An exploratory analysis of the transient and long term behaviour of small 3D perturbations in the circular cylinder wake
AAn exploratory analysis of the transient and long termbehavior of small 3D perturbations in the circularcylinder wake
S. Scarsoglio (cid:93) , D. Tordella (cid:93), ∗ and W. O. Criminale (cid:91) (cid:93) Dipartimento di Ingegneria Aeronautica e Spaziale, Politecnico di Torino, 10129Torino, Italy, (cid:91)
Department of Applied Mathematics, University of Washington, Seattle, WA 98195-2420, USA
Abstract
An initial-value problem (IVP) for arbitrary small three-dimensional vorticityperturbations imposed on a free shear flow is considered. The viscous perturbationequations are then combined in terms of the vorticity and velocity, and are solvedby means of a combined Laplace-Fourier transform in the plane normal to thebasic flow. The perturbations can be uniform or damped along the mean flowdirection. This treatment allows for a simplification of the governing equations suchthat it is possible to observe long transients, that can last hundreds time scales.This result would not be possible over an acceptable lapse of time by carryingout a direct numerical integration of the linearized Navier-Stokes equations. Theexploration is done with respect to physical inputs as the angle of obliquity, thesymmetry of the perturbation and the streamwise damping rate. The base flow a r X i v : . [ phy s i c s . f l u - dyn ] M a r s an intermediate section of the growing two-dimensional circular cylinder wakewhere the entrainment process is still active. Two Reynolds numbers of the orderof the critical value for the onset of the first instability are considered. The earlytransient evolution offers very different scenarios for which we present a summaryfor particular cases. For example, for amplified perturbations, we have observedtwo kinds of transients, namely (1) a monotone amplification and (2) a sequenceof growth - decrease - final growth. In the latter case, if the initial condition isan asymmetric oblique or longitudinal perturbation, the transient clearly shows aninitial oscillatory time scale. That increases moving downstream, and is differentfrom the asymptotic value. Two periodic temporal patterns are thus present in thesystem. Furthermore, the more a perturbation is longitudinally confined the moreit is amplified in time. The long-term behavior of two-dimensional disturbancesshows excellent agreement with a recent two-dimensional spatio-temporal multiscalemodal analysis and with laboratory data concerning the frequency and wave lengthof the parallel vortex shedding in the cylinder wake. Keywords:
Initial-value problem (IVP) - perturbation - early transient - spa-tially developing flows -instability - absolute - periodic temporal patterns - growth- three-dimensional. * corresponding author, e-mail: [email protected]
Recent shear flows studies ([1], [2], [3]) have shown the importance of the early time dy-namics, that in principle can lead to non-linear growth long before an exponential modeis dominant. The recognition of the existence of an algebraic growth, due - among otherreasons - to the non-orthogonality of the eigenfunctions ([4]) and a possible resonance be-tween Orr-Sommerfeld and Squire solutions ([5]), recently promoted many contributionsdirected to study the early-period dynamics. For fully bounded flows works by [3], [6],[7], [8], [9], [10], and for partially bounded flows works by [11], [12], [13], [14] can be cited.2s for free shear flows, the attention was first aimed in order to obtain closed-form so-lutions to the initial-value inviscid problem ([15], [16]) and was successful by consideringpiecewise linear parallel basic flow profiles.An interesting aspect observed in the intermediate periods is that the maximal ampli-fication is generally associated to oblique disturbances, that, as a consequence, potentiallycan promote early transition, see e.g. [13]. In fact, these perturbations, which are asymp-totically stable at all Reynolds numbers, are the perturbations best exploiting the energytransient amplification.In this work we consider as a prototype for free shear flow the two-dimensional wakepast a bluff body. The wake stability has been widely studied by means of modal anal-yses (e.g. [17], [18], [19], [20]). However, in this way only the asymptotic fate can bedetermined, regardless of the transient behavior and the underlying physical cause of anyinstability.In this work we adopt the velocity-vorticity formulation to evaluate the general initial-value perturbative problem. This method was proposed by Criminale, Drazin and co-authors in the years 1990-2000 ([2], [14], [16], [6], [11]). In