arXiv subjects

Eusebio Valero

Publications and source records attributed to Eusebio Valero.

At least 37 records · Page 2Linked to original sources

arXiv2022

Data-driven modal decomposition methods as feature detection techniques for flow problems: a critical assessment

Modal decomposition techniques are showing a fast growth in popularity for their good properties as data-driven tools. There are several modal decomposition techniques, yet Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) are considered the most demanded methods, especially in the field of fluid dynamics. Following their magnificent performance on various applications in several fields, numerous extensions of these techniques have been developed. In this work we present an ambitious review comparing eight different modal decomposition techniques, including most established methods: POD, DMD and Fast Fourier Trasform (FFT), extensions of these classical methods: based on time embedding systems, Spectral POD (SPOD) and Higher Order DMD (HODMD), based on scales separation, multi-scale POD (mPOD), multi-resolution DMD (mrDMD), and based on the properties of the resolvent operator, the data-driven Resolvent Analysis (RA). The performance of all these techniques will be evaluated on three different testcases: the laminar wake around cylinder, a turbulent jet flow, and the three dimensional wake around cylinder in transient regime. First, we show a comparison between the performance of the eight modal decomposition techniques when the datasets are shortened. Next, all the results obtained will be explained in details, showing both the conveniences and inconveniences of all the methods under investigation depending on the type of application and the final goal (reconstruction or identification of the flow physics). In this contribution we aim on giving a -- as fair as possible -- comparison of all the techniques investigated. To the authors knowledge, this is the first time a review paper gathering all this techniques have been produced, clarifying to the community what is the best technique to use for each application.

physics.flu-dyn↗

arXiv2022

A modified equation analysis for immersed boundary methods based on volume penalization: applications to linear advection-diffusion and high-order discontinuous Galerkin schemes

The Immersed Boundary Method (IBM) is a popular numerical approach to impose boundary conditions without relying on body-fitted grids, thus reducing the costly effort of mesh generation. To obtain enhanced accuracy, IBM can be combined with high-order methods (e.g., discontinuous Galerkin). For this combination to be effective, an analysis of the numerical errors is essential. In this work, we apply, for the first time, a modified equation analysis to the combination of IBM (based on volume penalization) and high-order methods (based on nodal discontinuous Galerkin methods) to analyze a priori numerical errors and obtain practical guidelines on the selection of IBM parameters. The analysis is performed on a linear advection-diffusion equation with Dirichlet boundary conditions. Three ways to penalize the immerse boundary are considered, the first penalizes the solution inside the IBM region (classic approach), whilst the second and third penalize the first and second derivatives of the solution. We find optimal combinations of the penalization parameters, including the first and second penalizing derivatives, resulting in minimum errors. We validate the theoretical analysis with numerical experiments for one- and two-dimensional advection-diffusion equations.

math.NA↗

arXiv2022

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

In this work, we extend the $τ$-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 $τ$-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.

math.NA↗

arXiv2022

Aeroacoustic airfoil shape optimization enhanced by autoencoders

We present a framework for airfoil shape optimization to reduce the trailing edge noise for the design of wind turbine blades. Far-field noise is evaluated using Amiet's theory coupled with the TNO-Blake model to calculate the wall pressure spectrum and fast turn-around XFOIL simulations to evaluate the boundary layer parameters. The computational framework is first validated using a NACA0012 airfoil at zero angle of attack. Particle swarm optimization is used to find the optimized airfoil configuration. The multi-objective optimization minimizes the A-weighted overall sound pressure level at various angles of attack, while ensuring enough lift and minimum drag. We compare classic parametrization methods to define the airfoil geometry (i.e., CST) to a machine learning method (i.e., a variational autoencoder). We observe that variational autoencoders can represent a wide variety of geometries, with only four encoded variables, leading to efficient optimizations, which result in improved optimal shapes. When compared to the baseline geometry, a NACA0012, the autoencoder-based optimized airfoil reduces by 3% (1.75 dBA) the overall sound pressure level (with decreased noise across the entire frequency range), while maintaining favorable aerodynamic properties in terms of lift and drag.

