This letter aims at understanding the dynamical process that leads to the onset of chaos in the flow past a blunt-based axisymmetric bluff body. On the basis of direct numerical simulations, conducted for Reynolds numbers ranging from 100 to 900, we show that the flow undergoes multiple transitions, successively giving rise to the SS, RSPa, RSPb, RSPc and RSB wake states. In particular, the RSPc state, revealed in this work via long-term computations, is characterized by intermittent vortex stretching denoting the onset of chaos before the symmetry breaking and the occurence of the RSB state

Bohorquez

Choi

Eckmann

Fabre

Jardin

Jeong

Thierry Jardin

Tomboulides

Yannick Bury

English

Crossref

Physics Letters A

Transitions to chaos in the wake of an axisymmetric bluff body

Bury, Yannick

Jardin, Thierry

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                             This is an author-deposited version published in: http://oatao.univ-toulouse.fr/  Eprints ID: 6503  To link to this article: DOI: 10.1016/j.physleta.2012.09.011 URL: http://dx.doi.org/10.1016/j.physleta.2012.09.011   To cite this version: Bury, Yannick and Jardin, Thierry Transitions to chaos in the wake of an axisymmetric bluff body. (2012) Physics Letters A, vol. 376 (n° 45). pp. 3219-3222. ISSN 0375-9601 Open Archive Toulouse Archive Ouverte (OATAO)  OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible.   Any correspondence concerning this service should be sent to the repository administrator: staff-oatao@inp-toulouse.fr  Transitions to chaos in the wake of an axisymmetric bluff bodyYannick Bury, Thierry JardinUniversite´ de Toulouse, ISAE, 10, avenue Edouard Belin, 31055 Toulouse, FranceAbstractThis letter aims at understanding the dynamical process that leads to the onset of chaos in the flow past a blunt-basedaxisymmetric bluff body. On the basis of direct numerical simulations, conducted for Reynolds numbers ranging from100 to 900, we show that the flow undergoes multiple transitions, successively giving rise to the SS, RSPa, RSPb, RSPcand RSB wake states. In particular, the RSPc state, revealed in this work via long-term computations, is characterizedby intermittent vortex stretching denoting the onset of chaos before the symmetry breaking and the occurence of theRSB state.Keywords: flow instabilities, chaos, wake, bluff body1. IntroductionThe thorough analysis of bluff body wakes is essentiallytwofold. First, it advances the knowledge of fundamentalmechanisms driving the destabilization of massively sep-arated flows, i.e. the transition between dominant flowinstabilities. Second, as an ultimate goal, it serves the def-inition of optimal flow control strategies through the ma-nipulation of such flow instabilities. As such, it has beenproven, for low Reynolds number flows, that forcing theflow at specific forcing frequencies/wavelengths can dras-tically modify the wake topology and the drag experiencedby the bluff body [1, 2, 3]. Moreover it is reasonable to as-sume that instabilities observed at low Reynolds numbersstill persist at much higher Reynolds numbers. An evi-dent but still fascinating example is the clear occurence ofthe von Ka´rma´n vortex shedding in the wake of geologicalobstacles, for Reynolds numbers O(109). Therefore onecan expect a flow control strategy, primarily optimized atlow Reynolds number, to be efficient at higher Reynoldsnumber. In this sense the effort to enhance knowledge ofthe flow destabilization towards chaotic states is a prereq-uisite for the potential definition of optimal flow controlstrategies at higher Reynolds numbers. In this letter, thewake instabilities behind a blunt-based axisymmetric bluffbody at various Reynolds numbers, ranging from 100 to900, are analysed. Wakes of axisymmetric bluff bodiesat low Reynolds number have gained interest due to thevariety of instabilities that develop as the Reynolds num-ber is increased, eventually leading to chaotic state. Nu-merous studies focused on spheres, revealing transitions ofthe wake structure from 1) an axisymmetric steady stateEmail address: yannick.bury@isae.fr,thierry.jardin@isae.fr (Yannick Bury, Thierry Jardin)(corresponding to the base flow) to 2) a reflectional