Effect of geometry and Reynolds number on the turbulent separated flow behind a bulge in a channel
Jean-Paul Mollicone, Francesco Battista, Paolo Gualtieri, Carlo Massimo Casciola
TThis draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics Effect of geometry and Reynolds number onthe turbulent separated flow behind a bulgein a channel
J.-P. Mollicone, F. Battista, P. Gualtieri and C.M. Casciola † Department of Mechanical and Aerospace Engineering, Sapienza University of Rome, viaEudossiana 18, 00184 Rome, Italy(Received xx; revised xx; accepted xx)
Turbulent flow separation induced by a protuberance on one of the walls of an oth-erwise planar channel is investigated using Direct Numerical Simulations. Differentbulge geometries and Reynolds numbers – with the highest friction Reynolds numbersimulation reaching a peak of Re τ = 900 – are addressed to understand the effect ofthe wall curvature and of the Reynolds number on the dynamics of the recirculatingbubble behind the bump. Global quantities reveal that most of the drag is due to theform contribution, whilst the friction contribution does not change appreciably withrespect to an equivalent planar channel flow. The size and position of the separationbubble strongly depends on the bump shape and the Reynolds number. The most bluffgeometry has a larger recirculation region, whilst the Reynolds number increase resultsin a smaller recirculation bubble and a shear layer more attached to the bump. Theposition of the reattachment point only depends on the Reynolds number in agreementwith experimental data available in the literature. Both the mean and the turbulentkinetic energy equations are addressed in such non homogeneous conditions revealing anon trivial behaviour of the energy fluxes. The energy introduced by the pressure dropfollows two routes: part of it is transferred towards the walls to be dissipated and partfeeds the turbulent production hence the velocity fluctuations in the separating shearlayer. Spatial energy fluxes transfer the kinetic energy into the recirculation bubble anddownstream near the wall where it is ultimately dissipated. Consistently, anisotropyconcentrates at small scales near the walls irrespective of the value of the Reynoldsnumber. In the bulk flow and in the recirculation bubble, isotropy is restored at smallscales and the isotropy recovery rate is controlled by the Reynolds number. Anisotropyinvariant maps are presented, showing the difficulty in developing suitable turbulencemodels to predict separated turbulent flow dynamics. Results shed light on the processesof production, transfer and dissipation of energy in this relatively complex turbulent flowwhere non-homogeneous effects overwhelm the classical picture of wall bounded turbulentflows which typically exploits streamwise homogeneity. Key words:
1. Introduction
Flow separation consists of fluid flow around bodies becoming detached, causing thefluid closest to the object’s surface to flow in reverse or different directions, most often † Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] D ec J.-P. Mollicone, F. Battista, P. Gualtieri and C.M. Casciola giving rise to turbulent fluctuations. The flow separation can be induced either bygeometrical singularities, for example in presence of sharp corners, or by smooth geometryvariations, as those occurring over a curved wall. The resulting adverse pressure gradientis sufficient to cause flow detachment.Given its importance from both theoretical and practical point of view, the study ofturbulence and separation has long been of interest to the fluid mechanics community,see e.g. Simpson (1989) for a review. However, due to its complexity, this classic subjectis still widely investigated. The separation of fluid flow from objects inevitably resultsin effects such as increased drag and mixing, momentum and energy transfer and vortexshedding. An understanding of such effects is helpful to improve road vehicle performance,in the study of fluid-structure interaction, to regulate air mixing with other substancessuch as pollutants or fuels, in the study of boundary layer control, see e.g. Marusic et al. (2014) and Bai et al. (2014). In modern bioengineering studies such as in hemodynamics,the nature of the flow and the intensity of the shear stresses helps to determine whetherlesions occur at particular vascular sites, as described by Epstein & Ross (1999).Turbulent boundary layers with pressure gradients are a common characteristic ofmany aerodynamic flows such as the flow past airfoils, gas turbine blades, sails anddiffusers. To correctly predict the behaviour and the efficiency of such components, theunderstanding of separation and reattachment mechanisms together with the associatedenergy behaviour is essential, see e.g. Harun et al. (2013) and references therein. Thefundamental physics is indeed complex and no entirely satisfactory turbulence modelsfor numerical simulation of high Reynolds number separated flows are nowadays available.This is mainly due to the complexity of the geometries inducing separation and to thedifficulty in obtaining sufficiently accurate experimental or numerical data for reliablestatistical analysis.Flow separation occurs in both external and internal flows. In external flows, boundarylayer separation is induced by strong curvature effects and the associated adverse pressuregradient (APG). The understanding of such complex interplay among flow curvature,APG and separation is considered one of the most challenging issues in fluid dynamicsboth for modelling, Wilcox (1998), and most recent DNSs, Soria et al. (2017). In theseconditions, the classical scaling of turbulent statistics is not valid since the flow separationmodifies the Reynolds shear stress distribution as discussed by Skaare & Krogstad (1994).In internal flows, such as channel or pipe flows, on average the pressure decreases in theflow direction. However, the pressure gradient may locally revert due to, for example, anabrupt change of section and/or the presence of curved walls. In this case, a localisedseparated flow occurs characterised by a statistically steady recirculation region and byan eventual reattachment downstream. In such conditions, turbulence develops in highlyanisotropic and non-homogeneous conditions. In addition to the non-homogeneous effectsinduced by the wall, it is fundamental to address the non-homogeneous effects in thestreamwise direction where the dynamics of turbulent fluctuations occurs under rapidlychanging conditions, see e.g. Chen et al. (2006); Gualtieri & Meneveau (2010) for a similarstudy in the context of turbulent flows subjected to rapid time variations of the meanflow.The statistical characterisation of separated flows in presence of adverse pressuregradients is challenging due to the difficulty to control the actual pressure gradient andthe ensuing separated flow in presence of curved flows, see Alam & Sandham (2000).The experimental generation of turbulent flow with an APG is not standardised and thedifferent approaches employed lead to substantially different configurations. Skaare &Krogstad (1994) and Krogstad & Skaare (1995) performed a detailed study of a turbulentboundary layer in presence of a strong APG and constant skin friction coefficient, ffect of geometry and Reynolds number on the separation behind a bulge et al. (2000). On the other hand,Castro & Epik (1998) study the separated flow at the leading edge of a flat plate in awind tunnel considering two different conditions: with and without added homogeneousisotropic turbulence. In Webster et al. (1996) the experimental data of an APG boundarylayer created by a bump in the wall are provided and a detailed analysis of turbulencestatistics is discussed. Dengel & Fernholz (1990) performed experimental measurements ofan APG turbulent boundary layer reporting different cases of pressure distributions, withand without reverse flow, showing the strong dependence of the near-wall flow propertieson the presence or absence of the recirculation region. To address the turbulent flowseparation on smooth geometry the ERCOFTAC test case 81 has been employed in theliterature. A period hill experiment has been designed by Manhart at TU Munich, Rapp& Manhart (2011). This experimental setup is made by nine consecutive 2D bumps toreproduce an infinite channel with periodic bumps in the streamwise direction. K¨ahler et al. (2016) carry out several measurements on this experimental setup to addressthe separated flow. High resolution particle image velocimetry and particle trackingvelocimetry highlight the crucial role of the spatial resolution close to the wall. As statedby the authors, the difficulties to perform these measurements can be compared to thoseencountered in obtaining reliable Large Eddy Simulations (LES), see e.g. the discussionin Gualtieri et al. (2007).A geometry similar to the ERCOFTAC test case 81 is also employed for validation ofdifferent numerical methods and