## Smagorinsky model to calculate the SGS turbulent viscosity,

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)

2.2. LES turbulent model

In this section, SGS turbulent models are introduced to model the unresolved terms ~s and ~q in the momentum equation and the energy equation, respectively.

SGS Reynolds stresses ~s could be expressed by Eq. (7) based on the Boussinesq hypothesis.

~sij ¼ �mt � 2Sij �

2 3 Skkdij

� (7)

Sij ¼ � vui=vxj þ vuj=vxi

� =2 (8)

In this paper, the standard Smagorinsky model is used to calculate the SGS turbulent viscosity, as shown in Eqs. (9e12).

mt ¼ rL2s jSj (9)

Ls ¼ CsD (10)

D ¼ V1=3 ¼ ðDxDyDzÞ1=3 (11)

jSj ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi 2SijSij

q (12)

In theory, the Smagorinsky constant Cs can be derived by assuming that the production and dissipation of subgrid-scale turbulent kinetic energy are in equilibrium [25]. In practice, it has been found that the best results could be obtained when the Smagorinsky constant is set as 0.1 [26].

For another unclosed term, SGS heat flux term ~q in the energy equation, it could be expressed by Eq. (13) based on the gradient hypothesis [27]. It has been found that for air-like forced flow in this paper, the simulation result is not sensitive to the value of turbulent Prandtl number, Prt [27]. According to the suggestions from the existing research [26], the turbulent Prandtl number Prt is set as 0.90 in this paper.

~qj ¼ �lt vT vxj

(13)

lt ¼ rmtPrt (14)

H. Zhang et al. / Nuclear Engineering and Technology 49 (2017) 1310e13171312

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