Mechanics of fluids (7.5 hec)

Prerequisites

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

Aim of the course

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

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. Student can use Python, Matlab or Octave. Both Octave and Python are 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 a Python/Matlab/Octave 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 solution to this complex function will be derived (flat plat boundary layer, stagnation flow, flow around a cylinder etc). These exact solution have many applications in real life such as Flettner rotors looping in table tennis and freekicks 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 need special attention. There are two options, either wallfunctions 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/Matlab/Octave format and the students will analyze the results in come detail.

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 pre-recorded lectures are available here and at Canvas. Instead of regular lectures, there will be five discussion seminars (Q&A) on Campus. At the discussion seminars, the questions in the appendix K 'TME226 Discussion seminars' in the eBook will be discussed. Students are encouraged to ask questions.
  • The students will register in six groups (approximately eight in each). Each group will have approximately one discussion seminar every week.
  • Each discussion seminar will last 30 minutes.
  • Suitable preparation for a discussion seminar
    • Watch the relevant recorded lecture(s)
    • Read relevant parts of the eBook
    • Study the questions in the appendix K 'TME226 Discussion seminars'
At other times, you can post a question on PIazza

Course literature

Examination

  • Grades. failed, grade 3, 4 or 5
  • Part 1: Two assignments including written presentations. This part is mandatory. Written presentations are part of the examination.
  • Part 2: Discussion seminars. This part is not mandatory.
  • Part 3: Either Quiz or oral exam is mandatory.
    • Part 3a: Quiz. Students who pass this quiz get grade 3 (passed). The Quiz exam will be similar to the Quiz 1-10 at Canvas related to the Lectures.
    • Part 3b: Oral exam based on the questions in the Discussion seminars 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 appox seven randomly choosen questions from the Discussion seminars. Grades: failed, 3, 4 or 5.
    • Students may do both the Quiz and Oral exam; the final grade is then the highest of the two.
    • To get grade 'passed' on the course, you must have grade 3 on either the Quiz or the Oral exam and handing in the two assignment reports.


Written presentations of the assignments are part of the examination. It should be uploaded to the student portal before deadline (see Calender at Canvas).

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

This page can be found on

https://www.tfd.chalmers.se/~lada/MoF



Course home page