E​RCOFTAC PC Iberia

The Flow Induced by a Rotating Disk: von Karman’s Turbulent Boundary Layer

Nicholas Morse, Mihai Mihaescu
(KTH Royal Institute of Technology, Stockholm, Sweden)

Authors: Guillermo Marin, Jeronimo Calderon, Miguel Esteban, Ivette Rodriguez, Ricard Montalà, Oriol Lehmkuhl
(Barcelona Supercomputing Center, Spain)

Ramis Orlu
(KTH Royal Institute of Technology, Stockholm, Sweden; OsloMet—Oslo Metropolitan University, Oslo, Norway)

Philipp Schlatter
(KTH Royal Institute of Technology, Stockholm, Sweden; Friedrich–Alexander–Universitat, Erlangen–Nurnberg, Germany)

The von Karman turbulent boundary layer induced by a rotating disk is a rare example of a canonical threedimensional turbulent boundary layer. Using direct numerical simulation, we visualize the temporal evolution of a passive scalar in this flow at unit Prandtl number on wall-parallel planes located approximately 10 and 100 plus units from the disk surface. Near the wall, we observe azimuthally aligned streaks reminiscent of two-dimensional turbulent boundary layers. Further from the wall, the intrinsic three-dimensionality of the flow emerges through an inclined, spiral organization of large-scale structures, which leave a subtle imprint on the near-wall streaks.

Scientific Abstract

Three-dimensional turbulent boundary layers (3DTBLs) are central to many engineering and geophysical flows, yet remain less well understood than their two-dimensional counterparts. The von K´arm´an turbulent boundary layer (vKTBL) developing on a rotating disk represents a rare canonical 3DTBL: it is intrinsically three-dimensional, statistically homogeneous along the azimuthal direction, and reduces to von K´arm´an’s classical similarity solution in the laminar limit (Alfredsson et al., 2024). Despite its fundamental nature, the vKTBL has been investigated in only a handful of experiments and a single previous direct numerical simulation (DNS), and its scalar transport characteristics remain largely unexplored.

In the present work, we perform a DNS of the vKTBL using Neko (Jansson et al., 2024), a spectral element method written in modern Fortran with backends for various accelerators. Leveraging GPU hardware acceleration, we resolve the incompressible flow over an annular sector of the disk using a grid of 3.75 million 7th-order spectral elements, comprising over one billion unique quadrature points. von K´arm´an’s similarity solution for the disk’s laminar boundary layer is provided as an inflow to the inner radius of the domain, which is tripped using a weak, random volume force spanning the periodic azimuthal direction. Finally, temperature transport from the isothermal disk surface is resolved as a passive scalar at unit Prandtl number.

Visualizations of this temperature field in wall-parallel planes in the buffer and logarithmic layers highlight the familiar near-wall streaks from 2DTBLs, but also serve as a jumping-off point for a detailed statistical and structural analysis of the flow. While the azimuthal streaks in the buffer layer align with the statistical similarity of the vKTBL to 2DTBLs, the outer layer of the vKTBL is characterized by the near absence of a wake component, which is nevertheless present in the mean temperature profile. Visualizations of the temperature field in the log layer and wake reveal structures inclined relative to the azimuthal direction. Only with these structures in mind does one notice their imprint on the near-wall streaks.

Our visualizations of this inclined, spiral organization of the outer layer flow structures and the marked dissimilarity between momentum and scalar transport in the vKTBL inspire our present work: to apply data/statistical analysis techniques to the same three-dimensional dataset used to produce these visualizations to develop a mechanistic understanding of turbulent momentum and scalar transport mechanisms in the vKTBL. Altogether, the present DNS not only illuminates the rich structure of the vKTBL but also establishes a high-fidelity numerical reference for future studies of canonical 3DTBLs.

References

Alfredsson, P. H., Kato, K., and Lingwood, R. J.: Flows over rotating disks and cones. Annual Review of Fluid Mechanics, 56, 45–68 (2024).

Jansson, N., Karp, M., Podobas, A., Markidis, S., and Schlatter, P.: Neko: A modern, portable, and scalable framework for high-fidelity computational fluid dynamics. Computers & Fluids, 275, 106243 (2024).