POD\u2013Galerkin reduced order methods for combined Navier\u2013Stokes transport equations based on a hybrid FV-FE solver

The purpose of this work is to introduce a novel POD\u2013Galerkin strategy for the semi-implicit hybrid high order finite volume/finite element solver introduced in Berm\ufadez et al. (2014) and Busto et al. (2018). The interest is into the incompressible Navier\u2013Stokes equations coupled with an additional transport equation. The full order model employed in this article makes use of staggered meshes. This feature will be conveyed to the reduced order model leading to the definition of reduced basis spaces in both meshes. The reduced order model presented herein accounts for velocity, pressure, and a transport-related variable. The pressure term at both the full order and the reduced order level is reconstructed making use of a projection method. More precisely, a Poisson equation for pressure is considered within the reduced order model. Results are verified against three-dimensional manufactured test cases. Moreover a modified version of the classical cavity test benchmark including the transport of a species is analysed

A semi-implicit hybrid finite volume / finite element scheme for all Mach number flows on staggered unstructured meshes

In this paper a new hybrid semi-implicit finite volume / finite element (FV/FE) scheme is presented for the numerical solution of the compressible Euler and Navier-Stokes equations at all Mach numbers on unstructured staggered meshes in two and three space dimensions. The chosen grid arrangement consists of a primal simplex mesh composed of triangles or tetrahedra, and an edge-based / face-based staggered dual mesh. The governing equations are discretized in conservation form. The nonlinear convective terms of the equations, as well as the viscous stress tensor and the heat flux, are discretized on the dual mesh at the aid of an explicit local ADER finite volume scheme, while the implicit pressure terms are discretized at the aid of a continuous $\mathbb{P}^{1}$ finite element method on the nodes of the primal mesh. In the zero Mach number limit, the new scheme automatically reduces to the hybrid FV/FE approach forwarded in \cite{BFTVC17} for the incompressible Navier-Stokes equations. As such, the method is asymptotically consistent with the incompressible limit of the governing equations and can therefore be applied to flows at all Mach numbers. Due to the chosen semi-implicit discretization, the CFL restriction on the time step is only based on the magnitude of the flow velocity and not on the sound speed, hence the method is computationally efficient at low Mach numbers. In the chosen discretization, the only unknown is the scalar pressure field at the new time step. Furthermore, the resulting pressure system is symmetric and positive definite and can therefore be very efficiently solved with a matrix-free conjugate gradient method. In order to assess the capabilities of the new scheme, we show computational results for a large set of benchmark problems that range from the quasi incompressible low Mach number regime to compressible flows with shock waves

Reduced order modelling in nuclear reaction thermal hydraulics

The context of the present thesis is to assess the potential of Reduced Order Models (ROMs) for nuclear reactor thermal hydraulics applications. ROMs constitute advanced modelling techniques aiming at fast high fidelity simulations. For the purposes of this research, two approaches have been selected and are investigated in depth: the Proper Orthogonal Decomposition (POD) with Galerkin projection (POD-Galerkin) and the hybrid method of Proper Orthogonal Decomposition with Interpolation using Radial Basis Functions, PODI - Galerkin, in the context of parametric model order reduction. Additionally, in terms of the POD method, two sampling techniques are presented and compared: the standard and the nested POD. The aforementioned methods are applied to a parametric case of non-isothermal mixing in a T-junction pipe for laminar and turbulent flow regimes. The flow is governed by the 3D, unsteady Navier - Stokes equations coupled with the energy equation. Furthermore, a ROM for modelling buoyancy driven flows with the Boussinesq approximation is discussed. Two cases are considered: a closed flow, where the method is applied to a benchmark case of a differentially heated square cavity, and an open flow, where a case of a "cold-trap" formation in a U-bend pipe is investigated. The suitability of the above techniques is assessed based on a comparison between the reduced order results and those obtained using high fidelity OpenFOAM solvers.Open Acces

A POD-Galerkin reduced order model of a turbulent convective buoyant flow of sodium over a backward-facing step

A Finite-Volume based POD-Galerkin reduced order modeling strategy for steady-state Reynolds averaged Navier--Stokes (RANS) simulation is extended for low-Prandtl number 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 reduced order model 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 is incorporated with a Neumann boundary condition in both the full order model and the reduced order model. The velocity and temperature profiles predicted with the reduced order model 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 $10^5$ times faster than the RANS simulations that are performed on eight cores.Comment: 26 pages, 15 figures, 3 table

Complexity reduction in parametric flow problems via Nonintrusive Proper Generalised Decomposition in OpenFOAM

