Student(s):
Ryan Thorpe and Nick Weinreich
Dates of Involvement:
Spring 2009 - Present
Faculty Advisor(s):
James Meiss
Graduate Mentor:
Brock Mosovsky
Finite Time Lyapunov Exponents
FTLE Background
At the basic level, FTLE's are used to characterize fluid flow. The FTLE itself is a scalar value that quantifies the amount of stretching between two particles flowing for a given time.
The illustration above shows how two particles can move far away from each other as time progresses. The corresponding FTLE scalar values of this particle pair will be high. Regions with high FTLE values correspond to particles that diverge faster than other particles in the flow field. These regions are considered Lagrangian Coherent Structures (LCS), which have little particle flux across their boundaries, and can be thought of as unstable manifolds. The LCS describes how a flow is moving, and provides useful insight to the actual dynamics of any sort of flow.
Theory
To better understand how these FTLE values are created, consider a particle located at [x(i,j,t), y (i,j,t)] that moves from one point in the flow to another point at [x(i,j,t+T), y (i,j,t+T)] over some given time interval T. This particle is surrounded by other particles that are at an infinitesimally small distance away. The flow is then adverted for some integration time, and the final location of the particles can then be measured. An idea of this occurrence is displayed below-
As this system of particles is flowing together, the orientation between each point is changing continuously. In order to produce FTLE values, the change in distance between the particle needs to be measured. To do this, some form of a derivative approximation in both the x and y directions is required. With the change in distance measured, the data is scaled down, shown in the equations below, to produce a FTLE value.
Above is the FTLE value for particle [x(i,j,t), y (i,j,t)] over the integration time T.
The above algorithm can be applied to all particles in a flow field to understand the fields global flow characteristics.
Finite Time Lyapunov Exponents of the Planetary Boundary Layer
Finite Time Lyapunov Exponents are being applied to characterize fluid movement patterns in nature. To this effect some degree of order and predictability can be found in what seems to be nature’s chaos. Finite Time Lyapunov Exponents (FTLE) have already been used by CalTech for studying activity of oceanic particles in Southern California’s Monterrey Bay [1].
Now another use for FTLEs is being applied to atmospheric velocity data [2]. Using free velocity data dating back to 1957. We are attempting to study the Lagrangian Coherent Structures (LCS) of the earth’s boundary layer. A possible result of discovering LCSs in the atmosphere deals with global warming. If for instance, a LCS was found to trap particles (CFC’s or GHG’s) in the atmosphere over a localized region on the world we might be able to study localized depletion of the ozone layer.
Currently we are trying to apply our FTLE algorithms to a spherical surface to study LCS formations on a global level. Although our FTLE algorithm is proven for a Cartesian coordinate system applying it to a spherical surface presents its own set of difficulties.
Now, a prepackaged MATLAB interpolation algorithm is used to interpolate for velocity values given a set of user specified tracer particles. However this interpolation function only works when interpolating over a flat surface. Now research is being done to develop a basic bilinear spherical interpolation (BISLERP) program that will interpolate for velocity values on a spherical surface. The basic interpolation code is based off of the findings of Ken Shoemake in the context of quaternion interpolation for the purpose of animating 3D rotation [3].
After an interpolation scheme is developed the user will be able to initialize a set of tracer particles and advect them in time, leading to a global FTLE field.
FTLE in Air Jet Data
Recently some velocity data from an air jet experiment has been collected from professor Hertzberg in the Mechanical Engineering department at the University of Colorado. This data represents a fluid flow that progresses in vortices's as it moves upward through its apparatus. The flow is controlled by a speaker that produces waves to create these vortices's and keeps the flow periodic in time. With this velocity data, FTLE calculations are being created to see what other information can be extracted from this experiment. It is interesting to see how LCS's evolve around vortices's and how they effect the particles next to them.
The air jet data is the first set of real world data that our FTLE calculations have been applied to. It is a great tool to work out some of the bugs in the code, and provides a way of answering some of the questions that arise when doing FTLE calculations. The biggest question when doing this sort of calculation is how long the integration time should be. Ideally, the flow should be developed, and a period and a half should be alloted for integration. This is an intrinsic value of each system being looked at, so getting a feel for some of the correct numbers is a bonus. Once the FTLE field is computed for the air jet, it will provide a unique perspective into what is actually happening in the flow, and allow us to deduce some interesting conclusions.
In the future, the FTLE calculations can be applied to a variety of systems, such as ocean currents, wind currents, blood flow and even air flow around an airfoil. This sort of calculation can provide new intuition in to some of life's most incredible events, and hopefully provide a tool in making them even more efficient.
About Ryan Thorpe and Nick Weinreich:
About Nick Weinreich
Nick Weinreich is currently a senior in Applied Mathematics with an emphasis in Mechanical Enineering. After this school year he hopes to obtain a job in the mechanical or mathematical field doing some sort of research, including alternative energy or composite technology.
About Ryan Thorpe
Currently Ryan pursuing a degree in Mechanical Engineering with a minor in Applied Mathematics. In the near future he hopes to obtain a masters degree in Mech E. while using applied mathematics in the Research and Development of Appropriate Technology.
References:
[1] S. C. Shadden, Lagrangian Coherent Structures Tutorial { Analysis of time- dependent dynamical systems using nite-time Lyapunov exponents, April 15, 2005, http://www.cds.caltech.edu/ shawn/LCS-tutorial/contents.html (Aug. 24, 2009).
[2] European Centre for Medium-Range Weather Forecasts, ECMWF 40 Year Re-analysis (ERA-40) Data Archive, August 4, 2008, http://www.ecmwf.int/products/data/archive/descriptions/e4/index.html#top (September 20,2009).
[3] SLERP from Wkipedia, the free encyclopedia, September 18, 2009, http://en.wikipedia.org/wiki/Slerp (September 208, 2009)