physics.flu-dyn↗

arXiv2022

An entropy stable spectral vanishing viscosity for discontinuous Galerkin schemes: application to shock capturing and LES models

We present a stable spectral vanishing viscosity for discontinuous Galerkin schemes, with applications to turbulent and supersonic flows. The idea behind the SVV is to spatially filter the dissipative fluxes, such that it concentrates in higher wavenumbers, where the flow is typically under-resolved, leaving low wavenumbers dissipation-free. Moreover, we derive a stable approximation of the Guermond-Popov fluxes with the Bassi-Rebay 1 scheme, used to introduce density regularization in shock capturing simulations. This filtering uses a Cholesky decomposition of the fluxes that ensures the entropy stability of the scheme, which also includes a stable approximation of boundary conditions for adiabatic walls. For turbulent flows, we test the method with the three-dimensional Taylor-Green vortex and show that energy is correctly dissipated, and the scheme is stable when a kinetic energy preserving split-form is used in combination with a low dissipation Riemann solver. Finally, we test the shock capturing capabilities of our method with the Shu-Osher and the supersonic forward facing step cases, obtaining good results without spurious oscillations even with coarse meshes.

math.NA↗

arXiv2022

A Novel Data-Driven Method for the Analysis and Reconstruction of Cardiac Cine MRI

Cardiac cine magnetic resonance imaging (MRI) can be considered the optimal criterion for measuring cardiac function. This imaging technique can provide us with detailed information about cardiac structure, tissue composition and even blood flow. This work considers the application of the higher order dynamic mode decomposition (HODMD) method to a set of MR images of a heart, with the ultimate goal of identifying the main patterns and frequencies driving the heart dynamics. A novel algorithm based on singular value decomposition combined with HODMD is introduced, providing a three-dimensional reconstruction of the heart. This algorithm is applied (i) to reconstruct corrupted or missing images, and (ii) to build a reduced order model of the heart dynamics.

eess.IV↗

arXiv2022

Higher Order Dynamic Mode Decomposition: from Fluid Dynamics to Heart Disease Analysis

In this work, we study in detail the performance of Higher Order Dynamic Mode Decomposition (HODMD) technique when applied to echocardiography images. HODMD is a data-driven method generally used in fluid dynamics and in the analysis of complex non-linear dynamical systems modeling several complex industrial applications. In this paper we apply HODMD, for the first time to the authors knowledge, for patterns recognition in echocardiography, specifically, echocardiography data taken from several mice, either in healthy conditions or afflicted by different cardiac diseases. We exploit the HODMD advantageous properties in dynamics identification and noise cleaning to identify the relevant frequencies and coherent patterns for each one of the diseases. The echocardiography datasets consist of video loops taken with respect to a long axis view (LAX) and a short axis view (SAX), where each video loop covers at least three cardiac cycles, formed by (at most) 300 frames each (called snapshots). The proposed algorithm, using only a maximum quantity of 200 snapshots, was able to capture two branches of frequencies, representing the heart rate and respiratory rate. Additionally, the algorithm provided a number of modes, which represent the dominant features and patterns in the different echocardiography images, also related to the heart and the lung. Six datasets were analyzed: one echocardiography taken from a healthy subject and five different sets of echocardiography taken from subjects with either Diabetic Cardiomyopathy, Obesity, SFSR4 Hypertrophy, TAC Hypertrophy or Myocardial Infarction. The results show that HODMD is robust and a suitable tool to identify characteristic patterns able to classify the different pathologies studied.

eess.IV↗

arXiv2020

High--order discontinuous Galerkin approximation for a three--phase incompressible Navier--Stokes/Cahn--Hilliard model

In this work we introduce the development of a three--phase incompressible Navier--Stokes/Cahn--Hilliard numerical method to simulate three--phase flows, present in many industrial operations. The numerical method is then applied to successfully solve oil transport problems, such as those found in the oil and gas industry. The three--phase model adopted in this work is a Cahn--Hilliard diffuse interface model, which was derived by Boyer and Lapuerta et al. 2006. The Cahn--Hilliard model is coupled to the entropy--stable incompressible Navier--Stokes equations model derived by Manzanero et al. 2019. The spatial discretization uses a high--order discontinuous Galerkin spectral element method which yields highly accurate results in arbitrary geometries, while an implicit--explicit (IMEX) method is adopted as temporal discretization. The developed numerical tool is tested for two and three dimensional problems, including a convergence study, a two--dimensional jet, a three--dimensional annular flow, and realistic geometries like T--shaped pipe intersections.

