As someone interested in pattern formation problems in general (I've always looked at things that produce complex patterns and wondered 'how?'), turbulence was immediately of interest, if only because of the wonderful complexity it presented. The structures of turbulent flows are not just cast-offs of the dynamics, but fundamentally tie into the macroscopic properties of the system and dominate things like the pipe friction and the velocity profile. This research concerns the connection between the spectrum of turbulence - the signature of its complex structure - and the friction factor and velocity profiles.
Gustavo Gioia and Pinaki Chakraborty published a paper [1] proposing a model that explained the complicated dependence of the friction factor on roughness and Reynolds number observed by Nikuradze in 1933 [2]. The research I am doing extends this model to 2D in order to test it with a different energy spectrum.
The results in 2D are quite interesting - a lot of flow properties that are indepdent of the details of the turbulence in 3D suddenly become dependent on how the 2D turbulence was generated. If the 2D turbulence was generated from rough walls, the results are the same as in 3D, but if the turbulence was generated by a grid you get a different velocity profile, friction factor scaling, and spectrum.
My simulations are done using a non-linear third order advection algorithm called SMART [3] and a spectral method to determine the pressure and satisfy the incompressibility criterion. The interesting wrinkle is that the spectral method (which is significantly faster than using conjugate gradient) only works on a rectangular domain, but we want to study rough walls. Fortunately, since this is in two dimensions, we can use conformal mapping to turn our complicated domain with rough walls into a rectangular domain.
I presented a talk on this research at the Understanding Complex Systems conference at UIUC in 2008. The talk is available here.
The above image is grid generated turbulence in a smooth pipe at a Reynolds number of 60000. The color scheme indicates positive (blue) and negative (red) vorticity.
This is roughness generated turbulence at a Reynolds number of 130000 in a rough pipe generated via conformal mapping.
Here is a movie of the development of turbulence from a short periodic 2D pipe segment with roughness.