sym-metry steady state (SS), 3) a reflectional symmetry pre-serving (RSP) oscillatory state, and 4) a reflectional sym-metry breaking (RSB) oscillatory state [4, 5]. Such tran-sitions and resulting wake structures exhibit similaritieswith wakes that develop downstream blunt-based axisym-metric bluff bodies [6, 7]. However this geometry is stillpoorly documented despite its practical interest and a lackof comprehension subsists regarding the scenario that leadsto a chaotic state.In order to decipher this scenario, we have conducted3D direct numerical simulations of the flow past a blunt-based axisymmetric bluff body of length-to-diameter ratioL/D = 7 for Reynolds numbers 100 < Re < 900, spanningthe transitions up to chaos.2. Numerical methodsThe three-dimensional time-dependent incompressibleNavier-Stokes equations around a blunt-based axisymmet-ric bluff body of diameter D and length L are directlysolved using an Eulerian finite volume method. The Reynoldsnumber Re, based on D and on the free stream velocityU∞, is varied from 100 to 900. Assuming incompressibleviscous flow, the equations read:~∇.~v = 0 (1)∂~v∂t+ (~v.~∇)~v = −1ρ~∇p+ ν∇2~v (2)Here ~v is the velocity, p the pressure, ρ and ν the fluiddensity and kinematic viscosity respectively.The geometry consists of a 4:1 semi-elliptic nose anda cylindrical aft section. It is enclosed in a cylindricalcomputational domain of diameter 20D and length 55D(a) (b)Figure 1: Computational domain (a) and illustration of the surfacic polyhedral mesh (b).aligned with the free stream direction ~x (figure 1(a)). Thecoordinates’ origin is located at the nose tip of the body,corresponding to the stagnation point. The inlet bound-ary is located at x/D = −8 and is subjected to a velocityDirichlet condition. A similar condition is imposed on thetubular surface of the computational domain. A zero dif-fusion flux condition is prescribed at the outlet boundarylocated at x/D = 47. The surface of the blunt-based ax-isymmetric bluff body is modelled as non-slip surface. Thedomain is composed of 2.3 × 106 polyhedral cells (figure1(b)).The spatial and temporal discretizations are achievedusing second-order upwind schemes and second-order im-plicit time-stepping method respectively. The pressure-velocity coupling is obtained using a SIMPLE algorithm.Time step is fixed in order to satisfy the CFL condition(Courant Number close to unity) for each Reynolds num-ber.3. ResultsFigure 2 illustrates a sequence of the flow topology ob-tained for Re numbers ranging from 100 to 450, in terms ofiso-surfaces of normalized λ+2 [8] (normalization is basedon the minimum value of λ2 for each Reynolds numberconsidered).At Re = 100, figure 2 depicts an opened torus of rev-olution at the base of the body. The associated recircu-lation region is axisymmetric. Between Re = 100 andRe = 400 the torus radially expands towards the exter-nal flow and progressively stretches along the streamwisedirection. This scenario results from the concomitant in-fluence of strong shear and suction effects on the torus.While the low-pressure region maintains the opened torusclose to the bluff body base, the external flow tends tostretch the periphery of the opened torus.As the Reynolds number is increased up to Re ≈ 450the torus progressively shifts away from the bluff bodyaxis of revolution. Hence, the upper part of the torus issubjected to stronger shear than the lower part, such thatit distorts into a pelvis-like structure. This is associatedwith the loss of axisymmetry of the torus, correspondingto the destabilization of the wake towards the reflectionalsymmetry steady state (SS). At this stage only a singlestreamwise planar symmetry is retained whose azimuthalposition fixes randomly.The streamwise vorticity fields provide additional in-formation on the topology of the flow where the asymme-try is reflected as a ‘double-threaded’ wake composed oftwo primary vorticity lobes distributed on both sides ofthe reflectional symmetry plane (figure 3). Secondary andternary vorticity lobes of alternate vorticity signs are alsoobserved in the near wake and on the body aft surface re-spectively. As the Reynolds number is further increased,the tails of the primary vorticity lobes elongate