
Comments from Attendees of the Short Course
 "An excellent overview of the current state of turbulence modeling. An equally excellent way to go from theory to application with a thorough understanding of the strengths and weaknesses of the models." Timothy Madden, National Academy of Sciences
 "Excellent speaker  able to keep your attention focused on the subject." Guy B. Spear, Atlantic Research Corp.
 "Excellent lecture!" Kenji Yoshida, Kawasaki Heavy Industries Ltd.
 "Dave's knowledge of the field was very impressive." Daniel Marcus, Lawrence Livermore National Lab
Course Description
Turbulence modeling is one of three key elements in Computational Fluid Dynamics (CFD). Very precise mathematical theories have evolved for the other two key elements, viz, grid generation and algorithm development. By its nature, i.e., creating a mathematical model which approximates the physical behavior of turbulent flows, far less precision has been achieved in turbulence modeling. This course addresses the problem of selecting and/or devising such a model for a given application. The fundamental premise of this short course is that, in the spirit of G. I. Taylor, a really good model should introduce the minimum amount of complexity while capturing the essence of the relevant physics.
The course begins with a careful discussion of turbulence physics in the context of modeling. The exact equations governing the extra turbulent (Reynolds) stresses, and the ways in which these equations can be closed, will be briefly outlined. We then discuss the behavior of length scales in turbulence, and the processes that govern the dissipation rate and other quantities used to provide model length (or time) scales. The section on modeling as such begins with the simplest turbulence models and charts a course leading to some of the most complex models that have been applied to a nontrivial turbulent flow problem.
Along the way, a systematic methodology involving use of similarity solutions, perturbation methods and numerical techniques, is presented for developing and/or analyzing a set of constitutive equations suitable for computation of turbulent flows. The course will stress the need to achieve a balance amongst the physics of turbulence, mathematical tools required to solve turbulencemodel equations, and common numerical problems attending use of such equations (i.e., what good is a model if it makes your program crash?). Several stateoftheart, userfriendly FORTRAN programs will be provided.
This short course was originally developed to satisfy a need identified by the NASA Johnson Space Center to help Lockheed and North American Rockwell engineers in their Computational Fluid Dynamics (CFD) activities. It has proven to be very popular and has been presented throughout the United States and Canada since June, 1993 as follows.
Short Course History
Date  Sponsor  Location 
June 1993 
Lockheed Engineering Company 
Houston, TX 
January, 1994 
AIAA 
Reno, NV 
June, 1994 
CERCA 
Montreal, Quebec, CANADA 
October, 1994 
National Program Office 
Palmdale, CA 
January, 1995 
AIAA 
Reno, NV 
June, 1995 
CFDCS Meeting 
Banff, BC, CANADA 
March, 1996 
AIAA 
Washington, DC 
March, 1996 
Knolls Atomic Power Lab 
Schenectady, NY 
June, 1997 
AIAA 
Snowmass, CO 
June, 1998 
AIAA 
Albuquerque, NM 
June, 1998 
Boeing 
Seattle, WA 
October, 1998 
Boeing North American 
Canoga Park, CA 
January, 1999 
AIAA 
Reno, NV 
February, 1999 
University of Adelaide 
Adelaide, AUSTRALIA 
June, 1999 
AIAA 
Norfolk, VA 
August, 1999 
NASA Langley 
Hampton, VA 
May, 2000 
NASA Glenn 
Cleveland, OH 
June, 2000 
NLR 
Amsterdam, Holland 
June, 2000 
AIAA 
Denver, CO 
June, 2001 
AIAA 
Anaheim, CA 
June, 2002 
AIAA 
St. Louis, MO 
June, 2007 
AIAA 
Miami, FL 
June, 2009 
AIAA 
Miami, FL 
May, 2010 
NASA Langley 
Hampton, VA 
June, 2011 
AIAA 
Honolulu, HI 
Intended Audience
The course is designed for all research engineers, programmers and managers engaged in turbulent flow CFD. Managers will gain an appreciation of what constitutes a suitable turbulence model for a given application and will gain the ability to interact effectively with specialists. Research engineers and programmers will learn the truths and myths of turbulence modeling along with a systematic methodology for testing and validating turbulence models and associated software.
