# Using vortex methods in CFD

Greetings from Saint Petersburg! My name is Artem Eliseev and I am a PhD student from Peter the Great Saint Petersburg Polytechnic University (SPbPU). In this post I would like to present some results of a research related to my Master’s thesis.

The research started at the end of 2016, when I participated in an ESGI-123 study group, which was held in Saint Petersburg in terms of BEES-Groups project (we wrote about this project in one of the recent posts: https://ecmiindmath.org/2019/03/21/trilateral-project-of-bees-groups-2016-2018/). Several industrial companies brought problems to this study group and the most interesting for me was the problem form the Robert Bosch company. It had the following formulation “Given the vector field v (then the formula followed). Find the corresponding scalar potential Ф”. Sounds not like a practical issue, isn’t it? However, this problem is just the tip of an iceberg which will be briefly described later.

For now, you may look at this photo from the study group which completely tells the atmosphere ruled there – atmosphere of hard math work (you can see me on the left hand side on the photo). The whiteboard brought by the organizers was totally occupied with equations and formulas for three days of tough considerations. At ESGI-123 problems were solved by groups of participants and there were also three members in our team – my groupmates Vasily (on the right hand side) and Danila and also prof. Thomas Götz (in the centre) from the University of Koblenz, Germany joined our team.

At the end of the study group we presented our results in front of the other participants of the group and also the Robert Bosch company representative Timofey Kruglov was there. He was delighted with our results and suggested me, Vasily and Danila to formalize and extend our work as an official research project promoted by the company. We agreed and the project was successfully finalized through the next few months.

The original problem formulated by the company indeed related to using so-called vortex methods in computational fluid dynamics (CFD) problems. Usually some mesh-using technics (FDM, FEM, FVM) are applied in such problems but in complex-shaped or time-dependent domains mesh construction could become very complicated. That’s why mesh-free methods could be considered and vortex methods is an important group among such methods because it has clear physical interpretation, which is important for practical applications.

This group of methods is not a part of mainstream in CFD, so practically important features of vortex methods in general and specific methods in particular are not fully described and investigated. That’s why in September, 2017 the Robert Bosch company suggested Sergey Lupuleac, head of the VIM Lab in our University, where I work up to date, to do the research on Viscous Vortex and Heat Domains method (VVHD). This offer was accepted and an emerged project became my Master’s thesis theme.

VVHD method could be used to simulate two-dimensional or axisymmetric three-dimensional flows of a viscous fluid. Also, heat transport could be resolved, that’s why the method has a word “heat” in the title. Here are some results of the research. Two-dimensional non-isothermal flow around stable circular cylinder was simulated as a test case. Instant flow structure after the cylinder is shown below compared with an experimental one from Van Dyke album. Instant temperature field after the uniformly heated cylinder is also demonstrated. Results show that vortex methods allow to simulate important effects occurring in flows (like fluid and heat von Karman vortex street) with good precision and further research in this direction could provide feasible solutions for practical issues.

The project finalized in a fascinating way – I have not only successfully defended the thesis in June 2018 but also did an oral presentation during the 25-th anniversary of Robert Bosch Research and Technology Office in Russia. I was invited there by the head of the Office, Dr. Uwe Iben, among other students conducted research promoted by the Robert Bosch company. This is an excellent demonstration of admiration and reception of the work done by myself and my scientific supervisor prof. Boris Grigor’ev.