Validation and uncertainty quantification of multiphase-CFD solvers: A data-driven Bayesian framework supported by high-resolution experiments

Nuclear Engineering and Design 354 (2019) 110200

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Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes

Validation and uncertainty quantification of multiphase-CFD solvers: A data-driven Bayesian framework supported by high-resolution experiments

T

Yang Liua, , Xiaodong Suna, Nam T. Dinhb ⁎

a b

Department of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, MI 48109, USA Department of Nuclear Engineering, North Carolina State University, Raleigh, NC 27606, USA

ABSTRACT

The two-fluid model-based Multiphase Computational Fluid Dynamics (MCFD) solvers are promising tools for a variety of engineering problems related to multiphase flows. Such a solver is a complex model system that consists of multiple closure relations with strong interactions. Furthermore, a typical engineering problem requires evaluation of multiple quantities of interest (QoIs) from the solver predictions. These features make the validation and uncertainty quantification (VUQ) of these MCFD solvers a challenging task. In this paper, we propose an approach for the VUQ of a two-fluid model-based MCFD solver based on a data-driven Bayesian framework. This framework relies on Bayesian inference to inversely quantify the parameter uncertainty and model form uncertainty and then propagate the obtained uncertainties through the solver to obtain the uncertainties of the QoI predictions. To make the Bayesian inference applicable to complicated MCFD simulations, two methods namely, parameter space reduction and surrogate modeling, are included in the proposed approach. Supported by high-resolution local measurements, this framework is able to simultaneously take multiple key QoIs into consideration. In the validation process, the area metric is extended to the Principal Component (PC) subspace so that the multiple correlated QoIs, in the form of spatial or temporal distributions, can be validated in a comprehensive manner. We demonstrate the applicability of the proposed approach with a case study of an adiabatic bubbly flow scenario.

1. Introduction Two-phase flow and boiling heat transfer are used for thermal management in various engineered systems including nuclear reactors. Essentially, two-phase flow and boiling heat transfer are complex processes, which involve various interactions among heated solid surface, liquid, and vapor, including bubble nucleation, evaporation, condensation, and interfacial mass/heat/momentum exchanges. Correspondingly, there are various quantities of interest (QoIs) or figures of merit of a given boiling system, such as the flow pattern, pressure drop, wall heat transfer, void fraction distribution, phasic temperature, and velocity distribution. In nuclear engineering applications, understanding the relevant phenomena and accurately predicting the QoIs involved in two-phase flow and boiling heat transfer are crucial for the system performance and safety of watercooled nuclear reactors. The design and safety analysis of two-phase flow and boiling systems are highly relied on the use of scientific simulation tools. Among various tools developed for simulation of two-phase flow and boiling

systems, the two-fluid model-based Multiphase Computational Fluid Dynamics (MCFD) solvers are regarded as promising tools. The major advantage of the MCFD solvers is that they average the interface information between the vapor and liquid phases and that they rely on closure relations to compensate the information loss in the averaging process, thus significantly reduce the computation requirement while being able to retain, to a certain degree, essential local information of the system. MCFD therefore attracts increasingly interest from the community (Bestion, 2010; D'Auria, 2018), from general bubbly flow and boiling simulation (Shams et al., 2018; Lucas et al., 2016; Krepper and Rzehak, 2011) to boiling crisis prediction (Yadigaroglu, 2014; Mimouni et al., 2016). On the other hand, it should be recognized that fundamental disconnection still exists between performing simulations using MCFD solvers and applying MCFD solvers for industrial applications. The main issue is related to the fact that the uncertainties of the MCFD solvers are not well quantified, especially for the uncertainty introduced by the closure relations. Considering the increasing interest in the MCFD solvers, their

Abbreviations: CDF, cumulative distribution function; CFD, computational fluid dynamics; DNS, direct numerical simulation; GP, Gaussian process; IR, infrared; LDV, laser Doppler velocimetry; MCFD, multiphase computational fluid dynamics; MCMC, Markov chain Monte Carlo; MSE, mean square error; PC, principal component; PIV, particle image velocimetry; PTV, particle tracking velocimetry; QoI, quantity of interest; SA, sensitivity analysis; STH, system thermal hydraulics; UQ, uncertainty quantification; V&V, verification and validation; VUQ, validation and uncertainty quantification ⁎ Corresponding author. E-mail address: [email protected] (Y. Liu). https://doi.org/10.1016/j.nucengdes.2019.110200 Received 13 June 2019; Received in revised form 6 July 2019; Accepted 8 July 2019 Available online 24 July 2019 0029-5493/ © 2019 Elsevier B.V. All rights reserved.

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Nomenclature

Aa Ab Cd Cl C wl Cp C vm Dd DS fd g h M Na nw Pr t p T t U yw y+

Greek symbols

interfacial area concentration, 1/m effective bubble area fraction drag coefficient lift coefficient wall lubrication coefficient specific heat at constant pressure, J/(kg K) virtual mass coefficient bubble departure diameter, m bubble diameter in bulk flow, m bubble departure frequency, 1/s Gravity vector, m/s2 specific enthalpy, J/kg interfacial force, N/m3 nucleation site density, 1/m2 unit vector normal to the wall turbulent Prandtl number pressure, Pa temperature, K time, s velocity, m/s near wall distance, m dimensionless wall distance

ki

t

µ

void fraction Evaporation/condensation rate per volume, kg/(m3 s) contact angle, rad surface tension, kg/s2 turbulent dispersion coefficient thermal conductivity, W/(m K) dynamic viscosity, Pa s kinematic viscosity, m2/s density, kg/m3 stress tensor, kg/(m s2) source term of interfacial area concentration, 1/m3

Subscripts

g r sat sub sup l

gas phase relative motion saturation subcooling superheat liquid phase

Superscript

t

turbulence

to such a problem. Last, high-resolution measurements of two-phase flows still remain a challenging task. For most cases, only limited measurements of the QoIs are available, while the data of intermediate closure relation outputs, such as the interfacial drag and lift, interfacial condensation, bubble-induced turbulence, etc., are missing. To summarize, the ultimate question that VUQ of the MCFD solvers aims to answer is: Given a complex model system that consists of interrelated closure relations, multiple empirical parameters, and multivariate QoIs, how should we perform a comprehensive VUQ for the model system as the best as one can, with the support of limited data sources. In this study, we present an approach that are mainly based on the modular Bayesian inference to address this question. There are several advantages for applying Bayesian inference to the VUQ of MCFD solver. First, the Bayesian inference is capable of simultaneously taking multiple QoIs into consideration, which makes it naturally compatible with the multivariate predictions of the MCFD solvers. Second, the Bayesian inference has strong flexibility with data. It can quantify the uncertainty of the system with limited data sources, while still update and improve the posterior uncertainties of the system if new data sources become available. In this regard, even with limited data, the uncertainties of the most influential parameters towards the solver predictions can be quantified and then the quantified uncertainties of these parameters can be propagated through the solvers to quantify the uncertainties of the QoIs. Furthermore, the modular Bayesian approach demonstrates good performance in quantification of both model parameter uncertainty and model form uncertainty (Liu et al., 2019; Wu et al., 2018). In addition to the modular Bayesian inference, our proposed comprehensive approach also includes methods for parameter dimension reduction, surrogate model construction, quantitative sensitivity analysis, and multivariate validation metrics calculation. In this paper, we first provide a review of VUQ practices on general computational models and discussions on how these practices can be related to MCFD solvers. Then we propose a six-step procedure based on the Bayesian framework that can efficiently utilize the high-resolution local measurements of two-phase flow systems for the VUQ of MCFD solvers. This paper is organized in the following structure: Section 2 provides a brief review of key concepts and major practices on