math.NA↗

arXiv2019

A Statically Condensed Discontinuous Galerkin Spectral Element Method on Gauss-Lobatto Nodes for the Compressible Navier-Stokes Equations

We present a static-condensation method for time-implicit discretizations of the Discontinuous Galerkin Spectral Element Method on Gauss-Lobatto points (GL-DGSEM). We show that, when solving the compressible Navier-Stokes equations, it is possible to reorganize the linear system that results from the implicit time-integration of the GL-DGSEM as a Schur complement problem, which can be efficiently solved using static condensation. The use of static condensation reduces the linear system size and improves the condition number of the system matrix, which translates into shorter computational times when using direct and iterative solvers. The statically condensed GL-DGSEM presented here can be applied to linear and nonlinear advection-diffusion partial differential equations in conservation form. To test it we solve the compressible Navier-Stokes equations with direct and Krylov subspace solvers, and we show for a selected problem that using the statically condensed GL-DGSEM leads to speed-ups of up to $200$ when compared to the time-explicit GL-DGSEM, and speed-ups of up to three when compared with the time-implicit GL-DGSEM that solves the global system. The GL-DGSEM has gained increasing popularity in recent years because it satisfies the summation-by-parts property, which enables the construction of provably entropy stable schemes, and because it is computationally very efficient. In this paper, we show that the GL-DGSEM has an additional advantage: It can be statically condensed.

physics.comp-ph↗

arXiv2019

Entropy-stable discontinuous Galerkin approximation with summation-by-parts property for the incompressible Navier-Stokes/Cahn-Hilliard system

We develop an entropy stable two-phase incompressible Navier--Stokes/Cahn--Hilliard discontinuous Galerkin (DG) flow solver method. The model poses the Cahn-Hilliard equation as the phase field method, a skew-symmetric form of the momentum equation, and an artificial compressibility method to compute the pressure. We design the model so that it satisfies an entropy law, including free- and no-slip wall boundary conditions with non-zero wall contact angle. We then construct a high-order DG approximation of the model that satisfies the SBP-SAT property. With the help of a discrete stability analysis, the scheme has two modes: an entropy conserving approximation with central advective fluxes and the Bassi-Rebay 1 (BR1) method for diffusion, and an entropy stable approximation with an exact Riemann solver for advection and interface stabilization added to the BR1 method. The scheme is applicable to, and the stability proofs hold for, three-dimensional unstructured meshes with curvilinear hexahedral elements. We test the convergence of the schemes on a manufactured solution, and their robustness by solving a flow initialized from random numbers. In the latter, we find that a similar scheme that does not satisfy an entropy inequality had 30% probability to fail, while the entropy stable scheme never does. We also solve the static and rising bubble test problems, and to challenge the solver capabilities we compute a three-dimensional pipe flow in the annular regime.

math.NA↗

arXiv2019

Entropy-stable discontinuous Galerkin approximation with summation-by-parts property for the incompressible Navier-Stokes equations with variable density and artificial compressibility

We present a provably stable discontinuous Galerkin spectral element method for the incompressible Navier-Stokes equations with artificial compressibility and variable density. Stability proofs, which include boundary conditions, that follow a continuous entropy analysis are provided. We define a mathematical entropy function that combines the traditional kinetic energy and an additional energy term for the artificial compressiblity, and derive its associated entropy conservation law. The latter allows us to construct a provably stable split-form nodal Discontinuous Galerkin (DG) approximation that satisfies the summation-by-parts simultaneous-approximation-term (SBP-SAT) property. The scheme and the stability proof are presented for general curvilinear three-dimensional hexahedral meshes. We use the exact Riemann solver and the Bassi-Rebay 1 (BR1) scheme at the inter-element boundaries for inviscid and viscous fluxes respectively, and an explicit low storage Runge-Kutta RK3 scheme to integrate in time. We assess the accuracy and robustness of the method by solving the Kovasznay flow, the inviscid Taylor-Green vortex, and the Rayleigh-Taylor instability.

