Fractal Modeling of Turbulence

For now this is just a place holder to discuss topics about fractals. For related topics see my post on Kolmogorov’s Turbulance. I would like to present the quote though to illustrate the difficulty of using numerical methods to solve Naiver Stokes equations:

1.1. Statement of the problem Many flows of interest in science and engineering display complex spatial and temporal structures (eddies) spanning a wide range of scales. The ratio between the largest (L) and smallest ( )  scale can easily exceed in typical engineering applications, and can be as high as or higher in geophysical applications. Since the nonlinear interaction between eddies of different sizes eludes even the most sophisticated analytical approaches, one must resort to either extensive experimentation or direct numerical simulation (DNS) of the governing equations. The latter approach has gained strength by the rapid increase in the power of digital computers during the past 20 years. Despite this fact, DNS of flows for which the ratio $latex  L/ \eta $ is much larger than are still prohibitive

http://www.me.jhu.edu/meneveau/pdf-papers/ScottiMeneveau99.pdf

More papers on fractals and turbulance can be found here:

http://www.me.jhu.edu/~meneveau/pubs-fractals.html

Kolmogorov’s Theory of Turbulence

It is impractical in climate models to give sufficient resolution in order to capture all of the inertial dynamics in the system. Inertial dynamics will dominate at large scales while on small scales viscous forces  will despite the energy associated with these dynamics.

The Reynolds number gives the ratio of inertial forces to viscous forces.  When viscous forces dominate it is known as laminar flow. Laminar flow occurs when Reynolds number is less then 10. If velocities is held constant and the length scale decreases then so does reynolds number and for this small region the flow will approach Laminar flow and average shear forces will be dominated by the viscosity.

When viscous forces dominate the shear forces are dominated by the momentum exchange though partial diffusion, when viscous forces don’t dominate then the bulk flow of fluid can give a momentum exchange that acts like an effective increase in viscosity.  Therefore when making large scale approximations it is important to know how the sub scale dynamics effect large scale properties such as the average shear forces.

One way we might do this is though Kolmogrov’s Theory of Turbulence:

A turbulent flow is characterized by a hierarchy of scales through which the energy cascade takes place. Dissipation of kinetic energy takes place at scales of the order of Kolmogorov length η, while the input of energy into the cascade comes from the decay of the large scales, of order L. These two scales at the extremes of the cascade can differ by several orders of magnitude at high Reynolds numbers. In between there is a range of scales (each one with its own characteristic length r) that has formed at the expense of the energy of the large ones. These scales are very large compared with the Kolmogorov length, but still very small compared with the large scale of the flow (i.e. ). Since eddies in this range are much larger than the dissipative eddies that exist at Kolmogorov scales, kinetic energy is essentially not dissipated in this range, and it is merely transferred to smaller scales until viscous effects become important as the order of the Kolmogorov scale is approached. Within this range inertial effects are still much larger than viscous effects, and it is possible to assume that viscosity does not play a role in their internal dynamics (for this reason this range is called “inertial range”).

http://en.wikipedia.org/wiki/Turbulence#Kolmogorov.27s_Theory_of_1941

Some other relevant links:

http://www.sjsu.edu/faculty/watkins/kolmo.htm
http://en.wikipedia.org/wiki/Kolmogorov_microscales

http://scienceworld.wolfram.com/physics/Turbulence.html

On a related curiosity, I am wondering how knowing the flow conditions at a discrete number of points effects the range of possible solutions to Navier Stokes Equations. I’ve posted a bit on this in the next two links:

http://www.climateaudit.org/phpBB3/viewtopic.php?f=4&t=760
http://www.physicsforums.com/showthread.php?t=331713