In fuid dynamics, the Reynold's number determines the relative importance of viscosity as compared to inertial effects. For very high Reynold's numbers, the viscosity term is negligible and the flow is dominated by inertia. In this regime, instabilities in the Navier Stoke's equation cause irregular flow referred to as turbulence. Beginning with the sketches of the turbulent flows of Leonardo Da Vinci, people repeatedly have put forth effort to describe this interesting behavior. 

A significant advance in the study of fast flows and turbulent systems is the measurement of fields. Due to the relatively fast timescales of turbulent dynamics, the fast sampling time required to measure these fields is difficult. Only recently have fast computers and high speed cameras been available to do the job. Velocity fields are often the primary measurement in fluid flows, but vorticity and scalar fields are also of interest. Vorticity is simply defined as the curl of the velocity field. The enstrophy is then determined by the integrated mean squared vorticity. These field used to determine 'coherent structures' in the system. Coherent structures are long lived coherently moving regions in the field. They are believed to be responsible for much of the dynamics occurring in turbulence. In two dimensions coherent structures correspond to whether a local area of flow is a saddle or a vortex or pure or nonlinear shear. Saddles are local areas of high pressure while vortices are low pressure areas.

For turbulent systems, the driving is often accomplished at a particular injection

length scale such as from the flow through a mesh screen. Dynamics often transport the energy away from this length scale through "cascades". In general, systems driven into a nonequilibrium state are always trying to return to equilibrium. There are two mechanisms that relax turbulent systems back toward equilibrium: The hydrodynamic modes dissipate energy into heat due to viscosity while the nonlinear terms give an exchange of energy among the hydrodynamic modes tending to bring the state of motion closer to equilibrium.

An important distinction in the study of turbulence is the difference between two

dimensional and three dimensional systems. This difference is mainly attributed to the difference in direction of the energy and entropy cascade. Also, the number of possible types of coherent structures is extremely large in three dimensions while in two dimensions it is much smaller (two). On the other hand, both two and three dimensional turbulence are similar in many aspects. They both form coherent structures, mix passive scalars and transport energy and enstrophy through cascades. In my work I plan to study the nature of turbulence in two dimensions. However, the tools previously and yet to be developed to study two dimensional dynamics will be applicable to three dimensional data as well.

Turbulence in two and three dimensions and high Reynold's number flows in

general are of great importance to a myriad of disciplines. Two dimensional turbulence in particular is of primary interest as a model of geophysical flows and flows controlled by magnetic fields and from strong rotation.

In general, phenomenological models are not adequate for a full description of physical systems. Phenomenological models give qualitative descriptions of a system such as scaling behavior in structure functions and spectra. Although the Navier Stokes equation may contain all of turbulence, scaling behavior in terms of Kolmogorov's 1941 theory is currently the most thorough description of the system. This theory on the nature of turbulent °ow is an example of a phenomenological model that although provides a useful description of the dynamics that previously had far poorer descriptions, goes into no further detail describing the fundamental components, interactions or processes that ultimately lead to such behavior. A deeper understanding of turbulence must extend beyond phenomenological models and develop a description based on the internal structure. The proposed "cohesive picture" of turbulence must focus on dynamics, the interaction of coherent structures and other local dynamics that lead to the global properties of turbulent systems.

Using measured velocity fields and individual particle tracks, I plan to measure

several things in this project. Macroscopically, this includes the dynamics and interaction of coherent structures, the dynamics of scale to scale energy and enstrophy transport as well as passive scalar mixing. This will be done using high speed digital imaging and particle tracking to extract fields as well as a variety of analysis techniques.

In Dr. Ecke's lab in the Center for Nonlinear Studies, the experimental set up consists of a rectangular container with electrodes attached

on opposite sides. The subphase consists of a thermally and electrically insulating fluid. The proposed studies will take place primarily in a stratified fluid layer that is electromagnetically forced by applying a voltage across the system.

When a voltage is applied, the fluid is driven. Frictional coupling of the turbulent

systemm to the subphase can be tuned to limit the extent of the energy cascade range. Particles on the order of a micron are suspended within the fluid and are used for particle tracking. Strong magnets underneath the system apply a magnetic force that can be used to stir the system to a state of turbulence. The system can be stirred without a mean flow which is experimentally desirable to observe the same region in the fluid over long periods of time. This system can explore both decaying and forced steady state turbulence.

Several things can be measured in the study of coherent structures: the interaction rate and interaction time of coherent structures, the byproducts of interactions as a function of initial strength and difference in size and finally a long range interaction potential could be deduced using the measured structure accelerations. These measurements could be used to build more realistic models.  Identifying coherent structures will be the initial step in identifying the fundamental processes that drive cascades. In two dimensional turbulence, the conjectured driving mechanisms behind cascades involves the idea of vortex straining and vortex merging. More investigation into understanding cascades will be discussed in the next section.

The next objective in this research project is to identify and characterize the mechanisms which drive the various cascade processes. The cascades include the inverse energy cascade which is a range of energy transport from the injection length scale to larger scales. It is thought to be driven by the merging of neighbor vortices with the same rotational direction. The second is the direct enstrophy cascade which exists at length scales smaller than the injection length scale but larger than length scales at which viscosity becomes important. It is thought that the transfer has much to do with the straining of small weak vortices by large stronger neighbors.  To do this an analysis technique for measuring scale to scale energy and enstrophy flux of scalar fields must be used. The filtering approach performs this job by separating the field into large and small length scale parts and then measures the coupling between the scales via equations that can be derived directly from the fluids’ equation of motion. This measured coupling yields information about the flux. Using the measured scale to scale °ux and time resolved measurements of structure interaction, we will be able to measure both byproducts of the interaction and important time scales with respect to the motion of energy and enstrophy.

Excitement in mixing in turbulent flow has been partially due to Kraichnan's theory which predicts intermittent statistics for passive scalar behavior being forced with non-intermittent velocity fields. In turbulence, the fixed points move chaotically in the flow. The motion of these points can be tracked with the time resolved measurements proposed. The rate of mixing can be established both near and far away from the manifold lines and ¯xed points. This work again could greatly simplify simulations if the majority of mixing occurs in the vicinity of unstable manifolds.

In my time spent working on this project, I plan to complete the proposed tasks

previously discussed. This includes investigating the dynamics and interaction of

coherent structures, the driving mechanisms for scale to scale transport of energy and enstrophy and determine in greater detail how mixing is achieved. A particularly useful outcome of this work will be in bridging the gap between experiment, theory and computer simulations. Many of the results will be useful for computer simulations and models of similar and more complex systems that will cut down on computation time and complexity. In a broader scope, this work will contribute significantly to one of the least well understood areas of fluid dynamics. Having a more fully realized and useable theory of fluids in all regimes is important for a large number of important questions today in everything from astrophysics, geology, meteorology and biology to name just a few.