subgrid models for LES and RANS, ˇSari´c et al. (2007),Hickel et al. (2008), Peller & Manhart (2006), Temmerman et al. (2003), Mellen et al. (2000), Breuer et al. (2009), Diosady & Murman (2014), Fr¨ohlich et al. (2005). Fromthis collection of works, separation and reattachment points or turbulence intensity inthe recirculation bubble are found to strongly depend on modelling and numerics, ˇSari´c et al. (2007), Temmerman et al. (2003) .Among the methods used in numerics to introduce an APG, one of the easiest waysis to use wall flow suction. Alternatively, the APG can be prescribed by a body force.Na & Moin (1998 a ) and Na & Moin (1998 b ) performed a Direct Numerical Simulation(DNS) of a separated boundary layer on a flat plate using suction and blowing velocitydistributions at the upper boundary. The inflow condition was taken from Spalart’stemporally evolving zero pressure gradient (ZPG) simulation, see Spalart (1988). Chong et al. (1998) used these data to analyse the topology of near-wall coherent structuresusing the invariants of the velocity gradient tensor. The comparison of experimental andDNS data is presented in Spalart & Watmuff (1993) for turbulent boundary layers withdifferent pressure gradients. The DNS was performed using a spectral code with a fringeregion to deal with periodic conditions in the non-homogeneous streamwise direction andthe friction velocity at the edge of the boundary layer was prescribed to reproduce thepressure gradient of the experiment. A similar numerical technique was used by Skote et al. (1998) for simulations of self-similar turbulent boundary layers in adverse pressuregradients prescribed by the freestream velocity. Skote & Henningson (2002) performedthe DNS of separated boundary layer flow with two different adverse pressure gradients,while Ohlsson et al. (2010) addressed the separation in a three dimensional turbulentdiffuser. On to relatively more complex geometries, Le et al. (1997 a ) concentrated on there-attachment location and the skin friction coefficient behind a backward facing step.Issues related to the separation control have been investigated by Neumann & Wengle(2004), by means of Large Eddy Simulations (LES). The LES performed by Wu & Squires J.-P. Mollicone, F. Battista, P. Gualtieri and C.M. Casciola (1998) was compared to the results by Webster et al. (1996) and it emerged that the useof a coarse resolution with an eddy viscosity model did not allow an accurate descriptionof the small coherent vortical structures in the near wall region which were observed inexperiments. LES has been performed by Kuban et al. (2012) to evaluate the consistencyand accuracy with respect to similar DNS simulations. Indeed, the sub-grid scale modelsneeded in any LES are expected to hamper the physics at the smallest scales, calling forthe use of DNS where no modelling assumptions are introduced. Simulations of channelflow with a lower curved wall were performed by Marquillie & Ehrenstein (2003) atrelatively low Reynolds numbers for a two dimensional case to study the onset of nonlinearoscillations. Marquillie et al. (2008) investigated the vorticity and kinetic energy budgetdownstream of such lower curved wall. Marquillie et al. (2011) and Laval et al. (2012)expanded on these simulations by studying the vorticity and streaks dynamics and linkingthe streaky structures to the kinetic energy production.The present work deals with the Direct Numerical Simulation (DNS) of a fully tur-bulent channel with a lower curved wall, or bump, which produces the flow separation.The simulations are based on the spectral element method, as implemented in Nek5000(Fischer et al. Re τ = 900 over the bump that is, presumably, oneof the highest friction Reynolds number achieved for such a configuration.The objective is to study the effects of the bump geometry and Reynolds number onflow separation. One of the global quantities available from experiments is the positionof the reattachment point, an elusive quantity to reproduce in numerical simulations,due to the need of using closure models to reach sufficiently high Reynolds numbers.In our case, we can directly compare the simulations with the experiments, observing agood agreement with the available data. Beside classical first and second-order statistics,the present DNSs provide access to high quality data concerning pressure and frictiondrag and to wall shear stress and pressure coefficient distributions at the walls. In thepresent flow geometry, the energetics of the flow is rather complex and needs an accuratediscussion. In particular, the shear layer at the boundary of the separation bubble acts assource of turbulent kinetic energy, which is spatially redistributed through the domain bythe associated energy fluxes. In the analysis a crucial role is played by the correspondingterms in the kinetic energy budget of the mean flow, which are usually trivial in absenceof separation. Locally the flow turns out to be strongly anisotropic, with anisotropypersisting down to the smallest scales. This effect was already discussed for the zeropressure gradient boundary layer, Jacob et al. (2008), and for the homogeneous shearflow, Casciola et al. (2007). In the present case, the analysis of the anisotropy of bothlarge and small scales is studied via the deviatoric components of the Reynolds stressesand the pseudo-dissipation tensor. Increasing the Reynolds number, isotropy recovery atsmall scale is found to occur in the recirculating bubble. However, the anisotropy persistsin the shear layer where the production of turbulent kinetic energy overwhelms the energycascade forcing the shear scale to approach the dissipative scales. The anisotropy invariantmaps of the Reynolds stresses are finally used to quantify the different anisotropic statesof the large turbulent scales. The results confirm that the present flow poses a significantchallenge for turbulence modelling due to the existence of the recirculating bubble behindthe bump and the adverse pressure gradient region along the opposite wall.The paper is organised as follows: the dataset is presented in section § § ffect of geometry and Reynolds number on the separation behind a bulge Simulation
Re Re τ Re τ ∆x + ∆z + ∆y + max/min a A1 2500 300 160 2.8 2.8 3.7/0.5 0.15B1 2500 300 160 2.8 2.8 3.7/0.5 0.25C1 2500 300 160 2.8 2.8 3.7/0.5 0.50A2 5000 550 280 4.4 5.0 6.0/0.7 0.15A3 10000 900 550 6.5 7.0 9.5/0.9 0.15
Table 1: Simulation matrix. The nominal Reynolds numbers is Re = h U b /ν where h and U b are the half nominal channel height and the bulk velocity respectively. Re τ = hu τ /ν is the maximum friction Reynolds number taken at the bump tip with u τ = (cid:112) τ w /ρ thelocal shear velocity ( τ w is the local mean shear stress and ρ is the density), and h half thelocal channel height. The average friction Reynolds number is denoted with Re τ whereaverages are performed on both the upper and lower walls. ∆x + , ∆z + and ∆y + max/min are the spatial resolution in the streamwise, spanwise and wall-normal directions madedimensionless with the average wall-unit. The parameter a determines the different bumpgeometries, see text.Figure 1: Sketch of the different bump geometries given by y = − a ( x − + 0 . a is reported in table 1 and localisation of the different stations where statistics areaddressed.which is divided into several subsections illustrating different topics, i.e. instantaneousflow fields, Reynolds stress tensor, budget of mean and turbulent kinetic energies andanisotropy analysis, including anisotropy invariant maps. The last section §
2. Simulations
Five different simulations, whose parameters are summarised in table 1, have beencarried out. Simulations A1, B1 and C1 have the same Reynolds number but differentbump geometries, going from the most streamlined (A1) to the most bluff (C1) profile, seefigure 1. A sketch of the whole three dimensional domain is shown in figure 3. SimulationsA2 and A3 have the same bump geometry as simulation A1 but are performed at higherReynolds numbers. Table 1 lists the nominal Reynolds number Re , the maximum frictionReynolds number Re τ on the bump, the friction Reynolds number averaged on the topand bottom walls Re τ and the grid spacing in all directions. The grid is uniform in thestreamwise and spanwise directions, and stretched in the wall normal direction to clustergrid nodes toward the walls, see table 1. The nominal Reynolds, Re = h U b /ν , is defined J.-P. Mollicone, F. Battista, P. Gualtieri and C.M. Casciola
