Article InternationalJournalofAeroacoustics Numerical analysis of 2016,Vol.15(8)734–756 !TheAuthor(s)2016 aerodynamic noise mitigation Reprintsandpermissions: sagepub.co.uk/journalsPermissions.nav via leading edge serrations DOI:10.1177/1475472X16672322 jae.sagepub.com for a rod–airfoil configuration Bharat Raj Agrawal and Anupam Sharma Abstract Noise produced by aerodynamic interaction between a circular cylinder (rod) and an airfoil in a tandem arrangement is investigated numerically using incompressible large eddy simulations. Quasi-periodic shedding from the rod and the resulting wake impinges on the airfoil to produce unsteadyloadsonthetwogeometries.Theseunsteadyloadsactassourcesofaerodynamicsound and the sound radiates to the far-field with a dipole directivity. The airfoil is set at zero angle of attack for the simulations and the Reynolds number based on the rod diameter is Re ¼48K. d Comparisons with experimental measurements are made for (a) mean and root mean square surfacepressureontherod,(b)profilesofmeanandrootmeansquarestreamwisevelocityinthe rodwake,(c)velocityspectrainthenearfield,and(d)far-fieldpressurespectra.Curle’sacoustic analogy is used with theairfoil surface pressuredatafrom thesimulations topredictthe far-field sound.Animprovedcorrectionbasedonobservedspanwisecoherenceisusedtoaccountforthe difference in span lengths between the experiments and the simulations. Good agreement with datais observed for thenear-field aerodynamics andthe far-field sound predictions. Thestraight leadingedgeairfoilisthenreplacedwithatestairfoilwithaserratedleadingedgegeometrywhile maintainingthemeanchord.Thisnewconfigurationisalsoanalyzednumericallyandfoundtogive a substantial reduction in the far-field noise spectra in the mid- to high-frequency range. Source diagnostics show that the serrations reduce unsteady loading on the airfoil, reduce coherence alongthespan,andincreasespanwisephasevariation,allofwhichcontributetonoisereduction. Keywords Rod-airfoil interaction, leading edge serrations, large eddy simulations Datereceived:31May2016;accepted:10September2016 DepartmentofAerospaceEngineering,IowaStateUniversity,Ames,IA,USA Correspondingauthor: AnupamSharma,DepartmentofAerospaceEngineering,IowaStateUniversity,2271HoweHall,Ames,IA50011,USA. Email:[email protected] Agrawal and Sharma 735 Introduction Aerodynamicnoiseisaby-productofmostengineeringmachines,e.g.,aircraft,gasturbines, andhouseholdfans.Noisecanbeeithertonal,inwhichcasetheacousticenergyislimitedto afewdiscretetones,orbroadband,inwhichcasetheenergyisspreadacrossawiderangeof frequencies.Flowturbulenceisoftenthesourceofbroadbandaerodynamicnoise.Thewide rangeoftimescalesofturbulenteddiesresultsinnoisethatisproducedoverawiderangeof frequencies.Untilrecently,suchbroadbandnoisesourceswereestimatedusingapproximate models for the flow turbulence energy spectrum, which is typically scaled using the turbu- lence kinetic energy and the integral length scale in the problem. These parameters are obtained by solving the Reynolds-averaged Navier–Stokes (RANS) equations, which arecomputationallymuchlessexpensivetosolvethansimulationsthatresolveeveryminute detail in the flow (e.g., direct numerical simulations). Large-scale computing has now become available to researchers, which allows direct computation of the full range of length and time scales important for sound generation and propagation. Such an approach gets rid of the modeling assumptions required in simpler models and thus provides more accuratepredictions.Thelargeeddysimulations(LES)techniqueisonesuchcomputational method that is becoming increasingly popular for noise prediction from engineering machines. Noise computation of a model engineering problem is presented here using LES. Themodelproblemistocomputethenoiseproducedduetotheaerodynamicinteraction between a circular cylinder (rod) and an airfoil (see Figure 1). The rod is placed upstream (in tandem) of the