UFR 3-30 Description
2D Periodic Hill
Semi-Confined Flows
Underlying Flow Regime 3-30
Description
The two-dimensional flow over periodically arranged hills separates from a curved surface, recirculates in the leeward side of the hill and reattaches naturally at the flat channel bottom. The challenge of this case is to predict the point of separation from that curved surface which has got a strong impact on the point of reattachment. The length and height of the main recirculation bubble varies with the Reynolds number. Furthermore a tiny recirculation zone has been detected on the hilltop at Re=10,595 and a minor one can be found for various Re at the windward foot of the hill. Fig. 2.1 depicts streamlines of the flow and shows the turbulent kinetic energy with its maximum right above the mean recirculation zone.
Review of UFR studies and choice of test case
Zilker et al. (1977) conducted experiments on small amplitude sinusoidal waves in a water channel. Zilker and Hanratty (1979) modified the channel and investigated the flow over large amplitude waves. A periodic behavior of the flow in the streamwise direction was assumed from the eighth out of ten wave trains. They recorded the wall shear stress by electro-chemical probes and measured velocities through thermal coated films. The same channel was used by Buckles et al. (1984) to investigate the flow phenomena separation from a curved surface, recirculation and reattachment with Laser Doppler Anemometry and high resolution pressure cells.
Almeida et al. (1993) published an article in 1993 on the flow over two-dimensional hills that correspond to the symmetry axis of a three-dimensional hill used by Hunt and Snyder (1980). The hills of height h (defined by the six polynomials shown above) were 3.857h long and confined the 6.07h channel by about one sixth. Almeida et al. chose an inter-hill distance of 4.5h and a lateral extent of the domain of 4.5h as well. The measurements with an LDA system were carried out at Re=6.0⋅104 between the hills seven and eight. These investigations became basis for a test case of the ERCOFTAC/IAHR-Workshop in 1995 [Rodi et al. (1995)]. However, the calculations carried out for this test case highlighted a number of serious problems and open questions, see Mellen et al. (2000). This concerns the unknown influence of the side walls in the experiment not taken into account in the predictions. Since the aspect ratio in the experiment was small (almost square cross-section), it was expected that the spanwise confinement provoked spanwise variations. Furthermore, the predictions at the workshop [Rodi et al. (1995)] have cast doubt on the true periodicity of the experimental setup leading to the fact that simulations and experiment were not comparable. Another critical point is the high Reynolds number. Based on the hill height h and the mean centerline velocity the Reynolds number was Re = 60,000. Since the channel height in the experiment was large (L_y = 6.071 h), the corresponding Reynolds number based on L_y is even about six times larger resulting in high computational costs for the configuration chosen. This problem even increases if the single hill case is considered for which suitable experimental data are available. The unknown effect of the side walls remains for this case. Therefore, a new configuration was defined by Mellen et al. (2000), which leans on the experimental setup by Almeida et al. (1993) but avoids the problems discussed above. The channel height was reduced to save computational time though the distance between the hills was doubled to achieve natural reattachment. Periodicity was applied in the streamwise and in the spanwise direction to keep the numerical cost affordable, however the Reynolds number had to be reduced to Re≈ O(104).
Several collaborative studies have followed because various research initiatives such as a DFG-CNRS group "LES of Complex Dlows" have chosen the case to benchmark their codes. For example, Temmerman and Leschziner (2001) investigated the periodic hill flow using LES. The emphasis was on the effectiveness of different combinations of subgrid-scale models and wall functions on relatively coarse grids. The accuracy was judged by reference to a wall-resolved simulation (lower wall only) on a grid with about 4.6 million nodes. It was demonstrated that even gross-flow parameters, such as the length of the separation bubble, are very sensitive to modeling approximations (SGS and wall models) and the grid quality. A similar investigation was carried out by Mellen et al. (2000) assessing the impact of different SGS models and the effect of grid refinement. In the succeeding study by Temmerman et al. (2003) the previous efforts of both groups were combined and a comparative investigation was carried out applying three grids, six SGS models and eight practices of approximating the near-wall region. Again the coarse-grid simulations were judged by wall-resolved simulations using the fine grid mentioned above and two independent codes. The simulations on coarse grids highlighted the outstanding importance of an adequate streamwise resolution of the flow in the vicinity of the separation line. The main reason is the high sensitivity of the reattachment position to that of the separation. Furthermore, the near-wall treatment was found to be more influential on the quality of the results obtained on coarse grids than the subgrid-scale modeling.
In the meantime several studies used this test case to evaluate the performance not only for coarse-grid LES predictions but also for different kinds of hybrid LES--RANS approaches including detached-eddy simulations (DES), see, e.g. Saric et al. (2007) and Breuer et al. (2005, 2006, 2008). The latter for example was a collaborative effort involving five different flow solvers used by five different groups in order to cover a broad range of numerical methods and implementations.
A detailed review of the flow physics was undertaken by Fröhlich et al. (2005) who conducted LES at Re=10,595. Mean and RMS-values, spectra and anisotropy measures are being presented whilst they found phenomena such as the so-called 'splatting effect' on the windward side of the hill. Moreover they studied the size of the largest structures by two-point correlations of the streamwise velocity component. Temmerman (2004) investigated the impact of the number of periods on the flow.
A recent publication comprises cross comparisons of numerical and experimental results up to a Reynolds number of 10,595 (Breuer et al. 2009). A Cartesian (MGLET) and a curvilinear code (LESOCC, Breuer and Rodi 1996, Breuer 2002) are checked with thoroughly validated PIV data. These data are presented here.
Contributed by: (*) Christoph Rapp, (**) Michael Breuer, (*) Michael Manhart, (*) Nikolaus Peller — (*) Technische Universitat Munchen, (**) Helmut-Schmidt University Hamburg
© copyright ERCOFTAC 2009