# General Remark

The experimental and numerical setups applied in this study were described in detail by Schanderl et al. (2017b) (PIV and LES). The experiment is further described by Jenssen (2019), the numerics in Schanderl & Manhart (2016), Schanderl et al. (2017a) and Schanderl & Manhart (2018). Thus, the following shall provide a brief overview only.

# Test Case Experiments

The experimental data were acquired by conducting planar monoscopic 2D-2C PIV in the vertical symmetry plane upstream of the cylinder. The PIV snapshots were evaluated by the standard interrogation window based cross-correlation of ${\displaystyle 16\times 16\mathrm {px} }$. Doing so, we achieved instantaneous velocity fields of the streamwise (${\displaystyle u}$) and the wall-normal (${\displaystyle w}$) velocity component. From these data the time-averaged turbulent statistics were calculated in the post-processing. We used a CCD-camera with a ${\displaystyle 2048\times 2048\mathrm {px} }$ square sensor. The size of a pixel was ${\displaystyle 36.86\mu \mathrm {m} }$, therefore the spatial resolution of the images was ${\displaystyle 2712\mathrm {px} /D}$. The size of the interrogation windows was ${\displaystyle 5.8976\cdot 10^{-3}D}$. The temporal resolution was ${\displaystyle 7.25\mathrm {Hz} }$, which is approximately twice as the macro time scale ${\displaystyle u_{\mathrm {b} }/D=3.9\mathrm {Hz} }$. The light sheet was approximately 2mm thick provided by a ${\displaystyle 532\mathrm {nm} }$ Nd:YAG laser. The f-number and the focal length of the lens was ${\displaystyle 2.8}$ and ${\displaystyle 105\mathrm {mm} }$, respectively.

At the measurement section, the flume had transparent walls. Therefore, the laser light, which entered the flow from above could pass with a minimum amount of surface reflections through the bottom wall. However, an acrylic glass plate had to be mounted at the water-air interface to suppress the bow waves of the cylinder and let the light sheet enter the water body perpendicularly. The influence of this device at the water surface was tested and considered to be insignificant at the cylinder-wall junction.

Hollow glass spheres were used as seeding and had a diameter of ${\displaystyle 10\mu \mathrm {m} }$. The corresponding Stokes number was ${\displaystyle 4.7\cdot 10^{-3}}$, and therefore, the particles were considered to follow the flow precisely.

The hydraulic boundary condition of a turbulent boundary layer developed naturally due to the ${\displaystyle 200D}$ long entry length and by the use of vortex generators as recommended by (Counihan 1969). The total number of time-steps was ${\displaystyle 27{,}000}$, the time-delay between two image frames of a time-step was ${\displaystyle 700\mu \mathrm {s} }$. Therefore, the total sampling time was ${\displaystyle 27{,}000/7.25=3724\mathrm {s} }$ or ${\displaystyle 1484D}$. During the experiment seeding and other particles accumulated along the bottom plate, which undermined the image quality by increasing the surface reflection. Therefore, the data acquisition was stopped after ${\displaystyle 1500}$ images to allow surface cleaning and to empty the limited capacity of the laboratory PC's RAM. The sampling time of such a batch was ${\displaystyle 1500/7.25=207\mathrm {s} }$ or ${\displaystyle 82D}$.

The data acquisition time and number of valid vectors was validated by the convergence of statistical moments. In the centre of the HV the number valid samples had its minimum. Therefore, the time-series at the centre of the HV was analysed as a reference for the entire flow field. The standard error of the mean was ${\displaystyle 0.0065}$ times the standard deviation, the corresponding error in the fourth central moment is ${\displaystyle 0.0545}$.

