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Interface Fluctuations in a Turbulent Binary Fluid using Data-Driven Methods

Samuel Z Khiangte, Triparna Sanyal, Sumantra Sarkar, Nairita Pal

Abstract

Interfacial fluctuations in a two-phase binary fluid mixture reveal signatures of underlying physical processes that occur within each phase and on a range of spatial and temporal scales. In this study, we investigate a model binary fluid system consisting of a single droplet of one phase moving in the background of the second phase. The binary fluid system is subjected to turbulent forcing. We perform extensive direct numerical simulations of the turbulent system to examine how quantities such as interfacial dynamics and droplet acceleration can be systematically decoded. Extensive simulations of binary fluid systems are computationally expensive and time-consuming. In contrast, data-driven models have shown promise in recent times in reducing computational cost. In this work, we build and compare the performances of four interpretable data-driven models, i.e., dynamic mode decomposition (DMD), Hankel DMD, sparse identification of nonlinear dynamics (SINDy), and Stochastic Langevin regression (SLR), each using dimensionality reduction via proper orthogonal decomposition, to identify simplified dynamical equations governing interfacial dynamics and center-of-mass acceleration. We show how these learned models can be generalized to encode physical properties, such as the interfacial surface tension and droplet size. In particular, we show that SLR predicts the underlying dynamical equations of the binary-fluid system with the greatest accuracy over a wide range of interfacial tension values and droplet sizes. In addition, SLR requires fewer terms compared to SINDy to capture the underlying dynamics, and is thus computationally the most efficient among the four methods. These data-driven techniques can be used in many practical applications, such as the dynamics of biological cell membranes, thin films, and other industrial applications.

Interface Fluctuations in a Turbulent Binary Fluid using Data-Driven Methods

Abstract

Interfacial fluctuations in a two-phase binary fluid mixture reveal signatures of underlying physical processes that occur within each phase and on a range of spatial and temporal scales. In this study, we investigate a model binary fluid system consisting of a single droplet of one phase moving in the background of the second phase. The binary fluid system is subjected to turbulent forcing. We perform extensive direct numerical simulations of the turbulent system to examine how quantities such as interfacial dynamics and droplet acceleration can be systematically decoded. Extensive simulations of binary fluid systems are computationally expensive and time-consuming. In contrast, data-driven models have shown promise in recent times in reducing computational cost. In this work, we build and compare the performances of four interpretable data-driven models, i.e., dynamic mode decomposition (DMD), Hankel DMD, sparse identification of nonlinear dynamics (SINDy), and Stochastic Langevin regression (SLR), each using dimensionality reduction via proper orthogonal decomposition, to identify simplified dynamical equations governing interfacial dynamics and center-of-mass acceleration. We show how these learned models can be generalized to encode physical properties, such as the interfacial surface tension and droplet size. In particular, we show that SLR predicts the underlying dynamical equations of the binary-fluid system with the greatest accuracy over a wide range of interfacial tension values and droplet sizes. In addition, SLR requires fewer terms compared to SINDy to capture the underlying dynamics, and is thus computationally the most efficient among the four methods. These data-driven techniques can be used in many practical applications, such as the dynamics of biological cell membranes, thin films, and other industrial applications.
Paper Structure (29 sections, 52 equations, 15 figures)

This paper contains 29 sections, 52 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Methods
  3. Direct Numerical Simulation
  4. Definition of Variables
  5. Predictive modeling using interpretable data-driven methods
  6. Proper Orthogonal Decomposition to identify reduced-order description:
  7. Effective model discovery for droplet fluctuation:
  8. Effective model discovery for the center-of-mass motion:
  9. Proper Orthogonal Decomposition to identify reduced-order description
  10. Identifying an effective model using dynamic mode decomposition
  11. Identifying effective model using Sparse Identification of Nonlinear Dynamics
  12. Set up of the SINDy model
  13. Identifying the effective model using Stochastic Langevin Regression
  14. RESULTS
  15. Effective DMD model for droplet interface dynamics in turbulent flow
  16. ...and 14 more sections

Figures (15)

  • Figure 1: Extracting measurable droplet properties from DNS: (A) Droplet under the influence of turbulent flow with a surface tension corresponding to $\Lambda=200$. The underlying flow is color-coded by the vorticity field, with red denoting the droplet interface. (B) Definition of height ($h_i$) field at different $\theta_i$ at a given time $t_0$. (C) Deformation parameter calculated from DNS. (D) Acceleration calculated from DNS.
  • Figure 2: Description of workflow
  • Figure 3: Schematic representation of the POD using SVD: (A, B, C) represent the components of the $\chi$, $\Sigma$, and $Q$ matrices in equation \ref{['svd1']}; (A) Singular values from $\Sigma_{rr}$ represent the degree of contribution of each mode to the dynamics; inset shows the mean squared error = $\frac{1}{T}\sum\limits_{i=1}^{T}|(h_i(\text{original}) - h_i(\text{truncated}))|$ when reconstructing the original $H$ matrix using mode truncation vs the number of modes used for reconstruction; (B) Shows the first three highest-energy spatial modes $\chi_r$; (C) Temporal modes $q_r(t)$ for the first three highest-energy modes.
  • Figure 4: Simple example of the implementation of SLR. (A) A representative stochastic trajectory of a one-dimensional particle evolving in a double-well potential. (B) Scatter plots illustrating the estimation of the drift term $f_{\mathrm{KM}}$ and diffusion coefficient $\sigma_{\mathrm{KM}}$ from the stochastic trajectory $x$ within each bin, computed using Eqs. \ref{['fkm']} and \ref{['akm']}. Dashed lines denote the corresponding model inferred via SINDy. (C) Comparison of the trajectory predicted by the inferred model with the testing dataset to assess statistical similarity; the model minimizing the cost function defined in Eq. \ref{['cost_SLR']} is selected.
  • Figure 5: DMD for interface fluctuations: (A) Discrete-Time Eigenvalues of DMD evolution matrix for DMD (red dots) and Hankel DMD (blue dots). $\mu=1$ (point on a circle) signifies purely oscillatory modes, $\mu>1$ signifies growing modes, and $\mu<1$ signifies decaying modes schmid2022dynamic. (B) Prediction of deformation parameter obtained from DMD models (red) and Hankel DMD (blue) compared to DNS (black) for the test dataset.
  • ...and 10 more figures