ICASE/LaRC Workshop on Adaptive Grid Methods

Solution-adaptive grid techniques are essential to the attainment of practical, user friendly, computational fluid dynamics (CFD) applications. In this three-day workshop, experts gathered together to describe state-of-the-art methods in solution-adaptive grid refinement, analysis, and implementation; to assess the current practice; and to discuss future needs and directions for research. This was accomplished through a series of invited and contributed papers. The workshop focused on a set of two-dimensional test cases designed by the organizers to aid in assessing the current state of development of adaptive grid technology. In addition, a panel of experts from universities, industry, and government research laboratories discussed their views of needs and future directions in this field

Numerical analysis for active control of drag over flat plate using sinusoidal travelling wave method

The drag is present due to flow around bodies such as vehicles, aircraft, bullet trains or ships etc. It plays a significant role in vehicle performance, rate of fuel consumption and stability. This drag depends mainly on flow, fluid and surface properties. Different methods are used for drag reduction like controlling surface roughness and injecting long-chain polymers etc. In the current research, a method based on the generation of a spanwise mean velocity gradient in a flat plate boundary layer by applying a sinusoidal travelling wave in the spanwise direction before the transition to turbulence occurrence is used for the drag reduction purpose. A solver is developed using the open-source CFD software OpenFOAM libraries. It consists of routines for generating synthetic isotropic homogeneous turbulence at the inlet plane of the channel flow and solving the Navier Stokes equations using the Monotonically Integrated Large Eddy Simulation method (MILES). The implemented inlet boundary condition showed improvements in the predictions of turbulence structures within a streamwise distance of approximately 5 times the half channel height (δ) from the inlet plane, a shorter distance than the other similar previous boundary conditions can predict. These improvements resulted in a considerable reduction in overall channel length required for numerical simulations and hence reduction in the associated computation costs. In addition, a prediction method for getting the required friction Reynolds number for setting up any simulation case is been developed and verified. The MILES solver and the inlet boundary condition are used together to generate the bypass transition through the channel flow. A travelling sinusoidal wave in the spanwise direction is applied by the effect of blowing and suction from the bottom wall of the channel before the bypass transition occurrence. This travelling wave leads to a drop in the value of the skin friction coefficient which means a drop in the drag. This may be due to the increase of the mixing effect of the eddies and the induced streamwise travelling wave. The bypass transition onset also appears to be delayed when an inclined sinusoidal wave is applied in the streamwise direction. However, grid independence was not thoroughly tested in this study

Semi-Analytical Semi-Lagrangian Discontinuous Galerkin Advection Scheme for the Compressible Linear Advection Equation