synthesis, the variables areLaplace-Fourier transformed in the plane normal to the basic flow. Afterward, the result-ing partial differential equations in time are integrated numerically. This procedure allowsfor completely arbitrary initial expansions by using a known set of functions (Schauderbasis in the L space) and yields the complete dynamics — the early time transients andthe asymptotic behavior (up to many hundred time scales) — for any disturbance. Thelong term dynamics would not, in fact, have been easily recovered by using the alternativemethod of the direct numerical integration of the linear equations since the integrationover a range of time larger than a few dozen basic time scales is not feasible.The base flow model that we employ includes the wake transversal velocity and thusthe nonlinear and diffusive dynamics that are responsible for the growth of the flow andthe associated mass entrainment. We consider the first two order terms of the analyticalNavier-Stokes expansion obtained by Tordella and Belan (2002, 2003) ([21], [22]), see § x , the longitudinal coordinate, and the Reynolds number Re .We use a complex wavenumber for the disturbance component aligned with the flowso that longitudinal spatially damped waves are represented. It should be observed thata longitudinal spatial growth could not be considered physically admissible as an initialcondition since the energy density of the initial perturbation would be infinite. In thecontext of the initial-value problem, this is an innovative feature adopted to introduce apossible spatial evolution (damping) of the perturbative wave in the longitudinal direc-tion. The perturbative equations are numerically integrated by the method of lines. Theequations formulation and initial and boundary conditions are presented in § , x = 10, of the intermediateregion of the flow where the entrainment process is active. A comparison with a base flowfar field configuration, x = 50, is also proposed. The normalized perturbation kineticenergy density is the physical quantity on which the transient growth is observed (see § § § §
4. 4
Initial-value problem
The base flow is considered viscous and incompressible. To describe the two-dimensionalgrowing wake flow, an expansion solution for the Navier-Stokes two-dimensional steadybluff body wake ([22], [21]) has been used. The x coordinate is parallel to the free streamvelocity, the y coordinate is normal. This approximated analytical Navier-Stokes solutionincorporates the effects due to the full non linear convection as well as the streamwiseand transverse diffusion. The solution was obtained by matching an inner Navier-Stokesexpansion in terms of the inverse of the longitudinal coordinate x ( x − n/ , n = 0 , , , . . . )with an outer Navier-Stokes expansion in terms of the inverse of the distance from thebody.Here we take the first two orders ( n = 0 ,
1) of the inner longitudinal component of thevelocity field as a first approximation of the primary flow. In the present formulation thenear-parallel hypothesis for the base flow, at a longitudinal position x = x , is made. Thecoordinate x plays the role of parameter of the steady system together with the Reynoldsnumber. The analytical expression for the profile of the longitudinal component is U ( y ; x , Re ) = 1 − aC x − / e − Re4 y x (1)where a is related to the drag coefficient ( a = 14 ( Re/π ) / c D ( Re )) and C is an integrationconstant depending on the Reynolds number. As said in the introduction, this two termsrepresentation is extracted from an analytical asymptotic expansion where the velocityvector and the pressure are determined to the forth order. It should be observed thatthe transversal velocity component V first appears at the third order ( n = 2), while thepressure only at the forth order ( n = 3). Up to the second order, the field is thus parallel.Beyond the second order the analytical expression becomes much more complex, specialfunctions as the confluent hypergeometric functions plays a role ([21]) associated to thedeviation from parallelism. By changing the x values, the base flow profile (1) will locally5igure 1: Wake schematic. Profile U f ( y ; x , Re ) in the intermediate ( x = 10) and far( x = 50) wake for different Reynolds numbers, U f is the free stream velocity. Thediameter of the cylinder is out of scale (three times) with respect to the wake profiles.approximate the behavior of the actual wake generated by the body. Here, the regionconsidered, if not otherwise specified, is fixed to a typical section, x = 10 D (where D isthe spatial scale of the wake) of the intermediate wake. The term intermediate is usedin the general sense as used by [28]: ’intermediate asymptotics are self-similar or near-similar solutions of general problems, valid for times, and distances from boundaries,large enough for the influence of the fine details of the initial/or boundary conditionsto disappear, but small enough that the system is far from the ultimate equilibriumstate...’. The distance beyond which the intermediate region is assumed to begin variesfrom eight to four diameters D for Re ∈ [20 ,
