2. OBSERVED ENERGY BALANCE AND INTEGRAL LENGTH SCALE

In the

limit, experiments and numerical simulations show that the average volume fraction profiles of the fluids in the TMZs are practically linear functions of the longitudinal coordinate, x (Read & Youngs, 1983; Burrows et al. 1984; Read, 1984; and the numerous collected references in Dimonte & Schneider, 1997; Llor, 2001a). Using the mass conservation equations, the average longitudinal fluid velocities in the TMZ are also found to be linear in x, and their difference is constant throughout the TMZ, given by δU_x(t) = (d/dt)L(t)/2 (Llor, 2001a).

These volume fraction and velocity profiles, together with a TMZ growth law, directly provide three important terms of the overall energy balance: K_I(t), the total input energy (converted from gravitational energy (RT) or mean transverse kinetic energy (KH)), K_M(t), the mean kinetic energy, and K_D(t), here called “directed” kinetic energy. K_M is calculated from the one-fluid (Favre) mean longitudinal velocity, and K_D is the excess of mean kinetic energy introduced when going from a one-fluid to a two-fluid average field description:

(+ and −: fluid indices, α^±, ρ^±, U_x^±: volume fractions, densities and longitudinal velocities of fluids, ρ: Reynolds average density, U_x: Favre average velocity, and 〈.〉: bulk average). For

, K_D is simply related to the TMZ growth law by K_D ≈ (δU_x)²/12 = (dL/dt)²/48 whereas K_M is vanishingly small as

(Llor, 2001a) and will be ignored here. The growth laws and the resulting K_I and K_D /K_I are given in Table 1 (Llor, 2001a).

Basic self-similar TMZ growth laws

The available data (Read & Youngs, 1983; Burrows et al. 1984; Read, 1984, and references in Dimonte & Schneider, 1997; Llor, 2001a) from experiments and numerical simulations can be summarized by two basic bulk parameters, as collected in Table 2: the coefficient in L(t),

for RT, IRM, or KH, respectively, and the turbulent kinetic energy, K. Using the results of Table 1, K_D is found to be negligible compared to K in the IRM and KH cases, whereas it is of similar order in the RT case. This reflects a striking difference of the energy distribution that motivates the need, already pointed out by various authors (Dimonte, 2000; Read, 1984; Youngs, 1989), of two-fluid modeling in the RT case instead of one fluid for IRM and KH. In the RT case, it also justifies distinguishing the (single-fluid) turbulent kinetic energy, as measured in experiments, from the two-fluid turbulent kinetic energy, defined as K_T = K − K_D. Furthermore, since K_D is associated to purely longitudinal velocity components, it has a major contribution to the overall Reynolds tensor anisotropy.

Bulk-averaged parameters of self-similar RT, IRM, and KH TMZs, as obtained from experiments and numerical simulations (estimated to ±20% from various sources), the modified k–ε model, and Youngs' two-fluid model

From the self-similar evolution of K_I − K_D − K_T, one can determine the bulk-averaged turbulent energy transfer to intermediate scales in the TMZs, E (also called spectral flux), or the closely related nondimensional parameter:

Λ_i is an average integral length scale (the typical size of the largest eddies) and κ_T thus stands as an effective Knudsen number of the turbulent transport (Tennekes & Lumley, 1972), previously called Von Kármán number in Llor, 2001b. E is characteristic of large scales and, for unsteady flows as considered here, it is close but not equal to the (small scale) energy dissipation rate. However, expression (2) assumes the existence of an inertial range to relate Λ_i and E, as is indeed observed even for the RT case (Dalziel et al., 1999). The effective Knudsen numbers found for the three TMZs, reported in Table 2, differ by a factor of up to 7 that reflects striking structural differences: if, as commonly accepted, the KH TMZ accommodates typically one large eddy in its width, the IRM TMZ accommodates two and the RT seven.

3. TEST OF ELEMENTARY MODELS ON BASIC BULK QUANTITIES

The energy balance is of utmost importance in testing the physical relevance and accuracy of models. As examples, two elementary Reynolds-Averaged Navier–Stokes (RANS)-based statistical turbulent mixing models have been examined: a modified k–ε model (Andronov et al., 1979; Gauthier & Bonnet, 1990; Bonnet et al., 1992), and a two-fluid turbulent model introduced by Youngs (1989). Here again, bulk averaging has been applied to simplify the model PDEs into ODEs (Llor, 2001a), as listed in Table 3, to reveal the basic phenomena that are captured. Self-similar solutions to the ODEs are then given by algebraic equations, in analytical form for the two models considered here (Llor, 2001a). With the recommended values of the coefficients, the modeled bulk-averaged quantities were obtained as listed in Table 2 (Llor, 2001a). The corresponding initial conditions, which must be compatible with the self-similarity constraint, are thus given by the listed solutions taken at a given finite time.

