Unified solver for fluid dynamics and aeroacoustics in isentropic gas flows

The high computational cost of solving numerically the fully compressible Navier–Stokes equations, together with the poor performance of most numerical formulations for compressible flow in the low Mach number regime, has led to the necessity for more affordable numerical models for Computational Aeroacoustics. For low Mach number subsonic flows with neither shocks nor thermal coupling, both flow dynamics and wave propagation can be considered isentropic. Therefore, a joint isentropic formulation for flow and aeroacoustics can be devised which avoids the need for segregating flow and acoustic scales. Under these assumptions density and pressure fluctuations are directly proportional, and a two field velocity-pressure compressible formulation can be derived as an extension of an incompressible solver. Moreover, the linear system of equations which arises from the proposed isentropic formulation is better conditioned than the homologous incompressible one due to the presence of a pressure time derivative. Similarly to other compressible formulations the prescription of boundary conditions will have to deal with the backscattering of acoustic waves. In this sense, a separated imposition of boundary conditions for flow and acoustic scales which allows the evacuation of waves through Dirichlet boundaries without using any tailored damping model will be presented.Peer ReviewedPostprint (author's final draft

Numerical simulation of aeroacoustics using the variational multiscale method : application to the problem of human phonation

The solution of the human phonation problem applying computational mechanics is covered by several research branches, such as Computational Fluid Dynamics (CFD), biomechanics or acoustics, among others. In the present thesis, the problem is approached from the Computational Aeroacoustics (CAA) point of view and the first main objective consists in developing numerical methods of general application that can take part in the solution of any scenario related to human phonation with a reasonable cost. In this sense, only the compressible Navier-Stokes equations can describe all flow and acoustic scales without any modeling, which is known as Direct Numerical Simulation (DNS), but its computational cost is usually unaffordable. Even in the case of a Large Eddy Simulation (LES), where the small scales are modeled, the cost can still be a handicap due to the complexity of the problem. This drawback gets worse in the low Mach regime due to the large disparity between flow velocity and sound speed, which leads to an ill-conditioning of the system of equations, specially for conservative schemes. At this point, it makes sense to move towards the incompressible flow approximation, bearing in mind the low velocities expected in human phonation problems. Incompressible flows do not yield any acoustics, for which a second problem containing the propagation of the sound sources needs to be modeled and solved. These are the so called hybrid methods, which allow a better conditioning of the problem by segregating flow and acoustic scales. Lighthill's analogy has been taken as starting point for the present work, but its restriction to free-field scenarios has motivated the extension of the method to arbitrary geometries and non-uniform flows. The first development in this direction consists in a splitting of Lighthill's analogy into a quadrupolar and dipolar component, which does not change the original problem but allows assessing the contribution of solid boundaries to the generation of sound. The second step consists in the development of a stabilized Finite Element (FEM) formulation for the Acoustic Perturbation Equations (APE) which account for non-uniform flows and perform a complete filtering of the acoustic scales. The final step assumes the compressible approach but omitting the energy equation and thus considering both flow and acoustic propagation as isentropic. In this case the solver is unified and hence a method for applying compatible boundary conditions for flow and acoustics has been developed. Moreover, the whole numerical framework has been extended to dynamic phonation cases, which require using an Arbitrary Lagrangian Eulerian (ALE) reference. Also, a novel remeshing strategy with conservative interpolation between meshes is presented. In the last chapter a challenging case in human phonation has been chosen for testing the developed computational framework: the fricative phoneme /s/. Unlike vowels, which are voiced sounds defined by a few characteristic frequencies, fricatives cannot be simulated as the propagation of a known analytic solution (glottal pulse) because the sound sources correspond to a wide range of turbulent scales. Therefore, a CFD calculation is mandatory in order to capture all relevant eddies behind the generation of sound. This problem is solved with an LES together with the Variational Multiscale (VMS) stabilization method as turbulence model, which is supplemented with several acoustic formulations when using incompressible flow. The analysis of the results focuses on the numerical representation of turbulence and the acoustic signal at the far-field, which has been compared to experimental recordings. Finally, the role of the upper incisors in the generation of the fricative sound has been evaluated. All simulations have been run with the parallel multiphysics FEM code FEMUSS, based on FORTRAN Object-Oriented-Programming land the