40] (see [21], [22]). Base flow configurationscorresponding to a Re of 50 ,
100 are considered. In Figure 1 a representation of the wakeprofile at differing longitudinal stations is shown.6 .2 Formulation
By exciting the base flow with small arbitrary three-dimensional perturbations, the con-tinuity and Navier-Stokes equations that describe the perturbed system are ∂ (cid:101) u∂x + ∂ (cid:101) v∂y + ∂ (cid:101) w∂z = 0 (2) ∂ (cid:101) u∂t + U ∂ (cid:101) u∂x + (cid:101) v ∂U∂y + ∂ (cid:101) p∂x = 1 Re ∇ (cid:101) u (3) ∂ (cid:101) v∂t + U ∂ (cid:101) v∂x + ∂ (cid:101) p∂y = 1 Re ∇ (cid:101) v (4) ∂ (cid:101) w∂t + U ∂ (cid:101) w∂x + ∂ (cid:101) p∂z = 1 Re ∇ (cid:101) w (5)where ( (cid:101) u ( x, y, z, t ), (cid:101) v ( x, y, z, t ), (cid:101) w ( x, y, z, t )) and (cid:101) p ( x, y, z, t ) are the perturbation velocitycomponents and pressure respectively. The independent spatial variables z and y aredefined from −∞ to + ∞ , while x is defined in the semispace occupied by the wake, from0 to + ∞ . All physical quantities are normalized with respect to the free stream velocity U f , the body scale D and the density. By combining equations (3) to (5) to eliminatethe pressure, the linearized equations describing the perturbation dynamics become( ∂∂t + U ∂∂x ) ∇ (cid:101) v − ∂ (cid:101) v∂x d Udy = 1 Re ∇ (cid:101) v (6)( ∂∂t + U ∂∂x ) (cid:101) ω y + ∂ (cid:101) v∂z dUdy = 1 Re ∇ (cid:101) ω y (7)where (cid:101) ω y is the transversal component of the perturbation vorticity field. By introducingthe quantity (cid:101) Γ, that is defined by ∇ (cid:101) v = (cid:101) Γ (8)we obtain three coupled equations (6), (7) and (8). Equations (6) and (7) are the Orr-Sommerfeld and Squire equations respectively, from the classical linear stability analysisfor three-dimensional disturbances. From kinematics, the relation (cid:101)
Γ = ∂ (cid:101) ω z ∂x − ∂ (cid:101) ω x ∂z (9)7hysically links together the perturbation vorticity components in the x and z directions( (cid:101) ω x and (cid:101) ω z respectively) and the perturbed velocity field. By combining equations (6)and (8) then ∂ (cid:101) Γ ∂t + U ∂ (cid:101) Γ ∂x − ∂ (cid:101) v∂x d Udy = 1 Re ∇ (cid:101) Γ (10)which, together with (7) and (8), fully describes the perturbed system in terms of vor-ticity. This formulation is a classical one. Alternative classical formulations, as thevelocity-pressure one, are in common use. We chose this formulation because the vortic-ity transport and diffusion is the principal phenomenology for the dynamics of a wakesystem. For piecewise linear profiles for U analytical solutions can be found. For con-tinuous profiles, the governing perturbative equations cannot be analytically solved ingeneral, but may assume a reduced form in the free shear case ([26]).Moreover, from the equations (7), (8) and (10), it is clear that the interaction of themean vorticity in z -direction (Ω z = − dU/dy ) with the perturbation strain rates in x and z directions ( ∂ (cid:101) v∂x and ∂ (cid:101) v∂z respectively) proves to be a major source of any perturbationvorticity production.The perturbation quantities are Laplace and Fourier decomposed in the x and z directions, respectively. A complex wavenumber α = α r + iα i along the x coordinateas well as a real wavenumber γ along the z coordinate are introduced. In order to havea finite perturbation kinetic energy, the imaginary part α i of the complex longitudinalwavenumber can only assume non-negative values. In so doing, we allow for perturbativewaves that can spatially decay ( α i >
