The role of geometry on the global instability of wakes behind streamwise rotating axisymmetric bodies

European Journal of Mechanics / B Fluids 76 (2019) 205–222

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European Journal of Mechanics / B Fluids journal homepage: www.elsevier.com/locate/ejmflu

The role of geometry on the global instability of wakes behind streamwise rotating axisymmetric bodies ∗

J.I. Jiménez-González a , , C. Manglano-Villamarín b , W. Coenen b,c a

Departamento de Ingeniería Mecánica y Minera, Universidad de Jaén, Campus de las Lagunillas, 23071, Jaén, Spain Departamento de Ingeniería Térmica y de Fluidos, Universidad Carlos III de Madrid, Leganés, Spain c Department of Mechanical and Aerospace Engineering, UC San Diego, 9500 Gilman Drive #0411 La Jolla, CA 92093, USA b

article

info

Article history: Received 24 August 2018 Received in revised form 22 February 2019 Accepted 5 March 2019 Available online 8 March 2019 Keywords: Global instability Wakes Sensitivity Sphere Axisymmetric body

a b s t r a c t We perform direct and adjoint global stability analyses of the wake behind streamwise rotating axisymmetric bodies of hemispherical nose and cylindrical trailing edge of different length-to-diameter ratios, ℓ, to evaluate the role of geometry on the unstable global modes. The study is limited to laminar Reynolds numbers, Re < 500, and moderate values of the rotation parameter, Ω ≤ 1, defined as the ratio between the azimuthal velocity at the body’s surface and the freestream velocity. As the aspect ratio, ℓ, is varied, important differences on the wake stability features are found as the rotation parameter Ω is increased. For short bodies such as hemispheres (ℓ = 0), axisymmetry breaking takes place through the destabilization of a low frequency (LF) mode, which is a modified version of the steady-state (SS) mode related to wakes without rotation, which becomes more unstable as Ω grows. This destabilizing effect of rotation vanishes for low Ω as the aspect ratio ℓ increases (bullet-like bodies, ℓ > 0), until a critical value is reached, ℓc1 , from which the LF mode is stabilized as Ω grows. In addition, increasing ℓ above two other critical thresholds, ℓc2 and ℓc3 , promotes, respectively, the destabilization of a high frequency (HF) mode, which is a modified version of the RSP mode at wakes without rotation, and a new medium frequency (MF) mode, whose origin remains unclear. Computation of sensitivity to base flow modifications shows that the LF mode is destabilized by rotation for short bodies (i.e. sphere or hemisphere) due to an increase in the shear along the separation line and a weakening of perturbation advection inside the recirculation bubble. Conversely, as ℓ grows, the larger angular momentum stabilizes the LF mode by strengthening the advection of perturbations in the near wake. Destabilization of HF and MF modes are also discussed in terms of sensitivity to base flow modifications. Finally, an analysis of the structural sensitivity shows that the destabilization of the MF mode is analogous to that behind the spiral vortex breakdown, in terms of conservation of angular momentum, being it mainly driven by an inviscid mechanism whose origin lies at the rear stagnation point of the recirculation bubble. © 2019 Elsevier Masson SAS. All rights reserved.

1. Introduction Laminar flows past rotating bluff-bodies have recently received great attention due to their relevance in microscale applications, e.g. micro-robots motion [1] or particle transport [2]. Different numerical and experimental studies have shed light on flow regimes developing in the wakes behind a streamwise rotating sphere [2–5], or rotating axisymmetric bodies with a blunt trailing edge [6,7], showing that the application of spin substantially modifies the stability properties of the wake, which are highly dependent on the body geometry. The origin of these ∗ Corresponding author. E-mail addresses: [email protected] (J.I. Jiménez-González), [email protected] (C. Manglano-Villamarín), [email protected] (W. Coenen). https://doi.org/10.1016/j.euromechflu.2019.03.003 0997-7546/© 2019 Elsevier Masson SAS. All rights reserved.

discrepancies based on geometry is unclear and does not seem to be related to any stability feature observed in the wake behind non-rotating bodies. In fact, wakes behind spherical and bluntbased bodies undergo the same type of bifurcations and unstable regimes as the Reynolds number Re increases for Ω = 0, where Ω is the rotation parameter, defined as the ratio between the maximum azimuthal velocity at the body wall and the free stream velocity. More precisely, they undergo, subsequently, a first regular axisymmetry-breaking bifurcation at a critical value Re = Rec1 , which leads the flow to sustain a planar-symmetric steady regime (SS mode), followed by a Hopf bifurcation, at Re = Rec2 , that sets the onset of unsteadiness and vortex shedding, while preserving the reflectional symmetry (RSP mode). As identified by Natarajan & Acrivos [8] by means of linear instability analysis of the steady axisymmetric flow around the sphere and disc, these bifurcations are related to the destabilization of global modes with azimuthal

