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

An Hybrid Finite Volume-Finite Element Method for Variable Density Incompressible Flows

International audienceThis paper is devoted to the numerical simulation of variable density incompressible flows, modeled by the Navier-Stokes system. We introduce an hybrid scheme which combines a Finite Volume approach for treating the mass conservation equation and a Finite Element method to deal with the momentum equation and the divergence free constraint. The breakthrough relies on the definition of a suitable footbridge between the two methods, through the design of compatibility condition. In turn, the method is very flexible and allows to deal with unstructured meshes. Several numerical tests are performed to show the scheme capabilities. In particular, the viscous Rayleigh-Taylor instability evolution is carefully investigate

Lagrangian FE methods for coupled problems in fluid mechanics

This work aims at developing formulations and algorithms where maximum advantage of using Lagrangian finite element fluid formulations can be taken. In particular we concentrate our attention at fluid-structure interaction and thermally coupled applications, most of which originate from practical “real-life” problems. Two fundamental options are investigated - coupling two Lagrangian formulations (e.g. Lagrangian fluid and Lagrangian structure) and coupling the Lagrangian and Eulerian fluid formulations. In the first part of this work the basic concepts of the Lagrangian fluids, the so-called Particle Finite Element Method (PFEM) [1], [2] are presented. These include nodal variable storage, mesh re-construction using Delaunay triangulation/tetrahedralization and alpha shape-based method for identification of the computational domain boundaries. This shall serve as a general basis for all the further developments of this work.Postprint (published version

Methods for stabilizing high Reynolds number Lattice Boltzmann simulations

The Lattice Boltzmann Method (LBM) is a simple and highly efficient method for computing nearly incompressible fluid flow. However, it is well known to suffer from numerical instabilities for low values of the transport coefficients. This dissertation examines a number of methods for increasing the stability of the LBM over a wide range of parameters. First, we consider a simple transformation that renders the standard LB equation implicit. It is found that the stability is largely unchanged. Next, we consider a stabilization method based on introducing a Lyapunov function which is essentially a discrete-time H-function. The uniqueness of an H-function that appears in the literature is proven, and the method is extended to stabilize some of the more popular LB models. We also introduce a new method for implementing boundary conditions in the LBM. The hydrodynamic fields are imposed in a transformed moment space, whereas The non-hydrodynamic fields are shifted over from neighboring nodes. By minimizing population gradients, this method exhibits superior numerical stability over other widely employed schemes when tested on the widely-used benchmark of incompressible flow over a backwards-facing step

Strong wave-mean-flow coupling in baroclinic acoustic streaming

The interaction of an acoustic wave with a stratified fluid can drive strong streaming flows owing to the baroclinic production of fluctuating vorticity, as recently demonstrated by Chini et al. (J. Fluid Mech., 744, 2014, pp. 329). In the present investigation, a set of wave/mean-flow interaction equations is derived that governs the coupled dynamics of a standing acoustic wave mode of characteristic (small) amplitude {\epsilon} and the streaming flow it drives in a thin channel with walls maintained at differing temperatures. Unlike classical Rayleigh streaming, the resulting mean flow arises at O({\epsilon}) rather than at O({\epsilon^2}). Consequently, fully two-way coupling between the waves and the mean flow is possible: the streaming is sufficiently strong to induce O(1) rearrangements of the imposed background temperature and density fields, which modifies the spatial structure and frequency of the acoustic mode on the streaming time scale. A novel Wentzel-Kramers-Brillouin-Jeffreys analysis is developed to average over the fast wave dynamics, enabling the coupled system to be integrated strictly on the slow time scale of the streaming flow. Analytical solutions of the reduced system are derived for weak wave forcing and are shown to reproduce results from prior direct numerical simulations (DNS) of the compressible Navier Stokes and heat equations with remarkable accuracy. Moreover, numerical simulations of the reduced system are performed in the regime of strong wave mean flow coupling for a fraction of the computational cost of the corresponding DNS. These simulations shed light on the potential for baroclinic acoustic streaming to be used as an effective means to enhance heat transfer.Comment: 29 pages, 7 figure