reynolds number are $a∼−1,Re∼106$⁠.^18,36 Our analysis achieves excellent agreement with the observed properties of the dynamics, including tendencies of RT mixing to keep order, retain memory of deterministic conditions, be dominated by the vertical scale, and exhibit intense small-scale structures due to the scale-invariance of velocity fluctuations at $a=−1$ (Fig. 1).^18,36 Our results can be further applied for better control fluid instabilities in high energy density plasmas, including inertial confinement fusion.^13,37

Rayleigh–Taylor mixing is the generic problem we inevitably encounter when expanding our knowledge of turbulence and when studying a broad range of processes in nature and technology, from supernovas and fusion to the fossil fuel industry and water flowing from an overturned cup.⁵ In this work, we analyzed the effect of variable acceleration on self-similar RT mixing and on its scaling, correlations, and spectra.

In full consistency with classical approaches,⁴ we applied group theory and scaling analysis to study properties of RT dynamics with power-law time-dependent acceleration. We elaborated the concept of invariance of the modified rate of loss of momentum, discovered the special class of self-similar solutions, and identified properties of RT mixing not discussed before. We found that dynamics of self-similar RT mixing can vary from superballistics to sub-diffusion depending on the acceleration's exponent and can retain memory of the deterministic conditions for any exponent. For superballistic, ballistic, and super-Kolmogorov dynamics, RT mixing has strong correlations and weak fluctuations, which can lead to laminarization and order. For quasi-Kolmogorov dynamics, in addition to proper scaling laws set by the acceleration, a source of turbulent energy and the flow homogenization are needed for keeping fluctuations strong and stochastic and the dynamics isotropic and uniform. For quasi- and sub-diffusive dynamics in RT/RM mixing, larger velocities correspond to smaller length scales. The fluctuations are more sensitive to the deterministic conditions for smaller acceleration exponents.

Our results achieve excellent agreement with available experiments and simulations in fluids and plasmas.^{15–20,37,47–50} These include the following: experiments in fluids detected laminarization of strongly accelerated turbulent flows; experiments in plasmas observed high degree of coherence and dominance of the vertical scale in RT mixing; experiments in fluids unambiguously found strong sensitivity of RT mixing to deterministic conditions; experiments and simulations observed anomalous spectral properties of RT/RM mixing. New experiments can be designed to quantify the interplay of acceleration, turbulent noise and deterministic conditions in RT mixing. For instance, one may employ narrow band initial conditions with various symmetries and study the effect of interference of the initial perturbation waves on the order and disorder in RT/RM mixing.^25,31 One can also employ a broadband incoherent perturbation with initially steep (gradual) spectra and observe the evolution of spectral properties of RT/RM mixing.^{20,24,49–55} One can further vary acceleration parameters and observe and quantify the mixing dynamics in ballistic, quasi-Kolmogorov, steady flex, superdiffusive, and sub-diffusion regimes.

Our results serve for better understanding and control of RT/RM relevant phenomena in nature and technology. For instance, for supernovas, our analysis explains the richness of structures observed in supernovas remnants and reveals the mechanism of energy localization at small scales required for nuclear synthesis.^12,33 For fusion, strong sensitivity of RT mixing to deterministic conditions suggests the opportunities of control of plasma flows by means of variable acceleration and deterministic conditions not explored before.^13,37 For turbulent flows, our analysis is fully consistent with experiments, simulations, and theory of turbulent flows and indicates that it is a highly formidable task to create canonical turbulence in a laboratory.^{1–3,50–55} For fundamentals, the existence of special class of self-similar dynamics is directly linked to the fractional analysis, dynamical systems, and boundary value problems,^{4,8,15,25,33,47} to be studied in the future. To conclude, the problem of water flowing from an overturned cup remains a source of inspiration for researchers in science, mathematics, and engineering and is well open for a curious mind.

S.I.A. was involved in conceptualization, methods, investigations, and writing.

The author thanks the University of Western Australia (AUS) and the National Science Foundation (USA).

The data that support the findings of this study are available within the article.