PhD - Efficient Algorithms for Finding Unstable Periodic Orbits in Turbulent Flows KU Leuven Applied Mechanics and Energy Conversion Section Belgium


This PhD position is based in the Turbulent Flow Simulation and Optimization (TFSO) research group headed by Prof. Dr. Johan Meyers, which is part of the department of Mechanical Engineering. The position is embedded in a large interdepartmental project on “Efficient methods for large-scale PDE-constrained optimization in the presence of uncertainty and complex technological constraints”, which is funded by the special research fund of KU Leuven. The project is a collaboration between the research group of Prof. Meyers and the Numerical Analysis and Applied Mathematics Section (NUMA) of Prof. Vandewalle at the department of Computer Science. The PhD research is one of 10 research positions in this project, and focusses on developing new fast numerical methods for finding unstable periodic orbits in turbulent flows.



Background: In recent years there has been a lot of interest in unstable periodic orbits in turbulent flows (see, e.g., Kawahara et al. 2012, Annu. Rev. Fluid Mech. 44, 203–225). These solutions are unstable, and thus do not occur of themselves in turbulence, but they are considered important building blocks to understand the dynamics of turbulence, and possibly find routes for control of turbulence.Moreover, unstable periodic orbits are much studied in chaotic systems in general (of which turbulent flows are an example), and may be used to determine ergodic averages of chaotic attractors, amongst others. Also, the sensitivity of such orbits in chaotic flows is well defined, in contrast to the sensitivity of chaotic trajectories themselves, making them potentially useful for the calculation of gradients of ergodic observables, such as, e.g., necessary in the context of gradient-based optimization. The numerical determination of unstable periodic orbits in turbulent flows is however very challenging.Current algorithms do not scale well with the number of degrees of freedom, so that results are limited to relatively simple flows at low Reynolds numbers and for low orbit periods. 
Research: the research focuses on the formulation of efficient and fast methods for the determination of unstable periodic orbits of Navier–Stokes equations in the turbulent regime. To this end, multiple shooting approaches and direct methods in time are considered, as well as different types of multigrid approaches. Algorithms are developed and tested on a number of simpler chaotic partial differential equations, such as the Kuramoto–Sivashinsky equations, before they are applied to the full Navier–Stokes equations. The research is well embedded within the research groups of J. Meyers (with large expertise on turbulent flow simulation and optimization) and S. Vandewalle (with large expertise in multigrid methods, multigrid in time).


Candidates have a master degree in one of the following or related fields: fluid mechanics,aerospace or mathematical engineering, numerical mathematics, or computational physics. They should have a good background or interest in fluid mechanics, optimization,simulation, and programming (Fortran, C/C++, Matlab, python, …). Proficiency in English is a requirement. The position adheres to the European policy of balanced ethnicity, age and gender. Both men and women are encouraged to apply.



Ideal start time is in spring or summer 2019. The PhD position lasts for the duration of four years, and is carried out at the University of Leuven. During this time, the candidate also takes up a limited amount (approx. 10% of the time) of teaching activities.The remuneration is generous and is in line with the standard KU Leuven rates. It consists of a net monthly salary of about 2000 Euro.



For more information please contact Prof. dr. Johan Meyers, tel.: +32 16 32 25 02, mail: or Prof. dr. ir. Stefan Vandewalle, tel.: +32 16 32 76 54, mail:
To apply, use the KU Leuven online application platform (applications by email are not considered). Please include:

- an academic CV and a PDF of your diplomas and transcript of course work and grades

- a statement of research interests and career goals, indicating why you are interested in this position

- a sample of technical writing, e.g. a paper with you as main author, or your bachelor or master thesis

- two recommendation letters

- a list of at least two additional references(different from recommendation letters): names, phone numbers, and email addresses

- some proof of proficiency in English (e.g.language test results from TOEFL, IELTS, CAE, or CPE)


Please send your application as soon as possible, and before May 1st, 2019 at the latest.

Decision: as soon as a suitable candidate applies

Starting Date: candidates can start immediately. Start preferable before (or on) October 1st, 2019.

You can apply for this job no later than May 01, 2019 via the online application tool

KU Leuven seeks to foster an environment where all talents can flourish, regardless of gender, age, cultural background, nationality or impairments. If you have any questions relating to accessibility or support, please contact us at

If you apply for this position please say you saw it on Engineeroxy


All Jobs




nottingham university



Uni Birmingham

Uni Copenhagen

Uni Geneve

Uni Gothenburg

Uni Kuwait

Uni Lausanne

Uni Laval

Uni Nus Singapore

Uni Seoul

Uni Unsw