As in any physical process [17], a dynamics of an ideal fluid is governed by the conservation of mass, momentum and energy represented in continuous approximation in an inertial frame of reference as

Here the spatial coordinates and time are (x_i, t) = (x, y, z, t); the fields of density, velocity, pressure and energy density are (ρ, v, P, E), with E = ρ(e+v²/2) and the specific internal energy e. The closure equation of state related the internal energy and pressure, with constant (P/ρe). In the presence of kinematic viscosity ν the momentum equation is augmented with the term (-ρν∂2vi/∂xj2) and the energy equation is also modified [10–12, 17].

For canonical Kolmogorov turbulence, the density field is uniform, ρ = ρ₀, the dynamics is the density independent, and the process is driven by an external source supplying energy at a constant rate per unit mass E₀. This specific power E₀, with the dimension m²s^-3, is the control parameter of the self-similar, isotropic and homogeneous turbulence [17–19, 40–46].

For Rayleigh-Taylor dynamics, the equations in the bulk are augmented with the boundary conditions at the interface and at the outside boundaries, so that the normal (tangential) component of velocity and pressure (enthalpy) are continuous (discontinuous) at the interface, and there are no external sources.

Here the jump of a quantity at the interface is […]; the normal and tangential unit vectors of the interface are n, τ with n = ∇θ/|∇θ|, (n·τ) = 0, and the function θ(x, y, z, t) is θ = 0 at the interface and is θ>0 (< 0) in the light (heavy) fluid sub-domain. The initial conditions prescribe the perturbations of the interface and the flow fields at some instance of time. The dynamics is specific and is driven by balance per unit mass, as follows from the independence of the boundary condition for the normal velocity from the fluid density.

Rayleigh-Taylor dynamics is driven by the acceleration g, g = (0, 0, −g), g = |g|. It is due to a body force, is directed from the heavy to the light fluid, and modifies the pressure field as P→P−ρgz. For constant acceleration, g = g₀, in the mixing regime the length scale in the acceleration direction (i.e., the amplitude of the interface perturbation) increases quadratic with time h~ g0t2. The acceleration strength g₀, with the dimension ms^-2, is the control parameter of the self-similar, anisotropic and heterogeneous Rayleigh-Taylor mixing [8–12, 23, 25, 61–63].

Symmetries of isotropic homogeneous turbulence include Galilean transformations, translations in space and time and spatial rotations and reflections. Self-similar canonical turbulence is also invariant under the scaling transformation of the length, L→LK, velocity v→vKⁿ, and time, t→tK¹⁻ⁿ, where n is an exponent and K>0 is a constant. In the governing equations in the limit of vanishing viscosity, ν/vL → 0, conditional on ν → νK¹⁺ⁿ, the exponent of the scaling transformation is n = 1/3. Its invariant form is the rate of energy dissipation, ε=ν(∂vi/∂xj),2, with ε~v³/L and ε → εK³ⁿ⁻¹. The energy dissipation rate and the energy power are similar quantities, ε~ϵ₀ [17–19, 40–47]:

Symmetries of non-inertial RT mixing include translations, rotations and reflections in the plane normal to the acceleration. Self-similar Rayleigh-Taylor mixing is also invariant with respect to the scaling transformation, L→LK, v→vKⁿ, , conditional on g0→g0K2n-1. In the governing equations in the limit of vanishing viscosity, ν/vL → 0, with ν → νK¹⁺ⁿ, the exponent of the scaling transformation is n = 1/2. Its invariant form is (the component of) the rate of loss of specific momentum μ in the acceleration direction, with μ~ v²/L and μ → μK²ⁿ⁻¹, where the vector of the rate of momentum loss is μi=ν(∂2vi/∂xj2). The rate of momentum loss and the acceleration strength are similar quantities, μ~ g₀. In RT mixing the rate of energy dissipation is scale-dependent, with ε~ μ2t~ g02t at time t and ε~ μ3/2L1/2~ g03/2L1/2 at length L [8–12, 48, 63]:

Distinct symmetries and invariant forms lead to substantial departures of properties of self-similar Rayleigh-Taylor mixing from those of Kolmogorov turbulence, including their scaling laws, spectral shapes, correlations and fluctuations [8–12, 48, 63].

The properties of Kolmogorov turbulence and Rayleigh-Taylor mixing are identified by the classical approaches and by the group theory approach, respectively, through analyzing symmetries and scaling transformations. We need to clarify whether the group theory approach and the classical approaches are consistent with one another, whether the distinctions in properties of Kolmogorov turbulence and Rayleigh-Taylor mixing are fully captured by their invariant forms, and whether the characteristics of these processes depend on their control dimensional parameters [8–12, 15, 17–19, 48].

Consider the velocity scaling law, with v (v_l) being the velocity scale at the length scale L (l).