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J.I. Jiménez-González, C. Manglano-Villamarín and W. Coenen / European Journal of Mechanics / B Fluids 76 (2019) 205–222

wavenumber |m| = 1. Interestingly, the instability thresholds for the sphere are found to be respectively Rec1 ≃ 212 and Rec2 = 273 [9,10], whereas for longer blunt-based bodies [11,12] higher values of Rec1 and Rec2 are reported as the aspect ratio grows, although unstable regimes are analogous. When streamwise rotation is applied however, the picture differs significantly for both bodies, as Fig. 1 depicts. For the wake behind the sphere, it has been shown [3] that, as Ω grows, the SS regime is modified and the wake rotates in a ‘‘frozen’’ way, as the vortical structures maintain their shape and strength while spinning around the axis. It was demonstrated for Ω ≤ 1 that this frozen state emerges through the destabilization of a ‘‘low frequency helical’’ regime (LF regime), which takes place at lower values of Rec1 as Ω grows [4] (see Fig. 1), indicating a destabilizing effect of rotation, which promotes transition to unsteadiness. For a slender blunt-based body of ellipsoidal nose [6,7] the symmetry breaking of the wake occurs through destabilization of three different unstable global modes (see Fig. 1), featuring azimuthal helical symmetry, i.e. m = −1 (the negative sign indicating that vortical structures wind in the direction opposite to the swirl motion). Thus, the modified SS regime, i.e. LF regime, occurs only at low values of Ω , whereas a new high frequency HF regime, which stems from deformation of the RSP mode and is characterized by an unsteady spiral frozen wake motion, breaks the wake symmetry for moderate values of Ω . Contrary to what is observed for a sphere, Rec1 increases with Ω , showing that slight rotation has a stabilizing effect for blunt-based bodies. Furthermore, another important feature, not reported for the wake of a sphere, is the existence, at large values of Ω and low Re, of the unstable spiral frozen medium frequency (MF) regime, whose nature does not seem to be related to any mechanism acting without rotation. These different stability scenarios suggest that both the nature of the unstable modes and the stabilizing or destabilizing effect of spin on wakes, might be related to instability mechanisms activated by the rotation of the blunt base or the trailing edge. In addition, recent studies on adjoint linear stability and sensitivity of global modes [see e.g. 13] have helped identifying the wavemaker, the latter being understood as the flow region where the instability emerges and the wake dynamics and frequency are set [14]. It can be argued that this location is also closely related to the region where the flow can be controlled through forcing or perturbation, so that its evaluation may help understanding how rotation acts on the wake. Consequently, a parametric sensitivity analysis aiming at identifying the core of instability and based on progressive body geometry modifications stands out as an interesting approach to unravel these different behaviors encountered in spinning axisymmetric bodies. The determination of regions where global modes are most sensitive to changes in the linearized Navier–Stokes operator, along with their role on the instability origin, has been extensively investigated for two dimensional wakes and recently, three-dimensional wakes. In particular, such a highly sensitive region can be identified as the location where feedback from velocity to force is most efficient, defining a map of sensitivity of the leading global mode to structural modifications in the momentum equation, i.e. the structural sensitivity [15]. This location does not necessarily correspond to the region where an external control force has the most influence on the growth rate and frequency of the unstable mode, since the forcing also acts through modifications of the base flow. More precisely, an adjoint formulation can be derived to compute sensitivity of eigenvalues to arbitrary base flow modifications, for which the eigenvalue is viewed as a function of the base flow, as described in [16]. In can be shown that, when the base flow modifications are a solution of the Navier–Stokes equations for the steadily forced flow, accurate predictions of eigenvalue drift are obtained using that sensitivity

distribution, providing with identical values than those resulting from sensitivity to a steady force [17]. Consequently, the use of sensitivity to base flow modifications arises as a powerful tool to evaluate how any steady flow modification affects the wake instability mechanisms. In particular, the recent application of three-dimensional direct-adjoint global instability techniques to the wake of a fixed sphere [10] has allowed to shed light on the origin of the bifurcations at Rec1 and Rec2 , and the controllability of SS and RSP regimes. It was shown that both the structural and base flow modifications sensitivities identify the core of the instability and highest receptive region for the RSP regime in the near wake region outside the asymmetric recirculation bubble. The same approach has been employed to evaluate the effect of transverse rotation on the first bifurcation [18], showing that under the application of small transverse rotation rates the first bifurcation verifies the form of an imperfect Pitchfork bifurcation, leading the flow to an asymmetric regime that enhances the lift as the body rotates. Therefore, such approach seems appropriate to analyze the effect of streamwise rotation, which is not yet fully understood, since it has been comparatively less investigated than transverse rotation. Following that idea, this work explores the use of different sensitivity distributions to elucidate the mechanisms originating the different wake instability scenarios for distinct spinning axisymmetric geometries. We perform a parametric analysis in the range Ω ≤ 1, for laminar wakes behind axisymmetric bodies whose aspect ratio is progressively increased, comprising the sphere, hemisphere, and blunt-based bodies of hemispherical nose. The problem formulation and numerical aspects are introduced in Section 2, whereas results are presented in subsequent sections. In particular, in Section 3, we analyze the basic steady flow for different geometries investigated and corresponding rotation-induced base flow modifications. Neutral diagrams, obtained through direct stability, along with unstable global modes and corresponding structural sensitivity distributions are included in Section 4. In Section 5, we then identify the mechanisms that promote wake stabilization or destabilization with increasing rotation, by analyzing results of sensitivity to base flow modifications. The nature of the unstable modes at large values of Ω is further discussed in Section 6, while main conclusions are finally included in Section 7. 2. Problem formulation and numerical aspects 2.1. Configuration We consider the incompressible flow around rotating axisymmetric bodies, such as the sphere or slender blunt-based bodies consisting of a hemispherical nose of diameter D followed by a cylindrical afterbody of variable length-to-diameter ratio ℓ = L/D. To describe the flow, a cylindrical coordinate system (x, r , θ ) is used, with the origin placed at the center of the base of the body, as depicted in Fig. 2; the velocity being denoted as u = (ux , ur , uθ ). Upstream, the flow is aligned with the centerline of the body, with velocity U = (U , 0, 0). The angular velocity with which the body rotates is γ . Using D, U, D/U and ρ U 2 as characteristic scales for length, velocity, time and pressure, respectively, the flow is governed by the nondimensional Navier–Stokes equations

∇ · u = 0, ∂u + u · ∇ u = −∇ p + Re−1 ∇ 2 u . ∂t

(1) (2)

The dimensionless parameters that characterize the flow are the Reynolds number Re = ρ UD/µ, the rotation parameter

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Fig. 1. Bifurcation diagrams for a sphere [4], and a slender blunt-based body of length-to-diameter aspect ratio equal to 2 [7].