OpenMPI parallel library.La solució del problema de la veu humana des de la mecànica computacional és objecte d'estudi per part de diverses disciplines, com per exemple la Dinàmica de Fluids Computacional (CFD), la biomecànica o l'acústica. En la present tesi s'encara el problema des de l'Aeroacústica Computacional (CAA) i el primer objectiu consisteix en desenvolupar mètodes numèrics d'aplicació general que puguin ser part de la solució, amb un cost computacional raonable, de qualsevol escenari relacionat amb la fonació humana. En aquest sentit, només les equacions de flux compressible de Navier-Stokes aconsegueixen descriure totes les escales alhora, tant les dinàmiques com les acústiques, sense recórrer a cap modelització, conegut com a Simulació Numèrica Directa (DNS), però el seu cost computacional és normalment inassumible. Fins i tot en el cas d'una Large Eddy Simulation (LES), on les escales petites són modelades, el cost pot resultar excessiu a causa de la complexitat del problema. Aquest fet encara és més accentuat en el règim de baix nombre de Mach donada la gran disparitat entre la velocitat del fluid i la del so i el conseqüent mal condicionament del sistema d'equacions, sobretot en esquemes conservatius. Per tant, tenint en compte les baixes velocitats de l'aire al tracte vocal, té sentit recórrer a l'aproximació de flux incompressible. Els fluids incompressibles no inclouen la part acústica, de manera que cal calcular un segon problema que descrigui la propagació de les fonts de so. Aquests són els anomenats mètodes híbrids, que permeten un millor condicionament del problema gràcies a la segregació de les escales acústiques de les dinàmiques. S'ha pres l'analogia de Lighthill com a punt de partida, però la seva restricció a casos en camp obert ha motivat l'extensió del mètode cap a geometries arbitràries i fluxos no uniformes. El primer desenvolupament en aquesta direcció consisteix en la divisió de l'analogia de Lighthill en una component quadrupolar i una altra de dipolar, fet que no altera el problema original però que permet analitzar la contribució de cossos sòlids en la generació de so. El segon pas consisteix en el desenvolupament d'una formulació estabilitzada en elements finits (FEM) de les Acoustic Perturbation Equations (APE), que incorporen la propagació en fluxos no uniformes i que realitzen un filtrat complet de les escales acústiques. El pas final assumeix la compressibilitat del fluid però omet l'equació d'energia, i per tant considera la dinàmica i l'acústica fenòmens isentròpics. En aquest cas el solver és unificat i per tant s'ha desenvolupat un mètode per imposar condicions de contorn compatibles entre ambdues escales del fluid. Finalment, les formulacions numèriques han estat adaptades a casos de fonació dinàmica mitjançant una referència Arbitrària Lagrangiana Euleriana (ALE). A més, es presenta una estratègia de remallat amb interpolació conservativa entre malles. En l'últim capítol es presenta un cas de fonació humana que suposa un repte per la seva complexitat i que ha servit per validar les formulacions numèriques presentades: la fricativa sorda /s/. A diferència de les vocals, que són sons sonors definits per unes poques freqüències característiques, les fricatives no poden ser simulades com la propagació d'una funció analítica coneguda (pols glotal) perquè les fonts de so corresponen a un rang ampli d'escales turbulents. Per tant és necessària una simulació CFD per tal de capturar-les. El problema se soluciona amb un model de turbulència LES amb el mètode d'estabilització Variational Multiscale. L'anàlisi se centra en la representació numèrica de la turbulència i en el senyal acústic al camp llunyà, tot comparant-lo amb dades experimentals. Finalment, s'avalua la contribució dels incisius superiors en la generació del so fricatiu sord /s/. Totes les simulacions han estat realitzades amb el codi FEM multi-físic en paral·lel FEMUSS, basat en programació orientada a objectes en FORTRAN i en OpenMPI

A fractional step method for computational aeroacoustics using weak imposition of Dirichlet boundary conditions

In this work we consider the approximation of the isentropic Navier–Stokes equations. The model we present is capable of taking into account acoustic and flow scales at once. After space and time discretizations have been chosen, it is very convenient from the computational point of view to design fractional step schemes in time so as to permit a segregated calculation of the problem unknowns. While these segregation schemes are well established for incompressible flows, much less is known in the case of isentropic flows. We discuss this issue in this article and, furthermore, we study the way to weakly impose Dirichlet boundary conditions via Nitsche’s method. In order to avoid spurious reflections of the acoustic waves, Nitsche’s method is combined with a non-reflecting boundary condition. Employing a purely algebraic approach to discuss the problem, some of the boundary contributions are treated explicitly and we explain how these are included in the different steps of the final algorithm. Numerical evidence shows that this explicit treatment does not have a significant impact on the convergence rate of the resulting time integration scheme. The equations of the formulation are solved using a subgrid scale technique based on a term-by-term stabilization.Peer ReviewedPostprint (author's final draft

Hydrodynamic/acoustic splitting approach with flow-acoustic feedback for universal subsonic noise computation

