UFR 3-30 Description

Front Page

Description

Test Case Studies

Evaluation

Best Practice Advice

References

Description of 2D Periodic Hill Flow

Introduction

This contribution presents a detailed analysis of 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.

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 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 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 ERCOFTAC/IAHR workshop held in Karlsruhe by 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 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 (see 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 re-definition of the test case also allows to meet a number of desiderata judged to be associated with 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 Redefined geometry of the test case

• 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.

• Fourth, the Reynolds number was reduced and set to Re = 10,595 where $Re=U_{B}h/\nu$ is based on the hill height h, the bulk velocity $U_{B}$ taken at the crest of the first hill and the kinematic viscosity $\nu$ of the fluid. Furthermore, the flow is assumed to be periodic in the streamwise direction which represents a simple way out of the dilemma to specify 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.

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 already a test case at various workshops, e.g. the ERCOFTAC/IAHR/COST Workshops on Refined Turbulence Modeling in 2001 (Jakirlic et al. 2001) and 2002 (Manceau et al. 2002), respectively. Consequently, a variety of predicted results using RANS as well as LES are available which can only be partially cited in the succeeding section.

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, Manceau et al. 2002) a few more studies on the modeling and simulation issue should be provided first emphasizing on LES. Temmerman and Leschziner (2001) investigated the periodic hill flow 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 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.

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 meaning. Similar to the plane channel the computational setup of the hill flow is simple owing to the possibility to apply periodic boundary conditions twice. 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 for coarse-grid LES predictions but also for different kinds of hybrid LES-RANS approaches including detached-eddy simulations (DES), see, e.g. Breuer et al. (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 a comprehensive investigation for the periodic hill flow at Re = 10,595 was carried out by Fröhlich et al. (2005) based on LES predictions with about 4.6 million nodes and two independent codes. By the arguments given above and especially the distinct post-reattachment-recovery region it is justified why 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). A detailed analysis was carried out including the evaluation of the budgets for all Reynolds stress components, anisotropy measures, and spectra. 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 that 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 and convected downstream towards the windward slope.

Conclusion and Objective

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

* it represents well-defined boundary conditions, 

* can be computed at affordable costs and 

* nevertheless inherits all the features of a flow separating from a curved surface and reattachment.

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 in order to be used for comparison.

Front Page

Description

Test Case Studies

Evaluation

Best Practice Advice

References

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

© copyright ERCOFTAC 2009