Fluid Dynamics

Course manual 2017/2018

Course content

Fluid dynamics, the study of the motion of fluids and plasmas, is critical to many different areas of physics, from astronomy to biophysics. This course will cover the basic principles underlying fluid flows, and illustrate their importance with reference to a range of cutting-edge research problems. Topics covered will include: the governing equations of hydrodynamics, hydrostatic and hydrodynamic equilibrium, waves and instabilities, turbulence, convection, shocks, and magnetohydrodynamics. Methods for solving fluid dynamics problems, including simple numerical techniques, will form an integral part of the course. Movies and demonstrations will be used throughout, to build intuition about fluid behaviour under different conditions.

Study materials

Literature

• Course textbook is Choudhuri's "The physics of fluids and plasmas: an introduction for astrophysicists"  https://www.cambridge.org/core/books/physics-of-fluids-and-plasmas/8A235D6F1D9DA51F05237D42BDFEFD06#

• Additional material on numerical methods in hydrodynamics will be posted online.

Objectives

Overall goal:  To understand the basic mechanisms governing fluid flow, and to be able to apply fluid dynamics methods to solve physical and astrophysical problems.

Detailed learning objectives per lecture

Lecture 1 - Introduction

At the end of this lecture, the students should understand that:

• Fluid dynamics is intimately connected to statistical mechanics and ‘particle’-based descriptions.
• Particles can be treated as a continuum (fluid) if: (a) collisions are frequent enough to keep distribution close to Maxwellian (Local Thermodynamic Equilibrium), and (b) mean free path << typical length scale
• In very dilute gases or regions where things change rapidly, fluid dynamics may not be useful.

Lecture 2 - Ideal Fluids

• Macroscopic derivation of inviscid hydrodynamics (HD) equations
  • Know and understand these derivations!
  • Understand the difference between Eulerian and Lagrangian viewpoints
• Be able to solve these equations under the following approximations, for simple problems.
  • Incompressibility (constant density, surprisingly useful - know why!)
  • Hydrostatic equilibrium (velocity, time derivatives zero)
  • Steady flows (time derivatives zero), including Bernoulli’s principle (pressure/velocity relation), which you should be able to derive from the HD equations and apply to simple problems.
• Understand the concept and consequences of Kelvin’s vorticity theorem and Potential flow, and when both are applicable.

Lecture 3  - Viscosity

• Understand and be able to describe the role that viscosity plays in transferring stress and momentum in a fluid.
• Navier-Stokes (NS) equation
  • Know the equation.
  • Understand the simplifications that are often made to it to solve practical problems.
  • Understand the effect of viscosity on the vorticity equation.
  • Recognise that additional boundary conditions are required to solve the NS equation (compared to Euler equations)
  • Be able to solve the NS equation for simple problems in different coordinate systems.
• Be able to derive and use Poiseuille’s formula for viscous flow through a circular pipe.
• Reynolds number:  be able to explain where the concept comes from and its importance.
• Viscous flow past solid bodies
  • Understand and be able to show why, in a system with a solid boundary, viscosity cannot be neglected.
  • Be able to describe qualitatively the effects of the boundary layer.

Lecture 4 - Compressibility

• Understand and be able to show that compressibility allows propagation of sound waves.
• Understand why compressibility is particularly important for gases: sound speed is lower than in liquids, so fluid velocities can be comparable to sound speed.
• Be able to explain the concept of Mach number.
• Be able to solve simple flow problems for compressible gases such as the de Laval nozzle.
• Basic shock physics: understand and be able to show the following.
  
  • What is a shock? A discontinuity in fluid variables.
  • When do they arise? When there is supersonic motion or when waves steepen.
  • Under what conditions are they steady? Need Mach number > 1.
  • How do we model them? Jump conditions.

Lecture 5 - Waves

• Oscillations and instabilities
  • Understand and be able to explain the relationship between the two.
  • Be able to apply linearized perturbation analysis to solve for oscillations/instabilities.
• In particular be able to explain and solve problems involving the following types of oscillation/instability:
  • Acoustic waves and the Jeans instability
  • Internal ‘gravity’ waves/convective instabilities
  • Interface waves/instabilities

Lecture 6 - Rotation

• Be able to explain what is differential rotation.
• Understand what is meant by the stability of rotating flows, and be able to derive and use Rayleigh’s criterion.
• Be able to write the fluid dynamics equations in the rotating frame and use them to solve simple dynamical problems.
• Understand and be able to explain and apply the concepts of effective gravitational potential, Rossby number, geostrophy, Bjerknes’ theorem and the Taylor-Proudman theorem.
• Be able to outline in principle how one calculates equilibrium states of rotating self-gravitating fluids.
• Understand the concept of the break-up rotation rate.

