Influence of the approach boundary layer on the flow over an axisymmetric hill at a moderate Reynolds number
IInfluence of the approach boundary layer on the flow over anaxisymmetric hill at a moderate Reynolds number
M. Garc´ıa-Villalba a , J. G. Wissink b and W. Rodi aa Karlsruhe Institute of Technology,Germany b Brunel University, United Kingdom
Abstract
Large Eddy Simulations of a flow at a moderate Reynolds number over and around a three-dimensional hill have been performed. The main aim of the simulations was to study the effectsof various inflow conditions (boundary layer thickness and laminar versus turbulent boundary lay-ers) on the flow behind the hill. The main features of the flow behind the hill are similar in allsimulations, however various differences are observed. The topology of the streamlines (frictionlines) on the surface adjacent to the lower wall was found to be independent of the inflow condi-tions prescribed and comprised four saddle points and four nodes (of which two are foci). In allsimulations a variety of vortical structures could be observed, ranging from a horseshoe vortex -that was formed at the foot of the hill - to a train of large hairpin vortices in the wake of the hill.In the simulation with a thick incoming laminar boundary layer also secondary vortical structures(i.e. hairpin vortices) were observed to be formed at either side of the hill, superposed on the legsof the horseshoe vortex. Sufficiently far downstream of the hill, at the symmetry plane the meanvelocity and the rms of the velocity fluctuations were found to become quasi-independent of theinflow conditions, while towards the sides the influence of the hill decreases and the velocity profilesrecover the values prevailing at the inflow.
Flow over obstacles occurs in many engineering applications. From a fundamental point of view,the flow over obstacles features a variety of phenomena and is particularly complex: it is three-dimensional (also in the mean), highly unsteady, involves separation and reattachment (possibly atseveral locations) and contains several interacting vortex systems. Typical examples are flow overaxisymmetric obstacles Hunt and Snyder [1980], wall-mounted prismatic obstacles Martinuzzi andTropea [1993] or finite-height circular cylinders Palau-Salvador et al. [2010]. In the case of obstacleswithout sharp edges, the separation location is not fixed by the geometry and is highly dependent onthe incoming flow characteristics, like Reynolds number or boundary layer thickness. Therefore, it isvery challenging to compute accurately this kind of flow.Recently, a series of experiments performed by Simpson and co-workers Simpson et al. [2002],Byun et al. [2004], Ma and Simpson [2005], Byun and Simpson [2006] renewed the interest in analysingand predicting three-dimensional separation. The configuration they considered was an axisymmetricthree-dimensional hill subjected to a turbulent boundary layer. The Reynolds number of the flow basedon the free-stream velocity and the height of the hill was relatively high ( Re = 130000). Since thena number of researchers have tried to reproduce the experimental results using various computationaltechniques: RANS Wang et al. [2004], Persson et al. [2006], hybrid RANS-LES Tessicini et al. [2007]and pure LES Persson et al. [2006], Patel and Menon [2007], Krajnovi´c [2008], Garc´ıa-Villalba et al.[2009]. It was shown that the RANS predictions were generally poor, while both the hybrid techniquesand the pure LES provided promising results although not completely satisfactory. In the simulationsreported in Garc´ıa-Villalba et al. [2009], it was observed that, in spite of the high resolution employed,the presence of a thin recirculation region made the flow very sensitive to the grid resolution.1 a r X i v : . [ phy s i c s . f l u - dyn ] A ug part from the grid resolution, one of the most important differences between all computationalstudies was the modelling of the incoming flow. In the experiment the hill was subjected to a turbulentboundary layer whose thickness was half of the hill height. Because the incoming boundary layer wasturbulent, the specification of the inlet conditions had to be done in an unsteady