and shiftaway from the axis of revolution of the bluff body (seealso [7]), in accordance with the pelvis-like structure dis-placement and distortion under free stream-induced shearstresses. At this stage the pelvis-like structure is subjectedto increasing shear stresses in conjunction with its dis-placement towards the external flow.Eventually as the Reynolds number reaches values closeto 590, the structure can no more sustain such stressesand is partially torn. This leads to the formation of un-steady hairpin structures periodically shed in the wakeof the blunt-based axisymmetric bluff body (figure 4(a:right)). The reflectional symmetry is still preserved, de-noting the transition from the SS state to the RSP os-cillatory state. Figure 4(a: left) depicts iso-surfaces ofstreamwise vorticity ω+ at Re = 600. It displays regu-larly spaced vorticity lobes shed at a non-dimensional fre-quency Sta = faD/U∞ ≈ 0.12. At this stage the flowexhibits a unique dominant frequency, revealed in figure5(a) by a single fundamental peak (and its first harmonic)on the FFT of the drag coefficient history. The funda-mental peak amplitude increases as Re is varied from 590to 680. Hereafter this single frequency RSP state will bedenoted RSPa.Beyond Re = 690 a second fundamental peak (and itsfirst harmonic) arises at Stb ≈ 0.02 (figure 5(b)), denot-ing the occurence of low frequency vorticity bursts thatpromote the transient waving of the primary lobes and ofthe associated hairpin structures in the reflectional sym-metry plane (figure 4(b)). The occurence of this secondfundamental peak is accompanied with beating frequen-cies Sta − Stb and Sta + Stb that result from its couplingwith the first fundamental peak. This double frequencyRSP state will be denoted RSPb. Note that the second tofirst fundamental peaks ratio increases with Re, which in-dicates that drag fluctuations, initially driven by the shed-ding of hairpin vortices at Sta are progressively dominatedby the vorticity bursts and the associated transient waving2(a)(b)(c)Re = 100 200 300 400 420 430 450Figure 2: Iso-surfaces of λ+2 = −0.02 obtained for Re numbers ranging from 100 to 450. Solid line depicts the base of the bluff body. (a), (b)and (c) correspond to rear, lower and side views respectively.(a) primary lobe↓secondary↓ternary↓(b)Figure 3: Normalized streamwise vorticity ω+ = ωD/U∞ obtained for Re numbers 450 (a) and 550 (b). Left: lower and side views ofiso-surfaces ω+ = ±0.05. Right: rear view of isolines |ω+| ∈ [0.05; 0.15] obtained at x/D = 14 (dark: positive, light: negative).(a)(b)(c)Figure 4: Lower and side views of the flow structure obtained for Re numbers 600 (a), 700 (b) and 800 (c). Left: iso-surfaces ω+ = ±0.05(dark: positive, light: negative). Right: iso-surfaces λ+2 < 0.3(a)St600aSt660aSt680a2St660a 2St680a(b)StaStbSta + StbSta − Stb(c)StaStb2StbSta + StbSta − Stb(d)Stb2Stbdamping↑• • • •Figure 5: FFT of the drag coefficient history obtained for Re numbers 600 (dotted line), 660 (dashed line), 680 (plain line) (a), 700 (b), 750(c) and 800 (inset picture depicts the drag coefficient history) (d).of the wake at Stb.Beyond Re = 750 a broadband noise appears on theFFT spectrum of the drag coefficient history (figure 5(c,d)), revealing the transition to chaos potentially inducedby the occurence of a third fundamental frequency, follow-ing the Ruelle-Takens-Newhouse scenario [9]. At this stagethe analysis of this chaotic state deserves a deeper atten-tion. To this avail a 12000 time units computation hasbeen conducted at Re = 800, which roughly correspondsto the upper bound of the RSPc state. Here the time unitis defined as t+ = tU∞/D. This long term computationallows both a better resolution of the broadband noise andthe capture of very low frequency instabilities. The promi-nent feature of the RSPc state is the transient stretchingof the hairpin vortices in the streamwise direction (figure4(c)), whose associated vorticity lobes remind the ‘double-threaded’ wake observed for the SS state. This could sug-gest a transient restabilization of the wake. It results inthe damping of the drag oscillations, clearly observed onthe drag coefficient history illustrated on the inset pictureof figure 5(d). It is