NACA 0012 airfoil. Wake/vorticity shed from the rod convects with the flowandimpingesonthedownstreamairfoil.Thisimpingement(oftentimescharacterizedby the upwash on the airfoil) produces unsteady lift on the airfoil, which radiates as noise, as seen in Figure 1(b). At Re ¼48,000, quasi-periodic vortex shedding is expected behind the d rod,whichgivesrisetotonesatthevortexsheddingfrequency(Strouhalnumber,St(cid:2)0.19) anditsharmonics.Inaddition,theturbulenceinthevorticesandthewakegeneratesbroad- band noise. The resulting noise spectrum has a broadband ‘‘floor’’ above which tones with broadened peaks are observed at the shedding frequency and its harmonics. This problem was experimentally investigated by Jacob et al.1 and has been widely used by various researchers to benchmark their codes’ capability and accuracy. The measurements1 include Figure 1. Snapshots from acompressible LES simulation byAgrawalandSharma2for the rod–airfoil problem: (a) hydrodynamic flow field illustrated usingiso-surfaces ofQ-criterion (Q¼25) with contours colored bythemagnitudeof density gradient, and(b)far-field acoustics shown usingfluid dilatation,r:v. 736 International Journal of Aeroacoustics 15(8) wakeandboundarylayerprofiles(meanandturbulentstatistics),near-fieldvelocityspectra, and far-field noise. Literature review A number of numerical studies have been carried out for the specific rod–airfoil configur- ation considered here. Casalino et al.3 was the first to numerically investigate this problem using unsteady RANS simulations. The simulations were two-dimensional (2D) and three- dimensional (3D) effects on noise were modeled using a statistical model coupled with the FfowcsWilliams–Hawkings (FW–H)acousticanalogy.Thestatistical modelwascalibrated using the experimental data. TheLEStechniquehasalsobeenusedtomodelthisproblem.Boudetetal.4reportedthe first LES computations for this benchmark problem. It used finite-volume, compressible LES on multi-block structured grids. Far-field noise was obtained by coupling the near- field data with a permeable FW–H solver. Berland et al.5 performed direct noise computa- tionsusinghigh-order,compressibleLESonoversetstructuredgrids.Theyalsoinvestigated the effect on noise of varying the spacing between the rod and the airfoil. Eltaweel and Wang6 used an incompressible LES solver coupled with a boundary element method to predict far-field noise. An unstructured mesh composed of 22.3 million cells was used. Their results showed very good agreement with data for near-field flow measurements as well as far-field acoustics. Giret et al.7,8 used a compressible LES solver with a fully unstructured grid. Far-field noise was predicted using an advanced-time formulation of the FW–H acoustic analogy.9 They used both porous and impermeable (on the rod and airfoil surface) boundary approachesforevaluatingtheFW–Hboundaryintegralandfoundlittledifferenceinthepre- dicted noise. They also numerically investigated the effect of offsetting the airfoil in the cross-stream direction by the small amount observed in the experiments. That however did not significantly improve the agreement with the measured wake and velocity profiles. Jiangetal.10carriedoutaparametricstudywithdifferentdistancesbetweentherodandthe airfoil using high-order implicit LES. The far-field noise was predicted using the FW–H acoustic analogy. This article presents an aeroacoustic analysis of the rod–airfoil problem using incom- pressible LES. Two different airfoil geometries are analyzed: one with a straight leading edge as in the experiments and the other with a serrated leading edge. Near-field hydro- dynamicsandfar-fieldacousticresultsarecomparedagainstmeasureddatawhereavailable for the straight-edge case. The pimpleFoam solver from OpenFOAM is used as the LES solver. Unsteady pressure on the airfoil surface is extracted from the simulations and used with Amiet’s formula,11 which extends Curle’s theory