Kelvin–Helmholtz instability
As Kelvin-Helmholtz instability is defined as the growth of small disturbances in the shear layer between two fluids of different flow rates.
A good example to the waves on a lake during a storm or the swirl of smoke from a joss stick in an otherwise quiet room.
Phenomenology
As a weather phenomenon may in strange clouds, which are available individually or can be seen in similar looking groups in the sky, are identified. They are formed by a swirl of two superposed layers of air that move with different speed and / or direction. Much like when wind passes over the water, waves arise at the boundary layer, with parts of the most humid lower air layer as far whirled up that her dew point is below and it comes to cloud formation.
Physical interpretation
Far from the boundary layer, the flow velocities are constant. Near the boundary layer, an air element must, however, move faster over the wave crest as a more distant (similar to a wing ). According to the Bernoulli equation, but the pressure on the shaft due to the higher wind speed is less than in the area. Consequently, there is a force that pulls the wave crest to the top. Similarly it behaves in a wave trough: the air flows more slowly over the surface of a wave trough than in the area, and therefore the ambient pressure is locally higher. The trough is pressed to the bottom.
Theory
A simple model for the Kelvin - Helmholtz instability is obtained when it is sought to answer the following question: Given a flow across a boundary layer. Under what conditions, the boundary layer is stable against small perturbations?
Perturbation theory
Thus, where is a liquid having a density which moves horizontally with the speed of a liquid with the density. Denote a coordinate along the shear layer and the coordinate perpendicular to it. Now you are a small perturbation in the shear layer and is denoted by. The associated disturbance to the pressure we denote with and the velocity field.
The pressure field can be written now as
And the velocity field as
Where the Heaviside function called. Substituted these two disorders is now in the simplest form of fluid dynamical equations, namely in the incompressible Euler equations: The Inkompressibilitätsgleichung is
And the Euler 's equation
There used are obtained for the perturbed quantities
And
These two equations providing for the disturbed printing the Laplace equation
Moreover it is according to a wave mode, which decreases exponentially with the distance from the interface. From the Laplace equation, we conclude that the pressure must apply:
Next, we substituted this result in the perturbed Euler equations. In this case, one obtains
And
For.
Now, the boundary conditions must be met: First, the vertical component of the disturbance at the shear layer must be continuous. Further, the pressure at the interface must be continuous. From these two conditions, these conditions result in the error:
Directly above the shear layer and
Directly below the shear layer. From this, a correlation between the density of the fluids, the wave modes and the relative velocity of the fluids produced:
Solving this equation with respect to on, then one obtains a dispersion relation for the linear Kelvin -Helmholtz modes:
Temporal growth of the disturbance
Now one can calculate the growth of the disturbance. If you move at the speed along the surface, we obtain for the velocity of the upper fluid. The fault is now developed as follows: