Mortar Board Turbulence Modeling Short Course
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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 turbulence-model equations, and common numerical problems attending use of such equations (i.e., what good is a model if it makes your program crash?). Several state-of-the-art, user-friendly 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
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.


Dr. David C. Wilcox is the President of DCW Industries, Inc., a California aerospace and book-publishing 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 three-day 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
    1. The ideal turbulence model
    2. Physics of turbulence
    3. Kolmogorov theory
    4. The law of the wall and the power-law controversy
    5. History of turbulence modeling
  • The Closure Problem
    1. Reynolds averaging
    2. Reynolds-averaged equations
    3. The Reynolds stress equation
    4. Length scales and their behavior
    5. Equations vs. unknowns
    6. The scales of turbulence
    7. Two-point statistics
  • Algebraic Models
    1. Molecular transport of momentum
    2. The mixing length hypothesis
    3. How molecules and eddies are different
    4. Free shear flows
    5. Cebeci-Smith and Baldwin-Lomax models
    6. Channel/pipe flow
    7. Attached boundary layers
    8. Separated flows
    9. The half-equation model
    10. Range of applicability

Day Two

  • Turbulence Energy Equation Models
    1. The turbulence energy equation
    2. One-equation models
    3. Two-equation models/generic
    4. k-omega and k-epsilon models
    5. Closure coefficients
    6. Free shear flows
    7. The role of cross diffusion
    8. Solution sensitivity to freestream conditions
    9. Surface boundary conditions
    10. Surface roughness and surface mass transfer
    11. Channel/pipe flow
    12. Perturbation analysis of the boundary layer
    13. Attached boundary layers
    14. Low-Reynolds-number corrections
    15. Transition prediction
    16. Separated flows
    17. The stress-limiter concept
    18. Range of applicability
  • Effects of Compressibility
    1. Favre-averaging
    2. Favre-averaged equations
    3. Compressible-flow closure approximations
    4. Compressible mixing layer
    5. Compressible law of the wall
    6. Shock induced separation
    7. More on the role of the stress limiter
    8. The reattachment-point heat-transfer anomaly
  • Beyond the Boussinesq Approximation
    1. Nonlinear constitutive relations
    2. Algebraic Stress Models
    3. Why the stress-limiter works so well
    4. Stress-transport models
    5. Pressure-strain correlation modeling
    6. LRR and Wilcox Stress-omega models
    7. Free shear flows
    8. Channel/pipe flow
    9. Attached boundary layers
    10. Streamline curvature
    11. Rotating channel flow
    12. Unsteady boundary layers
    13. Separated flows
    14. Range of Applicability

Day Three

  • Numerical Considerations
    1. Multiple time scales and stiffness
    2. Near-wall solution accuracy
    3. Turbulent/nonturbulent interfaces
    4. Parabolic marching methods
    5. Elementary time-marching methods
    6. Block-implicit methods
    7. Iteration and grid convergence
  • New Horizons
    1. Direct Numerical Simulation
    2. Large Eddy Simulation
    3. Detached Eddy Simulation
    4. Chaos
  • Special Topics
    1. Special Presentation
    2. Extended Discussion
    3. Open Forum Discussion

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