Numerical Simulations of Low-Reynolds-Number Flow Past ...

Numerical Simulations of Low-Reynolds-Number Flow Past FiniteWings with Leading-Edge Protuberances

A. Esmaeili,∗ H. E. C. Delgado,∗ and J. M. M. Sousa†

Instituto Superior Técnico, 1049-001 Lisbon, Portugal

DOI: 10.2514/1.C034591

The use of various modeling approaches for the numerical simulation of low-Reynolds-number flow past wings of

finite span with leading-edge protuberances is studied at pre- and poststall operating conditions. Variants of

Reynolds-averaged Navier–Stokes simulations (including transitional modeling) and detached-eddy simulations

(involving different options for shielding the boundary layer from the detached-eddy simulation limiter), as well as a

low-Reynolds-number correction, are considered. An assessment of the capabilities exhibited by these modeling

approaches is carried out, based on detailed comparisons against whole-flowfield and aerodynamic force

measurements in awind tunnel, at aReynolds number ofRe � 1.4 × 105 forwingswith aspect ratios of 1 and 1.5. Theresults show that the use of detached-eddy simulation with an improved description of the boundary layer is required

to reproduce the flow physics accurately at prestall conditions, where separation and subsequent reattachment are

observed. At poststall regimes, massive flow separation occurs, and a choice of conservative shielding produces the

best outcome with detached-eddy simulation. In addition, the present experimental dataset provides valuable

information toward understanding the passive stall controlmechanismgenerated by the use of leading-edge tubercles

and its application to low-aspect-ratio wings.

I. Introduction

L EADING-EDGE protuberances have been extensively studiedduring the last two decades as a passive means to achieve some

sort of control over aerodynamic stall. The idea was inspired byobservations in the 1990s [1–3] of the peculiar design exhibited bythe pectoral flippers of humpback whales, described as winglikegeometries with a high aspect ratio and rounded tubercles distributedalong their leading edges. It was then conjectured that suchmorphology resulted from an adaption for high maneuverability, andhence suggested that the scalloped appearance of these surfaceswas adirect consequence of the natural development of some kind ofenhanced lift mechanism. Altogether, the leading-edge protuber-ances were thought to have the primary function of generatingstreamwise vortices intended to maintaining lift and preventing stallat high incidence [4].The aforementioned observations and hypothesis subsequently

triggered a series of experimental investigations aimed at thesimulation of the effects of leading-edge tubercles on idealizedwhale flippers. Miklosovic et al. [5,6] performed wind-tunnel tests,employing both semi- and full-span models at Reynolds numbers(Re ≈ 3–6 × 105) claimed to be well within the typical operatingrange of those mammals. They concluded that, although theaerodynamicmechanisms associated to scalloped leading edgesweresimilar in both infinite and finitewings, therewas a substantial benefitfrom three-dimensional effects. By coupling flow visualizationwith force measurements, Johari et al. [7] examined the influenceof parametric characteristics (amplitude and wavelength) of thetubercles (modeled using a sinusoidal function) in quasi-two-dimensional experiments conducted at a somewhat lower Reynoldsnumber (Re � 1.8 × 105) in a water-tunnel facility. These authorsreported the larger significance of the amplitude of the protuberanceson the performance of the airfoils, and they noted the occurrence ofpremature flow separation at the troughs of the sinusoids; yet, they

also acknowledged marked differences with respect to the behaviorof three-dimensional models previously tested by others. In theirstudies carried out at even lowerReynoldsnumbers (Re ≈ 5 × 103 forhydrogen-bubble flow visualization and Re � 1.2 × 105 for forcemeasurements in awind tunnel), Hansen et al. [8] further investigatedthe impact of the geometric features of the tubercles for two21-percent-thick NACA airfoils with different aerodynamiccharacteristics. They observed a dependence of lift performance inpre- and poststall regimes on the airfoil geometry and conjecturedthat leading-edge protuberances behaved in a similar fashion tocounter-rotating vortex generators, but they were unsurprisinglyunable tooffer any contribution regarding finitewing effects.However,by explicitly targetingmicro air vehicle (MAV) applications,Guerreiroand Sousa [9] specifically addressed the latter issue via wind-tunneltesting of wings with small aspect ratios ( =1 and 1.5) at lowReynolds numbers (Re � 7 × 104 and 1.4 × 105). Their forcemeasurement analysis demonstrated that the use of sinusoidal leadingedges was still effective as a passive stall control technique inwings offinite span with an aspect ratio as low as 1.5, although they weredependent on a proper choice of the amplitude and wavelength of thesinusoids. Except for the work of Custodio et al. [10], where(nevertheless) onlymodelswith ≥4 were studied, later experimentalinvestigations [11–14] have been merely restricted to infinite wings,therefore curtailing their potential for practical applications.The scenario is not at all radically different as to what concerns

numerical simulations exploring the concept of scalloped leading edgesto improve aerodynamic performance. Pedro and Kobayashi [15]employed detached-eddy simulations (DESs) based on the Spalart–Allmaras (SA) model to scrutinize a few operating conditions of thehigh-aspect-ratio idealized flipper in [5,6]. The selected approachseemed to produce reasonable agreement between computations andavailable experimental data, with the advantage of granting easy accessto some of the intricacies of the three-dimensional separated flow. Adeeper examination of the foregoing set of experiments was laterconducted by Weber et al. [16], making use of the same turbulencemodel but giving preference to a Reynolds-averaged Navier–Stokes(RANS) formulation instead, on the grounds of the extremecomputational cost ofDESas comparedwithRANS;however,whereasacceptable lift predictions were still obtained in nonstall regimes, theseauthors struggled todealwith the effects of detachedvortices at poststallconditions. Malipeddi et al. [17] also chose the DES-SA approach[interchanged with the variant based on the shear stress transport (SST)k − ωmodel at high angles of attack] to perform simulations of infinitewing configurations, yet they did not confront their results withexperimental data. Conversely, Dropkin et al. [18] retrieved the use of

Received 24 May 2017; accepted for publication 30 August 2017;published online 6October 2017.Copyright© 2017 by theAmerican Instituteof Aeronautics and Astronautics, Inc. All rights reserved. All requests forcopying and permission to reprint should be submitted to CCC atwww.copyright.com; employ the ISSN 0021-8669 (print) or 1533-3868(online) to initiate your request. See also AIAA Rights and Permissionswww.aiaa.org/randp.