Figure 2: Kolmogorov scale with respect to the local grid spacing, πη/∆ , for simulationA3, where ∆ = √ ∆x∆y∆z .in terms of half the channel height, h , of the bulk velocity, U b , and of the kinematicviscosity, ν . The friction Reynolds number is Re τ = hu τ /ν , with u τ = (cid:112) τ w /ρ the shearvelocity ( τ w is the local mean shear stress and ρ is the density), and h half the localchannel height. DNS is employed to solve the incompressible Navier-Stokes equations, ∂u i ∂t + u j ∂u i ∂x j = − ∂p∂x i + 1 Re ∂ u i ∂x j ∂u i ∂x i = 0 , (2.1)where u i is the i th velocity component, and p is the hydrodynamic pressure. Henceforthall length scales are made dimensionless with the nominal channel half-height, h , timescales with h /U b and pressures with ρU b .The simulations are carried out using Nek5000, see Fischer et al. (2008), which isan open-source code that can simulate unsteady incompressible and low Mach numberflows. The discretisation is based on the Spectral Element Method (SEM), see Patera(1984), whose formulation allows for Direct Numerical Simulations. Highly accuratenumerical approaches for the simulation of wall bounded turbulent flows are crucialsince it is desirable that the numerical error does not contaminate the multi-scale non-linear interactions. This feature is fulfilled by the SEM approach, which reconciles thehigh accuracy, typical of a spectral method, and the flexibility (in terms of geometricalconfiguration), typical of finite element approaches.The grid spacing in the wall normal direction y at the centre of the domain and atthe walls is given by ∆y + max and ∆y + min , respectively. The superscript + denoteswall units referred to the average friction Reynolds number. The uniform grid spacingin the streamwise x and spanwise direction z in inner units is denoted by ∆x + and ∆z + respectively. These values, reported in table 1, are well within the grid resolutionsuggested by Kim et al. (1987) for well resolved DNS of wall bounded turbulent flows.Comparison of the local grid spacing, ∆ = √ ∆x∆y∆z , with the local Kolmogorov scale, η = (cid:0) ν /(cid:15) T (cid:1) / where (cid:15) T is the turbulent kinetic energy dissipation rate, is shown infigure 2, in particular the quantity πη/∆ is reported for the highest Reynolds numbercase. The resolution requirement for a classical spectral method is k max η = πη/∆ > πη/∆ ranges between 1 in the recirculation bubble and almost 3 inthe bulk of the flow, values adequate for the high fidelity reconstruction of the smallscale dynamics of the flow, given the accurate dispersion characteristics of the spectralelement method.The bump profile on the lower wall is generated using the equation y = − a ( x − +0 . a is reported in table 1 for each simulation. The mesh for the lowerReynolds number case contains approximately 120 million grid points and the simulationwas run on 8192 cores, using approximately 6 million core hours. The mesh for thehigher Reynolds number case was run with approximately 400 million grid points on32768 cores using approximately 30 million core hours. All simulations were run witha spectral element order N = 9 except for the high Reynolds number simulation (A3) ffect of geometry and Reynolds number on the separation behind a bulge x , and spanwise, z , direction.No slip and impermeability are enforced on the top and bottom walls. (a) (b) (c) Figure 4: Temporal auto-correlation of velocity and pressure signals, for simulation A1,probed just after the bump at x = 5 . x = 24 in panel (b). Panel(c) shows the spatial correlation of axial velocity fluctuations in the spanwise directionat x = 5 . x = 24. All panels refer to the same distance from the upper planarwall, d = 2 − y = 1 . τ is normalised with h /U b .which was run at a spectral element order N = 11. The reason for changing the spectralorder is purely technical, motivated by the need of optimising the machine performanceat changing dimensions of the simulation. All simulations were run on the FERMI BlueGene/Q Tier0 system at the CINECA supercomputer centre in Bologna, Italy.The geometry is shown in figure 3. The domain has dimensions ( L x × L y × L z ) =(26 × × π ) to avoid flow confinement at high Reynolds number, see Lozano-Dur´an &Jim´enez (2014) for similar issues in the context of planar channel flows. In the pictures,the flow is from left to right in the x direction with periodic boundary conditions inboth the x and z directions. No slip and zero normal velocity boundary conditions areimposed at the top and bottom walls. Accounting for periodicity, the actual geometryconsists of an infinite channel with periodic bumps in the streamwise direction that arespaced by approximately 44 bump heights. The distance between consecutive bumps isenough to allow flow reattachment and to minimise the effects that the separation behindthe fore bump may have on the aft bump. In this way, the use of inflow conditions, eithersynthetic or provided by companion channel simulations, is avoided. The flow is sustainedby an overall pressure drop ∆p ( t ) in the x direction that is modulated in time to keepthe same constant flow rate for all simulations.Approximately 500 statistically uncorrelated fields, separated by a time interval of ∆t stat = 6, were collected for each simulation in order to obtain properly convergingstatistics. ∆t stat is normalised with h /U b . Defining the “flow-through time”, t ft , as thetime needed for a turbulent structure to travel all along the channel length, see K¨ahler et al. (2016), the simulation time is T tot = 3000 (cid:39) t ft , which makes sure that thevelocity statistics converge (K¨ahler et al. J.-P. Mollicone, F. Battista, P. Gualtieri and C.M. Casciola (a) (b) (c)
Figure 5: Plot of mean stream-wise (x-direction) velocity normalised with frictionvelocity, (cid:104) u x (cid:105) + = (cid:104) u x (cid:105) /u τ , against y + at x = 24 for simulations A1, A2 and A3 inpanels (a), (b) and (c), respectively. The top and bottom wall velocities are representedby the blue and red lines, respectively. The dashed black line is the theoretical prediction, (cid:104) u x (cid:105) + = y + , in the viscous sub-layer. The solid black line is the theoretical prediction, (cid:104) u x (cid:105) + = 1 /k log ( y + ) + A , in the log-layer region with k = 0 .
41 and A = 5 . temporal auto-correlation of velocity, R ii ( x, y, τ ) = lim T →∞ πT u i (cid:90) T (cid:90) π u (cid:48) i ( x, y, z, t ) u (cid:48) i ( x, y, z, t + τ ) dzdt = (cid:104) u (cid:48) i ( x, y, z, t ) u (cid:48) i ( x, y, z, t + τ ) (cid:105) u i , no sum , and pressure signals, R pp ( x, y, τ ) = (cid:104) p (cid:48) ( x, y, z, t ) p (cid:48) ( x, y, z, t + τ ) (cid:105) /p , probed at differentlocations inside the domain with the local root mean square fluctuation defined tonormalise to one the correlation at zero time separation, i.e. R ii ( x, y,
0) = 1. Angularbrackets indicate averages over the homogeneous coordinates, z and t , while a primeindicates the fluctuation with respect to the local mean value. Some probes are locatedjust beyond the bump (panel (a) of figure 4) and some others in the fully reattached flow(panel (b) of figure 4). The correlations confirm that fields separated by ∆t stat are uncor-related. For the same points, the spatial correlation of streamwise velocity fluctuations, R xx ( x, y, ζ ) = (cid:104) u (cid:48) x ( x, y, z, t ) u (cid:48) x ( x, y, z + ζ, t ) (cid:105) /u x , in the spanwise direction, z , is shownin panel (c) of figure 4. The solid line refers to the correlation just beyond the bumpwhilst the dashed line refers to the correlation in the fully reattached flow. The spatialseparation is normalised with (nominal) wall units ζ + = ζ Re τ . In the reattached flowregion, the minimum correlation occurs at ζ + (cid:39)
100 assuring that the spanwise lengthis suitable to avoid confinement effects on the turbulent structures. Close to the bump,the correlation minimum occurs at ζ + (cid:39)
80. The inset in the figure reports the samequantities as a function of the spanwise separation normalised with the external unit.Figure 5 shows plots of dimensionless mean stream-wise (x-direction) velocity, (cid:104) u x (cid:105) + = (cid:104) u x (cid:105) /u τ , against y + = y Re τ at x = 24 averaged in the top half of the channel (blue)and in the bottom half (red), for simulations A1, A2 and A3. At this station, the flowalmost entirely recovers the structure of a canonical turbulent channel flow. The plotsshow a well resolved viscous sub-layer, the buffer layer and the expected log-law region.The theoretical prediction for the viscous region close to the wall is represented by thedashed black line, (cid:104) u x (cid:105) + = y + . The symbols in the plots denote actual data points,showing the high resolution achieved in the simulation. The solid black line representsthe log-law, (cid:104) u x (cid:105) + = 1 /k log y + + A . The figure shows that, both at the bottom and topwall, this law is approached by the present data with better accuracy as the Reynolds ffect of geometry and Reynolds number on the separation behind a bulge (A1)(A2)(A3)(B1)(C1) Figure 6: Instantaneous stream-wise velocity contour plots in x-y planes for all fivesimulations.number is increased, see the caption of the figure for the values of Von-Karman constantand intercept which are in agreement with those found in, e.g., Nagib & Chauhan (2008);Marusic et al. (2013).