In canonical turbulence, the energy dissipation rate is ε~ v³/L with υ ~ (εL)^1/3 at the length scale L, and it is εl~ vl3/l with v_l ~ (εll)1/3 at the length scale l. The invariance of the rate of energy dissipation, ε = ε_l with ε~ ϵ₀, leads to the velocity scaling law vl3/v3~ l/L [17–19, 41, 46, 47].

In Rayleigh-Taylor mixing, the rate of energy dissipation is scale-dependent, with ε~ v³/L~ μ^3/2L^1/2 and with εl~ vl3/l~ μl3/2l1/2, where the rate of momentum loss is μ~ v²/L at the length scales L and it is μl~ vl2/l at the length scale l. The rate of momentum loss is an invariant quantity, μ = μ_l with μ = g₀, leading to the velocity scaling law vl2/v2~ (μl/μ)(l/L)~ l/L [10–12, 48].

We directly link the velocity scaling laws in canonical turbulence and Rayleigh-Taylor mixing as:

This reveals that the velocity scaling laws in Rayleigh-Taylor mixing and in Kolmogorov turbulence are consistent with each other. Due to their distinct invariant forms—μ and ε – the velocity correlations are stronger in Rayleigh-Taylor mixing than in Kolmogorov turbulence [10–12, 23, 48].

Consider the Reynolds number scaling and the viscous scale [17–19].

The Reynolds number is Re = vL/ν at the length scale L, and the Reynolds number is Re_l = v_ll/ν at the length scale l. Since v~ (εL)^1/3 and vl~ (εll)1/3, we obtain Re~ ε^1/3L^4/3/ν and Rel~ εl1/3l4/3/ν. For the viscous length scale l~ l_ν, the local Reynolds number is Re_l~ 1 [17–19, 47].

In canonical turbulence, the invariance of the energy dissipation rate, ε = ε_l with ε~ ϵ₀, leads to the scaling law for the Reynolds number Rel /Re~ (l/L)4/3 and determines the viscous scale lν~ (ν3/εl)1/4~ (ν3/ε)1/4~ (ν3/ϵ0)1/4 [17–19, 47].

In Rayleigh-Taylor mixing, with account for the scale-dependence of the energy dissipation rates ε~ μ^3/2L^1/2 and εl~ μl3/2l1/2, we obtain Re ~ μ^1/2L^3/2/v and Rel~ μl1/2l3/2/ν. The invariance of the rate of momentum loss, μ = μ_l with μ~ g₀, leads to the Reynolds number scaling Rel /Re~ (l/L)3/2 and the viscous scale lν~ (ν2/μl)1/3~ (ν2/μ)1/3~ (ν2/g0)1/3 [10–12, 23, 48].

We directly link these quantities in canonical turbulence and in Rayleigh-Taylor mixing as:

This reveals that the Reynolds number scaling and the viscous scale in Rayleigh-Taylor mixing are consistent with those in Kolmogorov turbulence. Due to their distinct invariant forms—μ and ε , respectively—the Reynolds number scaling is steeper in Rayleigh-Taylor mixing than in canonical turbulence, whereas the viscous scale is set by the acceleration g₀ in Rayleigh-Taylor mixing and by the energy power ϵ₀ in turbulence [10–12, 23, 48].

Consider the spectral shape for fluctuations of the velocity (the specific kinetic energy) [17–19].

In canonical turbulence the spectral density of the velocity fluctuations is E(k). It is defined by the invariance of the energy dissipation rate ε and its independence of the wavevector k, leading to the exponent −5/3 of the k spectrum, with E(k)~ ε2/3k-5/3~ ϵ02/3k-5/3 [17–19, 47].

In RT mixing, the energy dissipation rate depends on the wavevector, ε~ μ^3/2k^−1/2. We obtain:

This demonstrates that the spectral shapes in Rayleigh-Taylor mixing and in Kolmogorov turbulence are consistent with one another. In Rayleigh-Taylor mixing the velocity fluctuations spectra are steeper than in canonical turbulence, due to the distinct invariant forms of these processes, μ and ε respectively [10–12, 23, 48].

In two-dimensional isotropic homogeneous turbulence, with account for invariance properties of the enstrophy Ω , with the dimension s^-2, and the rate of enstrophy Ω~, with the dimension s^-3, the spectral density for the kinetic energy fluctuations v² has the form E(k)~ Ω~2/3k-3 [64, 65]. In Rayleigh-Taylor mixing the enstrophy Ω and the rate of enstrophy Ω~ depend on the wavevector as Ω~ μk and Ω~~ (μk)3/2. We derive

and find that the spectral shapes in Rayleigh-Taylor mixing and in two dimensional turbulence are consistent with one another. In Rayleigh-Taylor mixing the velocity fluctuations spectra are more gradual than in two-dimensional turbulence, due to distinct invariant forms of these processes, μ and Ω~, respectively [10–12, 23, 48, 64–66].