*Graduate Student, IDMEC, Universidade de Lisboa, Department ofMechanical Engineering, Avenida Rovisco Pais.

†Associate Professor, IDMEC, Universidade de Lisboa, Department ofMechanical Engineering, Avenida Rovisco Pais. Associate Fellow AIAA.

Article in Advance / 1

JOURNAL OF AIRCRAFT

RANS-SA to carryout broadlysuccessful computations formanyof theairfoil experiments published earlier in [7], but significant discrepancieswere still observed with respect to wind-tunnel data, especially at andpast the stall point. As in the latter two investigations, subsequentcomputational studies only contemplated infinite wing geometries,with the sole exception of the work by Câmara and Sousa [19] andDelgado et al. [20], who resorted once again to DES formulations withthe objective of simulating pre- and poststall operating conditions forthe low-aspect-ratio finitewings reported in [9].Thoseother researchersconsidered additional numerical approaches, as well as a variety ofReynolds numbers and regimes, and also faced different degrees ofsuccess.Namely, direct numerical simulations (DNSs)were adopted byGross and Fasel [21] atRe � 6.4 × 104, as well as by Serson et al. [22]at Re � 1000; large-eddy simulations (LESs) were the choiceof Skillen et al. [23], as well as Pérez-Torró and Kim [24], atRe � 1.2 × 105 in both cases. Such diversity of methods andcircumstances kept the eyes of numericists on the problem but did littleto clarify the advantages and shortcomings of thevariousmethodologiesfor practical applications. Nevertheless, all of the aforementioned effortsdefinitely contributed to advancing our knowledge on the mattersassociated with the use of leading-edge protuberances in aerodynamics,as was comprehensively discussed in a recent review [25].As wisely put by Weber et al. [16], not all predictions have been

found to be entirely accurate when compared with experiments;additionally, the desired accuracy, the complexity of the flowfield (i.e.,stalled vs nonstalled regimes), and the computational resourcesrequired must all be taken into account when deciding the numericalmethod to use. In the present study, whole-flowfield and aerodynamicforce measurements in a wind tunnel are employed with the goal of

assessing the suitability of RANS and DES approaches for studies ofthe effect of leading-edge tubercles on the performance of finitewings.The operating Reynolds number and the low aspect ratio of the liftingsurfaces have been chosen as congruent with those found in typicalfixed-wing MAVapplications.

II. Wind-Tunnel Testing

A. Wing Models

Rectangular wing models were computer numerical control(CNC) machined from duralumin blocks and hand polished, thusachieving a surface finish quality of less than 1 μm rms in roughnessheight, as described in detail in [9]. Briefly, the section of baseline(i.e., straight leading edge) wing models was designed to match theNASA LS(1)–0417 airfoil, whereas this profile was modified alongthe (spanwise) z direction to generate tubercle wing models with asinusoidal leading edge using the following expression:

x � c� A sin

�2π

�z

λ−

λ

2c

��(1)

where A and λ denote the amplitude and the wavelength of theprotuberances, and x is the chordwise length as illustrated in Fig. 1a. Allbaseline and modified wings exhibit the same value of the mean chordc � 232 mm, hence sharing a common planform area if their aspectratio (either =1or 1.5) is also the same. The values prescribed for theaforementioned geometrical parameters resulted from an earlieroptimization studywhere a larger set was analyzed [9], viz.,A∕c�0.12and λ∕c�0.5 in nondimensional form. The protuberances in themodified wings smoothly blend into the baseline profile, shown inFig. 1b by a dashed line. Past the chordwise location of maximumthickness,modified andbaseline cross sections remain indistinguishablealong the whole span. It must be noted that, taking into account the lowaspect ratios used, the configuration of the wingtips must not bedisregarded. However, in order to avoid the introduction of thesupplementary parameters required for the exact geometric definition ofround caps, flat (sharp) edges at the wingtips were preferred.

B. Experimental Apparatus

All experiments were carried out at the low-speed open-circuit windtunnel of the Department of Mechanical Engineering at InstitutoSuperior Técnico. The test section of thiswind tunnel has a (rectangular)cross-sectional area of 1.35 × 0.8 m2, where freestream velocities up toU∞ � 10 m∕s may be achieved with a turbulence intensity of about0.3% in the potential core of the open jet. More details concerning thefull characteristics of this facility were given in [9]. Data acquisition, aswell as adjustment of the angle of attack and freestream velocity [so thata constant Reynolds number Re � 1.4 × 105 (based on U∞ and c)could be maintained throughout the whole test campaign], wasaccomplished via the in-house-developed software AeroIST. The

Fig. 1 Geometrical definition of tubercle leading-edge wing models ina) planform view and b) side view (the dashed line indicates the baselinewing section).

Fig. 2 Schematic diagram of the instrumentation in the wind tunnel.

2 Article in Advance / ESMAEILI, DELGADO, AND SOUSA

instrumentation, schematically depicted in Fig. 2, included continuousmonitorization of the air temperature and operating dynamic pressure,

together with a custom-made six-component Schenck compact balancefor themeasurement of aerodynamic forces (andmoments) employing a

single-strut model support connected to the load cells.A stereo particle image velocimetry (SPIV) system by Dantec

Dynamics further allowed conducting whole-field measurements ofthe flow around the wing models in the wind tunnel [20]. Its basic

characteristics are as follows:1) A DualPower 200-15 yttrium aluminium garnet (YAG) laser

by Litron, operated as a frequency of 10 Hz with pulse energy of2 × 190 mJ at 532 nm; the laser pulse separation was typically set to0.1–0.2 ms, following optimization of the image correlation inviews from both cameras; the thickness of the light sheet wasapproximately 3 mm.2) Two FlowSense 4M digital cameras with a resolution of

2048 × 2048 pixels, equipped with Macro Zeiss 50 mm objectives.3) Data analysis was performed by DynamicStudio software,

version 4.1, typically using 32 × 32-pixel interrogation windowswith an image overlap of 50%; two levels of multipass processingwere applied with the aim of extending the dynamic range of themeasurements.With the objective of maximizing the field of view when the test

modelswere positioned atmoderate to high incidence, the setup shown

in Fig. 3 was arranged. Accordingly, one camera was installedperpendicularly to the laser light, which in turn was at a right angle tothe planform of the wings, whereas the other camera was tilted by an

angle of 34 deg. In-plane velocity measurements were made using thefirst camera only, although the combined use of both cameras was

required in tests where the measurement of the out-of-plane velocitycomponent was also desired. Adequate camera mounts and a fully

computer-controlled four-axis translation system by Isel Automationmade it possible to keep the system aligned and focused in the

Scheimpflug condition, even during a spanwise field survey. Flowseeding was provided upstream of the wind-tunnel plenum by a1500Wcommercial smokegenerator with aDigitalMultiplex control.

