Uncertainty Quantification for Multiphase Computational Fluid Dynamics Closure Relations with a Physics-Informed Bayesian Approach

Abstract

Multiphase Computational Fluid Dynamics (MCFD) based on the two-fluid model is considered a promising tool to model complex two-phase flow systems. MCFD simulation can predict local flow features without resolving interfacial information. As a result, the MCFD solver relies on closure relations to describe the interaction between the two phases. Those empirical or semi-mechanistic closure relations constitute a major source of uncertainty for MCFD predictions.

In this paper, we leverage a physics-informed uncertainty quantification (UQ) approach to inversely quantify the closure relations’ model form uncertainty in a physically consistent manner. This proposed approach considers the model form uncertainty terms as stochastic fields that are additive to the closure relation outputs. Combining dimensionality reduction and Gaussian processes, the posterior distribution of the stochastic fields can be effectively quantified within the Bayesian framework with the support of experimental measurements. As this UQ approach is fully integrated into the MCFD solving process, the physical constraints of the system can be naturally preserved in the UQ results. In a case study of adiabatic bubbly flow, we demonstrate that this UQ approach can quantify the model form uncertainty of the MCFD interfacial force closure relations, thus effectively improving the simulation results with relatively sparse data support.

Keywords:

• Two-fluid model
• uncertainty quantification
• Kalman filtering
• physics-informed machine learning

Nomenclature

${\mathrm{C}}_d$	=	= drag coefficient
${\mathrm{C}}_l$	=	= lift coefficient
${\mathrm{C}}_{vm}$	=	= virtual mass coefficient
${\mathrm{C}}_{wl}$	=	= wall lubrication coefficient
$D_S$	=	= Sauter mean diameter (m)
$\mathit{g}$	=	= gravity vector (m/s2)
$h$	=	= specific enthalpy (J/kg)
$\mathit{M}$	=	= volumetric interfacial force (N/m3)
$\mathrm{P}{\mathrm{r}}^t$	=	= turbulent Prandtl number
$p$	=	= pressure (Pa)
$T$	=	= temperature (K)
$t$	=	= time (s)
$\mathit{V}$	=	= velocity (m/s)
Greek	=
$\alpha$	=	= void fraction
${\mathrm{\Gamma }}_{ki}$	=	= evaporation/condensation rate per volume [kg/(m3‧s)]
${\sigma }^t$	=	= turbulent dispersion coefficient
$\lambda$	=	= thermal conductivity [W/(m‧K)]
$\mu$	=	= dynamic viscosity (Pa‧s)
$\rho$	=	= density (kg/m3)
$\tau$	=	= stress tensor [kg/(m‧s2)]
Subscripts	=
$g$	=	= gas phase
$r$	=	= relative motion
$l$	=	= liquid phase
Superscript	=
$t$	=	= turbulence

Acknowledgments

This material is based upon work supported by Laboratory Directed Research and Development funding from Argonne National Laboratory, provided by the Director, Office of Science, of the U.S. Department of Energy under contract number DEAC02-06CH11357.

Disclosure Statement

No potential conflict of interest was reported by the author(s).