Consider the spectral shape for the density fluctuations [10, 17, 53].

In canonical turbulence, the spectral shape of the density fluctuations is E(k)~ ρ0ε0k-1, since the energy dissipation rate ε is independent of the fluid density ρ₀. In Rayleigh-Taylor mixing, the independence of the rate of momentum loss on the fluid density ρ₀ leads to the spectral shape E(k)~ ρ0μ0k-1 [10, 53].

We obtain

For the density fluctuations, the exponent −1 of the k spectrum is the same in the anisotropic and heterogeneous Rayleigh-Taylor mixing and in the isotropic and homogeneous Kolmogorov turbulence. In either case the dynamics is specific and is balanced per unit mass (rather than per unit volume), as displayed in the independence of the invariant forms of these processes—μ or ε —on the fluid density ρ₀ [10, 53].

We further illustrate in step-by-step derivations that our results on Rayleigh-Taylor mixing are consistent with other models of realistic turbulent processes and with some empirical models [53, 67–75].

In modeling realistic turbulent processes, the spectral density E(k) is often related to the energy dissipation rate ε , the wavevector k, and the process time scale τ as ε~ τk⁴E² [73].

In canonical turbulence the time-scale is τ ~ (k³E)−1/2, leading to ε~ (k³E)−1/2k⁴E²~ k^5/2E^3/2 and, due to the invariance of the energy dissipation rate ε , to the spectral density E(k)~ ε^2/3k^−5/3. In RT mixing, the time-scale is τ~ (g0k)-1/2, leading to ε~ (g0k)-1/2k4E2~ g0-1/2k7/2E2 and the spectral density E~ ε1/2g01/4k-7/4. By further accounting for the scale-dependence of the energy dissipation rate ε~ g03/2k-1/2~ μ 3/2k-1/2, we obtain the spectral density in Rayleigh-Taylor mixing with constant acceleration as E~ g0k-2~ μk-2, similarly to the foregoing [73, 74]:

By considering the Rayleigh-Taylor time scale τ~ (g0k)-1/2 and by formally treating the energy dissipation rate ε , the empirical model [74] identifies the spectral density as E~ k^−7/4. We derive this result from the spectral density E~ μk⁻² in Rayleigh-Taylor mixing, by accounting for the scale dependence of the energy dissipation rate ε~ μ^3/2k^−1/2, the invariant form of the rate of momentum loss μ~ g₀ and the time-scale τ~ (g0k)-1/2 as:

The phenomenological model [75] postulates that in Rayleigh-Taylor mixing the spectral density is the same as in canonical turbulence E~ k^−5/3. We reproduce this prospect from the spectral density defined by the invariant form of Rayleigh-Taylor mixing E~ μk⁻², with μ~ g₀, and with relations of the rates of momentum loss and energy dissipation as μ~ ε^2/3k^1/3:

The model further states that in Rayleigh-Taylor mixing the viscous scale vanishes with time [75]. For testing this statement, we consider the local Reynolds number set by the invariant form of Rayleigh-Taylor mixing, Rel~ μl1/2l3/2/ν, with μ_l~ μ~ g₀, relate the rates of momentum loss and energy dissipation, μl~ εl2/3l-1/3, and derive:

The model's result can be further reproduced with a formal replacement εl ~ ε and substitution ε~ g02t.

The results of empirical models of Rayleigh-Taylor mixing can be obtained within group theory approach by formally treating the energy dissipation rate and by masking its scale-dependence [53, 74, 75].

To conclude this section, we consider the velocity structure function, S_n, of the order n, n∈N, with the dimension (ms^-1)n [17–19]. It scales with the length scale l as Sn~ (εll)n/3 in Kolmogorov turbulence, and as Sn~ (μll)n/2 in Rayleigh-Taylor mixing [10–12, 17–19, 23, 48]. Since in Rayleigh-Taylor mixing the energy dissipation rate is scale-dependent, εl~ μl3/2l1/2, we obtain

The structure functions in Rayleigh-Taylor mixing and in Kolmogorov turbulence are consistent with one another and are set by their respective invariant forms. In Rayleigh-Taylor mixing the structure function has a steeper dependence on the order number when compared to Kolmogorov turbulence [10–12, 17–19, 23, 48].

In isotropic homogeneous turbulence in realistic environments the structure function is known to depart from the Kolmogorov scenario: It exhibits intermittency and multi-fractality mathematically, is influenced by the flow structures physically, and has remarkable statistics [17, 66, 76–79]. We believe that the approaches developed for canonical turbulence [66, 76, 78] can be generalized to the case of Rayleigh-Taylor mixing with variable accelerations [8–11], to be done in the future.