Ferrero, Andrea

Germano, Massimo

Larocca, Francesco

Scovazzi, Guglielmo

English

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

08 November 2022POLITECNICO DI TORINORepository ISTITUZIONALEA Numerical Study of the Spanwise Turbulence Past a Cylinder Flow / Ferrero, Andrea; Larocca, Francesco; Scovazzi,Guglielmo; Germano, Massimo. - 267:(2021), pp. 91-96. ((Intervento presentato al convegno iTi Conference inTurbulence 2021 tenutosi a Online nel February 25th–26th, 2021 [10.1007/978-3-030-80716-0_12].OriginalA Numerical Study of the Spanwise Turbulence Past a Cylinder FlowSpringer postprint/Author's Accepted ManuscriptPublisher:PublishedDOI:10.1007/978-3-030-80716-0_12Terms of use:openAccessPublisher copyrightThis version of the article has been accepted for publication, after peer review (when applicable) and is subject toSpringer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements,or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/978-3-030-80716-0_12(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/2934981 since: 2021-12-02T12:07:47ZSpringer VerlagA numerical study of the spanwise turbulencepast a cylinder flowAndrea Ferrero, Francesco Larocca, Guglielmo Scovazzi and Massimo GermanoAbstractMany flows of industrial interest and many important benchmark turbulentflows are statistically stationary in time and are provided with a spanwise directionof homogeneity. The numerical simulation of such flows is conditioned by thediscretization in space and time, and the statistical analysis of the data is biased bythe finite extent of the produced dataset. In this work the flow around a circularcylinder at Reynolds 3900 is numerically investigated by an implicit Large EddySimulation. The computations are performed by a modal Discontinuous Galerkinfinite element solver and the produced database is analysed in order to quantify thetemporal and spanwise contribution to the estimation of the statistics. The goal of thework is to investigate a procedure which allows to quantify the statistical efficiencyof the operators which are used to perform the average in time and in the spanwisedirection. Finally, the hierarchical nature of the modal basis used in each elementis exploited to perform a local element-wise filtering operation which allows toquantify the contributions given by the smallest resolved scales to the statistics.Andrea FerreroPolitecnico di Torino, Corso Duca degli Abruzzi 24, Torino, Italy, e-mail: andrea_ferrero@polito.itFrancesco LaroccaPolitecnico di Torino, Corso Duca degli Abruzzi 24, Torino, Italy, e-mail:francesco.larocca@polito.itGuglielmo ScovazziDuke University, 2080 Duke University Road, Durham, United States, e-mail:guglielmo.scovazzi@duke.eduMassimo GermanoDuke University, 2080 Duke University Road, Durham, United States, e-mail: mg234@duke.edu12 Andrea Ferrero, Francesco Larocca, Guglielmo Scovazzi and Massimo Germano1 IntroductionMany flows of industrial interest and many important benchmark turbulent flowsare statistically stationary in time and are provided with a spanwise direction ofhomogeneity. Typical examples are the flow in blade cascades and more simply theflow around a cylinder. The numerical simulation of such flows is conditioned bythe discretization in space and time, and the statistical analysis of the data is biasedby the finite extent of the produced dataset. In a previous paper the authors haveexamined in detail the statistical error in the time averages [1], but we remark that,due to the spanwise homogeneity, statistics can be also computed with a combinedaverage in time and in the spanwise direction. As remarked in [2], an insufficientspanwise