C. Experimental Uncertainty

The sensors used in themeasurement of the angleof attackof thewingmodels are the absolute angle encoders, indicated in Fig. 2.According tothe manufacturer information, rotations are measured by way ofreductiongearing,with a total resolutionof about 16million steps and anangular error of 10−3 rad. However, a larger uncertainty is associatedwith the installation of the models in the strut support, thus estimated tobeof theorder of�0.2 deg. Theposition uncertainty, resulting from thejoint use of the automated linear motion systems sketched in Fig. 3, isestimated to be �0.2 mm. The low tolerances involved in the use ofCNC machining to fabricate the wing models allowed us to minimizesize errors with respect to nominal dimensions; a postproduction surveydemonstrated that the resulting relative differences were altogetherlimited to less than�0.4%.As indicated inFig. 2, the operating dynamicpressurewasmeasured

by employing a micromanometer (model FC012 from FurnessControls Limited) with a maximum error of 0.02 mm of water columnwithin the tested range. Negligible errors in the air properties resultedfrom the temperature measurements, thus leading to an uncertaintyestimate in the freestream velocity of �0.02 m∕s. Together with theaforementioned uncertainty inmodel chord length, this value gave riseto a maximum uncertainty in the Reynolds number below 1%.The procedure outlined by Coleman and Steel [26] was used to

determine the bias uncertainties affecting the force coefficients foroverall 95% confidence limits, yielding maximum uncertainties of�5% in lift and drag, although thesemay be twice as large in the lattercoefficient for low values of incidence and aspect ratio [9]. Wind-tunnel corrections were also applied to the measured forces, takinginto account weight tares, load component interactions, and open jetoperation [27]; and a dedicated study provided additional correctionsto the experimental data regarding model–mount interactions.It must be further mentioned that the accuracy of the SPIV

measurements is influenced by many factors, namely, the particleimage density, the magnitude of in-plane and out-of-plane motions,as well as that of the spatial gradients present in the flow. All thesefactors have been taken into account during the design of theexperiments, and the system parameters have been optimized duringthe tests as a function of the operating conditions. Reported time-averaged velocity fields were typically obtained by ensembleaveraging of 100 independent images. By following the principlesdiscussed by Raffel et al. [28], uncertainties of �4 and �7% wereestimated for in-plane and out-of-plane SPIV velocities, respectively.

III. Numerical Simulations

A. Computational Mesh

The computational meshes used in the present simulations of flowpast thewings of finite span previously describedwere generated fromthe originalCNCmilling files employing a structuredC-H topology, asillustrated in Fig. 4. Far-field boundaries were set at a distance of

°

Fig. 3 Experimental setup used with the SPIV system.

Fig. 4 Computational meshes used in the simulations of flow past the wings of finite span with sinusoidal leading edges for a) =1 and b) =1.5.

Article in Advance / ESMAEILI, DELGADO, AND SOUSA 3

approximately 20c around the wings, except in the spanwise directionwhere a distance of 13c from the wingtips was prescribed instead. Inaddition, a simple open boundary condition was used at the outflowsection [29], imposed at a distance of 12c downstream of the trailingedge of thewings, and no-slip conditions were applied at all solid wallsurfaces. The limited size of the numerical domainwas established as atradeoff between computational effort and solution quality.Generating a structured mesh for such complex geometries

involving curved surfaces was a challenging task, especially for thewings with leading-edge protuberances. However, the alternative useof an unstructured mesh has proven to be a less accurate option whenwall boundary layersmust be resolved.Away out of the dilemmawasnevertheless found, which consisted of splitting the computationaldomain into four (smaller) volumes, and thus providing a bettercontrol over the density of a structured mesh. The first two volumeswere generated for the C contour encircling thewings, from upstreamto downstream, where mesh cells were adequately clustered near thesolid walls [30], using a geometric expansion, in order to resolve theviscous boundary layers. The remaining two volumes were locatedon the left and right sides of the finite span wings. Although it was atime-consuming procedure to mesh multiple volumes, all of thesewere meshed individually so that the desired cell density could beobtained in each region of the domain. In addition, aiming to ensurethe quality of themesh, the skewness and aspect ratio of the cellswerealso controlled, thus minimizing the occurrence of extreme values ofthese parameters throughout the whole computational domain. Forexample, the presence of leading-edge protuberances increased thecomplexity of the meshing procedure for the hexahedral elementsnear the surface of the wings; hence, to avoid inverted cell volumesand high skewness, the least possible distance of the first cell nearestthe solid walls was applied, while preserving a maximum value ofthis wall-normal length expressed in wall units close to unity, as abasic requirement for the DES approach [31].The samestrategywas followed for both the sinusoidal leading-edge

wings and the baseline wings, irrespective of their aspect ratio.Ultimately, the total number of mesh cells used was 10.3 million and13.5 million, respectively, for the finite wings with =1 and 1.5. Theuse of either DNS or LES in numerical simulations to be carried out atthe experimental value of the Reynolds number would still requirefurther refinement of the meshes. Due to limited computationalresources, attempts exploiting any of the foregoing approaches werediscarded, as two different values of aspect ratio and various operatingconditions of the finite span wings are to be investigated in this work.

