M.BELAN

S.SCARSOGLIO

TORDELLA D.

English

TORDELLA, Daniela

SCARSOGLIO, STEFANIA

M. BELAN

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

05 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEAnalysis of the convective instability of the two-dimensional wake / TORDELLA D.; S.SCARSOGLIO; M.BELAN. -1(2005), pp. 77-77. ((Intervento presentato al convegno 22nd IFIP TC7 CONFERENCE ON SYSTEM MODELING ANDOPTIMIZATION tenutosi a TURIN, ITALY nel JULY 18-22,2005.OriginalAnalysis of the convective instability of the two-dimensional wakePublisher:PublishedDOI:Terms of use:openAccessPublisher copyright(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: 11583/1418570 since:Analysis of the convective instability of the two-dimensional wakeD.Tordella1, S.Scarsoglio1, M.Belan21) Politecnico di Torino, Dipartimento di Ingegneria Aeronautica e Spaziale, Corso Duca degliAbruzzi 24, 10129 Torino daniela.tordella@polito.it2) Politecnico di Torino, C.so duca degli Abruzzi 24, 10129 Torinostefania.scarsoglio@studenti.polito.it3) Politecnico di Milano, Dipartimento di Ingegneria Aeronautica e Spaziale, Via La Masa 34,20156 Milano belan@aero.polimi.itKey words: instability, convective, nonparallelAbstractWe present an instability study that exploits, as basic-flow to be per-turbed, a new asymptotic expansion solution for the intermediate and far2D bluff body wake (Belan & Tordella, 2002; Tordella & Belan, 2003).The solution comes from the matching between an inner Navier-Stokesflow and an outer Navier-Stokes flow. The adoption of the Navier-Stokesmodel also in the outer region is a new approach, which takes into accountthe fact that the nonlinear convection and the diffusion are comparablein the lateral far field. The matching criteria are based on the vorticity,entrainment and pressure gradient matching. This solution yields an ac-curate representation of the cross momentum and pressure distributions.These last are properties of the mean-flow never included – to our knowl-edge – into a linear analysis of the convective instability that is framed onthe parametric Orr-Sommerfeld problem. Here, through a spatio-temporalmultiscaling which yields to an inhomogeneous Orr-Sommerfeld equation,the hypothesis of near parallelism is explored. The variation, with thestreamwise coordinate and the Reynolds number, R, of the instabilitycharacteristics is obtained. This allows to discuss the limits of the nearparallel theory, see, e.g., Tordella & Belan, 2004, where the quasi-steadyanalysis of the instabilty was performed through a multiscaling limited tothe space variable, and the review by Huerre & Monkewitz, 1990).1In the wake flow solution here considered as basic flow (Belan &Tordella, 2002; Tordella & Belan, 2003), the lateral decay results tobe algebraic at high orders in the inner expansion solution. Usingthe quasi-similar transformation x = x, η = x−1/2y the velocitycomponents, up to O(x−3/2), can be written asu = 1 + x−1/2φ1(η) + x−1φ2(η) + x−3/2φ3(η) (1)v = x−1χ1(η) + x−3/2χ2(η) (2)The multiscale approach, that introduces the slow spatial and tem-poral variables, x1 = εx, t1 = εt with ε = 1/R, yields η =(Rx1)−1/2y , u = ∂yΨ = u0(x1, y) + εu1(x1, y) + · · · , v = −∂xΨ =−ε∂x1Ψ = εv1(x1, y) + · · ·, where Ψ is the stream function of themean flow. This leads to the the perturbation hypothesisψ = [ϕ0(x1, y, t1) + εϕ1(x1, y, t1) + · · ·] eiθ(x,t;ε). (3)According to the Whitham theory (Whitham, 1974), ∂xθ = h0 =k0 + is0, ∂tθ = −σ = −(ω + ir), where h0 is the complex dimen-sionless wave number and σ is the complex dimensionless pulsation.The imaginary parts of h0 and σ are the spatial and temporal growthrates, respectively. In terms of x1 and θ the spatial and temporalderivatives transform like∂x → h0∂θ + ε∂x1 , ∂y → ∂y, ∂t → −σ0∂θ + ε∂t1 . (4)Following Bouthier (1972), by applying the previous derivative tran-formations to the linearized equations for the perturbation of thestream function, a hierarchy of ODE is obtained. The zero orderequation is the parametric Orr-Sommerfeld equation, where x1 andthe Reynolds numberR are parameters,Aϕ0 = σBϕ0 , (5)with A = {(∂2y − h20)2 − ih0R [u0(∂2y − h20)− u′′0]} , B = −iR(∂2y −h20). By writing ϕ0(x1, t1, y) = A(x1, t1)ζ0(x1, y) , (A: slow spatio-temporal modulation), (5) remains parametric in x1. The unknownamplitude A describes the transmission of the instability wave fromone near parallel region to the next and is determined at the nextorder. The first order inhomogenous multiscaling equation is:Aϕ1 = σBϕ1 +Mϕ0. (6)The operator M isM ={ [R(2h0σ − 3h20u0 − u′′0) + 4ih30]∂x1+(Ru0 − 4ih0)∂3x1yy −Rv1(∂3y − h20∂y) +Rv′′1∂y+ih0R[u1(∂2y − h20)− u′′1]+R(∂2y − h20)∂t1}. (7)Notice that M is function of the basic flow, in particular of thetransversal momentum v1 and thus of the entrainment, as well of the2zero order dispersion relation and eigenfunction. To avoid