Truncation Error-Based Anisotropic pp-Adaptation for Unsteady Flows for High-Order Discontinuous Galerkin Methods

In this work, we extend the $\tau$-estimation method to unsteady problems and use it to adapt the polynomial degree for high-order discontinuous Galerkin simulations of unsteady flows. The adaptation is local and anisotropic and allows capturing relevant unsteady flow features while enhancing the accuracy of time evolving functionals (e.g., lift, drag). To achieve an efficient and unsteady truncation error-based $p$-adaptation scheme, we first revisit the definition of the truncation error, studying the effect of the treatment of the mass matrix arising from the temporal term. Secondly, we extend the $\tau$-estimation strategy to unsteady problems. Finally, we present and compare two adaptation strategies for unsteady problems: the dynamic and static $p$-adaptation methods. In the first one (dynamic) the error is measured periodically during a simulation and the polynomial degree is adapted immediately after every estimation procedure. In the second one (static) the error is also measured periodically, but only one $p$-adaptation process is performed after several estimation stages, using a combination of the periodic error measures. The static $p$-adaptation strategy is suitable for time-periodic flows, while the dynamic one can be generalized to any flow evolution. We consider two test cases to evaluate the efficiency of the proposed $p$-adaptation strategies. The first one considers the compressible Euler equations to simulate the advection of a density pulse. The second one solves the compressible Navier-Stokes equations to simulate the flow around a cylinder at Re=100. The local and anisotropic adaptation enables significant reductions in the number of degrees of freedom with respect to uniform refinement, leading to speed-ups of up to $\times4.5$ for the Euler test case and $\times2.2$ for the Navier-Stokes test case

Robust and adaptive high-order discontinuous Galerkin methods for Multiphase flows

