AAE 41600 Viscous Flows


Navier-Stokes equations, boundary layer theory, frictional drag and heating, separation and transition. Turbulent flows. Computational methods for laminar and turbulent flows.

Format 3 hrs lecture per week
Credit hours 3
Status Elective
Offered Spring
Pre-requisite AAE 333
Co-requisite None
Staff Professor Schneider
Text Viscous Fluid Flow, Frank M. White, 2nd edition, McGraw-Hill, New York, 1991, ISBN 0-07-069712-4
Assessment Method Midterm Exam 25%, Homework 40%, Computer Project 25%, Final Report 10%. Grading Policy is an instructor option and may vary

Course Goal & Objectives:


To extend the theoretical framework built in AAE 333 to include viscous flow. Apply the theory to the calculation of boundary layers in two dimensions. Introduction to transonic, supersonic, and hypersonic boundary layers, laminar-turbulent transition, and turbulence. Apply a standard finite-difference boundary-layer code by comparing to 2D experimental data.

Objectives include developing abilities to:

  • Understand the concepts of molecular diffusion, viscous flow, boundary layers, separation, laminar-turbulent transition, and turbulence
  • Calculate laminar incompressible 2D boundary layers using integral methods such as Thwaites
  • Calculate exact solutions to the Navier-Stokes equations for simple flows
  • Calculate exact self-similar solutions to the boundary-layer or Navier-Stokes equations, such as the Falkner-Skan solutions. Use these to develop a qualitative understanding of parametric effects such as those due to pressure gradient.
  • Compute the laminar and turbulent boundary layer on a 2D body using a standard finite-difference computer code. Learn the proper use of a canned code, including an understanding of the method, the limits of its applicability, the effect of grid resolution, and the sensitivity of the results to the input parameters. Compare to experimental data.
  • Understand the phenomena observed in compressible viscous flow, including compressibility effects on boundary layers, and an introduction to shock-shock and shock-boundary-layer phenomena.
  • Understand the phenomena of laminar-turbulent transition, it's importance, when it occurs, some of the parameters that affect it, and some means for estimating it.
  • Understand some basics of turbulence, including the Reynolds averaged equations, the closure problem, some computational methods of predicting turbulent flows, along with their limitations.

Necessary Background:

  1. Vector calculus and differential equations through PDEs
  2. Thermodynamics
  3. Introductory Fluid Mechanics
  4. Inviscid flow; incompressible required and compressible desirable.
  5. Ability to write a computer program. Experience with FORTRAN desirable.


  1. Introduction: Significance of viscous phenomena. Review of thermodynamics. Molecular diffusion of momentum and heat. Newtonian viscous shear. The Navier-Stokes equations. Boundary-layer concepts and boundary-layer equations. Boundary-layer separation. Turbulence. (6 classes).
  2. Integral Analysis of Laminar Boundary Layers: Boundary layer integral equations. Karman-Pohlhausen technique. Thwaite's method. Applications of Thwaite's method. (3 classes).
  3. Differential Equations for Laminar Flow: Continuity, momentum and energy conservation. (3 classes).
  4. Exact Solutions for Laminar Flow: Parallel flow solutions. Self-similar solutions. Methods for obtaining numerically exact solutions. (6 classes).
  5. Introduction to Compressible Laminar Viscous Flow: Exact solutions: compressible Couette flow. Compressible boundary layers. Introduction to shock-boundary layer interaction. (6 classes).
  6. Introduction to Laminar-Turbulent Transition: Linear instability theory. e**N methods. Experimental results in bounded and free shear flows, both incompressible and compressible. Effects of roughness, turbulence, vibration, noise, curvature, and so on. (12 classes).
  7. Introduction to Turbulent Boundary Layers: Reynolds averaged equations of motion. The Kolmogorov scale. The law of the wall and the law of the wake. Integral models. Differential equation models. (12 classes).

Exams (2 classes).

Relation to Program Objectives:

This is an elective for seniors, usually those majoring in aerodynamics. All course objectives contribute to the program objective of providing technical competence in aerodynamics (1). The ability to formulate and solve engineering problems (2a) is emphasized in the homework assignments and projects. Teamwork (2b) is emphasized in the computer project and in the group homework assignments leading up to it. For these projects, the students work in teams of 2 or 3. The ability to communicate effectively in writing (2c) is covered in the projects and homeworks, many of which are informal reports. Professional conduct (2d), life-long learning (3) and society impact (4) are discussed by the instructor through anecdotal stories throughout the course.

Prepared by: Steven Schneider

Date: February 16, 2001