to predict noise from distributed dipole sources over a thin airfoil.ItshouldbeemphasizedthatCurle’sanalogyisusedinthemostgeneralsenseandno approximations, e.g., thin-airfoil theory for computing lift and isotropic turbulence, typic- ally associated with Amiet’s theory are made here. A frequency-based correction given by Seo and Moon12 is utilized to account for the difference in the airfoil span lengths between the simulation and the experiment. Curle’s analogy to predict far-field noise for this rod– airfoil configuration has not been utilized before in available literature. Previousexperimentalandnumericalinvestigations13–17haveshownsubstantialreduction in inflow turbulence (broadband) noise with the use of leading edge serrations. Almost all Agrawal and Sharma 737 Figure 2. A2D schematicshowing thenon-dimensional size,positionsof therod andtheairfoil, and thenear-field locations (pointsandlines inblue)wherecomparisons are madewithexperimental data. these investigations have used homogeneous and grid-generated turbulence. In this article, weanalyzetheeffectivenessofleadingedgeserrationsinmitigatingnoisefortherod–airfoil configuration. The use of leading serrations for this configuration has not been numerically investigatedbefore.Thenumericalapproachtoanalyzetheserratedcaseisthesameasthat used for the straight-edge case. Numerical setup Figure2showsanon-dimensionalschematicoftherod–airfoilproblemwherelengthisnon- dimensionalized bytheairfoilchord. Also,velocity anddensityarenon-dimensionalized by the speed of sound and the freestream density, respectively. The rod and the airfoil are placed in tandem along the x direction, the span direction is along the z axis, and the y direction is given by the right-hand rule. In the experiments by Jacob et al.,1 two different rod diameters were tested. In this article, we focus on the experiment with the rod diameter, d¼0.1(cid:3)c, where c is the airfoil chord. Measurements were made for several Reynolds numbers and we limit our focus to Re ¼48K (based on d) since at that Re, broadband noise contribution is apparent in the d data. The distance between the rod trailing edge and the airfoil leading edge is equal to the airfoil chord c. Theairfoilissetatzeroangleofattackinthesimulations, aswasintendedintheexperi- ments. However, based on the measured data, Jacob et al.1 suspect that in the experiments, the airfoil might have been at a slight ((cid:2)2(cid:4)) angle of attack and slightly offset in the y direction. These geometric anomalies are not incorporated in the numerical model as a previous study8 has shown that their effect on the far-field OASPL is less than 1dB at 90(cid:4). Computational mesh Figure 3 shows a cross-sectional view of the mesh with the gridlines shown in the bottom half and only the block boundaries shown in the upper half. The far-field computational boundary(notshown)isnearlycircularwitharadiusofapproximately11(cid:3)c.Thegeometry is essentially 2D and is extruded in the third, spanwise direction to obtain a 3D mesh. A reduced span length of 0.3(cid:3)c is used in the simulations to reduce the mesh size and computation time. This choice is guided by previous works,4–6 which also used partial span (0.3(cid:3)c) domain in their simulations. ‘‘Frequency-Dependent Correction’’ section of this 738 International Journal of Aeroacoustics 15(8) Figure 3. Cross-sectional (x–y) viewofthenear-field computationaldomain showing theblock bound- ariesin theupperhalfandthe gridlines inthelowerhalf. paper discusses the aeroacoustic implications of this choice. In order to compare with the measurements,thepredictednoisespectraarescaledusingtheapproachbySeoandMoon12 to account for the difference in span lengths between the experiments and simulations. A fully structured mesh generated using Pointwise (www.pointwise.com/pw) is used for the simulations. A planar mesh is first generated in the z¼0 plane, which is then repeated