Course Materials
You will be provided with course notes. Optionally, you may purchase (at a discounted price) a copy of the hardback text, Turbulence Modeling for CFD, by David C. Wilcox, which includes a compact disk containing both source and executable code for the programs documented in the text. The book is now in its third edition and is used at universities all over the world. Course attendance also entitles you discounts on all DCW Industries publications for a limited time.
Instructor
Dr. David C. Wilcox is the President of DCW Industries, Inc., a California aerospace and bookpublishing firm he founded in 1973. He is also a Lecturer at UCLA and USC. Dr. Wilcox did his undergraduate studies at the Massachusetts Institute of Technology and received his PhD in Aeronautics from the California Institute of Technology in 1970. He served as an AIAA Journal Associate Editor from 1989 to 1992.
Course Outline
The short course touches on highlights of each chapter of the text. The course involves approximately 20 hours of lectures with provision for discussion and software demonstration. It will be presented over a threeday period. The course also includes provision for focusing on topics of particular interest to your organization. Such topics are agreed to in advance, and Dr. Wilcox will make a special presentation followed by extended discussion. The general outline is as follows.
Day One
 Introduction
 The ideal turbulence model
 Physics of turbulence
 Kolmogorov theory
 The law of the wall and the powerlaw controversy
 History of turbulence modeling
 The Closure Problem
 Reynolds averaging
 Reynoldsaveraged equations
 The Reynolds stress equation
 Length scales and their behavior
 Equations vs. unknowns
 The scales of turbulence
 Twopoint statistics
 Algebraic Models
 Molecular transport of momentum
 The mixing length hypothesis
 How molecules and eddies are different
 Free shear flows
 CebeciSmith and BaldwinLomax models
 Channel/pipe flow
 Attached boundary layers
 Separated flows
 The halfequation model
 Range of applicability
Day Two
 Turbulence Energy Equation Models
 The turbulence energy equation
 Oneequation models
 Twoequation models/generic
 komega and kepsilon models
 Closure coefficients
 Free shear flows
 The role of cross diffusion
 Solution sensitivity to freestream conditions
 Surface boundary conditions
 Surface roughness and surface mass transfer
 Channel/pipe flow
 Perturbation analysis of the boundary layer
 Attached boundary layers
 LowReynoldsnumber corrections
 Transition prediction
 Separated flows
 The stresslimiter concept
 Range of applicability
 Effects of Compressibility
 Favreaveraging
 Favreaveraged equations
 Compressibleflow closure approximations
 Compressible mixing layer
 Compressible law of the wall
 Shock induced separation
 More on the role of the stress limiter
 The reattachmentpoint heattransfer anomaly
 Beyond the Boussinesq Approximation
 Nonlinear constitutive relations
 Algebraic Stress Models
 Why the stresslimiter works so well
 Stresstransport models
 Pressurestrain correlation modeling
 LRR and Wilcox Stressomega models
 Free shear flows
 Channel/pipe flow
 Attached boundary layers
 Streamline curvature
 Rotating channel flow
 Unsteady boundary layers
 Separated flows
 Range of Applicability
Day Three
 Numerical Considerations
 Multiple time scales and stiffness
 Nearwall solution accuracy
 Turbulent/nonturbulent interfaces
 Parabolic marching methods
 Elementary timemarching methods
 Blockimplicit methods
 Iteration and grid convergence
 New Horizons
 Direct Numerical Simulation
 Large Eddy Simulation
 Detached Eddy Simulation
 Chaos
 Special Topics
 Special Presentation
 Extended Discussion
 Open Forum Discussion
