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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 . 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.
¼ VV$ðb� uÞ (1)
¼ V$½rðb� uÞ� (2)
¼ V$½ruðb� uÞ� � Vpþ V$sþ rg� V$~s (3)
Internal energy equation
¼ V$½rIðb� uÞ� � pV$u� V$q� V$~q (4)
General thermodynamic equation of state
p ¼ Zðr; TÞr R M
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
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
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 . In practice, it has been found that the best results could be obtained when the Smagorinsky constant is set as 0.1 .
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 . 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 . According to the suggestions from the existing research , the turbulent Prandtl number Prt is set as 0.90 in this paper.
~qj ¼ �lt vT vxj
lt ¼ rmtPrt (14)
H. Zhang et al. / Nuclear Engineering and Technology 49 (2017) 1310e13171312