# Description of 2D Periodic Hill Flow

## Motivation

This contribution presents detailed LES,DNS and experimental data for the flow over smoothly contoured constrictions in a plane channel. This configuration represents a generic case of a flow separating from a curved surface with well-defined flow conditions which makes it especially suited as benchmark case for computing separated flows and testing RANS and Hybrid LES-RANS methods.

Flow separation from curved surfaces and subsequent reattachment is a flow phenomenon often appearing in engineering applications. Its prediction is complicated by several phenomena including irregular movement of the separation and reattachment lines in space and time, strong interactions with the outer flow, transition from a boundary layer type of flow to a separated shear layer with failure of the law-of-the wall and standard model assumptions for either attached flows or free shear layers. The improvement of flow prediction by Reynolds-averaged Navier-Stokes (RANS) simulation or large-eddy simulation (LES) in such flows is dependent on reliable data of generic test cases including the main features of the respective flow phenomena.

The 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 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 top of the hill 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 time-averaged flow and the turbulent kinetic energy with its maximum in the free shear layer right above the mean recirculation zone.

Fig. 2.1 Time-averaged flow over periodic hills

## Review of UFR studies and choice of test case

In order to motivate why this case is especially useful for basic investigations of the performance of turbulence models - not only subgrid-scale (SGS) models but also statistical models in the RANS context -, and other issues such as wall modeling, the history of how this test case was established is briefly sketched.

Almeida et al. (1993) experimentally investigated the flow behind two-dimensional model hills. Two different configurations were considered, i.e. the flow over a single hill and the flow over periodic hills. In 1995 these experiments were chosen as the basis of a test case at the 4th ERCOFTAC/IAHR workshop held in Karlsruhe in 1995 (Rodi et al. 1995). In order to select the least demanding configuration, the periodic arrangement without side walls was considered. 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 and hence 3D effects in the experiment not taken into account in the 2D 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 (see Rodi et al. 1995) have cast doubt on the true periodicity of the flow achieved in the experiment, leading to the conclusion that simulations and experiment were not really comparable. As a test case for LES, 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 is even greater for the single hill case for which suitable experimental data are available. Theunknown effect of the side walls remains for this case. Therefore, intended as a test case for LES,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 re-definition of the test case allows to meet a number of desiderata for this to be a good test case for LES studies (Mellen et al. 2000, Temmerman et al. 2003). The flow has to contain the key generic phenomena of interest, whilst being amenable to a simulation at economically tolerable cost. The new geometry is sketched in Fig. 2.2. The shape of the hill is taken from the study of Almeida et al. (1993). An accurate geometric specification is available in form of a polynomial ansatz (see Section "Test Cases Studies").

Fig. 2.2 Re-defined geometry of the test case

The new configuration differs from the original setup in five aspects:

• First, compared with Almeida et al.'s configuration the distance between two hill crests in streamwise direction was doubled. This increased distance allows the flow to reattach naturally between successive hills, providing a significant post-reattachment-recovery region on the flat plate between the two hills prior to the re-acceleration over the next hill. From the numerical and modeling point of view this modification means that reattachment is now strongly influenced by wall modeling, SGS modeling, and grid arrangement issues. This aspect was not obvious in the original configuration since reattachment was dictated by the presence of the windward face of the consecutive hill.
• Second, the original channel height was halved. This measure reduces the computational effort and allows a higher aspect ratio L_z / L_y.
• Third, the side walls existing in the original experimental setup of Almeida et al. (1993) are removed and instead periodicity in the spanwise direction is assumed. Based on additional investigations by Mellen et al. (2000) a spanwise extension of the computational domain of L_z = 4.5 h was recommended for LES or hybrid LES-RANS predictions. For a detailed discussion on this issue we refer to the sections "Test Case Studies" and "Best Practice Advice".
• Fourth, the Reynolds number was reduced and set to Re = 10,595 where ${\displaystyle Re=U_{B}h/\nu }$ is based on the hill height h, the bulk velocity ${\displaystyle U_{B}}$ taken at the crest of the first hill and the kinematic viscosity ${\displaystyle \nu }$ of the fluid.
• Fifth, the flow is assumed to be periodic in the streamwise direction which represents a simple way out of the dilemma of specifying appropriate inflow boundary conditions for LES or DNS. For that purpose the increase of the distance between two consecutive hills described above is beneficial too, since it enhances the streamwise decorrelation. Thus a well-defined flow state independent of inflow conditions is achieved. For a detailed discussion on this issue we also refer to the sections "Test Case Studies" and "Best Practice Advice".

As a consequence the resulting geometrically simple test case offers a number of important features challenging from the point of view of turbulence modeling and simulation. The pressure-induced separation takes place from a continuous curved surface and reattachment is observed at the flat plate (see Fig. 2.1). Hence these flow features are sensitive to numerical and modeling aspects. Therefore, this configuration was chosen as a test case at the subsequent ERCOFTAC/IAHR/COST Workshops on Refined Turbulence Modeling held in Darmstadt in 2001 (Jakirlic et al. 2001,2002) and in Poitiers in 2002 (Manceau et al. 2002, Manceau and Bonnet 2003), respectively and in the European ATAAC project. From these workshops, predicted results using a wide variety of RANS models as well as some LES results are available which can only be partially cited in the following section. Results from the ATAAC project are available and described in D3.2-36_Jakirlic-ST01-ERCOFTAC-WIKI.pdf.

