UFR 3-30 Description
2D Periodic Hill
Semi-Confined Flows
Underlying Flow Regime 3-30
Description
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.
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. . The shape of the hill is taken from the study of Almeida et al.~\cite{almeida93}. An accurate geometric specification is available in form of a polynomial ansatz~\cite{web1}.
%----------------------------------------------------------------------------- \begin{figure} \begin{center}
\epsfig{file=./eps/re5600-ui-bw-measure2.eps,width=0.70\textwidth,clip=}
\caption{Sketch of the flow geometry and snapshot of instantaneous
streamwise velocity at $Re = 5600$. \label{fig:eyecatcher}}
\end{center} \end{figure} %-----------------------------------------------------------------------------
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.~\cite{almeida93} are removed and instead periodicity in the spanwise direction is assumed. Based on additional investigations by Mellen et al.~\cite{mellen_00} 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.~\ref{fig:eyecatcher}). 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 \textit{ERCOFTAC/IAHR/COST Workshops on Refined
Turbulence Modeling} in 2001~\cite{workshop2001} and
2002~\cite{workshop2002}, 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.
%----------------------------------------------------------------------------
\subsection{Previous studies}
The periodic hill flow test case has been studied so far pursuing two main objectives, either the \textit{modeling and simulation} issue or the \textit{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 \textit{physical} mechanisms of separation on curved surfaces in more detail.
%--------------------------------------------------------------------------- % Modeling and simulation issue %--------------------------------------------------------------------------- \subsubsection{Modeling and simulation issue}
Besides the workshops mentioned above \cite{workshop2001,workshop2002} a few more studies on the \textit{modeling and simulation} issue should be provided first emphasizing on LES. Temmerman and Leschziner al.~\cite{temmerman_01} 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 \cite{temmerman_01} 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.~\cite{mellen_00} assessing the impact of different SGS models and the effect of grid refinement. In the succeeding study by Temmerman et al.~\cite{temmerman_03} 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.\ \cite{euromech469b,dles-6b,breuer2007a,cemracs2006b}. The latter for example was a collaborative effort %%% by Manhart et al.~\cite{for507} involving five different flow solvers used by five different groups in order to cover a broad range of numerical methods and implementations. All simulations were conducted on the same grid with approximately one million cells and compared to a highly resolved LES by Breuer~\cite{breuer2005}. Overall the DES predictions and also LES predictions on the same grid were found to be in good agreement with the reference data. Further coarsening of the grid did not alter the performance of DES substantially unless the LES--RANS interface moves outside the boundary layer on the crest of the hill. In that situation a massive deterioration of the results was detected.
To evaluate the performance of wall models for LES of attached flows the turbulent plane channel flow \cite{kim87,moser99} 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.~\cite{temmerman_03} 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. E.g.\ Manhart et al.~\cite{manhart_06a} used this flow to evaluate a modified law of the wall for the viscous sublayer which accounts for the effect of both, the wall shear stress and the pressure gradient in the streamwise direction which plays an important role for separated flows. They analyzed the performance of this new formulation based on DNS data of the hill flow at $Re = 5600$. In similar investigations Breuer et al.~\cite{dles-6a,breuer2007b} developed a new wall modeling strategy for separated flows. It allows to derive enhanced wall models which also take the streamwise pressure gradient into account. Moreover, the concept of artificial viscosity used for that purpose makes an accurate description of the physics of the flow in the wall-nearest region possible. Statistical evaluations of highly resolved LES data for the hill flow case using nonlinear stochastic estimation were carried out in order to determine the important physical dependence involved in the new wall model, namely the ratio of the thickness of the viscous sublayer to the height of the wall-nearest cell. The results show that the enhanced wall model yields reliable predictions for separated flows. The agreement with the reference data was found to be much better than the results obtained by no-slip boundary conditions or classical wall models such as those by Schumann~\cite{schumann75} and Werner and Wengle~\cite{werner93}.
%--------------------------------------------------------------------------- % Physical issue %--------------------------------------------------------------------------- \subsubsection{Physical issue}
Concerning the \textit{physical} issue a comprehensive investigation for the periodic hill flow at $Re = 10,595$ was carried out by Fr\"ohlich et al.~\cite{froehlich_05} based on LES predictions with about 4.6 million nodes and two independent codes. By the arguments given in Sect.~\ref{subsect:def} 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. 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. That explains why RANS simulations even when applying second-moment closures can not capture the flow field accurately. As stated by the authors \cite{froehlich_05}, the identification of other structures by means of structure--identification methods turned out to be difficult, mainly because of the high Reynolds number.
In \cite{dles-6b} a preliminary study was carried out in which the Reynolds number effect was investigated for the first time. Interesting flow features such as the variation of the reattachment length were found. Furthermore, the existence of a tiny recirculation at the foot of the windward face of the hill reported in \cite{froehlich_05} was confirmed for $Re = 10,595$. That and other findings motivated to study the flow under varying Reynolds numbers more deeply.
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