validation becomes a vital task. Generally speaking, any computational model should be validated before its prediction can be trusted to for real engineering applications. By definition, model validation is a comprehensive process that determines the degree to which a model is an accurate representation of the real world from the perspective of the intended use of the model (Oberkampf and Roy, 2010). Validation, as a general engineering topic, has attracted intense research interest over the past decades. Significant progress has been made in different aspects related to the validation, from high-level validation framework (Roy and Oberkampf, 2011), validation metrics (Ferson and Oberkampf, 2009; Li and Lu, 2018), to various methods and practices for the uncertainty quantification (UQ) of general computational models (Kennedy and O'Hagan, 2001; Higdon et al., 2008; Wu et al., 2018; Smith, 2014). A comprehensive validation practice should cover the following basic items: 1). Identifying major uncertainty sources related to the model; 2). Based on the uncertainty sources, evaluating the uncertainty of the interested model predictions, i.e., QoIs; and 3). Quantitatively evaluating the agreement between predicted QoIs and experimental measurements through validation metrics. In this regard, validation and uncertainty quantification (VUQ) is often considered in an integrated manner for a comprehensive validation practice of a computational model (Roy and Oberkampf, 2011; Skorek, 2019). However, it is noticed that, despite a few efforts that aim to evaluate the prediction accuracy of MCFD solvers (Krepper et al., 2007; Colombo and Fairweather, 2016; Marfaing et al., 2018), many of the “validation” practices with MCFD solvers are still limited to a “graphical comparison” level. Based on the authors’ best knowledge, there still lacks a comprehensive validation practice for MCFD solvers due to several challenges. First, these MCFD solvers can be regarded as a complex model system, with multiple closure relations that have strong interactions among them. Such strong interactions make the traditional divide-and-conquer approach widely applied for complex systems challenging. Furthermore, typical engineering problems in a two-phase flow and boiling system require evaluations of multiple QoIs from the solver predictions, such as heat transfer, temperature, phasic velocities, void fraction, etc. Such QoIs also have interactions among them, thus the validation metrics developed for a single QoI is not directly applicable 2

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VUQ. Section 3 introduces a general overview of two-fluid model-based MCFD solvers; Section 4 introduces the proposed VUQ approach for MCFD solver; Section 5 provides a brief review of high-resolution experiments that can provide data for the VUQ of MCFD solver; Section 6 includes a case study to demonstrate the applicability of this proposed approach; and Section 7 summarizes and concludes the study.

Bayesian inference is one of the widely used approaches for Inverse UQ. It combines a prescribed prior uncertainty and the data-dependent likelihood function to generate the posterior uncertainty of the parameter. In this approach, the modular Bayesian inference (Liu et al., 2019) is applied which considers both the model parameter uncertainty and model form uncertainty through the inverse UQ process. A comprehensive UQ will involve both processes. In our proposed approach, the sensitive empirical parameters of the MCFD solver will go through the inverse UQ through modular Bayesian inference. These parameters with quantified posterior uncertainties, along with the quantified model form uncertainty and other parameters with assumed known uncertainty, will go through the forward UQ process to determine the uncertainties of QoIs predicted by the MCFD solver.

2. Review of concepts and methodology related to VUQ practices 2.1. VUQ related terminology To better clarify the scope of this work, a few key concepts related to the VUQ practices are discussed in this section. 2.1.1. Validation One of the most widely accepted definition of validation in scientific simulation can be found in (Oberkampf and Roy, 2010), which defines validation as: The process of determining the degree to which a model is an accurate representation of the real world from the perspective of the intended uses of the model. In practice, researchers use different approaches to validate a model with experimental measurement, from the most simple and straightforward “graphic comparison” to various types of validation metrics. It should also be noted that the verification is often discussed along with validation, as the term “V&V”. The verification aims to evaluate the accuracy of mapping a mathematical model to a computational code. The verification of MCFD solver is not within the scope of this work.

2.1.6. Sensitivity analysis (SA) SA aims to quantify the uncertainty contributions of different parameters towards the QoIs and determine how variations in parameters affect the QoIs. SA initially started with adjoint-based local analysis (Cacuci and Ionescu-Bujor, 2004), then the global based analysis have been applied in more recent years (Morris, 1991; Sobol, 2001). SA plays an essential role in this proposed VUQ approach in two aspects. First, SA is used to reduce the number of parameters in the modular Bayesian inference, which otherwise would be suffered from the “curse of high dimensionality”. Furthermore, a quantitative global SA is helpful to determine how to allocate the limited resources to the study of key sensitive parameters in future modeling work and experiments. There are already successful applications of global SA on the MCFD simulations (Marfaing et al., 2018; Liu et al., 2017). The relationship between the aforementioned concepts within the context of MCFD solver can be summarized as depicted in Fig. 1.

2.1.2. Classification of uncertainty Uncertainty can be characterized into three types based on its source. In this work, the following two types of uncertainties are considered.

2.2. Bayesian inference for UQ practice For the VUQ practice for model with empirical parameters, the Bayesian inference is widely used. Multiple methodologies of applying Bayesian inference to UQ problems have been developed, including coupling Bayesian inference with surrogate model of the original solver to reduce the computational cost (Wu et al., 2017; Liu and Dinh, 2019), applying modular Bayesian (Liu et al., 2019) or full Bayesian (Higdon et al., 2008) to quantify the model form uncertainty (Saleem and Kozlowski, 2019; Wang et al., 2018). These methodologies are widely used in the application of Bayesian inference for VUQ practices. Similar to other data-driven approaches (Bao et al., 2019; Liu et al., 2018; Wang et al., 2017), Bayesian inference relies on experimental measurements to update the posterior uncertainties of parameters under investigation. On the other hand, applying Bayesian inference for complex models remains a challenging task. The major issue is the “parameter identifiability”. When the number of parameters is large, Bayesian inference would be difficult to identify the posterior uncertainty of individual parameters due to the strong correlations between parameters. In this proposed approach, methods will be used to reduce the dimension of parameter space for the Bayesian inference; the basic idea is to select only the influential parameters for the Bayesian inference while keeping the remaining non-influential parameters with their prior uncertainty.

2.1.3. Model parameter uncertainty A computational model inevitably contains parameters that need to be specified before the model can be used for prediction. Those parameters, whether denoting certain input physical quantities such as inlet velocity or wall heat flux, or representing the empirical description of a closure relation, have uncertainties that influence the prediction. The uncertainty introduced by these parameters are termed as model parameter uncertainty. 2.1.4. Model form uncertainty The model form uncertainty is also termed as model bias, model inadequacy, or model discrepancy. The model form uncertainty is embedded in the formulation of the model, which usually includes approximations and simplifications for certain complicated physical process, as well as ignorance of some physical interactions between different phenomena. Besides these two sources of uncertainty, the numerical uncertainty also exists in MCFD solver. As a topic of verification, the numerical uncertainty is not discussed in this work. Instead, it is assumed that the numerical uncertainty is negligible compared to the model parameter uncertainty and the model form uncertainty in MCFD simulations. 2.1.5. Uncertainty quantification (UQ) The uncertainty quantification (UQ) process can be characterized as two different types: the forward UQ and the inverse UQ. Forward UQ is the process to determine the uncertainties of QoIs by propagating all known uncertainties of the parameters through the solver. Forward UQ can be done by sampling parameters according to their uncertainty distributions and run simulations based on the sampled parameters multiple times. Inverse UQ is the process to inversely quantify the uncertainty of the parameters of a model with the support of experimental observation.