3. Results
Instantaneous flow fields
Figure 6 shows instantaneous contour plots of streamwise velocity in the x-y planefor all the simulations. As expected, investing the bump, the flow velocity increases atthe channel restriction and separates behind the bump with the formation of an intenseshear layer between the bulk flow and the separation bubble close to the bottom wall.With increasing Reynolds number, cases A1, A2 and A3 progressively, structures atsmaller scales appear. The separated region behind the bump becomes smaller, moreattached to the bump and less protruded in the streamwise direction. It is interesting tocompare, at least qualitatively, the flow snapshots for cases A1 and A3 with the smokepatterns used to visualise the boundary layer on a convex wall as shown in pg. 91 of theclassical album by Van Dyke (1982). Indeed, in case A3 the flow impinging the bumpis clearly characterised by small scale structures while case A1, even though nominallyturbulent, appears smoother. Under this respect, case A3 can be regarded as producinga turbulent boundary layer between the external turbulent stream and the wall able tobetter withstand the adverse pressure gradients before separation. The separated regionin front of the bump is also smaller at high Reynolds number. On the other hand, theseparated region behind the bump becomes larger as the bump becomes bluffer. At thetop wall, the boundary layer thickens after the constriction due to the adverse pressure0
J.-P. Mollicone, F. Battista, P. Gualtieri and C.M. Casciola (a)(b)(c)(d)(e)(f)
Figure 7: Instantaneous stream-wise velocity contour plots in x-z planes at y + = 15 inpanels (a) and (b), y = 0 . y + = 15 (from the top wall) inpanels (e) and (f). Simulation A1 in panels (a), (c), (e) and simulation A3 in panels (b),(d) and (f). ffect of geometry and Reynolds number on the separation behind a bulge y + = 15. This wall distance corresponds tothe classical buffer layer of the channel flow. The empty region in the plot representsthe intersection of the plane with the bump. The recirculation region behind the bumpis characterised, at this distance from the wall, by negative velocity and small-scalestructures. The size of the region where reverse flow occurs decreases when increasing theReynolds number. Downstream, the small scale structures elongate in the flow directionand resemble the streaky structures found in turbulent planar channel flow. The xz-planesfor simulations A1 and A3, in panels (c) and (d), respectively, just touch the top of thebump which is indicated by the continuous zero velocity line at x = 4 in the contourplot. The acceleration of the flow just before the bump and its deceleration just behindcan be observed. The small scale structures far away from the bump increase their lengthdownstream. For simulation A3, the flow structures generated in the shear layer appearconfined in a smaller region since the flow is more attached to the bump’s surface andthe separated region protrudes less in the streamwise direction. Panels (e) and (f) showthe xz-planes close to the top wall, at y + = 15. The effect of the bump on the velocityis still present and a clear velocity increase is seen at the bump’s location, due to thecross-section restriction. This is followed by a low velocity region corresponding to theend of the separation bubble at the opposite wall. In this region, the turbulent structuresmaintain a streamlined, streaky shape and no separated region is present.3.2. Mean velocity and turbulent fluctuations
In the statistical analysis to follow, the average of a generic quantity q is indicated withthe angular brackets, (cid:104) q (cid:105) , or with the capital letter, Q , according to convenience, whilethe fluctuation is indicated with the apex, q (cid:48) . Details of the recirculation bubble for allthe simulations are shown in figure 8, providing the contour plot of the mean streamwisevelocity (cid:104) u x (cid:105) normalised with the bulk velocity. The black solid line highlights the zeroisolines to better appreciate the mean flow reversal.By progressively restricting the section, the first part of the bump makes the flowvelocity increase consistently with the pressure decrease which is mechanically responsiblefor the acceleration. After the top of the bump the flow decelerates and a strong adversepressure gradient occurs. This produces a backward flow near the bottom wall, givingrise to flow detachment. The bubble becomes smaller and more attached to the bumpstarting from the lower Reynolds number (A1) to the higher Reynolds number (A3). Theprofiles recuperate positive average velocity at the bottom wall behind the bump at anearlier x-position for simulation A3 compared to simulation A2 or A1. Concerning theeffect of the geometry, the bubble becomes larger starting from the streamlined bump insimulation A1 to the more bluff geometry in simulation C1. However, the mean positionof the reattachment point in the streamwise direction is basically independent of thebump width, at x = 7 for all three geometries at Re = 2500, while it clearly depends onthe Reynolds number.The above discussion is confirmed in detail by considering the mean velocity profilesextracted from figure 8 and reported in figures 9 and 10 at the downstream positions2 J.-P. Mollicone, F. Battista, P. Gualtieri and C.M. Casciola (A1)(A2)(A3)(B1)(C1)
Figure 8: Average streamwise velocity (cid:104) u x (cid:105) contour plots with isoline at (cid:104) u x (cid:105) = 0 for allfive simulations.shown in figure 1. Simulations A1, B1 and C1 are represented by solid, dashed and dashed-dotted blue lines respectively whilst simulations A2 and A3 are represented by solid greenand red lines respectively. Figure 9 shows mean streamwise velocity profiles for all fivesimulations. For the station shown in panel (a), the profiles are almost independent of theReynolds number (solid lines) while they depend strongly on the geometry (broken lines),suggesting that the mean flow has already achieved a sort of asymptotic state. The profilefor case C1 extends down to the bottom wall with a slightly negative velocity, indicatinga small recirculating region ahead of the bump. At the tip of the bump, panel (b), all theprofiles essentially exhibit the same behaviour. The recirculation is already well developedat the station of panel (c) for the bluffest bulge, case C1. Further downstream, panel (d),the wall-normal extension of the backward flow are well evident for all cases except forthe highest Reynolds number (case A3), where the recirculation is more attached to thewall and extends less streamwise. Since the extension of the bubble is larger for the lowerReynolds number cases, it is still present at the station corresponding to panel (e). In ffect of geometry and Reynolds number on the separation behind a bulge (a) (b) (c)(d) (e) (f) Figure 9: Mean axial velocity, (cid:104) u x (cid:105) , at six positions (a) to (f) corresponding to thestations in figure 1. Simulations A1, B1 and C1 are represented by solid, dashed anddashed-dotted blue lines respectively. Simulations A2 and A3 are represented by solidgreen and red lines respectively.contrast, reattachment already occurred for the higher Reynolds number (cases A2-A3).Further downstream, panel (f), the flow is attached for all conditions.Figure 10 shows the mean wall-normal