manner. For instance,in Garc´ıa-Villalba et al. [2009] a precursor simulation was used, while in Patel and Menon [2007] aboundary layer profile plus random noise was employed. Because of the high Reynolds number of theflow, it is computationally very expensive to perform parametric studies and, hence, it is very difficultto assess the impact on the results of the modelling of the incoming flow.In the present paper, we report simulations of flow over the same hill as considered in the previousstudies, but at a significantly lower Reynolds number. The aim is to study the influence of the flowcharacteristics of various approach flows: two simulations with incoming laminar boundary layers ofdifferent thicknesses and one simulation with an incoming turbulent boundary layer. The LES were performed with the in-house code LESOCC2 (Large Eddy Simulation On CurvilinearCoordinates). The code has been developed at the Institute for Hydromechanics. It is the successorof the code LESOCC developed by Breuer and Rodi Breuer and Rodi [1996] and is described in Hin-terberger [2004]. The code solves the Navier-Stokes equations on body-fitted, curvilinear grids using acell-centered Finite Volume method with collocated storage for the cartesian velocity components andthe pressure. Second order central differences are employed for the convection as well as for the dif-fusive terms. The time integration is performed with a predictor-corrector scheme, where the explicitpredictor step for the momentum equations is a low-storage 3-step Runge-Kutta method. The correc-tor step covers the implicit solution of the Poisson equation for the pressure correction (SIMPLE). Thescheme is of second order accuracy in time because the Poisson equation for the pressure correction isnot solved during the sub-steps of the Runge-Kutta algorithm in order to save CPU-time. The Rhieand Chow momentum interpolation Rhie and Chow [1983] is applied to avoid pressure-velocity decou-pling. The Poisson equation for the pressure-increment is solved iteratively by means of the ’stronglyimplicit procedure’ Stone [1968]. Parallelization is implemented via domain decomposition, and ex-plicit message passing is used with two halo cells along the inter-domain boundaries for intermediatestorage.The configuration mentioned above consists of the flow over and around an axisymmetric hill ofheight H and base-to-height ratio of 4. The hill shape is described by: y ( r ) H = − . (cid:104) J (Λ) I (cid:16) Λ r H (cid:17) − I (Λ) J (cid:16) Λ r H (cid:17)(cid:105) , (1)where Λ = 3 . J is the Bessel function of the first kind and I is the modified Bessel function ofthe first kind Simpson et al. [2002]. The approach-flow boundary-layer has a thickness δ which variesdepending on the simulation (see Table 1). The Reynolds number of the flow based on the free-streamvelocity U ref and the hill height H is Re = 6650.The size of the domain is 22 H × H × H in streamwise, wall-normal and spanwise directions,respectively. The grid consists of 504 × ×
400 cells in these directions. The choice of the compu-tational mesh is based on the experience gained in performing an LES of a similar flow problem at asignificantly higher Re Garc´ıa-Villalba et al. [2009], with results compared to experimental data. Inorder to minimize numerical errors, the grid is quasi-orthogonal close to the hill’s surface and the gridpoints are concentrated in the boundary layer.As in Garc´ıa-Villalba et al. [2009], the dynamic Smagorinsky subgrid-scale model, first proposedby Germano et al. Germano et al. [1991], has been used in the simulations. The model parameteris determined using an explicit box filter of width equal to twice the mesh size and smoothed bytemporal under-relaxation Breuer and Rodi [1996]. The impact of the subgrid-scale model on the2ase Inflow δ/H Re θ C D Line colorS1 Laminar 1 900 0.305 BlueS2 Laminar 0.1 90 0.493 GreenS3 Turbulent 1 670 0.407 RedTable 1: Parameters of the simulations.results is likely to be small. For instance, the maximum values of the ratio of the time-averagededdy viscosity and the molecular (kinematic) viscosity is (cid:104) ν sgs (cid:105) /ν ∼ (cid:104) ν sgs (cid:105) /ν ∼ δ/H = 1 and in Simulation S2, δ/H = 0 .