2.3. Validation metrics The purpose of calculating validation metrics is to quantitatively evaluate the agreement between the solver predictions and the experimental measurements for the QoIs. For the validation for a single QoI, the validation metrics can be characterized into three categories. The first type is based on hypothesis testing (Liu et al., 2011), the validation metrics is in the form of a “Yes or No” statement regarding whether the model is accurate or not. The second type of validation 3

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Fig. 1. Relationship between key V&V-UQ concepts within the context of MCFD solver.

metrics is in the form of confidence interval (Oberkampf and Barone, 2006), an obtained 95% confidence interval of the discrepancy between simulation and experiment can be interpreted as “we have (1 ) × 100% confidence that the true discrepancy between model and observed data is within the interval”. The third type of validation metrics is based on the area metric (Ferson et al., 2008), which measures the area between the Cumulative Distribution Functions (CDFs) of the solver predictions and the experimental measurement. The area metric can be further extended to p-box (Ferson and Oberkampf, 2009) and u-pooling (Ferson et al., 2008). The former treats the QoI as interval-valued probability distributions, and the latter integrates the evidence of multiple validation sites and generate a single measure of overall mismatch. The aforementioned validation metrics are only applicable to single QoI. For problems with multiple correlated QoIs such as the VUQ of MCFD solver, multivariate validation metrics are needed. It is worth noting that some validation metrics for the multivariate QoIs are proposed recently (Li and Lu, 2018; Li et al., 2014; Wu et al., 2015; Zhao et al., 2017), which reflect the state-of-the-art development in this field.

(

k k)

+

t

·(

k k Uk)

=

ik ,

ki

(1)

where the two terms on the left-hand side represent the rate of change and convection, the two terms on the right-hand side represent the rate of mass exchanges between phases due to condensation and evaporation, which are modeled by closure relations. The k-phasic momentum equation is given by

(

k k Uk )

+

t =

·(

pk +

k

k k Uk Uk)

·[

k( k

t k )]

+

+

k kg

+

ki Ui

ki Uk

+ Mki

(2)

where i represents the interphase between two phases, Mki represents the term of averaged interfacial momentum exchange, which are modeled by a set of interfacial force closure relations. The k-phasic energy conservation equation in terms of specific enthalpy can be given as

(

k k hk )

t

3. Two-fluid model based MCFD solver

=

A typical two-fluid model based MCFD solver can be regarded as a complex model system. The system is based on three ensemble averaged conservation equations, which are supported with multiple closure relations. Some closure relations are in the form of sophisticated partial differential equations, while some other relations are empirical expressions or semi-mechanistic correlations with empirical parameters. The whole model system needs to be solved numerically, commonly using a finite-volume or finite-element method. The convergence and accuracy of the solution depend on numerical techniques and temporal and spatial resolutions needed to capture the dynamics and scales of governing physical processes. A brief glance of the twofluid model is provided in this section.

·

k

+

·( k

Tk

k k hk Uk)

µk

Pr tk

hk

+

k

Dp + Dt

ki hi

ik hk

+ qk ,

(3)

where the terms on the right-hand side represent heat transfer in phase k, work done by pressure, enthalpy change due to evaporation and condensation, and heat flux from the wall. The wall boiling heat transfer is modeled by a set of closure relations. 3.2. Characterization of closure relations in two-fluid model The ensemble average process in deriving the conservation equations of the two-fluid model results in loss of information at the interface between the two phases, thus closure relations need to be introduced to describe the unresolved phenomena. As can be seen from Eqs. (1)–(3), closure relations are required as source terms to make the equations solvable. Although the specific closure relations could vary among different MCFD solvers, they can be characterized into five categories: wall boiling, interfacial momentum exchange, interfacial mass/heat transfer, bubble size, and turbulence. The detailed discussion of these closure relations can be found in multiple references (Liu and Dinh, 2018).

3.1. Conservation equations The two-fluid model based MCFD solver relies on solving three ensemble averaged conservative equations for mass, momentum and energy (Ishii and Hibiki, 2011). The k-phasic mass conservation equation can be written as 4

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Running simulation cases with an MCFD solver for two-phase flow and boiling problems require setting up 30–40 parameters. The situation becomes further complicated as strong connections exist among certain phenomena related to the flow dynamics and interfacial interactions, which in turn results in tight coupling between the corresponding closure relations. For some cases, closure relations share the same parameters. For example, convective heat transfer and turbulence wall function, as they both about the boundary layer formed in the near wall region. The interaction and connection between different closure relations is depicted in Fig. 2. Compared to the single-phase CFD, the closure relations of twofluid model consist major sources of uncertainty for MCFD solvers. The uncertainties of these closure relations, including the model parameter uncertainty and model form uncertainty are the focus of this work.

the prior uncertainty of the parameters should also be specified in this step based on expert-opinion. In the second step, the parameter space should be reduced to avoid the “curse of high dimensionality”, we propose to use Sensitivity Analysis (SA) methods to identify and select influential parameters towards the QoIs under investigation. In the third step, surrogate model is constructed with takes the parameters identified in Step.1 as input and the QoIs under investigation as output. The surrogate model should be computationally efficient while also be able to accurately approximate the original MCFD solver. In this work, we use Gaussian Process (GP) to construct the surrogate models. In the fourth step, the model parameter uncertainty and model form uncertainty are evaluated with the Bayesian inference through a modular treatment. We suggest two UQ strategies based on the problem and the availability of data, the obtained uncertainties are propagated through the solver to obtain the corresponding uncertainties of QoIs under investigation. Last, the QoIs are validated against experimental measurements through a validation metric. We adopt the idea of extending the MCFD predictions and the experimental measurements to the Principal Component space so that the validation metric can take into

4. VUQ for MCFD solvers within the Bayesian framework The workflow of the proposed VUQ approach can be summarized in Fig. 3, in five steps. Firstly, evaluation should be performed to identify the relevant closure relations to be investigated based on the scenario,

Fig. 2. Closure relation structures in a typical MCFD solver. 5

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Y. Liu, et al.

Fig. 3. VUQ workflow for flow dynamics prediction in MCFD solver.

account the correlated QoIs. In the following of this section, the workflow will be discussed in detail.

availability issue within the Bayesian framework. On the other hand, the VUQ results would be more accurate with the support of detailed measurement from validation experiments. Also, the framework takes measurements from different conditions (e.g. different mass flow rate, heat flux, etc.) for the VUQ work simultaneously. This would generate robust VUQ results that can be extended to untested conditions. As discuss in Section 2, there are many closure relations of different categories in a MCFD solver. Some of the closure relations have very limited influence on certain QoIs or are even not activated in certain scenarios. Thus, it is impractical and unnecessary to evaluate the parameters of all closure relations for a given scenario. The evaluation of closure relations aims to identify closure relations that are relevant to the QoIs of a scenario. Once the relevant closure relation is evaluated, the uncertainties of the corresponding parameters are then inversely quantified through the Bayesian inference. To perform Bayesian inference, the empirical parameters are treated as random variables with given prior distribution that base on “expert judgment”.

