Data-Driven RANS Turbulence Closures for Forced Convection Flow in Reactor Downcomer Geometry

Abstract

Recent progress in data-driven turbulence modeling has shown its potential to enhance or replace traditional equation-based Reynolds-averaged Navier-Stokes (RANS) turbulence models. This work utilizes invariant neural network (NN) architectures to model Reynolds stresses and turbulent heat fluxes in forced convection flows (when the models can be decoupled). As the considered flow is statistically one dimensional, the invariant NN architecture for the Reynolds stress model reduces to the linear eddy viscosity model. To develop the data-driven models, direct numerical and RANS simulations in vertical planar channel geometry mimicking a part of the reactor downcomer are performed. Different conditions and fluids relevant to advanced reactors (sodium, lead, unitary-Prandtl-number fluid, and molten salt) constitute the training database. The models enabled accurate predictions of velocity and temperature, and compared to the baseline $\mathit{k}-\mathit{\tau }$ turbulence model with the simple gradient diffusion hypothesis, do not require tuning of the turbulent Prandtl number. The data-driven framework is implemented in the open-source graphics processing unit–accelerated spectral element solver nekRS and has shown the potential for future developments and consideration of more complex mixed convection flows.

Keywords:

• Machine learning
• turbulence modeling
• forced convection
• low- and high-Prandtl fluids
• data-driven modeling

Nomenclature

$a_{\mathit{ij}}$	=	= anisotropic tensor
$b_{\mathit{ij}}$	=	= scaled anisotropic tensor
$d_{\mathit{ij}}=\mathrm{D}$	=	= diffusivity tensor
$f_n$	=	= vector basis functions
$g_n$	=	= tensor basis functions
$h_i$	=	= scaled THF
$\mathcal{I}$	=	= invariants for TBNN
$\mathcal{J}$	=	= additional invariants for VBNN
$\mathit{k}$	=	= TKE
$k_T$	=	= temperature variance
$L_i=\left\{L_x,L_y,L_z\right\}$	=	= channel dimensions
$\mathcal{L}$	=	= loss
$N_t$	=	= number of tensor bases
$N_v$	=	= number of vector bases
$\mathit{p}$	=	= pressure
$\mathit{q}{ }^{\mathrm{\prime \prime }}$	=	= wall heat flux
$s_{\mathit{ij}}=\mathrm{S}$	=	= scaled symmetric tensor
$\mathit{t}$	=	= time
$t_{ij}^{(n)}={\mathrm{T}}^{(\mathit{n})}$	=	= tensor bases
$\mathit{T}$	=	= temperature
$u_i=\left\{u_x,u_y,u_z\right\}$	=	= velocity vector
$\overline{u_j^{\prime }\mathit{T}{ }^{\mathrm{\prime }}}$	=	= THF vector
$\overline{u_i^{\prime }{u}_j^{\prime }}$	=	= RS tensor
$v_i^{(n)}$	=	= vector bases
$v_{ij}^{(n)}$	=	= vector bases (with postponed multiplication by ${\vartheta }_j$)
$w_{\mathit{ij}}=\mathrm{W}$	=	= scaled antisymmetric tensor
$x_i=\left\{\mathit{x},\mathit{y},\mathit{z}\right\}$	=	= coordinate vector
$y_w$	=	= distance to the wall

Greek

${\delta }_{\mathit{ij}}$	=	= Kronecker delta
$\mathit{\epsilon }$	=	= rate of dissipation of TKE
${\epsilon }_T$	=	= thermal dissipation rate
${\vartheta }_j=\mathit{\vartheta }$	=	= scaled temperature gradient
$\mathit{\tau }$	=	= turbulent timescale

Nondimensional criteria

$\mathrm{Re}$	=	= Reynolds number
$\mathrm{Pr}$	=	= Prandtl number
$\mathrm{Ri}$	=	= Richardson number

Subscripts

$\mathit{a}$	=	= antisymmetric
$\mathit{in}$	=	= inlet
$\mathit{s}$	=	= symmetric
$\mathit{t}$	=	= turbulent

Other notations

$\mathrm{tr}(\cdot )$	=	= trace of matrix
$\cdot {}^{\mathrm{\top }}$	=	= transposed matrix
$\bar{\cdot }$	=	= Reynolds-averaged component
$\cdot { }^{\mathrm{\prime }}$	=	= fluctuating components
$\hat{\cdot }$	=	= predicted quantity

Acknowledgments

The authors are grateful to all the IRP NEAMS-1.1 Challenge Problem 1 participants for valuable discussions and recommendations.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Notes

^a See an example on GitHub (https://github.com/aiskhak/nekRS_use_NN).

Additional information

Funding

This work was funded by the U.S. Department of Energy’s IRP entitled, “Center of Excellence for Thermal-Fluids Applications in Nuclear Energy: Establishing the Knowledgebase for Thermal-Hydraulic Multiscale Simulation to Accelerate the Deployment of Advanced Reactors—IRP-NEAMS-1.1: Thermal-Fluids Applications in Nuclear Energy.”