Fig. 2. (a) Schematic diagram of the problem configuration and of the computational domain in the (x, r)-plane, which is bounded by the wall Σw of the body, the axis of rotational symmetry, Σa , and the upstream, downstream and lateral boundaries Σi , Σo and Σf . The round numbered markers indicate the typical cell size that is imposed on these boundaries and on auxiliary lines in order to create a computational mesh that becomes finer with decreasing distance to the body. (b) Image of the computational mesh, showing the different levels of refinement. (c) Zoomed image of the computational mesh around the base of the body.

Ω = γ D/(2U), defined as the ratio between the azimuthal velocity at the surface of the body and the free stream velocity, and the aspect ratio ℓ of the cylindrical part of the body. The flow domain D under consideration, shown in Fig. 2, is delimited by the wall Σw of the body, by the axis Σa , and by the upstream, downstream and lateral boundaries Σi , Σo and Σf , located at a distance lu = 30, ld = 50 and r∞ = 10 away from the body. It was ensured that larger values for lu , ld and r∞ did not change the results. 2.2. Base flow The base flow (u¯ x , u¯ r , u¯ θ , p¯ ) around which we will perform a linear stability analysis is a steady axisymmetric solution of the governing Eqs. (1)–(2), subject to the boundary conditions u¯ x − U = u¯ r = u¯ θ = 0

on Σi ,

(3)

u¯ θ − Ω = u¯ x = u¯ r = 0

on Σw ,

(4)

∂r u¯ x = u¯ r = u¯ θ = 0 on Σa ,

(5)

− p¯ nˆ + Re−1 nˆ · ∇ u¯ = 0 on Σl , Σo ,

(6)

where nˆ is the outward normal vector on the boundary. The numerical methodology employed in the present work is similar to that used recently to analyze the onset of self-sustained oscillations in low-density jets [19]. Eqs. (1)–(2) are solved numerically with a finite element formalism in combination with a Newton–Raphson algorithm. The software FreeFem++ [20] is used for the P2–P1 discretization of the equations; the linear system being solved in FreeFem++ with the MUMPS solver [21,22]. The refinement of the unstructured mesh is controlled through the distance h between discretization points on the boundaries of the domain, and on auxiliary lines, as indicated in Fig. 2. In the computations we have used the values lu1 = 2.5, lu2 = 1, ld1 = 5, ld2 = 2.5, r0 = 4.5, r1 = 2.5, where the discretization points are tightly clustered to properly resolve the large physical gradients characterizing the near wake flow. Thus, to test the implementation and convergence of the base flow, we use three different meshes: #1 (coarse), #2 (medium) and #3 (fine). In particular, the leading direct eigenmode with respect to the mesh

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refinement was ensured, and is illustrated in Table 1 for the body ℓ = 1.5 and case Ω = 0.45 and Re = 330. In view of such results, mesh #2 is used in the following for computations as the best compromise between accuracy and computational time, since relative differences between medium and fine meshes are very small. 2.3. Direct eigenmodes

ˆ , r)]ei(mθ−ωt) , (7) (ux ′ , ur ′ , uθ ′ , p′ ) = [uˆx (x, r), uˆr (x, r), uˆθ (x, r), p(x where m ∈ Z is the azimuthal wavenumber of the perturbations, and ω = ωr + iωi contains the frequency ωr and the growth rate ωi of the perturbations. In this work we consider the case m = −1, since other azimuthal wavenumbers are stable for the range (Ω , Re) investigated. Substitution of the normal modes in the linearized equations of motion yields the generalized eigenvalue problem

− iωBqˆ = Lqˆ ,

(8)

where qˆ = [uˆ , pˆ] = [uˆx , uˆr , uˆθ , pˆ] contains the perturbation quantities. The operators B and L are fully specified by Eqs. (A.1)–(A.4) in Appendix. The perturbations are subjected to the boundary conditions T

uˆx = uˆr = uˆθ = 0 uˆx = ∂r uˆr = ∂r uˆθ = pˆ = 0

on Σi , Σw , on

Σa (for |m| = 1),

− pˆ nˆ + Re nˆ · ∇ uˆ = 0 on Σl , Σo . −1

(9) (10) (11)

The eigenvalue problem given in Eq. (8) is tackled numerically using the same finite element discretization in FreeFem++ that was used for the base flow. P2 elements are employed for uˆx , ˆ The operators L and B are uˆr and uˆθ and P1 elements for p. discretized in FreeFem++, yielding the discrete system matrices L and B, which are exported to MATLAB, where the ARPACK library is employed to solve the discretized eigenvalue problem

−iωBq˜ = Lq˜ ,

(12)

where q˜ denotes the discretized eigenvector. The direct eigenvector is normalized so that the perturbation kinetic energy 2

2

To obtain sensitivity distributions we need to compute the † adjoint eigenvector qˆ associated with a certain eigenvalue ω of the direct problem. We define the inner product between two perturbation states qˆ 1 and qˆ 2 as

⟨ˆq1 , qˆ 2 ⟩ =

A linear stability analysis is performed by adding small perturbations (ux ′ , ur ′ , uθ ′ , p′ ) to the base flow described before, as (ux , ur , uθ , p) = (u¯ x , u¯ r , u¯ θ , p¯ ) +ϵ (ux ′ , ur ′ , uθ ′ , p′ ). The evolution of the perturbations is governed to leading order by Eqs. (1)–(2), linearized around the base flow. We assume temporal normal-mode solutions of the form

∫

2.4. Adjoint eigenmodes and structural sensitivity

2

H

(|uˆx | + |uˆr | + |uˆθ | ) r dr dx = q˜ Qu q˜ = 1 .