alongthespanwithauniformspacingtoobtainthe3Dmesh.Theplanarmeshisgenerated in three steps. The first step involves extruding the curves that define the rod and airfoil surfaces inthesurfacenormaldirections. Thisprocessyieldshigh-quality orthogonal quad- rilateral elements which are suitable for resolving wall boundary layers. The second step is to create quadrilateral elements between the rod and airfoil that are fine enough to capture the rod wake accurately. This is done by creating two parabolic curves, one each on the upper and lower sides, between the outer boundaries of the earlier extruded domains. Theseparaboliccurvesarethenfilledwithquadrilateralelementswithaspectratioofnearly 1.Thefinalsteprequiresaclosedcurveencompassingthethreedomains:therodandairfoil boundary layer regions and the rod wake region. In the final step, this closed curve is extruded normally until the outer radius is about 11(cid:3)c. This process gives a good qual- ity mesh throughout the domain. Figure 3 shows a zoom view of the final 2D mesh in the z¼0 plane. The blocking structure and the grid density are designed to resolve (a) the turbulence in therodwakeinthegapregion,(b)theboundarylayerontherod,and(c)theboundarylayer on the airfoil. The first cell height on the airfoil and the rod is chosen such that p yþ ¼y=ð(cid:2) ffi(cid:3)ffiffi=ffiffi(cid:4)ffiffiffiffiffiÞ¼1, where v is the fluid kinematic viscosity, (cid:4) is the wall shear stress, w w and (cid:3) isthe fluid density. This isa conservative estimate, since such small first cell height is required for resolving wall boundary layers. The problem under investigation is the inter- action of the turbulence in the rod wake with the airfoil. Hence, accurate resolution of the turbulence generated in the airfoil boundary layers is not of paramount importance. This conservative approach was still taken however with the intent that in the future, the same Agrawal and Sharma 739 grid could be used to study ‘‘self’’ (trailing edge) noise from this airfoil and a comparison could be made between ‘‘self’’ noise and inflow (coming from rod wake) turbulence noise. The total number of cells in the computational mesh is approximately 19 million. There arefiveboundarysurfaces:rod,airfoil,twoperiodicandonefar-field.Meshqualitymetrics areas follows: themaximum cellaspect ratio is169, themaximum mesh non-orthogonality is 30.5(cid:4), and the maximum skewness is 0.65. Flow conditions and non-dimensionalization The simulations are setup in non-dimensional variables denoted by the overhead tilde and the freestream values are used for non-dimensionalization. Therefore, the non-dimensional freestream density ((cid:3)~ ), speed of sound (a~ ), and temperature (T~ ) are all unity. Using 1 01 1 (cid:3)~ ¼1 and a~ ¼1, we get p~ ¼1=(cid:5) ¼0:7143. The freestream velocity is obtained as 1 01 1 u~ ¼0:2 from a~ ¼1 and M ¼0.2. The length scale is normalized w.r.t. the airfoil 1 01 1 chord length, and hence, the diameter of the rod in the non-dimensional units is d~¼d=c¼0:1. The required Reynolds number of the flow based on the rod diameter, i.e., Re ¼48,000 is obtained by setting the dynamic viscosity to (cid:6)~ ¼(cid:3)~ u~ d~=Re equal to d 1 1 d 4:2(cid:3)10(cid:5)7. Inphysicalunits,thefreestreamconditionsare(cid:3) ¼1.226kg/m3,a ¼360:0m/s(hence- 1 01 forthdenotedbya ),u ¼72m/s,andp ¼113,500Pa.Theratiosofphysicalunitstonon- 0 1 1 dimensional units are required for direct comparisons with measurements. The ratio of dimensional to non-dimensional time is t=t~¼u =c¼720 s, where c¼0.1m. All spectral 1 results are plotted w.r.t. Strouhal number based on the rod diameter, St¼fd=u . 