Development of a coupling between a system thermal–hydraulic code and a reduced order CFD model

https://doi.org/10.1016/j.anucene.2020.108056Get rights and content

Highlights

  • Coupling between RELAP5-MOD3.3 and a reduced order model of the CFD solver Open-FOAM.

  • The coupled model is 3–5 times faster than coupling RELAP5 with the CFD solver.

  • The coupled simulation results are close to those of coupled RELAP5/CFD simulations.

  • Methodology works well on open and closed pipe flow configurations.

  • Tested and evaluated the coupled model on parametric problems.

Abstract

The nuclear community has coupled several three-dimensional Computational Fluid Dynamics (CFD) solvers with one-dimensional system thermal–hydraulic (STH) codes. This work proposes to replace the CFD solver by a reduced order model (ROM) to reduce the computational cost. The system code RELAP5-MOD3.3 and a ROM of the finite volume CFD solver OpenFOAM are coupled by a partitioned domain decomposition coupling algorithm using an implicit coupling scheme. The velocity transported over a coupling boundary interface is imposed in the ROM using a penalty method. The coupled models are evaluated on open and closed pipe flow configurations. The results of the coupled simulations with the ROM are close to those with the CFD solver. Also for new parameter sets, the coupled RELAP5/ROM models are capable of predicting the coupled RELAP5/CFD results with good accuracy. Finally, coupling with the ROM is 3–5 times faster than coupling with the CFD solver.

Introduction

The Generation IV innovative nuclear systems cooled by heavy liquid metal are the subject of an ongoing interest demonstrated by a large number projects in progress. One of them is MYRRHA, an experimental fast-spectrum irradiation facility featuring a pool-type primary cooling system operating with molten Lead–Bismuth Eutectic (LBE), that is currently being developed by SCK·CEN, a nuclear research institution in Belgium (Abderrahim, 2012).

For the design and safety assessment of a new generation of nuclear reactors, computer codes have been developed for the thermal–hydraulic analyses of the reactor’s primary system in operational and accident conditions. There are two main types of numerical codes for thermal–hydraulic analyses used in the nuclear industry: the system codes, also called the lumped parameter codes, based on one-dimensional (1D) models of physical transport phenomena and the field codes, based on three-dimensional (3D) Computational Fluid Dynamics (CFD) models (OECD/NEA, 1999). The system codes are, in general, based upon the solution of six balance equations for liquid and vapor. In addition, they use (quasi-steady state) heat transfer correlations to model the heat transfer between a solid, such as tubes or structures, and its surrounding fluid (Petruzzi and D’Auria, 2008, Bertolotto et al., 2009).

The flow in many reactor primary components exhibit phenomena as natural circulation, mixing and stratification that cannot be modeled by system codes adequately. CFD codes are therefore used to numerically simulate these types of transient flows to accurately quantify the system behavior in accident conditions and to handle complex geometries (Toti et al., 2018). However, the number of nuclear reactor simulations in a safety analysis is, in the majority of cases, beyond the possibilities of present hardware if a CFD code is used alone.

Thus, to get the best out of both worlds, coupling between system and CFD codes has been postulated as a new method for thermal–hydraulic analyses. The nuclear community has performed extensive research on interfacing CFD codes with the traditional system codes. Gibeling and Mahaffy (2002) were among the first to study the transition between the 1D and 3D descriptions at an interface.

A well recognized system thermal–hydraulic (STH) code by many nuclear authorities for safety analyses is the RELAP5 series. The RELAP5 series have been coupled with several computer codes as the sub-channel code COBRA-TF (Lee et al., 1992, Jeong et al., 1997), the containment analysis code GOTHIC (Grgić et al., 2004, Huang and Ma, 2016) and CFD codes ANSYS-CFX (previous called CFDS-FLOW3D) (Aumiller et al., 2001, Burns et al., 1988), ANSYS Fluent (Schultz and Weaver, 2003, Feng et al., 2017, Li et al., 2014, Anderson et al., 2008, Angelucci et al., 2017) and Star-CCM+ (Jeltsov et al., 2013). Recently, work has been conducted in the framework of the THINS project of the 7th Framework EU Program on nuclear fission safety (Bandini et al., 2015, Pialla et al., 2015).

