Theory and Methodology

The two-step procedure is a common approach for multiphysics full-core nuclear reactor analysis. These steps are:

Step 1: Generate neutron group constant data with a lattice or full-core reactor model on a high-fidelity, continuous/fine-energy neutronics software at various reactor states.

Step 2: Use an intermediate-fidelity, computationally cheaper neutronics software with the neutron group constant data to perform multiphysics reactor analysis.

Moltres falls under Step 2 of the two-step procedure. Moltres solves the conventional multigroup neutron diffusion and delayed neutron precursor equations. The precursor equations include precursor drift terms to account for precursor movement under molten salt flow. Moltres is built on the MOOSE finite element framework, enabling highly flexible and scalable reactor simulations.

Multigroup Neutron Diffusion

More information to be added in future.

Heat Transfer and Fluid Flow

Moltres compiles with the MOOSE Navier-Stokes and Heat Transfer modules for flow modeling capabilities by default. Moltres couples with these modules natively because they are all built on the MOOSE framework.

Past work with Moltres have modeled incompressible salt flow and temperature advection-diffusion with the INS or INSAD implementations of the Navier-Stokes equations (Peterson et al., 2018). Coupling with the compressible or finite volume implementations within the Navier-Stokes module are likely possible, but have not been demonstrated yet.

For turbulence modeling, Moltres has a Spalart-Allmaras (SA) turbulence model (Spalart and Allmaras, 1992) implementation with streamline-upwind Petrov-Galerkin (SUPG) stabilization. On balance the SA model is a complete (does not require prior knowledge of the actual turbulence behavior) and computationally efficient turbulence model for approximating wall-bounded turbulent flows. The SA model implementation in Moltres is compatible with the INSAD implementation only. Additionally, Moltres contains turbulent diffusion physics kernels for temperature and the delayed neutron precursors.

The SA model in Moltres follows the (SA-noft2-R) implementation as described on the NASA Turbulence Modeling Resource website. The $var element = document.getElementById("moose-equation-69d98fe9-e250-4afd-863c-379a9b6ddb80");katex.render("f_{t2}", element, {displayMode:false,throwOnError:false});$ term is togglable using the use_ft2_term input parameter (false by default). Moltres' implementation solves for the modified dynamic viscosity $var element = document.getElementById("moose-equation-23af8de0-5504-436d-92b9-48553e606f03");katex.render("\\tilde{\\mu}", element, {displayMode:false,throwOnError:false});$ which can be easily obtained using $var element = document.getElementById("moose-equation-01a508a3-c47b-41f3-b921-a078b655ecb3");katex.render("\\nu=mu/\\rho", element, {displayMode:false,throwOnError:false});$ to convert $var element = document.getElementById("moose-equation-80b27d41-2df3-4108-9d80-57a65f733926");katex.render("\\nu", element, {displayMode:false,throwOnError:false});$ to $var element = document.getElementById("moose-equation-8ea90396-26cc-4ddc-9c3e-50f284ebc3bd");katex.render("\\mu", element, {displayMode:false,throwOnError:false});$ as follows:

(1)var element = document.getElementById("moose-equation-6b60fca4-bb8b-4945-b20f-11bfe2c3fa88");katex.render("\\rho \\frac{\\partial\\tilde{\\mu}}{\\partial t} + \\rho \\mathbf{u}\\cdot\\nabla\\tilde{\\mu} = \\rho c_{b1} \\left(1-f_{t2}\\right)\\tilde{S}\\tilde{\\mu} + \\frac{1}{\\sigma}\\{\\nabla\\cdot\\left[\\left(\\mu+ \\tilde{\\mu}\\right)\\nabla\\tilde{\\mu}\\right] + c_{b2}|\\nabla\\tilde{\\mu}|^2\\} - \\left(c_{w1}f_w - \\frac{c_{b1}}{\\kappa^2}f_{t2}\\right)\\left( \\frac{\\tilde{\\mu}}{d}\\right)^2", element, {displayMode:true,throwOnError:false});