1. In Simulation S3, the approachingboundary layer is turbulent. The mean inflow profile corresponds to one of the cases reported inSpalart [1988]. As in Garc´ıa-Villalba et al. [2009], the time-dependent inflow conditions are obtainedby performing simultaneously a separate periodic LES of channel flow in which the mean velocity isforced to assume the desired vertical distribution using a body-force technique Pierce [2001]. By usingthis technique, the distribution of turbulent stresses obtained at the inflow plane is very similar to thestandard distribution Spalart [1988] in a fully developed turbulent boundary layer. In this precursorcalculation, the number of cells in streamwise direction is 72. The cost of this simulation is, therefore,1 / Re θ , for the three cases is also provided in Table 1. After discarding initial transients, statistics have been collected for a time span of roughly 250
H/U ref .This corresponds approximately to 11 flow-through times of the computational domain.3 .1 Pressure distribution
The mean drag coefficient of the hill, C D = D/ (0 . ρU ref S ), where D is the drag including pressureand viscous terms, ρ is the fluid density, and S is the frontal surface, is reported in Table 1 for thethree cases. Note that using the free stream velocity U in the definition of C D might not be idealin the present case because of the different amounts of momentum present in the incoming flow for y/H < C p = ( p − p ∞ ) / (0 . ρU ref ) along the hill centreline are shown inFig. 2. A similar trend is observed in the three cases. Upstream of the hill the pressure coefficientincreases as the hill is approached, reaching a local maximum shortly after the windward slope of thehill starts. The maximum is more pronounced in case S2, which is the case in which more momentumis present below y/H = 1 (Fig. 1). As a consequence, this is the case with the highest drag coefficientof the three. Thereafter, the flow accelerates and the pressure drops significantly reaching a localminimum near the top of the hill. Further downstream, the flow decelerates and the pressure recoverssomewhat but due to the adverse pressure gradient the flow separates soon, producing a typical plateauin the profile of C p upto x/H ∼ .
5. The peak pressure at reattachment occurs around x/H ∼ Closely connected with the pressure distribution is the mean flow topology map displayed in Fig 3.This figure shows streamlines of the mean flow projected onto a wall-parallel surface at a distance tothe wall of y/H = 0 .
01. In addition, the blue patches indicate the region where backflow is present(
U < d for case S3). Foursaddle points, which are all located along the centreline, and four nodes, two located on the centreline,and two foci, located on both sides of the hill, around x/H ∼ . z/H ∼ ± .
5. As expected, these4 ) d ) a ) b )Figure 3: Streamlines of the mean flow projected onto a wall-parallel surface at a distance to the wall y/H = 0 .
01. The blue patches indicate the region where backflow is present
U <
0. a) S1. b) S2. c)S3. d) Zoom of S3 identifying saddles (orange) and nodal points (green).5igure 4: Streamlines of the mean flow in the midplane. Top, S1. Middle, S2. Bottom, S3. The redsquare identifies the zoomed-in region shown in Fig. 5topological features satisfy the conditions provided by Hunt et al. Hunt et al. [1978] for flow overobstacles (same number of saddle and nodal points).In all cases, there are two main areas of backflow, one in the windward part of the hill and asecond one in the rear part. The backflow in the windward part of the hill is located between a saddlepoint and a nodal point on the centreline. The appearance of this region is related to a well-knowphenomenon in the flow around wall-mounted obstacles: the formation of a horseshoe vortex at thefoot of the obstacle. This has been observed in flow over wall-mounted cubes, cylinders, etc. However,this is not observed for the present geometry at significantly higher Reynolds numbers Garc´ıa-Villalbaet al. [2009]. It is noticeable that this region is largest in Simulation S1, while in S2 and S3 the smallerregions are of comparable size. The streamlines arriving at the first saddle point from upstream, aredeviated to both sides to go around the hill. In the Simulation S1 they are deviated as far as | z/H | ∼ | z/H | ∼ Figure 4 shows a comparison of streamlines of the mean flow in the symmetry plane of Simulations S1,S2 and S3. The main recirculation region behind the hill is longer and higher than the one observed forthis same geometry at a significantly higher Reynolds number Garc´ıa-Villalba et al. [2009]. The centreof