B. Governing Equations and Turbulence Modeling

The equations governing the flow past the finite wings are theunsteady Navier–Stokes equations, expressing conservation of massand momentum for an incompressible fluid. In the (unsteady) RANSapproach, nonturbulent unsteadiness is resolved in the mean flow viafinite-time ensemble averaging, whereas a closure model is used todescribe the turbulent fluctuations. Among the plethora of availablechoices, the SST k − ω turbulence model [32] has been used in thepresent study because of its superior performance in the prediction ofadverse pressure gradients; other reasons are discussed in thefollowing in the context of DES. This two-equation model comprisesadditional transport equations for the turbulent quantities k and ω,and it makes use of an eddy viscosity μt as follows:

∂�ρk�∂t

� ∇ ⋅ �ρUk� � ∇ ⋅ ��μ� σkμt�∇k� � Pk −ρ

�����k3

p

L(2)

∂�ρω�∂t

�∇ ⋅ �ρUω� �∇ ⋅ ��μ� σωμt�∇ω� � 2�1−F1�ρσω2

∇k ⋅ ∇ωω

� αρ

μtPk − βρω2 (3)

μt � ρk

ω ⋅max�1∕α; F2S∕a1ω�(4)

where U stands for the velocity vector, ρ and μ are properties of thefluid, Pk denotes the turbulence production, and S is the strain rateinvariant. Because turbulence models are generally designed for highReynolds numbers, a low-Reynolds-number correction (LRC) isincluded inEq. (4). This effect is embodied in the damping coefficientα, which is derived based on asymptotic consistency, as described indetail in [32]. In addition, the so-called SST blending functions F1

and F2, equaling one inside the boundary layer and falling rapidly tozero at its edge, read as follows:

F1 � tanh�arg41�;arg1 � min

�max

� ��k

pCμωdw

; 500μρωd2w

�; 4ρσω2

kDkωd

2w

�;

Dkω � max

�2ρσω2

∇k⋅∇ωω ; 10−10

�

(5)

F2 � tanh�arg22�; arg2 � max

�2

���k

p

Cμωdw;500μ

ρωd2w

�(6)

where dw indicates the distance to the nearest solid wall. The valuesof the model constants such as a1 and Cμ in Eqs. (2–6), where F1

blending has been applied for the determination of σk, σω, α, and β,were as given by Gritskevich et al. [33].The same set of governing equations was also used in this work for

the DES approach, thus switching explicitly between the RANS andthe LESmodel formulations based on the local mesh spacing and theturbulent length scale L appearing in Eq. (2), which is calculated foreach case of the foregoing cases from the following expressions:

LRANS ����k

p

Cμω(7)

LLES � CDEShmax; CDES � CDES1⋅ F1 � CDES2

⋅ �1 − F1� (8)

where hmax corresponds to the maximum edge length of the meshcell. TheRANSmode is activated in near-wall regionswhere the ratioLRANS∕LLES < 1, thus retaining the lower computational cost of thiskind of simulation when the boundary layer must be resolved,together with the ability of LES to capture the features of themassively separated flow occurring at high incidence. However,switching from the RANS to the LES formulation inside wallboundary layers should be avoided to prevent the undesirable effectsof an imbalance between the consequent reduction in eddy viscosityand the contribution from resolved turbulent contents. The blendingfunctions F1 and F2 readily available in the SST turbulence modeloffer straightforward options to shield the boundary layer from theDES limiter. The relatively conservative choice proposed by Menterand Kuntz [34], and denoted here by DES-F2, achieves this purposeby modifying the aforementioned length scale ratio to the product(1 − F2) LRANS∕LLES. Inspired by the same concept, a more generic(though still empiric) shielding function fd was later suggested bySpalart et al. [35], leading to a formulation termed as “delayed”DES(with the acronym DDES), and is considered here in the followingform:

LDDES � LRANS − fd max�0; LRANS − LLES� (9)

fd � 1 − tanh��20rd�3�; rd � μ

ρd2wκ2

�����������������������������1∕2�S2 � Ω2�

p (10)

where κ is the vonKármán constant,Ω stands for themagnitude of thevorticity tensor, and μ is given by the sum of molecular and eddyviscosities. The constants appearing in Eqs. (8–10) were also chosenas reported in [33]. In contrast to DES-F2, the present DDESapproach results in the earliest onset of the LES mode occurring justoutside the boundary layer, thus avoiding what may be deemed asexcessive shielding.

4 Article in Advance / ESMAEILI, DELGADO, AND SOUSA

The unintentional use of (nonshielded) DES as awall-modeled LESmodel is problematic because it generally produces a significantmismatch in the logarithmic layer, between the inner RANS and theouter LES regions.Nevertheless,Shur et al. [36] remedied this issueviatheir proposal of an “improved” DDES (with the acronym IDDES),which involves the implementation of an elaborate series of blendingand shielding functions. In the case of IDDES, the turbulent lengthscale L in Eq. (1) is calculated here from the following expression:

LIDDES � ~fd�1� fe�LRANS � �1 − ~fd�LLES (11)

with the empiric blending function given in Eq. (10) reformulated asfollows:

~fd�max��1−fd�;fb�; fb�min�2e−9τ2 ;1�; τ�1

4−

dwhmax

(12)

and a so-called elevating function, reading as follows:

fe � fe2 max��fe1 − 1�; 0�;

fe1 ��2e−11.09τ

2

; τ ≥ 0

2e−9τ2

; τ < 0;

fe2 � 1 −max�ft; fl�;ft � tanh��1.87rdt�3�;fl � tanh��5rdl�10�

(13)

where rdt and rdl arevariants of the quantity rd inEq. (10), obtained byalternatively making μ equal to the eddy viscosity and the molecularviscosity, respectively. Additionally, LLES in Eq. (11) is reformulatedwith respect to its previous definition in Eq. (8), using a different LESlength scale in the case of IDDES, by the following:

LLES � CDESΔ; Δ � min�0.15max�dw; hmax�; hmax� (14)

All the these improvements to the standard DES approachconsidered in this study, although particularly relevant concerning thequality of the simulations, have a minor impact on the correspondingcomputational cost. However, the same observation does not apply tothe enhanced version of RANS, employed here as well, and developedfor transitional flow. Following the modeling procedure proposedby Langtry and Menter [37], the SST k − ω turbulence model iscoupled with two additional transport equations, respectively, for theintermittency γ and the transition onsetmomentum-thicknessReynoldsnumber Reθt. The transport equation for the intermittency reads asfollows:

∂�ργ�∂t

�∇ ⋅ �ρUγ� � ∇ ⋅ ��μ� μt�∇γ� � Pγ − Eγ (15)

whereEγ is a destruction/relaminarization source, andPγ describes thetransition sources, which are controlled by empirical correlationsestablished for the length of the transition region and the transition onsetthat is based on a critical value of the local Reynolds number whereintermittency first starts to increase in the boundary layer. Both the latterparameter and the lengthof the transition region are, in turn, functions ofthe transition onset momentum-thickness Reynolds number, which iscalculated as a transported scalar using the following equation:

∂�ρReθt�∂t

� ∇ ⋅ �ρUReθt� � ∇ ⋅ �2�μ� μt�∇Reθt � � Pθt (16)

where the source term Pθt is fundamentally driven by the differencebetween the transported quantity and its local value as calculated froman additional empirical correlation. However, in order to turn off thissource term in the boundary layer, thus allowing the transported scalar todiffuse in fromthe freestream, ablending function is also employedhere.The interaction of the transition model with the SST turbulence modeltakes place via modification of the original terms in Eq. (2) standing fortheproductionanddestruction (the last term in this equation)of turbulent

kinetic energy using an “effective” intermittency that also takes intoaccount separation-induced transition. For a more comprehensivedescription of all the steps and definitions in this modeling approach,denoted in the present study by T-RANS, the reader is referred againto [37].

