^{1}

^{1}

^{2}

^{1}

^{1}

This article was submitted to Nuclear Energy and Policies, a section of the journal Frontiers in Energy Research

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Computational fluid dynamics (CFD) has become an effective method for researching two-phase flow in reactor systems. However, the uncertainty analysis of Computational fluid dynamics simulation is still immature. The effects of uncertainties from two-phase models and boundary conditions have been analyzed in our previous work. In this work, the uncertainties from a turbulence model on the prediction of subcooled boiling flow were analyzed with the DEBORA benchmark experiments by a deterministic sampling method. Seven parameters in the standard _{
μ
}, _{
μ,g
}, _{
1ε
}, _{
2ε
}, _{
k
}, _{
ε
}, and Pr_{
t
}. Radial parameters were calculated to study the effects of uncertainties from the turbulence model. The contributions of each uncertainty source on void fraction and liquid temperature were also analyzed. It was found that the models can simulate subcooled boiling flow accurately and uncertainty analysis by deterministic sampling can give a reference interval to increase the reliability of results. The _{
2ε
} and _{
1ε
}, parameters in the production term and dissipation term of transport equations, dominate the radial distributions of void fraction and liquid temperature.

Subcooled boiling flow has received attention from industrial designers due to its high heat transfer coefficient. However, if the heat flux reaches a critical value a transient vapor film will appear on the heated wall due to the polymerization of bubbles. It may hinder the heat transfer and cause the wall temperature to rise.

In most Computational fluid dynamics (CFD) analysis on subcooled boiling flow, the boundary conditions and models are all treated as deterministic values. However, there might be some uncertainties from boundary condition measuring or the simplification of the model that needs to be considered for the numerical simulation of boiling flow (

In two-phase flow simulations uncertainty analysis is significantly important due to the deficiency in two-phase flow theory and the measuring technique. Compared to the reality or truth value of interest, the errors in simulation results are divided into three parts by ASME V&V 20-2009 standards, which are simulation inputs, numerical methods, and modeling assumptions (

The uncertainty brought by turbulence models for single phase flow and heat transfer has already attracted the attention of researchers (_{
2ε
} in the dissipation term is usually derived from the experimental values of decay exponent, which was obtained in single-phase experiments (

The uncertainty of two-phase models and boundary conditions have been analyzed in our previous work (

The Eulerian two-fluid model along with the RPI model has been widely used in subcooled boiling flow simulations. Their equations and the closure auxiliary models of the RPI model can be found in our previous work (

The standard _{
k
} and _{
b
} represent the turbulent kinetic energy generated by mean velocity gradients and buoyancy, respectively:_{
i
} is the gravitation vector in the _{
1ε
} concerns the production term with a default value of 1.44 and the _{
2ε
} influences the calculation of the dissipation term. The parameter _{
k
} is the Prandtl number of turbulent kinetic energy. It represents the ratio of turbulent viscosity to turbulent kinetic energy diffusion. The turbulent viscosity is used to calculate the turbulent stress term in the momentum equation. Thus, the parameter _{
k
} interrelates momentum and turbulent kinetic energy in the _{
ε
} denotes the ratio of turbulent viscosity to turbulent dissipation rate diffusion and the energy Prandtl number Pr_{
t
} represents the ratio of turbulent viscosity to the thermal diffusion induced by turbulence.

The _{
t
} in transport equations is the turbulent viscosity. As mentioned before, it is a crucial step to determine the value of _{
t
}, since it will be used to calculate the turbulent stress, which is an addition item induced by turbulent fluctuation in the momentum equation. It is different from single-phase flow where the bubbles induce additional turbulence in subcooled boiling flow. This phenomenon can be described by adding a term to the turbulent viscosity (_{
μ
} and _{
μ,g
} are the empirical coefficient achieved by experimental results. Besides, the standard wall function is applied for the near-wall region.