1 Incompressible flow solver, pimpleFoam Equation (1) gives the filtered Navier–Stokes equation for incompressible LES computa- tions,18 where, ð Þ and (cid:4)SGSð¼uu (cid:5)u^u^ Þ represent a homogeneous LES filter and subgrid b ij di j i j stress, respectively. The governing equations are the continuity and momentum equations, written in differential form here as @u^ @u^ @u^u^ 1 @p^ @(cid:4)SGS i ¼0; iþ i j ¼(cid:5) þ(cid:2)r2u^ (cid:5) ij ð1Þ @x @t @x (cid:3)@x i @x i j i j The subgrid stress ((cid:4)SGS) cannot be computed directly and requires modeling. Equation ij (2) gives the standard Smagorinsky19 model which is an eddy viscosity type model to com- pute the subgrid stresses, where S ð¼ð@u^=@x þ@u^=@xÞ=2Þ is the rate-of-strain tensor that ij i j j i can be directly computed. qffiffiffiffiffiffiffiffiffiffiffiffiffi (cid:4)SGS ¼(cid:5)2ðC (cid:2)Þ2S^ 2S^ S^ ð2Þ ij s ij ij ji Thetransient,incompressibleflowsolver,pimpleFoamisusedinLESmodewithsubgrid stresses computed using equation (2). The continuity and momentum equations (equation (1))aresolvedusingthePIMPLEalgorithm,whichisacombinationofthepressure-implicit split-operator (PISO) algorithm20 and the semi-implicit method for pressure linked equations (SIMPLE) algorithm.21 The PIMPLE algorithm allows the Courant Friedrichs Lewy (CFL) number to be greater than unity while still maintaining numerical stability. Asecond-orderimplicitschemeisusedfortimemarchingandatimestepof0.005ischosen. 740 International Journal of Aeroacoustics 15(8) Theflowfieldisinitializedwiththenon-dimensionalvelocity,u~ ¼0:2,gaugepressuresetto 1 zero, and kinematic viscosity specified as (cid:2)~¼4:2(cid:3)10(cid:5)7. Attheouterboundary,thevelocityswitchesbetweenzerogradientforoutflowandfixed (prescribed) value for inflow. The boundary condition for pressure is zero gradient, which fixes the flux across the boundary using the freestream velocity. Gaussian integration with linear central differencing interpolation is used to compute gradients, Laplacian, and divergence terms. Divergence for the convective term is computed using linear upwind differencing interpolation. Far-field noise prediction Time-resolvedpressuredataontheairfoilsurfaceisusedwithCurle’sanalogytopredictfar- field noise. Curle’s analogy can be used to predict noise radiation due to surface (dipole) sources.DuetothesmallflowMachnumberinthisproblem,thenoisesourcesareprimarily theunsteadyforces(dipoles)ontheairfoilandtherod.Hence,theuseofCurle’sanalogyto compute far-field sound is justified. The contribution to far-field noise from off-surface (quadrupole) sources has been shown to be significant only at very high (St>1) and very low (St<0.05) frequencies.8 Giret et al.8 showed that the quadrupole sources have little effect on the overall sound pressure level (OASPL). Eltaweel and Wang6 also ignored the volume sources in their prediction methodology. Far-field noise prediction Pressure data are collected on the airfoil surfaces at a high sampling rate after the initial transients have been removed from the simulations. Using this surface pressure data, the unsteady lift per unit area (difference in pressures, (cid:2)pðx,z,tÞ between the upper and the lower surfaces of the airfoil) is computed for all points (x, z) on the blade (airfoil) planform at each time, t. Note that x is along the chord and z is along the span. Amiet11 usedtheideathatfar-fieldacousticresponsecanbecomputedbyassumingdipolesourcesin place of unsteady surface loads and gavethe following expression forsound power spectral density (S ) for acoustic pressure at any given point ðx0,y0,z0Þ in the far-field pp (cid:3) !y0 (cid:4)2ZZZZ S ðx0,y0,z0,!Þ¼ S ðx ,x ,z ,z ,!