C. Numerical Solution and Procedure

The numerical procedure employed a SIMPLE (which stands forsemi-implicit method for pressure linked equations) pressure–velocity coupling and a second-order-accurate spatial discretizationfor the pressure. In addition, the quadratic interpolation formula ofthe QUICK (which stands for quadratic upwind interpolation forconvective kinematics) scheme [38] was used in the finite volumediscretization of the transport equations. However, the timeintegration was performed here by employing a second-order-accurate implicit method to alleviate numerical stability restrictions.The time step used in every simulation reported herein was 0.0025 s,although this value has been halved at a preliminary stage with theobjective of confirming time-step independence.The same values of wing chord and operating Reynolds number

used in the experiments were also specified while setting up thenumerical simulations, corresponding in this case to a freestreamvelocity of 8.815 m∕s. Calculations conducted at different angles ofattack were accomplished by adjusting the vector components of theforegoing velocity. Different numericalmesheswere employed in theanalysis of finite wings with distinct values of the aspect ratio, butthose meshes were kept unchanged, irrespective of the turbulencemodel applied. Hence, the computational cost of DES and RANSsimulations did not differ significantly. On the other hand, as aconsequence of the additional transport equations to be solved inT-RANS, this approachwas approximately 30%more expensive thanthe former ones. Often, the simulations had to be carried out longerthan 40 s in order to reach a statistically converged unsteady stateunaffected by initial transients. Time-averaged results were typicallycomputed from 400 instantaneous realizations of the flow, coveringabout 10 s of simulation time.

IV. Results and Discussion

A. Overview of the Effects of the Leading-Edge Modification andWing Aspect Ratio

The overall effects of the modification of the leading edge of thefinite wings can be gauged based on the changes produced in theevolution of the lift coefficientCL as a function of the angle of attack.This is depicted in Fig. 5 for the investigated values of the aspect ratio

Fig. 5 Measured lift coefficients for baseline andmodifiedwings of finitespan, as a function of the angle of attack, illustrating the effect of theaspect ratio at Re � 1.4 × 105.

Article in Advance / ESMAEILI, DELGADO, AND SOUSA 5

at Re � 1.4 × 105. A significant benefit is obtained by the use of the

modified geometry with =1.5, resulting in a milder aerodynamic

stall as well as substantially higher lift generated at the poststall

regime.However, the reduction by 3 deg in thevalue of the stall angle,

with respect to that observed for the baseline geometry, must not be

overlooked.

As may be anticipated, and will be discussed later in further detail,

the velocity field generated by the leading-edge protuberances is

strongly variable along the spanwise direction, especially at high

incidence. Nevertheless, a simple comparison between the (in-plane)

flow patterns measured at the midspan plane (z � 3∕4c) of the

baseline and modified wings with =1.5 and α � 25 deg already

provides ample justification for the aforementioned differences in

poststall lift. As shown in Fig. 6, where the time-averaged flow is

represented by streamlines as well as contours of the streamwise

velocity component (normalized by the freestream velocity), both the

size and the strength of the flow separation observed at the foregoing

spanwise location have been considerably reduced for the modified

geometry.

To understand the earlier occurrence of stall in the case of the

modified wing with =1.5, one must first take notice of the

associated decrease in the lift slope, at prestall conditions, observed in

Fig. 5. Once again, time-averaged (in-plane) flow patterns measured

for both the baseline andmodified geometries are compared, but now

withα � 15 deg. These are first shown in Fig. 7 at themidspan plane

(z � 3∕4c), i.e., at the same spanwise location portrayed in Fig. 6.

However, the minor differences between the two flow maps at this

location do not offer an irrefutable explanation for themeasured force

data in Fig. 5. The fundamental reasons behind the smaller lift

generated by awing with a tubercle leading edge, at prestall angles of

attack, can only be clarified when the (in-plane) time-averaged flow

at one of the two adjacent trough sections (e.g., z � 1∕2c) is alsodisplayed. At this location, the presence of a long, although limited in

span (cf. Sec. IV.C), separation bubble is evidenced in Fig. 8 for the

modified geometry, whereas the boundary layer remains fully

attached over the baseline wing. This bounded flow separation at the

troughs for a moderate wing incidence has been described in earlier

experimental [6,7] and numerical [18,19] investigations, irrespective

of the aspect ratio.

The same flow feature was observed as well for a wing with =1,thus explaining the similar loss in lift at prestall conditions seen in

Fig. 5 when a tubercle leading edgewas also used with a lower aspect

ratio. This matter will be revisited in the paper, showing that, in this

case, a separation bubble of identical characteristics is formed at

themidspan trough (z � 1∕2c) instead.Oncemore in reference to the

data in Fig. 5, the lift slope (in the linear part of the prestall region) is,

not surprisingly, smaller with =1 than with =1.5 due to strongerfinitewing effects in the former, but themost striking consequence of

varying the aspect ratio is revealed in the poststall regime. In contrast

to the baseline wing with =1.5, the occurrence of stall is hardly

perceived for its counterpart with unity aspect ratio. The stronger

global impact of the tip vortices in the latter case, energizing a much

larger fraction of the flow over the wing, has been suggested earlier

[9,39] as an explanation for the near absence of stall within the

investigated range of the angle of attack. This is illustrated in Fig. 9,

showing the measured (in-plane) time-averaged flow pattern past the

baseline wing with =1 and α � 25 deg at two different spanwisesections only 1/4-chord apart. It can be seen that, whereas the

Fig. 6 SPIV measurements of flow over the wings with =1.5 and α � 25 deg, at z � 3∕4c, for a) baseline, and b) modified geometries.

Fig. 7 SPIV measurements of flow over the wings with =1.5 and α � 15 deg, at z � 3∕4c, for a) baseline, and b) modified geometries.