where

var element = document.getElementById("moose-equation-9da6c51e-3391-471e-9e92-a4095c740fd2");katex.render("\\mu_t = \\tilde{\\mu}f_{v1} = \\text{turbulent eddy viscosity}, \\hspace{4cm} f_{v1} = \\frac{\\chi^3}{\\chi^3 + c_{v1}^3}, \\hspace{4cm} \\chi = \\frac{\\tilde{\\mu}}{\\mu}, \\hspace{2cm} \\\\ \\hspace{1cm} \\tilde{S} = \\Omega + \\frac{\\tilde{\\nu}}{\\kappa^2 d^2} f_{v2}, \\hspace{6cm} f_{v2} = 1 - \\frac{\\chi}{1+\\chi f_{v1}}, \\hspace{2cm} \\Omega = \\sqrt{2W_{ij}W_{ij}} = \\text{vorticity magnitude}, \\\\ W_{ij} = \\frac{1}{2}\\left(\\frac{\\partial u_i}{\\partial x_j} - \\frac{\\partial u_j}{\\partial x_i} \\right), \\hspace{5cm} f_w = g\\left(\\frac{1 + c_{w3}^6}{g^6 + c_{w3}^6}\\right)^{1/6}, \\hspace{3cm} g = r + c_{w2}\\left(r^6 - r\\right), \\\\ r = \\text{min}\\left(\\frac{\\tilde{\\nu}}{\\tilde{S}\\kappa^2d^2}, 10\\right), \\hspace{5cm} f_{t2} = c_{t3} \\exp{\\left(-c_{t4}\\chi^2\\right)}, \\hspace{6cm}", element, {displayMode:true,throwOnError:false});

and the constants are:

var element = document.getElementById("moose-equation-4be9580a-2436-404b-a142-a98659f47210");katex.render("\\sigma = \\frac{2}{3}, \\ c_{b1} = 0.1355, \\ c_{b2} = 0.622, \\ \\kappa = 0.41, \\ c_{w1} = \\frac{c_{b1}}{\\kappa^2} + \\frac{1+c_{b2}}{\\sigma}, \\ c_{w2} = 0.3, \\ c_{w3} = 2, \\ c_{v1} = 7.1, \\ c_{t3} = 1.2, \\ c_{t4} = 0.5 \\ \\text{.}", element, {displayMode:true,throwOnError:false});

We performed simple verification and validation tests of the SA model in Moltres using reference problems for channel, pipe, and backward-facing step flow. The Moltres test dataset is available on Zenodo. The following subsections show plots comparing results from Moltres to reference data from literature.

Turbulent Channel Flow Verification Test

The channel flow test is based on the direct numerical simulation (DNS) of turbulent channel flow with $var element = document.getElementById("moose-equation-4fe10ccd-8eaa-4caf-9f3b-5d4f92cb840b");katex.render("Re_\\tau\\approx395", element, {displayMode:false,throwOnError:false});$ by Moser et al. (1999). The Moltres input file for this problem may be found in moltres/problems/2023-basic-turbulence-cases/channel-flow/.

Figure 1: Normalized velocity distribution across the channel.

Figure 2: Dimensionless velocity vs dimensionless wall distance.

Figure 3: Normalized stress distribution across the channel.

Turbulent Pipe Flow Validation Test

The pipe flow test is based on the turbulent pipe flow experiment with $var element = document.getElementById("moose-equation-3d3079f9-9608-4e8b-9ab1-8f2543b07c72");katex.render("Re\\approx 40,000", element, {displayMode:false,throwOnError:false});$ by Laufer (1954). The Moltres input file for this problem may be found in moltres/problems/2023-basic-turbulence-cases/pipe-flow/.

Figure 4: Normalized velocity distribution across the pipe.

Figure 5: Dimensionless velocity vs dimensionless wall distance.

Figure 6: Normalized turbulent shear stress distribution across the channel.

Backward-Facing Step Flow Validation Test

The backward-facing step (BFS) flow test is based on the BFS flow experiment with $var element = document.getElementById("moose-equation-f542ec30-3773-452f-854f-1b3b64c61a1d");katex.render("Re\\approx 36,000", element, {displayMode:false,throwOnError:false});$ by Driver and Seegmiller (1985). The Moltres input file for this problem may be found in moltres/problems/2023-basic-turbulence-cases/pipe-flow/. The SA model implementation in Moltres performs largely similarly to the reference SA model results provided on the Turbulence Modeling Resource website.

