The standard hyperbolic methods used to solve the compressible Euler equations are not effective in the limit of incompressible flow. The sound waves dominate the system and it becomes poorly conditioned for numerical solution. For steady flow governed by the incompressible Euler equations, artificial compressibility is a technique that removes the troublesome sound waves. It leads to a hyperbolic system of equations that we solve by finite-volume differences centred in space, and explicit multistage time-stepping. The stability of this novel system is analysed, its allowable discontinuities are described, and appropriate far-field and solid-wall boundary conditions are introduced. Results are presented for both two- and three-dimensional flows, including vorticity shed from a delta wing. Whether vorticity is produced or not depends very strongly on the body geometry, the accuracy of the solution method, and the transient discontinuities that evolve in the flow field. The results are analysed for the total-pressure losses in the flow fields, and for the diffusion of the vortex sheets.

Approximation von Punktmengen durch implizite Funktionen

Computation of inviscid incompressible flow with rotation

Inviscid transonic flows containing either strong shock waves or complex vortex structure call for the Euler equations as a realistic model. We present here a computational procedure, termed WINGA2, for solving the Euler equations for transonic flow around aircraft upon a 0–0 mesh generated by transfinite interpolation. An explicit time-marching finite-volume technique solves the flow equations and features a non-reflecting far-field boundary condition and an internal mechanism for temporal damping together with a model for artificial viscosity. The method's convergence to a steady state is studied, and results computed on the CYBER 205 vector processor are presented. The Euler equation model is found to predict the existence of a tip vortex created by flow separating from the downstream region of the tip of the ONERA M6 wing where the radius of curvature approaches zero.

Computation of flow around wings based on the Euler equations

In this paper we revisit the problem of implementing wall boundary conditions for the Euler equations of gas dynamics in the context of unstructured meshes. Both (a) strong formulation, where the zero normal velocity on the wall is enforced explicitly and (b) weak formulation, where the zero normal velocity on the wall is enforced through the flux function are discussed. Taking advantage of both approaches, mixed procedures are defined. The new wall boundary treatments are accurate and can be applied to any approximate Riemann solver. Numerical comparisons for various flow regimes, from subsonic to supersonic, and for various approximate Riemann solvers point out that the mixed boundary procedures drastically improve the accuracy. © 1998 John Wiley & Sons, Ltd.

Wall boundary conditions for inviscid compressible flows on unstructured meshes

Transonic wing-body calculations using Euler equations

An approximately factored, implicit multigrid algorithm is developed to calculate three-dimensional compressible viscous flows. This scheme solves the full Reynolds-averaged Navier-Stokes equations with a k-e model of turbulence. The flow equations are solved within the framework of the multigrid method using a four grid level W-cycle, while the k-e equations are solved only on the finest grid and uncoupled from the flow calculations

Diagonally inverted lower-upper factored implicit multigrid scheme for the three-dimensional navier stokes equations

Die numerische Mathematik gehort zu denjenigen mathematischen Diszi- plinen, deren Ergebnisse in besonderem MaBe in anderen Wissenschaften wirksam werden. Sie schHigt die Briicke zwischen den gnmdlegenden Be- griffen und Modellen der Analysis und ihren Anwendungen in Naturwissen- schaft, Technik, Okonomie und in anderen Bereichen. Immer starker erhebt sich deshalb die Forderung breiter Kreise nach besseren und leistungs- fahigeren numerischen Verfahren. Gleichzeitig vollzieht sich ein grund- legender Wandel von der "manuellen numerischen Mathematik" zur "computer-gesttitzten numerischen Mathematik". Alles das tragt dazn bei, daB Bedeutung und Anziehungskraft der numerischen Mathematik standig zunehmen. Wir versuchen in unserem Buch, dieser Entwicklung Rechnung zu tragen. Das Buch ist aus V orlesungen entstanden, die beide Verfasser in den letzten ftinf Jahren an der Universitat Rostock fliT Studenten der Mathematik und der technischen Wissenschaften gehalten haben. Es ist flir Student en der ersten Studienjahre gedacht und setzt lediglich Grund- kenntnisse aus der Analysis, der linearen Algebra und der Rechentechnik voraus. Es wendet sich besonders an die Studenten der Mathematik und entspricht in seinem Umfang den Anforderungen der Grundausbildung nach dem Studienplan der Grundstudienrichtung Mathematik an den Universi- taten und Hochschulen der DDR. Es kann aber ebenso interessierten Stu- denten der N aturwissenschaften, der technischen und 6konomischen Wissen- schaften ftir ein tieferes Eindringen in die Grundlagen der numerischen Mathematik empfohlen werden. Auch der Praktiker, der im Zusammenhang mit dem Einsatz der Rechentechnik numerische Verfahren benotigt, wircl aus cler Lekttirc clieses Buches Nutzen ziehen und Anregungen fill die Aus- wahl geeigneter numerischer Verfahren erhalten.

Approximation von Punktmengen durch implizite Funktionen

Citations

Computation of inviscid incompressible flow with rotation

Computation of flow around wings based on the Euler equations

Wall boundary conditions for inviscid compressible flows on unstructured meshes

Transonic wing-body calculations using Euler equations

Diagonally inverted lower-upper factored implicit multigrid scheme for the three-dimensional navier stokes equations

References

Elementare Methoden der numerischen Mathematik

Lineare numerische Analysis

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Numerical implementation of solid-body boundary conditions for the euler equations