SCK·CEN uses the RELAP5-3D (The RELAP5-3D Code Development Team, 2018) version for MYRRHA safety studies that allows the use of LBE as a working fluid. Moreover, SCK·CEN has developed a numerical algorithm to couple RELAP5-3D with ANSYS Fluent for multi-scale transient simulations of pool-type reactors (Toti et al., 2018). Another CFD code that has been coupled already to several STH codes (Pialla et al., 2015, Fiorina et al., 2015, Zhang, 2020), but to the best of the authors’ knowledge not yet with RELAP5, is the open source code OpenFOAM (OF) (Jasak et al., 2007).

Even though coupled systems require considerably less computational resources and time than stand-alone CFD codes, the gain in computational effort is still limited by the CFD part (Bury, 2017). To overcome this burden, this work proposes to couple the system code with a reduced order model (ROM) of the high fidelity CFD code.

The basic idea of reduced order CFD modeling is to retain the essential physics and dynamics of a high fidelity CFD model by projecting the (discretized) equations describing the fluid problem onto a low-dimensional basis (Hesthaven et al., 2016, Quarteroni et al., 2015, Veroy et al., 2003, Rozza et al., 2008). This basis contains only the essential features of a number of solutions of the high-fidelity simulations. Therefore, the reduced order model contains a lower number of degrees of freedom than the high fidelity models. That way, they are computationally more efficient, but have generally a lower accuracy than the high fidelity models (Lassila et al., 2014, Grepl et al., 2007). Parametric ROMs can then be used for evaluating solutions on new sets of parameter values or for time evolution that are different from those of the original simulations (Benner et al., 2015, Gunzburger et al., 2007, Fick et al., 2017). Therefore, reduced order models are suitable for control purposes or sensitivity analyses that require results of a large number of simulations for different parameter values.

In this work, the STH code RELAP5-MOD3.3 (Nuclear Safety Analysis Division, 2003) and a reduced order CFD model that is constructed using the libraries of the open source code OpenFOAM 6 are coupled, which is called the RELAP5/ROM model hereafter. The codes are coupled using a partitioned domain decomposition coupling algorithm, which is explained in Section 2. The exchange of the hydraulic quantities between the coupled domains at the coupling interfaces is explained in Section 3. The CFD and ROM formulations for an incompressible Newtonian fluid are described in Sections 4 The coupled codes’ governing equations and models, 5 POD-Galerkin reduced order model for incompressible turbulent flow, respectively. In Section 6, some challenges for coupling STH codes with reduced order models are presented. The set-up of three numerical test cases, the open pipe flow test, the open pipe flow reversal test and the closed pipe flow test, are described in Section 7. Then in Section 8, the coupling methodology is first evaluated by comparing the results of a coupled RELAP5/CFD model with RELAP5 stand-alone results. Consecutively, the coupled RELAP5/ROM model is tested on a series of parametric problems that are evaluated against the coupled RELAP5/CFD model and the results are discussed in Section 9. Finally, conclusions are drawn and an outlook for further improvements is provided in Section 10.

Section snippets

Coupling methodology

A methodology is developed for a partitioned coupling approach (Matthies and Steindorf, 2003) together with a domain decomposition method (Smith et al., 2004) in which the different domains are resolved separately by independent solvers. The whole simulation domain is split into sub-domains; where the one-dimensional approximation is deemed accurate enough for the given problem, the sub-domain is allocated to the STH code and if not to the CFD code. The number of coupling faces between the

Transport of hydraulic quantities over the coupling interfaces of a coupled RELAP5/CFD model

The transport of hydraulic quantities over the coupling interfaces of a coupled RELAP5 with OpenFOAM (RELAP5/CFD) model is explained in this section. The procedure is the same when RELAP5 is coupled with the reduced order model.

