Hydrodynamic stability

Kelvin–Helmholtz instability on Saturn, caused by the interaction between two bands of the planet's atmosphere.

In fluid dynamics, hydrodynamic stability is the field which analyses the stability and the onset of instability of fluid flows. The study of hydrodynamic stability aims to find out if a given flow is stable or unstable, and if so, how these instabilities will cause the development of turbulence. The foundations of hydrodynamic stability, both theoretical and experimental, were laid most notably by Helmholtz, Kelvin, Rayleigh and Reynolds during the nineteenth century. These foundations have given many useful tools to study hydrodynamic stability. These include Reynolds number, the Euler equations, and the Navier–Stokes equations. From the 1980s, computational methods are also being used to analyse such flows.

Stable and unstable flows

To distinguish between the different states of fluid flow one must consider how the fluid reacts to a disturbance in the initial state. These disturbances will relate to the initial properties of the system, such as velocity, pressure, and density. James Clerk Maxwell expressed the qualitative concept of stable and unstable flow nicely when he said:

"when an infinitely small variation of the present state will alter only by an infinitely small quantity the state at some future time, the condition of the system, whether at rest or in motion, is said to be stable but when an infinitely small variation in the present state may bring about a finite difference in the state of the system in a finite time, the system is said to be unstable."

That means that for a stable flow, any infinitely small variation, which is considered a disturbance, will not have any noticeable affect on the initial state of the system and will eventually die down in time. For a fluid flow to be considered stable it must be stable with respect to every possible disturbance. This implies that there exists no mode of disturbance for which it is unstable.

On the other hand, for an unstable flow, any variations will have some noticeable affect on the state of the system which would then cause the disturbance to grow in amplitude in such a way that the system progressively departs from the initial state and never returns to it. This means that there is at least one mode of disturbance with respect to which the flow is unstable, and the disturbance will therefore distort the existing force equilibrium.

Determining flow stability

Reynolds number

A key tool used to determine the stability of a flow is the Reynolds number (Re), first put forward by George Gabriel Stokes at the start of the1850's. Associated with Osborne Reynolds who further developed the idea in the early 1880's. This dimensionless number gives the ratio of inertial terms and viscous terms. In a physical sense, this number is a ratio of the forces which are due to the momentum of the fluid (inertial terms), and the forces which arise from the relative motion of the different layers of a flowing fluid (viscous terms). The equation for this is given by:

$R_{e}={\frac {\text{inertial}}{\text{viscous}}}={\frac {\rho u^{2}}{\frac {\mu u}{L}}}={\frac {\rho uL}{\mu }}={\frac {uL}{\nu }}$

where

$\rho ={\text{density}}$

${\text{u}}={\text{velocity of the fluid flow}}$

$\mu ={\text{dynamic viscosity}}$ - measures the fluids resistance to shearing flows

${\text{L}}={\text{characteristic length}}$

$\nu ={\text{kinematic viscosity}}={\frac {\mu }{\rho }}$ - measures ratio of dynamic viscosity to the density of the fluid

The Reynolds number is useful because it can provide cut off points for when flow is stable or unstable. As it increases, the amplitude of a disturbance which would then lead to instability gets smaller. At high Reynolds numbers it is agreed that fluid flows will be unstable. High Reynolds number can be achieved in several ways, e.g if $\mu$ is a small value or if $\rho$ and ${\text{u}}$ are high values. This means that instabilities will arise almost immediately and the flow will become one which is unstable or turbulent.

