PX284 - M1 - turbulence

• considering two symmetric vortices rotating oppositely
• after a while, the flow becomes asymmetric (eg: thermal fluctuations)

• a street of vortices, known as von karmen street, is formed
• the vortices rotating in alternate sense are shed from either side of the cylinder
• the size of each vortex near the cylinder is similar to the diameter of the cylinder
• the parameters of the process:
  • $D:$ diameter of cylinder (and vortices)
  • $u:$ flow speed upstream
  • $P:$ time period of vortex shedding
• a dimensionless combination of these parameters can be formed, which is called the 'storuhal number':

• for a very broad range of applications (air, water, mercury, plasma), $\text{St}\approx 0.25$
• this is a very robust feature of the oscillating vortex shedding phenomenon
• as vortices have mass and momentum, the periodic vortex shedding imparts a force on the cylinder

• the force changes direction periodically
• therefore, the cylinder oscillates in the direction perpendicular to the flow velocity

• further downstream, the vortices are observed to break up into smaller vortices, in a process called 'nonlinear cascade'
• they can also collide and merge, giving a 'gas' or a 'soup' of vortices of different sizes, giving rise to turbulence
• it is a dynamic process, but if a snapshot is taken and the sizes of the vortices are plotted as a histogram,
• the fourier transform in spatial domain (s):

where, $\lambda$ is the wavelength, or the size of the vortex

• the energy of vortices against the wavelength in a logarithmic plot:

• this is called the kolmogorov spectrum of turbulence