extent can produce a overprediction of the streamwise and normal turbu-lent intensities and corrispondingly artificial suppression of the spanwise turbulentintensity. These considerations can be used to define a minimum spanwise extensionrequired to correctly describe the evolution of the largest structures. However, thisprovides only a lower bound to the spanwise extension: in a general flow it is notclear a-priori whether it is more convenient to collect the statistics by performinga simulation with a relatively large spanwise extension and a small time windowor the contrary. In order to find a numerical criterion which allows to quantify theefficiency of computing statistics in time and in the spanwise direction, the potentialof some indices of statistical resolution is investigated.In particular a database of the turbulent flow past a circular cylinder at Reynolds 3900is produce by means of an implicit Large Eddy Simulation (LES). The computationsare carried out by means of a modal Discontinuous Galerkin (DG) finite elementsolver based on the use of a modal orthonormal basis implemented following theguidelines of [3]. The solver is compressible but in this study a low far field Machnumber (𝑀∞ = 0.2) is assumed in order to make comparisons with available exper-imental results in almost incompressible conditions [4].Finally, the hierarchical nature of the modal orthonormal basis used inside eachelement allows to perform a local element-wise filtering operation: in this way it ispossible to quantify the contributions given by the smallest resolved scales.2 Indices of statistical resolutionThe turbulent flow past a cylinder is characterised by two homogeneities, the time 𝑡and the spanwise spatial direction 𝑧, and we can average along one or both of them inorder to estimate average fields and Reynolds stresses. Given the turbulent velocityfield 𝑢𝑖 (𝑥, 𝑦, 𝑧, 𝑡) we will introduce the following space and time statistical filteringoperators T andZA numerical study of the spanwise turbulence past a cylinder flow 3Z{𝑢𝑖} ≡ 〈𝑢𝑖〉𝑧 ≡1𝑍∫ 𝑍0𝑢𝑖 (𝑥, 𝑦, 𝑧′, 𝑡)𝑑𝑧′T {𝑢𝑖} ≡ 〈𝑢𝑖〉𝑡 ≡1𝑇∫ 𝑇0𝑢𝑖 (𝑥, 𝑦, 𝑧, 𝑡 ′)𝑑𝑡 ′ (1)and we define as E the product of the twoE{𝑢𝑖} ≡ TZ ≡ ZT ≡ 〈𝑢𝑖〉𝑒 ≡1𝑍𝑇∫ 𝑍0∫ 𝑇0𝑢𝑖 (𝑥, 𝑦, 𝑧′, 𝑡 ′)𝑑𝑧′𝑑𝑡 ′ (2)where 𝑍 and 𝑇 are the extents of the domain in the spanwise direction 𝑧 and in time𝑡. The turbulent stresses can be defined by means of the operators T andZ𝜏𝑧 (𝑢𝑖 , 𝑢 𝑗 ) ≡ 〈𝑢𝑖𝑢 𝑗〉𝑧 − 〈𝑢𝑖〉𝑧 〈𝑢 𝑗〉𝑧𝜏𝑡 (𝑢𝑖 , 𝑢 𝑗 ) ≡ 〈𝑢𝑖𝑢 𝑗〉𝑡 − 〈𝑢𝑖〉𝑡 〈𝑢 𝑗〉𝑡𝜏𝑒 (𝑢𝑖 , 𝑢 𝑗 ) ≡ 〈𝑢𝑖𝑢 𝑗〉𝑧𝑡 − 〈𝑢𝑖〉𝑧𝑡 〈𝑢 𝑗〉𝑧𝑡𝜏𝑧 (〈𝑢𝑖〉𝑡 , 〈𝑢 𝑗〉𝑡 ) ≡ 〈〈𝑢𝑖〉𝑡 〈𝑢 𝑗〉𝑡 〉𝑧 − 〈𝑢𝑖〉𝑧𝑡 〈𝑢 𝑗〉𝑧𝑡𝜏𝑡 (〈𝑢𝑖〉𝑧 , 〈𝑢 𝑗〉𝑧) ≡ 〈〈𝑢𝑖〉𝑧 〈𝑢 𝑗〉𝑧〉𝑡 − 〈𝑢𝑖〉𝑡 𝑧 〈𝑢 𝑗〉𝑡 𝑧 (3)We note that due to E = ZT = TZ we have the two identities𝜏𝑒 (𝑢𝑖 , 𝑢 𝑗 ) ≡ 𝜏𝑧 (〈𝑢𝑖〉𝑡 , 〈𝑢 𝑗〉𝑡 ) + 〈𝜏𝑡 (𝑢𝑖 , 𝑢 𝑗 )〉𝑧≡ 𝜏𝑡 (〈𝑢𝑖〉𝑧 , 〈𝑢 𝑗〉𝑧) + 〈𝜏𝑧 (𝑢𝑖 , 𝑢 𝑗 )〉𝑡 (4)and we can define two measures of turbulence resolution, the first related to the timeaverage and the second to the spanwise average, given by𝑀𝑡 (𝑥, 𝑦) =〈𝜏𝑡 (𝑢𝑖 , 𝑢𝑖)〉𝑧𝑅𝑖𝑖= 1 − 𝜏𝑧 (〈𝑢𝑖〉𝑡 , 〈𝑢𝑖〉𝑡 )𝑅𝑖𝑖𝑀𝑧 (𝑥, 𝑦) =〈𝜏𝑧 (𝑢𝑖 , 𝑢𝑖)〉𝑡𝑅𝑖𝑖= 1 − 𝜏𝑡 (〈𝑢𝑖〉𝑧 , 〈𝑢𝑖〉𝑧)𝑅𝑖𝑖(5)where 𝑅𝑖 𝑗 ≡ 𝜏𝑒 (𝑢𝑖 , 𝑢 𝑗 ). The indices 𝑀𝑡 (𝑥, 𝑦) and 𝑀𝑧 (𝑥, 𝑦) allow to quantify therelative contribution given by time and spanwise direction to the evaluation of thestatistics. For example, where 𝑀𝑡 −→ 0 the statistics are mainly captured by samplingin the spanwise direction while where 𝑀𝑡 −→ 1 the statistics are mainly captured bysampling in time. The index 𝑀𝑧 shows an opposite behaviour.3 Numerical resultsThe simulations are performed on an unstructured mesh with approximately 3 · 105elements by a third order accurate DG scheme (𝑝 = 2): 3 ·106 degrees of freedom areemployed for each