(13)

D

Here, Qu is a weight matrix whose elements reflect the area of the individual mesh elements, as well as the factor r from the integral, and selects only the degrees of freedom corresponding to the velocity components. The numerical schemes and the eigenvalue solver have been validated by computation of the neutral curve (leading eigenvalue with ωi = 0) for the rotating sphere (see Fig. 7), and the comparison between our results and those obtained by Pier [4] by means of an immersed boundary method. In particular, a quantification of differences on computed frequencies and growth rates corresponding the leading eigenvalue for selected cases (Re, Ω ) in the vicinity of the neutral stability curve, is presented in Table 2. As observed, relative errors are in general very small (being them slightly larger for the growth rate), finding then a very good overall agreement between studies for cases with low and large rotation rates (see also neutral curves in Fig. 7 for a deeper comparison between results). Finally, an additional validation of the present numerical techniques for open flows can be found in [19].

∫

∗

∗

∗

∗

H

(uˆx 1 uˆx2 + uˆr 1 uˆr 2 + uˆθ 1 uˆθ 2 + pˆ1 pˆ2 ) r dr dx = q˜ 1 Qq˜ 2 . D

(14) where H and ∗ denote respectively the transconjugate and the complex conjugate, and Q is the complete weight matrix, including degrees of freedom related to the perturbation pressure. The discrete adjoint eigenvalue problem then is †

†

iω† Q−1 BH Qq˜ = Q−1 LH Qq˜ .

(15)

†

Each adjoint eigenvalue ω is the complex conjugate of an associated direct eigenvalue ω. The adjoint eigenvectors are normalized so that †H

†

q˜ Qu Bq˜ = 1 .

(16)

The sensitivity of an eigenvalue measures how much the eigenvalue is affected by variations of the associated operator. A spatial map of the sensitivity of ω with respect to internal feedback interactions can be obtained by measuring the local overlap between the direct and adjoint eigenfunctions. The original formulation [15] provides an upper-bound estimation of the eigenvalue drift due to modified velocity–velocity coupling. If U is a matrix that extracts the velocity components from a vector q˜ , then the structural sensitivity in the present context is characterized by the scalar quantity †

S(x) = ∥UQq˜ ∥ ∥Uq˜ ∥.

(17)

It is argued [15] that flow regions with a large value of S influence strongly the eigenvalue selection, and thus represent the origin of instability or wavemaker of the eigenmode. 2.5. Sensitivity to base-flow modifications The aforementioned structural sensitivity allows to quantify the receptivity of the global mode to a local forcing with the same frequency, being the force proportional to the local perturbation velocity. Nevertheless, if we pursue an identification of local changes of the base flow which may have a more efficient effect on the eigenvalue growth drift, and want to relate them to the overall base flow modifications induced by body rotation, then the concept of sensitivity to generic base flow modifications may be more adequate. Here, we employ the definition of [16], in ¯ is not required to be a which the base flow modification δ u steady solution of Eqs. (1)–(2). The global mode eigenvalue drift is considered to be a function of the base flow velocity modification,

δωu¯ = ⟨∇u¯ ω, δ u¯ ⟩,

(18)

where the complex-valued gradient ∇u¯ ω is the sensitivity to generic base flow modifications, and can be separated into a real part (growth rate sensitivity R[∇u¯ ω]), and an imaginary part (frequency sensitivity I [∇u¯ ω]). Eq. (18) can be evaluated using a Lagrangian functional [16], providing

ˆ H · uˆ † + ∇ uˆ † · uˆ ∗ , ∇u¯ ω = −(∇ u)

(19)

where the first term corresponds to the sensitivity to changes in the base flow advection (∇u¯ ,A ω), and the second term is related to production of perturbations by the base flow, i.e. energy extracted from the gradients of base flow (∇u¯ ,P ω). This decomposition

J.I. Jiménez-González, C. Manglano-Villamarín and W. Coenen / European Journal of Mechanics / B Fluids 76 (2019) 205–222

209

Table 1 Convergence of the leading eigenvalue with respect to the mesh refinement for the body of ℓ = 1.5 and case Ω = 0.45, Re = 330, where n represents the number of grid triangles, ωi the leading eigenvalue growth rate and ωr its frequency. Also shown are the relative error ϵj,j+1 (%) = |vj+1 − vj |/vj × 100, where v refers to variables ωr and ωi . Mesh

n

h1

h2

h3

h4

h5

ωr

ϵj,j+1 (%)

ωi

ϵj,j+1 (%)

#1 #2 #3

29 453 43 841 194 809

1.000 1.000 0.500

0.200 0.200 0.100

0.200 0.100 0.050

0.050 0.050 0.025

0.050 0.025 0.010

0.65068 0.65070 0.65066

0.004 0.006 –

−0.0093 −0.0102 −0.0105

8.823 2.857 –

allows an interpretation of instability mechanisms within the frame of local theory [14], whereby a stronger local production of perturbations is an indication of a local absolute instability in the flow region. The magnitudes ∇u¯ ,A ω and ∇u¯ ,P ω can be further separated into three complex components, related to the individual contributions of the modifications of the base flow velocity components, namely