Table 1: Flow reattachment length estimates (normalized by step height $var element = document.getElementById("moose-equation-45ffbbae-3329-4ec8-b2c7-94d857723008");katex.render("H", element, {displayMode:false,throwOnError:false});$ )

Reference experimental data	Reference SA model	Moltres
$var element = document.getElementById("moose-equation-cba48658-e2f4-475a-a47b-4bbcbdc43f12");katex.render("6.26 \\pm 0.10", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-3a59da49-137e-49bd-8d25-efda184ebc49");katex.render("6.1", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-5388fa4d-7a87-4ca1-a2b3-86f85ff02101");katex.render("6.36", element, {displayMode:false,throwOnError:false});$

Figure 7: Normalized velocity distribution at $var element = document.getElementById("moose-equation-d3844f83-66ff-468f-840e-c5da61b4eb9a");katex.render("x/H", element, {displayMode:false,throwOnError:false});$ equal -4 (upstream of step).

Figure 8: Normalized velocity distribution at various $var element = document.getElementById("moose-equation-c341f770-4baf-49f7-94ce-bdb4f8a0e8af");katex.render("x/H", element, {displayMode:false,throwOnError:false});$ locations (downstream of step).

Figure 9: Skin friction coefficient along the bottom wall.

Figure 10: Skin pressure coefficient along the bottom wall.

Figure 11: Normalized turbulent shear stress distribution at various $var element = document.getElementById("moose-equation-c2832324-82b4-4dbe-ae24-ebc3f030210c");katex.render("x/H", element, {displayMode:false,throwOnError:false});$ locations (downstream of step).

Figure 12: Close-up of the velocity magnitude and streamlines around the backward-facing step

References

David M. Driver and H. Lee Seegmiller. Features of a reattaching turbulent shear layer in divergent channelflow. AIAA Journal, 23(2):163–171, February 1985. URL: https://arc.aiaa.org/doi/10.2514/3.8890 (visited on 2022-01-17), doi:10.2514/3.8890.

@article{driver_features_1985,
    author = "Driver, David M. and Seegmiller, H. Lee",
    title = "Features of a reattaching turbulent shear layer in divergent channelflow",
    volume = "23",
    issn = "0001-1452, 1533-385X",
    url = "https://arc.aiaa.org/doi/10.2514/3.8890",
    doi = "10.2514/3.8890",
    language = "en",
    number = "2",
    urldate = "2022-01-17",
    journal = "AIAA Journal",
    month = "February",
    year = "1985",
    pages = "163--171",
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John Laufer. The structure of turbulence in fully developed pipe flow. Technical Report, National Bureau of Standards, January 1954. NTRS Author Affiliations: NTRS Report/Patent Number: NACA-TR-1174 NTRS Document ID: 19930092199 NTRS Research Center: Legacy CDMS (CDMS). URL: https://ntrs.nasa.gov/citations/19930092199 (visited on 2023-10-30).

@techreport{laufer_structure_1954,
    author = "Laufer, John",
    title = "The structure of turbulence in fully developed pipe flow",
    url = "https://ntrs.nasa.gov/citations/19930092199",
    abstract = "Measurements, principally with a hot-wire anemometer, were made in fully developed turbulent flow in a 10-inch pipe at speeds of approximately 10 and 100 feet per second. Emphasis was placed on turbulence and conditions near the wall. The results include relevant mean and statistical quantities, such as Reynolds stresses, triple correlations, turbulent dissipation, and energy spectra. It is shown that rates of turbulent-energy production, dissipation, and diffusion have sharp maximums near the edge of the laminar sublayer and that there exist a strong movement of kinetic energy away from this point and an equally strong movement of pressure energy toward it.",
    urldate = "2023-10-30",
    institution = "National Bureau of Standards",
    month = "January",
    year = "1954",
    note = "NTRS Author Affiliations: NTRS Report/Patent Number: NACA-TR-1174 NTRS Document ID: 19930092199 NTRS Research Center: Legacy CDMS (CDMS)",
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Robert D. Moser, John Kim, and Nagi N. Mansour. Direct numerical simulation of turbulent channel flow up to Reτ=590. Physics of Fluids, 11(4):943–945, April 1999. URL: https://doi.org/10.1063/1.869966 (visited on 2023-10-30), doi:10.1063/1.869966.

