Deep learning interfacial momentum closures in coarse-mesh CFD two-phase flow simulation using validation data

Highlights

• Interfacial momentum exchange is recovered using deep learning and validation data.

• A data-driven approach is developed to correct coarse-mesh CFD results.

• Local physical features are identified to represent underlying local patterns.

• High similarity leads to better predictive performance of deep learning model.

Abstract

Multiphase flow phenomena have been widely observed in the industrial applications, yet it remains a challenging unsolved problem. Three-dimensional computational fluid dynamics (CFD) approaches resolve of the flow fields on finer spatial and temporal scales, which can complement dedicated experimental study. However, closures must be introduced to reflect the underlying physics in multiphase flow. Among them, the interfacial forces, including drag, lift, turbulent-dispersion and wall-lubrication forces, play an important role in bubble distribution and migration in liquid-vapor two-phase flows. Development of those closures traditionally rely on the experimental data and analytical derivation with simplified assumptions that usually cannot deliver a universal solution across a wide range of flow conditions. In this paper, a data-driven approach, named as feature-similarity measurement (FSM), is developed and applied to improve the simulation capability of two-phase flow with coarse-mesh CFD approach. Interfacial momentum transfer in adiabatic bubbly flow serves as the focus of the present study. Both a mature and a simplified set of interfacial closures are taken as the low-fidelity data. Validation data (including relevant experimental data and validated fine-mesh CFD simulations results) are adopted as high-fidelity data. Qualitative and quantitative analysis are performed in this paper. These reveal that FSM can substantially improve the prediction of the coarse-mesh CFD model, regardless of the choice of interfacial closures. It demonstrates that data-driven methods can aid the multiphase flow modeling by exploring the connections between local physical features and simulation errors.

Introduction

As a carbon-free energy source, nuclear power plants (NPPs) play an important role in the reliable supply of energy, national security, and environmental impact. However, since the 1970s, the number of NPPs under construction in the United States has gradually dropped due to high capital and construction costs and increased safety standards. It is imperative to leverage advanced numerical models and simulations, which can provide advanced suggestions for reactor design and complement dedicated experimental testing, to reduce the design and construction cost of NPPs. Recently, three-dimensional thermal hydraulics methods, in the form of computational fluid dynamics (CFD), have made tremendous advancements. They promise to transform the way in which we approach the design of more efficient and reliable systems. One of the most challenging, yet widely encountered phenomena extant in NPPs is multiphase flow—i.e., liquid and vapor in complex phase interactions of mass, momentum, and energy.

Various modeling approaches have been introduced and applied on multiphase flow based on specific research goals and on-hand computational costs. Among them, on a spectrum toward first-principle methods is direct numerical simulation (DNS) coupled with interface tracking (IT) (Hirt and Nichols, 1981; Sussman et al., 1994; Unverdi and Tryggvason, 1992), where bubble-liquid interfaces are resolved on a sufficiently fine scale. While the DNS/IT methods can be leveraged to study separate effects on single bubble (Bunner and Tryggvason, 2003; Feng and Bolotnov, 2017a; 2017b; 2017c), practical two-phase flow simulations with on the order of hundreds of bubbles are still not computationally affordable (Fang et al., 2018). Compared to the DNS/IT methods, the most general framework for multiphase flow is the Eulerian-Eulerian two-fluid model, which combines Reynolds-averaged Navier-Stokes (RANS)-based models to model turbulence with a variety of interfacial closure laws to account for the exchange of mass, momentum, and energy between individual phases. The Eulerian–Eulerian two-fluid approach assumes that all phases coexist inside each computational cell. For each fluid, the full set of conservation equations is solved; therefore, each fluid has a different velocity field.

Based on ensemble-averaging techniques, the Eulerian-Eulerian two-fluid model can operate on a relatively coarse mesh and is comparatively computationally efficient for industrial applications. It is important to explore the options of adopting coarse-mesh CFD methods for the practical industrial applications.