∇u¯ x ω = ∇u¯ x ,A ω + ∇u¯ x ,P ω ) ( ( ∗ ∗ ∗ ) ∂ uˆ †x ∗ ∂ uˆ †x ∗ ∂ uˆ x † ∂ uˆ r † ∂ uˆ θ † , uˆ − uˆ − uˆ uˆ + uˆ + = − ∂x x ∂x r ∂x θ ∂x x ∂r r

Table 2 Comparison of the results obtained with the present numerical schemes and those reported by Pier [4] for a streamwise rotating sphere, showing the leading eigenvalue frequency ωr and growth rate ωi for unstable cases (Re, Ω ) = (215, 0.05) and (150, 0.75), lying close to the neutral curve (see Fig. 7). The relative errors ϵ (%), computed taking results from [4] as reference, are also listed. Present computations are obtained using the mesh parameters #2. Source

Re

Ω

ωr

ϵ (%)

ωi

ϵ (%)

Present Present Pier [4] Pier [4]

215 150 215 150

0.05 0.75 0.05 0.75

0.0347 0.5019 0.0346 0.4978

0.289 0.823 – –

0.0658 0.0182 0.0670 0.0187

1.791 2.673 – –

(20)

∇u¯ r ω = ∇u¯ r ,A ω + ∇u¯ r ,P ω ( ∗ ∗ ∗ ) ∂ uˆ x † ∂ uˆ r † ∂ uˆ θ † = − uˆ − uˆ − uˆ ∂r x ∂r r ∂r θ ( ) ∂ uˆ †r ∗ ∂ uˆ †r ∗ uˆ †θ ∗ + uˆ + uˆ − uˆ , ∂x x ∂r r r θ ∇u¯ θ ω = ∇u¯ θ ,A ω + ∇u¯ θ ,P ω ) ) ( † ( ∗ uˆ θ † ∂ uˆ θ ∗ ∂ uˆ †θ ∗ uˆ †r ∗ uˆ ∗r † + = uˆ − uˆ uˆ + uˆ − uˆ . r r r θ ∂x x ∂r r r θ

(21)

(22)

Therefore, the total eigenvalue drift due to the base flow modification, δωu¯ , can obtained by means of the following decomposition:

δωu¯ = δωu¯ x + δωu¯ r + δωu¯ θ = ⟨∇u¯ x ω, δ u¯ x ⟩ + ⟨∇u¯ r ω, δ u¯ r ⟩ + ⟨∇u¯ θ ω, δ u¯ θ ⟩.

wake is characterized by a region of high swirl, which develops within the core of recirculation and extends downstream of it. The larger angular driving force of the ℓ = 1.5 body leads to a stronger swirl velocity in the wake, whose radial distribution remains barely constant in the range x ∈ (1, 4). The overall effect of body rotation in the wake for the two mentioned aspect ratios is summarized in Fig. 4. In particular, we plot, for Ω = 0.1, 0.5 and 1: the pressure along the x-axis, p¯ (x, r = 0) (Fig. 4a); the radial distribution of axial circulation, Γ (r) (Fig. 4b), at the rear stagnation point of the recirculation bubble xrec (whose location as a function of Ω is depicted in Fig. 4d); and the axial velocity profile u¯ x (r) at x = 1 (Fig. 4c). The circulation Γ (r) is computed as

Γ (r) =

∫ 0

(23)

In order to evaluate the sensitivity of the eigenvalue growth rate δωi (resp. frequency δωr ) to modifications of each base-flow velocity component, only the real (resp. imaginary) part of Eq. (23) must be considered. Besides, in order to avoid numerical singularities in the vicinity of the axis, 1/r terms in Eqs. (20)–(22) have been treated appropriately by means of power series expansions. Finally, it must be pointed out that, since the base flow modifications δ u¯ considered here are the physical outcome of a small increment in body rotation, δ Ω (i.e. the solution of the steady Navier–Stokes equations), the latter may be viewed as a forcing, in line with the validation study carried out for a control cylinder [16]. As it was proven therein, the use of the sensitivity to generic base flow modifications is a valid procedure to compute eigenvalue drifts. 3. Base flow features The steady base flows past a rotating hemisphere (ℓ = 0) and past a slender body of ℓ = 1.5 are respectively depicted in Fig. 3(a) and (b), for Ω = 0.5 and Reynolds numbers Re = 170 and 330, which represent approximately their corresponding instability thresholds without rotation, Rec1 (Ω = 0). Streamlines show a recirculation bubble forming behind the bodies, whose length and radial thickness depends on the body aspect ratio and the swirling motion in the wake. Contours depict the distribution of azimuthal velocity u¯ θ , whose value around the body is prescribed by means of the rotation boundary condition. The near

2π

r

∫

ξx rdrdθ = 2π 0

∫ r( 0

∂ u¯ θ − r ∂r

u¯ θ

)

rdr ,

(24)