4.1. Input parameter and prior uncertainty evaluation The MCFD solver deals with many different scenarios related to multiphase flow, from adiabatic bubbly flow to subcooled boiling flow. For different problems, the closure relations used for simulation could be different and the Quantities of Interest (QoIs) would also vary. In this sense, the initial step in the procedure is evaluating the solver based on scenarios and collecting relevant experimental data to support the VUQ process. Based on the studied scenario, several items should be addressed in this step:

• Evaluation of QoIs • Collection of available experimental datasets • Evaluation of closure relations • Evaluation of input parameters

4.2. Parameter dimension reduction

The QoIs of an MCFD simulation is scenario dependent and needs to be specified in the first place. For example, for boiling flow simulation, the wall superheat is considered as a QoI since it closely relates to safety, while the heat partitions are also QoIs since it relates to the heat transfer efficiency. For adiabatic bubbly flow, the interface distribution (characterized by void fraction) and the phasic flow field are QoIs. Once the QoIs for the given scenario is determined, the experimental measurement for the QoIs should be collected. For most cases, the resolution of the experimental measurement is coarse than the simulation results whose resolution can be easily controlled through mesh setup. The proposed VUQ framework can deal with such limited data

As demonstrated in Fig. 2, setting up a simulation case for a typical two-phase flow system with MCFD solver usually require tens of parameters of many closure relations to be specified. Directly inference all these parameters in the Bayesian framework will result in the issue of parameter identifiability. This issue stems from the fact that there could exist different values of the input parameter sets that produce very similar results and fit the data equally well. The Bayesian inference would in turn face convergence difficulty. In this work, we do not attempt to quantify all the parameter uncertainty of the solver. This is neither practical nor necessary, as not all parameters will have strong influence on the solver prediction over an interested scenario. Rather, 6

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the parameter’s input dimension is reduced first to select a set of parameters that has significant influence on the simulation results. Such dimension reduction is based on qualitative sensitivity analysis to evaluate the importance of parameters towards the prediction of QoIs. The general idea is to select parameters with high impact to the QoIs which are identifiable in the sense that they can be uniquely determined by the data. As demonstrated in the case study shown in Section 6, the Morris screening method (Morris, 1991) is an tool to evaluate the sensitivity of parameters with acceptable computational efficiency. Morris Screening method evaluates local sensitivity approximations, termed elementary effects, over the input space. Morris Screening method can rank parameters according to their importance, but cannot quantify how much one parameter is more important to another. The major advantage of Morris Screening is its low computational cost. To construct the elementary effect, one partitions (Bestion, 2010) into l levels. Thus, the elementary effect associated with the ith input can be calculated as y M (x , v , [ 1,

di ( ) =

,

i 1,

i

+

,

i + 1,

p ]) dx

y M (x , v, ) dx

generally valid for the considered problems. To avoid the potential issue of “curse of high dimensionality” and reducing the sampling simulations for constructing the surrogate model, we choose to build the surrogate model based on the parameters in reduced dimension only. The general form of a GP model can be expressed as m

y M (q , ) =

cov[Z (q(i) ), Z (q(j ) )] =

,

1 1

}

(5)

For r sample points, the sensitivity measurement for xi can be represented by the sampling mean µi , standard deviation i2 , and mean of absolute values µi*, which can be calculated respectively.

µi* =

µi =

2 i

=

1 r 1 r

r

r

(6)

di j (q)

(7)

j=1 r

1 r

di j (q)

j=1

1

(di j (q)

µi ) 2

j=1

+ f (q)=h (q)T + Z (q),

(9)

2K

(q(i) , q(j) ).

(10)

here K (q(i) , q(j) ) is the kernel function, which has a fixed functional form with several hyperparameters. This kernel function ensures that two inputs with close distance will produce outputs that are also close together. The hyperparameters in the kernel function can be evaluated through point estimate method such as Maximum Likelihood Estimate (MLE). A GP model needs to be trained by a number of runs from the original model before it can be used for prediction. The mathematical details of GP can be found in (Liu et al., 2019). In MCFD simulations, we are often deal with QoIs in the form of spatial distributions such as the void fraction and gas-phase velocity in the computational domain. Constructing surrogate models for each elements in the distribution would be a formidable task and completely neglect the correlations between these elements in the distribution. In this work, we proposed to employ the Principal Component Analysis (PCA) method to overcome this issue. The PCA uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables termed principal components. The number of principal components is less than or equal to the smaller of the number of original variables or the number of observations. The Singular Value Decomposition (SVD) is a robust algorithm for PCA. Given an MCFD simulation case, its results can be output as a vector. The output vector with N different MCFD simulations can form a matrix:

which means the QoIs are integrated over the whole input condition space for evaluation. The step size is chosen from the set

l

i

where q = [q1, q2 , ...,qp] is the p input variables, which can be empirical parameters or boundary conditions and y M (q) is the model output with the given input. The first term on the right side is a deterministic trend function, which is the product of regression coefficients and the basis function, which has known form, usually set to be a constant or polynomial function. The second term Z(q) is a GP error model with zero mean, variance σ2 and non-zero covariance cov [Z (q1), Z (q2)], which can be modeled as

(4)

{ l 1 1 , ...,1

hi (q)T i=1

(8)

The mean µ* represents the effect of the specific parameter on the output, while the variance 2 represents the combined effects of the input parameters due to nonlinearities or interactions with other inputs. In other words, a parameter with high µ* means it has strong influence on the QoIs prediction, while a parameter with high 2 means it has strong non-linear interaction with other parameters. Thus, the parameters with high standard deviation i2 and high absolute value of sampling mean µi* are considered as influential.

Y = (y M (q(1) ), y M (q(2) ), ...,y M (q(N ) )).

(11)

Based on Y , SVD can be performed:

Y = U V T,

4.3. Surrogate model construction

(12)

where Σ is a diagonal matrix whose diagonal entries are non-negative and ordered from largest to smallest. U is a unitary matrix whose column vectors are called the Principal Components (PCs). Since U is unitary matrix, it has the property for such manipulation:

The VUQ process requires many solver evaluations, considering the relative expensiveness of running a MCFD simulation, a surrogate model, also known as response surface model or emulator, is a necessary. In this step, a surrogate model is constructed based on the outputs of a limited number of runs of the original solver. The following UQ and quantitative sensitivity analysis will be performed based on the surrogate model with computational efficiency. There are multiple statistical and numerical methods that can be used for constructing a surrogate model. Each has various applications, such as polynomial response surface (Hosder et al., 2001), stochastic collocation (Wu et al., 2017), and Gaussian Process (Constantine et al., 2014), etc. In this work, we chose the Gaussian Process (GP) regression for surrogate construction, which has been widely used in the area of data-driven modeling and optimization. The only assumption for GP model is the QoIs are smooth over the whole input domain, which is

U TY = U TU V T = V T = S,

(13)

where S is termed PC scores. The PC scores are the transformed representations of the original input matrix Y. For most cases, the singular values of Y decrease very quickly which means the first few PCs can quantify the structures of the Y. So the manipulations in Eq. (13) can be modified as:

U *TY = S*,

(14)

where U * is a matrix with first few PCs, S*is the matrix with the corresponding PC scores. In practice, one usually retains the first few PCs to ensure the corresponding variances can account for 95%−99.9% of the 7

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total variance. In this way, the dimension of outputs can be significantly reduced. In training GP models for the PC scores, each column in S*serve as a sample. The GP model give predictions for the corresponding PC scores. S* can be transformed back from the PC subspace to the original space.