0) or remain constant in amplitude ( α i = 0). Theperturbation quantities ( (cid:101) v, (cid:101) Γ , (cid:101) ω y ) involved in the system dynamics are now indicated as(ˆ v, ˆΓ , ˆ ω y ), where ˆ g ( y, t ; α, γ ) = (cid:90) + ∞−∞ (cid:90) + ∞ (cid:101) g ( x, y, z, t ) e − iαx − iγz dxdz (11)indicates the Laplace-Fourier transform of a general dependent variable in the α − γ phase space and in the remaining independent variables y and t . The governing partialdifferential equations are 8 ˆ v∂y − ( k − α i + 2 ikcos ( φ ) α i )ˆ v = ˆΓ (12) ∂ ˆΓ ∂t = − ( ikcos ( φ ) − α i ) U ˆΓ + ( ikcos ( φ ) − α i ) d Udy ˆ v + 1 Re [ ∂ ˆΓ ∂y − ( k − α i + 2 ikcos ( φ ) α i )ˆΓ] (13) ∂ ˆ ω y ∂t = − ( ikcos ( φ ) − α i ) U ˆ ω y − iksin ( φ ) dUdy ˆ v + 1 Re [ ∂ ˆ ω y ∂y − ( k − α i + 2 ikcos ( φ ) α i )ˆ ω y ] (14)where φ = tan − ( γ/α r ) is the angle of obliquity with respect to the x - y physical plane, k = (cid:113) α r + γ is the polar wavenumber and α r = kcos ( φ ), γ = ksin ( φ ) are the wavenumbersin x and z directions respectively. The imaginary part α i of the complex longitudinalwavenumber represents the spatial damping rate in the streamwise direction. In figure 2the three-dimensional perturbative geometry scheme is depicted.From equations (12)-(14), it can be noted that there can neither be advection nor pro-duction of vorticity in the lateral free stream. The vorticity can only be diffused sinceonly the diffusive terms remains in the limit when y → ∞ . Perturbation vorticity van-ishes in the free stream, regardless if it is initially inserted there (if inserted, vorticity isfinally dissipated in time when y → ∞ ). This means that the velocity field is harmonicas y → ∞ .Governing equations (12), (13) and (14) need proper initial and boundary conditionsto be solved. Among all solutions, those whose perturbation velocity field is bounded inthe free stream are sought. Periodic initial conditions forˆΓ = ∂ ˆ v∂y − ( k − α i + 2 ikcos ( φ ) α i )ˆ v (15)can be cast in terms of a set of functions in the L Hilbert space, asˆ v (0 , y ) = e − ( y − y ) cos( n ( y − y )) or ˆ v (0 , y ) = e − ( y − y ) sin( n ( y − y )) , n is anoscillatory parameter for the shape function, while y is a parameter which controlsthe distribution of the perturbation along y (by moving away, or bringing nearer, theperturbation maxima from the axis of the wake). The trigonometrical system is aSchauder basis in each space L p [0 , < p < ∞ . More specifically, the system(1 , sin ( n y ) , cos ( n y ) , . . . ), where n = 1 , , . . . , is the Schauder basis for the space ofsquare-integrable periodic functions with period 2 π . This means that any element of thespace L , where the dependent variables are defined, can be written as an infinite linearcombination of the elements of the basis.The transversal vorticity ˆ ω y is chosen initially equal to zero throughout the y domain inorder to ascertain which is the net contribution of three-dimensionality on the transver-sal vorticity generation and temporal evolution. However, it can be demonstrated that10he eventual introduction of an initial transversal vorticity does not actually affect theperturbation temporal evolution.Once initial and boundary conditions are properly set, the partial differential equa-tions (12)-(14) are numerically solved by the method of lines. The spatial derivativesare centre differenced and the resulting system is then integrated in time by an adap-tative multi-step method (variable order Adams-Bashforth-Moulton PECE solver). Thetransversal computational domain is large thirty body scales. By enlarging the com-putational domain to 50 and 100 body scales the results vary on the third and fourthsignificant digit, respectively. One of the salient aspects of the initial-value problem is to observe the early transientevolution of various initial conditions. To this end, a measure of the perturbation growthcan be defined through the disturbance kinetic energy density in the plane ( α, γ ) (see e.g[25],[26]) e ( t ; α, γ, Re ) = 12 12 y d (cid:90) + y d − y d ( | ˆ u | + | ˆ v | + | ˆ w | ) dy = 12 12 y d | α + γ | (cid:90) + y d − y d ( | ∂ ˆ v∂y | + | α + γ || ˆ v | + | ˆ ω y | ) dy, (16)where 2 y d is the extension of the spatial numerical domain. The value y d is defined so thatthe numerical solutions are insensitive to further extensions of the computational domainsize. Here, we take y d = 15. The total kinetic energy can be obtained by integratingthe energy density over all α r and γ . The amplification factor G ( t ) can be introduced interms of the normalized energy density G ( t ; α, γ ) = e ( t ; α, γ ) e ( t = 0; α, γ ) . (17)This quantity can effectively measure the growth of a disturbance of wavenumbers ( α, γ )11t the time t , for a given initial condition at t = 0 (Criminale et al. etal. rr ( t ; α, γ ) = log | e ( t ; α, γ ) | t , t > r asymptotes to a constant value( dr/dt < (cid:15) , where (cid:15) is of the order 10 − ). The angular frequency (pulsation) ω of theperturbation can be introduced by defining a local, in space and time, time phase ϕ ofthe complex wave at a fixed transversal station (for example y = 1) asˆ v ( y, t ; α, γ, Re ) = A t ( y ; α, γ, Re ) e iϕ ( t ) (19)and then computing the time derivative of the phase perturbation ϕ ([29]) ω ( t ) = dϕ ( t ) dt . (20)Since ϕ is defined as the phase