6 Article in Advance / ESMAEILI, DELGADO, AND SOUSA

established recirculation occupies a wide area over the upper side ofthe wing at the midspan plane (z � 1∕2c), with high reversed flowvelocities, both its size and strength have been drastically reduced,with reattachment taking place considerably before of the leadingedge in the plane located at z � 1∕4c. As a result, the influence of thetubercle leading edge is much less felt in this case than with =1.5;still, a change to a monotonic evolution of the lift coefficient with theangle of attack was achieved for the modified wing with unity aspectratio, as seen in Fig. 5.It has been suggested that this passive stall control mechanism is

related to the generation by the turbercles of a net downwash at thespanwise regions of the peaks [16], hence producing a smaller effectiveangle of attack there. The principal agents of this process are likely tobe (downstream travelling) “hairpin-like” vortices, first identified inthe numerical simulations carried out by Câmara and Sousa [19] asemanating from the vicinity of low-pressure zones occurring at thetroughs. Although the accuracy of quantities requiring the evaluationof spatial derivatives of SPIV velocity data is generally modest, areconstruction of the three-dimensional vortical field past themodifiedwing with =1 (to minimize the amount of measurements involved)and α � 15 deg has been performed based on 13 planar flow mapsdistributed along one half-span (assuming flow symmetry on the timeaverage [40]). Naturally, the unsteady characteristics of the mentionedvortices cannot be reproduced by the present time-averaged field, yetthe signature of these highly coherent flow structures can still beascertained from Fig. 10. In this representation, and to facilitate theirvisualization, contours of the spanwise component of vorticity areshown using solid lines, whereas (gray) filled contours have been usedto depict the streamwise component in planar cuts at two differentstreamwise locations. Both cuts correspond to y–z planes: the first

located at x∕c � 0.7 (i.e., 0.18c aft of the leading-edge trough; notingthe wing reference frame in Fig. 1), and the second at x∕c � 0.3(i.e., 0.4c downstream of the previous one). As expected, a sheet of

spanwisevorticity (solid lines) is attached to the surface of thewingdue

to the development of a boundary layer, but this sheet is seen lifting up

from the wall in the region directly downstream of the trough as a

consequence of the upwash induced there by a counter-rotating pair of

streamwisevortices (filled contours). In turn, these are also responsible

for the generation of a downwash (arrows in Fig. 10) in the port and

starboard sections of the wing adjacent to the central region that

ultimately contributes to counteracting boundary-layer separation.

Fig. 8 SPIV measurements of flow over the wings with =1.5 and α � 15 deg, at z � 1∕2c, for a) baseline, and b) modified geometries.

Fig. 9 SPIV measurements of flow over the baseline wing with =1 and α � 25 deg, at a) z � 1∕2c, and b) z � 1∕4c.

Fig. 10 Signature of vortical structures in flow past the modified wingwith =1 and α � 15 deg, from SPIV measurements.

Article in Advance / ESMAEILI, DELGADO, AND SOUSA 7

B. Preliminary Simulations at Low Angle of Attack

At low incidence, and for the investigated Reynolds number, the

boundary-layer flow over both the original and modified wings

remains completely attached to the upper surface, irrespective of the

value of the aspect ratio [9]. This is illustrated here in Fig. 11, showing

the time-averaged (in-plane) flow at the midspan section (z � 1∕2c)of the scallopedwingwith =1andα � 0 deg, as obtained from the

SPIV measurements. The chosen spanwise location crosses the

trough region, where the most intense streamwise pressure gradients

may be expected to develop. However, at Re � 1.4 × 105 and

α � 0 deg, those pressure gradients do not seem to be sufficiently

strong to cause boundary-layer separation anywhere in the uppersurface.The flow case depicted in Fig. 11 is used in this study for a

preliminary assessment of the performance of the turbulence modelsused in the simulations. The ability of correctly reproducing anexperimental operating condition characterized by fully attachedflow on the wing can be seen as a minimum prerequisite for themodeling. Provided that themesh resolution near thewall is adequate(cf. Sec. III.A), previous studies have demonstrated that RANSsimulations are accurate enough at low angles of attack [16,18]. Onthe other hand, concerning the DES approach, questions arise aboutwhether the correction for low-Reynolds-number effects, or theblending improvement to avoid the mismatch in the logarithmiclayer, is indeed required for a suitable representation of the flowphysics at a relatively simple state. Aiming to clarify the foregoingissues, Fig. 12 portrays the results obtained from numericalsimulations corresponding to the same scenario depicted in Fig. 11,employing RANS, DDES, DDES with LRC, and IDDES. The mostimportant conclusion from this test is that the LRC should not be usedin this study togetherwithDDESbecause it leads to the formation of alarge separated flow region at the trailing edge of the wing. Thisphenomenon was noted in the flow visualizations of [9] at half thevalue of the present Reynolds number only. The remaining modelingapproaches represented in Fig. 12 generated flowfields exhibitinggood agreement with their experimental counterpart. A slightly moreaccurate prediction of the extent of the region of accelerated flowoverthe upper surface seems to be obtained with IDDES, hinting that theassociated improvement in the description of the boundary-layerstructure may be especially important in the presence of attached(or reattached) flow.

C. Simulations at Prestall Operating Conditions

As noted in Sec. IV.A, at moderate incidences before theoccurrence of stall, the flow over the upper surface of the modifiedwings does not remain fully attached anymore. A long separationbubble appearing in the region(s) directly downstream of the trough

Fig. 11 SPIV measurements of flow over the modified wing with =1and α � 0 deg, at z � 1∕2c.

Fig. 12 Numerical results of flow over the modified wing with =1 and α � 0 deg, at z � 1∕2c, using a) RANS, b) DDES, c) DDES with LRC, andd) IDDES.

8 Article in Advance / ESMAEILI, DELGADO, AND SOUSA

(s) is themain flow feature, which is common to both values of aspect

ratio, at prestall operating conditions. This is illustrated here in

Fig. 13, using the three-dimensional reconstruction of the time-

averaged flow fromSPIV data, for =1 (cf. Sec. IV.A), to visualize aclosed zone of reversed streamwise velocity (shown in dark gray)

protruding from the wing surface past the central trough. Additional

flow separation seems to take place past the reattachment point as

the trailing edge is approached. However, the magnitude of the

streamwisevelocities in the immediate vicinity of thewall is too small

to be accurately quantifiedwith the present setup of the SPIV system.

