2D Periodic Hill

Semi-Confined Flows

Underlying Flow Regime 3-30

Description

The flow over periodically arranged hills at Reynolds numbers from 5,600 to 37,000 has been investigated experimentally and numerically. Since Mellen et al. (2000) presented this flow case its geometry has been used for various studies and has served as a benchmark case in different research groups. The flow features phenomena such as separation from a curved surface, recirculation and natural reattachment. It can be simulated at relatively low computational cost because of the periodic boundary conditions and the two-dimensionality. DNS have been performed up to Re=5,600 (based on the bulk velocity above the hill and the hill height) [Peller and Manhart (2005)] whereas numerous LES works were published for Re=10,600 [e.g. Fröhlich et al. (2005), Breuer et al. (2009), Temmerman and Leschziner (2001)]. Recently LES results were presented for Re=37,000.

A physical experiment has been thoroughly set up in the Laboratory for Hydromechanics of the Technische Universität München to investigate the flow experimentally and provide reliable reference data (Rapp, 2009). Recently, a complementary numerical and experimental study was carried out for this flow case (Breuer et al. (2009). The focus of this study was twofold: (i) Numerical and experimental data are presented which can be referred to as reference data for this widely used standard test case. Physical peculiarities and new findings of the case under consideration are described and confirmed independently by different codes and experimental data. Mean velocity and pressure distributions, Reynolds stresses, anisotropy-invariant maps, and instantaneous quantities are shown. (ii)Extending previous studies the flow over periodic hills is investigated in the wide range of Reynolds numbers covering 100 <= Re <= 10,595. Starting at very low Re the evolution and existence of physical phenomena such as a tiny recirculation region at the hill crest are documented. The limit to steady laminar flow as well as the transition to a fully turbulent flow stage are presented. For 700 <= Re <= 10,595 turbulent statistics are analyzed in detail. Carefully undertaken DNS and LES predictions as well as cross-checking between different numerical and experimental results build the framework for physical investigations on the flow behavior. New interesting features of the flow were found.

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 testcase of the ERCOFTAC/IAHR-Workshop in 1995 [Rodi et al. (1995)]. It turned out that the comparability of this experiment and the 2D numerical simulations was limited because of strong 3D effects in the experiment.

Mellen et al. (2000) used the advantages of this geometry and adjusted it to meet numerical needs. 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≈ 10⁴.

Several collaborative studies have followed because various research initiatives such as a DFG-CNRS group have chosen the case to benchmark their codes. Temmerman et al. (2003) state that the challenge of this case is to prediction the separation point, which has a strong impact on the recirculation length. According to the authors the grid resolution of the LES in the vicinity of the separation point has got a not negligible influence on the recirculation length.

A detailed review of he flow physics can be found in 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, whereas 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 — (*) Technische Universitat Munchen, (**) Helmut-Schmidt University Hamburg