@article{moser_direct_1999,
    author = "Moser, Robert D. and Kim, John and Mansour, Nagi N.",
    title = "Direct numerical simulation of turbulent channel flow up to {Reτ}=590",
    volume = "11",
    issn = "1070-6631",
    url = "https://doi.org/10.1063/1.869966",
    doi = "10.1063/1.869966",
    abstract = "Numerical simulations of fully developed turbulent channel flow at three Reynolds numbers up to Reτ=590 are reported. It is noted that the higher Reynolds number simulations exhibit fewer low Reynolds number effects than previous simulations at Reτ=180. A comprehensive set of statistics gathered from the simulations is available on the web at http://www.tam.uiuc.edu/Faculty/Moser/channel.",
    number = "4",
    urldate = "2023-10-30",
    journal = "Physics of Fluids",
    month = "April",
    year = "1999",
    pages = "943--945",
    file = "Full Text PDF:C\:\\Users\\Sun Myung\\Zotero\\storage\\R6IZTYHZ\\Moser et al. - 1999 - Direct numerical simulation of turbulent channel f.pdf:application/pdf;Snapshot:C\:\\Users\\Sun Myung\\Zotero\\storage\\NN9Z4VLF\\Direct-numerical-simulation-of-turbulent-channel.html:text/html"
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John Peterson, Alexander Lindsay, and Fande Kong. Overview of the Incompressible Navier-Stokes simulation capabilities in the MOOSE Framework. Advances in Engineering Software, 119:68–92, May 2018. doi:10.1016/j.advengsoft.2018.02.004.

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    author = "Peterson, John and Lindsay, Alexander and Kong, Fande",
    title = "Overview of the {Incompressible} {Navier}-{Stokes} simulation capabilities in the {MOOSE} {Framework}",
    volume = "119",
    doi = "10.1016/j.advengsoft.2018.02.004",
    abstract = {The Multiphysics Object Oriented Simulation Environment (MOOSE) framework is a high-performance, open source, C++ finite element toolkit developed at Idaho National Laboratory. MOOSE was created with the aim of assisting domain scientists and engineers in creating customizable, high-quality tools for multiphysics simulations. While the core MOOSE framework itself does not contain code for simulating any particular physical application, it is distributed with a number of physics "modules" which are tailored to solving e.g. heat conduction, phase field, and solid/fluid mechanics problems. In this report, we describe the basic equations, finite element formulations, software implementation, and regression/verification tests currently available in MOOSE's navier\\_stokes module for solving the Incompressible Navier-Stokes (INS) equations.},
    journal = "Advances in Engineering Software",
    month = "May",
    year = "2018",
    keywords = "65N30, Mathematics - Numerical Analysis",
    pages = "68--92",
    file = "arXiv\:1710.08898 PDF:C\:\\Users\\Sun Myung\\Zotero\\storage\\PBHHJKCD\\Peterson et al. - 2017 - Overview of the Incompressible Navier-Stokes simul.pdf:application/pdf;arXiv.org Snapshot:C\:\\Users\\Sun Myung\\Zotero\\storage\\3RRL7ED2\\1710.html:text/html;Full Text PDF:C\:\\Users\\Sun Myung\\Zotero\\storage\\YGQ9TDMU\\Peterson et al. - 2018 - Overview of the Incompressible Navier-Stokes simul.pdf:application/pdf"
}

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@inproceedings{spalart_one-equation_1992,
    author = "Spalart, P. and Allmaras, S.",
    title = "A one-equation turbulence model for aerodynamic flows",
    url = "https://arc.aiaa.org/doi/abs/10.2514/6.1992-439",
    urldate = "2023-11-02",
    booktitle = "30th {Aerospace} {Sciences} {Meeting} and {Exhibit}",
    publisher = "American Institute of Aeronautics and Astronautics",
    month = "January",
    year = "1992",
    doi = "10.2514/6.1992-439",
    note = "\\_eprint: https://arc.aiaa.org/doi/pdf/10.2514/6.1992-439"
}

Multigroup Neutron Diffusion
Heat Transfer and Fluid Flow
References