As can be seen in Eq. (18), the model form uncertainty term (x ) is included in the likelihood function as a known term. In this proposed approach, we apply a modular treatment for the model form uncertainty and model parameter uncertainty evaluation. The first module is for model form uncertainty (x ) , which is evaluated with a GP (independent to the GP used for surrogate model). The input for this GP is the state variable x , while the output is the discrepancy between QoI measurements and MCFD simulations with nominal . In practice, the available datasets are decomposed into three parts. One used for evaluating model form uncertainty, one used for evaluating parameter uncertainty, the last one used for testing the UQ results. This modular treatment is illustrated in Fig. 4. With the model form uncertainty (x ) resolved, the next modular is to evaluate the model parameter uncertainty through Bayesian inference. For most cases with complex models, the posterior distributions cannot be analytically obtained through the Bayes formula. A widely used approach is Markov Chain Monte Carlo (MCMC) sampling. This approach constructs a Markov Chain through Monte Carlo sampling. The Markov Chain is constructed in the way that when it converges to steady state, the obtained samples from the chain will follow the posterior distributions of the parameter. There are multiple algorithms for MCMC sampling, in this work, we employ the Hamiltonian Monte Carlo (HMC) algorithm (Neal, 2011) supported in Pyro (Bingham et al., 2018), an open source probabilistic programming language written in python. The samples drawn from MCMC can be regarded as the posterior distribution of the parameters under investigation. Moreover, by the definition of Markov Chain, the state of sample at t + 1 only dependent on the present sample state at t. One way to measure the correlation of the sample traces is the autocorrelation. By definition, the autocorrelation rk between θt and θt+k can be calculated as:

4.4. Uncertainty quantification for MCFD using data-driven Bayesian framework Once the surrogate model is constructed and validated to achieve accurate approximation of the original MCFD simulations, we can use it for following UQ and SA work. We rely on the Bayesian inference for the UQ task. The formulation of the Bayesian inference can be summarized as follows. For a general computational model, the relationship between the outputs and experimental measurements can be expressed as (15)

y E (x) = y M (x, ) + (x) + (x),

where stands for empirical parameters and x stands for design variables or locations such as inlet mass flux, radial location, wall heat flux, etc. y E (x) is the experimental measurement, y M (x , ) is the model prediction, (x) is the model form uncertainty, which is usually caused by missing physics, simplified assumptions or numerical approximations in the model. (x) is the measurement uncertainty which is assumed to be i.i.d (independent and identically distributed) normal distributions with zero means and know variance σ2 in this work:

(x)~N (0,

(16)

2 I)

The goal in the inverse UQ process is to evaluate the uncertainty of the parameter based on the data. In the framework of Bayesian inference, which treats the parameter as random variables, the prior knowledge for the parameter is also considered. The prior knowledge usually comes from previous simulations, other experimental observations or purely expert opinion. The Bayes formula is the foundation of Bayesian inference:

p ( y E) =

p (y E ) p0 ( ) p (y E )

p (y E ) p0 ( ),

1 T

(17)

(2

2) n 2

n

exp i=1

(y E (x i )

y M (x i , ) 2 2

(x i )) 2

.

1 T

T t=1

t+k 2

.

t

(19)

In the MCMC sampling, we need to ensure the autocorrelation between samples at different steps at minimum. The aforementioned modular Bayesian inference is an inverse UQ process. Within this process, the model form uncertainty and model parameter uncertainty are quantified. The next step is a forward UQ process that propagates the obtained uncertainties through the MCFD solver for the uncertainties of QoIs. Combing the inverse and forward UQ processes we could develop a comprehensive UQ practice for the

) is the likelihood function, and p0 ( ) is the prior diswhere tribution of θ. Under the assumption that (x, v)~N (0, 2I) , the likelihood function has the form 1

t

rk =

p (y E

L ( y E) =

T k t=1

(18)

Fig. 4. Modular treatment for model form uncertainty and parameter uncertainty. 8

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MCFD solver. In practice, we perform the modular Bayesian inference for the influential parameters of all closure relations in an integrated and simultaneous manner. The obtained model form uncertainty and parameter uncertainty are propagated through the solver for the uncertainty of QoIs. The detailed implementation of this modular approach is illustrated in Fig. 5. There are several advantages of the proposed UQ process. First, it is modular and non-intrusive. This ensures the approach’s flexibility for different scenarios as well as its extendibility to different MCFD solvers. Moreover, the approach has strong flexibility with data availability. It is capable of simultaneously taking into account multiple QoIs measurement under different conditions. It can also provide a reasonable VUQ result with limited data availability, while providing a more accurate result with better data support. Last, the UQ approach ensures the possible interactions among closure relations in the MCFD solver can be captured since all closure relations are considered in an integrated manner.

extending the area metric to the PC subspace (as discussed in Section 4.3) of the simulation results (Li and Lu, 2018). The weighted area metrics based on PCs can generate a quantitative evaluation of the agreement between simulation results and the experimental measurements. This practice maps the experimental measurements to the PC space which is created based on the MCFD predictions with the posterior parameter uncertainties and/or model form uncertainties. The workflow is illustrated in Fig. 6. There are several advantages of applying area metric to the PC subspace. As the PCs are combinations of the distributions of multiple QoIs, it naturally captures the correlation between the multiple QoIs as well as the correlations among their distributions. Furthermore, we can use a few PCs to represents all the QoIs in multiple spatial locations thus significantly reduced the number of validation sites. Last, we can combine the area metrics with the weights of each PC to generate onesinge value as a quantitative evaluation of the agreement between data and prediction.

4.5. Validation metrics

5. High-resolution local measurements of two-phase flow systems to support VUQ of MCFD solvers

The advantage of area metric is that it has several properties as suggested by (Ferson et al., 2008) that make it mathematically well behaved and well understood. Such properties can be expressed as following for CDFs of two random variables X and Y :

As discussed in Section 4, the proposed approach relies on experimental measurements to inversely quantify the uncertainties associated with the MCFD simulations. As the MCFD simulations usually involve multi-dimensions with high resolution, correspondingly, the measurements for the VUQ of MCFD solvers also need to be of high-resolution and reflect the local detailed information of the QoIs. Recent development of measurement technologies has enabled high-resolution local measurements for multiple QoIs of two-phase flow systems. Such measurements serve as the foundation of successful VUQ of MCFD solvers.

• Non-negativity: d (X , Y ) 0 • Symmetry: d (X , Y ) = d (Y , X ) • Triangle inequality: d (X , Y ) + d (Y , Z ) d (X , Z ) • Identity of indiscernible: d (X , Y ) = 0 if and only if X = Y However, as discussed in Section 2.4, applying validation metrics to correlated QoIs is a challenge task, especially these QoIs are in the form of spatial or temporal distributions. In this work, we adopt the idea of

Fig. 5. UQ approach based on modular Bayesian treatment for MCFD solver. 9

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Fig. 6. Validation metric calculation based on PC subspace.

5.1. Experimental techniques for high-resolution two-phase flow measurements

simultaneous manner thus are ideal tools to provide data support for the VUQ of MCFD solvers. In addition to PIV/PTV, the hot-film anemometry (Abel and Resch, 1978) and Laser Doppler Velocimetry (LDV) (Kried, et al., 1979) can also be used to measure the liquid-phase velocity in two-phase flow systems, but the measurements are of point-bypoint nature. Traditionally, the measurements of boiling heat transfer are carried out with thermocouples that are attached to a heated test section. The local wall temperature can be measured and the heat transfer can be derived. More recently, high-speed infrared (IR) cameras have been applied to measure the detailed boiling process. The pioneering work is the UCSB-BETA experiment (Theofanous, 2002; Theofanous et al., 2002), which measured the detailed nucleation and the corresponding nucleate boiling heat transfer of pool boiling and thin liquid film boiling. Another representative work is reported by Richenderfer et al. (2018), which couples the high-speed IR camera and high-speed video camera to estimate the wall heat flux partitioning in nucleate boiling heat transfer. Some major experimental techniques for two-phase flow measurements are summarized in Table 1. Information extracted from these high-resolution experiments can significantly expand available twophase flow and boiling databases, which are usually limited to measurements at a certain points in the whole flow domain. It should also be noted that high-fidelity two-phase flow and boiling simulations based on Direct Numerical Simulation (DNS) have made significant progress over the past decade (Sato and Niceno, 2013; Sato et al., 2018; Feng and Bolotnov, 2017; Fang et al., 2018; Tryggvason et al., 2016). These high-fidelity simulation results could serve as a potential new data sources for the VUQ of MCFD simulations (Liu et al., 2018a,b).