the main recirculation bubble behind the crest of the hill is identified by the label RB . While HS identifies the small upstream area of recirculation (vortex) obtained at the foot of the hill. Because ofthe strong accelerating mean flow along the foot of the hill, the upstream vortex HS is stretched inthe streamwise direction. As it is wrapped partially around the foot of the hill the horse shoe vortexmentioned above is formed. The shape of the horse shoe vortex is reflected in the lateral streamlinesthat originate from the upstream saddle point, shown in Figure 3. Because of the larger Re θ (Tab.1) of the incoming (laminar) boundary layer, compared to Simulations S2 and S3, in Simulation S1 asignificantly larger upstream separation bubble HS is generated. Also, the streamlines - originatingfrom the crest of the hill - that bound the wake-like wall-parallel flow show that the height of the wakeincreases with increasing Re θ of the inflow profile (see Table 1).Below the main recirculation region, a very shallow secondary bubble is obtained in all threesimulations. Evidence of this can be seen in Fig. 3. However, this bubble is not visible in Fig. 4. Azoomed view of simulation S3, presented in Fig. 5, illustrates the shape of the secondary bubble. Thisthin region is resolved in the simulation with 8 to 10 grid points in wall-normal direction.Figure 6 shows the streamwise velocity profiles of the three simulations at x/H = − , − , , , , x/H = − HS at the foot of the hillin all simulations. The presence of the main recirculation bubble RB is clearly reflected in all threesimulations by the reverse flow in the profiles shown at x/H = 2. Despite the difference in the pro-files and the state (laminar/turbulent) of the flow at the inflow plane, for x/H = 2 , . . . ,
11 the meanprofiles of all three simulations do not present significant differences. Furthermore, the profiles for x/H = 5 , ,
11 all exhibit the characteristic full turbulent boundary layer profile at the bottom witha wake-like region on top.Figure 7 shows a comparison of the mean u -velocity profiles at various stations z/H = 0 , , , x/H = 5. While at the symmetry plane ( z/H = 0) the velocityprofiles almost collapse, towards the edges of the computational domain gradually more and more7igure 6: Profiles of mean streamwise velocity in the midplane z/H = 0, at various streamwiselocations: x/H =-4, -2, 2, 5, 8, 11. Line colors defined in Table 1.Figure 7: Profiles of mean streamwise velocity at x/H = 5, at various spanwise locations: z/H =0, 1,2, 3, 4. Line colors defined in Table 1. 8igure 8: Contours of turbulent kinetic energy in the midplane. Top, S1. Middle, S2. Bottom, S3.Lines are k/U ref =0.025,0.05, 0.075 and 0.1.differences can be observed. At z/H = 4, finally, for each simulation the shape of the mean u -velocityprofile is found to be very similar to the mean inflow velocity profile. Hence, the downstream influenceof the hill (in the form of a wake) is only noticeable in a spanwise region of limited size. Figure 8 shows the turbulent kinetic energy, k , in the symmetry plane for Simulations S1, S2 andS3. Immediately upstream of the hill, at x/H ≈ −
2, in each simulation a small patch with anelevated k -level can be observed that coincides with the upstream separation bubble labelled HS in Figure 4. The production of kinetic energy leading to increased values of k in the re-circulationzone of a separation bubble was observed earlier in Wissink [2003], Wissink et al. [2006] and wasaccounted for by an elliptic instability of the rolled-up shear layer. Downstream of the small separationbubble, the streamwise pressure gradient turns favourable and the energized boundary layers in allsimulations re-attach. Also, at the centre of the circulation bubble, downstream of the crest of the hill- labelled RB in Figure 4 - production of kinetic energy is observed in all simulations. Of the threesimulations, the momentum thickness of the incoming boundary layer in Simulation S S S
1. Because of this, in the inflow region the wall-shear9n Simulation B is significantly larger than in Simulations S1 and S2. As the boundary layer separatesfrom the crest of the hill, the mean shear in the free-shear layers generates turbulence. Because inSimulation S2 the mean shear is much stronger than in Simulations S1 and S3, the production of k inSimulation S2 is significantly higher (as confirmed in Figure 8). Similarly, because of the difference inwall-shear strength, in Simulation S1 the production of k is found to be lower than in Simulation S3.Profiles of the rms values of the u -, v - and w -velocity components of the three simulations areshown in Figure 9. Close to the inflow plane, at x/H = −
