The problem of turbulent fluid flow remains the outstanding unsolved problem of classical physics. Motivated by the fact that turbulence is the rule rather than exception in flows of technological importance, there has been extensive theoretical and experimental work to explain the underlying physical processes. Osborne Reynolds first introduced the idea of decomposing the velocity into time mean and fluctuating components. The so-called Reynolds-averaged Navier-Stokes equation has traditionally been the starting point for many investigations.
Contrasting the idea that the turbulence is completely random, numerous experiments over the last several years have shown the existence of large-scale quasi-organized vortical structures in a variety of free and wall bounded turbulent flows. These so-called coherent structures appear to be dynamically important and play a key role in determining the macrocharacteristics of the flow such as mass, momentum and energy transport. Many basic questions remain concerning the origin of coherent structures, their evolution and mutual interaction. The attraction of the coherent structure concept is the hope that if they are dynamically important and organized, then the essential aspects of the turbulent flow may be captured by low order dynamical models. To date, however, there are no turbulence models in routine use that utilize the coherent structure concept.
The first rigorous theoretical approach to investigating coherent structures was developed by John Lumley and is termed the Proper Orthogonal Decomposition ( POD ). Briefly, the idea is to find a minimum number of modes to describe a given statistical ensemble. Let u(t) be a random generalized process with t as a parameter (spatial or temporal). We would like to find a deterministic function k(t) with a structure typical of the members of the ensemble in some sense. In other words, a functional in the following form needs to be maximized :
where 
is the autocorrelation function of u(t) and u(t'), 
is the average or expected value of f(u), 
is a scalar product and the asterisk denotes a complex conjugate. The classical methods of the calculus of variations gives the final result for k, if R(t,t') is an integrable function
The solution of (2) forms a complete set of a square-integrable orthogonal functions 
with associated eigenvalues 
. It was shown that any ensemble of random generalized functions can be represented by a series of orthonormal functions with random coefficients, the coefficients being uncorrelated with one another:
These functions are the eigenfunctions of the autocorrelation with positive eigenvalues. The eigenvalues are the energy of the various eigenfunctions (modes). Moreover, since the modes were determined by maximizing 
(the energy of a mode), the series (3) converges as rapidly as possible. This means that it gives rise to an optimal set of basis functions from all possible sets. In the theory of turbulence the modes k(t) are often called characteristic eddies. An example of the extraction of coherent eddies via POD from a fully turbulent 2-D jet is presented in the figure below.
Figure 1: An example of a coherent structure in fully turbulent flow.
Once a complete set of optimum orthogonal modes is extracted from the turbulent flow, one can use them as basis functions and expand the velocity field u according to (3). Unlike standard mathematical basis functions, the POD modes represent real statistical averaged structures which exist in the flow. Once obtained, the Navier-Stocks (N-S) equations can be reduced to a n infinite number of ODE's (by Galerkin procedure). Because of experimental extraction, the set of modes (2) is finite and therefore we are faced with solving a finite number of ODE's. The influence of the rest of terms, which involve incoherent turbulence can be modeled by adding extra parametric terms into the N-S equations. These terms take into account an interaction between incoherent fine-scale turbulence, coherent structures and the mean flow. The obtained ODE's can be analyzed theoretically. Implicitly, it is a triple decomposition of the flow into a time mean component, a finite number of coherent modes or structures and incoherent turbulence.
This approach requires intensive mathematical work using diverse tools such as theory of dynamical systems, bifurcation theory, stability theory, wavelet theory as well as other mathematical disciplines to investigate the obtained system of ODE's and to answer basic questions about coherent structure evolution and dynamics. The results will provide a better understanding of the dynamical processes occurring in both wall bounded and free turbulent shear flows and lead to a rational physical model for turbulent flow, that embodies the coherent structure concept.
Specifically, the objective of the research will be to answer the following questions:
1. How do large-scale structures arise and evolve in a turbulent flow ? Is their origin a consequence of global or local flow instability ?
2. Do turbulent flow structures tend to a dynamical equilibrium that is independent of the details of their generation ? Could there be multiple metastable equilibria ?
3. How do the coherent structures alter the classical concept of an energy cascade ? How do the coherent structures interact with the mean flow and turbulence in the dissipative range ?
4. Are there universal structures for wakes, jets and boundary layers ? Do these turbulent flows have a unique attraction state or multiple attractors depending upon the initial conditions? The answer of these questions has important implications for flow control strategies.
