Kelvin-Helmholtz instability
Encyclopedia
The Kelvin–Helmholtz instability, after Lord Kelvin and Hermann 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 and air). 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's bands, waves in the ocean, and in the sun's corona.
The theory can be used to predict the onset of instability and transition to turbulent flow in fluid
s of different densities
moving at various speeds. Helmholtz studied the dynamics
of two fluids of different densities when a small disturbance such as a wave is introduced at the boundary connecting the fluids.
For some short enough wavelengths, if surface tension can be ignored, two fluids in parallel motion with different velocities and densities will yield an interface that is unstable for all speeds. The existence of surface 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 over water.
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 described by the Taylor–Goldstein equation
and its onset is given by a suitably defined Richardson number, Ri. Typically the layer is unstable for Ri<0.25. These effects are quite common in cloud layers. Also the study of this instability becomes applicable in plasma physics, e.g. inertial confinement fusion
and the plasma
–beryllium
interface.
The classic textbooks by Chandrasekhar
and Drazin & Reid consider the KH and RT instabilities in much detail.
From a numerical point of view, the KH instability is simulated either in a temporal or a spatial way. In the temporal approach, one considers the flow in a periodic (cyclic) box "moving" at the mean speed (absolute instability). In the spatial approach, one tries to simulate a lab experiment with natural inlet and outlet conditions (convective instability).
Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz was a German physician and physicist who made significant contributions to several widely varied areas of modern science...
, can occur when velocity shear
Shear velocity
Shear velocity, also called friction velocity, is a form by which a shear stress may be re-written in units of velocity. It is useful as a method in fluid mechanics to compare true velocities, such as the velocity of a flow in a stream, to a velocity that relates shear between layers of flow.Shear...
is present within a continuous fluid
Continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...
, 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 and air). 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's bands, waves in the ocean, and in the sun's corona.
The theory can be used to predict the onset of instability and transition to turbulent flow in fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
s of different densities
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
moving at various speeds. Helmholtz studied the dynamics
Dynamics (mechanics)
In the field of physics, the study of the causes of motion and changes in motion is dynamics. In other words the study of forces and why objects are in motion. Dynamics includes the study of the effect of torques on motion...
of two fluids of different densities when a small disturbance such as a wave is introduced at the boundary connecting the fluids.
Stability
Surface tension
Surface tension is a property of the surface of a liquid that allows it to resist an external force. It is revealed, for example, in floating of some objects on the surface of water, even though they are denser than water, and in the ability of some insects to run on the water surface...
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 over water.
Taylor–Goldstein equation
The Taylor–Goldstein equation is an ordinary differential equation used in the fields of geophysical fluid dynamics, and more generally in fluid dynamics, in presence of quasi-2D flows. It describes the dynamics of the Kelvin–Helmholtz instability, subject to buoyancy forces , for stably-stratified...
and its onset is given by a suitably defined Richardson number, Ri. Typically the layer is unstable for Ri<0.25. These effects are quite common in cloud layers. Also the study of this instability becomes applicable in plasma physics, e.g. inertial confinement fusion
Inertial confinement fusion
Inertial confinement fusion is a process where nuclear fusion reactions are initiated by heating and compressing a fuel target, typically in the form of a pellet that most often contains a mixture of deuterium and tritium....
and the plasma
Plasma (physics)
In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...
–beryllium
Beryllium
Beryllium is the chemical element with the symbol Be and atomic number 4. It is a divalent element which occurs naturally only in combination with other elements in minerals. Notable gemstones which contain beryllium include beryl and chrysoberyl...
interface.
The classic textbooks by Chandrasekhar
Subrahmanyan Chandrasekhar
Subrahmanyan Chandrasekhar, FRS ) was an Indian origin American astrophysicist who, with William A. Fowler, won the 1983 Nobel Prize for Physics for key discoveries that led to the currently accepted theory on the later evolutionary stages of massive stars...
and Drazin & Reid consider the KH and RT instabilities in much detail.
From a numerical point of view, the KH instability is simulated either in a temporal or a spatial way. In the temporal approach, one considers the flow in a periodic (cyclic) box "moving" at the mean speed (absolute instability). In the spatial approach, one tries to simulate a lab experiment with natural inlet and outlet conditions (convective instability).
See also
- Rayleigh–Taylor instability
- Richtmyer–Meshkov instability
- Mushroom cloudMushroom cloudA mushroom cloud is a distinctive pyrocumulus mushroom-shaped cloud of condensed water vapor or debris resulting from a very large explosion. They are most commonly associated with nuclear explosions, but any sufficiently large blast will produce the same sort of effect. They can be caused by...
- Plateau–Rayleigh instability
- Kármán vortex street
- Taylor–Couette flowTaylor–Couette flowIn fluid dynamics, the Taylor–Couette flow consists of a viscous fluid confined in the gap between two rotating cylinders. For low angular velocities, measured by the Reynolds number Re, the flow is steady and purely azimuthal. This basic state is known as circular Couette flow, after Maurice...
- Fluid mechanicsFluid mechanicsFluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...
- Fluid dynamicsFluid dynamicsIn physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...