As introduced previously, the coupling method is based on a domain decomposition technique. In this work, the computational domain Ω is divided into several non-overlapping sub-domains: the STH sub-domain(s), ΩSTH, attributed to RELAP5 and the CFD sub-domain(s), ΩCFD,

The coupled codes’ governing equations and models

This section presents a brief description of the best-estimate system thermal–hydraulic code RELAP5-MOD3.3. Furthermore, the governing equations that are discretized and solved with the CFD code OpenFOAM are described as those equations are projected onto a reduced basis in order to construct the ROM.

POD-Galerkin reduced order model for incompressible turbulent flow

The reduced order model for the full order CFD code is constructed using a POD-Galerkin technique. POD stands for Proper Orthogonal Decomposition and is used to reduce the dimensionality of a system by transforming the original set of Nx degrees of freedom into a new set of Nr degrees of freedom, so-called modes, where Nr<Nx. These modes are ordered in such a way that the first few modes retain most of the energy present in the original solution (Lassila et al., 2014). For more details about

Challenges for coupling system thermal–hydraulic codes with a reduced order model

In the previous section and in Section 2, we noted that POD-Galerkin reduced order models are, in general, sensitive to numerical instabilities (Akhtar et al., 2009, Sirisup and Karniadakis, 2005, Bergmann et al., 2009). This is one of the main challenges of reduced order modeling for fluid flow problems (Lassila et al., 2014). Therefore, only an implicit coupling scheme is considered in this work, which is numerically more stable than the explicit coupling schemes (Toti et al., 2016).

The

Set-up numerical test cases

In this section, the set-ups for three different configurations are described: the open pipe flow test, the open pipe flow reversal test and the closed pipe flow test. All tests are carried out for single-phase water flow with kinematic viscosity ν=1.0·10−6 m2/s.

For the coupled models, the computational domain is divided into a CFD sub-domain and an STH sub-domain. For all configurations, the CFD sub-domain consists of a circular pipe of length LCFD=0.5 m and diameter D = 0.1 m. A mesh with

Results

For each of the flow configurations, the coupled model is first evaluated against the corresponding STH stand alone model in order to evaluate the implicit coupling methodology. Thereafter, the coupled RELAP5/ROM models are tested and compared with the coupled RELAP5/CFD models.

Discussion

The RELAP5/CFD models exhibit numerical perturbations in the form of oscillations throughout the coupled simulations. As concluded in previous works on coupled CFD codes with 1D system codes (Toti et al., 2018, Grunloh and Manera, 2016), these perturbations are caused by the overestimation of the mass flow rate at the coupling interfaces in the first few time steps of the simulations. A change in pressure drop will immediately affect the whole solution domain as the fluid density does not

Conclusions and outlook

The best-estimate system thermal–hydraulic code RELAP5 is coupled with the finite volume CFD solver OpenFOAM and its reduced order model. The codes are coupled implicitly by a partitioned domain decomposition coupling algorithm in which the hydraulics variables are exchanged between the sub-domains at the coupling boundary interfaces.

The ROM is constructed with a finite volume based POD-Galerkin projection method. The average velocity determined at the single junction of the STH sub-domain at

CRediT authorship contribution statement

S. Kelbij Star: Conceptualization, Methodology, Visualization, Writing - original draft. Giuseppe Spina: Methodology, Software, Formal analysis, Investigation, Visualization, Writing - review & editing. Francesco Belloni: Writing - review & editing, Project administration. Joris Degroote: Writing - review & editing, Project administration, Funding acquisition.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

The authors would like to thank the ITHACA-FV developers and contributors for their input and insightful discussions on the development of reduced order models. In particular, Giovanni Stabile, Saddam Hijazi and Matteo Zancanaro from SISSA mathLab, Umberto Morelli from ITMATI and Sokratia Georgaka from Imperial College London.

References (73)

  • M.D. Gunzburger et al.

    Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data

    Computer Methods in Applied Mechanics and Engineering

    (2007)
  • Z. Huang et al.

    Performance evaluation of passive containment cooling system of an advanced PWR using coupled RELAP5/GOTHIC simulation

    Nuclear Engineering and Design

    (2016)
  • R.I. Issa

    Solution of the implicitly discretised fluid flow equations by operator-splitting

    Journal of Computational Physics

    (1986)
  • B.E. Launder et al.

    Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc

    Letters in Heat and Mass Transfer

    (1974)
  • D. Lazzaro et al.