variation in time of the perturbative wave, it is reasonableto expect constant values of frequency, once the asymptotic state is reached. We present a summary of the most significant transient behavior and asymptotic fate ofthe three-dimensional perturbations. The temporal evolution is observed in the interme-diate asymptotic region of the wake, which is the region where the spatial evolution ispredominant. It can be demonstrated that changing the number of oscillations n and12he parameter y that controls the perturbation distribution along the y direction canonly extend or shorten the duration of the transient, while the ultimate state is not al-tered. More specifically, if the perturbation oscillates rapidly or is concentrated mainlyoutside the shear region of the basic flow, for a stable configuration, the final damping isaccelerated while, for an unstable configuration, the asymptotic growth is delayed. Thus,these two parameters are not crucial, because their influence can be recognized a priori.Therefore, in the following we use the two reference values, n = 1 and y = 0, and focusthe attention mainly on parameters such as the obliquity, the symmetry, the value of thepolar wavenumber and the spatial damping rate of the disturbance. In particular, thepolar wavenumber k changes in a range of values reaching at maximum the order of mag-nitude O (1), according to what is suggested by recent modal analyses ([24], [23]). Theorder of magnitude of the spatial damping rate α i varies around the polar wavenumbervalue. Figure 3 takes into account the influence, on the early time behavior, of the perturbationsymmetry and of the wake region considered in the analysis, which is represented bythe parameter x . All the configurations considered are asymptotically amplified, butthe transients are different. The asymmetric cases (a) present, for both the intermediateposition x = 10 (solid curve) and the far field position x = 50 (dashed curve), twotemporal evolutions. For x = 10 a local maximum, followed by a minimum, is visible inthe energy density, then the perturbation is slowly amplifying and the transient can beconsidered extinguished only after hundreds of time scales. For x = 50 these featuresare less marked. It can be noted that the far field configuration ( x = 50) has a fastergrowth than the intermediate field configuration ( x = 10) up to t = 400. Beyond thisinstant the growth related to the intermediate configuration will prevail on that of thefar field configuration. In the symmetric cases (b) the growths become monotone afterfew time scales ( t = 20) and the perturbations quickly reach their asymptotic states13around t = 50). The intermediate field configuration ( x = 10, solid curve) is alwaysgrowing faster than the far field configuration ( x = 50, dashed curve). This particularcase shows a behavior that is generally observed in this analysis, that is, asymmetricconditions lead to transient evolutions that last longer than the corresponding symmetricones, and demonstrates that the transient growth for a longitudinal station in the farwake can be faster than in the intermediate wake. It should be noted that, even if theasymmetric perturbation leads to a much slower transient growth than that observedfor the symmetric case, the growth rate become equal when the asymptotic states arereached (see for example fig. 9). The temporal window shown in fig.3 ( t = 500) does notyet capture the asymptotic state of the asymmetric input. However, we observed thatfurther in time the amplification factor G reaches the same order of magnitude of thesymmetric perturbation.The more noticeable results presented in fig. 3 are that the asymmetric growths inthe early transient are much less rapid than the symmetric ones and that the function G ,in the case of asymmetric perturbations only, shows a modulation which is very evidentin the first part of the transient, and which corresponds to a modulation in amplitudeof the pulsation of the instability wave, see fig.4. In fact, the pulsation varies: in theearly transient it oscillates around a mean value with a regular period, which is the samevisible on G, and with an amplitude which is growing until this value jumps to a newvalue around which oscillates in a damped way. This second value is the asymptoticconstant value. This behavior is always observed in the case of asymmetric longitudinalor oblique instability waves. Instead, it is not shown by transversal ( φ = π/