Using coarse-mesh CFD methods is expected to have two major sources of error: physical-model error and mesh-induced error. Physical-model error arises from physical assumptions and mathematical approximations in the form of closure models and errors in assigning values for model parameters or calibration coefficients. For adiabatic two-phase flow, one of the most-significant approximations is the formulation of interfacial forces. Drag force originates from the balance between buoyancy and gravitational forces. Despite the difficulty in modeling these physical complexities, two phase modeling is not possible wihtout interfacial drag closure. Lateral distribution of the bubble is a combination effect of lift, turbulent-dispersion, and wall-lubrication forces where those three forces are model treatments based on relevant physical observations. Although driven by dedicated experiments (Dijkhuizen et al., 2010; Tomiyama et al., 2002) and analytical derivation (Antal et al., 1991; Lubchenko et al., 2018), the expression of those forces may not be consistent and universal across a wide range of flow conditions. Physical experiments have high construction cost for full-size prototype facilities and scaling issues in using scaled-down facilities. These costs limit the validation and uncertainty quantification of the CFD models. While new techniques are continuously developed and deployed, the availability of an experimental database is still limited by the state-of-the-art measurement techniques. For example, measurements of bubbles size and distribution are not available for reactor operating pressures.

Distinct from physical-model errors, mesh-induced error comes from the solution discretization in space and time and the approximation used in over-cell integration of variables that are non-uniformly distributed in the cell. Ideally, adopting finer mesh would approach the realistic representations of physical phenomena, but the computational cost requires attention. However, when the mesh size is much smaller than the bubble size, the ensemble-averaging assumption for bubbly flow will be no longer hold. Physical-model and mesh-induced error are tightly connected. For example, the calculation of flow variable gradients depends on the mesh resolution between two adjacent cells, thus inaccuracies in gradients from truncation error will augment the physical model errors of a RANS turbulence closure model.

Recently, machine-learning (ML)-based methods have emerged as a valuable approach to aid the development and application of CFD methods. Machine learning provides new avenues for dimensionality reduction and reduced-order modeling in fluid mechanics by providing a concise framework that complements and extends existing methodologies. A comprehensive review of machine learning for fluid mechanics is recently published (Brunton et al., 2020) where the latest progress on flow modeling, optimization, and control of fluids are summarized. Among them, the use of machine learning to aid multiphase flow modeling is barely studied while the development of turbulence closures is a popular research direction. Different ML algorithms, such as neural networks and random forests, have been widely applied to predict relevant parameters or source terms in turbulence-transport equations (Milani et al., 2019; Xiao et al., 2020). These ML applications mainly focus on the improvement of RANS turbulence modeling for the single phase, without considering mesh-induced numerical uncertainties and resultant errors. Turbulence-model error is quantified in different validation domains, but the applicability of these data-driven models needs further demonstration. Another pioneering work that utilizes local physical features from high-fidelity (HF) DNS simulation to predict boiling heat transfer is conducted by Liu et al. (Liu et al., 2018); therein, mesh-induced numerical error is assumed to be similarly small. Aiming at correcting mesh-induced numerical error without considering the model errors, a coarse-gird CFD approach was proposed by Hanna et al. (Hanna et al., 2020) for the prediction of local simulation errors. These existing efforts estimate model error and mesh-induced numerical error separately to achieve a better predictive performance; however, they ignore the tight connection between these two main simulation-error sources, as described in the last paragraph. For two-phase flow using CFD or other system codes, local mesh size is treated as one of the key model parameters, and fine-mesh convergence is somehow not expected. The uncertainty propagation resulting from scaling distortions makes it more difficult to estimate and reduce simulation errors for modeling and simulation of realistic system-level NPP analysis. Machine learning algorithms need to be guided by the local physics to assist the multiphase flow modeling.

In this paper, a recently proposed data-driven approach, feature-similarity measurement (FSM), takes model error, mesh-induced numerical error, and scaling distortions into consideration by treating the physical correlations, coarse mesh sizes, and numerical solvers as an integrated model. This FSM approach uses deep learning to explore the relationship between specific physical feature groups and simulation variables. The well-trained deep-learning model can be considered as a surrogate for governing equations and closure correlations of coarse-mesh CFD. Coarse-mesh cases with both a mature set of interfacial force closures, i.e., bubbly and moderate void fraction (BAMF) (Sugrue et al., 2017), and a simplified set of interfacial-force closures are trained with the neutral network with limited experimental data and HF CFD models. The research presented here paves the path for resolving the challenges of modeling interfacial-force closures with the aid of FSM.