A generalized approach to decompose the compressible Navier-Stokes equations into an equivalent set of coupled equations for flow and acoustics is introduced. As a significant extension to standard hydrodynamic/acoustic splitting methods, the approach provides the essential coupling terms, which account for the feedback from the acoustics to the flow. A unique simplified version of the split equation system with feedback is derived that conforms to the compressible Navier-Stokes equations in the subsonic flow regime, where the feedback reduces to one additional term in the flow momentum equation. Subsonic simulations are conducted for flow-acoustic feedback cases using a scale-resolving run-time coupled hierarchical Cartesian mesh solver, which operates with different explicit time step sizes for incompressible flow and acoustics. The first simulation case focuses on the tone of a generic flute. With the major flow-acoustic feedback term included, the simulation yields the tone characteristics in agreement with reference results from K\"uhnelt based on Lattice-Boltzmann simulation. On the contrary, the standard hybrid hydrodynamic/acoustic method with the feedback-term switched off lacks the proper tone. As the second simulation case, a thick plate in a duct is studied at various low Mach numbers around the Parker-beta-mode resonance. The simulations reveal the flow-acoustic feedback mechanism in very good agreement with experimental data of Welsh et al. Simulations and theoretical considerations reveal that the feedback term does not reduce the stable convective flow based time step size of the flow equations.Comment: Submitted to Journal of Computational Physic

Towards industrial large eddy simulation using the FR/CPR method

NASA’s 2030 CFD Vision calls for the development of accurate and efficient scale-resolving simulations for turbulent flow, such as large eddy simulation (LES) and direct numerical simulation (DNS). This is primarily because the Reynolds-averaged Navier-Stokes (RANS) approach has failed to predict vortex-dominated flow involving large flow separations, e.g., flow through a jet engine or over aircraft near the edge of the flight envelope, i.e., during take-off and landing at high angles of attack. Although the DNS approach resolves all turbulence scales, it is too expensive in the foreseeable future for real world flow problems because of the disparate length and time scales in the flow. LES resolves the energetic large scales while modeling the smaller scales, so it provides a good compromise between accuracy and cost. As a result, LES is widely considered to be the method of choice for next generation CFD design tool. The major obstacle for LES is its considerable computational cost since unsteady 3D simulations need to be performed to obtain the mean flow quantities such as the drag and lift coefficients. In order to resolve the dominant scales in a turbulent flow, numerical methods used for LES should have low dissipation and dispersion errors. This means standard second order finite-volume methods are usually not accurate or efficient enough for LES applications. High-order methods (order of accuracy 2) have demonstrated their potential for LES and DNS in the past decade because of their low embedded numerical dissipation and dispersion errors. In the present study, we develop and demonstrate a recently developed high-order method, called flux reconstruction (FR) or correction procedure via reconstruction (CPR), for industrial LES. A major advantage of the FR/CPR method is its capability to handle unstructured mixed meshes, and its compactness and scalability, which is particularly desired on modern super-computers. We therefore address the following major pacing items in industrial LES in the present study: High-order methods Geometric flexibility Efficient time integration Efficient implementation on modern super computers Demonstration for real world application

ADER-WENO Finite Volume Schemes with Space-Time Adaptive Mesh Refinement

We present the first high order one-step ADER-WENO finite volume scheme with Adaptive Mesh Refinement (AMR) in multiple space dimensions. High order spatial accuracy is obtained through a WENO reconstruction, while a high order one-step time discretization is achieved using a local space-time discontinuous Galerkin predictor method. Due to the one-step nature of the underlying scheme, the resulting algorithm is particularly well suited for an AMR strategy on space-time adaptive meshes, i.e.with time-accurate local time stepping. The AMR property has been implemented 'cell-by-cell', with a standard tree-type algorithm, while the scheme has been parallelized via the Message Passing Interface (MPI) paradigm. The new scheme has been tested over a wide range of examples for nonlinear systems of hyperbolic conservation laws, including the classical Euler equations of compressible gas dynamics and the equations of magnetohydrodynamics (MHD). High order in space and time have been confirmed via a numerical convergence study and a detailed analysis of the computational speed-up with respect to highly refined uniform meshes is also presented. We also show test problems where the presented high order AMR scheme behaves clearly better than traditional second order AMR methods. The proposed scheme that combines for the first time high order ADER methods with space--time adaptive grids in two and three space dimensions is likely to become a useful tool in several fields of computational physics, applied mathematics and mechanics.Comment: With updated bibliography informatio

High Order Cell-Centered Lagrangian-Type Finite Volume Schemes with Time-Accurate Local Time Stepping on Unstructured Triangular Meshes