math.NA↗

arXiv2019

A free-energy stable nodal discontinuous Galerkin approximation with summation-by-parts property for the Cahn-Hilliard equation

We present a nodal Discontinuous Galerkin (DG) scheme for the Cahn-Hilliard equation that satisfies the summation-by-parts simultaneous-approximation-term (SBP-SAT) property. The latter permits us to show that the discrete free-energy is bounded, and as a result, the scheme is provably stable. The scheme and the stability proof are presented for general curvilinear three-dimensional hexahedral meshes. We use the Bassi-Rebay 1 (BR1) scheme to compute interface fluxes, and an IMplicit-EXplicit (IMEX) scheme to integrate in time. Lastly, we test the theoretical findings numerically and present examples for two and three-dimensional problems.

math.NA↗

arXiv2018

Coanda Flow Characterization on Base Bleed Configurations Using Global Stability Analysis

Base pressure control is often employed on drag reduction design, but the interaction of the jet with the base region flow topology can generate undesired or uncontrolled flow configurations. The ejected flow can drive pressure bifurcations at the trailing edge, that without a correct optimization can affect the aerodynamic performance of the system. The purpose of this work is to fully understand the physical mechanisms related with the use of injected base bleed employing global stability analysis and adjoint methodologies. Moreover, the results of the sensitivity analysis are used to identify the regions more receptive to passive and active flow control methodologies. This work has a particular relevance in turbine research, where coolant flow is ejected at the trailing edge to ensure an adequate thermal protection of the blade and the downstream turbine stages.

physics.flu-dyn↗

arXiv2018

A p-Multigrid Strategy with Anisotropic p-Adaptation Based on Truncation Errors for High-Order Discontinuous Galerkin Methods

High-order DG methods have become a popular technique in computational fluid dynamics because their accuracy increases spectrally in smooth solutions with the order of the approximation. However, their main drawback is that increasing the order also increases the computational cost. Several techniques have been introduced in the past to reduce this cost. On the one hand, local mesh adaptation strategies based on error estimation have been proposed to reduce the number of degrees of freedom while keeping a similar accuracy. On the other hand, multigrid solvers may accelerate time marching computations for a fixed number of degrees of freedom. In this paper, we combine both methods and present a novel anisotropic p-adaptation multigrid algorithm for steady-state problems that uses the multigrid scheme both as a solver and as an anisotropic error estimator. To achieve this, we show that a recently developed anisotropic truncation error estimator [\textit{Rueda-Ramírez et al., Truncation Error Estimation in the p-Anisotropic DGSEM, J. of Scientific Computing}] is perfectly suited to be performed inside the multigrid cycle with negligible extra cost. Furthermore, we introduce a multi-stage p-adaptation procedure which reduces the computational time when very accurate results are required. The proposed methods are tested for the compressible Navier-Stokes equations, where we investigate two cases: the 2D flow on a flat plate is studied to assess accuracy and computational cost of the algorithm, where a speed-up of 816 is achieved compared to the traditional explicit method; and the 3D flow around a sphere is simulated and used to test the anisotropic properties of the proposed method, where a speed-up of 152 is achieved compared to the explicit method. The proposed multi-stage procedure achieved a speed-up of 2.6 in comparison to the single-stage method in highly accurate simulations.

math.NA↗

arXiv2018

Pressure Bifurcation Phenomenon on Supersonic Blowing Trailing Edges

Turbine blades operating in transonic-supersonic regime develop a complex shock wave system at the trailing edge, a phenomenon that leads to unfavorable pressure perturbations downstream and can interact with other turbine stages. Understanding the fluid behavior of the area adjacent to the trailing edge is essential in order to determine the parameters that have influence on these pressure fluctuations. Colder flow, bled from the high-pressure compressor, is often purged at the trailing edge to cool the thin blade edges, affecting the flow behavior and modulating the intensity and angle of the shock waves system. However, this purge flow can sometimes generate non-symmetrical configurations due to a pressure difference that is provoked by the injected flow. In this work, a combination of RANS simulations and global stability analysis is employed to explain the physical reasons of this flow bifurcation. Analyzing the features that naturally appear in the flow and become dominant for some value of the parameters involved in the problem, an anti-symmetrical global mode, related to the sudden geometrical expansion of the trailing edge slot, is identified as the main mechanism that forces the changes in the flow topology.

