Introduction to hydrodynamic stability

This notebook has been written in Mathematica by

Mark J. McCready
Professor and Chair of Chemical Engineering
University of Notre Dame
Notre Dame IN 46556
USA

Mark.J.McCready.1@nd.edu
http://www.nd.edu/~mjm/


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Version:  5/14/99
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Motivation: Importance of waves

Waves form as the result of wind on natural bodies of water and contribute to increased gas transfer and droplet production.  Despite the high Reynolds numbers involved, these originate as a hydrodynamic instability that requires a fully-viscous solution of the linearized Navier-Stokes equations to characterize.  You may also observe waves on the windshield of   your car during a rain storm or in the car wash or on the wing of an airplane -- perhaps as the deicing fluid is sheared of.  These examples involve much lower liquid Reynolds numbers and  (with some additional simplifications) can be analyzed with wave equations similar to the example used in the first part below.  

A big area of interest in interfacial waves are those that occur in gas-liquid (i.e., two-phase flow) pipelines and process equipment.  In these situations they greatly increase the pressure drop and interface transport rates, and can lead to flow regime transitions.  Some examples of these kinds of waves are available as still pictures and movies at http://www.nd.edu/~mjm/waves.descrip.html.        

Conclusions

1.  We first believe or observe (or learn from a combination of mathematical analysis and observation) that fluid dynamical instabilities arise when the flow is such that very small amplitude disturbances are amplified by the main flow.   (click    here to read again)

2.  Because of this infinitesimal origin of instabilities, the problems are amenable to linear analysis.  That is, the nonlinear governing equations can be linearized before they are solved.  This makes the problems much more tractable.   (click   here to read again)

3.  Analysis of instability requires us to choose the mathematical form of the disturbance.  For waves on a liquid interface this would be a spatially and temporally periodic form.  We recognize that waves could grow in space and or time and they can travel in space.   It is important to use a form that  can be used as the basis of a Fourier Series so that any initial shape of "noise" could be represented.  Recall that because the equations are linear, the modes are independent.  

4.  A complex exponential is a good form for a periodic disturbance as it can be divided back out of any derivation (for linear equations) even after arbitrary derivatives have been taken.  We see that this introduction of an imaginary part of an expression does not lead to anything unphysical.  The presence of an  ⅈ just denotes a phase shift.  

5.  We find that waves can grow in either space or time and there is a relation between these,  as recognized by  Gaster.

6.  It is important to understand (and not later forget!) that the linearization  that was done means that the solutions are not valid once the wave amplitude becomes large enough that the quadratic terms in amplitude can no longer be neglected.   

7.  The Kelvin-Helmholtz example shows the manipulations necessary to deal with separate governing equations for each fluid (because the fluid properties are different), and boundary conditions.  

8.  The Orr-Sommerfeld equation is the general starting point for stability analysis of Newtonian fluids with constant properties.  The derivation from the linearized Navier-Stokes equations using the definition of a disturbance stream function (obtained from the continuity equation) and a general  traveling, periodic disturbance  is shown.