4, the rms-values of all velocity componentsof the two Simulations S1 and S2 are observed to be zero. In the upstream separation bubble, HS ,at x/H = −
2, all three velocity components exhibit fluctuations as reflected by the locally increasedrms-values. On the lee side of the hill, at x/H = 2, the presence of the large recirculation bubble RB induces significant fluctuations in all simulations. Because of the increased momentum thickness ofSimulation S1’s incoming boundary layer,the fluctuation in the rms-values is found to be slightly lessthan in Simulations S2 and S3. For x/H ≥
5, the profiles of the rms values of the velocity componentsbecome similar, indicating that the significant differences in the shape and state (laminar/turbulent)of the boundary layers at the inflow plane is no longer identifyable by studing differences in the rmsvalues in the symmetry plane. Compared to the Simulations S1 and S3, the only significant differenceis observed in the w rms values of Simulation S2 at x/H = 5 which, close the the lower wall, aresignificantly higher than in Simulations S1 and S3. This is likely related to the increased mass flux inSimulation S2. In the wake of the hill, the boundary layer recovers with a combination of streamwise acceleration(Fig. 6) and transverse (secondary) circulation. The secondary motion is relatively weak with apeak velocity around 10-15% of U ref and it originates from the realignment of the vorticity generatedupstream of the hill (horseshoe vortex) and additional vorticity shed from the surface of the hill. Thevorticity generation at the wall and its subsequent re-orientation is discussed in § x/H = 1, 2 and 5.At x/H = 1, the legs of the horseshoe vortex are clearly visible around | z/H | ∼
2. It is interestingto note that in Simulation S1 two vortices ( | z/H | ∼ | z/H | ∼ .
8) are present with a counter-rotating region in between. On the other hand, in Simulations S2 and S3 only one such vortex isvisible. Further downstream for these two simulations there is a trace of a second vortex (labeled HS ) although much weaker than in Simulation S1.At x/H = 2, a secondary vortex labelled HP appears in all three simulations at | z/H | <
1. Furtherdownstream, at x/H = 5, for Simulations S1 and S3 this vortex remains present, with the eye slightlydisplaced upwards. In Simulation S2, this secondary vortex seems to have collapsed in the midplane.Contours of turbulent kinetic energy k are also included in Fig. 10. The development of theturbulent kinetic energy in the wake occurs at a later streamwise position in Simulation S1 comparedto Simulations S2 and S3. At x/H = 1 patches of k can be observed in the region | z/H | < x/H = 2, the peak of k isstronger in Simulations S2 and S3. The turbulent kinetic energy is concentrated in the shear layerbetween the free stream and the recirculation region, in all three simulations. Additionally, thereis a patch further outwards, which is related to the recirculation in the horseshoe vortex. Furtherdownstream, at x/H = 5, the decay of k with respect to the previous location is clearly visible, atypical phenomenon of wakes. In order to illustrate in a more quantitative manner the differencesbetween the three simulations at this streamwise location ( x/H = 5), vertical profiles of k at variousspanwise locations are shown in Fig. 11. At the midplane ( z/H = 0), k is largest in Simulation S2,by a factor of about 20%, while Simulations S1 and S3 present similar values. At z/H = 1, k ofSimulation S1 has decreased significantly compared to Simulation S3, indicating that the width of the10igure 9: Profiles of rms-velocity components in the midplane z/H = 0, at various streamwise lo-cations: x/H =-4, -2, 2, 5, 8, 11. Top, streamwise velocity u rms /U ref . Middle, vertical velocity v rms /U ref . Bottom, spanwise velocity w rms /U ref . Line colors defined in Table 1.11igure 10: Secondary motions. Top, S1. Middle, S2. Bottom, S3. Left column, x/H = 1. Middlecolumn, x/H = 2. Right column, x/H = 5. Color represents turbulent kinetic energy.12igure 11: Profiles of turbulent kinetic energy at x/H = 5, at various spanwise locations: z/H =0, 1,2, 3, 4. Line colors defined in Table 1.wake is smaller in Simulation S1. Further outwards, at z/H = 2 and 3, k of Simulation S1 presentslarger values than Simulations S2 and S3, an in particular at larger heights. This indicates that thehorseshoe vortex is strongest in Simulation S1 and transition to turbulence happens later. Finally, at z/H = 4, all simulations have recovered the values specified at the inlet, S1 and