For a two-phase flow system, its measurements can be characterized into three major categories: measurement for liquid-phase, measurement for gas-phase, and measurement for boiling/condensation heat transfer. A variety of measurement techniques can be used for measurement of the gas-phase velocity, bubble size, and void fraction in two-phase flow systems. High-speed camera systems can non-intrusively capture the bubble dynamics in two-phase flows. Obtaining the gas-phase information from the raw images captured by high-speed cameras requires robust algorithms (Fu and Liu, 2016). On the other hand, the multi-sensor conductivity probes can provide detailed local measurements on the bubble size, void fraction, interfacial area concentration and gas-phase velocity (Wang et al., 2018; Kim et al., 2000; Qiao and Kim, 2018), but their measurements are of point-by-point nature and thuse can be time consuming to obtain the distributions of the measured quantities. Furthermore, wire-mesh sensors can also serve as a measurement technique for the void fraction and bubble size at a certain cross section of the test section (Manera et al., 2009; Prasser et al., 2001). For liquid-phase measurement, the Particle Image Velocimetry (PIV) and Particle Tracking Velocimetry (PTV) have been used in the measurement of the liquid phase velocity of two-phase flow systems. In the representative works (Estrada-Perez and Hassan, 2010; EstradaPerez et al., 2015; Zhou et al., 2013), promising results are demonstrated for quantitative measurements of the detailed bubble dynamics and flow velocity fields. The turbulence characteristics can also be derived from the fluctuations of velocity fields (Zhou et al., 2016). The PIV/PTV systems are capable of measuring the 2-D or 3-D flow field in a 10

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Table 1 Summary of experimental techniques for high-resolution measurements in two-phase flows. Techniques

Measurable QoIs

Gas-phase measurements

High-speed camera Fu and Liu (2016) Conductivity probe Wang et al. (2018); Kim et al. (2000); Qiao and Kim (2018) Wire-mesh sensor Manera et al. (2009); Prasser et al. (2001)

Bubble velocity, Bubble size, Void fraction Gas-phase velocity, Bubble size, Interfacial area concentration, Void fraction Void fraction, Bubble size

Liquid-phase measurements

PIV/PTV Estrada-Perez and Hassan (2010); Estrada-Perez et al. (2015); Zhou et al. (2013) Hot-film anemometry Abel and Resch (1978) LDV Kried, et al. (1979)

Liquid-phase velocity, Turbulence characteristics Liquid-phase velocity Liquid-phase velocity

Infrared camera Theofanous (2002); Theofanous et al. (2002); Richenderfer et al. (2018)

Temperature at boiling surface, Nucleation information, Heat transfer information

Boiling measurements

QoIs from one same test facility to ensure consistent boundary conditions and eliminate any distortion that might be introduced by scaling from one facility to another. Obtaining multiple QoIs from one same facility could be achieved by either adopting multiple techniques in the measurements (Wang et al., 2018; Hassan et al., 2014) or using advanced algorithms to derive multiple QoIs from the one single measurement of the corresponding physical quantities (Richenderfer et al., 2018; Liu and Dinh, 2016). Overall, it is vital for simulation analysts to work closely with experimentalists in designing and conducting experiments that provide data for the VUQ of MCFD solvers. In Section 6, we will use data extracted from the literature to perform a simplified case study to demonstrate the applicability of this proposed approach.

5.2. Desired features of experimental measurements for the VUQ of MCFD solvers On the other hand, the VUQ of MCFD solvers has raised more requirements on the experimental measurements. We note that strict criteria of “validation experiment” for CFD community have been discussed by Oberkampf and Smith (2017). For validation experiments, their measurements should cover all relevant physical features related to the QoIs of the system, while also form a full hierarchy of system responses from globally integrated quantities to local quantities. Furthermore, the uncertainties of the measurements should be well-quantified with distinguishable random and systematic measurement uncertainties. Although these desired features are of important value, we have to admit that strictly following the criteria of validation experiment is still very technically and/or economically challenging. In this work, we propose three desired features for the experimental measurements for the VUQ of MCFD solvers, which follow the guidance of validation experiments but are not as strict. The experiment should clearly capture its boundary and inlet conditions. The inlet conditions (e.g., the gas-phase and liquid-phase velocities and their mean fluctuations, void fraction, and bubble size at the injection of the experiment facility) and boundary conditions (e.g., wall conditions) serve as inputs in the MCFD simulations. To ensure the simulation is an accurate representation of the experiment, the spatial or temporal distributions of the inlet and boundary conditions should be accurately measured. The measurement should be consistent with the dimensionality of the simulation, i.e., if the simulation is three-dimensional, the measurements should be three-dimensional as well and serve as accurate inputs for the simulation. The experiment should have a comprehensive uncertainty analysis. The uncertainties of the experimental measurements influence the UQ process as well as the calculation of validation metrics. It is vital to evaluate the accuracy of the measurement techniques so that a basic understanding of the uncertainty limit of the measurements can be formed. Furthermore, the uncertainties can be evaluated with a statistical analysis of multiple observations. It is also desired to distinguish the measurement uncertainty due to the accuracy of techniques and that due to system fluctuations, the latter serves as one important source of aleatory uncertainty, which should be treated differently in the UQ process. Measurements of multiple QoIs in one test facility. As a two-phase flow system usually involves multiple QoIs, it is desired to obtain multiple

6. Case study: VUQ on adiabatic bubbly flow simulations 6.1. Case setup and closure relation evaluation We develop a simplified case study to demonstrate the applicability of the proposed VUQ approach. The scenario of the case study is the adiabatic upward bubbly flows in a circular pipe. Two physical features that are important in two-phase flow systems are considered as QoIs in this case study: the void fraction and gas-phase velocity. The experimental data regarding these two QoIs are extracted from literature (Leung et al., 1995). A total eight datasets under different flow conditions are used in this case study. The experimental conditions are summarized in Table 2. The uncertainty of the experimental measurements is important for the UQ process and the following validation metrics calculation. However, the measurement uncertainty is not comprehensively discussed in the literation, thus based on the general description of the measurement error, we conservatively assume a 10% uncertainty for each measurement. Experimental investigation has confirmed that the interfacial force and turbulence has significant influence on the radial distribution of the phasic velocity and void fraction (Sato and Sekoguchi, 1975). In this sense, the uncertainties of parameters in the closure relations of interfacial forces and turbulence model are considered in this case study. For a typical MCFD solver, five interfacial forces are modeled. The drag force is modeled to describe the resistance of relative motion between the two phases. The lift force is modeled to describe the force that exerts by continuous fluid flow past the bubble. The turbulent dispersion force is modeled to describe the effect of liquid turbulence on the bubble. The wall lubrication force is designed as an artificial force

Table 2 Summary of experimental measurement. Scenario

Geometry

QoIs

Adiabatic upward bubbly flow in a round tube

Diameter = 25.4 mm, Inlet location: z/D = 12, Outlet location: 62 Leung (1997)

spatial distributions of , U1

11

Varying input Condition

jg from 0.09 to 0.48 m/s, jl from 0.64

to 2.0 m/s

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Table 3 Summary of the studied interfacial force closure relations. Force type

Expression

Drag force Ishii and Hibiki (2011)

MD g =

Lift force Ishii and Hibiki (2011)

MLg

Wall lubrication force Antal et al. (1991)