Bulk-averaged equations of the modified k–ε and Youngs' two-fluid models for RT type mixing in the incompressible, limits (ζ ≈ 3/2: bulk correction factor). In the KH case, not explicitly given here, Π becomes the Reynolds stress production, Cε0 is replaced by Cε1, and an equation in Λ must also be added to Youngs' model (Llor, 2001a)

Most deviations between modeled and measured results fall in a ±30% range and can be attributed to experimental and numerical errors (for instance, K in the IRM case is estimated from a single numerical study), bulk-averaging bias (these are not expected to exceed ±10%; Llor, 2001a), and limited “universality” of the models (k–ε is typically optimized to capture boundary layers). Within this margin, reasonable estimates of all the basic quantities in the IRM and KH cases are obtained from both models, but significant discrepancies are observed in the RT case:

is twice too low in the modified k–ε, and κ_T is overestimated by 15- and 7-fold factors, respectively, with the k–ε and two-fluid models. Particularly striking is the κ_T value for the k–ε that would indicate typical eddies' sizes over twice larger that the TMZ width!

For the modified k–ε, these inconsistencies result from the purely diffusive modeling of the TMZ growth as visible on the L(t) equation in Table 3. The IRM and KH cases are therefore properly captured, but for the RT case, the directed transport is replaced by an unphysically enhanced turbulent diffusion, obtained by increasing Λ_i (or κ_T): After substitution with the measured RT quantities in Table 2, the L(t) equation is incompatible with any reasonable values of the standard k–ε model constants C_μ and σ_c. The analytical results also show that no combination of σ_ρ and C_ε0 values can make the model to capture at once

, κ_T, and K/K_I (Llor, 2001a).

As in any two-fluid approach, the TMZ growth in Youngs' model is simply driven by directed transport as shown by the L(t) equation in Table 3. The model specificity stands in the two evolution equations for quantities k_T and λ (besides the usual four equations for mass and momentum, with (δU)² drag force, turbulent diffusion, and added-mass effects). λ is used to provide a closure of the characteristic length scales for both turbulence and drag between the fluids: λ_i = C_i λ/C_μ and λ_d = λ/C_d, respectively (notations k_T, λ, C_d, and C_i, respectively, stand here for k, L, c₁, and c₂ in Youngs publications). The production term in the λ equation dominates the other terms and is proportional to δU_x in the RT and IRM cases. This means that λ, as δU_x, is almost uniform across the TMZ and its bulk average, Λ = 〈λ〉, behaves as L. Simple calculations (Llor, 2001a) show that, in any gravitationally induced instability such as RT and IRM, Λ = L/2, hence, κ_T = C_i /(2C_μ) ≈ 0.58, and in the KH case, this value is corrected by a factor close to 1 which depends only on C_d. Here again, the sevenfold span of experimental κ_T values observed between the RT and KH cases cannot be captured.

The strong overestimation of κ_T can generate major modeling errors whenever changes in turbulent structure or viscosity become essential, as in the case of combined RT and KH. However, in the RT case, the κ_T distortions of the k–ε and two-fluid models play different roles in connection with the modeled TMZ growth processes: In the former, transport proceeds by enhanced turbulent diffusion, thus forcing large κ_T values, whereas in the latter, directed transport dominates, and the κ_T value is incidental.

4. VARIABLE ACCELERATION TO EXPLORE DIRECTED VERSUS TURBULENT EFFECTS

Regardless of turbulence structure, that is, of κ_T, the competition between the two transport processes becomes more visible in self-similar variable acceleration RT (SSVARTs) flows, although experimental or numerical data are not yet available. They are defined by

(n = 0 is standard RT). Such flows were previously considered for n > −1 only by Neuvazhaev (1983) who gave their TMZ growth rates as captured by his specific model. The extension for n values below −2 is dictated by the unsuitable decreasing behavior of L(t) as tⁿ⁺² for n < −2 if g ∝ tⁿ. The TMZ would then experience an unphysical complete demixing down to L = 0 under a positive, instability-inducing gravity field. Although experimentally very demanding or even out of reach, SSVARTs at nonnegligible values of n can be investigated through numerical simulations. It will only be assumed here that SSVARTs do exist a priori as Reynolds (ensemble) averages, yielding

in the incompressible,

limits, where

is positive.

