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= 2D Periodic Hill =


==  Semi-Confined Flows ==


=== Underlying Flow Regime 3-30 ===  
= 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.
= 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
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.
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.
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.
 


[[Image:Str3.jpg|centre|thumb|400px|Fig. 2.1 Time-averaged flow over periodic hills]]
[[Image:Str3.jpg|centre|thumb|400px|Fig. 2.1 Time-averaged flow over periodic hills]]
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investigations of the performance of turbulence models - not only
investigations of the performance of turbulence models - not only
subgrid-scale (SGS) models but also statistical models in the RANS
subgrid-scale (SGS) models but also statistical models in the RANS
context -, and other issues such as wall modeling, the history how
context -, and other issues such as wall modeling, the history of how
this test case was established is briefly sketched.
this test case was established is briefly sketched.


Almeida et al. (1993) experimentally investigated the
Almeida et al. (1993) experimentally investigated the
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configurations were considered, i.e. the flow over a single hill and
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 flow over periodic hills. In 1995 these experiments were chosen as
the basis of a test case at the ERCOFTAC/IAHR workshop held in
the basis of a test case at the 4th ERCOFTAC/IAHR workshop held in
Karlsruhe by Rodi et al. (1995). In order to select the
Karlsruhe in 1995 (Rodi et al. 1995). In order to select the
least demanding configuration, the periodic arrangement without side
least demanding configuration, the periodic arrangement without side
walls was considered. However, the calculations carried out for this
walls was considered. However, the calculations carried out for this
test case highlighted a number of serious problems and open questions,
test case highlighted a number of serious problems and open questions,
see Mellen et al. (2000). This concerns the unknown influence of the side
see Mellen et al. (2000). This concerns the unknown influence of the side
walls in the experiment not taken into account in the predictions.
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
Since the aspect ratio in the experiment was small (almost square
cross-section), it was expected that the spanwise confinement provoked
cross-section), it was expected that the spanwise confinement provoked
spanwise variations.  Furthermore, the predictions at the
spanwise variations.  Furthermore, the predictions at the
workshop (see Rodi et al. 1995)  have cast doubt on the true periodicity of
workshop (see Rodi et al. 1995)  have cast doubt on the true periodicity of
the experimental setup leading to the fact that simulations and
the flow achieved in the experiment, leading to the conclusion that simulations and
experiment were not comparable. Another critical point is the high
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
Reynolds number.  Based on the hill height h and the mean centerline
velocity the Reynolds number was Re = 60,000.  Since the channel
velocity the Reynolds number was Re = 60,000.  Since the channel
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corresponding Reynolds number based on L_y is even about six times
corresponding Reynolds number based on L_y is even about six times
larger resulting in high computational costs for the configuration
larger resulting in high computational costs for the configuration
chosen.  This problem even increases if the single hill case is
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
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
leans on the experimental setup by Almeida et al. (1993) but
avoids the problems discussed above.
avoids the problems discussed above.


The re-definition of the test case also allows to meet a number of
The re-definition of the test case allows to meet a number of
desiderata judged to be associated with a good test case for LES
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
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
generic phenomena of interest, whilst being amenable to a simulation
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specification is available in form of a polynomial ansatz (see Section "Test Cases Studies").
specification is available in form of a polynomial ansatz (see Section "Test Cases Studies").


[[Image:Hill3d.jpg|centre|thumb|400px|Fig. 2.2 Redefined geometry of the test case]]


[[Image:Hill3d.jpg|centre|thumb|400px|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
* 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.
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
* Second, the original channel height was halved. This measure reduces the computational effort and allows a higher aspect ratio L_z / L_y.
the computational effort and allows a higher aspect ratio L_z / L_y.


Third, the side walls existing in the original experimental setup of
* 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".
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
* Fourth, the Reynolds number was reduced and set to Re = 10,595 where <math>Re = U_B  h / \nu </math> is based on the hill height h, the bulk velocity  <math>U_B </math> taken at the crest of the first hill and the kinematic viscosity <math>\nu</math> of the fluid.   
<math>Re = U_B  h / \nu </math> is based on the hill height h, the bulk
 
velocity  <math>U_B </math> taken at the crest of the first hill and the kinematic
* 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".
viscosity <math>\nu</math> 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
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.
LES or DNS. For that purpose the increase of the distance between two
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.
consecutive hills described above is beneficial too, since it enhances
Results from the ATAAC project are available and described in [[media:D3.2-36_Jakirlic-ST01-ERCOFTAC-WIKI.pdf|D3.2-36_Jakirlic-ST01-ERCOFTAC-WIKI.pdf]].
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 ===
=== Previous studies ===


The periodic hill flow test case has been studied so far pursuing two
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
main objectives, either the ''modeling and simulation issue'' or
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.
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 ====
==== Modeling and simulation issue ====