being ξx the axial vorticity. Fig. 4(a) shows an overall base pressure decrease with rotation, due to centrifugal effects, which becomes more acute along the axis in the case of the ℓ = 1.5 body. This longer axial extension of negative pressure translates into a larger recirculation bubble for the longest body as Ω increases (Fig. 4d), which is initially shorter at Ω = 0. The corresponding axial pressure gradient becomes also slightly smoother with spin, what leads to a weaker backflow close to the axis (see Fig. 4c), although a stronger shear. Conversely, the hemisphere shows a steeper negative pressure gradient in the nearest wake, creating a stronger axial backflow in x = 1 as rotation increases (Fig. 4c). Moreover, the stronger swirling motion in the near wake as ℓ increases is also clearly observable in Fig. 4(b). In view of Eq. (23), it is also interesting to evaluate the general base flow wake modifications, δ u¯ = (δ u¯ x , δ u¯ r , δ u¯ θ ), induced by slight increments of body rotation, δ Ω . Fig. 5 presents maps of velocity components and pressure variation: (a,b) δ u¯ θ , (c,d) δ u¯ x , (e,f) δ u¯ r and (g,h) δ p¯ ; for bodies of ℓ = 0 and ℓ = 1.5, at their respective instability thresholds without rotation, i.e. Re = 170 and Re = 330, when δ Ω = (Ω2 − Ω1 ) = (0.05 − 0). As shown previously, the prescribed rotation at the wall entails an azimuthal rotation of the fluid in the near wake (Fig. 5a,b), which is considerably larger behind the ℓ = 1.5 body. Besides, both bodies presents weaker axial velocity backflow in the very near wake (see positive values of δ u¯ x in Fig. 5c and d), as a consequence of a lower negative pressure gradient in the axis since the pressure becomes more negative and uniform along the axial coordinate (Fig. 5g,h). The effect is more pronounced for

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Fig. 3. Base flow for Ω = 0.5: (a) ℓ = 0, Re = 170 ≃ Rec1 (Ω = 0); and (b) ℓ = 1.5, Re = 330 ≃ Rec1 (Ω = 0). Contours represent the azimuthal velocity u¯ θ , while streamlines are computed using axial and radial velocities, u¯ x and u¯ r .

Fig. 4. Base flow features for Ω = 0.1, 0.5 and 1, for ℓ = 0 (solid lines) and for ℓ = 1.5 (dashed lines) bodies: (a) axial distribution of pressure on the axis, p(x, r = 0); (b) radial distribution of circulation, Γ (r), at the rear stagnation point of the recirculation bubble; (c) axial velocity profile u¯ x (r) at x = 1 and (d) length of recirculation bubble, xrec , as a function of Ω . Arrows indicates growing sense of Ω .

Fig. 5. Base flow velocity modifications induced by a rotation change δ Ω = (Ω1 − Ω2 ) = (0.05 − 0) for bodies of ℓ = 0 (left column) and ℓ = 1.5 (right column), at Re = 170 and 330 respectively: (a,b) δ u¯ θ , (c,d) δ u¯ x , (e,f) δ u¯ r and (g,h) δ p¯ .

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Fig. 6. Bifurcation diagrams in the (Ω , Re) parameters space (left), together with spectra showing the evolution of the LF, HF and MF eigenvalues (right), for blunt-based bodies with increasing afterbody length. From top to bottom: bodies of (a) ℓ = 0 (hemisphere), (b) ℓ = 0.35, (c) ℓ = 0.5, (d) ℓ = 1 and (e) ℓ = 1.5. Note that the arrows in the spectra indicate the eigenvalue evolution along a path following the corresponding neutral curve Rec1 = Rec1 (Ω ), from Ω = 0 to Ω = 1, in the (Ω , Re) space. Bold triangles in (e) define the neutral curve for the blunt-based body of ellipsoidal nose, defined in [7].

ℓ = 1.5 due to a stronger pressure decay for x ≥ 0.5. Regarding the radial velocity change (Fig. 5e,f), both wakes feature fluid entrainment towards the base (negative δ u¯ r ) as a consequence of the lower base pressure, which in the case of the hemisphere, is concomitant with a noticeable radial ejection at the edge. Last, note that δ u¯ and δ p¯ are one order of magnitude larger in the wake behind the ℓ = 1.5 body. 4. Global instability 4.1. Neutral diagrams We present now results from direct global stability analyses in the range Ω ≤ 1 (with ∆Ω = 0.05), aimed at identifying the global modes that trigger the instability, and characterizing neutral stability curves, defined by the axisymmetry-breaking bifurcation for increasing Ω . Comparison of neutral curves for different ℓ will provide with a clear indication of the stabilizing or destabilizing nature of spin for the geometries under study, and

consequently, it will allow a better understanding of the role of geometry. Fig. 6 depicts the bifurcation diagrams (left column) for the bodies under consideration: (a) ℓ = 0, (b) ℓ = 0.35, (c) ℓ = 0.5, (d) ℓ = 1 and (e) ℓ = 1.5; together with the spectra (right column) obtained approximately at the critical Reynolds numbers for each Ω evaluated, Rec1 (Ω ), hence following the neutral curve defined by the axisymmetry-breaking bifurcation. As mentioned in Section 1, only perturbations with azimuthal wavenumber m = −1 are unstable for Ω ≤ 1. For the sake of clarity, branches of highly damped modes are not plotted in the spectra, showing exclusively the evolution of those distinguished eigenvalues that are significant for the stability problem (LF, HF and MF modes). In fact, arrows in Fig. 6 track the eigenvalues along the neural curves, from Ω = 0 to Ω = 1, showing how the different modes become unstable or stable as Ω and Re change. The spatial structure of these dominant modes will be later discussed along with their corresponding structural sensitivity. As mentioned earlier, the transition to instability from axisymmetry in the wake past a sphere is governed by the destabilization

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of the so-called LF mode, whose threshold occurs at lower Re as

