NFYK14024U Turbulence

Volume 2024/2025

MSc Programme in Physics

MSc Programme in Physics with a minor subject


The governing fluid flow equations, the Navier-Stokes Equations are introduced and analyzed with respect to scaling and symmetries. The role of conservation laws and energy budgets is examined. A probabilistic theory is developed and dimensional analysis and Kolmogorov’s 1941 theory is described. Introducing and using tensor analysis the cornerstone 4/5'th law for the correlations function is derived. Modern developments as multi-scaling models and models for intermittency are introduced. The students will perform numerical simulations using shell models.

Learning Outcome

The students will have a good overview of the phenomenon of turbulence, energy cascades, predictability and the Navier-Stokes equation. They will understand the phenomenon of scale invariance and how statistical relations are derived using scaling techniques.


The students will be able to do dimensional analysis of the governing equation, derive the Reynolds number and the Kolmogorov scale. They will be able to manipulate the statistical quantities, such as correlation functions using basic tensor analysis, Fourier transform and scaling transformations.


The students will mature in understanding the physical implications given by the Navier-Stokes equations, and acquire mathematical routine to derive statistical relations from the governing equations.

See Ansalon for final course material. The following is an example of expected course littereature:

Turbulence and Shell Models, Peter Ditlevsen, Cambridge University Press

Mathematics courses completing the Physics Bachelor.
The Bachelor should be completed. However, exceptionally skillful third year students can follow the course.

Academic qualifications equivalent to a BSc degree is recommended.
Lectures, exercises and numerical work.
The course is offered every second year.
  • Category
  • Hours
  • Lectures
  • 30
  • Preparation
  • 145,5
  • Exercises
  • 30
  • Exam
  • 0,5
  • Total
  • 206,0
7,5 ECTS
Type of assessment
Oral examination, 30 minutes, including 5 minutes voting.
Type of assessment details
Oral examination without preparation. A subject is drawn at the beginning of the exam, and the student is expected to give a 15 min presentation of the subject followed by a 10 minutes session of questioning to the subject and the rest of the course material. The subjects will be announced ten days prior to the exam. The student’s report, presented as relevant graphs or similar in connection with the numerical work, will be shortly presented and discussed at the exam.
Exam registration requirements

A report should be approved in order for the student to take the oral eam.

Only certain aids allowed

1 page of notes, prepared for each subject, can be used to support the presentation

Marking scale
7-point grading scale
Censorship form
No external censorship
Several internal examiners (the course responsible and one more)

Same as ordinary exam.

If the student did not fulfill the prerequisite for the exam, a new report should be handed in no later than two weeks before the re-exam.

Criteria for exam assesment

see learning outcome