3 Cd 4 Ds l

= Cl

MWL g

Parameter studied

=

l

Ug

(Ug

Ul (Ug

Virtual mass force Auton et al. (1988)

fWL (Cwl, yw )

MTD g =

t 3 CD l 4 Ds tPr t l l

MVM g =

Cvm

Table 4 Prior uncertainties of interfacial force coefficients. Nominal value

Lower bound

Upper bound

Cµ Cd Cl Cwl Cvm

0.09

0.05

0.15

t

0.44 −0.05 0.03 0.3 0.6

(

µkT =

Parameters

0.6 0.03 0.05 0.5 0.9

l

(

l

Ur

(Ur ·nw ) n2w nw Ds

0.2Cwl + Ug

D Ug Dt

k Cµ

k k kk )

t

1.0 0.1 0.08 1.0 1.2

Cl

Ul) ×( ×Ug)

fWL (Cwl, yw ) = max Turbulent dispersion force Gosman et al. (1992)

Cd

Ul)

Cwl yw

Cwl

,

Ds , 0 t

Ul DUl Dt

Cvm

)

k2

+

(20)

·(

k k Uk kk )

=

k

k

(

k k k)

t

+

·(

k k Uk k )

+

= C2

to move the bubble away from the wall to describe. The virtual mass force is modeled to describe the inertia of bubble acceleration or deceleration. Table 3 summarizes the expressions of those interfacial forces. These interfacial forces involve several coefficients in the expression, among which the force coefficients Cd and Cl can either be a constant or be calculated from semi-empirical correlations (Tomiyama, 1998; Tomiyama et al., 1998; Ishii and Zuber, 1979). In this work, we choose Cd and Cl to be constants as a reasonable simplification. The k turbulence model solves two differential transport equations in order to determine the turbulent kinetic energy k and the turbulent dissipation ε for the liquid phase (Launder and Spalding, 1974):

k

kk

kk

kk +

kP

k k

(21)

k

k

k

µk +

+

µkT

µk +

k

µkT k k

k

+

kC 1

k

kk

P

k k

(22)

In Eqs. (30) and (31), P is the production of the shear-induced turbulent kinetic energy; k and k are the source terms due to the effects of the dispersed phase on the turbulence, which are modeled by the bubble-induced turbulence closure relation (Troshko and Hassan, 2001). The k model relies on several empirical coefficients to be specified in the simulation case. In this case study, we consider the coefficient Cµ in the UQ process. The prior uncertainties of these coefficients are assumed to be uniformly distributed. We conservatively assume our knowledge on these studied parameters are poor, so we assume “non-informative” prior for these parameters. Given more knowledge on these parameters,

Fig. 7. Morris screening measures for parameters related to bubbly flow simulations. 12

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other kind of priors can be prescribed. The ranges of these distributions are determined based on literatures (Krepper et al., 2007; Liu et al., 2019; Sugrue et al., 2017) and the authors’ research experience, as summarized in Table 4. The simulations are performed with a quasi-cylindrical geometry. Mesh study is performed to ensure the mesh-independent solution in the bulk flow region is reached. The inlet boundary is set at z/D = 12 so that the measurements at this location can be used to define accurate inlet conditions. The QoIs are extracted along the radial direction of the tube at z/D = 62. For each QoI, there are 13 outputs at different radial locations, which are consistent with the experimental measurement. 6.2. Parameter dimension reduction In the Morris screening step for parameter selection, the solver predictions are averaged to generate a global response for each QoI. Thus two globally averaged quantities are analyzed: the averaged void fraction and averaged gas-phase velocity. The obtained results are displayed in Fig. 7. In this figure, the parameter with high value of µi* indicates high impact on the QoI, while parameter with high value of σ indicates strong correlation with other parameters. So in the figure the parameter in the upper right is deemed as influential. We can find that for both QoIs, there are four influential parameters, i.e. Cl, Cwl, Cd,and Cµ . The virtual mass force coefficient Cvm and turbulence dispersion coefficient t have less influence on both QoIs. In this sense, these four parameters are selected for the following surrogate model construction and VUQ process.

Fig. 9. Surrogate model construction based on PCA.

combinations of the three interfacial force coefficients are generated for the MCFD solver simulations. The simulation results of void fraction and gas-phase velocity are centered and scaled according to the inlet conditions, then stacked to create a matrix to which PCA is applied. The results of the PCA applied to the combined void fraction and gas-phase velocity predictions are shown in Fig. 8. We can observe that up to 99.5% of the total variances can be explained by the first 8 principal components (PCs), where each PC is a linear combination of the 208 QoIs. Thus, 8 surrogate models are constructed based on these 8 PCs, reducing the number of surrogate models from 208 to 8. The 208 QoIs can be reconstructed based on these 8 PCs. This process can be illustrated in Fig. 9. To minimize the issue of overfitting, we employ the cross-validation in training the GP based surrogate model (Wu et al., 2018). In the practice, we perform a fifth-fold cross validation, i.e., 80 simulation cases are used for training the GP surrogate model, while the rest 20 cases are used to test the accuracy of it. The accuracy of the trained surrogate model can be evaluated through Mean Square Error (MSE),

6.3. Surrogate model construction The surrogate model is constructed using GP-based regression. A comprehensive VUQ process should be able to take multiple correlated QoIs under various conditions into simultaneous consideration. In this sense, the surrogate model should be able to predict the radial distributions of void fraction and gas-phase velocity under all eight conditions corresponding to the experimental measurements. Thus, the total number of QoIs to be considered in the surrogate model is 8 × 13 × 2 = 208. It is very inefficient to construct 208 separate surrogate models for these QoIs, so we introduce PCA in this case study to reduce the dimension of QoI outputs. The samples of MCFD simulations are performed with perturbed interfacial force coefficients from LHS. A total 100 samples of different

MSE =

1 N

N

(yi

yiM ) 2

(23)

i=1

Table 5 MSE of surrogate model for each principal component. Surrogate model for PC score

MSE

1st score 2nd score 3rd score 4th score 5th score 6th score 7th score 8th score

1.4 × 10−3 4.2 × 10−3 3.7 × 10−3 4.8 × 10−4 8.8 × 10−4 6.3 × 10−4 4.3 × 10−3 1.2 × 10−4

Table 6 Flow dynamics datasets decomposition for modular Bayesian inference.

Fig. 8. Accumulative percentage of variances explained by PCs.

13

Datasets decomposition

Inlet phasic superficial velocities (jg , jl ) (m/s)

Model form uncertainty evaluation datasets Parameter uncertainty evaluation datasets Testing dataset

(0.09, 0.64), (0.09, 1.1), (0.16, 2.0), (0.48, 2.0)

(0.09, 2.0), (0.29, 1.1), (0.29, 2.0)

(0.16, 0.64)

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Fig. 10. MCMC sample traces of parameters for inverse UQ.

The MSE of the 8 PCs are summarized in Table 5. It can be found that the MSE of these 8 surrogate models demonstrate that the average difference between surrogate predictions and the original MCFD solver predictions are less than 1%. In this sense, we can reasonably conclude that the uncertainties introduced by statistical methods are trivial compared to the model parameter uncertainty and model form uncertainty. 6.4. Uncertainty quantification Following the modular Bayesian treatment, we decompose the eight datasets into three parts, as shown in Table 6. As the first module of the modular Bayesian inference, the model form uncertainty term is modeled by a multi-variant GP: (r /R, jg , jl ) GP (r /R, jg , jl ) . The trained (r /R, jg , jl ) serve as a known term in the second module: inference of model parameter uncertainty through MCMC. The samples drawn from MCMC can be regarded as the posterior distribution of the parameters under investigation. The sample traces of the four parameters are depicted in Fig. 10. It can be observed that the

Fig. 11. Autocorrelations of MCMC sample traces.

