Works of General Interest in Fluid Mechanics and Computational Fluid Dynamics (CFD)

Aerodynamic Design Optimization on Unstructured Grids with a Continuous Adjoint Formulation

W. Kyle Anderson (Texas A&M Class of ’81) was the professor for my last course for my PhD. He is a brilliant man; unfortunately, I didn’t do well in this course. But optimization in general was crucial for my PhD disssertation.

W. Kyle Anderson and V. Venkatakrishnan
NASA Langley Research Center Contract NASI-19480
January 1997

A continuous adjoint approach for obtaining sensitivity derivatives on unstructured grids is developed and analysed. The derivation of the co-state equations is presented, and a second-order accurate discretization method is described. The relationship between the continuous formulation and a discrete formulation is explored for inviscid, as well as for viscous flow. Several limitations in a strict adherence to the continuous approach are uncovered, and an approach that circumvents these difficulties is presented. The issue of grid sensitivities, which do not arise naturally in the continuous formulation, is investigated and is observed to be of importance when dealing with geometric singularities. A method is described for modifying inviscid and viscous meshes during the design cycle to accommodate changes in the surface shape. The accuracy of the sensitivity derivatives is established by comparing with finite-difference gradients and several design examples are presented.

An Efficient Method for Solving the Steady Euler Equations

Liou Meng-Sing
NASA Technical Memorandum 87329
May 1986

The present paper shows an efficient numerical procedure for solving a set of non-linear 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 non-linear 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.

CCM CONTINUITY CONSTRAINT METHOD: A Finite-Element Computational Fluid Dynamics Algorithm for Incompressible Navier-Stokes Fluid Flows

Oak Ridge National Laboratory ORNL/TM-12389
September 1993

As tile field of computational fluid dynamics (CFD) continues to mature, algorithms are required to exploit the most recent advances in approximation theory, numerical mathematics, computing architectures, and hardware. Meeting this requirement is particularly challenging in incompressible fluid mechanics, where primitive-variable CFD formulations that are robust, while also accurate and efficient in three dimensions, remain an elusive goal. This dissertation asserts that one key to accomplishing this goal is recognition of the dual role assumed by the pressure, i.e., a mechanism for instantaneously embracing conservation of mass and a force in tile mechanical balance law for conservation of momentum. Proving this assertion has motivated the development of a new, primitive-variable, incompressible, CFD algorithm called tile Continuity Constraint Method (CCM). The theoretical basis fbr the CCM consists ofa finite-element spatial semi-discretization of a Galerkin weak statement, equal-order interpolation for all state-variables, a O-implicit time-integration scheme, and a quasi-Newton iterative procedure extended by a Taylor Weak Statement (TWS) formulation for dispersion error control. Original contributions to algorithmic theory include: (a) formulation of the unsteady evolution of the divergence error, (b) investigation of the role of non-smoothness in the discretized continuity-constraint function, (c) development of a uniformly Ht Galerkin weak statement for the Reynolds-averaged Navier-Stokes pressure Poisson equation, (d) derivation of physically and numerically well-posed boundary conditions, and (e) investigation of sparse data structures and iterative methods for solving the matrix algebra statements generated by the algorithm. In contrast to tile general family of”pressure-relaxation” incompressible CFD algorithms, the CCM does not use the pressure as merely a mathematical device to constrain the velocity distribution to conserve mass. Rather, the mathematically smooth and physically-motivated genuine pressure is an underlying replacement for tile non-smooth continuity-constraint function to control inherent dispersive-error mechanisms. Dominated by this dispersive-error mode, the non-smoothness of the discrete continuity-constraint function is proven to play a critical role in its ability to remove the divergence error in tile discrete velocity distribution. The genuine pressure is calculated by the diagnostic pressure Poisson equation, evaluated using the verified solenoidal velocity field. This new separation of tasks also produces a genuinely clear view of the totally distinct boundary conditions required for the continuity-constraint function and genuine pressure. A broad range of 3-dimensional verification, benchmark, and validation lest problems, as computed by the code CFDI,.PHI3D, completes this dissertation.

Comparison of NACA 0012 Laminar Flow Solutions: Structured and Unstructured Grid Methods

R. C. Swanson, NASA Langley Research Center, Hampton, Virginia
S. Langer, DLR, Deutsches Zentrum für Luft- und Raumfahrt, Institut für Aerodynamik und Strömungstechnik, Braunschweig, Germany
January 2016

In this paper we consider the solution of the compressible Navier-Stokes equations for a class of laminar airfoil flows. The principal objective of this paper is to demonstrate that members of this class of laminar flows have steady-state solutions. These laminar airfoil flow cases are often used to evaluate accuracy, stability and convergence of numerical solution algorithms for the Navier-Stokes equations. In recent years, such flows have also been used as test cases for high-order numerical schemes. While generally consistent steady-state solutions have been obtained for these flows using higher order schemes, a number of results have been published with various solutions, including unsteady ones. We demonstrate with two different numerical methods and a range of meshes with a maximum density that exceeds 8,000,000 grid points that steady-state solutions are obtained. Furthermore, numerical evidence is presented that even when solving the equations with an unsteady algorithm, one obtains steady-state solutions.

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.

Harvard Lomax, Thomas H. Pulliam and David W. Zingg

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, analysing, 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 concise manner.

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

Click here for “Proof of Harten’s Lemma re the Convergence of TVNI Finite-Difference Schemes”

H. C. Yee, R. F. Warming,and A. Harten
NASA Technical Memorandum 84342
March 1983

We examine the application of a new implicit unconditionally stable 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 non-linear 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 highly resolved 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.

Pressure Distribution on Buildings

Edwin Sereno Holdredge taught Mechanical Engineering at Texas A&M University for many years. His use of dimensionless analysis, illustrated in this work, became a hallmark of his teaching.

A native of Lenoir City, TN and graduate of the University of Tennessee at Knoxville, he had a dry sense of humour that frequently eluded many of his students. Concerning this report, he stated in class that “The Army told me they didn’t know anything about fluid flow around buildings.” He paused and then said, “They were right, they didn’t.”

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 pattern 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

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 non-linear scalar second-order schemes and the methods of extending these non-linear scalar schemes to non-linear 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 non-equilibrium 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.