velocity profiles. Wall-normal velocities areparticularly intense as the flow invests the bump, panel (a). At the tip of the bump,panel (b), the positive (away from the bottom wall) wall-normal velocity peak is higherfor the less streamlined bump (C1), indicating that the flow is strongly converging towardsthe opposite wall, leading to a contraction of the effective section ( vena contracta ). Thewall-normal velocity is progressively reduced downstream, to eventually become negative.At intermediate stations, see e.g. panel (c), the wall-normal velocity is negative in theouter flow, indicating the trend toward reattachment to the lower wall. Approaching thewall, (cid:104) u y (cid:105) becomes zero at the edge of the recirculation bubble. Inside the bubble (cid:104) u y (cid:105) ispositive, indicating that the profile is traversing the fore part of the bubble, recirculatingclockwise. Moving further downstream, the external flow still moves toward the lowerwall, but now the aft part of the bubble is reached, implying a negative wall-normalvelocity also inside the bubble. Finally, the reattachment point is reached and the wall-normal average velocity tends to vanish in the entire channel section, starting to recovernearly parallel-flow conditions expected far away from the bump. The shorter bubblelength leads to a more abrupt reattachment, as seen by the large negative wall-normalaverage velocity in the red plot of panel (c). The bluffest configuration induces an evidentsecondary recirculation bubble just below the primary one where the bump ends, see alsofigure 8. Note that in this figure a small bubble is also apparent just ahead of the bump.The next figures present the mean profiles for the fluctuating quantities, i.e. (cid:104) u (cid:48) x (cid:105) , (cid:104) u (cid:48) y (cid:105) and (cid:104) u (cid:48) x u (cid:48) y (cid:105) , at the same stations addressed above for the mean velocity profiles. Figure 11shows mean streamwise velocity fluctuation profiles (cid:104) u (cid:48) x (cid:105) for all five simulations. Theseare particularly strong close to the walls or inside the shear layer above the recirculating4 J.-P. Mollicone, F. Battista, P. Gualtieri and C.M. Casciola (a) (b) (c)(d) (e) (f)
Figure 10: Mean wall-normal velocity, (cid:104) u y (cid:105) , at six positions (a) to (f) corresponding tothe stations in figure 1. Simulations A1, B1 and C1 are represented by solid, dashed anddashed-dotted blue lines respectively. Simulations A2 and A3 are represented by solidgreen and red lines respectively. (a) (b) (c)(d) (e) (f) Figure 11: Mean streamwise velocity fluctuations, (cid:104) u (cid:48) x (cid:105) , at six positions (a) to (f)corresponding to the stations in figure 1. Simulations A1, B1 and C1 are representedby solid, dashed and dashed-dotted blue lines respectively. Simulations A2 and A3 arerepresented by solid green and red lines respectively. ffect of geometry and Reynolds number on the separation behind a bulge (a) (b) (c)(d) (e) (f) Figure 12: Mean wall-normal velocity fluctuations, (cid:104) u (cid:48) y (cid:105) , at six positions (a) to (f)corresponding to the stations in figure 1. Simulations A1, B1 and C1 are representedby solid, dashed and dashed-dotted blue lines respectively. Simulations A2 and A3 arerepresented by solid green and red lines respectively. (a) (b) (c)(d) (e) (f) Figure 13: Reynolds stress, (cid:104) u (cid:48) x u (cid:48) y (cid:105) , at six positions (a) to (f) corresponding to the stationsin figure 1. Simulations A1, B1 and C1 are represented by solid, dashed and dashed-dottedblue lines respectively. Simulations A2 and A3 are represented by solid green and redlines respectively.6 J.-P. Mollicone, F. Battista, P. Gualtieri and C.M. Casciola
Figure 14: Dependence of the reattachment point normalised with the height of theobstacle x r /h b , on the Reynolds number based on h b , Re h b = Re h b . h b is the bumpheight normalised with the nominal channel half-height, h . Closed black symbols referto experimental measurements, open black symbols to numerical simulations (mainlyLES). Red dashed line is the exponential fit of the experimental data K¨ahler et al. (2016). Open red symbols are extracted from the present DNSs. Breuer et al. (2009),K¨ahler et al. (2016), Diosady & Murman (2014), Fr¨ohlich et al. (2005), Hickel et al. (2008), Mellen et al. (2000), Peller & Manhart (2006), Rapp & Manhart (2011), ˇSari´c et al. (2007), Temmerman et al. (2003)region, for all cases. The fluctuations are maximum in the high Reynolds number case(A3) and in the most bluff case (C1). In the former, the high velocity fluctuations aredue to the higher Reynolds number, whilst in the latter, the fluctuations are fed bythe strong flow separation induced by the bump shape. In the lower Reynolds numbersimulations, the regions with higher velocity fluctuations are larger, corresponding to alarger shear layer. The presence of the bump also results in an increase in fluctuations atthe top wall, particularly evident for the lower Reynolds number simulations. The changein geometry also affects the maxima reached by the fluctuations. In particular, due tothe increased size of the recirculating region, simulation C1 exhibits peak values whichare comparable to simulation A3 in the shear layer, see panel (c). In correspondence ofthe recirculating region, the peaks are higher for case C1 with respect to the cases at thesame Reynolds number, B1 and A1. The profiles for the wall-normal velocity fluctuations (cid:104) u (cid:48) y (cid:105) and the Reynolds stress, (cid:104) u (cid:48) x u (cid:48) y (cid:105) , follow a similar trend. The profiles are shown infigure 12 and figure 13 for all five simulations. The positive values of Reynolds stressconfirm the presence of the small recirculation bubble ahead of the bump, panel (a) infigure 13.The position of the reattachment point clearly does not depend on the dimension of theobstacle in the streamwise direction, but on the Reynolds number. These observationsare in agreement with both numerical and experimental data already available in theliterature. Figure 14 shows the reattachment point normalised with the height of thebump, x r /h b as a function of the Reynolds number based on the bulk velocity and on theheight of the bump, Re h b = Re h b (note that h b is the dimensionless bump height). Datahave been collected in K¨ahler et al. (2016) from several experiments (closed symbols) ffect of geometry and Reynolds number on the separation behind a bulge . x/h b for large Reynolds numbers.The reattachment location scales with Re h b to the power − .
4. The reattachment positionmoves further upstream with increasing Reynolds number due to the stronger turbulentmixing, i.e. due to a higher turbulent momentum transfer towards the wall. The presentDNS data agree well with the experimental results of K¨ahler et al. (2016) and of Rapp& Manhart (2011). Overall, this compilation of data shows a significant scatter of dataobtained by turbulence modeling and a certain inability of the models to capture thebubble reattachment position, see also the discussion in K¨ahler et al. (2016) for moredetails. Concerning the present DNS, the slight differences with the experiments can beattributed to details of the turbulence investing the bump and the confinement effect ofthe upper wall. 3.3.