Besides the workshops mentioned above (Jakirlic et al. 2001, Manceau et al. 2002)  
Besides the workshops mentioned above (Jakirlic et al. 2001,2002, Manceau et al. 2002)  
a few more studies on the ''modeling and simulation issue''
a few more studies on the ''modeling and simulation issue''
should be provided first emphasizing on LES. Temmerman and Leschziner
should be reviewed first emphasizing on LES. Temmerman and Leschziner
(2001) investigated the periodic hill flow at Re =
(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
10,595 using LES.  The emphasis was on the effectiveness of different
combinations of subgrid-scale models and wall functions on relatively
combinations of subgrid-scale models and wall functions on relatively
coarse grids.  The accuracy was judged by reference to a wall-resolved
coarse grids.  The accuracy was judged by reference to a wall-resolved
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parameters, such as the length of the separation bubble, are very
parameters, such as the length of the separation bubble, are very
sensitive to modeling approximations (SGS and wall models) and the
sensitive to modeling approximations (SGS and wall models) and the
grid quality.  A similar investigation was carried out by Mellen et
grid quality.  A similar investigation had been carried out by Mellen et
al. (2000) assessing the impact of different SGS models and
al. (2000)at Karlsruhe University  assessing the impact of different SGS models and
the effect of grid refinement.  In the succeeding study by Temmerman
the effect of grid refinement.  In the succeeding study by Temmerman
et al. (2003) the previous efforts of both groups were
et al. (2003) the previous efforts of both groups (Imperial College/Karlsruhe) were
combined and a comparative investigation was carried out applying
combined and a comparative investigation was carried out applying
three grids, six SGS models and eight practices of approximating the
three grids, six SGS models and eight practices of approximating the
near-wall region. Again the coarse-grid simulations were judged by
near-wall region. Again the coarse-grid simulations were judged by
wall-resolved simulations using the fine grid mentioned above and two
wall-resolved simulations carried out by both groups using the fine grid mentioned above and two
independent codes. The simulations on coarse grids highlighted the
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
outstanding importance of an adequate streamwise resolution of the
flow in the vicinity of the separation line. The main reason is the
flow in the vicinity of the separation line. The main reason is the
high sensitivity of the reattachment position to that of the
high sensitivity of the reattachment position to that of the
separation.  Furthermore, the near-wall treatment was found to be more
separation.  Furthermore, the near-wall treatment was found to have more influence on the quality of the results obtained on coarse grids
influential on the quality of the results obtained on coarse grids
than the subgrid-scale modeling.
than the subgrid-scale modeling.


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completely. For the development and investigation of wall models for
completely. For the development and investigation of wall models for
separated flows, the channel flow with periodic constrictions has
separated flows, the channel flow with periodic constrictions has
nearly reached an equivalent status and meaning. Similar to the plane
nearly reached an equivalent status and importance. Similar to the plane
channel the computational setup of the hill flow is simple owing to
channel the computational setup of the hill flow is simple owing to
the possibility to apply periodic boundary conditions twice. However,
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
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
back-flow region emerges. Further downstream the flow reattaches and is
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wall-nearest computational point is located outside the viscous
wall-nearest computational point is located outside the viscous
sublayer.  This renders the case as a sensitive platform to develop
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).
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
In the meantime several studies used this test case to evaluate the
performance not only for coarse-grid LES predictions but also for
performance not only of LES on coarse grids but also of
different kinds of hybrid LES-RANS approaches including detached-eddy
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
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.
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.
 


==== Physical issue ====
==== Physical issue ====


Concerning the ''physical'' issue a comprehensive investigation
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
for 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.
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
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<ref>managed by C. Rumsey</ref> (http://turbmodels.larc.nasa.gov/Other_LES_Data/2dhill_periodic.html).
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
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 and convected downstream towards the windward slope.  
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).


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.
==== Test case study ====


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
In conclusion, the flow over periodically arranged hills described above is a very useful benchmark test case since
Re=6.0&sdot;10<sup>4</sup>
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&asymp; O(10<sup>4)</sup>.


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
* it represents well-defined boundary conditions,  
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.
* can be computed at affordable costs and  


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.
* nevertheless inherits all important features of a flow separating from a curved surface, reattachment and recovery of the reattached 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.'''
''' 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 [[media:D3.2-36_Jakirlic-ST01-ERCOFTAC-WIKI.pdf|D3.2-36_Jakirlic-ST01-ERCOFTAC-WIKI.pdf]].
----
<references/>


{{UFRHeader
{{UFRHeader
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© copyright ERCOFTAC 2009
© copyright ERCOFTAC 2010

Latest revision as of 13:36, 12 February 2017

Front Page

Description

Test Case Studies

Evaluation

Best Practice Advice

References


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 is based on the hill height h, the bulk velocity taken at the crest of the first hill and the kinematic viscosity 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


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


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