physics.flu-dyn↗

arXiv2018

Truncation Error Estimation in the p-Anisotropic Discontinuous Galerkin Spectral Element Method

In the context of Discontinuous Galerkin Spectral Element Methods (DGSEM), $τ$-estimation has been successfully used for p-adaptation algorithms. This method estimates the truncation error of representations with different polynomial orders using the solution on a reference mesh of relatively high order. In this paper, we present a novel anisotropic truncation error estimator derived from the $τ$-estimation procedure for DGSEM. We exploit the tensor product basis properties of the numerical solution to design a method where the total truncation error is calculated as a sum of its directional components. We show that the new error estimator is cheaper to evaluate than previous implementations of the $τ$-estimation procedure and that it obtains more accurate extrapolations of the truncation error for representations of a higher order than the reference mesh. The robustness of the method allows performing the p-adaptation strategy with coarser reference solutions, thus further reducing the computational cost. The proposed estimator is validated using the method of manufactured solutions in a test case for the compressible Navier-Stokes equations.

math.NA↗

arXiv2018

On the role of numerical dissipation in stabilising under-resolved turbulent simulations using discontinuous Galerkin methods

We analyse numerical errors (dissipation and dispersion) introduced by the discretisation of inviscid and viscous terms in energy stable discontinuous Galerkin methods. First, we analyse these methods using a linear von Neumann analysis (for a linear advection-diffusion equation) to characterise their properties in wave-number space. Second, we validate these observations using the 3D Taylor-Green Vortex Navier-Stokes problem to assess transitional/turbulent flows. We show that the dissipation introduced by upwind Riemann solvers affects primarily high wave-numbers. This dissipation may be increased, through a penalty parameter, until a critical value. However, further augmentation of this parameter leads to a decrease of dissipation, reaching zero for very large values. Regarding the dissipation introduced by second order derivatives, we show that this dissipa- tion acts at low and medium wave-numbers (lower wave-numbers compared to upwind Riemann solvers). In addition, we analyse the Spectral Vanishing Viscosity (SVV) technique, previously used in continuous discretisations (e.g. Fourier), to find that with an appropriate kernel (which damps selected modes) it is possible to control the amount of dissipation introduced in the low and medium wave-number range. Combining these ideas, we finally propose a DG-SVV approach that uses a Smagorinsky model to compute the numerical viscosity. This DG-SVV approach is tested in an isotropic laminar/turbulent under-resolved scenario. Combining the SVV technique with a low dissipation Riemann solver, we obtain a scheme capable of maintaining low dissipation levels for laminar flows, whilst providing the correct dissipation for all wave-number ranges in turbulent regimes.

math.NA↗

arXiv2017

Insights on aliasing driven instabilities for advection equations with application to Gauss-Lobatto discontinuous Galerkin methods

We analyse instabilities due to aliasing errors when solving one dimensional non-constant advection speed equations and discuss means to alleviate these types of errors when using high order discontinuous Galerkin (DG) schemes. First, we compare analytical bounds for the continuous and discrete version of the PDEs. Whilst traditional $L^2$ norm energy bounds applied to the discrete PDE do not always predict the physical behaviour of the continuous version of the equation, more strict elliptic norm bounds correctly bound the behaviour of the continuous PDE. Having derived consistent bounds, we analyse the effectiveness of two stabilising techniques: over-integration and split form variations (conservative, non-conservative and skew-symmetric). Whilst the former is shown to not alleviate aliasing in general, the latter ensures an aliasing-free solution if the splitting form of the discrete PDE is consistent with the continuous equation. The success of split form de-aliasing is restricted to DG schemes with the summation-by-parts simultaneous-approximation-term (SBP-SAT) properties (e.g. DG with Gauss-Lobatto points). Numerical experiments are included to illustrate the theoretical findings.

math.NA↗