2D Periodic Hill

Underlying Flow Regime 3-30

Test Case Study

Brief Description of the Study Test Case

The measures of the geometry introduced by Mellen et al. (2000) relate to the hill height h. The hill constricts the channel height of 3.035h by about one third whereas the inter hill distance is 9h. The contour of the 3.857h long two dimensional hill is described by the following six polynomials. $x\;\epsilon \;\left[0;\;0.3214h\right];\quad y(x)=\min(1;1+0\cdot x+2.420\cdot 10^{-4}\cdot x^{2}-7.588\cdot 10^{-5}\cdot x^{3});$ $x\;\epsilon \;\left[0.3215h;\;0.5h\right];\quad y(x)=0.8955+3.484\cdot 10^{-2}\cdot x-3.629\cdot 10^{-3}\cdot x^{2}+6.749\cdot 10^{-5}\cdot x^{3};$ $x\;\epsilon \;\left[0.5h;\;0.7143h\right];\quad y(x)=0.9213+2.931\cdot 10^{-2}\cdot x-3.234\cdot 10^{-3}\cdot x^{2}+5.809\cdot 10^{-5}\cdot x^{3};$ $x\;\epsilon \;\left[0.7144h;\;1.071h\right];\quad y(x)=1.445-4.927\cdot 10^{-2}\cdot x+6.950\cdot 10^{-4}\cdot x^{2}-7.394\cdot 10^{-6}\cdot x^{3};$ $x\;\epsilon \;\left[1.071h;\;1.429h\right];\quad y(x)=0.6401+3.123\cdot 10^{-2}\cdot x-1.988\cdot 10^{-3}\cdot x^{2}+2.242\cdot 10^{-5}\cdot x^{3};$ $x\;\epsilon \;\left[1.429h;\;1.929h\right];\quad y(x)=\max(0;2.0139-7.180\cdot 10^{-2}\cdot x+5.875\cdot 10^{-4}\cdot x^{2}+9.553\cdot 10^{-7}\cdot x^{3});$

At x/h=0 the hill height is maximal whereas the boundary is flat in the range between x/h=1.929 and x/h=7.071. Between x/h=7.071 and x/h=9 the contour follows the above equations in axis-symmetry. Besides the geometry, the following figure shows streamlines at a Reynolds number, that is based on the hill height h and the bulk velocity above the crest, of 5,600 [Rapp 2009)].

$Re={\frac {u_{b}h}{\nu }}$

$u_{b}={\frac {1}{2.035}}\int _{h}^{3.035h}u\left(y\right)dy$

Streamlines from PIV measurements at Re=5,600

The mean flow separates at the curved hill crown. In the wake of the hill the fluid recirculates before it attaches naturally at about x/h=4.5.

Test Case Experiments

A water channel has been set up in the Laboratory for Hydromechanics of the Technische Universität München to investigate the flow experimentally. In total 10 hills with a height of 50 mm were built into the rectangular channel to accomplish periodicity whilst the measurement range lies between hills seven and eight. To achieve homogeneity in the spanwise direction an extent of 18 hill heights was appointed. The following figure sketches the experimental setup.

Sketch of the experimental setup.

The 2D PIV measurements were undertaken between hills seven and eight - and to investigate the periodicity of the flow - between the hill pair six and seven through vertical laser light sheets. The homogeneity in the spanwise direction was controlled by 2D PIV measurements in horizontal planes. The PIV field data was thoroughly validated through 1D LDA measurements. Experiments were done at four Reynolds numbers: Re=5,600; Re=10,600; Re=19,000 and Re=37,000.