The main objective of this thesis is to develop a numerical method, known as the SASLDG method, that solves the compressible linear advection equation in a semi-analytical way using a semi-Lagrangian approach and applying ideas from the discontinuous Galerkin (DG) methods. In order to achieve physically meaningful solutions, mass conservation is required. For the same reason, the option of slope limiting and thus positivity preservation are also necessary. Furthermore, high-order accuracy and no time step restriction are required. First, the basic methods underpinning the SASLDG method are introduced. We describe a method developed independently by Prather and van Leer. This so-called second-order moments method is a preliminary stage of the SASLDG method. Both methods are equivalent for CFL numbers less than or equal to one and with constant velocity. We then introduce DG methods to generalize the concept of high-order accuracy of numerical solutions using a polynomial representation for each grid cell. One drawback of this class of methods is the strict time step restriction. In contrast, semi-Lagrangian methods allow time steps of arbitrary size without violating stability properties. We also introduce and analyze the numerical scheme multidimensional positive definite advection transport algorithm (MPDATA). Following the semi-Lagrangian notion, the advected quantity is tracked analytically along the trajectories. It is represented as piecewise polynomials of degree two. A condition for this approach is the restriction of the given velocity field to a piecewise linear distribution. This enables the derivation of the exact solution, which is in general not in polynomial form. Therefore, much like with DG methods, a projection onto the polynomial space is carried out. A deviation from the solution only occurs in the projection step. The SASLDG method involves rigorously computing every step of the solution to the linear advection equation analytically. However, this concept places many conditions on the algorithm, which require the application of different branches of the SASLDG method. To prevent the cancellation of significant digits, further exceptions have to be made and resolved by alternative computations. A thorough analysis of the SASLDG method proves its consistency and stability. Third-order accuracy for advection with constant velocity and second-order accuracy for variable velocity is demonstrated analytically and confirmed by numerical convergence tests. Test cases in one- and two- space dimensions are conducted to show the performance of the SASLDG method. High accuracy is shown in tests using smooth and discontinuous initial values. The method’s ability to handle an irregular grid and arbitrary CFL numbers is assessed. Tests in 1D show that an increase of the CFL number results in smaller maximum- and $l_1$-errors. This feature cannot be transferred to the 2D case since the extension is done via operator splitting, where accuracy increases with decreasing time step sizes. Numerous physical phenomena, e.g.\ in the dynamics of the atmosphere or the ocean, can occur on different scales of the space dimensions. This can lead to computational grids with higher resolution and grid cells sizes in the vertical direction than in the horizontal direction. To address this scenario, we develop a hybrid method which employs MPDATA and a modified version of the SASLDG method. This scheme can compute the solution to the advection equation on grids with high aspect ratio without strict time step restrictions in the vertical direction. Overall, the SASLDG method shows promising results and is worthy of further development.Das Ziel dieser Arbeit ist die Entwicklung einer numerischen Methode, die die kompressible lineare Advektionsgleichung auf eine semi-analytische Weise mit einem semi-Lagrangian Ansatz unter Verwendung von Ideen der Discontinuous Galerkin (DG) Methoden löst, hier als SASLDG Methode bezeichnet. Eine Vorgabe für das Verfahren ist die Massenerhaltung, um eine physikalisch sinnvolle Lösung zu garantieren. Aus diesem Grund soll auch “slope limiting” möglich sein und die Positivität der Lösung erhalten bleiben. Hohe Genauigkeit und unbeschränkte Zeitschrittgröße sind weitere Bedingungen. Zunächst werden die Methoden erläutert, die der SASLDG Methode zugrunde liegen. Eine von Prather und van Leer unabhängig voneinander entwickelte Methode wird beschrieben. Diese sogennante Second-Order Moments Methode ist eine Vorstufe zur SASLDG Methode. Für CFL Zahlen, die kleiner gleich eins sind, und für eine konstante Advektionsgeschwindigkeit sind die Verfahren identisch. DG Methoden verallgemeinern das Konzept der hohen Genauigkeit durch Verwendung von Polynomen höherer Ordnung. Diese haben jedoch den Nachteil strikter Zeitschrittbeschränkung. Semi-Lagrangian Verfahren lassen Zeitschritte beliebiger Größe zu, ohne Stabilitätsbedingungen zu verletzen. Schließlich wird die Methode Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) vorgestellt und analysiert. Im Sinne des semi-Lagrangian Ansatzes wird die advektierte Größe analytisch \mbox{exakt} entlang von Trajektorien transportiert. Sie wird durch ein Polynom zweiten Grades dargestellt. Eine Bedingung für dieses Vorgehen ist die Beschränkung des Geschwindigkeitsfeldes auf stückweise lineare Funktionen. So wird die Herleitung der analytischen Lösung ermöglicht, die im Allgemeinen nicht polynomiell ist. Durch einen Projektionsschritt, ähnlich dem Vorgehen bei DG Verfahren, wird die Lösung auf ein Polynom projiziert. Nur dadurch entstehen Abweichungen zur analytischen Lösung. Die Idee der SASLDG Methode besteht darin, alle Schritte rigoros analytisch durchzuführen. Diese Methodik stellt allerdings komplexe Bedingungen an den Algorithmus und führt stellenweise zum Problem der Auslöschung. Dadurch sind Verzweigungen notwendig sowie alternative Berechnungswege. Konsistenz und Stabilität wird durch eine Analyse des SASLDG Verfahrens gezeigt. Das Verfahren konvergiert mit zweiter Ordnung für variable Geschwindigkeit und mit dritter Ordnung für konstante Geschwindigkeit. Dies wird analytisch hergeleitet und mithilfe numerischer Konvergenztests bestätigt. Die hohe Genauigkeit des SASLDG Verfahrens wird durch numerische Tests in 1D und 2D gezeigt. Dabei werden glatte Anfangswerte sowie Daten mit Diskontinuitäten verwendet. Tests in 1D liefern Ergebnisse auf regel- und unregelmäßigem Gitter sowie mit verschiedenen CFL Zahlen. Daraus resultiert, dass mit größerer CFL Zahl $l_{\infty}$- und $l_1$-Fehler sinken. Dieses Ergebnis gilt jedoch nicht für 2D-Testfälle aufgrund der Verwendung von ``operator splitting'', für das die Genauigkeit bei kleineren Schrittweiten steigt. Zahlreiche physikalische Phänomene, beispielsweise im Bereich der Dynamik der Atmosphäre und des Ozeans, erstrecken sich über unterschiedliche Raumskalen. Das kann zu Rechengittern führen, die in vertikaler Richtung eine viel höhere Auflösung mit kleinerer Gitterweite als in horizontaler Richtung erfordern. Eine dafür entwickelte hybride Methode, eine Kombination aus MPDATA und einer angepassten Variante des SASLDG Verfahrens, kann auf diesen Gittern ohne strikte Zeitschrittbeschränkung in vertikaler Richtung Lösungen berechnen. Insgesamt zeigt die in dieser Arbeit hergeleitete SASLDG Methode vielversprechende Resultate, die eine Weiterentwicklung des Verfahrens erstrebenswert machen

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

Aeronautical engineering: A continuing bibliography with indexes (supplement 291)

This bibliography lists 757 reports, articles, and other documents introduced into the NASA scientific and technical information system in May. 1993. Subject coverage includes: design, construction and testing of aircraft and aircraft engines; aircraft components, equipment, and systems; ground support systems; and theoretical and applied aspects of aerodynamics and general fluid dynamics

Bibliography of Lewis Research Center technical publications announced in 1992

This compilation of abstracts describes and indexes the technical reporting that resulted from the scientific and engineering work performed and managed by the Lewis Research Center in 1992. All the publications were announced in the 1992 issues of STAR (Scientific and Technical Aerospace Reports) and/or IAA (International Aerospace Abstracts). Included are research reports, journal articles, conference presentations, patents and patent applications, and theses

Aeronautical engineering: A continuing bibliography with indexes (supplement 301)

This bibliography lists 1291 reports, articles, and other documents introduced into the NASA scientific and technical information system in Feb. 1994. Subject coverage includes: design, construction and testing of aircraft and aircraft engines; aircraft components, equipment, and systems; ground support systems; and theoretical and applied aspects of aerodynamics and general fluid dynamics