Course unit

Hydrodynamic instabilities: theory and numerical analysis

Last updated: 22/02/2024

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Course Director(s):

DUMAZER Guillaume

General Description:

In this course, we focus on hydrodynamic instabilites.

Many of these instabilities exist in natural settings and can be leveraged in industrial contexts (below some examples).

The theoretical analysis uses concepts from dynamical systems : phase space, phase portrait, attractor/fixed point as well as linear stability analysis. Thanks to these tools, we can identify the conditions for the system to be stable. If unstable, the dispersion relationship gives the wavelentgh with the highest growth rate. A detailed demonstration of the Rayleigh-Bénard instability is given.

  The numerical analysis if hydrodynamic instabilities can be done with any fluid solver. Here, we will adapt the LBM code developed in part I so as to model coupled phenomena such as mass, momentum and heat transfers. Then, the growth rate of an infinitesimal perturbation is calculated and helps finding the marginal stability curve. Practically, we can make use of it tu calculate the minimal temperature difference to set in motion a fluid initially at rest.

Examples of instabilities :

Thermo-convective instability of Rayleigh-Bénard : a fluid at rest starts moving when heated from below and with a minimal temperature gradient (see moving water in a pan).

Rayleigh-Plateau instability : a liquid jet will spontaneously fragment into droplets under the action of surface tension and/or shear by surrounding gas phase (cf. fuel injection in combustion chamber)

Kelvin-Helmholtz instability :  the interface between two sheared fluids does not stay plane when there is a density difference (by example driven by a temperature difference)

von Karman instability : an obstacle in a flow gives rise to vortices that detach in the wake.

Key words:

Fluid mechanics Stability and transition to chaos Hydrodynamics instabilities Rayleigh-Bénard, Rayleigh-Plateau, ...

Number of teaching hours

20

Fields of study

Teaching language

French English

Intended learning outcomes

On completion of the unit, the student will be capable of:	Classification level	Priority
Basic knowledge in dynamical systems	1. Knowledge	null
Perform linear stability analysis for a given system / PDE	3. Apply	null
Knowledge of some hydrodynamic instabilities	1. Knowledge	null
Use a CFD code to compute the dispersion relationship and the marginal stability curve	null	null

Learning assessment methods

Percentage ratio of individual assessment			Percentage ratio of group assessment
Written exam:	100	%	Project submission:	100	%
Individual oral exam:	100	%	Group presentation:	100	%
Individual presentation:	100	%	Group practical exercise:	100	%
Individual practical exercise:	100	%	Group report:	100	%
Individual report:	100	%
Other(s): 100 %

Programme and content

Type of teaching activity	Content, sequencing and organisation
Lecture	The part devoted to dynamic systems (~10 hours) is a classical lecture, with handout and exercises.
Tutorial	The part devoted to numerical analysis (~10 hours) is a hands-on tutorial aimed at programming in matlab. It is done in groups of two or three.