Course syllabus

Prerequisites

A basic course in fluid mechanics. The course at Mechanical Enginnering at the Bacherlor level (3rd year) is based on the book: Fluid Mechanics, Frank M. White, McGraw-Hill, New York, 8th ed., 2016

Contact details

• Examiner:/Lecturer Gaetano Sardina (sardina@chalmers.se, 0737176687)
• Teaching assistants: Dario Maggiolo (maggiolo@chalmers.se),  Andhika Pratama (andhikap@chalmers.se)

Aim of the course

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

Schedule

Scehdule available in TimeEdit.

Course content

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 (i.e. zero viscosity) and potential flow will be introduced. The transport equation for the vorticity vector will be derived from Navier-Stokes equations.

In the first assignment, fully developed channel flow or developing boundary layer flow will be analyzed in detail. Students will use Python to analyze the flow field. B Python is an open-source software. Many large Swedish industries prefer engineers to use Python instead of Matlab due to Matlab's high license fees. The results from a numerical solution is provided to the students. In the assignment, the students will compute different quantities such as the decrease in the streamwise 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.

An introduction to potential flow will be given. A complex function is defined where the real part is the velocity potential and the imaginary part is the streamfunction. Exact solutions to this complex function will be derived (flat plate boundary layer, stagnation flow, flow around a cylinder etc). These exact solutions have many applications in real life, such as Flettner rotors, looping in table tennis, and free kicks in football. 

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.

In the last part of the course, we will work with the time-averaged Navier-Stokes equations. They include an unknown tensor -- the Reynolds stress tensor -- which must be modeled. We will derive the k-eps model, which is the most common turbulence model in industry. The treatment of walls needs special attention. There are two options: either wall functions or low-Reynolds number turbulence models. Both options will be discussed.

In the second assignment, the commercial code STAR-CCM+ will be used. The students will be given instructions on how to compute the flow over a 2D hill. The influence of using different turbulence models will be investigated. The results will be exported to Python format, and the students will analyze the results in more details.

Learning outcome

After completion of this 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
• Derive analytical solutions to the inviscid Navier-Stokes equations
• Characterize turbulence
• Understand and explain the energy spectrum for turbulence and the cascade process
• Derive the exact transport equations for the turbulence kinetic energy
• 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
• Derive the k-eps turbulence model
• Understand the different between wallfunctions and low-Reynolds number turbulence models

Organization

10 frontal lectures and two extra lectures for repetitions and preparations to the oral exam. Students are encouraged to ask questions. 10 computer room sessions with teaching assistants and the students that can receive help for the two Assignments.

Course literature

• eBook: Fluid dynamics, turbulent flow and turbulence modeling, Lars Davidson

The book is available to download on Canvas

Important dates/Deadlines

-Assignment 1: 4th October 2026

-Assignment 2: 25th October 2026

-Oral Exam: approximately 19th-22nd October 2026 and 2nd-13th November 2026

Examination

• Grades. failed, grade 3, 4 or 5
• Part 0: One assignment (A1-A13) in continuum mechanics including written report. This part is mandatory.
• Part 1: Two assignments including written reports. This part is mandatory.
• Part 2: Oral exam is mandatory.

Oral exam based on the questions linked to the Learning Outcomes and the Assignments. The teachers will also ask follow-up questions. There we try to test if the student has understood the topic or if he/she only has memorized it. A good understanding gives grade 4 or 5.
Three students at the time (90 minutes). Each student will get six randomly choosen questions available on Canvas and the questions of the Assignments. Grades: failed, 3, 4 or 5. No aids except the one provided by the teachers.

To get grade 'passed' on the course, you must have grade 3 on the Oral exam and hand in the two assignment reports.

Written reports of the assignments are part of the examination. They should be uploaded to the student portal before the deadlines (see Calendar on Canvas).

To get a grade 4 or 5 in the course, the written projects must be handed in on time.