Tesi en modalitat cotutela: Universitat Politècnica de Catalunya i Swansea University. Programa Erasmus Mundus en Simulació en Enginyeria i Desenvolupament de l'Emprenedoria (SEED)The present thesis explores the viability of the proper generalised decomposition (PGD) as a tool for parametric studies in a daily industrial environment. Starting from the equations modelling incompressible flows, the separated formulation of the equations, the development of a parametric solver, the implementation in a commercial computational fluid dynamics (CFD) software, OpenFOAM, and a numerical validation are presented. The parametrised Stokes and Oseen flows are used as an initial step to test the applicability of the PGD to flow problems. The rationale for the construction of a separable approximation is described and implemented in OpenFOAM. For the numerical validation of the developed strategy analytical test cases are solved. Then, the parametrised steady laminar incompressible Navier-Stokes equations are considered. The nonintrusive implementation of PGD in OpenFOAM is formulated, focusing on the seamless integration of a reduced order model (ROM) in the framework of an industrially validated CFD software. The proposed strategy exploits classical solution strategies in OpenFOAM to solve the PGD spatial iteration, while the parametric one is solved via a collocation approach. Such nonintrusiveness represents an important step towards the industrialisation of PGD-based approaches. The capabilities of the methodology are tested by applying it to benchmark tests in the literature and solving a parametrised flow control problem in a realistic geometry of interest for the automotive industry. Finally, the PGD framework is extended to turbulent Navier-Stokes problems. The separable form of an industrially popular turbulence model, namely Spalart-Allmaras model, is formulated and a PGD strategy for the construction of a parametric turbulent eddy viscosity is devised. Different implementation possibilities in the nonintrusive PGD for parametrised Navier-Stokes equations are explored and the proposed strategy is applied to well-documented turbulent flow control benchmark cases in both two and three dimensions.La tesis explora la viabilidad del método de reducción de modelos Proper Generalised Decomposition (PGD) como herramienta habitual en un entorno industrial para obtener soluciones de problemas de flujo viscoso incompresible que dependan de parámetros. En este documento, partiendo de las ecuaciones que modelan el flujo viscoso e incompresible, se describe en detalle la formulación en forma separada, espacio-parámetros, de las ecuaciones para el método PGD, se desarrolla el algoritmo de resolución teniendo en cuenta los parámetros, se detalla como realizar la implementación en OpenFOAM, que es un software comercial de dinámica de fluidos computacional (CFD por sus siglas en inglés) y se discuten las validaciones numéricas correspondientes. Como paso previo para probar la viabilidad de la PGD a problemas de interés, se estudian flujos de Stokes y Oseen con datos parametrizados. De esta forma, se desarrollan las bases para la construcción de una aproximación separada, espacio-parámetros, de la solución numérica velocidad-presión, todo ello implementado en OpenFOAM. Para estas formulaciones se valida la aproximación numérica de la estrategia desarrollada con ejemplos cuya solución analítica es conocida, lo que permite analizar los errores cometidos, y se presentan ejemplos numéricos de referencia ampliamente estudiados en la literatura para mostrar su viabilidad. Seguidamente se consideran las ecuaciones de Navier-Stokes para flujo incompresible, estacionario y laminar de nuevo dependiendo de parámetros de diseño. La implementación no intrusiva de la PGD en OpenFOAM está formulada para obtener integración perfecta de un modelo de orden reducido (ROM por sus siglas en inglés) con un software CFD validado industrialmente. La metodología propuesta explota las estrategias de solución clásicas ya existentes en OpenFOAM para resolver la iteración espacial de la PGD, mientras que la iteración de las funciones que dependen de los parámetros se realiza de forma externa a OpenFOAM (empleando formulaciones basadas en la colocación puntual). La no-intrusividad es crítica para una cualquier estrategia que pretenda emplear la formulación PGD en la práctica diaria de la producción y diseño industrial. Para justificar la metodología propuesta así como su viabilidad, se muestra la solución de problemas de referencia clásicos y habituales en la literatura así como la resolución de un problema de control de flujo parametrizado en una geometría realista de interés para la industria de la automoción. Finalmente, es importante resaltar que se extiende a flujos turbulentos la metodología propuesta para trabajar con la PGD de manera no-intrusiva. Más concretamente, las ecuaciones de Navier-Stokes se complementan con un modelo de turbulencia habitual en aplicaciones industriales: el modelo de Spalart-Allmaras. En este caso, se propone una extensión de la estructura separada de las aproximaciones (velocidad y presión), y se diseña una estrategia PGD para la construcción de una viscosidad turbulenta paramétrica. Se exploran diferentes posibilidades de implementación de la PGD no intrusiva para las ecuaciones de Navier-Stokes para flujo turbulento y dependiendo de parámetros. La estrategia propuesta se aplica a casos de referencia de control de flujo turbulento bien documentados en dos y tres dimensiones.Postprint (published version