remarkable that this damping occursintermittently. In that sense, this phenomenon is a markerof the chaotic state of the flow associated with the poten-tial emergence of a third instability.The further increase of the Reynolds number finallybreaks the reflectional symmetry and triggers the RSBstate. For Re = 900, phases of ‘apparent’ reflectional sym-metry (figure 6(a)) whose flow topology ressembles thatof the RSPb state are randomly interrupted by dramaticchanges of the symmetry plane azimuthal position (figure6(c)). Such changes, also characterized by strong inter-mittent oscillations of the lateral loads (figure 7(b)), oc-cur immediately after a transient phase of wake twisting(figure 6(b)). In addition the long-term, 9000 time unitscomputation achieved for Re = 900 reveals the persistenceof the intermittent damping of the drag coefficient (figure7(a)) and of the associated vorticity lobes stretching (fig-ure 6(d)), previously observed at Re = 800.4. ConclusionThe transition to chaos of the flow past a blunt-basedaxisymmetric bluff body can thus be deciphered as a suc-cession of bifurcations associated with distinct wake pat-terns. In particular the dynamics of the recirculation re-gion from an axisymmetric torus (base flow) to a pelvis-likestructure (SS state) subsequently leads to the oscillatoryRSP state. This RSP state can be subdivided into threesubstates, whether the flow unsteadiness is characterizedby a single fundamental frequency (RSPa), two distinctfundamental frequencies (RSPb) or a chaotic dynamics(RSPc). The latter exhibits flow intermittencies associatedwith the transient stretching of the hairpin structures andthe damping of the drag oscillations. The scenario revealedin this work shows strong similarities with the Ruelle-Takens-Newhouse scenario, although the precise identifi-cation of the ‘chaos triggering’ frequency is still unclearand requires deeper investigation. In addition these inter-mittencies still persist beyond the bifurcation towards theRSB state whose wake topology consists in a succession ofphases of ‘apparent’ reflectional symmetry, wake twistingand dramatic reorientation of the ‘apparent’ reflectionalsymmetry.4(a)(c)(b)(d)Figure 6: Iso-surfaces ω+ = ±0.05 (dark: positive, light: negative) of the flow obtained for Re = 900 at different phases of the RSB state.‘Apparent’ reflectional symmetry (a), wake twisting (b), reorientation of the ‘apparent’ reflectional symmetry (c), vorticity lobes stretching(d).(a)Stb2Stb damping↑•(b)Figure 7: FFT of the drag coefficient history obtained for Re = 900 (inset picture depicts the drag coefficient history) (a) and lateral forcecoefficient history (b).AcknowledgementsThe authors gratefully acknowledge the French Min-istry of Defence and the DGA for supporting this work.References[1] J. Kim and H. Choi 2005, Distributed forcing of the flow over acircular cylinder. Phys. Fluids, 17:033103.[2] H. Choi, W.P. Jeon and J. Kim 2008, Control of flow over abluff body. Annu. Rev. Fluid Mech., 40:113-139.[3] T. Jardin and Y. Bury 2012, Spectral and lagrangian analysis ofthe forced flow past a circular cylinder using pulsed tangentialjets. J. Fluid Mech., 696:285-300.[4] A. G. Tomboulides and S.A. Orszag 2000, Numerical investiga-tion of transitional and weak turbulent flow past a sphere. J.Fluid Mech., 416:51-73.[5] D. Fabre, F. Auguste and J. Magnaudet 2008, Bifurcations andsymmetry breaking in the wake of axisymmetric bodies. Phys.Fluids, 20:051702.[6] V. Schwarz, H. Bestek and H. Fasel 1994, Numerical simulationof nonlinear waves in the wake of an axisymmetric bluff body.In 25th AIAA Fluid Dynamics Conference, AIAA-94:2285.[7] P. Bohorquez, E. Sanmiguel-Rojas, A. Sevilla, J.I. Jimenez-Gonzalez and C. Martinez-Bazan 2011, Stability and dynamicsof the laminar wake past a slender blunt-based axisymmetricbody. J. Fluid Mech., 676:110-144.[8] J. Jeong and F. Hussain 1995, On the identification of a vortex.J. Fluid Mech., 285:69-94.[9] J.P. Eckmann 1981, Roads to turbulence in dissipative dynam-ical systems. Rev. Mod. Phys., 53:643-654.5

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Transitions to chaos in the wake of an axisymmetric bluff body

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