Two cut planes across the span of the wing are also shown in Fig. 13,

displaying views of the (in-plane) time-averagedvelocity field above the

upper surface, at approximately the midlength of the separation bubble,

and just before the trailing edge. The development of the powerful tip

vortices along the streamwise direction is readily recognized, but these

are still incipient in the region where the aforementioned bubble is

formed. Based on this observation, it is reasonable to assume that the

fundamental characteristics of the (trough) separation bubble at this

angle of attack are mostly unaffected by the tip vortices. For this reason,

and taking into account the associated savings in computational time, the

capabilities of the turbulence modeling for the numerical simulations at

the prestall regime are carried out in this study at the unity aspect ratio

as well.

In accordance with the preliminary tests conducted in the previous

section, the RANS, DDES (without LRC), and IDDES modeling

approacheswere considered for the subsequent numerical investigation.

Furthermore, in an attempt to improve significantly the RANS

predictions at these operating conditions, the set of models was

completed with the T-RANS variant. This idea was justified by the

hypothesis that, although flow separation past the trough takes place in

the laminar flow regime, reattachment to the wing surface follows the

occurrence of transition to turbulent flow. The results of the numerical

simulations are portrayed in Fig. 14, revealing the three-dimensional

shape of the surfaces (in dark gray) encircling regions of time-averaged

reversed streamwise velocity.

Despite the moderate incidence of the wing, RANS predicts

massively separated flow, startingat the troughandextending far beyond

the trailing edge. The use of T-RANS reduces the size of the main

separated flow region drastically, leading to a shape clearly resembling

Fig. 13 Velocity fields in flow past the modified wing with =1 andα � 15 deg, from SPIV measurements.

Fig. 15 Comparison of flow over themodifiedwingwith =1andα � 15 deg, at z � 1∕2c, froma) SPIVmeasurements, andb) numerical simulationsusing IDDES.

Fig. 14 Regions of reversed streamwise velocity over themodifiedwingwith =1andα � 15 deg, computedusing a)RANS, b)T-RANS, c)DDES, andd) IDDES.

Article in Advance / ESMAEILI, DELGADO, AND SOUSA 9

that observed in the experiments immediately downstream of the trough

(cf. Fig. 13). However, flow reattachment never truly happens in this

case, and the surface indicating reversed streamwise velocity heightens

considerably toward the trailing edge.An unphysically broad separation

is also produced by the DDES approach, but this outcome is

substantially improved by the use IDDES instead. Hence, a better

description of the turbulent boundary-layer structure in the modeling

proves to be essential, so that the flow reattachment occurring at the

midspan section may be successfully predicted. Very low reversed

streamwise velocities are still encountered near the wall past that point,

displaying a pattern quite similar to that also observed in Fig. 13.Amore

detailed comparison between the results of the latter numerical approach

and the corresponding SPIV flow map taken at z � 1∕2c is shown in

Fig. 15. Altogether, fairly good agreement is found between the two

time-averaged velocity fields at this cut plane.A comparison between measured and computed (time-averaged)

values of the aerodynamic force coefficients is given in Table 1. The

large underprediction of the experimental value of CL by the RANS

approach is expected due to thevastly separated flow region observed

for the latter in Fig. 14. However, it is rather surprising that all the

remaining simulations have produced lift coefficients in good

agreement with the experiments, regardless of the fact that some of

these failed completely to reproduce the flowphysics correctly for the

present operating conditions. The major discrepancy found in the

velocity field results fromDDES is nonetheless reflected by the larger

overprediction of the experimental value of the drag coefficientCD in

the set. On the other hand, the computed value of the latter coefficient

from RANS is in line with those obtained via the modeling

approaches that have shown to perform better in the prediction of the

velocity field. This demonstrates that an analysis based solely on

integral parameters of the flow (such as the force coefficients), rather

than on detailed comparisons of the flowfield, may be highlydeceiving concerning the true performance of a particular numericalmodel employed in the study of the problem under consideration.

D. Simulations at Poststall Operating Conditions

As the angle of attack is further increased, the adverse pressuregradient following the suction peak on the upper surface of thewingsbecomes stronger as well. Consequently, boundary layers will bemore prone to separation, and reattachment to thewall may no longerbe possible. In addition, eventual transition from a laminar to aturbulent flow regime is expected to occur rapidly in separated shearlayers subjected to such adverse pressure gradients. Unless the aspectratio of the wings is too small (e.g., unity), so that the supplement ofmomentum from thewingtip vortices is felt by the flow along (nearly)thewholewing span, a nonnegligible loss of lift will inevitably ensue,as previously shown in Fig. 5 for =1.5 (cf. Sec. IV.A). Naturally,this is the case of real interest for the application of the present passivestall controlmethod, in order to recoverwing lift at poststall operatingconditions.Previous numerical studies have demonstrated that the use of

protuberances in the leading edges is accompanied by the formationof low-pressure pockets at the troughs [18,19]. At moderate to highangles of attack, the pattern exhibited by those pockets of fluid isknown to change from periodic to biperiodic (in infinite wings).Together with the downwash mechanism described earlier in thepaper, the persistence of some regions of concentrated lowpressure athigh incidences contributes to the intended recovery of wing lift. Thephenomenon is investigated here for themodifiedwings with =1.5at an angle of attack α � 22 deg, employing numerical simulations.Anticipating a very minor relevance of the transition process at thisoperating condition, and due to the higher computational cost ofthose simulations, the T-RANS approach was discarded; for reasonsdiscussed in the next paragraphs, it was replaced in the set byDES-F2. The time-averaged pressure contours over the upper surfaceof the wing, obtained by the use of the various modeling approachesfor the aforementioned conditions, are portrayed in Fig. 16. Peaksuction intensities vary significantly among these maps, as describedby the indicated ranges of the pressure coefficientCp, using the sameset of contour levels in all cases.The results of the RANS simulation indicate a marked asymmetry

in the flow past the troughs that is consistent with the predictions of

Table 1 Aerodynamic force coefficients fromsimulations and experiments obtained for the modified

wing with =1 and α � 15 deg

RANS T-RANS DDES IDDES Experiments

CL 0.441 0.632 0.623 0.636 0.61CD 0.154 0.153 0.181 0.151 0.13

Fig. 16 Pressure contours over the upper surface of themodifiedwingwith =1.5andα � 22 deg, computed using a)RANS, b)DDES, c) IDDES, andd) DES-F2.