The experimental parameters are listed in the following Table:

Experimental parameters
Description Value Unit
Cylinder diameter ${\displaystyle D}$ ${\displaystyle 0.1}$ ${\displaystyle [\mathrm {m} ]}$
Flow depth ${\displaystyle z_{0}}$ ${\displaystyle 0.15}$ ${\displaystyle [\mathrm {m} ]}$
Channel width ${\displaystyle b}$ ${\displaystyle 1.17}$ ${\displaystyle [\mathrm {m} ]}$
Flow rate ${\displaystyle Q}$ ${\displaystyle 0.069}$ ${\displaystyle [\mathrm {m} ^{3}\mathrm {s} ^{-1}]}$
Depth-averaged velocity of approach flow ${\displaystyle u_{\mathrm {b} }}$ ${\displaystyle 0.3986}$ ${\displaystyle [\mathrm {m} \,\mathrm {s} ^{-1}]}$
Kinematic viscosity ${\displaystyle \nu }$ ${\displaystyle 1.0502\cdot 10^{-6}}$ ${\displaystyle [\mathrm {m} ^{2}\mathrm {s} ^{-1}]}$
Reynolds number ${\displaystyle Re_{D}}$ ${\displaystyle 37954}$ ${\displaystyle [-]}$

The laboratory infrastructure is sketched in the following Figure:

The PIV set-up is given in detail here, including the qualitative size of the filed-of-vies (FOV) for investigating the approaching boundary layer as well as the flow in front of the wall-mounted cylinder.

# CFD Code and Methods

We applied our in-house finite volume code MGLET with a staggered Cartesian grid. The grid was equidistant in the horizontal directions and stretched in the vertical direction by a factor smaller than ${\displaystyle 1.01}$. The horizontal grid spacing was four times as large as the vertical one. The time integration was done by applying a third order Runge-Kutta sheme, the spatial approximation by second order central differences and the maximum of the CFL number was in the range of 0.55 to 0.82. To model the cylindrical body, a second order immersed boundary method was applied (Peller et al.2006; Peller 2010). The sub-grid scales were modelled using the Wall-Adapting Local Eddy-Viscosity (WALE) model (Nicoud & Ducros 1999). Around the cylinder, the grid was refined by three locally embedded grids (Manhart 2004), each reducing the grid spacing by a factor of two. A grid study shows the results to be converged over grid spacing (Schanderl & Manhart 2016). The resulting grid spacing in the vertical direction at the bottom plate around the cylinder was smaller than approximately 1.6 wall units at the cylinder and based on the local wall-shear stress (Schanderl & Manhart 2016). The fraction of the modelled dissipation is about 30% of the total dissipation rate (Schanderl & Manhart 2018).

The setup simulated was intended to be identical to the experimental one. To model the bottom and side walls, we applied no-slip boundary conditions, whereas the free surface was modelled by a slip boundary condition. Therefore, the Froude number in the LES was infinitesimal and no surface waves occurred. By conducting a precursor simulation a fully-developed turbulent open-channel flow was achieved as inflow condition. The streamwise boundary conditions were periodic, and the precursor domain had a length of 30D. The wall resolution of the precursor grid was 7.5 wall units.

Grid arrangement of the LES (Schanderl 2018)

Applied grids in the LES
Grid Level of refinement Cells per diameter

horizontal / vertical

Grid spacing

${\displaystyle \Delta {x}^{+}/\Delta {y}^{+}/\Delta {z}_{\mathrm {wall} }^{+}}$

Number of grid cells
Precursor 0 ${\displaystyle 60/60/15}$ ${\displaystyle 44\cdot 10^{6}}$
Base 0 ${\displaystyle 31.25/125}$ ${\displaystyle 60/60/15}$ ${\displaystyle 35\cdot 10^{6}}$
Grid 1 1 ${\displaystyle 62.5/250}$ ${\displaystyle 30/30/7.5}$ ${\displaystyle 80\cdot 10^{6}}$
Grid 2 2 ${\displaystyle 125/500}$ ${\displaystyle 15/15/3.7}$ ${\displaystyle 64\cdot 10^{6}}$
Grid 3 3 ${\displaystyle 250/1000}$ ${\displaystyle 7.5/7.5/1.9}$ ${\displaystyle 177\cdot 10^{6}}$

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