kelvin helmhotz instability

8/12/2019 Kelvin Helmhotz Instability

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The KelvinHelmholtz instability(afterLord Kelvin andHermann von Helmholtz)can occur

when velocity shear is present within a continuous fluid, or when there is sufficient velocity

difference across the interface between two fluids. One example is wind blowing over a water

surface, where the wind causes the relative motion between the stratified layers (i.e., water andair). The instability will manifest itself in the form of waves being generated on the water

surface. The waves can appear in numerous fluids and have been spotted in clouds, Saturn'sbands, waves in the ocean, and in the sun's corona.

The theory can be used to predict the onset of instability and transition toturbulent flow influidsof differentdensities moving at various speeds. Helmholtz studied thedynamics of two fluids of

different densities when a small disturbance such as a wave is introduced at the boundary

connecting the fluids.

Stability

A KH instability rendered visible by clouds overMount Duval in Australia

For some short enough wavelengths, if surface tension can be ignored, two fluids in parallelmotion with different velocities and densities will yield an interface that is unstable for allspeeds. The existence ofsurface tension stabilises the short wavelength instability however, and

theory then predicts stability until a velocity threshold is reached. The theory with surface

tension included broadly predicts the onset of wave formation in the important case of wind overwater.

A KH instability on the planet Saturn, formed at the interaction of two bands of the planet's

atmosphere.

In presence of gravity, for a continuously varying distribution of density and velocity, (with the

lighter layers uppermost, so the fluid is RT-stable), the dynamics of the KH instability is
http://en.wikipedia.org/wiki/Lord_Kelvinhttp://en.wikipedia.org/wiki/Hermann_von_Helmholtzhttp://en.wikipedia.org/wiki/Shear_velocityhttp://en.wikipedia.org/wiki/Continuum_mechanicshttp://en.wikipedia.org/wiki/Turbulent_flowhttp://en.wikipedia.org/wiki/Fluidhttp://en.wikipedia.org/wiki/Densityhttp://en.wikipedia.org/wiki/Dynamics_%28mechanics%29http://en.wikipedia.org/wiki/Mount_Duval_%28New_South_Wales%29http://en.wikipedia.org/wiki/Surface_tensionhttp://en.wikipedia.org/wiki/Rayleigh%E2%80%93Taylor_instabilityhttp://en.wikipedia.org/wiki/Rayleigh%E2%80%93Taylor_instabilityhttp://en.wikipedia.org/wiki/File:Saturn_Kelvin_Helmholtz.jpghttp://en.wikipedia.org/wiki/Surface_tensionhttp://en.wikipedia.org/wiki/Mount_Duval_%28New_South_Wales%29http://en.wikipedia.org/wiki/File:Wavecloudsduval.jpghttp://en.wikipedia.org/wiki/Dynamics_%28mechanics%29http://en.wikipedia.org/wiki/Densityhttp://en.wikipedia.org/wiki/Fluidhttp://en.wikipedia.org/wiki/Turbulent_flowhttp://en.wikipedia.org/wiki/Continuum_mechanicshttp://en.wikipedia.org/wiki/Shear_velocityhttp://en.wikipedia.org/wiki/Hermann_von_Helmholtzhttp://en.wikipedia.org/wiki/Lord_Kelvin


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described by the TaylorGoldstein equation and its onset is given by a suitably defined

Richardson number, Ri. Typically the layer is unstable for Ri


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using the vortex sheet model (Chen and Forbes 2011)and found the roll-up of a thin viscous

layer in a weakly compressible system (Forbes 2011). These results suggest that viscosity acts as

a regularization effect in the roll-up.

TollmienSchlichting wave

Influid dynamics,a TollmienSchlichting wave(often abbreviated T-S wave) is a streamwise

instability which arises in aviscousboundary layer.It is one of the more common methods by

which a laminar boundary layer transitions to turbulence. The waves are initiated when somedisturbance (sound, for example) interacts with leading edge roughness in a process known as

receptivity. These waves are slowly amplified as they move downstream until they mayeventually grow large enough that nonlinearities take over and the flow transitions to turbulence.

These waves, originally discovered byLudwig Prandtl,were further studied by two of his former

students,Walter Tollmien andHermann Schlichting for whom the phenomenon is named.

copyright ONERA 1996-2006

Transitional flow along a sheet without longitudinal pressure gradient.

Tollmien Schlichting waves on the left of the picture.

Physical Mechanism

In order for a boundary layer to be absolutely unstable (have an inviscid instability), it must

satisfy Rayleigh's criterion; namely D2U= 0 Where D represents the y-derivative and U is thefree stream velocity profile. In other words, the velocity profile must have an inflection point to

be unstable.