secularterms in the solution of (6), a solvability condition has to be satisfied.This condition requires that the inhomogeneous term is orthogonalto every solution of the adjoint homogenous problem. In short form,by means of the substitution A(x1, t1) = ea(x1,t1), Bouthier (1972),and going back to the original coordinates:∂ta+K2∂xa+ εK1 = 0 (8)where coefficients K1 e K2 are not singular. This modulation equa-tion is solved numerically, specifying the accessory conditions on thedomain of validity of the perturbative theory (from a few body-scalesd downstream of the body, x = xb, to the far field, xf À d. Hav-ing one boundary condition only, it is better to use it to imposethe far field uniformity to the solution. With regards to the ini-tial condition, it is opportune to write a = eθ1. By recalling thatϕ1(x1, y, t1) = A(x1, t1)ζ1(x1, y) , and that the complete solution(order 0 + order 1) isψ = (ϕ0 + εϕ1)eiθ = A((x1, t1)(ζ0 + εζ1)eiθ = (ζ0 + εζ1)eiθ+iθ1 , (9)the complete phase will then be Θ = θ + θ1. In this way the initialcondition can be given in terms of a distribution of phases. We con-sidered the phase distribution corresponding to zero order spatial xsequence of saddle points, ∂h0σ = 0. Due to the multiscaling, thewave number is h = ∂Θ/∂x = h0∂Θ/∂θ+ε∂Θ/∂x1 = h0+ε∂θ1/∂x1and the pulsation is −σ = −∂Θ/∂t = −σ0∂Θ/∂θ + ε∂θ1/∂t1 =−σ0 + ε∂θ1/∂t1. The first order correction of the comples insta-bility characterisitcs are thus obtained as h1 = ∂θ1/∂x1 and σ1 =−∂θ1/∂t1. In fig.1 the distribution of the module of the coefficientsof the modulation equation is shown. Fig.2 shows the first ordercorrection of the instability charactheristics, which increases withR and moving towards the body. From this, fixing a threshold forthe maximum accettable corrections proportional to ε, it is possibleto discuss both the limits of validity of the locally parallel theory,and the portion of the wake where the extension to the second ordercan be consider meaningful. It is observed, that even if the position(x ≈ 10) where the absolute instability (r > 0) is reached wouldbe accettable on the base of the actual laboratory knowledges, theperturbative theory there is no more reliable.Belan, M.; Tordella, D. 2002, ZAMM, 82 4, 219-234.Bouthier, M. 1972, J. Mec., 11, 4, 599-621.Huerre, P.: Monkewitz.P.A. 1990, Annu. Rev. Fluid Mech., 22, 473-537.Tordella, D.; Belan, M. 2003, Phys. Fluids 15 (7), 1897-1906.Tordella, D.; Belan, M., to appear, ZAMM, 84, December 2004.Whitham, G. B. 1974 John Wiley & Sons.30 20 40 6001234x |εΚ1|  |Κ2|   ε K1| |K2| Re 35 50 100 | Figure 1: Coefficients of the modulation equation.List of results– The position where the wave number and spatial growth rate of thelocal saddle point coincide with those of the pertubation is called x0: x0is thus the parameter of the perturbation, let us call it the center of theperturbation.– R is the flow control parameter◦ Moving downstream the center of the perturbation, the absoluteinstability condition is never reached in the nearby of the near wake, evenrising R to the value 100.• Placing x0 close to the near wake x0 = 10, the first order (supposingto accept the first order correction without restriction) yields absoluteinstability (AI, i.e. r=0) at x = 15 and R=100, nearly AI at x ∼ 6 atR=50 (extrapolate data)◦ k variation:large in the left region of the domain (next to the near wake), k(x),when R=const, decreases with x0, k(x), when x0 =const, increases withR.• ω, pulsation:increases with R, decreases with x0the order one correction are larger close to the near wake, their inten-sity grows with R◦ regions of minimal (less) stability (regions about the points x = xlswhere r = rmax)the maximum of r(x) is always placed upstream with respect to theperturbation center (x0)when x0 is low, in the points of less stability, rmax grows with Rwhen R=35 and x0 rises, the position of less stability settle near x ∼ 15for higher R, the less stable points are placed in locations where x < x0(e.g., at R=50, xls ∼ 10 for x0 ∼ 21, xls ∼ 18 for x0 ∼ 40; at R=100,xls ∼ 14 for x0 ∼ 21, xls ∼ 25 for x0 ∼ 40)the maximum values of r decreases with x0 and, beyond a point abso-lute maximum (presumably near Rcr = 40,) decreases with R•• analysis of the limits of the near-parallel stability theory.40 20 40 6000.20.40.60.8xWave numberk0 k Re 35 50 100 k 0 20 40 600123xFrequencyω ω0 ω Re 35 50 100 0 20 40 60−1012xTemporal growth rater0 r Re 35 50 100 r 0 20 40 60−2.2−1.8−1.4−1−0.6xSpatial growth rates0 s Re 35 50 100 s Figure 2: Instability characterisitcs downstream of the body, x is the distancefrom the body; - - - zero order, —– first order5

Analysis of the convective instability of the two-dimensional wake

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Analysis of the convective instability of the two-dimensional wake

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