    Radial basis functions for the multivariate interpolation of large scattered data sets

    Journal of Computational and Applied Mathematics

    (2002)
  • W. Li et al.

    Preliminary study of coupling CFD code FLUENT and system code RELAP5

    Annals of Nuclear Energy

    (2014)
  • J.-G. Liu et al.

    Stable and accurate pressure approximation for unsteady incompressible viscous flow

    Journal of Computational Physics

    (2010)
  • S. Lorenzi et al.

    POD-Galerkin method for finite volume approximation of Navier-Stokes and RANS equations

    Computer Methods in Applied Mechanics and Engineering

    (2016)
  • G. Rozza et al.

    On the stability of the reduced basis method for Stokes equations in parametrized domains

    Computer Methods in Applied Mechanics and Engineering

    (2007)
  • S. Sirisup et al.

    Stability and accuracy of periodic flow solutions obtained by a POD-penalty method

    Physica D: Nonlinear Phenomena

    (2005)
  • G. Stabile et al.

    Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier-Stokes equations

    Computers & Fluids

    (2018)
  • S. Star et al.

    A POD-Galerkin reduced order model of a turbulent convective buoyant flow of sodium over a backward-facing step

    Applied Mathematical Modelling

    (2021)
  • A. Toti et al.

    Improved numerical algorithm and experimental validation of a system thermal-hydraulic/CFD coupling method for multi-scale transient simulations of pool-type reactors

    Annals of Nuclear Energy

    (2017)
  • A. Toti et al.

    Coupled system thermal-hydraulic/CFD analysis of a protected loss of flow transient in the MYRRHA reactor

    Annals of Nuclear Energy

    (2018)
  • K. Veroy et al.

    Reduced-basis approximation of the viscous Burgers equation: rigorous a posteriori error bounds

    Comptes Rendus Mathematique

    (2003)
  • Z. Wang et al.

    Proper orthogonal decomposition closure models for turbulent flows: a numerical comparison

    Computer Methods in Applied Mechanics and Engineering

    (2012)
  • H.A. Abderrahim

    Multi-purpose hYbrid Research Reactor for High-tech Applications a multipurpose fast spectrum research reactor

    International Journal of Energy Research

    (2012)
  • I. Akhtar et al.

    On the stability and extension of reduced-order Galerkin models in incompressible flows

    Theoretical and Computational Fluid Dynamics

    (2009)
  • M. Angelucci et al.

    STH-CFD codes coupled calculations applied to HLM loop and pool systems

    Science and Technology of Nuclear Installations

    (2017)
  • P. Benner et al.

    A survey of projection-based model reduction methods for parametric dynamical systems

    SIAM Review

    (2015)
  • Burns, A., Jones, I., Kightley, J., Wilkes, N. Harwell, 1988. FLOW3D, Release 2.1: User Manual, UKAEA Report AERE-R...
  • T. Bury

    Coupling of CFD and lumped parameter codes for thermal-hydraulic simulations of reactor containment

    Computer Assisted Methods in Engineering and Science

    (2017)
  • F. Chinesta et al.

    A short review on model order reduction based on proper generalized decomposition

    Archives of Computational Methods in Engineering

    (2011)
  • M. Couplet et al.

    Intermodal energy transfers in a proper orthogonal decomposition-Galerkin representation of a turbulent separated flow

    Journal of Fluid Mechanics

    (2003)
  • Ferziger, J.H., Perić, M., 2002. Computational methods for fluid dynamics, vol. 3, Springer....
  • S. Georgaka et al.

    Parametric POD-Galerkin model order reduction for unsteady-state heat transfer problems

    Communications in Computational Physics

    (2018)
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      In recent years, POD has been widely utilized [10,11]. In the field of fluid dynamics [12], without pretending to be exhaustive, we may cite recent research in the fields of thermal-hydraulics [13], heat transfer [14], Reynolds-averaged Navier–Stokes equations [15] and even applications to industry problems [16,17]. Finally, when our construct will be ready, we will apply our model for the investigation of the transient phenomena of natural convection [21], trying to validate our creation reproducing the results of a CFD.

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