Þ pp 4(cid:7)a0(cid:8)2 qq 1 2 1 2 ð3Þ (cid:3)eai!0½(cid:9)(cid:5)2ðx1(cid:5)x2ÞðM1(cid:5)x0=(cid:8)Þþz0(cid:10)=(cid:8)(cid:6) dx1 dx2 dz1 dz2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where (cid:9)¼ 1(cid:5)M2 , (cid:8) ¼ x2þ(cid:9)2ðy2þz2Þ, (cid:10) is the spanwise separation, and S is the 1 qq cross power spectral density (PSD) of predicted unsteady pressure difference ((cid:2)p) between anytwopoints(x ,z )and(x ,z )ontheplanform.S iscomputedusingWelch’saverage 1 1 2 2 qq periodogram method22 for all point pairs and is then used with equation (3) to numerically compute the far-field S . Theoretical background on the spectral analysis used here is pp provided in Appendix 1. Kato’s correction The span length of the rod–airfoil assembly simulated in the present calculations is smaller than the experimental model. The predicted far-field noise therefore has to be corrected before comparing with the measured data. The correction that needs to be applied to the Agrawal and Sharma 741 predictedspectradependsonspanwisecoherence.Ifwedenotespanwisecoherencelengthby L and use subscripts s and e for the simulations and the experiment respectively, then c equation (4) can be used for comparing the measured and the predicted spectra. (cid:5) (cid:6) (cid:5) (cid:6) S ð!Þ ¼ S ð!Þ þ20logðL =L Þ8L 5L 5L , pp e pp s e s s e c (cid:5) (cid:6) (cid:5) (cid:6) S ð!Þ ¼ S ð!Þ þ10logðL =L Þ, 8L 5L 5L , ð4Þ pp e pp s e s c s e (cid:5) (cid:6) (cid:5) (cid:6) S ð!Þ ¼ S ð!Þ þ20logðL =L Þþ10logðL =L Þ8L 5L 5L pp e pp s c s e c s c e Equation (4) assumes that there is perfect correlation over the span length of L , outside c ofwhich,thecorrelationdropsidenticallytozero.This‘box-car’simplificationbyKatoand Ikegawa23 is often used. The span length of the rod and airfoil assembly in the simulations (L)isthreetimestheroddiameter(d),i.e.,L ¼3d,whichisone-tenthofthespanlengthof s s themodelusedintheexperiment,i.e.,L ¼10L.AssumingthatthecorrelationlengthL is e s c less than L, the correction required is s (cid:5) (cid:6) (cid:5) (cid:6) S ð!Þ ¼ S ð!Þ þ10logðL =L Þ, or, pp scorr pp s e s ð5Þ (cid:5) (cid:6) (cid:5) (cid:6) S ð!Þ ¼ S ð!Þ þ10logð10Þ pp scorr pp s The spanwise correlation length in this problem is a strong function of frequency: the correlation length is very large at the peak shedding frequency and its harmonics but small at other frequencies. This highlights the need for a frequency-dependent span- correction for noise prediction. This issue and a potential resolution are discussed in ‘‘Frequency-Dependent Correction’’ section. Results and data comparisons The phenomena of interestin theproblem underinvestigation are unsteady butstatistically stationary.Theinterestisnotintransientphenomenasuchasinstantaneous/impulsive start of the rod/airfoil combination. In the experiments, the wind tunnel was started and the rig allowed to reach a statistically stationary state before measurements were taken. Similarly, thecomputationshavetoreacha statistically stationarystatebefore anyunsteadydatacan be gathered from the simulations. Removal of initial transients from the computational domain is therefore required before meaningful results can be sampled. The time period of wake shedding from the cylinder for Re ¼48,000 is approximately 2.6 non-dimensional d timeunits.Thedatacollectionbeganafter40timeunitsandthensampledforapproximately 40 shedding periods (approximately 104 time units) in the simulation. These data are used for the statistical analysis presented in the following sections. Mesh sensitivity study A mesh sensitivity study is carried out with three mesh sizes comprising of 10-, 19-, and 64-million cells. The different meshes are generated by refining the grid in the wall-normal and streamwise directions while maintaining the spanwise grid count at 80. The first cell height is maintained to give a y+of unity when refining in the wall-normal direction. Figure 4 plots the results of the mesh sensitivity study. The PSD of the x-component of velocity (S ð!Þ) at point A ð(cid:5)0:87c,0:05cÞ is shown in subplot (a). Subplots (b) and (c), uu respectively, show the predicted mean (C(cid:3) ) and root mean square (rms) (C ) of the P P,rms coefficient of pressure on the rod surface. All these quantities are averaged along the span 742 International Journal of Aeroacoustics 15(8) (a) 10 (b) 1.0 (c) 0.6 10M 10M 19M 19M SωB/Hz()duu−−−3210000 1109MM ¯CMeanpressurecoeff.()P --0001....0550 64M Crms pressurecoeff.