The parameters, including _{
μ
}, _{
μ,g
}, _{
1ε
}, _{
2ε
}, _{
k
}, _{
ε
}, and Pr_{
t
}, interrelated momentum, energy, turbulent kinetic energy, and dissipation rate in the standard k-ε model. The values significantly influence the applicability and accuracy of the _{
μ
} is obtained by the experiments which have a dynamic equilibrium between the production and the dissipation of pulsation kinetic energy in the boundary layer, while it may be not applicable to the flow that deviates from the dynamic equilibrium (

In our previous work, the effects of two-phase model uncertainties and boundary condition uncertainties were analyzed by the Latin Hypercube sampling (LHS) method, respectively (

Thus, the DS method, which can reduce the number of samples substantially, was applied in this paper to predict the effects of turbulence model uncertainties on the simulation results. Unlike random sampling which characterizes the continuous probability density function of the sources, the DS method tries to represent them with a number of deterministic locations, known as sigma points (

In the current work, the DS method with fourth order statistical moments, which is abbreviated to DS4, will be used and its results will be compared with the experimental data to provide a confidence interval for parameters. Besides, the contribution of each uncertainty source can be obtained by data analysis. As described in _{
i2
} need to be reset due to the fact that the sum of all weights should be one. The central point weight will then become:

Then the fifteen samples extracted by DS4 are set as inputs in functions or codes, respectively. The outputs will be analyzed to quantify the effects propagated from the input uncertainty sources by statistical parameters, including mean value _{
n
} represents the _{
i
} is the contribution of the

The DEBORA experiment (

In order to avoid the error introduced by mesh, eight meshes with different radial and axial nodes were used to analyze the grid sensitivity with Case4. The radial distributions of void fraction and liquid temperature were compared and the results are presented in

Grid sensitivity analysis.

The samples of turbulence model parameters were set as computational inputs to analyze the uncertainty transition. The uncertainties of radial parameters in the DEBORA benchmark are presented and discussed in

The distribution of radial parameters with uncertainties, including void fraction, liquid temperature, phase velocity, and bubble diameter, are presented in

Effects of model uncertainties on radial parameters (the experiment data of Case1 are extracted from

According to the results in

Based on the assumption that the parameters are independent from each other, the contributions of each uncertainty source to the radial void fraction and radial liquid temperature can be calculated by using

As shown in _{
2ε
} and _{
1ε
} have a significant influence on the calculation of wall maximum void fraction. In addition, compared with the pipe center, the source contribution of _{
μ,g
} to void fraction increases in the near-wall area. This means that the parameters _{
2ε
}, _{
1ε
}, and _{
μ,g
} must be treated seriously in critical heat flux prediction. What is more, no matter the amount of void fraction or liquid temperature, the parameters _{
2ε
} and _{
1ε
} are always the most influential parameters in radial positions. This means that, to reasonably simulate subcooled boiling flow, the accurate value or probability density functions of _{
2ε
} and _{
1ε
} in the standard _{
t
} contributes more in liquid temperature calculation. This is because turbulence will enhance the properties of thermal conduction, while Pr_{
t
} is an important parameter for heat conductivity coefficient calculation.

Uncertainty sources contribution on void fraction.

Uncertainty sources contribution on liquid temperature.

The uncertainty of two-phase models and boundary conditions has been analyzed using Latin Hypercube sampling in our previous work (

Compared with the experiment data, a reasonable result within the confidence interval could be obtained. The results suggested that the models in our work can reasonably simulate the DEBORA experiments and the deterministic sampling can be a powerful tool for uncertainty quantification.

The uncertainty band of radial parameters, including void fraction, liquid temperature, phase velocity, and bubble diameter, were obtained, which produced the sensitive parameter regions of the turbulence model.

The contribution of each uncertainty source on different radial positions was analyzed. The parameters _{
2ε
} and _{
1ε
} always played predominant roles on the radial distributions of void fraction and liquid temperature, which specifies that these two parameters need to be treated carefully in further boiling flow simulations.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

XZ contributed to the CFD simulation work and uncertainty analysis. GX contributed to the guidance of the deterministic sampling method. TC contributed to the guidance of Fluent. MP contributed to the guidance of the theory of numerical heat transfer. ZW contributed to the data processing

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This work is supported by the National Natural Science Foundation of China (No. 11705035) and the Natural Science Foundation of Heilongjiang (No. LH2019A009), which are gratefully acknowledged.