Scalings and simulation requirements in two-phase flows

TL;DR

This paper addresses the challenge of determining grid-point and time-step requirements for direct numerical simulations of two-phase turbulence, linking resolution to key non-dimensional groups $Re$, $We$, and $Ca$. It derives scalings for interface-to-turbulence length scales, introducing Kolmogorov-Hinze and Kolmogorov-viscous scales and the regime-diagnostic Ris, to predict computational cost across inertia- and viscous-dominated flows. The authors verify their guidelines with high-fidelity HIT simulations using a diffuse-interface method, showing that $k_{\max}\eta_{KH}\approx 60$ adequately resolves interfacial area, size distributions, SMD, and curvature, while capturing intermittency-driven extreme events. The work yields practical a priori estimates of mesh and time-step requirements, and highlights the limitations of current resources in the viscous-dominated regime, offering a path toward accelerated physics discovery and model development for two-phase flows.

Abstract

In this work, important two-phase flow scalings are derived, which enable the quantification of grid-point and time-step requirements as functions of Re, We, and Ca numbers. The adequate grid resolution is determined in the inertia-dominated regime with the aid of high-fidelity simulations of stationary two-phase homogeneous isotropic turbulence by evaluating convergence of total interfacial area, size distribution, SMD, and curvature distribution. Although standards for DNS for single-phase turbulence flow exist, there is a lack of similar guidance in two-phase flows. Therefore, length scale ratios of the Kolmogorov-Hinze to the Kolmogorov scale of η_{KH}/η\sim We_{L}^{-3/5}Re_{L}^{3/4} in the inertia-dominated regime and the Kolmogorov-viscous to Kolmogorov scale of η_{KV}/η\sim Ca_{L}^{-1}Re_{L}^{3/4} for the viscous-dominated regime, are constructed. These scalings imply a computational cost increase like We_{L}^{12/5} and Ca_{L}^4, in the inertia-dominated and viscous-dominated regimes, respectively. A novel dimensionless number, coined as the ratio of interface scales (Ris), is proposed to aid in the classification of the turbulence regimes in the presence of an interface. Convergence of the total interfacial area, size distribution, SMD, and curvature distribution are observed for grid resolutions of k_{\max} η_{KH} \geq 60$ for second-order schemes. Furthermore, it is observed that this lower bound is the minimum required to capture intermittent events responsible for the increase of instantaneous total interfacial area. This criterion will be a valuable tool for determining grid resolution and time-step requirements a-priori for DNS of two-phase flows and for estimating the corresponding computational cost. This work provides guidelines and best practices for numerical simulations of two-phase flows, which will accelerate physics discovery and model development.

Figures (11)

• Figure 1: Scales of interest in single-phase (a), two-phase inertia-dominated (b), and two-phase viscous-dominated turbulence (c). Reference length scales highlighted by black arrows. In the inertia-dominated regime (b), super-Kolmogorov-Hinze scale is denoted by the blue blob and the sub-Kolmogorov-Hinze by the orange sphere. In the viscous-dominated regime (c), the orange ellipsoid represents the Kolmogorov-viscous scale deformed by viscous stresses below the Kolmogorov scale.
• Figure 2: Regime demarcations of interface scales in turbulence. (a) Inertia-dominated, (b) viscous-dominated, (c) unified regimes. Regimes of interest are highlighted in the two shades of blue, defining an interface in the inertial range, $\eta \ll \eta_{KH} \ll \mathcal{L}$, and, in the two shades of orange, defining a sub-Kolmogorov scale interface, $\eta_{KH} \ll \eta$. Reference scalings for order of unity ratios in dashed blue line for $\eta_{KH}/\eta \sim 1$, in dash-dot orange line for $\eta_{KV}/\eta \sim 1$, and in dotted black line for $\eta_{KH}/\eta_{KV} \sim 1$.
• Figure 3: Interfacial area evolution as a function of grid resolution. (a) Cases A, $Re_\lambda = 55$, $We_\mathcal{L} = 6.5$, (b) cases B, $Re_\lambda = 87$, $We_\mathcal{L} = 60$. Transient state is highlighted for (b) cases B on the inset for $t/\tau_e \in [0, 4]$.
• Figure 4: Average interfacial area at steady state comparison for two pairs of dimensionless number as a function of grid resolution. One standard deviation band based on finest resolution added for reference.
• Figure 5: Three-dimensional visualizations of instantaneous volume fraction contours, $\phi = 0.5$, over slices of enstrophy, $\Omega = \omega_i \omega_i$, at $t/\tau_e = 10$ for different grid resolutions. Cases A (top row) and B (bottom row). Solid blue line rectangles indicate the expected converged resolutions for both cases.