# Cylinder-wall junction flow

## Key Physics

To cover this highly complex flow situation, a high spatial resolution is required for both the CFD as well as the experiment. 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, to study the horseshoe vortex in detail and in a generic way, the development of a fully developed turbulent boundary layer approaching the cylinder should be ensured in the first place. Furthermore, the wall-shear stress is highly sensitive with respect to the spatial resolution of the data. To cover the strong velocity gradients, in particular of the the near-wall jet, high spatial resolution in the range of the Kolmogorov length scale is required (${\displaystyle \Delta x\propto \eta _{\mathrm {K} }}$).

## Numerical Modelling Issues

We conducted the Large Eddy Simulation using our in-house code MGLET as a finite volume code with the WALE model to represent the subgrid scale stresses without a damping function. In this model, the subgrid scale viscosity decreases towards the wall following the correct limiting behaviour of ${\displaystyle \nu _{\mathrm {t} }\propto z^{3}}$.

### Discretisation method

The spatial and temporal discretization were done by a second order central differences and a thrid order Runge-Kutta method, respectively.

### Grids and grid resolution

We used a staggered Cartesian grid fo the entire domain, and modelled the surface of the cylinder using a conservative second order immersed boundary method. The grid resolution in thevertical direction was ${\displaystyle \Delta z^{+}=0.95}$ using the wall-shear stress of the precursor simulation, and ${\displaystyle \Delta z^{+}=1.6}$ based on the local shear stress. The grid resolution was, therefore, sufficently fine for a LES. In case of further refinement, it would become a Direct Numerical Simulation (DNS).

### Boundary conditions and computational domain

The computaitional domain containing the cylinder had a rectangular cross-section with the following size: ${\displaystyle L\times B\times H=25D\times 12D\times 1.5D}$. The bottom and side boundaries were treated with a no-slip condition, while the top boundary was modelled using a slip condition. As inflow condition a fully developed turbulent boundary layer was used stemming from a precursor simulation.

The precursor domain had a size of ${\displaystyle L\times B\times H=30D\times 12D\times 1.5D}$ with periodic boundary conditions and no-slip condition at the bottom and the side walls. The surface was also modelled as slip condition.

## Physical Modelling

The horseshoe vortex system is a complex three dimensional flow configuration. The two dimensional data acquisition method is, therefore, 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 LES considering the surface waves by a level set method would improve the numerical results. Aiming to reduce the portion of modelled stresses, a further grid refinement could be implemented. 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 set-up 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