Pressure, drag and friction coefficients
The instantaneous pressure is decomposed as the sum of a contribution linearlydecreasing in the streamwise direction, which is associated to the instantaneous pressuredrop ∆p ( t ) across the channel, plus a departure ˜ p from the linear law, p ( x , t ) = − ∆pL x x + ˜ p ( x , t ) . In the present simulations ∆p ( t ) does not significantly fluctuate in time, oscillating within1% at most, although its value is in principle continuously adjusted to keep the flow raterigorously constant. The drag coefficient in terms of the present dimensionless variablesis C d = 4 ∆PL x = − L x (cid:90) walls (cid:104) t (cid:105) · e x dl , where ∆P = (cid:104) ∆p (cid:105) is the average pressure drop, t is the (dimensionless) traction at thewall (pressure plus shear force) and e x is the unit vector in the streamwise direction.Wherever needed (see e.g. § ∆p (cid:48) ( t ).The drag increases moving from the most streamlined (A1) to the most bluff profile(C1), see table 2. On the other hand, the drag coefficient decreases with the increase inReynolds number, from simulation A1, A2 to A3. For purpose of comparison, the drag ina planar channel at the same flow rate is C channeld = 4 (cid:0) Re τ / Re b (cid:1) . The drag coefficientcan be decomposed in form and friction components, namely C formd = 2 L x e x · (cid:90) walls P n dl , C frictiond = − L x e x · (cid:90) walls µ ∂ U ∂n dl , where n is the unit normal exiting the fluid domain. The form drag coefficient increasesby 50% going from the most streamlined to most bluff shape. The observed decrease ofthe form drag coefficient with increasing Reynolds number is more than compensatedby the larger velocity for flows in the same geometry implying, as obvious, the increaseof the corresponding contribution to the resistance, D form ∝ U b C formd . In the presentgeometry, the friction component is dominated by the straight part of the channel and itsvalue is not significantly different from the one expected in a planar channel, see the small8 J.-P. Mollicone, F. Battista, P. Gualtieri and C.M. Casciola
Case C d C formd C channeld C frictiond − C channeld A . · − . · − . · − . · − B . · − . · − . · − . · − C . · − . · − . · − . · − A . · − . · − . · − . · − A . · − . · − . · − . · − Table 2: Drag coefficient decomposed into form and friction contributions andcomparison against an equivalent planar channel. See text for definitions.difference C frictiond − C channeld in table 2. This confirms that most of the bump-induceddrag should be interpreted as form drag.Given its relevance in determining the drag, the mean pressure field is shown in figure 15for all simulations. Roughly, the qualitative behaviour is similar for all cases. A highpressure region occurs just before the bump where a stagnation point occurs. Furtherdownstream the pressure decreases reaching its minimum in the high velocity region,at the top of the bump. The pressure field behind the bump is strongly influencedby the shape and dimension of the recirculation bubble. When the separation regionis significant, cases A1, B1 and C1, a second pressure minimum develops inside therecirculation bubble. The high Reynolds number simulations where performed on themost streamlined geometry, which produces a smaller separation compared to the othergeometries. With increasing Reynolds number, cases A2 and A3, the flow more easilyfaces the adverse pressure gradient due to enhanced turbulent mixing, resulting in delayedseparation. As a consequence, pressure recovery behind the bump is more effective.Given the average pressure drop along the channel, the effect of the bump on the wallpressure is better addressed in terms of a departure-pressure coefficient C t/bp ( x ) = 2 (cid:104) ˜ p (cid:105)| y =2 /y =0 , where the superscripts ‘ t ’ and ‘ b ’ refer to the top and bottom wall respectively. Addingthe linear term − x∆P/L x recovers the standard definition of C p . Figure 16 showsthe departure-pressure at both walls. The effect of the bump extends to the oppositewall, with C tp presenting a trough just after the bump tip (at x = 4) and recoveringdownstream. The trough is smaller and shifted closer to the bump tip with increasingReynolds number. At constant Reynolds number, the trough is more pronounced andfarther away from the tip when the geometry is bluffer, i.e. simulations B1 and C1. Atthe bottom wall, as the flow reaches the bump, C bp initially increases producing a smallrecirculation just ahead of the bump. The pressure then decreases to its minimum slightlyahead of the tip. These trends become stronger for the bluffer geometries. Immediatelydownstream of the tip, the pressure abruptly rises reaching the separation point. Inthe recirculation bubble the pressure remains almost constant, with the extension ofthe plateaux becoming smaller at increasing Reynolds number (red line). The extensionof the plateaux is almost independent of the geometry, where C bp decreases for blufferbumps.Figure 17 shows the mean skin friction coefficient C f = 2 τ w . At the top wall, C f isalways positive, showing that the adverse pressure gradient (see figure 16) is too mildto induce average flow separation on the wall opposite to the bump. The maximum and ffect of geometry and Reynolds number on the separation behind a bulge (A1)(A2)(A3)(B1)(C1) Figure 15: Average pressure (cid:104) p (cid:105) contour plots for all five simulations.minimum C f occur just before the bump tip and at the end of the bubble, respectively.At the bottom wall, the skin friction coefficient ahead of the bump becomes slightlynegative due to the small recirculation bubble and reaches a positive peak as the flowapproaches the tip of the bump. For the different Reynolds numbers, the position ofthe peak coincides but the maximum reduces with increasing Reynolds number. Thepeak is shifted towards the tip with increasing bluffness of geometry. A skin frictionplateaux is observed behind the tip, along the bump where the cross-stream section ofthe channel increases. At the bubble, consistently with the backward flow at the wall, theskin friction is negative, with increasing absolute value for bluffer geometries and lowerReynolds numbers.3.4. Mean kinetic energy and turbulent kinetic energy budgets
The Reynolds decomposition entails the splitting of the total kinetic energy in twoparts, K = K M + k T , where K M = 1 / (cid:104) u i (cid:105)(cid:104) u i (cid:105) is the kinetic energy of the meanflow and k T = 1 / (cid:104) u (cid:48) i u (cid:48) i (cid:105) is the turbulent kinetic energy. In literature, little attention istypically paid to the kinetic energy of the mean field and most interest is focused on the0 J.-P. Mollicone, F. Battista, P. Gualtieri and C.M. Casciola
Figure 16: Modified pressure coefficient at the top and bottom walls of the domain,top and bottom panels respectively. Simulations A1, B1 and C1 are represented by solid,dashed and dashed-dotted blue lines respectively. Simulations A2 and A3 are representedby solid green and red lines respectively.Figure 17: Mean skin friction coefficient along x at the top and bottom walls of the domainin the top and bottom panels respectively. Simulations A1, B1 and C1 are representedby solid, dashed and dashed-dotted blue lines respectively. Simulations A2 and A3 arerepresented by solid green and red lines respectively. ffect of geometry and Reynolds number on the separation behind a bulge K M is trivial. In the present case both mean and turbulentkinetic energy need to be dealt with explicitly. The reason is that the bump breaks thestreamwise homogeneity and induces strong mean wall-normal velocities. This gives riseto non-trivial mean and turbulent spatial energy fluxes, dissipation, and turbulent kineticenergy production.3.4.1. Mean kinetic energy
The (stationary) mean flow kinetic energy equation reads ∂Φ M j ∂x j = − ε M − Π + ∆PL x U x , (3.1)where ε M = 1 / Re ( ∂U i /∂x j )( ∂U i /∂x j ) is the mean flow energy dissipation rate per unitvolume and Π = −(cid:104) u (cid:48) i u (cid:48) j (cid:105) ∂U i /∂x j is the turbulent kinetic energy production. U x ∆P /L x is the external average power input. The spatial flux, Φ M j = U j K M + U j (cid:104) ˜ p (cid:105) − Re ∂K M ∂x j + U i (cid:104) u (cid:48) i u (cid:48) j (cid:105) , (3.2)redistributes energy across the flow, overall providing zero net contribution to the power.Figure 18 shows the terms in equation (3.1) normalised with the total average powerinjection per unit volume, i.e. the total dissipation rate, (cid:82) ( ε M + ε T ) dV where ε T = (cid:104) / Re ( ∂u (cid:48) i /∂x j )( ∂u (cid:48) i /∂x j ) (cid:105) is the turbulent dissipation rate density. In the figure, theturbulent kinetic energy production rate, − Π , is shown in the background colour plotwhilst solid isolines (mostly concentrated near the bump wall) represent the mean fielddissipation, ε M . Vectors correspond to the spatial flux Φ Mj .Given the behaviour of the mean streamwise velocity, see figure 8, the mean energyinput, U x ∆P /L x , is largest at the bump tip. On the other hand, the production − Π ,is concentrated in the detaching shear layer well behind the bump, where the largestfluctuation intensities are attained, as discussed in the previous section. This region actsas a sink of mean energy and is fed by the mean energy