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Syllabus for

Academic year
TME225 - Mechanics of fluids
Syllabus adopted 2011-02-21 by Head of Programme (or corresponding)
Owner: MPAME
7,5 Credits
Grading: TH - Five, Four, Three, Not passed
Education cycle: Second-cycle
Major subject: Mechanical Engineering
Department: 42 - APPLIED MECHANICS

Teaching language: English
Open for exchange students
Block schedule: A

Course module   Credit distribution   Examination dates
Sp1 Sp2 Sp3 Sp4 Summer course No Sp
0111 Examination 7,5c Grading: TH   7,5c   21 Oct 2013 pm M,  13 Jan 2014 pm V,  26 Aug 2014 am M

In programs



Professor  Lars Davidson

Course evaluation:

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For single subject courses within Chalmers programmes the same eligibility requirements apply, as to the programme(s) that the course is part of.

Course specific prerequisites

A basic course in fluid mechanics


The course provides an introduction to continuum mechanics and turbulent fluid flow.

Learning outcomes (after completion of the course the student should be able to)

  • Confidently manipulate tensor expressions using index notation, and use the divergence theorem and the transport theorem.

  • Derive the Navier-Stokes equations and the energy equation using tensor notation

  • Analytically solve Navier-Stokes equations for a couple of simple fluid flow problems and analyze and understand these flows

  • Characterize turbulence

  • Understand and explain the energy spectrum for turbulence and the cascade process

  • Derive the exact transport equations for the turbulence kinetic energy and the turbulent Reynolds stresses

  • Identify the various terms in these equations and describe what role they play

  • Derive the linear velocity law and the logarithmic velocity law for a turbulent boundary layer

  • Recognize the difference between wall-bounded and free shear flows


The students will initially learn the basics of Cartesian tensors and the index notation.

A strong focus is placed on deriving and understanding the transport equations in three dimensions.These equations provide a generic basis for fluid mechanics, turbulence and heat transport. In continuum mechanics we will discuss the strain-rate tensor, the vorticity tensor and the vorticity vector. In connection to vorticity, the concept of irrotational flow, inviscid flow and potential flow will be introduced. The transport equation for the vorticity vector will be derived from Navier-Stokes equations.

Developing channel flow will be analyzed in detail. The results from a numerical solution is provided to the students. In a Matlab assignment, the students will compute different quantities such as the increase in the centerline velocity, the decrease of the wall shear stress, the vorticity, the strain-rate tensor, the dissipation, the eigenvectors and the eigenvalues of the strain-rate tensor.

In the larger part of the course the students will learn the basics of turbulent flow. Turbulence includes short-lived eddies of different size and frequency. The larger the Reynolds number, the larger the difference in size and frequency between the largest and the smallest eddies. This is the very reason why there is no computer large enough at which we can numerically solve the Navier-Stokes equations at high Reynolds number.

The energy spectra and the energy cascade process will be discussed in some detail; in the energy cascade it is assumed that turbulent kinetic energy is transferred from the largest energy-rich turbulent eddies to the smallest eddies. At the small eddies the kinetic energy is transformed into internal energy, i.e. an increase in temperature.

From the instantaneous Navier-Stokes equations the time averaged momentum equation will be derived. In this equation appears a new term, the turbulent Reynolds stress tensor. The students will learn how to derive the exact Reynolds stress transport (RST) equation, as well as the equations for turbulent kinetic energy and mean kinetic energy. The physical meaning of the different terms in the RST equation, such as the production term, the pressure-strain term and the dissipation, will be discussed in some detail.

In a second Matlab assignment, the students will be given a database of instantaneous turbulent flow for fully developed channel flow. The students will compute and analyze various turbulent quantities such as the turbulent Reynolds stresses, the pressure strain term in the RST equation, the two-point correlation, the integral turbulent length scale and the dissipation.

The Department has extensive experimental facilities for fluid dynamics. Two or three visualizations
of turbulent flow will be presented to the students.

.For more information

.Lecturer's homepage


Lectures, workshops using Matlab, assignments including written reports, demonstrations in the fluid dynamics laboratory


Lecture notes


Assignments and written examination

Published: Mon 28 Nov 2016.