10 Article in Advance / ESMAEILI, DELGADO, AND SOUSA

Dropkin et al. [18] for an infinite wing. Additional complexities

observed in the surface pressure contours for the present case arise as

a direct consequence of the finite span considered here. Surprisingly,

the DDES (and, namely, the IDDES approach that performed

very satisfactorily at prestall conditions) produced flow solutions

exhibiting almost complete symmetry between both (central)

troughs. The worse performance of these DES models when

compared to RANS suggests that, at higher angles of attack, their

shielding of the near-wall region from the DES limiter is inadequate.

Hence, the more conservative shielding strategy implemented in the

DES-F2 approach was subsequently tested, and improved results

were indeed obtained.The foregoing idea is further substantiated in Table 2, where a

comparison betweenmeasured and computed (time-averaged) values

of the aerodynamic force coefficients is presented for the investigated

poststall regime. The equally strong suction at both troughs resulting

from the IDDES is responsible for the large overprediction of the

experimental lift, as well as for the higher value of the corresponding

drag coefficient. In contrast, the most accurate prediction of lift at

poststall operating conditions is produced by the DES-F2 approach,although overall good results can also be obtained in this case bysimply selecting the RANS model.The aforementioned flow asymmetry was also observed in the

course of the experiments, with eventual side switching if a majordisturbance was temporarily introduced into the flow. Velocitymeasurements from SPIV were taken at α � 25 deg and, aiming toconduct a more detailed assessment of the capabilities of the DES-F2

modeling at the poststall regime, additional simulations were carriedout for that angle of attack by employing only this particularapproach. Comparisons with SPIV data are first shown in Figs. 17and 18 for the modified wing and at the spanwise locations of the(central) troughs, i.e., at z � 1∕2c and z � c, respectively. Theseresults confirmed the ability of DES-F2 to predict the exceedinglyasymmetric flow generated by the presence of the protuberances inthe leading edge of a finite wing in deep stall, although computedreversed streamwise velocities were slightly larger than those foundin the experiments (in agreement with the force data in Table 2). Anadditional comparison of numerical results with SPIV flow maps isdepicted in Fig. 19 for the baseline geometry and at the samespanwise section used in Fig. 17 for the modified wing. Once again,good agreement is found between the experiments and computations.By coupling the velocity field in Fig. 19 with that previously given inFig. 6 (cf. Sec. IV.A) for the midspan section (z � 3∕4c) of the samegeometry, one may conclude that the size and strength of the flowseparation vary little along (at least) the central one-third of the totalspan. This is markedly in contrast to the modified wing, and it shedsnew light over the origin of the significant gains in lift resulting fromthe use of the present passive stall control technique.

Table 2 Aerodynamic force coefficients fromsimulations and experiments obtained for the modified

wing with =1.5 and α � 22 deg

RANS DDES IDDES DES-F2 Experiments

CL 0.695 0.906 1.075 0.685 0.67CD 0.280 0.354 0.327 0.292 0.28

Fig. 17 Comparison of flow over the modified wing with =1.5 and α � 25 deg, at z � 1∕2c, from a) SPIV measurements, and b) numericalsimulations using DES-F2.

Fig. 18 Comparison of flow over the modified wing with =1.5 and α � 25 deg, at z � c, from a) SPIVmeasurements, and b) numerical simulationsusing DES-F2.

Article in Advance / ESMAEILI, DELGADO, AND SOUSA 11

V. Conclusions

In this study, numerical simulations of the flow past wings of finite

span with leading-edge protuberances have been conducted at low

Reynolds numbers (Re � 1.4 × 105). The performance of a variety of

modeling approaches was investigated, with emphasis on the detailed

prediction of the flow features exhibited at pre- and poststall operating

conditions. Based on previous studies and taking into account the

computational resources available, Reynolds-averaged Navier–Stokes

(RANS) and DES approaches employing the shear stress transport

k − ω turbulence model were selected. As an attempt to improve the

results produced by the RANS simulations at prestall conditions, an

enhanced version for transitional flow was considered as well, despite

the significant increase in computational cost. The relevance of the

procedure to shield the boundary layer from the DES limiter in DES

simulations at both pre- and poststall conditions was analyzed by

considering different shielding options, including a variant with

correction of the logarithmic layer mismatch between the RANS and

DES regions. In addition, the need of a low-Reynolds-number

correction in the DES approach was examined, demonstrating that its

use led to unphysical results at the investigated flow regime.

Time-averaged results from the numerical simulations were

compared with wind-tunnel measurements of aerodynamic force

coefficients, as well as detailed experimental maps of the velocity field

past the baseline and modified wings obtained using a stereo particle

image velocimetry system. The experiments have shown that the

baseline wings with a NASA LS(1)–0417 cross section exhibit distinct

stall characteristics: an almost imperceptible stall for =1 contrasted

with a major drop in lift for =1.5. Hence, the modification of these

wings via the incorporation of leading-edge tubercles had a substantial

impact in the poststall regimes for the larger aspect ratio only. On the

other hand, the lift characteristics of modified wings were negatively

affected at prestall conditions, irrespective of their aspect ratio, due to the

establishment of long separation bubbles in the regions immediately

downstream of the troughs. A clear signature of streamwise vortices

emanating from the vicinity of the troughs was also detected at these

moderate incidences, therebydisclosing themainmechanismbehind the

passive stall control technique. The downwash induced by thesevortical

structures in adjacent regionscurbedboundary-layer separationathigher

angles of attack, ultimately leading to poststall lift recovery and flow

asymmetry in the modified wing with =1.5.Numerical simulations following the DES approach have proved

their capabilities to reproduce the flow features experimentally

observed at both pre- and poststall regimes. At the former, the

improved description of the boundary-layer structure in IDDES was

crucial to deal with the separation and reattachment phenomena

responsible for the formation of the long separation bubbles. However,

at the latter operating conditions, the occurrence of massive flow

separation required the use of the conservative F2 shielding in DES to

better approximate the experimental data.

Acknowledgments

This work has been supported by Fundação para a Ciência e aTecnologia (FCT), through IDMEC(InstitutodeEngenhariaMecânica),under LAETA (Laboratório Associado de Energia, Transportes eAeronáutica), project UID/EMS/50022/2013. The financial support viaFCT scholarships SFRH/BD/100554/2014 and SFRH/BSAB/114588/2016 is acknowledged as well.

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