Within the MYRRHA project, which stands for Multi-purpose hYbrid Research
Reactor for High-tech Applications, the Belgian Nuclear Research Center SCK
CEN is developing and designing a multi-functional experimental fast-spectrum
irradiation facility. The MYRRHA design features a compact pool-type primary
system cooled by molten Lead-Bismuth Eutectic, i.e. a heavy metal. Reliable computational methods are required to accurately quantify the reactor’s primary system behavior in operational and accidental conditions and to handle complex geometries.
However, the number of nuclear reactor simulations in a safety analysis is,
in the majority of cases, beyond the possibilities of present hardware if a computational fluid dynamics solver is used alone. This has motivated the development of reduced order modeling techniques that reduce the number of degrees of freedom of the high fidelity thermofluids models. Mathematical techniques are used to extract “features” of the complex model in order to replace them by a more simplified model. In that way, the required computational time and computer memory usage is reduced.
Despite the potential and increasing popularity of reduced order models for all sorts of flow applications, they tend to have issues with accuracy and exhibit numerical instabilities. Challenges regarding velocity-pressure coupling
and satisfying the boundary conditions at the reduced order level make it difficult to generalize the methods such that they can be applied to any problem.
The complex fluid dynamics problems are generally solved numerically using
discretization methods. In this work, we focus on the finite volume discretization
method for the numerical solution of incompressible fluid flows on collocated
grids. To obtain a computationally efficient reduced order model (ROM), the procedure is ideally split into a so-called offline stage and an online stage. In the
offline stage, solutions of the high fidelity model are collected at several time instances and/or for different parameter values. They are used to generate a reduced basis of a much smaller order than the full order model (FOM). In this work, the reduced basis spaces are spanned by basis functions, or so-called modes, which are computed using the proper orthogonal decomposition (POD) technique. POD is commonly used for reduced-order modeling of incompressible flows. Reduced matrices (linear terms) and tensors (nonlinear terms) of the ROM associated with the terms of the full order model are determined during the offline stage, for which two techniques are developed and investigated in this work. The first technique is a non-intrusive reduction method that identifies the system matrix of linear fluid dynamical problems with a least-squares technique. The main advantage of nonintrusive methods is that they do not require access to the solver’s discretization and solution algorithm. The second technique is the intrusive Galerkin projection approach for which the full order equations are projected onto the reduced POD basis spaces. In the online stage, the reduced system of equations are solved for the same or new values of the parameters of interest at a lower computational cost compared to solving the full order systems.
The non-intrusive reduction method that identifies the system matrix of linear
fluid dynamical problems with a least-squares technique is presented in the
first part of the thesis. The methodology is applied to the linear scalar transport
convection-diffusion equation for a two-dimensional square cavity problem with a heated lid. The (time-dependent) boundary conditions are imposed in the reduced order model with a penalty method. The results are compared and the accuracy of the reduced order models is assessed against the full order solutions. It is shown that the reduced order model can be used for sensitivity analysis by controlling the non-homogeneous Dirichlet boundary conditions. For nonlinear problems, the required number of snapshots scales with the cube of the number of POD basis functions and at least as many reduced matrices are to be identified as the number of basis functions used. Therefore, it is not feasible to use the POD-based identification method for nonlinear problems. However, for the simulation of fluid flows in (nuclear) engineering applications, it is necessary to develop reduced order models for nonlinear problems, such as convective flows and buoyancy-driven flows. Therefore, the main part of the thesis is dedicated to the intrusive PODbased Galerkin projection approach due to its applicability to nonlinear problems.
POD-Galerkin reduced order models are developed of which the (timedependent)
boundary conditions are imposed at reduced order level using two different
boundary control strategies: the lifting function method, whose aim is to
obtain homogeneous basis functions for the reduced basis spaces and the penalty method where the boundary conditions are imposed in the reduced order model using a penalty factor. The penalty method is improved by using an iterative solver for the determination of the penalty factor rather than tuning the factor with a sensitivity analysis or numerical experimentation. The boundary control methods are compared and tested for two cases: the classical lid driven cavity benchmark problem and a Y-junction flow case with two inlet channels and one outlet channel. The results show that the boundaries of the reduced order model can be controlled with the boundary control methods and the same order of accuracy is achieved for the velocity and pressure fields. However, computing the ROM solutions takes more time in the case of the lifting function method as the reduced basis spaces contain additional modes, namely the lifting functions, compared to the penalty method.
