An Accurate and Robust Numerical Scheme for Transport Equations

En esta tesis se presenta una nueva técnica de discretización para ecuaciones de transporte en problemas de convección-difusión para el rango completo de números de Péclet. La discretización emplea el flujo exacto de una ecuación de transporte unidimensional en estado estacionario para deducir una ecuación discreta de tres puntos en problemas unidimensionales y cinco puntos en problemas bidimensionales. Con "flujo exacto" se entiende que se puede obtener la solución exacta en función de integrales de algunos parámetros del fluido y flujo, incluso si estos parámetros son vari- ables en un volumen de control. Las cuadraturas de alto orden se utilizan para lograr resultados numéricos cercanos a la precisión de la máquina, incluso con mallas bastas.Como la discretización es esencialmente unidimensional, no está garantizada una solución con precisión de máquina para problemas multidimensionales, incluso en los casos en que las integrales a lo largo de cada coordenada cartesiana tienen una primitiva. En este sentido, la contribución principal de esta tesis consiste en una forma simple y elegante de obtener soluciones en problemas multidimensionales sin dejar de utilizar la formulación unidimensional. Además, si el problema es tal que la solución tiene precisión de máquina en el problema unidimensional a lo largo de las líneas coordenadas, también la tendrá para el dominio multidimensional.In this thesis, we present a novel discretization technique for transport equations in convection-diffusion problems across the whole range of Péclet numbers. The discretization employs the exact flux of a steady-state one-dimensional transport equation to derive a discrete equation with a three-point stencil in one-dimensional problems and a five-point stencil in two-dimensional ones. With "exact flux" it is meant that the exact solution can be obtained as a function of integrals of some fluid and flow parameters, even if these parameters are variable across a control volume. High-order quadratures are used to achieve numerical results close to machine- accuracy even with coarse grids. As the discretization is essentially one-dimensional, getting the machine- accurate solution of multidimensional problems is not guaranteed even in cases where the integrals along each Cartesian coordinate have a primitive. In this regard, the main contribution of this thesis consists in a simple and elegant way of getting solutions in multidimensional problems while still using the one-dimensional formulation. Moreover, if the problem is such that the solution is machine-accurate in the one-dimensional problem along coordinate lines, it will also be for the multidimensional domain.<br /

Revisiting and Extending Interface Penalties for Multi-Domain Summation-by-Parts Operators

General interface coupling conditions are presented for multi-domain collocation methods, which satisfy the summation-by-parts (SBP) spatial discretization convention. The combined interior/interface operators are proven to be L2 stable, pointwise stable, and conservative, while maintaining the underlying accuracy of the interior SBP operator. The new interface conditions resemble (and were motivated by) those used in the discontinuous Galerkin finite element community, and maintain many of the same properties. Extensive validation studies are presented using two classes of high-order SBP operators: 1) central finite difference, and 2) Legendre spectral collocation

Vortex dynamics of accelerated flow past a mounted wedge

This study is concerned with the simulation of a complex fluid flow problem involving flow past a wedge mounted on a wall for channel Reynolds numbers $Re_c=1560$, $6621$ and $6873$ in uniform and accelerated flow medium. The transient Navier-Stokes (N-S) equations governing the flow has been discretized using a recently developed second order spatially and temporally accurate compact finite difference method on a nonuniform Cartesian grid by the authors. All the flow characteristics of a well-known laboratory experiment of Pullin and Perry (1980) have been remarkably well captured by our numerical simulation, and we provide a qualitative and quantitative assessment of the same. Furthermore, the influence of the parameter $m$, controlling the intensity of acceleration, has been discussed in detail along with the intriguing consequence of non-dimensionalization of the N-S equations pertaining to such flows. The simulation of the flow across a time span significantly greater than the aforesaid lab experiment is the current study's most noteworthy accomplishment. For the accelerated flow, the onset of shear layer instability leading to a more complicated flow towards transition to turbulence have also been aptly resolved. The existence of coherent structures in the flow validates the quality of our simulation, as does the remarkable similarity of our simulation to the high Reynolds number experimental results of Lian and Huang (1989) for the accelerated flow across a typical flat plate. All three steps of vortex shedding, including the exceedingly intricate three-fold structure, have been captured quite efficiently.Comment: 28 pages, 27 figures, 2 table

A provably stable and high-order accurate finite difference approximation for the incompressible boundary layer equations

In this article we develop a high order accurate method to solve the incompressible boundary layer equations in a provably stable manner. We first derive continuous energy estimates, and then proceed to the discrete setting. We formulate the discrete approximation using high-order finite difference methods on summation-by-parts form and implement the boundary conditions weakly using the simultaneous approximation term method. By applying the discrete energy method and imitating the continuous analysis, the discrete estimate that resembles the continuous counterpart is obtained proving stability. We also show that these newly derived boundary conditions removes the singularities associated with the null-space of the nonlinear discrete spatial operator. Numerical experiments that verifies the high-order accuracy of the scheme and coincides with the theoretical results are presented. The numerical results are compared with the well-known Blasius similarity solution as well as that resulting from the solution of the incompressible Navier Stokes equation

A provably stable and high-order accurate finite difference approximation for the incompressible boundary layer equations

In this article we develop a high order accurate method to solve the incompressible boundary layer equations in a provably stable manner.~We first derive continuous energy estimates,~and then proceed to the discrete setting.~We formulate the discrete approximation using high-order finite difference methods on summation-by-parts form and implement the boundary conditions weakly using the simultaneous approximation term method.~By applying the discrete energy method and imitating the continuous analysis,~the discrete estimate that resembles the continuous counterpart is obtained proving stability.~We also show that these newly derived boundary conditions removes the singularities associated with the null-space of the nonlinear discrete spatial operator.~Numerical experiments that verifies the high-order accuracy of the scheme and coincides with the theoretical results are presented.~The numerical results are compared with the well-known Blasius similarity solution as well as that resulting from the solution of the incompressible Navier Stokes equations

Boundary Closures for Fourth-order Energy Stable Weighted Essentially Non-Oscillatory Finite Difference Schemes

A general strategy exists for constructing Energy Stable Weighted Essentially Non Oscillatory (ESWENO) finite difference schemes up to eighth-order on periodic domains. These ESWENO schemes satisfy an energy norm stability proof for both continuous and discontinuous solutions of systems of linear hyperbolic equations. Herein, boundary closures are developed for the fourth-order ESWENO scheme that maintain wherever possible the WENO stencil biasing properties, while satisfying the summation-by-parts (SBP) operator convention, thereby ensuring stability in an L2 norm. Second-order, and third-order boundary closures are developed that achieve stability in diagonal and block norms, respectively. The global accuracy for the second-order closures is three, and for the third-order closures is four. A novel set of non-uniform flux interpolation points is necessary near the boundaries to simultaneously achieve 1) accuracy, 2) the SBP convention, and 3) WENO stencil biasing mechanics