# Test Case

## 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 16x16px. Doing so, we achieved instantaneous velocity fields of the streamwise (U) and the wall-normal (W) velocity component. From these data the time-averaged turbulent statistics were calculated in the post-processing. We used a CCD-camera with a 2048x2048px square sensor. The size of a pixel was 36.86µm, therefore the spatial resolution of the images was 2712 px/D, of the PIV data however, it was 5.8976e-3 D. The temporal resolution was 7.25Hz, which is approximately twice as the macro time scale ub/D = 3.9 Hz. The light sheet was approximately 2mm thick provided by a 532nm Nd:YAG laser. The f-number and the focal length of the lens was 2.8 and 105mm, 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. 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 of minor importance for the cylinder-wall junction.

Hollow glass spheres were used as seeding and had a diameter of 10µm. The corresponding Stokes number was 4.7e-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 200D long entry length and by the use of vortex generators as recommended by (Counihan 1969). The total number of time-steps was 27,000, the time-delay between two image frames of a time-step was 700µs. Therefore, the total sampling time was 27,000/7.25 = 3724s or 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 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 1500/7.25 = 207s or 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 0.0065 times the standard deviation, the corresponding error in the fourth central moment is 0.0545.

The experimental parameters are listed in Table 1:

Table1: Experimental parameters
Description Value Unit
Cylinder diameter D 0.1 [m]
Flow depth z0 0.15 [m]
Channel width b 1.17 [m]
Flow rate Q 0.069 [m^3 s^-1]
Depth-averaged velocity of approach flow ub 0.3986 [m s^-1]
Kinematic viscosity 1.0502e-6 [m^2 s^-1]
Reynolds number Re_D 37954 [-]

## CFD Codes and Methods

As the numerical details of our large eddy simulation (LES) can be found in (Schanderl & Manhart2016), we provide a brief summary here. The set-up was intended to be identical to the experimental infrastructure. To model the bottom and side walls, we set the boundary conditions to no-slip, 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.

We used our in-house finite volume code MGLET with a staggered non-equidistant Cartesian grid. The Runge-Kutta time-integration was of third order, the spatial approximation of second order 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. The sub-grid scales were modelled using the Wall-Adapting Local Eddy-Viscosity (WALE) model, and the portion of the modelled dissipation is about 30% of the total dissipation rate.

By conducting a precursor simulation a fully developed turbulent boundary layer was generated. The streamwise boundary conditions were periodic, and the precursor domain had a length of 30D to prevent the flow from superstructures. The wall resolution of the precursor grid was 7.5 wall units; thus, no wall model was applied. When the statistics of the precursor simulation converged, the fully developed the turbulent boundary layer was fed into the main simulation domain as inflow condition. Around the cylinder, the grid was refined in three steps, each with a factor of two. Schanderl & Manhart (2016) showed while performing a grid study that three refinement levels were enough to achieve 0.95 wall units at the cylinder and based on the oncoming wall-shear stress. When using the local shear stress, the spatial resolution slightly decreased to 1.6 wall units. Furthermore, the sensitivity of the HV system regarding the inflow conditions was also investigated by Schanderl & Manhart (2016).

File:UFR3-35 gridLVL.png (Schanderl 2018)

Table2: Applied grids
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