Turbulent Jet Flows: DNS and Symmetry Theory

• Applicant: Professor Dr.-Ing. Martin Oberlack
• Subject Area: Fluid Mechanics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 577383178

Turbulent jets have been intensively researched for decades, and yet theoretical questions have not been conclusively resolved and even new related applications such as chevrons have been developed. The key proposal starting points are (i) the hypothesis of George (1989) that turbulent jets eventually become self-similar but scale individually due to different inflow conditions, and (ii) the publications (Nguyen, Oberlack 2024a,b, 2025, Nguyen, Benedikt, Oberlack 2025) based on the largest jet DNS calculations to date and the extension of symmetry theory. Therein, using the infinite multi-point moment equations and statistical scaling symmetry, we obtained a one-parameter family of similarity solutions, thus proving George's thesis at least theoretically. The DNS using an upstream turbulent pipe flow led to very fast jet self-similarity up to the 10th moment. This specific DNS showed a breaking of the statistical scaling symmetry. We therefore only know one DNS data point on the one-parameter solution manifold that emerges from the statistical scaling symmetry. From the high potential of the theory three objectives are stated: (i) By varying the inflow condition using (a) fractal grids and (b) chevrons, the above-mentioned set of similarity solutions from the symmetry theory is to be verified using new DNS. We note that independent of the inflow conditions, the theory predicts invariance of the 2nd moment with respect to the parameter in the one-parameter family of solutions. (ii) In the above publications, we found an almost perfect Gaussian profile for arbitrary axial velocity moments as a function of the radius. Statistical symmetries seem to be responsible for this. From the moments an approximation of the probability density function (PDF) is to be constructed, and in turn employed in the next task. (iii) Although the statistical scaling symmetry of intermittency was not visible in the scaling of the DNS data, intermittency was very clearly evident in the PDF of the velocity as it had heavy tails and strong skewness, and this increased with the radial distance from the jet axis. The symmetry theory will be extended to PDF on the basis of the infinite Lundgren hierarchy (ILH). Symmetry-invariant solutions as well as solutions for a new finite-dimensional nonlinear eigenvalue equation are constructed. This is based on a new approach to solving the ILH, which solves both the ILH and all but the coincidence side condition. The latter is to be fulfilled by a superposition of eigenfunctions, because, interestingly, this central property of linear equations is not broken by the nonlinearity of the eigenvalue equation. The aforementioned new approach forms the basis for a theory of intermittency in the PDF.

• DFG Programme: Research Grants