Navier-Stokes equation and the continuity equation

In order to analytically find the stability of fluid flows, it is useful to note that hydrodynamic stability has a lot in common with stability in other fields, such as magnetohydrodynamics, plasma physics and elasticity; although the physics is different in each case, the mathematics and the techniques used are similar. The essential problem is modeled by nonlinear partial differential equations and the stability of known steady and unsteady solutions are examined. The governing equations for almost all hydrodynamic stability problems are the Navier-Stokes equation and the continuity equation. The Navier-Stokes equation is given by:

${\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} -\nu \nabla ^{2}\mathbf {u} =-\nabla p_{0}+\mathbf {b} .$

where

• $\mathbf {u} ={\text{velocity field of fluid}}$

• $p_{0}={\text{pressure of fluid}}$

• $\mathbf {b} ={\text{body force acting on fluid e.g gravity}}$

• $\nu ={\text{kinematic viscosity}}$

• ${\frac {\partial \mathbf {u} }{\partial t}}={\text{partial derivative of the velocity field with respect to time}}$

• $\nabla =({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}})$

Here$\nabla$ is being used as an operator acting on the velocity field on the left hand side of the equation and on the pressure on the right hand side.

and the continuity equation is given by:

${\frac {D\mathbf {\rho } }{Dt}}+\rho \nabla \cdot \mathbf {u} =0$

where

• ${\frac {D\mathbf {\rho } }{Dt}}={\text{material derivative of the density}}$

Once again $\nabla$ is being used as an operator on $\mathbf {u}$ and is calculating the divergence of the velocity.

but if the fluid being considered is incompressible, which means the density is constant, then ${\frac {D\mathbf {\rho } }{Dt}}=0$ and hence:

$\nabla \cdot \mathbf {u} =0$

The assumption that a flow is incompressible is a good one and applies to most fluids travelling at most speeds.

Finding the solutions to the these equations under different circumstances and determining their stability will, in turn, help determine the stability of the fluid flow itself.

Analysing flow stability

The way in which we analyse the stability of flows and their transition to turbulence can be categorised into four different classes:

Natural phenomena and laboratory experiments

There would be no need to study hydrodynamic stability if we were unable to observe it's effects in natural phenomena, man-made processes and laboratory experiments. Physically seeing how the flow changes over time can provide a lot of information without having to use a lot of mathematical technique. All findings achieved through theory and analysing the governing equations must relate to any observations and help to understand and explain them. Such examples of observed phenomena include the Kelvin-Helmholtz instability, the Rayleigh-Taylor instability and the Rayleigh-Bernard Convection.

Numerical experiments

The use of computer analysis and computational fluid dynamics has grown in recent years and become a very helpful way for studying problems. Due to the improvement in computer programming it has enabled us to use them to integrate the Navier-Stokes equations even more accurately for various types of flow. The use of computers is now at the stage where it is being used to simulate experiments as it is much easier to control and vary the variables, that are being investigated, in this way.

Linear and weakly nonlinear theory

When considering a basic flow, the linearisation for small disturbances is the first method used when studying hydrodynamic stability and has been the main one used up until the development of weakly nonlinear theory. This develops the theory of linearisation further by considering the leading nonlinear effects of small disturbances. This theory has grown increasingly since the 1960's.

Qualitative theory of bifurcation and chaos

By the use of differential equations it can be shown that flows can evolve as the dimensionless parameter, e.g the Reynolds number, increase. The succession of bifurcations of one flow to another flow as a parameter increases cannot be explained quantitatively without numerical calculations, although the route a flow takes to instability and then turbulence can be identified using the qualitative mathematical theory. This means the theory of Dynamical systems, as well as weakly nonlinear analysis can be very important when trying to understand numerical and laboratory experiments.

Applications

Kelvin–Helmholtz instability

The Kelvin–Helmholtz instability (KHI) is an application of hydrodynamic stability that can be seen in nature. It occurs when there are two fluids flowing at different velocities. The difference in velocity of the fluids causes a shear velocity at the interface of the two layers. The shear velocity of one fluid moving induces a shear stress on the other which, if greater than the restraining surface tension, then results in an instability along the interface between them. This motion causes the appearance of a series of overturning ocean waves, a characteristic of the Kelvin–Helmholtz instability. Indeed, the apparent ocean wave-like nature is an example of vortex formation, which are formed when a fluid is rotating about some axis, and is often associated with this phenomenon.

