246 research outputs found

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

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    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

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    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

    Discontinuous Galerkin Methods for inviscid low Mach number flows

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    In this work we present two preconditioning techniques for inviscid low Mach number flows. The space discretization used is a high-order Discontinuous Galerkin finite element method. The time discretizations analyzed are explicit and implicit schemes. The convective physical flux is replaced by a flux difference splitting scheme. Computations were performed on triangular and quadrangular grids to analyze the influence of the spatial discretization. For the preconditioning of the explicit Euler equations we propose to apply the fully preconditioning approach: a formulation that modifies both the instationary term of the governing equations and the dissipative term of the numerical flux function. For the preconditioning of the implicit Euler equations we propose to apply the flux preconditioning approach: a formulation that modifies only the dissipative term of the numerical flux function. Both these formulations permit to overcome the stiffness of the governing equations and the loss of accuracy of the solution that arise when the Mach number tends to zero. Finally, we present a splitting technique, a proper manipulation of the flow variables that permits to minimize the cancellation error that occurs as an accumulation effect of round-off errors as the Mach number tends to zero

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

    Get PDF
    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

    A Continuous/Discontinuous FE Method for the 3D Incompressible Flow Equations

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    A projection scheme for the numerical solution of the incompressible Navier-Strokes equation is presented. Finite element discontinuous Galerkin (dG) discretization for the velocity in the momentum equations is employed. The incompressibility constraint is enforced by numerically solving the Poisson equation for pressure using a continuous Galerkin (cG) discretization. The main advantage of the method is that is does not require the velocity and pressure approximation spaces to satisfy the usual inf-sup condition, thus equal order finite element approximations for both velocity and pressure can be used. Furthermore, by using cG discretization for the Poisson equation, no auxiliary equations are needed as it is required for dG approximations of second order derivatives. In order to enable large time steps for time marching to steady-state and time evolving problems, implicit scheme is used in connection with high order implicit RK methods. Numerical tests demonstrate that the overall scheme is accurate and computationally efficient

    Accelerated Temporal Schemes for High-Order Unstructured Methods

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    The ability to discretize and solve time-dependent Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) remains of great importance to a variety of physical and engineering applications. Recent progress in supercomputing or high-performance computing has opened new opportunities for numerical simulation of the partial differential equations (PDEs) that appear in many transient physical phenomena, including the equations governing fluid flow. In addition, accurate and stable space-time discretization of the partial differential equations governing the dynamic behavior of complex physical phenomena, such as fluid flow, is still an outstanding challenge. Even though significant attention has been paid to high and low-order spatial schemes over the last several years, temporal schemes still rely on relatively inefficient approaches. Furthermore, academia and industry mostly rely on implicit time marching methods. These implicit schemes require significant memory once combined with high-order spatial discretizations. However, since the advent of high-performance general-purpose computing on GPUs (GPGPU), renewed interest has been focused on explicit methods. These explicit schemes are particularly appealing due to their low memory consumption and simplicity of implementation. This study proposes low and high-order optimal Runge-Kutta schemes for FR/DG high-order spatial discretizations with multi-dimensional element types. These optimal stability polynomials improve the stability of the numerical solution and speed up the simulation for high-order element types once compared to classical Runge-Kutta methods. We then develop third-order accurate Paired Explicit Runge-Kutta (P-ERK) schemes for locally stiff systems of equations. These third-order P-ERK schemes allow Runge-Kutta schemes with different numbers of active stages to be assigned based on local stiffness criteria, while seamlessly pairing at their interface. We then generate families of schemes optimized for the high-order flux reconstruction spatial discretization. Finally, We propose optimal explicit schemes for Ansys Fluent finite volume density-based solver, and we investigate the effect of updating and freezing reconstruction gradient in intermediate Runge-Kutta schemes. Moreover, we explore the impact of optimal schemes combined with the updated gradients in scale-resolving simulations with Fluent's finite volume solver. We then show that even though freezing the reconstruction gradients in intermediate Runge-Kutta stages can reduce computational cost per time step, it significantly increases the error and hampers stability by limiting the time-step size
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