Cylinder-wall junction flow

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 $16\times 16\mathrm {px}$ . 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 $2048\times 2048\mathrm {px}$ square sensor. The size of a pixel was $36.86\mu \mathrm {m}$ , therefore the spatial resolution of the images was $2712\mathrm {px} /D$ , of the PIV data however, it was $5.8976\cdot 10^{-3}D$ . The temporal resolution was $7.25\mathrm {Hz}$ , which is approximately twice as the macro time scale $u_{\mathrm {b} }/D=3.9\mathrm {Hz}$ . The light sheet was approximately 2mm thick provided by a $532\mathrm {nm}$ Nd:YAG laser. The f-number and the focal length of the lens was $2.8$ and $105\mathrm {mm}$ , 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. 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 $10\mu \mathrm {m}$ . The corresponding Stokes number was $4.7\cdot 10^{-3}$ , 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 $200D$ 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 $700\mu \mathrm {s}$ . Therefore, the total sampling time was $27{,}000/7.25=3724\mathrm {s}$ 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 $1500/7.25=207\mathrm {s}$ 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 the following Table:

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

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 investigatig the approaching boundary layer as well as the flow in front of the wall-mounted cylinder.

CFD Code and Methods

As the numerical details of our large eddy simulation (LES) can be found in (Schanderl & Manhart 2016), 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).

Grid arrangement of the LES (Schanderl 2018)

Applied grids in the LES
Grid	Level of refinement	Cells per diameter horizontal / vertical	Grid spacing $\Delta {x}^{+}/\Delta {y}^{+}/\Delta {z}_{\mathrm {wall} }^{+}$	Number of grid cells
Precursor	0		$60/60/15$	$44\cdot 10^{6}$
Base	0	$31.25/125$	$60/60/15$	$35\cdot 10^{6}$
Grid 1	1	$62.5/250$	$30/30/7.5$	$80\cdot 10^{6}$
Grid 2	2	$125/500$	$15/15/3.7$	$64\cdot 10^{6}$
Grid 3	3	$250/1000$	$7.5/7.5/1.9$	$177\cdot 10^{6}$