Ω grows [4]. When the stability of the hemisphere, i.e. the ℓ = 0

body, is evaluated (see Fig. 6a), a similar picture is observed, with two distinguished eigenmodes in the spectrum. When Ω = 0, these two modes correspond to a steady mode and an unsteady mode (with Strouhal number St = −ωr /2π ≃ 0.128), whose subsequent destabilizations lead respectively to the nonlinear steady planar symmetric regime and the time-periodic asymmetric regime described in [23]. When rotation is applied, the value of St increases for both modes, as the arrows in the spectrum show (see left Fig. 6a). Since now both eigenvalues are unsteady, we adopt the notation used in former works, denoting them LF and HF respectively, based on their frequency values. The LF mode destabilization governs the axisymmetry-breaking, and increasing values of rotation trigger this bifurcation at lower Re, so apparently, a blunt base does not have a great qualitative effect on the wake stability when compared to that of the sphere (Fig. 1). However, when the length is increased by adding a cylindrical afterbody, the picture differs. For a body of ℓ = 0.35 (Fig. 6b) there is a slight wake stabilization when the spin increases within Ω < 0.4. This LF-mode stabilization at low Ω becomes more pronounced as ℓ grows, as can be observed in Fig. 6(c–d). Besides, the HF eigenvalue becomes progressively less stable, when tracking the neutral limits towards larger Re, until eventually, for a critical value of ℓc2 ≃ 0.5, its destabilization governs the axisymmetry breaking at moderate values of spin, for a range of Ω that widens with the aspect ratio. More than a consequence of a HF-destabilization with rotation (note that Rec1 barely remains constant for this intermediate values of Ω ), this scenario is the outcome of a drastic damping of the LF eigenvalue for ℓ > 0.5, as observed in Fig. 6(d–e). As ℓ grows, a third distinguished MF eigenvalue stands out (see Fig. 6c), whose destabilization for bodies of ℓ ≥ 1 (Fig. 6d) delimits the axisymmetry region at low values of Re and increasing Ω . Based on that observation, it is evident that the instability mechanisms promoting the MF mode is associated to the afterbody rotation, since it requires a critical body length to be triggered. Interestingly, its instability is independent from the nose geometry, as Fig. 6(e) shows, where the bifurcation diagram for a ℓ = 1.5 body is compared with that obtained for a body with ellipsoidal nose [7] and a ℓ = 1 afterbody (bold triangles), both bodies featuring a total length equal to 2. Notice that the stability borders defined by the MF mode for both bodies nearly overlap, although the former body exhibits a more stable behavior at low Ω . This fact may be related to the larger afterbody aspect ratio (ℓ = 1.5), which allows the boundary layer to develop over a longer distance and consequently be thicker at detachment, thus fostering the stabilization of LF and HF modes, as for non-rotating wakes [11]. The effect of body geometry on the wake axisymmetry breaking is summarized in Fig. 7, which shows neutral curves for all bodies under investigation, together with that of the sphere. As expected, for Ω = 0, the first bifurcation occurs at higher values of Rec1 as the afterbody aspect ratio ℓ increases. Moreover, the LF mode is highly affected by rotation at low values of Ω , and it can be either stabilized or destabilized depending on the body aspect ratio. In particular, for ℓ > ℓc1 ≃ 0.25 the mode is stabilized, showing that a critical level of swirling motion in the wake, induced by the body’s angular driving force, is required. As an outcome, the HF mode controls the wake symmetry breaking at intermediate values of Ω for ℓ = ℓc2 ≃ 0.5. Now the bifurcation threshold Rec1 (Ω ) remains barely constant and unaffected by rotation, suggesting that the HF mode destabilization mechanism is somehow decoupled from the swirling motion. At larger values of Ω , a third MF mode emerges when ℓ is larger than a third critical value, 0.5 < ℓc3 < 1, whereas the LF mode drives

the wake destabilization for ℓ < ℓc3 . It must be remarked that at low Re, the neutral instability curves are very similar for all bodies under study, independently of which of these two mode governs the transition. This might indicate that the nature of such instabilities is similar at high Ω . In view of these results, we will focus in the following sections on the nature of the modifications introduced by body rotation in the instability mechanisms governing the LF and MF modes, by analyzing sensitivity distributions. 4.2. Global modes and structural sensitivity As shown in Fig. 7, the wake instability at low values of rotation is mainly characterized by the stabilization or destabilization of the LF mode, depending on the body geometry. Interestingly, although its controllability differs when ℓ varies, the global mode spatial structure and the origin of its instability, identified by means of the structural sensitivity (Eq. (17)), are fundamentally similar for all bodies at Rec1 . To illustrate that, we plot in Fig. 8 the real components of direct and adjoint streamwise velocities, † uˆx and uˆx , along with the corresponding structural sensitivity S(x), for bodies of ℓ = 0 (Fig. 8a–c) and ℓ = 1.5 (Fig. 8d– f), at their respective Re ≃ Rec1 , namely Re = 170 and 330, for Ω = 0.1. Direct perturbations uˆx (Fig. 8a, d) are mainly characterized by axially elongated structures – resembling the classical streamwise vortices of a SS mode [23] – that extend downstream featuring some waviness. Conversely, the adjoint † perturbation uˆx is concentrated in the recirculation bubble and shows non-negligible magnitudes upstream of the bodies, due to the convective non-normality of the linearized Navier–Stokes operator L in Eq. (8) [24]. The different spatial location of direct and adjoint modes leads to a small overlapping region of structural sensitivity (Fig. 8c, f), whose maximum lies within the center of the recirculation bubble and along the separation line, indicating that these areas are most receptive to localized forcing with the frequency of the global mode, and may control the intrinsic flow dynamics and mode frequency [13,15]. The higher magnitude obtained for ℓ = 1.5 indicates an increasing receptivity as the geometry is enlarged. In general, the LF mode retains the major characteristics of the SS mode in the wake of non-rotating axisymmetric bodies [25], whose core of sensitivity basically coincides with the loci of recirculation, identified previously as the origin of instability of the SS mode in the wake of the sphere [26]. Interestingly, this structural sensitivity map is similar to those observed in three-dimensional elliptic instabilities of two dimensional flows characterized by elliptic streamlines, as for instance in flows over a backward-facing step [27]. For bodies with a sufficiently large cylindrical afterbody (ℓ > ℓc2 ≃ 0.5), at intermediate values of Ω , the axisymmetry breaking bifurcation occurs due to the HF mode destabilization, which is a rotation-modified version of the second least damped oscillatory mode in wakes without rotation, i.e. the so-called RSP mode [11]. Fig. 9(a–c) depict, respectively, the real components † of direct and adjoint streamwise velocities, uˆx and uˆx , and their corresponding structural sensitivity S(x), for the dominant HF mode at Ω = 0.2 and Re = 410 ≃ Rec1 (Ω = 0.2). As expected, the direct mode displays an oscillatory pattern, with the magnitude of uˆx growing as the mode is convected downstream. The adjoint mode however, reaches its maximum inside the recirculation region and extends upstream, displaying the highest magnitude along the separation line and rear base edge. Again, the spatial separation between direct and adjoint modes leads to a structural sensitivity map (Fig. 9c) that identifies the wavemaker inside the recirculation bubble, although the most sensitive region is located below the separation line, in a similar manner to the sensitivity maps featured by the RSP mode of the