It is clear that in a typical boundary layer with a zero pressure gradient, the flow will be

unconditionally stable; however, we know from experience this is not the case and the flow does
http://www.scholarpedia.org/article/Kelvin-Helmholtz_Instability_and_Roll-up#forbes1http://www.scholarpedia.org/article/Kelvin-Helmholtz_Instability_and_Roll-up#forbes2http://en.wikipedia.org/wiki/Fluid_dynamicshttp://en.wikipedia.org/wiki/Hydrodynamic_stabilityhttp://en.wikipedia.org/wiki/Viscoushttp://en.wikipedia.org/wiki/Boundary_layerhttp://en.wikipedia.org/wiki/Boundary_layerhttp://en.wikipedia.org/wiki/Laminar_flowhttp://en.wikipedia.org/wiki/Turbulencehttp://en.wikipedia.org/wiki/Ludwig_Prandtlhttp://en.wikipedia.org/wiki/Walter_Tollmienhttp://en.wikipedia.org/wiki/Hermann_Schlichtinghttp://en.wikipedia.org/wiki/Hermann_Schlichtinghttp://en.wikipedia.org/wiki/Walter_Tollmienhttp://en.wikipedia.org/wiki/Ludwig_Prandtlhttp://en.wikipedia.org/wiki/Turbulencehttp://en.wikipedia.org/wiki/Laminar_flowhttp://en.wikipedia.org/wiki/Boundary_layerhttp://en.wikipedia.org/wiki/Viscoushttp://en.wikipedia.org/wiki/Hydrodynamic_stabilityhttp://en.wikipedia.org/wiki/Fluid_dynamicshttp://www.scholarpedia.org/article/Kelvin-Helmholtz_Instability_and_Roll-up#forbes2http://www.scholarpedia.org/article/Kelvin-Helmholtz_Instability_and_Roll-up#forbes1


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transition. It is clear, then, that viscosity must be an important factor in the instability. It can be

shown using energy methods that

The rightmost term is a viscous dissipation term and is stabilizing. The left term, however, is the

Reynolds stress term and is the primary production method for instability growth. In an inviscidflow, the u' and v' terms are orthogonal, so the term is zero, as one would expect. However, with

the addition of viscosity, the two components are no longer orthogonal and the term becomes

nonzero. In this regard, viscosity is destabilizing and is the reason for the formation of T-Swaves.

Transition Phenomena

Initial Disturbance

In a laminar boundary layer, if the initial disturbance spectrum is nearly infinitesimal and

random (with no discrete frequency peaks), the initial instability will occur as two-dimensional

Tollmien-Schlichting waves, travelling in the mean flow direction if compressibility is notimportant. However, three-dimensionality soon appears as the Tollmein-Schlichting waves rather

quickly begin to show variations. There are known to be many paths from Tollmein-Schlichting

waves to turbulence, and many of them are explained by the non-linear theories of flow

instability.

Final Transition

A shear layer develops viscous instability and forms Tollmien-Schlichting waves which grow,

while still laminar, into finite amplitude (1 to 2 percent of the freestream velocity) three-dimensional fluctuations in velocity and pressure to develop three-dimensional unstable waves

and hairpin eddies. From then on, the process is more a breakdown than a growth. The

longitudinally stretched vortices begin a cascading breakdown into smaller units, until therelevant frequencies and wave numbers are approaching randomness. Then in this diffusively

fluctuating state, intense local changes occur at random times and locations in the shear layer

near the wall. At the locally intense fluctuations, turbulent 'spots' are formed that eventually

coalesce into fully turbulent flows that burst forth in the form of growing and spreading spots.
http://en.wikipedia.org/wiki/Reynolds_stresshttp://en.wikipedia.org/wiki/Hydrodynamic_stabilityhttp://en.wikipedia.org/wiki/Hydrodynamic_stabilityhttp://en.wikipedia.org/wiki/Hydrodynamic_stabilityhttp://en.wikipedia.org/wiki/Eddyhttp://en.wikipedia.org/wiki/Eddyhttp://en.wikipedia.org/wiki/Vortexhttp://en.wikipedia.org/wiki/Frequencyhttp://en.wikipedia.org/wiki/Wavelengthhttp://en.wikipedia.org/wiki/Wavelengthhttp://en.wikipedia.org/wiki/Frequencyhttp://en.wikipedia.org/wiki/Vortexhttp://en.wikipedia.org/wiki/Eddyhttp://en.wikipedia.org/wiki/Hydrodynamic_stabilityhttp://en.wikipedia.org/wiki/Hydrodynamic_stabilityhttp://en.wikipedia.org/wiki/Reynolds_stress

kelvin helmhotz instability

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