Kelvin Helmholtz Instability

In this presentation, I will be talking about Kelvin-Helmholtz instability, a hydrodynamic instability process named after scientists William Thomson, Lord Kelvin, and Hermann von Helmholtz, who first studied this phenomena in the mid-19th century. Helmholtz originally came across this phenomena when he was studying the formation of billow clouds such as the ones on the left, and around the same time, Lord Kelvin studied this problem when modeling the wind-driven formation of ocean surface waves. The basic setup of the problem is Consider two layers of fluid with the less dense fluid on top of the heavier one. Think of a pond in the summer or wind blowing over water. Suppose there is a small perturbation, small deformation of the interface, for example caused by gravity waves moving along the interface between these two layers. If the velocity between the two layers is distinct, then there is shear at this interface. The shear can cause the amplitude of the small disturbance to spontaneously grow. As it grows, the relative velocity between the two layers causes the waves to break and overturn, forming breaking internal waves, or billows. This spontaneous growth of a small disturbance at the interface separating two fluid layers of different densities and different velocities is referred to as Kelvin-Helmholtz instability. This spontaneous growth of wave amplitude and the destabilizing effect of shear can be understood physically through mass conservation. After a wave-like perturbation arises, by conservation of mass, the fluid must accelerate at the peaks or troughs of the perturbation. As it is traveling faster at the peaks or troughs, Bernoulli's equation tells us that the pressure at these points must decrease to keep the equation constant along a streamline. Since the pressure below a trough or above a peak at the interface decreases and the pressure above a trough or below a peak increases, the force will be in a direction that causes the amplitude of the perturbations to grow. As the amplitude grows, the motion of the two layers causes the boundary to twist. While the above description and the original investigation of the two scientists focuses on uniform velocity and density in each layer with a discontinuity at the boundary, Kelvin-Helmholtz instability more generally also refers to the case of continuously varying density and velocity distributions. This instability phenomena is commonly observed in clouds, aptly named Kelvin-Helmholtz clouds, when two layers of air are moving at different speeds. It is also present in this Cassini spacecraft photograph of Saturn, where it is caused by velocity differences between two latitudinal bands in the planet's atmosphere. This process is not exclusive to the atmosphere, however. Kelvin-Helmholtz instability was first observed in water in a diode release experiment performed by scuba divers in the thermocline of the Mediterranean. After initial release, the interface curls up into the billow shape expected from this instability. Of course, talking about examples of Kelvin-Helmholtz instability and the onset of turbulent flow, I am obligated to also mention Van Gogh's Starry Night painting for its accurate depiction of fluid turbulence. To understand the onset of this instability, we can study the response of the equations of motion, continuity, and momentum to small plane wave perturbations. To start, we are going to assume that the fluids in the two layers are irrotational incompressible and inviscid. This will allow us to use Bernoulli's equation. u1 is the background fluid velocity in the first layer and u2 is the velocity in the second. Assuming irrotational flow, we can define the velocities in terms of potentials vi, i indicating the fluid layer and can be either 1 or 2. To find the conditions under which instability occurs, we will perturb the flow by adding a perturbation psi to the background. The goal of the analysis is to see if the equations of motion predict the growth or decay of the perturbation. Using our assumptions, the continuity equation simplifies to Laplace's equation and the momentum equation simplifies to the unsteady Bernoulli equation. There are three boundary conditions in this case. First, we must require the pressure across the interface to be equal on either side. We must also require the velocity of the flow in each layer normal to the boundary to be equal to the velocity of the boundary, us. Finally, the perturbations must vanish far away from the interface. To keep the video short and sweet, I am going to forgo the details of the mathematical analysis. The highlights are that, since we assume the perturbations are small, we can linearize the equations of motion. Then, assuming plane wave solutions were the perturbations, we arrive at an equation describing the conditions under which the exponential solution grows or decays with time. Requiring our solutions to grow with time, we arrive at the following inequality indicating the condition under which amplitudes of the oscillations grow, the condition under which we have instability. Here, rho1 and rho2 are the densities of the two layers and u1 and u2 are the velocities, and k is the wave number of the perturbation. Therefore, in order to have instability, we either need a large enough velocity difference or shear between the two layers or a small enough density difference between the two layers. Furthermore, if there is a shear between the two layers, the flow is always unstable to small enough wavelengths or large enough wave number. Therefore, Kelvin-Helmholtz instability is caused by the destabilizing effect of a velocity shear overpowering the stability provided by a density gradient. It is important to note the limitations of the above analysis, where we assume that the perturbations are small, allowing us to linearize the equations of motion. Thus, the above analysis only describes the response of our system to very small perturbations over a small period after the onset of the perturbation. To gain insight on the late-time behavior of the system, we must turn to simulations and numerically solve Navier-Stokes equation. This is a simulation of Kelvin-Helmholtz instability adapted from code written in Python. As you can see, the shear at the interface results in the formation of billows after an initial perturbation very similar to the example. We can also observe this effect experimentally. To perform the experiment, I stacked two layers of fluid in a glass baking dish. I tried different sizes to see if I can observe the effect more clearly. I used water and salt water, the denser on the bottom, and added food coloring to the bottom layer to make the effect observable. Tilting the tank causes the denser fluid to move downhill and the lighter uphill due to the buoyancy difference. This results in a velocity gradient between the two fluid layers, thus causing shear that leads to formation of billows at the interface. Out of many attempts, the following demonstrates the effect most clearly. Due to the small scale of the experiment, the effect is not as obvious as one would observe in clouds or an experiment using very large rectangular tanks such as this experiment performed by the Department of Applied Math and Physics at Cambridge. This is because at the scale of this experiment, the velocity gradient established between the two layers through tilting is small. I also performed the experiment in a fish tank. Here is the result. Not as amazing as the Cambridge experiment, but you can see one of the eddy-like structures. To conclude, Kelvin-Helmholtz instability refers to instability at the interface of two fluids of different densities and relative velocities. Instability in this case means that a small disturbance of the interface, for example a wave propagating at the interface, spontaneously grows in amplitude. Using linear stability analysis, we concluded that this instability is caused by shear at the boundary between the two fluids. Thank you for listening.