This dissertation focuses on efficient and robust computation methods for multiphase flow applications in the context of high–order (HO) methods. The flow modelling is done through a modified form of the compressible Navier–Stokes equations. The first addition is to include the artificial compressibility method which allows the solver to simulate flows with very low Mach number and alleviates the stiffness of the system. The second modification is the use of the Cahn–Hilliard model which allows to model multiphase flows of two or more immiscible fluids. This type of modelling of multiphase flows falls into the category of phase–field methods. The numerical framework used to discretize the system of equations is the Discontinuous Galerkin Spectral Element Method (DGSEM). The focus of this work is to exploit and expose some of the numerical and algorithmic characteristics that a DGSEM discretization offers to increase the robustness and the efficiency of the framework without compromising each of those two characteristics. Both of these characteristics are critical towards the goal of establishing HO methods as an industry–ready tool, which has been one of the main pillars of the project that this work has been part of. To increase the efficiency of the solver, reduce the computational overhead and the consumed energy, we exploit one basic characteristic of DGSEM which can also be generalised towards other high–order methods such as Flux–Reconstruction or Continuous Galerkin methods. That is p–adaptation. DGSEM offers the flexibility to locally refine or coarsen the polynomial order approximation in each element and in each direction independently. This technique allows to reduce the total number of degrees of freedomand thus reduce the associated computational cost. While doing so, it is important to retain or increase the level of accuracy of the original solution. Within this work, p–adaptation has been initially applied to RANS and LES simulations to verify its applicability and benefits and showcase that it can retain the original level of accuracy for both steady and unsteady cases. Since the goal of this thesis is to apply these techniques to unsteady multiphase flows, p–adaptation has been initially applied to the Cahn–Hilliard equation for two–phase flow problems. This initial assessment shows the benefits of p–adaptation in terms of computational cost. Then, the same technique has been applied to the full Navier–Stokes/Cahn– Hilliard system of equations showing that substantial computational savings can be achieved while retaining the quality of the original solution. One of the major novelties of this work is that the choice of p–adaptation for multiphase flows had not been considered before and only h– and r– adaptation had been applied. The second desideratum is to have a robust solver. This is particularly of interest within the industrial setting as the requirements dictate that the solver should be able to output a solution for cases such as under–resolved flows. One method with growing popularity, particularly in the HO research community, are entropy–stable schemes. This refers to the concept of the existence of a mathematical entropy which allows to prove that the stability of the discretized scheme is in par with the stability analysis of the conservation law. We use a particular flavour of the DGSEM with Gauss–Lobatto nodes, a skew–symmetric form and the summation– by–parts simultaneous–approximation–term (SBP–SAT) to have an entropy–stable discretization of the Cahn–Hilliard equation and the Navier–Stokes/Cahn–Hilliard system. One of the novelties of this work is that the discretisation has beenmodified to retain the entropy–stability properties for p–non–conforming elements. This allows to have a robust solver and exploit the performance benefits of p–adaptation. Through several numerical tests of increasing complexity, for both the Cahn-Hilliard and the Navier–Stokes/Cahn–Hilliard system, we show the robustness and the increased efficiency of the new framework. RESUMEN Esta tesis se centra en métodos de cálculo eficientes y robustos para aplicaciones de flujo multifásico en el contexto de los métodos de alto orden (HO). El modelado del flujo se realiza mediante una forma modificada de las ecuaciones compresibles de Navier–Stokes. El primer ingrediente es el método de compresibilidad artificial que permite al solucionador simular flujos con un número de Mach muy bajo y alivia la rigidez del sistema. El segundo ingrediente es el modelo de Cahn–Hilliard que permite modelar flujos multifásicos de dos o más fluidos inmiscibles. Este tipo de modelización de los flujos multifásicos entra en la categoría de los métodos de campo de fase. El marco numérico utilizado para discretizar el sistema de ecuaciones es el método de elementos espectrales de Galerkin discontinuo (DGSEM). El objetivo de este trabajo es explotar y exponer algunas de las características numéricas y algorítmicas que ofrece una discretización DGSEM para aumentar la robustez y la eficiencia del esquema sin comprometer cada una de esas dos características. Ambas características son críticas hacia el objetivo de establecer los métodos de HO como una herramienta lista para la industria, que ha sido uno de los principales pilares del proyecto en el que se ha enmarcado este trabajo. Para aumentar la eficiencia del solucionador, reducir la sobrecarga computacional y la energía consumida, explotamos una característica básica de DGSEM que también se puede generalizar hacia otros métodos de alto orden como Flux–Reconstruction o métodos de Galerkin continuos. Se trata de la adaptación de p. El DGSEM ofrece la flexibilidad de refinar o engrosar localmente la aproximación de orden polinómico en cada elemento y en cada dirección independientemente. Esta técnica permite reducir el número total de grados de libertad y, por tanto, reducir el coste computacional asociado. Al mismo tiempo, es importante mantener o aumentar el nivel de precisión de la solución original. En este trabajo, la adaptación p se ha aplicado inicialmente a simulaciones RANS y LES para verificar su aplicabilidad y beneficios y demostrar que puede mantener el nivel de precisión original tanto para casos estables como no estables. Dado que el objetivo de esta tesis es aplicar estas técnicas a flujos multifásicos no estacionarios, la adaptación p se ha aplicado inicialmente a la ecuación de Cahn-Hilliard para problemas de flujo bifásico. Esta evaluación inicial muestra los beneficios de la adaptación–p en términos de coste computacional. A continuación, se ha aplicado la misma técnica al sistema completo de ecuaciones de Navier–Stokes/Cahn–Hilliard, mostrando que se puede conseguir un ahorro computacional sustancial manteniendo la calidad de la solución original. Una de las principales novedades de este trabajo es que la elección de la adaptación–p para los flujos multifásicos no se había considerado antes y sólo se había aplicado la adaptación h– y r–.-. El segundo desiderátum es disponer de un solucionador robusto. Esto es especialmente interesante en el ámbito industrial, ya que los requisitos exigen que el solucionador sea capaz de ofrecer una solución para casos como los flujos insuficientemente resueltos. Un método que goza de creciente popularidad, especialmente en la comunidad de investigación de la HO, son los esquemas de entropía estable. Esto se refiere al concepto de la existencia de una entropía matemática que permite demostrar que la estabilidad del esquema discretizado está a la par con el análisis de estabilidad de la ley de conservación. Usamos un tipo particular de DGSEM con nodos de Gauss–Lobatto, una forma sesgada–simétrica y el término de aproximación simultánea por partes (SBP–SAT) para tener una discretización estable en entropía de la ecuación de Cahn–Hilliard y del sistema de Navier–Stokes/Cahn– Hilliard. Una de las novedades de este trabajo es que la discretización se ha modificado para mantener las propiedades de estabilidad de la entropía para elementos p–no conformes. Esto permite tener un solucionador robusto y explotar las ventajas de rendimiento de la adaptación de p. Mediante varias pruebas numéricas de complejidad creciente, tanto para el sistema de Cahn-Hilliard como para el de Navier–Stokes/Cahn–Hilliard, mostramos la robustez y la mayor eficiencia del nuevo marco

towards a robust detection of viscous and turbulent flow regions using unsupervised machine learning

We propose an invariant feature space for the detection of viscous dominated and turbulent regions (i.e., boundary layers and wakes). The developed methodology uses the principal invariants of the strain and rotational rate tensors as input to an unsupervised Machine Learning Gaussian mixture model. The selected feature space is independent of the coordinate frame used to generate the processed data, as it relies on the principal invariants of strain and rotational rate, which are Galilean invariants. This methodology allows us to identify two distinct flow regions: a viscous dominated, rotational region (boundary layer and wake region) and an inviscid, irrotational region (outer flow region). We test the methodology on a laminar and a turbulent (using Large Eddy Simulation) case for flows past a circular cylinder at $Re=40$ and $Re=3900$. The simulations have been conducted using a high-order nodal Discontinuous Galerkin Spectral Element Method (DGSEM). The results obtained are analysed to show that Gaussian mixture clustering provides an effective identification method of viscous dominated and rotational regions in the flow. We also include comparisons with traditional sensors to show that the proposed clustering does not depend on the selection of an arbitrary threshold, as required when using traditional sensors