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## Numerical methodology in GASFLOW-MPI

H. Zhang et al. / Nuclear Engineering and Technology 49 (2017) 1310e1317 1311

subspace method, the conjugate gradient (CG) method, was pro- posed by Hestenes and Stiefel [20]; it is used for solving the sym- metrical linear system. Then,many variantmethods for symmetrical linear equations were presented, such as the minimal residual (MINRES)method [21] and symmetric LQ (SYMMLQ)method [21]. At the same time, the Krylov subspace method also extended to the asymmetrical cases, such as the generalized minimal residual (GMRES) method [22]. Besides the choice of Krylov subspace methods, how to construct the preconditioning matrix is another important technology for the linear solver performance. In this pa- per, several advanced Krylov subspace methods and scalable pre- conditioningmethods are implemented and compared. The optimal combination of the Krylov subspace solver and preconditioning method is presented to improve computational performance.

In order to accurately predict the turbulent behavior, a suitable turbulent model should be used. Several turbulent models had been developed and validated in GASFLOW-MPI, including the algebraic turbulent model [23], the keepsilon turbulentmodel [23], and the LES turbulent model [24]. The LES turbulent model can resolve the large-scale turbulent fluctuations directly, wherein only the unresolved subgrid scale fluid motion should be modeled. Meanwhile, for the Reynolds-averaged NaviereStokes (RANS)- based turbulent models, such as the algebraic model and the keepsilon model, all turbulent fluctuations are unresolved. With the help of the powerful parallel computational capability, the LES turbulent model is used in this paper to resolve more detailed turbulence information. The standard Smagorinsky subgrid scale (SGS) model [25] is used to model the effects of unresolved small- scale fluidmotions in the LES turbulentmodel. The turbulent inflow boundary based on white noise is used to consider the turbulent information at the inlet. The LES turbulent model in GASFLOW-MPI had been validated by turbulent jet flows, which are the free shear turbulence [24]. In this study, we simulate a backward-facing step turbulent flow by the LES turbulent model to study the wall- bounded turbulent flows, a widespread turbulence phenomenon in scientific and engineering applications.

This paper is organized as follows. The physical model in GASFLOW-MPI is described in Section 2. The conservation equation, LES turbulent model, and numerical methods are discussed here. The parallel linear solver and parallel computing capability are discussed in Section 3. The turbulent simulation is presented and discussed in Section 4. The conclusions are presented in Section 5.

- Numerical methodology in GASFLOW-MPI

2.1. Governing equations

GASFLOW-MPI is a powerful CFD numerical tool used to simu- late the complicated thermalehydraulic behavior in NPP contain- ments where a three-dimensional (3-D) transient compressible multicomponent NaviereStokes equation system is solved [11]. However, as only single-species isothermal gas flow is carried out in this paper, the radiation transfer model, combustion model, and mass/heat transfer model are therefore not considered in the following conservation equations, which include the volume equation, mass equation, momentum equations, and internal en- ergy equation [Eqs. (1e4)]. General thermodynamic equation of state, Eq. (5), and the general caloric equation of state, Eq. (6), are also used to close the governing equation system.

Volume equation

vV vt

¼ VV$ðb� uÞ (1)

Mass equation

vr vt

¼ V$½rðb� uÞ� (2)

Momentum equations

vðruÞ vt

¼ V$½ruðb� uÞ� � Vpþ V$sþ rg� V$~s (3)

Internal energy equation

vðrIÞ vt

¼ V$½rIðb� uÞ� � pV$u� V$q� V$~q (4)

General thermodynamic equation of state

p ¼ Zðr; TÞr R M

T (5)

General caloric equation of state

I ¼ Iðr; TÞ (6)