An Efficient Method for Solving the
Steady Euler Equations

Meng-Sing Liou

NASA Technical Memorandum 87329
May 1986

The present paper shows an efficient numerical procedure for solving a set of nonlinear partial differential equations, specifically the steady Euler equations. Solutions of the equations were obtained by Newton's linearization procedure, commonly used to solve the roots of nonlinear algebraic equations. In application of the same procedure for solving a set of differential equations we give a theorem showing that a quadratic convergence rate can be achieved. While the domain of quadratic convergence depends on the problems studied and is unknown a priori, we show that first- and second-order derivatives of flux vectors determine whether the condition for quadratic convergence is satisfied. The first derivatives enter as an implicit operator for yielding new Iterates and the second derivatives Indicates smoothness of the flows considered. Consequently flows involving shocks are expected to require larger number of iterations. First-order upwind discretization In conjunction with the Steger-Warming flux-vector splitting is employed on the implicit operator and a diagonal dominant matrix is resulted. However the explicit operator is represented by first- and second-order upwind differencings, using both Steger-Warming's and van Leer's splittings. We discuss treatment of boundary conditions and solution procedures for solving the resulting block matrix system. With a set of test problems for one- and two-dimensional flows, we show detailed study as to the phlyctenae, accuracy, and convergence of the present method.

Fundamentals of Computational Fluid Dynamics (Sample Portion)

This is but a sample portion of the book released by one of the authors. To order the entirety of this excellent book, click here or on the link to the right.

Harvard Lomax, Thomas H. Pulliam and David W. Zingg
Springer-Verlag
2003

This book is intended as a textbook for a first course in computational fluid dynamics and will be of interest to researchers and practitioners as well. It emphasizes fundamental concepts in developing, analyzing, and understanding numerical methods for the partial differential equations governing the physics of fluid flow. The linear convection and diffusion equations are used to illustrate concepts throughout. The chosen approach, in which the partial differential equations are reduced to ordinary differential equations, and finally to difference equations, gives the book its distinctiveness and provides a sound basis for a deep understanding of the fundamental concepts in computational fluid dynamics.

Webmaster's Note: this is an outstanding book, gets to the core of the subject in a clear and consise manner.

Implicit Total Variation Diminishing (TVD) Schemes for Steady-State Calculations

H. C. Yee, R. F. Warming,and A. Harten

NASA Technical Memorandum 84342
March 1983

We examine the application of a new implicit unconditionallystable high-resolution TVD scheme to steady-state calculations. It is a member of a one-parameter family of explicit and implicit second-order accurate schemes developed by Harten for the computation of weak solutions of one-dimensional hyperbolic conservation laws. This scheme is guaranteed not to generate spurious oscillations for a nonlinear scalar equation and a constant coefficient system. Numerical experiments show that this scheme not only has a fairly rapid convergence rate, but also generates a highlyresolved approximation to the steady-state solution. A detailed implementation of the implicit scheme for the one- and two-dimensional compressible inviscid equations of gas dynamics is presented. Some numerical computations of one- and two-dimensional fluid flows containing shocks demonstrate the efficiency and accuracy of this new scheme.

Edwin S. Holdredge and Bob H. Reed, Texas Engineering Experiment Station
August 1956

This report describes the principal work performed for the period 15 June 1954-31 August 1956 by the Texas Engineering Experiment Station under contract between the Texas Agricultural and Mechanical College System as Contractor and the Department of the Army. The referenced contracts cover investigations to determine air flow patterm and infiltration characteristics of military-type buildings by means of scale models thereof. The following items were studied:

1. Air flow patterns around model buildings at wind speeds ranging from approximately one to twenty mph.
2. Amount of infiltration of particulate substances such as smokes into model buildings at wind speeds ranging from one to twenty mph.
3. Ths minimum pressure required to prevent infiltration of particulate substances into the model buildings at wind speeds of approximately one to twenty mph and gusts up to approximately twenty mph.

Solution Methods In Computational Fluid Dynamics

Thomas H. Pulliam
NASA Ames Research Centre
1985

Implicit finite difference schemes for solving two dimensional and three dimensional Euler and Navier-Stokes equations will be addressed. The methods are demonstrated in fully vectorized codes for a CRAY type architecture. We shall concentrate on the Beam and Warming implicit approximate factorization algorithm in generalized coordinates. The methods are either time accurate or accelerated non-time accurate steady state schemes. Various acceleration and efficiency modifications such as matrix reduction, diagonalization and flux split schemes will be presented. Examples for 2-D inviscid and viscous calculations (e.g. airfoils with a deflected spoiler, circulation control airfoils and unsteady buffeting) and also 3-D viscous flow are included.

Upwind and Symmetric Shock-
Capturing Schemes

H.C. Yee

NASA Technical Memorandum 89464
May 1987

The development of numerical methods for hyperbolic conservation laws has been a rapidly growing area for the last ten years. Many of the fundamental concepts and state-of-the-art developments can only be found in meeting proceedings or internal reports. This review paper attempts to give an overview and a unified formulation of a class of shock-capturing methods. Special emphasis will be on the construction of the basic nonlinear scalar second-order schemes and the methods of extending these nonlinear scalar schemes to nonlinear systems via the exact Riemann solver, approximate Riemann solvers, and flux-vector splitting approaches. Generalization of these methods to efficiently include real gases and large systems of nonequilibrium flows will be discussed. The performance of some of these schemes is illustrated by numerical examples for one-, two- and three-dimensional gas-dynamics problems.