equation. Time integration is performed by means of an explicit4 Andrea Ferrero, Francesco Larocca, Guglielmo Scovazzi and Massimo GermanoRK3 method. The spanwise extension of the domain along the direction 𝑧 is setequal to three diameters 𝐷. The simulation is carried out for several hundreds ofconvective times 𝐶𝑇 , as shown in Figure 1a which reports the time evolution of thedrag coefficient 𝐶𝑑 . A snapshot of the vorticity magnitude field is reported in Figure1b which puts in evidence the development of turbulence structures in the wake.The spectrum of the streamwise and normal velocity in a station located at 𝑥/𝐷 = 4and 𝑦/𝐷 = 0 is reported in Figure 2. The spectrum is computed from a time windowwith 400𝐶𝑇 and it is averaged along the spanwise direction 𝑧. The plots show apeak at 𝑓 𝐷/𝑈 = 0.4 for the streamwise velocity and two peaks at 𝑓 𝐷/𝑈 = 0.2 and𝑓 𝐷/𝑈 = 0.6 for the normal velocity: this is in line with the results reported by [4].The 𝑅11 component of the Reynolds stress tensor is evaluated in the station at𝑥/𝐷 = 2.02 according to Eq. 4. The results are reported in Figure 3a for twodifferent choices of the average windows 𝑇 and 𝑍 and they are compared with theexperimental results from [4]. The numerical simulation is performed for𝑇 = 400𝐶𝑇with a spanwise extension 𝑍 = 3𝐷 and it is used to generate a global database. Twosubsets are extracted from this database by selecting different extensions in timeand spanwise direction: (𝑇 = 100𝐶𝑇, 𝑍 = 3𝐷/20) and (𝑇 = 5𝐶𝑇, 𝑍 = 3𝐷). Inthis problem the computational cost is directly proportional to both the spanwiseextension and the duration of the simulated time window. This means that the samecomputational cost could be associated to the two subsets. However, the results re-ported in Figure 3a show that the estimation of the Reynolds stresses performed fromthe dataset with (𝑇 = 100𝐶𝑇, 𝑍 = 3𝐷/20) is significantly closer to the experimentalresults. In contrast, the results obtained from the dataset with (𝑇 = 5𝐶𝑇, 𝑍 = 3𝐷)appear to be far from statistical convergence since they are characterised by a stronglyasymmetric distribution. This first test suggests that in the considered test case timeaveraging is more efficient with respect to averaging in the spanwise direction. Thiscan be explained by the fact that in this problem the flow field is dominated bythe streamwise velocity and so the transport of turbulent structures in the stream-wise direction determines strong temporal fluctuations for a fixed control station.Furthermore, the streawise fluctuations are significantly stronger with respect to thespanwise fluctuations, as confirmed by the distribution reported in Figure 3b.In order to verify the possibility to link the statistical efficiency of time and spanwisedirections to a measurable quantity the indices𝑀𝑡 and𝑀𝑧 are evaluated and reportedin Figure 4a: the plot shows clearly that the index 𝑀𝑡 is systematically higher thanthe index 𝑀𝑧 for all the values of 𝑦. This result suggests a general strategy whichcan be applied for statistically steady flows with a direction of spatial homogene-ity. First of all, a preliminary simulation can be done by choosing the minimumspanwise extension which is necessary to allow the correct evolution of the largeststructures and running the simulation for a few convective times. Then the resultsof this preliminary simulation can be used to estimate the indices 𝑀𝑡 and 𝑀𝑧 whichprovide insight in the statistical efficiency of time and spanwise direction: finally, afull simulation can be set up by increasing the runtime or the spanwise extension,according to the indications provided by 𝑀𝑡 and 𝑀𝑧 .Finally, the plot in Figure 4b