S2 are laminar whileS3 is turbulent. Therefore, at this location the boundary layer is undisturbed by the hill. Vorticity production at a solid boundary can be described in terms of vorticity flux. For three-dimensional flows, the mean vorticity flux can be defined Panton [1984], Andreopoulos and Agui[1986] as (cid:126)σ = − ν ( n · ∇ (cid:126)ω ) w , (2)where (cid:126)σ is the mean vorticity-flux vector, (cid:126)n is the normal vector to the surface, towards the fluid and (cid:126)ω is the mean vorticity vector.As an illustration, Fig. 12 displays the three components of the mean vorticity flux at the wall fromcase S2 (the distributions of the other two cases are similar). There is production of spanwise vorticityeverywhere in the flow, while the production of vertical and streamwise vorticity is concentrated inthe hill region. Obviously, in the absence of the hill, ω x and ω y would be zero. In the vicinity ofthe hill, σ z (Fig. 12 c ) reverses sign as a consequence of the reverse flow which occurs both upstreamand downstream of the hill. The negative vorticity flux peaks in the region where the flow stronglyaccelerates. Production of ω y occurs when the oncoming flow is deviated sidewards to pass along theleft and the right of the hill (only the deviation to the right is illustrated). On the half-domain displayedin Fig. 12 b , only negative ω y is generated. Finally, production of ω x (Fig. 12 a ) is concentrated towardsthe side of the hill: Negative ω x is generated upstream of the crest due to the flow moving upwards andto the right, while downstream of the crest positive ω x is generated due to the flow moving downwardsand to the left.It is interesting to compare the magnitude of the mean vorticity flux σ = √ σ i σ i for the threecases. This is done in Fig. 13. Away from the hill, the values of σ are higher in cases S2 and S3compared to S1. This is due to the much lower wall-shear in case S1 (see Fig. 1). In the hill region,the maximum values of σ are of the same order in all cases ( σ ∼ U ref /H ). These high values are13igure 12: Components of the mean vorticity flux at the wall, case S2. Blanking has been used for | σ | < . σ x . Middle, σ y . Bottom, σ z . 14igure 13: Magnitude of the mean vorticity flux at the wall. Top, S1. Middle, S2. Bottom, S3.15igure 14: Iso-surfaces of mean vertical vorticity from S2. Blue, ω y = − .
5. Red, ω y = 0 . ω z ), and towards the sides: where the flow is deviated to the left andto the right (which leads to production of ω x and ω y ). It can be seen that, while in all three casessimilarly shaped contours are obtained, case S2 has the highest flux, followed by S3 and, finally, S1.The reason for this is that in S2 more momentum is present below y/H = 1 (Fig. 1), than in S3,while S3 has more momentum below y/H = 1 than S1. The trend (and the argument) is the same asfor the pressure coefficient discussed above. By comparing σ in the region of the hill ( σ H ) to its valueat the inflow ( σ ), it is possible to quantify the relative influence of the upstream vorticity and thevorticity generated over the hill. Due to the low σ in case S1, σ H /σ reaches values as high as 50,while in the other two cases this ratio is lower; σ H /σ ∼ σ H /σ ∼ b shows that, for the region considered, only negative ω y is produced at the wall. The blue iso-surface ω y = − . U ref /H originates exactly in the region of production and is then convected intothe wake. The red iso-surface ω y = 0 . U ref /H is, however, not produced at the wall. Instead, thisregion corresponds to the horseshoe vortex and is formed through re-orientation of spanwise vorticityfrom the incoming boundary layer. Fig. 15 displays the values of ω x and ω y in the wake of thehill. These two components, which are generated as the flow passes over and around the hill, canbe seen to be gradually dissipated in the downstream direction. The vanishing non-spanwise meanvorticity indicates that, with increasing distance from the hill, the wake-flow is becoming more andmore homogeneous in the spanwise direction. This figure is related to the secondary motions thatwere displayed in Fig. 10. In particular, the patches of ω x and ω y , which are located in the region | z/H | (cid:46)
1, are related to the secondary vortex labelled HP in Fig. 10. The outer patches are relatedto the horse-shoe vortex labelled HS1 and HS2 in the same figure. Therefore, we can conclude that thesecondary vortex HP is a direct consequence of the vorticity production at the surface of the hill whilethe horse-shoe vortices HS1 and HS2 are only indirectly generated by the hill through re-orientation16f vorticity.