Furthermore, a parametric reduced order model for buoyancy-driven flow is
introduced. The Boussinesq approximation is used for modeling the buoyancy.
Therefore, there exists a two-way coupling between the incompressible Boussinesq equations and the energy equation. To obtain the reduced order model, a Galerkin projection of the governing equations onto the reduced POD bases spaces is performed. The ROM is tested on a two-dimensional differentially heated cavity of which the side wall temperatures are parametrized. The parametrization is done using a lifting function method. The lifting functions are obtained by solving a Laplacian function for temperature. Only one unsteady full order simulation was required for the creation of the reduced bases. The obtained ROM is efficient and stable for different parameter sets for which the temperature difference between the walls is smaller than for the set in the FOM used for the POD basis creation.
In addition, the POD-Galerkin reduced order modeling strategy for steadystate
Reynolds averaged Navier–Stokes (RANS) simulation is extended for low-
Prandtl number fluid flow. The reduced order model is based on a full order model for which the effects of buoyancy on the flow and heat transfer are characterized by varying the Richardson number. The Reynolds stresses are computed with a linear eddy viscosity model. A single gradient diffusion hypothesis, together with a local correlation for the evaluation of the turbulent Prandtl number, is used to model the turbulent heat fluxes. The contribution of the eddy viscosity and turbulent thermal diffusivity fields are considered in the reduced order model with an interpolation based data-driven method. The ROM is tested for buoyancy-aided turbulent liquid sodium flow over a vertical backward-facing step with a uniform heat flux applied on the wall downstream of the step. The wall heat flux boundary condition is incorporated in both the full order model and the reduced order model.
The velocity and temperature profiles predicted with the ROM for the same and
new Richardson numbers inside the range of parameter values are in good agreement with the RANS simulations. Also, the local Stanton number and skin friction distribution at the heated wall are qualitatively well captured. Finally, the reduced order simulations, performed on a single core, are about 105 times faster than the full order RANS simulations that are performed on eight cores.
The final part of the thesis is dedicated to the development of a novel nonparametric reduced order model of the incompressible Navier-Stokes equations on collocated grids. The reduced order model is developed by performing a Galerkin projection based on a fully (space and time) discrete full order model formulation.
This ‘discretize-then-project’ approach requires no pressure stabilization
technique (even though the pressure term is present in the ROM) nor a boundary
control technique (to impose the boundary conditions at the ROM level). These
are two main advantages compared to existing and previously applied approaches.
The fully discrete FOM is obtained by a finite volume discretization of the incompressible Navier-Stokes equations with a forward Euler time discretization.
Two variants of the velocity-pressure coupling at the fully discrete level, the inconsistent and consistent flux method, have been investigated. The latter leads to divergence-free velocity fields, also on the ROM level, whereas the velocity fields are only approximately divergence-free in the former method. For both methods,
stable and accurate results have been obtained for test cases with different types
of boundary conditions: a lid-driven cavity and an open cavity with an inlet and
outlet. The ROM obtained with the consistent flux method, having divergence-free velocity fields, is slightly more accurate but also slightly more expensive to solve compared to the inconsistent flux method due to an additional equation for the flux.
The speedup ratio of the ROM and FOM computation times strongly depends on
which method is used, the number of degrees of freedom of the full order model
and the number of modes used for the reduced basis spaces.
Finally, an application with the coupling between a system thermal hydraulics
code and a reduced order model of a computational fluid dynamics solver is presented in the appendix of this work. The system code and the ROM are coupled by a domain decomposition algorithm using an implicit coupling scheme. The velocity transported over a coupling boundary interface is imposed in the ROM using a penalty method. The coupled models are evaluated on open and closed pipe flow configurations. The results of the coupled simulations with the ROM are close to those with the CFD solver. Also for new parameter sets, the coupled models with the ROM are capable of predicting the results of the coupled models with the FOM with good accuracy. The coupled simulations with the ROM are about 3-5 times faster than those with the FOM.