# 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 be aware which flow profile approaches the cylinder 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, the key physics takes place in a very thin layer in front of the cylinder. A simulation or a measurement not resolving this layer might strongly miss the key features -- 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 in the upstream-directed wall jet between the cylinder and the horseshoe vortex. This means that the wall shear stress in this region might not be captured by conventional wall models, such as the law of the wall for turbulent boundary layers.

## Numerical Modelling Issues

A numerical simulation should first take care of the inflow condition and best use a precursor simulation to get realistic mean velocity distributions and fluctuations.

A high spatial resolution is required to capture the wall shear stress, the streamwise fluctuations and the horseshoe vortex dynamics. 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.

For numerical methods based on the Reynolds Averaged Navier Stokes (RANS) equations several challenges appear important. The production term is negative in the wall jet due to the strong local acceleration, the transport by turbulent fluctuations and by pressure are dominating the tke balance in the wall jet under the horseshoe vortex.

The near-wall velocity distribution in the region of interest (the wall jet) is hard to be modelled or approximated and thus has to be resolved. An evaluation of every single term of the stress balance reveals various terms to contribute significantly to the wall shear stress (Schanderl et al. 2017a). Further, at the investigated Reynolds number the dissipation of turbulent kinetic energy is not necessarily isotropic (Schanderl & Manhart 2018). Thus, the turbulence model should be chosen with care or its contribution to the momentum balance kept small.

## Physical Modelling

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 corresponding 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

The standard error of the mean value of the measured velocities was determined as follows:

${\displaystyle \varepsilon _{\mathrm {std} }(\langle u\rangle )={\frac {\sigma (u)}{\sqrt {N_{\mathrm {samples} }}}}={\frac {\sqrt {M_{2}(u)}}{\sqrt {N_{\mathrm {samples} }}}}}$,

the standard error of the higher central moments ${\displaystyle M_{n}=\langle u'^{n}\rangle }$ was obtained likewise:

${\displaystyle \varepsilon _{\mathrm {std} }\left(\langle u'^{n}\rangle \right)={\frac {\sigma \left(u'^{n}\right)}{\sqrt {N_{\mathrm {samples} }}}}={\frac {\sqrt {M_{\mathrm {2n} }-M_{\mathrm {n} }^{2}}}{\sqrt {N_{\mathrm {samples} }}}}}$.

The standard errors with respect to the standard deviation ${\displaystyle \sigma }$ were quantified as follows:

 ${\displaystyle \varepsilon _{\mathrm {std} }(\langle u\rangle )/\sigma }$ ${\displaystyle \varepsilon _{\mathrm {std} }\left(M_{2}\right)/\sigma ^{2}}$ ${\displaystyle \varepsilon _{\mathrm {std} }\left(M_{3}\right)/\sigma ^{3}}$ ${\displaystyle \varepsilon _{\mathrm {std} }\left(M_{4}\right)/\sigma ^{4}}$ ${\displaystyle 0.0065}$ ${\displaystyle 0.0089}$ ${\displaystyle 0.0237}$ ${\displaystyle 0.0545}$

## Recommendations for Future Work

Performing a converged Direct Numerical Simulation would end all discussions about models and is - in our opinion - not out of reach. Further, considering surface roughness might give additional insight in 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