A visualization of the instantaneous coherent structures of the flow is displayed in Fig. 16. Thefigure (and corresponding animations) shows an iso-surface of pressure fluctuations for the value p − (cid:104) p (cid:105) = − .
02. This visualization technique has been often used in the past Fr¨ohlich et al. [2005],Garc´ıa-Villalba et al. [2006]. Coherent structures are observed to form in the lee of the hill and areconvected downstream. Many of them have the shape of a hairpin vortex. Similar structures havebeen also observed at high Reynolds number, although in that case, they were more irregular Garc´ıa-Villalba et al. [2009]. It is also well-known that at lower Reynolds number, a hemisphere protuberancein a laminar boundary layer generates a train of very regular hairpin vortices Acarlar and Smith [1987].It was shown in the cited study that the behaviour of the wake was quite regular upto Re H ∼ Re H = 6650) lies already inthe irregular regime.An important difference with respect to the high Reynolds number case is the visibility of structuresoriginating upstream of the hill at the reduced Reynolds number considered. These structures aresecondary vortical structures that appear on top of the horseshoe vortex. In the present case, thesehairpin vortices are more clearly visible in the cases with laminar inflow, in particular, in SimulationS1, see Fig. 16(top) and Animation S1. In the middle, a train of large hairpin vortices can be observed,while at both sides of the hill a sequence of much smaller hairpin vortices can be seen. In SimulationsS2 and S3, (Fig. 16 middle and bottom, respectively) the large hairpin vortices in the middle can stillbe observed, but the smaller vortices at each side of the hill have become more difficult to identify(Simulation S2 and Animation S2) or almost completely vanished (Simulation S3 and Animation S3).It appears that the reduced wall-shear in Simulation S1 provides ideal conditions for the generation of ahorseshoe vortex on which secondary vortical structures are formed that, in the downstream direction,turn into hairpin vortices. Note that the primary horsehoe vortex can only be indirectly detected bystudying iso-surfaces of the pressure fluctuations. The turbulence, combined with the increased wall-shear stress in Simulation S3 virtually prevent the formation of secondary vortical structures on thehorseshoe vortex. In this paper, results of three LES of flow over and around a three-dimensional hill at moderateReynolds numbers have been presented. The Reynolds number is lower than in previous investigationsGarc´ıa-Villalba et al. [2009] and this has a significant impact in the wake region. While in the presentinvestigation the flow is massively separated, leading to a large recirculation region behind the hill, athigh Re the recirculation region is very shallow Garc´ıa-Villalba et al. [2009]. Therefore, no quantitativecomparison with the previous case is reported in this paper. Two of the simulations have incominglaminar boundary layers of different thicknesses and the third one has an incoming turbulent boundarylayer. It has been shown that the main features of the flow behind the hill are very similar in all threesimulations. For instance, the similar size of the main recirculation bubble behind the hill, the presenceof a horseshoe vortex originating inmediately upstream of the hill, a similar wall-topology map and avirtual collapse of the mean streamwise velocity profiles in the midplane beyond x/H = 5. In spiteof this, there are various differences which need to be pointed out: 1) The height of the wake, whichincreases with increasing Re θ of the inflow profile 2) The maximum level of kinetic energy in thewake varies from 20% to 30% depending on the simulation. 3) The horseshoe vortex is observed tobe significantly affected by the inflow characteristics. 4) The main secondary motion in the centralregion is found to be quite sensitive to the actual inflow condition prescribed, which might havea significant impact on heat and mass transport. 5) The instantaneous coherent structures show17igure 15: Contours of mean streamwise (left) and vertical (right) vorticity from S2. From top tobottom, x/H = 2,3,4,5 and 6. 18igure 16: Iso-surface of pressure fluctuations. Top, S1- Middle, S2. Bottom, S3.19ignificant variations as well. All these fine details indicate that, when trying to reproduce physicalexperiments, special care has to be taken concerning the modelling of the inflow conditions in orderto avoid observable differences in the region of interest. Acknowledgments
The authors are grateful to the steering committee of the supercomputing facilities in Stuttgart forgranting computing time on the NEC SX-8. MGV acknowledges the financial support of the GermanResearch Foundation (DFG).
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