For a SSVART to be stable, it must eventually “forget,” for late t, any added perturbation on its initial state at time t₀ taken as origin. A necessary condition is provided by K_I(t), which behaves as a(± t)²ⁿ⁺² + b(± t)⁻⁽ⁿ⁺²⁾: The first term is the expected self-similar law produced by the time integration of gδU, whereas the second is due to the dilution as 1/L of any initial deviation of K_I(t₀). a and b being positive, the contribution of K_I(t₀) is thus forgotten when t → ∞ or −t → 0 unless −2 < n < −4/3. This result does not take into account the spectral transfer and dissipation of K_I(t₀). Therefore, a critical value n_c exists between −2 and −4/3 such that, for −2 < n < n_c, the behavior of SSVARTs is dominated by K_I(t₀), and L(t) grows necessarily as t^n_c+2 at late times. This is exactly the situation of IRM mixing; hence the result n_c = n₀ − 2.

The bulk-averaged k–ε and two-fluid models in Table 3 estimate

as listed in Table 4 (Llor, 2001a). Also given are the results from scaled free fall and Read's formula (Read & Youngs, 1983; Burrows et al. 1984; Read, 1984), which are the simplest approximations of any two-fluid description such as Youngs' in Table 3: the former by assuming Π = 0 and scaling g into

(thus

), and the latter (explicitly solved as

) by assuming that the directed transport dominates diffusion, the production and destruction of K_D are balanced, and the ratio Λ_D /L is independent of g(t) (Llor, 2001a). Unscaled free fall (if

) provides an upper bound to TMZ growth, and Read's formula has been found to retrieve the TMZ growth to within 2% on the presently available variable acceleration experiments (Burrows et al., 1984; Dimonte, 2000).

Analytical expressions of , as obtained from the bulk-averaged models (Table 3) and their approximations

The k–ε modeled

displays two poles at n₁ = n₀^kε − 2 = −(4C_ε2 − 3)/(3C_ε2 − 3) ≈ −1.7 and n₂ = −(4C_ε0 − 3)/(3C_ε0 − 3) ≈ 0.89.

is thus negative outside the −1.7 to 0.89 interval, but the range of physically acceptable n values is certainly much wider. Again, the model is unable to capture the directed transport effects that are enhanced when increasing n.

can be made positive for all n < −2 and n > n₁ by setting C_ε0 = 3/2, a value reassuringly close to C_ε1 that causes turbulence to be produced at the integral length scale. However, it is then found that

for the recommended value of σ_ρ = 2, or that

for σ_ρ ≈ 0.017, both extremes being physically unacceptable.

The two-fluid modeled

also displays two simple poles, at n₁^Y = n₀^Y − 2 and n₂^Y = −2, being positive outside the poles, over the widest range of physically acceptable n values. Scaled free fall and Read's approximation again display two simple poles: n₂ is always preserved at −2, but n₁ is respectively increased to −4/3 and decreased to −2.

The plot of

in Figure 1, as scaled by (1 − n/n₁^Y)⁻¹(1 − n/n₂^Y)⁻¹, shows the existence of three continuously connected growth regimes, loosely defined by:

1. n₀ − 2 < n [lsim ] − 1; directed transport is dominated by turbulent diffusion (minimal for IRM at n = n₀),
2. −1 [lsim ] n < +∞ and −∞ < n [lsim ] −5/2; directed transport is controlled by turbulent viscosity,
3. −5/2 [lsim ] n < −2; directed transport is limited by free fall.

Scaled representation of (Table 4), as obtained from Youngs' two-fluid model (solid line), scaled free fall (dot dashed line), and Read's approximation (dashed line). SSVARTs are not stable in the shaded interval of n.

Single-fluid models, such as the modified k–ε, appear unnatural to describe directed transport and are limited to the first regime. In the second regime, the widest, both scaled free fall and Read's approximations provide surprisingly good accuracies of about ±30% compared to a full two-fluid model, but fail in the first and third, where they either diverge or vanish. They limit the range of acceptable responses because their RM poles take the extreme values −2 and −4/3. Only a full two-fluid model, such as Youngs', is able to capture all three regimes at once in a physically acceptable manner, even though it misses some important features such as dissipation of density fluctuations (Linden & Redondo, 1991; Youngs, 1995; Dalziel et al. 1999), effective Knudsen number (present work).