flux, Φ M j , that is crucial inredistributing energy from the external input to the turbulent production. With respectto case A1, taken as a basis for comparison, the maximum turbulent production increasesby almost 50% for the bluffer geometry and by 400% at the maximum Reynolds number.By definition, turbulent production is the product of mean flow gradients and Reynoldsstresses. For the given geometry, the mean gradients in the shear layer slightly dependon Reynolds number, as shown in panel (c) of figure 9 which corresponds to the sectionof maximum production. This suggests that the mean field already attained an almostReynolds independent state. On the other hand, turbulent stresses increase significantlyat this section, see panel (c) of figure 13, resulting in the increased peak productionapparent in figure 18. In general, the position of the energy production region depends,through the shear layer, on the dimensions and the position of the separation bubbles.Changing geometry at fixed, lower Reynolds number, the strength of the mean gradientsin the same section, now in panel (d) of figure 9, are only marginally affected by thechange in geometry. On the other hand, the Reynolds stresses are greatly enhancedpassing from a streamlined to a bluff configuration, panel (d) in figure 13, consistentwith the increasing peak energy production from case A1 to case C1 in figure 18.Although hardly apparent in figure 18, for the considered cases the mean flow dissi-pation rate is not irrelevant, and contributes order 40% of the total dissipation in thesystem, consistently with significant mean velocity gradients, observed at the bump wallwhere the flow is abruptly accelerated.2
J.-P. Mollicone, F. Battista, P. Gualtieri and C.M. Casciola (A1)(A2)(A3)(B1)(C1)
Figure 18: Mean kinetic energy balance equation: turbulent kinetic energy production − Π (background colour), mean energy dissipation ε M (solid isolines) and mean energyspatial flux Φ M (vectors).3.4.2. Turbulent kinetic energy
The balance equation for the turbulent kinetic energy reads ∂Φ T j ∂x j = − ε T + Π + (cid:104) ∆p (cid:48) ( t ) L x u (cid:48) x (cid:105) , (3.3)where, as anticipated, ε T is the turbulent kinetic energy dissipation rate, Π , here withthe opposite sign with respect to eq. (3.1), is the production and (cid:104) ∆p (cid:48) u (cid:48) x /L x (cid:105) is theexternal source of fluctuating energy. The spatial flux, Φ T j = U j k T + 12 (cid:104) u (cid:48) i u (cid:48) i u (cid:48) j (cid:105) + (cid:104) ˜ p (cid:48) u (cid:48) j (cid:105) − Re ∂k T ∂x j , (3.4)contributes zero net power when integrated over the whole domain. The energy locallyprovided by the fluctuations of pressure difference between inlet and outlet (cid:104) ∆p (cid:48) u (cid:48) x /L x (cid:105) is negligible, max x,y (cid:104) ∆p (cid:48) u (cid:48) x /L x (cid:105) (cid:39) − max x,y Π . ffect of geometry and Reynolds number on the separation behind a bulge (A1)(A2)(A3)(B1)(C1) Figure 19: Turbulent kinetic energy balance equation: turbulent kinetic energy production Π (background colour), turbulent energy dissipation ε M (solid isolines) and turbulentenergy spatial flux Φ M (vectors).Figure 19 shows the turbulent kinetic energy production, turbulent energy dissipationand spatial fluxes for all the simulations. The terms are normalised with the overallpower injected in the system which, in the statistically steady state, is balanced by thetotal dissipation rate, (cid:82) ( ε M + ε T ) dV . The production term injects most energy in theshear layer behind the bump. From the shear layer the energy follows different paths, seethe vector field in the figure where local energy release is associated with the (positive)divergence of the energy flux. The turbulent energy is transferred towards the centre ofthe channel, into the separation bubble or towards the wall, in particular behind thebump under the separation bubble. From the analysis of the dissipation field, ε T , partof the energy is found to be locally dissipated in the shear layer and in the separationbubble. Most of the energy is dissipated at the bottom wall after the bump (note theisolines of dissipation concentrated in that region).4 J.-P. Mollicone, F. Battista, P. Gualtieri and C.M. Casciola (A1)(A2)(A3)(B1)(C1)
Figure 20: Norm of the deviatoric component of the Reynolds stress tensor, || b || = (cid:112) b ij b ij with b ij = (cid:104) u (cid:48) i u (cid:48) j (cid:105) / (cid:104) u (cid:48) k u (cid:48) k (cid:105) − / δ ij , and δ ij the components of the identity tensor.3.4.3. Large and small scale anisotropy
The anisotropy of the large turbulent scales is described by the deviatoric componentof the Reynolds stress, b ij = (cid:104) u (cid:48) i u (cid:48) j (cid:105) / (cid:104) u (cid:48) k u (cid:48) k (cid:105) − / δ ij , where δ ij denotes the Kroneckersymbol. Note that in isotropic conditions, b ij is identically zero. An overall measure ofanisotropy is given by the norm || b || = (cid:112) b ij b ij , figure 20. The anisotropy is particularlysignificant in the near wall region and in the shear layer, while the bulk flow and therecirculation bubble are almost isotropic, consistent with the Reynolds stress profiles offigures 11, 12 and 13. Increasing the Reynolds number, the anisotropic regions becomeprogressively smaller, squeezed closer to the wall, on one side, and more concentrated inthe shear layer, on the other. The extension of the isotropic region in the bulk widens,whilst it shrinks with the recirculation bubble in the separated region. As a result of thechange in geometry, the bluffest bump produces the highest anisotropic content.Figure 21 reports the norm, || d || = (cid:112) d ij d ij , of the deviatoric component, d ij = (cid:15) ij /(cid:15) kk − / δ ij , of the pseudo-dissipation tensor, (cid:15) ij = 2 / Re (cid:104) ( ∂u (cid:48) i /∂x k )( ∂u (cid:48) j /∂x k ) (cid:105) . || d || ffect of geometry and Reynolds number on the separation behind a bulge (A1)(A2)(A3)(B1)(C1) Figure 21: Norm of the deviatoric component of the pseudo dissipation tensor, || d || = (cid:112) d ij d ij , where d ij = (cid:15) ij /(cid:15) kk − / δ ij and (cid:15) ij = 2 / Re (cid:104) ( ∂u (cid:48) i /∂x k ) (cid:0) ∂u (cid:48) j /∂x k (cid:1) (cid:105) is the pseudodissipation tensor.provides a measure of the small scale anisotropy content (Antonia et al. et al. || d || approaches zero, isotropic behaviour of the smallestscales is achieved. The small scales in the recirculation bubble and in the bulk of the floware isotropic, consistently with the isotropy of the large scales in the same regions. Stronganisotropy persists in the near wall regions and in the shear layer. The behaviour of || d || isstrongly dependent on the Reynolds number which ultimately sets the separation betweenthe largest and the smallest scales. The regions of small-scale isotropy progressivelyincrease with the Reynolds number, basically due to the shrinking of the large scaleanisotropy regions. However, anisotropy still persists at small scales, irrespectively ofthe Reynolds number, near the walls and in the shear layer. This behaviour can beexplained and understood by addressing the dynamics of a turbulent flow in presenceof strong shear. In isotropic conditions the turbulence is forced at the largest scalescomparable with the integral scale L = (2 k T ) / /(cid:15) T and is dissipated by viscosity at6 J.-P. Mollicone, F. Battista, P. Gualtieri and C.M. Casciola the Kolmogorov scale η = ( ν /(cid:15) T ) / . In the inertial range ( η (cid:28) r (cid:28) L ) the energy issimply transferred from the large to the small scales. In turbulent shear flows the shearscale L S = (cid:112) (cid:15) T /S , extensively discussed in Casciola et al. (2003), where S is the shearrate, plays a crucial role in explaining the dynamics. Basically the shear scale identifiesthe range of scales L S < r < L where the turbulence is driven by the (anisotropic)production of turbulent kinetic energy due to the Reynolds stresses. In the range ofscales below L S , η < r < (cid:32)L S , the dynamics of the turbulent fluctuations are driven by theprocess of energy cascade typical of isotropic flows, see e.g. Marati et al. (2004); Cimarelli et al. (2013) for a detailed analysis of the energy paths in a planar channel. It follows thatthe dynamics of a shear flow is described by two dimensionless parameters. The first oneis the shear intensity S ∗ = S (2 k T ) /(cid:15) T , see e.g. Lee et al. (1990), that can be recast interms of the shear scale as S ∗ = ( L /L S ) / , (Casciola et al. S c = S (cid:112) ν/(cid:15) T (Corrsin 1958) that can berecast in terms of the shear scale as S c = ( η/L S ) / . The Corrsin parameter measures theextension of the range between the shear scale and the Kolmogorov scale where the flowis driven by the inertial cascade. Clearly, only in the range of scales below the shear scaleisotropisation of turbulent fluctuations can take place. The shear scale can be evaluatedin spatially non homogeneous flows by considering the norm of the local mean velocitygradient L S = (cid:113) (cid:15) T / ( ∂ j U i ∂ j U i ) / . In our case the shear scale is a field L S ( x, y ). In asimilar way the local integral scale L = (2 k T ) / /(cid:15) T and the local Kolmogorov scale η = ( ν /(cid:15) T ) / can be considered. The shear strength