### Previous studies

The periodic hill flow test case has been studied so far pursuing two main objectives, either the modeling and simulation issue or the physical issue. Regarding the first, it is used as a benchmark case to investigate the ability of RANS and LES to resolve separation from a curved geometry. Furthermore, the flow is also an interesting case to study the physical mechanisms of separation on curved surfaces in more detail.

#### Modeling and simulation issue

Besides the workshops mentioned above (Jakirlic et al. 2001,2002, Manceau et al. 2002) a few more studies on the modeling and simulation issue should be reviewed first emphasizing on LES. Temmerman and Leschziner (2001)at Imperial College, London, investigated the periodic hill flow configuration set up by Mellen et al (2000) at Re =10,595 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 had been carried out by Mellen et al. (2000)at Karlsruhe University 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 (Imperial College/Karlsruhe) 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 carried out by both groups using the fine grid mentioned above and two independent codes, these simulations to be published later by Fröhlich et al (2005). 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 have more influence on the quality of the results obtained on coarse grids than the subgrid-scale modeling.

To evaluate the performance of wall models for LES of attached flows the turbulent plane channel flow is the standard test case. That is due to its geometrical simplicity including two homogeneous directions which allow the application of periodic boundary conditions avoiding inflow and outflow boundary conditions completely. For the development and investigation of wall models for separated flows, the channel flow with periodic constrictions has nearly reached an equivalent status and importance. Similar to the plane channel the computational setup of the hill flow is simple owing to the possibility to apply periodic boundary conditions in 2 directions. However, for the hill case the flow separates from a curved surface and a large back-flow region emerges. Further downstream the flow reattaches and is accelerated at the windward side of the hill. Therefore, the separation and reattachment process can be studied in detail and wall models developed for attached and separated flows can be evaluated based on this flow.

As mentioned above, Temmerman et al. (2003) investigated the predictive accuracy of different wall models based on this case. It was clearly shown that the predictions provided by classical wall models developed for attached flows are not satisfactory if the wall-nearest computational point is located outside the viscous sublayer. This renders the case as a sensitive platform to develop and improve wall models, see e.g. Manhart et al. (2008) and Breuer et al. (2007).

In the meantime several studies used this test case to evaluate the performance not only of LES on coarse grids but also of different kinds of hybrid LES-RANS approaches including detached-eddy simulations (DES), see, e.g. Breuer et al. (2008), Jaffrezic and Breuer (2008) and Saric et al. (2007, 2010). The latter, for example, was a collaborative effort of the DFG-CNRS group "LES of Complex Flows" involving five different flow solvers used by five different groups in order to cover a broad range of numerical methods and implementations.

#### Physical issue

Concerning the physical issue, based on the wall-resolved LES by 2 groups (Imperial College/Karlsruhe) using about 4.6 million nodes and two independent codes, a comprehensive investigation of the periodic hill flow at Re = 10,595 was carried out by Fröhlich et al (2005). For reasons given above, especially because of the existence of a distinct post-reattachment-recovery region, the chosen configuration stands out from the crowd of investigations on flows over wavy-terrain geometries (e.g. Zilker et al. 1977 or Zilker and Hanratty 1979). Fröhlich et al (2005) carried out a detailed analysis including the evaluation of the budgets for all Reynolds stress components, anisotropy measures, and spectra. Note that these budgets are available in digital form from the Imperial College work. They can be obtained from L. Temmerman (lionel.temmerman@numeca.be) or M. Leschziner (mike.leschziner@imperial.ac.uk) or downloaded from the NASA LARC database[1] (http://turbmodels.larc.nasa.gov/Other_LES_Data/2dhill_periodic.html). The emphasis was on elucidating the turbulence mechanisms associated with separation, recirculation and acceleration. The statistical data were supported by investigations on the structural features of the flow. Based on these interesting observations such as the very high level of spanwise velocity fluctuations in the post-reattachment zone on the windward hill side were explained. This phenomenon revealed to be a result of the `splatting' of large-scale eddies originating from the shear layer, convecting downstream towards the windward slope and finally impinging on the wall.

#### Experimental investigation

In addition to the numerical investigations a physical experiment has been set up in the Laboratory for Hydromechanics of the Technische Universität München to study the flow experimentally and provide reliable reference data (Rapp, 2009). In the experimental setup periodicity is achieved by an array of 10 hills in streamwise direction and a large spanwise extent of the channel. The assumption of periodicity in the experiment was checked by the pressure drop between consecutive hill tops and PIV measurements. Experimental data in a 2D cross-section were provided by PIV measurements. For further details we refer to Rapp (2009).

#### Test case study

In conclusion, the flow over periodically arranged hills described above is a very useful benchmark test case since

• it represents well-defined boundary conditions,
• can be computed at affordable costs and
• nevertheless inherits all important features of a flow separating from a curved surface, reattachment and recovery of the reattached flow.

The test case study comprises new well-resolved DNS and LES obtained with curvilinear and cartesian-grid codes for the described test case geometry for various Reynolds numbers and a comparison with the recently obtained detailed experimental results (Rapp 2009). The main results for Reynolds numbers up to 10,595 are published in Breuer et al (2009). Results obtained with RANS and Hybrid LES-RANS methods in the ATAAC project for the Reynolds numbers 10,595 and 37,000 are available in D3.2-36_Jakirlic-ST01-ERCOFTAC-WIKI.pdf.

1. managed by C. Rumsey

Contributed by: (*) Christoph Rapp, (**) Michael Breuer, (*) Michael Manhart, (*) Nikolaus Peller — (*) Technische Universitat Munchen, (**) Helmut-Schmidt University Hamburg