()P,rms 00..24 64M 64M −40 -1.5 0.0 f1 f2 f3f4 70 180 80 180 St=fd/U∞ Anglefromupstream(degrees) Anglefromupstream(degrees) Figure 4. Resultsof themesh sensitivity study: (a)PSDof thex-componentofvelocityatpointA, andtime- andspan-averaged(b) C -and(c) C distributionsonthe cylindersurface. P P,rms rms: rootmeansquare. in the plots. Grid convergence is observed beyond the 19-M mesh in the hydrodynamic spectra in the cylinder wake as well as in the mean surface pressure on the cylinder surface (seeFigure4(a)and(b)).Thereis,however,aperceptibledependenceonmeshinpredicting thermsofaerodynamicpressureonthecylindersurface(Figure4(c)).Sincethenoisesource of interest here is due to the cylinder wake interacting with the airfoil, and the fact that no statisticallysignificantchangeisobservedinthecylinderwakevelocityspectrabyincreasing themeshbeyond19Mcells,the19-Mmeshisdeemedsufficienttostudyaerodynamicnoise for this problem and to investigate aeroacoustics effects of serrations on the airfoil. Furthermore, the wall-clock times for simulating a unit non-dimensional flow time using 128processorsforthe10M,19M,and64Mcellmeshesare759,2315,and34,569s,respect- ively.Thewall-clocktimeincreasesbyafactorof15asthemeshsizeincreasesfrom19Mto 64M. This large increase coupled with limited parallel scalability of the flow solver (OpenFOAM does not scale beyond 512 processors) make it intractable to attempt a larger mesh size. Based on the mesh sensitivity study, the 19-M mesh simulation results are used for experimental validation described in the following section. Experimental validation Rod surface pressure statistics. Pressure distributions on the rod and airfoil surfaces are obtained by averaging the time-accurate data (sampled over 40 shedding periods) in time aswellasinspace(alongthespandirection).Themeanpressurecoefficient,C(cid:3) ,andtheroot P mean squared pressure coefficient, C , for the rod are obtained using this averaging P,rms procedure and compared against measurements in Figure 5 for the rod. The pressure coef- ficients are plotted w.r.t. angle measured from upstream. Thus, 0(cid:4) and 180(cid:4) denote the rod leading and trailing stagnation points, respectively. Figure 5(a) shows that the expected value of 1 is obtained for C(cid:3) at the rod leading P stagnation point after which C(cid:3) drops steadily until the peak negative value is reached at P 70(cid:4). The peak location predicted by OpenFOAM matches with the experimental data from Norberg.24TwosetsofdatafromNorberg24areshown,whichcorrespondtoRe of20,000 d and 60,000. The agreement between the predictions and the measured data is very good. Agrawal and Sharma 743 (a) 1.0 (b) 0.6 ¯Ccient()P 0.5 OEEpxxeppneeFrriiOmmAeennMtt::,RRReeeddd===264008KKK C()P,rms 0.4 OEEpxxeppneeFrriiOmmAeennMtt::,RRReeeddd===264008KKK ffi 0.0 ff. oe oe C c eanPressure --10..05 rms pressure 0.2 M -1.5 0.0 70 180 80 180 Anglefromupstream(degrees) Anglefromupstream(degrees) Figure 5. Mean andrmsaerodynamic pressurecoefficients ontherod.Experimentaldata intheseplots isfrom Norberg.24rms: rootmeansquare. (a)Mean pressurecoeff.,C(cid:3) .(b) RMSpressure coeff.,C . P P,rms (a) 0.25 (b) 0.06 Experiment Experiment OpenFOAM OpenFOAM 0.20 0.04 a0 /a0 u/¯˜ u˜rms 0.15 0.02 0.10 0.00 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 y/c y/c Figure 6. Profiles ofmeanandrmsof thexcomponentof velocity intherod wakeatstation x¼(cid:5)0.255c.(a) u mean,(b) u rms. Figure 5(b) shows that C starts from zero at the rod leading edge and then mono- P,rms tonicallyincreaseswithanglefromupstreamuntil80(cid:4),afterwhichitdropsrapidly.Theflow separates approximately between 75(cid:4) and 80(cid:4) as observed by Achenbach.25 X-velocity statistics in rod wake. Figure 6 compares profiles of the mean and the rms of the x-component of velocity in the rod wake at x¼(cid:5)0.255c. The momentum deficit and tur- bulenceintensityinthewakeareslightlyoverpredicted.Themeasuredprofilesshowashift