2) waves orby symmetric waves, see fig. 4, where, on the one hand, the asymptotic value, nearlyequal to that of the asymmetric perturbation, is rapidly reached after a short monotonegrowth and where, on the other, the growth is many orders of magnitude faster, and as aconsequence, a modulation would not be easily observable. Thus, we may comment hereon the fact that two time scales are observed in the transient and long term behavior oflongitudinal and oblique perturbations: namely, the periodicity associated to the average14alue of the pulsation in the early transient, clearly visible in the asymmetric case only,and the final asymptotic pulsation. The asymptotic value of the pulsation is higher thanthe initial one, typically is about 2 . G is larger, nearly (1.4 - 1.7) higher, than the average period of theoscillating wave in the early transient, because G is a square norm of the system solution.Thus the evolution of the system exhibits two periodic patterns at different frequencies:the first, of transient nature, and the other of asymptotic nature. When the averagedamping of the energy density in the early transient is not strong and is then followed bya monotone asymptotic growth, the change of the frequency of the oscillation is evident,see fig.4.This kind of behavior is often observed in the study of linear systems with an oscillat-ing norm, a problem that naturally arises in the context of the linearized formulation ofconvection-dominated systems over finite length domains. The occurrence of oscillatingpatterns in the energy evolution of the solutions is linked to the non-normal character ofthe linear operator which describes the system, see e.g. Coppola and De Luca (2006)[30].Figure 5 illustrates an interesting comparison between two-dimensional and three-dimensional waves (note that a logarithmic scale on the ordinate is used in part (a) of Fig.5). The purely two-dimensional wave (solid curve) is rapidly reaching a first maximum ofamplitude (at about t = 15), then the perturbation decreases while oscillating and reachesa minimum around t = 150. Afterwards, the disturbance slowly grows up to t ≈ G occurs. It should be noted that thisbehavior is controlled solely by the evolution of the ω z , that is by the vorticity componentpresent in the basic flow only. Then, the growth becomes faster and the perturbation ishighly amplified in time. The purely orthogonal perturbation (dashed curve) is insteadimmediately amplified. The trend is monotone, and does not present visible fluctuationsin time. The initial growth is actually rapid and an inflection point of the amplificationfactor G can be found around t = 50. Beyond this point, the growth changes its velocityand becomes slower, but still destabilizing. Both cases have asymmetric initial conditions15nd are ultimately amplified. In agreement with Squire theorem, the two-dimensional caseturns out to be more unstable than the three-dimensional one, as the 2D asymptoticallyestablished exponential growth is more rapid than the 3D one (see solid and dashed curvesin Fig. 5(a) for t > t ≈ k = γ ). In the case displayed inthis figure, this happens when the wave number is increased beyond the value 1. Beforethe asymptotic stable states are reached, these configurations yield maxima of the energydensity (e.g. when k = 1 . , G ∼ t ∼
30) in the transients. This trend is also typicalof oblique and longitudinal waves, and it can be considered a universal feature in thecontext of the stability of near parallel shear flows. It should be noted that, in fig. 6, theperturbation is symmetric and again the amplitude modulation is not observed in theearly transient, even in the asymptotically stable situations. However, this example oftransient behavior also contains a feature which is specific of orthogonal, both amplifiedor damped, and symmetric or asymmetric, perturbations, namely, the fact the mostamplified component of the vorticity is the ω y , see also fig.5, part d.In fig. 7 a significant phenomenon is observed for a longitudinal wave. By changingthe order of magnitude of α i , it can be seen that perturbations that are more rapidlydamped in space (see, in fig. 7b, the longitudinal spatial evolution of the wave) yielda faster growth in time. In fact, for nearly uniform waves in x direction ( α i → .2 Physical interpretation of the different growth rate of sym-metric and asymmetric disturbances The dramatically increased growth rate of the symmetric mode with respect to the asym-metric mode can be understood from the induced velocity of the vorticity field. Forsimplicity, imagine that the wake consists of two parallel shear layers of opposite vortic-ity, as shown in figure 8(a). Further assume that the vorticity is discretized into a finitenumber of identical vortices. Suppose the upper shear layer is perturbed into a sinusoidalshape, as shown in figure 8(b). The induced velocity at the