14

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Fig. 12. Marginal and pairwise joint distributions of selected interfacial force coefficients.

sample traces demonstrate good mixing patterns and fully explore the parameter space. To ensure small autocorrelations between the samples, we keep every 10th samples for the posterior distribution construction. The autocorrelation of the sample traces is depicted in Fig. 11. The fast decay of auto-correlations for all parameters can be observed. This indicates that the Markov Chain has reached its stationary state, and the samples in the chain can be regarded as generated from the posterior distributions of these parameters. The obtained marginal and point-wise distributions of these four parameters are plotted in Fig. 12. We can observe that there is a positive correlation between Cl and Cwl . Generally speaking, large value of Cl will result a large value of Cwl . This indicates there are interaction between different phenomena relevant to the bubbly flow. We can also find that compared to the uniform prior uncertainties, the posterior uncertainties of parameters demonstrate a normal distribution pattern and have much narrower ranges. This indicates with the data support; the uncertainties of the studied parameters are significantly reduced. In turn, the uncertainties of MCFD predictions can also be reduced.

The obtained parameter uncertainty can be propagated through the surrogate model to obtain the uncertainties of QoIs. Two separate propagations, one with model form uncertainty considered and one without, are conducted. In Fig. 13, we demonstrate an example with the dataset for testing, i.e., the dataset that is not used in the UQ process. For the predictions of the void fraction, we observe that the MCFD predictions can capture the near wall void fraction peak for all the conditions. The peak location can also be identified with acceptable accuracy. We can also find that if we do not consider the model form uncertainty, there would be relatively large discrepancies between experimental data and solver prediction for the values of void fraction. It can be further observed that the MCFD simulation results tend to overestimate the gas phase velocity at the central region of the tube, while underestimate the values at the near wall regions. The MCFD predictions with the posterior uncertainties considered have a much narrower uncertainty range compared to the predictions with the prior uncertainties, but the discrepancies between the measurements and simulations still cannot be addressed. Such discrepancies can only be reduced with the consideration of the model form uncertainty.

15

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Fig. 13. Uncertainty quantification of MCFD predictions, the effect of model parameter uncertainty and model form uncertainty, jg = 0.16 m/s, jl = 0.64 m/s.

However, we also observe some unphysical fluctuations in the gas phase velocity in the results with the model form uncertainty considered. Such fluctuations may not have physical meanings but may be caused by a GP process trained with insufficient samples. Such discrepancies in both QoIs indicate that there are significant model form uncertainties in the studied closure relations. In fact, it is consistent with our expectation since the drag and lift coefficients studied in this case are set to be constant. Such treatment simplifies the complicated interaction between bubble and fluid. In the mechanistic interfacial force model, these two coefficients have sophisticated expressions which are dependent on bubble shape, relative velocity, bubble size, and surface tension (Tomiyama, 1998; Liu et al., 2008).

the distribution of experimental measurements, forming an experimental observation matrix Y E . The corresponding area metric of each PC is summarized in Figs. 14 and 15, with Fig. 14 only considers the model parameter uncertainty while Fig. 15 considers both model parameter uncertainty and model form uncertainty. It can be clearly found that the area metric is smaller when we consider the model form uncertainty. A comprehensive evaluation that covers all the validation sites and all the conditions under investigation can be formed through a weighted sum of the area metrics of each PC, which can be calculated as: K

wk· dk (FkM , DkE )

d (F M , S E ) = k=1

6.5. Validation of MCFD predictions

Based on the Eq. (25), we can conclude that the weighed area metric when we only consider the model parameter uncertainty is 0.493. When we take the model form uncertainty into consideration, this value can be reduced to 0.301. Although being a single-value evaluation, such a validation metric takes into account correlated QoIs in spatial distributions as well as in a variety of conditions, thus can be trusted to comprehensively evaluate the simulation performance with the support of experimental measurements.

Through perturbing the parameters according to their posterior uncertainties, we can obtain multiple results in the form of distributions of the two QoIs, i.e., void fraction and gas-phase velocity. As discussed in Section 4.5, we combine these distributions of void fraction and gasphase velocity and transform the concatenated distributions to the PC subspace. The same methodology can be applied to the experimental measurements. In here, we assume the uncertainty of each measurement follows the normal distribution, with standard deviation equals to 10% of the measurement value. The area metric of each PC can be calculated through the differences of CDFs of simulations and experimental measurements as Eq. (24).

dk (FkM , DkE ) =

+

|FkM (s )

DkE (s )| dh

(25)

7. Conclusions The two-fluid model based MCFD solver has demonstrated strong potential for the safety and reliability analysis of complex two-phase flow and boiling systems for industrial applications. On the other hand, it should also be admitted that fundamental disconnection still exists between performing simulations with MCFD solver and using the simulation results to guide the decision-making process. To overcome such a disconnection, it is essential to have a comprehensive Validation and Uncertainty Quantification (VUQ) practice for the MCFD solver, so

(24)

As a demonstrating example, we draw 100 samples from the posterior uncertainties of the parameters and propagate them through the surrogate model, obtaining a MCFD prediction matrix Y M that includes all eight conditions. Correspondingly, we draw 100 samples based on

16

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Fig. 14. Area metric between experimental measurements and MCFD predictions, without the consideration of model form uncertainty.

that the accuracy of the solver predictions can be assessed quantitatively and objectively. In this work, we propose a data-driven approach that based on Bayesian inference for the VUQ of MCFD solver. The proposed approach evaluates model form uncertainty and parameter uncertainty in a modular manner. This approach is able to take multiple QoIs and all relevant closure relations into simultaneous consideration. We adopt the idea to transform MCFD predictions and experimental measurements to the Principal Component (PC) subspace, based on which the area metric for each PC can be calculated. A comprehensive metric that covers correlated QoIs under multiple conditions can be evaluated through weighted sum of the area metric for each PC. A case study on the adiabatic bubbly flow scenario is developed to demonstrate the applicability of this approach. This approach should also be compatible with other complex models with multiple closure relations and empirical parameters. There are still some unresolved issues with the proposed approach. One of the most important issues is the effect of scaling. The experimental study of two-phase flow and boiling for large, industrial scales are technically challenging and expensive, thus the current available data are mainly obtained from down-scaled facilities. In other words, the application domain would be larger than the validation domain

when performing MCFD simulations for most real industrial problems. Additional uncertainty or bias will be introduced when extrapolating the VUQ results obtained in the validation domain to the outside application domain. Such uncertainty requires further efforts to evaluate. In addition, another additional uncertainty source would be introduced when applying the surrogate model in the inverse UQ process. In this work, the Gaussian process (GP) is used for the surrogate model construction. Given proper hyperparameters setup, the additional uncertainty introduced by the GP would be small, but such uncertainty is not negligible. Additional effort is required for the evaluation of this new uncertainty source. Be that as it may, this proposed approach can still be regarded as a significant improvement compared to current VUQ practices for MCFD solver that mainly relies on expert-opinion based uncertainty characterization and graphical-comparison-based validation. The proposed approach should be able to apply to the VUQ of other computational models, especially System Thermal-Hydraulic (STH) codes (e.g. RELAP5, CTF, etc.) which also involve multiple empirical parameters with large uncertainties. Furthermore, simulations based on STH codes usually have a much smaller computational cost compare to the MCFD simulations, so the surrogate modeling step may not be necessary.

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Fig. 15. Area metric between experimental measurements and MCFD predictions, with the consideration of model form uncertainty.

Acknowledgments

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