shows a comparison between the experimental valuesof 𝑅11 and the numerical predictions evaluated according to Eq. 4 for the full orderA numerical study of the spanwise turbulence past a cylinder flow 5results (𝑝 = 2) and for a filtered solution (𝑝 = 0) obtained by truncating the results ofthe 𝑝 = 2 simulation to the first term in each element. This makes it possible to quan-tify the contributions given by the turbulent structures whose size is comparable tothe size of the element. The results show that this contribution is small in the presentsimulation and so it is possible to assume that the contribution of the subgrid scalesis even smaller. This explains why the implicit LES approach provides reasonableresults. However, for problems characterised by higher values of Reynolds number,the contributions associated to the subgrid scales can be more important and so anexplicit LES with a subgrid model would be more suitable. In that case, the modalnature of the DG solution can be exploited to develop dynamic approaches for thesubgrid scale model, following for example the guidelines provided by [5]. As afinal remark, in this work the extensions of the spanwise and temporal windows arenormalised with respect to the diameter and the convective time: it would be possibleto get more physical insight by normalising with respect to the spatial and temporalturbulence scales. However, the use of diameter and convective time simplifies theinvestigation since these values are known a-priori.21.61.20.80.40Cd100 300 500 700 900t/T(a) (b)Fig. 1 Drag coefficient history (a) and instantaneous vorticity magnitude field (b).10-1100101102fD/U10-1010-5100Power(a)10-1100101102fD/U10-1010-5100Power(b)Fig. 2 Spectrum for the streamwise (a) and normal (b) velocity at 𝑥/𝐷 = 4 and 𝑦/𝐷 = 0.6 Andrea Ferrero, Francesco Larocca, Guglielmo Scovazzi and Massimo Germano-2 -1 0 1 200.050.10.150.20.25Exp100T - Z/205T - Z(a)-2 -1 0 1 200.10.20.30.4(b)Fig. 3 Effects of the choice on the spanwise extension and time window size on 𝑅11 (a) andcomparison of the Reynolds stresses in the three directions (b) for the control station at 𝑥/𝐿 = 2.02.-2 -1 0 1 2y/D00.20.40.60.81Mt, MzMtMz(a)-2 -1 0 1 200.050.10.150.2Expp=2p=0(b)Fig. 4 Indices of statistical resolution (a) and element-wisemodal filtering (b) for the control stationat 𝑥/𝐿 = 2.02.Acknowledgements We aknowledge the CINECA award under the ISCRA initiative for the avail-ability of high performance computing resources and support.References1. A Ferrero, F Larocca,MGermano andGScovazzi (2020) A study on the statistical convergenceof turbulence simulations around a cylinder, AIP Conference Proceedings, 2293(1):2000022. MR Mankbadi and NJ Georgiadis (2015) Examination of parameters affecting large-eddysimulations of flow past a square cylinder, AIAA J 53:1706–17123. F Bassi, L Botti, A Colombo, DADi Pietro and P Tesini (2012) On the flexibility of agglomera-tion based physical space discontinuous Galerkin discretizations, J Comput Phys 231(1):45–654. P Parnaudeau, J Carlier, D Heitz and E Lamballais (2008) Experimental and numerical studiesof the flow over a circular cylinder at Reynolds number 3900, Phys Fluids 20:0851015. A Abbà, L Bonaventura, M Nini and M Restelli (2015) Dynamic models for Large EddySimulation of compressible flows with a high order DG method, Comput Fluids 122:209–222

A Numerical Study of the Spanwise Turbulence Past a Cylinder Flow

https://iris.polito.it/bitstream/11583/2934981/1/paper.pdf

A Numerical Study of the Spanwise Turbulence Past a Cylinder Flow

Abstract

Similar works

Full text

Available Versions

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)