crest of the sinusoid at point 1due to the other vortices in the upper shear layer alone is up and to the right as indicatedby the arrow, corresponding to the classical Kelvin-Helmholtz instability.Now consider the asymmetric mode, so that the lower shear layer is a sinusoid inphase with the upper one (figure 8(c)). The induced velocity at the crest of the lowershear layer at point 2 due to the other vortices in the lower shear layer alone is exactlythe opposite of that at point 1. Thus the growth in the asymmetric mode is due to onlyhigher order gradients in the induced velocity field of one shear layer on the other.In contrast, with a symmetric perturbation, the lower shear is the mirror image of theupper. In figure 8(d), the induced velocity at point 2 is necessarily the mirror image ofthat at point 1. Both points move in concert to the right, without the low order velocitycancellation of the asymmetric mode. Thus the symmetric mode grows much faster.This linear absolute instability takes place in the intermediate and far field and acts asa source of excitation for the pair of steady recirculating eddies in the lee of the cylinder.The onset of a time periodic flow, a supercritical Hopf bifurcation ([31], [32]) indicatesthat in the end the vortices are shed alternatively from the separated streamlines aboveand below the cylinder forming the Von Karman vortex street. The vortex street isthe stable configuration after the bifurcation has taken place. It has the symmetry of atraveling sinuous mode, which indicates asymmetry up-and-down of the cylinder. Thus,it can be observed that, also in the context of the vortex shedding, asymmetry shows17igher stability properties with respect to symmetry.
Figure 9 presents a comparison between the initial-value problem and the asymptotictheory results represented by the zero order Orr-Sommerfeld problem ([24]) in terms oftemporal growth rate r and pulsation ω .In fig. 9 the imaginary part α i of the complex longitudinal wavenumber is fixed, and differ-ing polar wavenumbers ( k = α r ) are considered. For both the symmetric and asymmetricarbitrary disturbances here considered, a good agreement with the stability characteristicsgiven by the multiscale near-parallel Orr-Sommerfeld theory can be observed. However,it should be noted that the wave number corresponding to the maximum growth factor inthe case of asymmetric perturbations is about 15% lower than that obtained in the caseof symmetric perturbations and that obtained by the normal mode analysis. When theperturbations are asymmetric, the transient is very long, of the order of hundreds timescales. This difference can be due either to the fact that the true asymptote was not yetreached, or to the fact that the extent of the numerical errors in the integration of theequations is higher than that obtained in the case of symmetric transients, which lastonly a few dozen time scales. Note that this satisfactory agreement is observed by usingarbitrary initial conditions in terms of elements of the trigonometrical Schauder basis forthe L space, and not by considering as initial condition the most unstable waves givenby the Orr-Sommerfeld dispersion relation. Moreover, a maximum of the perturbationenergy (in terms of r ) is found around k = 0 . Re = 50 (see part a of fig. 9) has a – theoretically18educed – frequency which is very close to the frequency measured in the laboratory. Atthis point, also the laboratory experimental uncertainty, globally of the order of a ± Re = 100. The three-dimensional stability analysis of the intermediate asymptotics of the 2D bluff-body viscous growing wake was considered as an initial-value problem. The velocity-vorticity formulation was used. The perturbative equations are Laplace-Fourier trans-formed in the plane normal to the growing basic flow. The Laplace transform allows forthe use of a damped perturbation in the streamwise direction as initial condition. Inthis regard, the introduction of the imaginary part of the longitudinal wavenumber (thespatial damping rate) was done to explicitly include in the perturbation, which other-wise would have been longitudinally homogeneous, a degree of freedom associated to thespatial evolution of the system.An important point is that the vorticity-velocity formulation, Fourier-Laplace trans-formation, allows (over a reasonable lapse of computing time) for the following of thetemporal evolution over hundred of basic flow time scales and thus to observe very longtransients. Such a limiting behaviour would not have been so easily reached by means ofthe direct numerical integration of the linearized governing equations of the motion.Two main transient