The detection of vortices, ripples and other singularities in a flowing liquid or gas has been the focus of attention of engineers and mathematicians for many decades. Their occurrence may indicate poor aerodynamic properties of the object being flown around or obstacles preventing smooth transport of the fluid. We present a method that detects structures of interest without the need to model the flow with the Navier-Stokes equations.
The flow of a fluid or air in an environment, flowing around given objects, is traditionally studied using a mathematical model given by the so-called Navier-Stokes partial differential equations. At the input of the model, we define the boundary conditions (i.e. the shape of the object being flown around), and at the output we get a solution describing the velocity vector at each point. In it, singularities can be detected by mathematical analysis methods. This approach has two basic shortcomings. Firstly, it is very computationally and theoretically demanding (not for nothing is the question of general solvability of Navier-Stokes equations one of the famous "Millennium Problems") and secondly, it cannot be applied to real data where, for example, we study an object in a wind tunnel and measure the flow velocity vectors directly on a discrete network of points. In this project, we proposed a method that completely omits any flow model, avoids the Navier-Stokes equations, and instead uses artificial intelligence methods to find structures in the data. To simplify, we can imagine the basic idea as follows: we have a database of structures that we want to search for in a flow. It is usually manually selected from some sample data and need not consist of just vortices; the method works for any regular and singular structures. On this database, the method "learns" what we are looking for and then explores the data from the actual flow (see Fig. 2). This procedure is known from 2D image analysis as "template matching". However, it has not yet been applied to vector fields. The problem is what characteristics to represent the data with. The training database is always limited, and we cannot expect it to contain all possible vortices. Only when characteristics that do not depend on the particular size and shape of the vortex can be found will the method work effectively. Discovering of such characteristics to describe structures in vector fields is the main achievement of the project.
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| Illustration of the main idea of the method. A sample object from the training database (top) is compared with all locations of the investigated vector field (bottom). The comparison is performed using originally designed mathematical descriptors. Achieving a high similarity is considered as finding the occurrence of the structure. |
In the Figures below we see examples of vortex detection in the flow around the obstacle and in a satellite image mapping the global world wind.
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| Detection of vortices in the so-called Kármán vortex street, which is created when flowing around rounded bodies such as wings, cars or smokestacks. |
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| Vortex detection in the NOAA global wind image |
Contact person: Jan Flusser
Related Publication:
- Kostkova J., Suk T., Flusser J.: „Affine Invariants of Vector Fields“, IEEE Trans. Pattern Anal. Mach. Intell., vol. 43, No. 4, pp. 1140-1155, 2021
- Yang B., Kostkova J,. Flusser J., Suk T., Bujack R. : "Rotation Invariants of Vector Fields from Orthogonal Moments", Pattern Recognition, vol. 74, pp. 110-121, 2018
