# Cylinder-wall junction flow

## Key Physics

The inflow condition is of major relevance for the vortex system. The horseshoe vortex dynamics are driven by the downflow in front of the cylinder. This downward directed flow is caused by a vertical pressure gradient, which in turn depends on the shape of the approaching inflow profile. Therefore, one has to know the flow profile approaching the cylinder in order to be able to interpret the results and compare it to other studies. In the presented study, we took special care to have a fully developed turbulent open-channel flow approaching the cylinder in both experiment and simulation.

The wall distance of the horseshoe vortex is about ${\displaystyle 0.06D}$, the thickness of the wall jet between the cylinder and the horseshoe vortex center is smaller than ${\displaystyle 0.01D}$. This means that the key physics take place in a very thin layer in front of the cylinder. A simulation or a measurement not resolving this layer might strongly miss some key features such as the levels of the wall shear stress and the large levels of the streamwise velocity fluctuations and their bimodal character under the horseshoe vortex.

It has been demonstrated that Reynolds shear stresses are small below the velocity maximum of the wall jet between the cylinder and the horseshoe vortex. This indicates a laminar-like behavior of the velocity profile below the jet's velocity maximum. As a consequence, conventional wall models relying on the logarithmic law of the wall will have difficulties to model the wall shear stress in this region.

## Numerical Modelling Issues

A fully turbulent open channel flow has to be modeled at the inflow plane which should be placed at least 10 diameters upstream of the cylinder. It is essential to have a realistic turbulence structure and secondary flow at the inflow plane. For eddy resolving simulation strategies (DNS, LES and DES) this implies modeling or generating a fully developed open channel flow which is best achieved by a precursor simulation.

The downstream extent of the domain largely depends on the region of interest and should be placed far enough from the recirculation downstream of the cylinder to avoid instabilities and upstream-propagated waves. According to the authors' estimation this might vary depending on the specific type of the numerical scheme and outflow boundary condition.

A high spatial resolution is required to capture the horseshoe vortex dynamics and the wall shear stress. For the Reynolds number considered, a wall-normal resolution of ${\displaystyle D/1000}$ was necessary to obtain converged wall shear stresses. This high resolution was necessary because the wall jet does not follow the law of the wall for turbulent boundary layers and therefore needs to be fully resolved for wall shear stress calculations. A realistic representation of the junction vortex also requires a fine resolution in the horizontal directions.

## Physical Modelling

This flow case represents a highly complex three-dimensional turbulent flow which shows several characteristics making it difficult to be represented by numerical methods based on the Reynolds Averaged Navier Stokes (RANS) equations. There is the dynamics of the horseshoe vortex which tumbles in horizontal direction and produces bimodal velocity distributions near the wall (as shown by many authors). The production of turbulent kinetic energy is far from a standard shear stress production observed in canonical boundary layers. In some important parts of the flow (in the wall jet where the largest wall shear stress values can be found) there is even a negative TKE production due to the strong flow acceleration. The levels of streamwise fluctuations under the horseshoe vortex seem to depend on the level of TKE in the wall jet in which a balance between the pressure transport and the negative normal stress production term is observed. In other regions (around the horseshoe vortex) it seems that the pressure and turbulent diffusive transport terms cancel each other and should be modeled as a whole. This all indicates that RANS modeling of this flow is a non-trivial task. The turbulent viscosity might be strongly overestimated by standard two equation models, especially in the wall jet.

For eddy-resolving turbulence modeling strategies, several observations might be of importance. The turbulence is far from an isotropic state or from equilibrium. However, around the horseshoe vortex, the gradients of fluctuations in the main flow direction contribute to the dissipation in the same way as gradients in the vertical direction (Schanderl & Manhart 2018). This implies that strongly anisotropic grids are probably not suited to capture the physics around the horseshoe vortex in a sufficiently accurate way. We observed a relatively strong grid dependence of the results, although the SGS contribution was relatively small.

According to our observation, a subgrid-scale model needs to be able to adapt itself to the laminar-like behaviour of the wall layer under the wall jet. Additionally, we have no hope that explicit wall models relying on the logarithmic law of the wall can give an accurate correlation of the wall shear stress with a velocity value in a wall distance in inner coordinate corresponding to the logarithmic layer (Schanderl et al. 2017a).

For designing a grid for a DNS, it is important to learn that estimating the Kolmogorov scale by ${\displaystyle D/Re^{3/4}}$ gives a conservative value and is in the correct order of magnitude (Schanderl & Manhart 2018).

## Measurement issues

The horseshoe vortex system is a complex three-dimensional flow configuration. Therefore, the two-dimensional data acquisition method is a limitation as the out-of-plane velocity component leads to a considerable loss of particles. The number of valid samples suffered from this issue in combination of the low seeding density resulting from the large size of the flume. To overcome this issue, we additionally evaluated the PIV images with a ${\displaystyle 32\times 32\mathrm {px} }$ grid. Whenever the instantaneous velocity fields based on a ${\displaystyle 16\times 16\mathrm {px} }$ grid revealed a missing vector, the corresponding vector of the coarser evaluation was taken as a substitute, if possible. In this way, we could improve the number of valid samples and still keep the spatial resolution high. However, the spatial resolution of the PIV data was too coarse to resolve the velocity gradient correctly. Therefore, a single pixel evaluation is recommended, in order to capture the wall-shear stress correctly.

## Application Uncertainties

When simulating this flow configuration, we address the largest uncertainties to the inflow conditions of the approach flow and to the representation of the water surface. Both, numerical and experimental approaches, face the challenge in generating a fully developed turbulent boundary layer. Even though we intended to reproduce identical flow conditions and validated both of our methods (PIV and LES) by comparison to results in the literature, we observed differences in our results concerning the size and location of the horseshoe vortex for example (see Fig. 6), which we attribute to the uncertainties in the structure of secondary flows or in modelling the water surface.

Another uncertainty is the roughness of the wall and its effect on the flow. It is our understanding that at the moment little is known on this issue.

## Recommendations for Future Work

Performing a converged Direct Numerical Simulation would end all discussions about models and is - in our opinion - not far out of reach to date (2020). Further, considering surface roughness might give additional insight into the interaction of the wall jet with the wall.

The experiments could be improved by stereoscopic or tomographic PIV to acquire three dimensional data sets. Furthermore, the temporal resolution could be increased, in order to analyse the time scales of the horseshoe vortex system. The experimental setup can be improved by providing the light sheet from below passing through the transparent bottom plate, while the PIV camera(s) are mounted at the side outside of the flume.

Contributed by: Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart — Technical University Munich