Lecture 7 - Turbulence

• Be able to explain the general properties of turbulent flows, and when they might arise.
• Be able to explain what is meant by a statistical theory of turbulence
• Be able to derive the properties of homogeneous isotropic turbulence
• Be able to derive and apply Kolmogorov’s theory of turbulence to simple problems.
• Understand and be able to describe, qualitatively, the transport properties of turbulent flows and the effect on the Navier-Stokes equations
• Be able to list some applications where turbulence is important

Lectures 8 and 9 - Magnetohydrodynamics (MHD)

• Be able to write down the basic equations of MHD and explain the differences from the HD equations.
• Understand the conditions under which the MHD equations are a valid description of charged fluids.
• Understand and be able to explain and apply the concept of Magnetic Reynolds number
• Understand what is meant by ideal MHD and flux-freezing, and when they are applicable.
• Be able to derive the MHD perturbation equations and use them to demonstrate the existence of  hydromagnetic, Alfvén, and magnetosonic waves.
• Convective instability in the presence of magnetic fields - be able to prove that magnetic fields can lead to suppression of convection, and describe qualitatively link to sunspots.
• Magnetic buoyancy - be able to demonstrate how it arises, and describe qualitatively the link to bipolar sunspots and clumpiness of the interstellar medium (the Parker instability).
• Be able to solve simple problems involving magnetic reconnection and current sheets, by analogy with the viscous problem from hydrodynamics.
• Be able to describe qualitatively why magnetic fields can transfer angular momentum.

Teaching methods

• Lecture
• Presentation/symposium
• Self-study
• Supervision/feedback meeting

Lectures explain the basic underpinning physics concepts.   Students are then set problem sheets to attempt at home, with answers being worked through in tutorial sessions led by the TA.  

Learning activities

Attendance

Requirements concerning attendance (OER-B).

Assessment

There are 8 homework problem sheets. Homework is not assessed, but critical for the exams and for mastery of the subject. Homework will be reviewed in the tutorials, and worked solutions posted on Blackboard after the tutorials.  Attempt the problems before the tutorial, to at least identify where you are getting stuck.

Assessment consists of the following items. 

• Midterm exam (20%).  This is an open book exam, consisting of a mix of concept (see below) and technical questions.
• Project presentation on a recent research paper (40%). Presentations in lecture/class slots.  See under 'Assignments' for more details.
• Final exam (40%).   No book, mix of concept and technical questions.

Concept questions require the students to give brief answers to a set of basic questions on fluid dynamics concepts.  The full list of potential questions is listed online at the start of the course, so that the students can work on their answers as they proceed through the course. A subset of these will be set in the midterm and final exams.

Assignments

Presentation

• Presentation of a recent Physical Review Letter paper.

Each student selects (from a set of options) a paper on fluid dynamics published over the last 2 years from Physical Review Letters. They are asked to give an individual presentation summarizing the research done (including motivation), connecting it to the basic fluid dynamics concepts learned during the course. 

The goal of the project is to learn about modern fluid dynamics research, since in the taught material we only get as far as work on turbulence from the early 1960s.  The presentation must answer the following questions:

• What is the research that has been done?
• Why is it important or interesting, and how does it relate to previous research in the area?
• How does it relate to the basic fluid dynamics principles that we have learned about in the class?
• Where is the research going next (or where has it gone for older papers - look at citing papers!)?

Then students must communicate this clearly in a presentation!  Presentation are given in class in Week 7, to the whole class. The presentation length is 15 minutes, with 5 minutes for questions. 

Assessment criteria:

• Science: Is research (methods, results) adequately explained? Is motivation behind research clear? Is the link to basic fluid dynamics principles clear? Is context of research (how it fits into existing picture, and where it goes next) clear?
• Quality of presentation: Are slides and speech easy to follow? Are figures/movies informative and necessary?
• Ability to answer questions: Can you answer questions from your peers about the presentation?

In order to prepare for the project presentation we do one practice session in the class, where we review a paper together as a group, and attempt to answer these questions. 

Fraud and plagiarism

The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl

Course structure

Timetable

The schedule for this course is published on DataNose.

Contact information

Coordinator

• dr. A.L. Watts