C8406 - VISCOUS FLOWS AND TURBULENCE

Learning outcomes

The course provides a comprehensive understanding of advanced concepts in fluid mechanics, bridging theoretical foundations with applications in aerodynamics and viscous flows. This module covers analytical solutions of canonical viscous flows and introduces advanced concepts of wall-bounded flows, including transition and turbulence.

Course contents

1. VISCOUS FLOW

Non-dimensional equations for incompressible flow. The steady solution in a plane duct (Poiseuille solution). Link between flow rate and pressure drop. The Couette solution. Basic concepts on the vectorial operators in non cartesian frame of reference. The equations of motion in cylindrical coordinates. The flow in axisymmetric ducts: the Hagen-Poiseuille flow. The Taylor-Couette flow. Compatibility conditions for the solution of the Navier-Stokes equations. The unsteady Stokes problems.

Weakly divergent flows: the boundary layer, jets and wakes. Non dimensional equations in weakly divergent problems. The Prandtl equations. Self-similar techniques for the solution of Prandtl equations. The flow on a flat-plate at zero angle of attack. The Blasius solution. Generalization of the Blasius solution. The Falkner-Skan solution. Approximate solutions in general boundary layers, application to a thin wake. The Von Karman integral equation. The method of Pohlhausen. Main results and limits of the method.

2. STABILITY

Introduction. Definition of stability. Rayleigh and Orr-Sommerfeld equations. Normal mode hypothesis. Rayleigh inflection point criterion. Tollmien-Schlichting waves in channel and boundary layer flows.

Eigenvalue problems. Solution of the Rayleigh equation for a piecewise liner profile (mixing layer). Convective and absolute stability. Transition prediction with the e-to-the-N-method. Introduction to by-pass transition. Transient growth and non-modal stability. Global stability and sensitivity. Transition to turbulence and effects of free stream turbulence.

3. TURBULENCE

Introduction. Dissipation in turbulent flows. Derivation of the Kolmogorov scales. Length scales and Reynolds number. Statistical methods, probability density distribution, mean, variance and higher moments. Derivation of the RANS equations. Reynolds stresses. Derivation of the turbulent kinetic energy equation. Production, dissipation and transport terms. Example of turbulent flows. The 2D turbulent jet. The 2D turbulent wake. Wall bounded flows: turbulent channel flow. Turbulent boundary layers. The logarithmic velocity distribution. Near-wall turbulence. The “Long Pipe” at CICLoPE. Wall shear stress measurements. Isotropic turbulence, Taylor scale, integral scale, correlation function. Kolmogorovs first and second hypothesis, The k-5/3 law. Spectral transfer. Wave number and frequency spectra. Taylors hypothesis of frozen turbulence.

Readings/Bibliography

VISCOUS FLOW:

Viscous fluid flow – F. White – Mc Graw Hill – ISBN 0070697124

Elements of Fluid Dynamics – G. Buresti – Imperial College Press

STABILITY - Lecture notes Prof. Schlatter

TURBULENCE - Lecture notes Prof. Örlü

Teaching methods

Lectures and exercises given by the docents.

Assessment methods

The exam consists of a single session in which the student should answer a written test. The student must show a sufficient skill in writing down and commenting the mathematical and physical models as well as the different theoretical techniques.

Teaching tools

Blackboard and power point presentations as well as hands-on tutorials.

Office hours

See the website of Ramis Örlü

See the website of Philipp Christian Schlatter