scenarios have been observed in the region of the wake wherethe entrainment is present, the region in between x = 10 diameters (intermediate) and x = 50 diameters (far field), for a Re number equal to 50 and 100. A long transient, wherean initial growth smoothly levels off and is followed either by an ultimate damping or by a19low amplification for both oblique or 2D waves, and a short transient where the growth orthe damping is monotone. The most important parameters affecting these configurationsare the angle of obliquity, the symmetry, the polar wavenumber and the spatial dampingrate. While the symmetry of the disturbance is remarkably influencing the transientbehaviour leaving inalterate the asymptotic fate, a variation of the obliquity, of the polarwavenumber and of the spatial damping rate can significantly change the early trendas well as the final stability configuration. Interesting phenomena are observed. A firstone is that, in the case of asymmetric longitudinal or oblique perturbations, the systemexhibits two periodic patterns, a first, of transient nature, and a second one of asymptoticnature. A second phenomenon is that, in the case of an orthogonal perturbation, albeitalways initially set equal to zero, the transversal vorticity component is the vorticitycomponent which grows faster. The third phenomenology is linked to the magnitude ofthe spatial damping rate. Perturbations that are more rapidly damped in space lead toa larger growth in time.For disturbances aligned with the flow, the asymptotic behaviour is shown to bein excellent agreement with the zero order results of spatio-temporal multiscale modalanalyses and with the laboratory determined frequency and wave length of the parallelvortex shedding at Re = 50 and 100. It should be noted that the agreement betweenthe initial value problem results and the normal mode theory is obtained not using asinitial condition the most unstable wave given by the Orr-Sommerfeld dispersion relationat any section of the wake, but arbitrary initial conditions in terms of elements of thetrigonometrical Schauder basis for the L space. Acknowledgement
The authors would like to recognize Robert Breidenthal for his insightful interpretationof the perturbation dynamics in the study for wake vorticity. His experience in this fieldwas a significant contribution. 20 eferences [1]
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J. Fluid Mech . 182: 1–22 (1987).23igure 3: Effect of the symmetry of the perturbation. (a) - (b): The amplification factor G , and (c)-(d) the perturbation vorticity components at y = 1 as a function of time.(a)-(c) asymmetric initial condition, (b)-(d) symmetric initial condition. Intermediate( x = 10, solid curves) and far field ( x = 50, dashed curves) wake configurations. Theperiods τ inter , τ far are the periods of the modulation visible on G , in the intermediateand far field, respectively. The values of the vorticity component in part (d) have nophysical meaning. The plot simply shows that, on the contrary of the asymmetric case inpart(c), the symmetric disturbance growth has a short transient after which it becomeshomogeneous in time. 24igure 4: Pulsation behavior. The wave parameters are those shown in the previousfigure, for x = 10. 25igure 5: Effect of the angle of obliquity φ . (a) The amplification factor G and (b)the temporal growth rate r as functions of time. Asymmetric initial condition, φ = 0(solid curves), φ = π/ y = 1, (c) φ = 0 and (d) φ = π/
2. The periods τ inter is the periodof the modulation visible on G , in the intermediate field.26igure 6: Effect of the polar wavenumber k . (a) The amplification factor G and (b) thetemporal growth rate r as function of time. Symmetric initial condition, k = 0 . , , . , α i . (a) The amplification factor G as functionof time and (b) the wave spatial evolution in the x direction for k = α r = 0 .
5. Asymmetricinitial condition, α i = 0 , . , . , .
1. 27igure 8: Induced velocity field of the symmetric and asymmetric modes. (a): Idealizationof the base flow of the wake into two shear layers of opposite sign. (b): Perturbed uppershear layer. (c): Induced velocities in the asymmetric mode. Note that they cancel tolowest order. (d): Induced velocities in the symmetric mode. Superposition enhances theamplitude at both points 1 and 2. 28igure 9: (a)
Temporal growth rate and (b) pulsation. Comparison among the asymp-totic results obtained by the IVP analysis (circles: symmetric perturbation; triangles:asymmetric perturbation) and the normal mode analysis (solid lines, see [24]. The asymp-totic values for the IVP analysis are determined when the condition dr/dt < (cid:15) ( (cid:15) ∼ −4