The Kelvin–Helmholtz instability can be seen in the bands in planetary atmospheres such as Saturn and Jupiter, for example in the giant red spot vortex. In the atmosphere surrounding the giant red spot there is the biggest example of KHI that is known of and is caused by the shear force at the interface of the different layers of Jupiter's atmosphere. There have been many images captured where the ocean-wave like characteristics discussed earlier can be seen clearly, with as many as 4 shear layers visible.

Weather satellites take advantage of this instability to measure wind speeds over large bodies of water. Waves are generated by the wind, which shears the water at the interface between it and the surrounding air. The computers on board the satellites determine the roughness of the ocean by measuring the wave height. This is done by using radar, where a radio signal is transmitted to the surface and the delay from the reflected signal is recorded, known as the "time of flight". From this meteorologists are able to understand the movement of clouds and the expected air turbulence near them.

Rayleigh–Taylor instability

The Rayleigh–Taylor instability is another application of hydrodynamic stability and also occurs between two fluids but this time the densities of the fluids are different. Due to the difference in densities, the two fluids will try to reduce their combined potential energy. The less dense fluid will do this by trying to force its way upwards, and the more dense fluid will try to force its way downwards. Therefore, there are two possibilities: if the lighter fluid is on top the interface is said to be stable, but if the heavier fluid is on top, then the equilibrium of the system is unstable to any disturbances of the interface. If this is the case then both fluids will begin to mix. Once a small amount of heavier fluid is displaced downwards with an equal volume of lighter fluid upwards, the potential energy is now lower than the initial state, therefore the disturbance will grow and lead to the turbulent flow associated with Rayleigh–Taylor instabilities.

This phenomenon can be seen in interstellar gas. This is pushed out of the Galactic plane by magnetic fields and cosmic rays and then becomes Rayleigh–Taylor unstable if it is pushed past its normal scale height. This instability also explains the "mushroom cloud" which forms in processes such as volcanic eruptions and atomic bombs.

Rayleigh–Taylor instability has a big effect on the Earth's climate. Winds that come from the coast of Greenland and Iceland cause evaporation of the ocean surface over which they pass, increasing the salinity of the ocean water near the surface, and making the water near the surface denser. This then generates plumes which drive the ocean currents. This process acts as a heat pump, transporting warm equatorial water North. Without the ocean overturning, Northern Europe would likely face drastic drops in temperature.

See also

• Fluid mechanics
• Fluid dynamics
• List of hydrodynamic instabilities
• Reynolds number
• Navier-Stokes equation
• Kelvin–Helmholtz instability
• Plasma stability
• Rayleigh–Taylor instability
• Taylor–Couette flow

References

• Drazin, P.G. (2002), Introduction to hydrodynamic stability, Cambridge University Press, ISBN 0-521-00965-0
• Chandrasekhar, S. (1961), Hydrodynamic and hydromagnetic stability, Dover, ISBN 0-486-64071-X
• Charru, F. (2011), Hydrodynamic instabilities, Cambridge University Press, ISBN 1139500546
• Godreche, C.; Manneville, P., eds. (1998), Hydrodynamics and nonlinear instabilities, Cambridge University Press, ISBN 0521455030
• Lin, C.C. (1966), The theory of hydrodynamic stability (corrected ed.), Cambridge University Press, OCLC 952854
• Swinney, H.L.; Gollub, J.P. (1985), Hydrodynamic instabilities and the transition to turbulence (2nd ed.), Springer, ISBN 978-3-540-13319-3
• Happel, J.; Brenner, H. (2009), Low Reynolds number hydrodynamics (2nd ed.), ISBN 9024728770
• Foias, C.; Manley, O.; Rosa, R.; Teman, R. (2001), Navier-Stokes Equations and Turbulence, Cambridge University Press, ISBN 978-8126509430
• Panton, R.L. (2006), Incompressible Flow (3rd ed.), Wiley India, ISBN 8126509430

External links

• "Flow instabilities". National Committee for Fluid Mechanics Films (NCFMF). Retrieved 9 March 2009.
• "Advanced Instability Methods (AIM) Network". various authors. Retrieved 12 May 2013.