Flow Topology
Topology has many applications, but one of the most common ones is in aerodynamics and flow. Maybe an image like one below comes to mind:
But how do engineers and designers make sense of complex flow dynamics such as the one above with multiple points of conflicting air/water flow? They have to determine what sections of the model cause turbulence and then represent them with manipulatable equations.
First, mathematicians identify what turbulence was. This was defined as “infinite-dimensional and strongly non-local and nonlinear.” This very definition is why parameterizing it is so challenging. Non-locality prevents outputs to be determined from general outputs since the entire topological turbulent surface can not be quantified at once. Instead of usually a general mathematical approach, one must instead partition the topological surface into different sub-units, basically reducing its complexity. By defining these sub-units using characteristics such as local maximums, minimums, and saddle points, simpler sections can be somewhat deconstructed into equations.
Since such a process takes multi-dimensional data and produces a 2d plot, it needs something similar to the Fourier transformation. One of these processes is “Wavelet analysis” which takes similarities over time and converts them to wavers over finite portions of time. Of course, with each transformation, the actual physical geometry is lost making it harder to predict physical results from potential inputs. And since each transformation only pertains to sub-units, it again prevents general classifications.
A paper titled “A new view of flow topology and conditional statistics in turbulence“ uses the extreme example of a “vortex tube,“ something which itself can be defined to a small degree of certainty but with also most likely not be “relevant to properties of turbulence“ as a whole. See an example below:
Such researchers instead developed “dissipation-element analysis“ and “vector tube segment analysis“ to maintain scalar and vector values in turbulence. The first analysis (DE), uses the same extreme points to define fully filled spacial regions and understands the molecular process of publication. Since all of the molecules are “bumping“ into each other in each sub-unit, this DE finds those monotonous regions. The “vector tube segment“ then analyzes those more complex examples that can’t be generalized.
Just like before, these researchers used a grid system and determined extrema in the turbulent regions. Turbulent zones were classified as places where a vector trajectory would connect a maximum and a minimum, and zones are separated via the same method. If a trajectory doesn’t connect any local extremal, it is in a “quasi laminar diffusion layer.” Basically, it travels unaffected through different turbulent regions, a way to define seperation. Using this vector analysis and system that preserves original geometry allowed for a better representation of turbulence. This also uses a form of statistical probability for vortex fields, something that seems immensely complicated, but helps define the enstrophy, or kinetic energy, of regions. This also preserves the rotating and moving aspect of vortexes and turbulence as a whole.
Again, some of the finer physics definitions are way beyond my comprehension, but these methods suggest there are better ways of understanding flow topology that preserve greater physical elements of the entire model.
References:
Wang, L., & Peters, N. (2013). A new view of flow topology and conditional statistics in turbulence. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 371(1982), 20120169. doi:10.1098/rsta.2012.0169
Elelwi, M., Botez, R., & Dao, T. (2021, September 26). Structural sizing and topology optimization based on weight minimization of a variable tapered span-morphing wing for aerodynamic performance improvements. Retrieved August 10, 2022, from https://www.mdpi.com/2313-7673/6/4/55/htm
Wavelet analysis. (n.d.). Retrieved August 15, 2022, from https://www.sciencedirect.com/topics/earth-and-planetary-sciences/wavelet-analysis#:~:text=Wavelet%20analysis%20is%20an%20alternative,(or%20frequency)%20and%20time.