S ∗ and the Corrsin parameter S c are position dependent. When S ∗ is large the whole range of scales is dominatedby production and there is no room left for isotropy recovery at small scales. On thecontrary, an isotropy recovery range is available where the Corrsin parameter is small.Figures 22 and 23 provide the fields S ∗ and S c respectively, for the different Reynoldsnumbers and geometries considered in this paper. A joint analysis of S ∗ and S c providesthe physical interpretation of the observed anisotropy (figures 20 and 21). In the bulkregion, S ∗ decreases, denoting weak production of turbulent kinetic energy since theshear scale approaches the integral scale. Concurrently, S c is small, indicating a largeseparation between shear and Kolmogorov scale. This behaviour is generic and the onlyrelevant changes are observed when the Reynolds number is increased, A1-A3. At largeReynolds number, the spatial region in the bulk where isotropisation occurs is broadened.The relative position of integral, shear and Kolmogorov scales in the bulk explains whythe small scales are isotropic (figure 21). The conditions are different near the wall andin the shear layer. In these regions, see figure 22, S ∗ is large and the whole range of scalesis now dominated by turbulent kinetic energy production. Concurrently S c is order one,i.e. the shear scale is forced on the Kolmogorov scale (figure 23). This behaviour is againgeneric for the cases we address. In a nutshell, near the wall and in the shear layer thereis no room for the formation of the inertial range where isotropisation can take place.The flow is driven by the anisotropic mechanisms of turbulent kinetic energy production,see figure 20 and the anisotropy persists down to the smallest scales (figure 21).3.4.4. Invariant Maps
The Anisotropy Invariant Map (AIM) originally introduced by Lumley & Newman(1977); Lumley (1979) provides a description of the different anisotropic states of thelarge turbulent scales. They are quantified in terms of the invariants of the anisotropytensor, i.e. the deviatoric component of the Reynolds stress, b ij , namely I = b ii = 0, ffect of geometry and Reynolds number on the separation behind a bulge (A1)(A2)(A3)(B1)(C1) Figure 22: Shear intensity S ∗ = S (2 k T ) /(cid:15) T = ( L /L S ) / in the flow domain for allsimulations. II = b ij b ji and III = b ij b jk b ki . The admissible states of the flow must lie within a(curvilinear) triangle of the II − III plane. This constraint comes from the requirementthat the eigenvalues of b ij should be real and the squared velocity fluctuation in theprincipal direction must be positive. The admissible region is delimited above by theline II = 2 / III corresponding to statistically two-dimensional turbulence, i.e. thefluctuation intensity in one of the eigen-directions vanishes. The other two limiting lines, II = 3 / (cid:0) / III (cid:1) / , represent axisymmetric turbulence, i.e. the fluctuation intensityin two eigen-directions are identical. In the left branch ( III <
III > II = 0 , III = 0), the two-component isotropic state ( II = − / , III = 1 / II =2 / , III = 2 / et al. (2006); Kumar et al. (2009)and the general discussion in Jovanovic (2013) where II and III are used to compute8
J.-P. Mollicone, F. Battista, P. Gualtieri and C.M. Casciola (A1)(A2)(A3)(B1)(C1)
Figure 23: Corrsin parameter S c = S (cid:112) ν/(cid:15) T = ( η/L S ) / in the flow domain for allsimulations. (a) (b) (c) Figure 24: Anisotropy Invariant Map for case C1 at: (a) the tip of the bump; (b) end ofthe bump; (c) inside the recirculation region. Note that (a), (b) and (c) correspond tostations b), c) and d) of figure1. The color legend represents the y coordinate.the length scale appearing in the eddy-viscosity thus including anisotropic effects in themodel.Figure 24 shows the AIM for simulation C1 at three stations. At the tip of the bump,panel (a), close to the bump wall at y = 0 .
5, turbulence is essentially two-dimensional. As y increases, the points in the plot approach the lower left branch indicating axisymmetric ffect of geometry and Reynolds number on the separation behind a bulge y coordinate increases, the initial axisymmetric state is hardly reached and thetrajectory in the II − III plane follows an inner path towards the opposite axisymmetricstate (
III >
0) as the shear layer is crossed (light blue). At the centreline, the flow isagain isotropic and follows an inner path towards the top wall. These results are typical ofseparated flows as found, e.g., in the backward facing step configuration (Le et al. b ).Panel (c) shows the trajectory at a station closer to the reattachment point. Flow close tothe lower wall is completely axisymmetric ( III <
0) until the shear layer is reached (lightblue). The turbulence shifts from axisymmetric contraction to axisymmetric expansion,similar to panel (b). The flow is isotropic at the centreline and follows the same trenddiscussed for panel (a) and (b) as the top wall is reached. The analysis of the AIMsuggests that the flow we are addressing is rather complex to model, due to the presenceof the recirculating region behind the bump and the adverse pressure gradient along thetop wall.
4. Conclusions
Turbulent separation behind a bump in channel flow is addressed using Direct Numer-ical Simulations (DNS) for different bump geometries and for Reynolds number rangingbetween Re = 2500 and Re = 10000. The latter is probably the largest Reynolds numberever achieved in the DNS of this specific configuration, corresponding to a maximumfriction Reynolds number of approximately Re τ = 900.The separation behind the bump generates small scale structures which grow down-stream, an intense shear layer and a recirculating region after the bump. Althoughthe recirculation size depends on geometry, the reattachment position is constant. Thereattachment point is controlled by the Reynolds number based on the bump height, asconfirmed by the available experimental data, K¨ahler et al. (2016).With increasing Reynolds number, a net decrease in drag coefficient is observed inassociation with the reduced dimensionless pressure drop needed to maintain the flowrate constant. The reduction is overwhelmed by the increase in the dimensional velocity,quadratically entering the expression for the drag force, leading to the expected increasein flow resistance. The drag increase with respect to that of an equivalent planar channel isalmost entirely due to form effects induced by the separation, even though a significantincrease in velocity, hence in local wall shear stress, is measured at the bump tip. Atlarger Reynolds number, the shear layer separating the recirculation bubble from theouter stream becomes more attached to the lower wall. Its fluctuations correspond tohigher turbulent kinetic energy production peaks. The DNS captures a small recirculationoriginated by the sudden change in slope at the bump leading edge. The separation atthe bottom wall affects the opposite near-wall region by inducing a significant adversepressure gradient which is not sufficient to separate the flow at the upper wall.Due to the strong non-homogeneity and the resulting mean gradients, the mean flowdraws energy from the local external energy source, namely the pressure drop multipliedby velocity. The uptake mostly occurs in the bulk. Fluxes move this energy to the shearlayer where it is partially dissipated but mostly intercepted by the production term to0 J.-P. Mollicone, F. Battista, P. Gualtieri and C.M. Casciola sustain the turbulent fluctuations. The dissipation in the mean flow is significant, giventhe strong mean gradients present at the walls and in the shear layer. Overall, the mostimportant feature is the peak of turbulent kinetic energy production localised in theshear layer. The path taken by this energy bifurcates, in part sustaining the turbulentfluctuations inside the bubble and in part feeding the turbulence of the external flowdownstream of the bubble. From the most streamlined to the bluffest bump, a 50%increase in peak energy production is observed. For fixed geometry, a fourfold Reynoldsnumber change leads to approximately 400% increase in peak production.In turbulence modelling, the level of anisotropy at both the large and small scales iscrucial. They can be characterised in terms of the deviatoric components of Reynoldsstress and pseudo-dissipation tensor, respectively. Apart from the near wall region,anisotropy at both large and small scales concentrates in the shear layer, irrespectiveof bump shape and Reynolds number. Interestingly, the small scales keep a high levelof anisotropy in the shear layer, even at larger Reynolds number. This is due to theintensity of the mean gradients which maintain the production active close to dissipationscales. Finally, the analysis of the anisotropy invariant maps shows that the separatedflow poses a significant difficulty for turbulence modelling due to the recirculating regionbehind the bump and the adverse pressure gradient along the top wall.The research has received funding from the European Research Council under theERC Grant Agreement no. 339446. We acknowledge PRACE for awarding us access tosupercomputing resource FERMI based in Bologna, Italy through PRACE project no.2014112647.
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