High-Resolution Numerical Analysis of Turbulent Flow in Straight Ducts with Rectangular Cross-Section

**Principal Investigator:**

Markus Uhlmann

**Affiliation:**

Karlsruhe Institute of Technology (Germany)

**Local Project ID:**

DNSDUCT

**HPC Platform used:**

Hornet of HLRS

**Date published:**

Pressure-driven fluid flow in a straight duct with rectangular cross-section exhibits turbulence-induced secondary motion of small amplitude, but with large consequences for momentum, heat, and mass transport. The corresponding open duct flow (featuring one free surface) is of particular practical relevance to environmental and civil engineers. If features such practically important effects as the so-called “dip phenomenon”, referring to the fact that the maximum of average streamwise velocity is not found at the fluid surface. At the origin of this property – as well as of many others – is the specific pattern of the secondary motion in the open duct geometry. While the closed duct features the well-known eightfold vortex pattern (cf. figure 1a), the presence of the free surface breaks the symmetry in the open duct configuration, leading to a modified distribution of the average motion in the cross-plane (figure 1b). Note that the maximum secondary flow intensity is obtained in the surface plane.

It has been recognized that the secondary motion in closed ducts is a statistical consequence of the fact that the instantaneous coherent flow structures are not homogeneously distributed along the perimeter (Uhlmann et al., 2007; Pinelli et al., 2010). For the same geometry it has also been shown that non-linear travelling-wave solutions exhibit secondary motion consistent with the statistics obtained in the low-Reynolds number turbulent regime (Uhlmann et al., 2010; Kawahara et al., 2012).

Thanks to the direct numerical simulations (DNS) which have been carried out on HPC system Hornet of the High Performance Computing Center Stuttgart (HLRS) in the framework of the present project, the researchers have been able to show that a similar mechanism is at the origin of the secondary motion in the rectangular duct with one free surface. In their DNS, the scientists employed a Chebyshev pseudo-spectral method, a fast diagonalisation technique and a parallel slice data model. In the present analysis, coherent vortices were extracted by means of various eduction criteria applied to hundreds of data fields. Analyzing the probability of finding vortex cores as a function of position in the duct cross-section lets the researchers establish a strong link between those structures and the secondary flow vorticity – analogous to the closed duct case.

Further exploiting the three-dimensional nature of the educed vortex structures, as well as tracking the propagation and evolution of these structures over a well-resolved time series enables the scientists to elucidate the particular effect of the junction between the solid side-walls and the free-slip surface upon the flow in its vicinity, which in turn leads to a global effect upon the secondary flow pattern.*y*

Using the presently allotted resources, it was also possible to shed light upon the scaling of secondary flow intensity with Reynolds number (for a given aspect ratio), as well as the corresponding scaling with aspect ratio (for fixed Reynolds number). The researchers have also found that in open ducts with wide aspect ratio the turbulent state on the wider wall can coexist with near-laminar flow on the narrow side-walls; the occurrence of this effect can be linked to the intrinsic length scale of the near-wall turbulence cycle, as has already been observed in transitional flows by Takeishi et al. (2015).

More details on the results can be found in the forthcoming Ph.D. thesis by Y. Sakai.

**Core Research Team:**

Markus Uhlmann and Yoshiyuki Sakai, Institute for Hydromechanics (IfH), Karlsruhe Institute of Technology (KIT)

**References:**

• M. Uhlmann, A. Pinelli, G. Kawahara, and A. Sekimoto. Marginally turbulent flow in a square duct. *J. Fluid Mech.*, 588:153–162, 2007. doi:10.1017/S0022112007007604 .

• Pinelli, M. Uhlmann, A. Sekimoto, and G. Kawahara. Reynolds number dependence of mean flow structure in square duct turbulence. *J. Fluid Mech.*, 644:107–122, 2010. doi:10.1017/S0022112009992242.

• M. Uhlmann, G. Kawahara, and A. Pinelli. Travelling-waves consistent with turbulence-driven secondary flow in a square duct. *Phys. Fluids,* 22(8):084102, 2010. doi:10.1063/1.3466661 .

• G. Kawahara, M. Uhlmann, and L. van Veen. The significance of simple invariant solutions in turbulent flows. *Ann. Rev. Fluid Mech.,* 44:203–225, 2012. doi:10.1146/annurev-fluid-120710-101228 .

• K. Takeishi, G. Kawahara, H. Wakabayashi, M. Uhlmann, and A. Pinelli. Localized turbulence structures in transitional rectangular-duct flow. *J. Fluid Mech.*, 782:368–379, 2015. doi:10.1017/jfm.2015.546.

**Scientific Contact:**

Markus Uhlmann

Computational Fluid Dynamics group

Institute for Hydromechanics

Karlsruhe Institute of Technology (KIT)

Kaiserstraße 12, D-76131 Karlsruhe (Germany)

e-mail: markus.uhlmann [@] kit.edu