We present a novel cell-centered direct Arbitrary-Lagrangian-Eulerian (ALE) finite volume scheme on unstructured triangular meshes that is high order accurate in space and time and that also allows for time-accurate local time stepping (LTS). The new scheme uses the following basic ingredients: a high order WENO reconstruction in space on unstructured meshes, an element-local high-order accurate space-time Galerkin predictor that performs the time evolution of the reconstructed polynomials within each element, the computation of numerical ALE fluxes at the moving element interfaces through approximate Riemann solvers, and a one-step finite volume scheme for the time update which is directly based on the integral form of the conservation equations in space-time. The inclusion of the LTS algorithm requires a number of crucial extensions, such as a proper scheduling criterion for the time update of each element and for each node; a virtual projection of the elements contained in the reconstruction stencils of the element that has to perform the WENO reconstruction; and the proper computation of the fluxes through the space-time boundary surfaces that will inevitably contain hanging nodes in time due to the LTS algorithm. We have validated our new unstructured Lagrangian LTS approach over a wide sample of test cases solving the Euler equations of compressible gasdynamics in two space dimensions, including shock tube problems, cylindrical explosion problems, as well as specific tests typically adopted in Lagrangian calculations, such as the Kidder and the Saltzman problem. When compared to the traditional global time stepping (GTS) method, the newly proposed LTS algorithm allows to reduce the number of element updates in a given simulation by a factor that may depend on the complexity of the dynamics, but which can be as large as 4.7.Comment: 31 pages, 13 figure

On adomian based numerical schemes for euler and navier-stokes equations, and application to aeroacoustic propagation

140 p.En esta tesis se ha desarrollado un nuevo método de integración en tiempo de tipo derivadas sucesivas (multiderivative), llamado ABS y basado en el algoritmo de Adomian. Su motivación radica en la reducción del coste de simulación para problemas en aeroacústica, muy costosos por su naturaleza transitoria y requisitos de alta precisión. El método ha sido satisfactoriamente empleado en ambas partes de un sistema híbrido, donde se distinguen la parte aerodinámica y la acústica.En la parte aerodinámica las ecuaciones de Navier-Stokes incompresibles son resueltas con unmétodo de proyección clásico. Sin embargo, la fase de predicción de velocidad ha sido modificadapara incluir el método ABS en combinación con dos métodos: una discretización espacial MAC devolúmenes finitos, y también con un método de alto orden basado en ADER. El método se ha validado respecto a los problemas (en 2D) de vórtices de Taylor-Green, y el desarrollo de vórticesde Karman en un cilindro cuadrado. La parte acústica resuelve la propagación de ondas descritaspor las ecuaciones linearizadas de Euler, empleando una discretización de Galerkin discontinua. El método se ha validado respecto a la ecuación de Burgers.El método ABS es sencillo de programar con una formulación recursiva. Los resultados demuestran que su sencillez junto con sus altas capacidades de adaptación lo convierten en un método fácilmente extensible a órdenes altos, a la vez que reduce el coste comparado con otros métodos clásicos

On Adomian Based Numerical Schemes for Euler and Navier-Stokes Equations, and Application to Aeroacoustic Propagation

In this thesis, an Adomian Based Scheme (ABS) for the compressible Navier-Stokes equations is constructed, resulting in a new multiderivative type scheme not found in the context of fluid dynamics. Moreover, this scheme is developed as a means to reduce the computational cost associated with aeroacoustic simulations, which are unsteady in nature with high-order requirements for the acoustic wave propagation. We start by constructing a set of governing equations for the hybrid computational aeroacoustics method, splitting the problem into two steps: acoustic source computation and wave propagation. The first step solves the incompressible Navier-Stokes equation using Chorin's projection method, which can be understood as a prediction-correction method. First, the velocity prediction is obtained solving the viscous Burgers' equation. Then, its divergence-free correction is performed using a pressure Poisson type projection. In the velocity prediction substep, Burgers' equation is solved using two ABS variants: a MAC type implementation, and a ``modern'' ADER method. The second step in the hybrid method, related to wave propagation, is solved combining ABS with the discontinuous Galerkin high-order approach. Described solvers are validated against several test cases: vortex shedding and Taylor-Green vortex problems for the first step, and a Gaussian wave propagation in the second case. Although ABS is a multiderivative type scheme, it is easily programmed with an elegant recursive formulation, even for the general Navier-Stokes equations. Results show that its simplicity combined with excellent adaptivity capabilities allows for a successful extension to very high-order accuracy at relatively low cost, obtaining considerable time savings in all test cases considered.Predoc Gobierno Vasc