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Fig. 7. Comparison between bifurcations diagrams in the (Ω − Re) space, for all bodies under investigation, including the sphere (see validation study in Table 2).

Fig. 8. Low-frequency (LF) mode for ℓ = 0 (a–c) and ℓ = 1.5 (d–f) bodies, at Ω = 0.1 and respective Re ≃ Rec1 : (a, d) real part of the streamwise velocity, R(uˆ x ); † (b, e) real part of the streamwise velocity of the adjoint mode, R(uˆ x ); and (c, e) structural sensitivity S(x). Solid lines indicate the separation line.

Fig. 9. High-frequency (HF) mode (a–c) and Medium-frequency (MF) mode (d–f) for ℓ = 1.5 body, at respective thresholds of instability, i.e. (Ω , Re) = (0.2, 410) and † (Ω , Re) = (0.5, 330): (a, d) real part of the streamwise velocity, R(uˆ x ); (b, e) real part of the streamwise velocity of the adjoint mode, R(uˆ x ); and (c, e) structural sensitivity S(x). Solid lines indicate the separation line.

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sphere [see e.g. Fig. 10 in 25]. Such strong resemblance (note that even the magnitudes of S(x) are similar) suggests that the same shear instability mechanism drives the development of RSP and HF modes, regardless of the application of rotation, and might explain the slight modification displayed (see Fig. 6c–e) by the value of the critical threshold of instability, Rec2 , associated with the HF mode, as the value of Ω increases. On the other hand, as mentioned, the destabilization of the MF mode is triggered at Ω > 0.45 for bodies of large aspect ratio, ℓ > ℓc3 . Spatial features of such unstable MF mode are displayed in Fig. 9(d–f) for Ω = 0.5 and Re = 330, where it is shown that the direct mode (Fig. 9d) shows a periodical pattern of vortices that extends downstream with a larger characteristic wavelength than that displayed by the HF mode. The adjoint mode (Fig. 9e) concentrates again around the rear base edge and separation line, and spreads upstream as a consequence of the convective non-normality. However, this adjoint mode displays a strong magnitude at the rear end of the separation line, around x ≃ 2. This region is precisely the most receptive to localized forcing, as the structural sensitivity distribution S(x) shows in Fig. 9(f), and will be further analyzed in Section 6. Finally, to conclude with the discussion of the characteristics of the global modes at large values of Ω , we focus now on the effect of rotation on the LF mode, which governs the flow destabilization for bodies with ℓ < ℓc3 , and for the sphere. As shown in Fig. 7, the neutral curves display similar profiles at low Re for all the bodies under investigation, regardless of the unstable mode governing the flow bifurcation, which may be an indication of LF and MF modes sharing some instability features. To evaluate this, Fig. 10 depicts the corresponding unstable LF modes for the ℓ = 0.5 body (Fig. 10a–c) and for the sphere (Fig. 10d–f), at values of (Ω , Re) in the vicinity of the neutral curves. The direct modes (Fig. 10b, d) display in both cases an oscillatory structure, showing that, as Ω increases, the progressive growth in frequency (see Fig. 6) is accompanied by the shortening of the characteristic mode axial wavelength (see the shorter scale in Fig. 10d). The adjoint modes (Fig. 10b, e) are again concentrated within the recirculation region and near the separation lines (slight differences between blunt-based body and sphere are due to the rear edge), but now feature important magnitudes close to the rear stagnation point of the recirculation bubble. The latter spatial distribution leads to sensitivity maps (Fig. 10c, f) that present two fundamental areas: a first core coinciding with the locus of recirculation, reminiscent of the LF mode at low Ω (see Fig. 8), and a second one, concentrated close to the rear stagnation point, in a similar manner to what was shown for the MF mode (Fig. 9). Such resemblance will be further discussed in Section 6. 5. Sensitivity to rotation-induced base flow modifications Once the global modes characterizing the flow destabilization for the bodies under investigation have been presented, we will next try to unveil the underlying physics behind the stabilizing or destabilizing nature of rotation regarding the LF mode as a function of the aspect ratio ℓ, along with the destabilization of the HF and MF modes with rotation for large values of ℓ. To that aim, we will identify regions in the near wake that play a role in the instability mechanisms, by analyzing the distributions of sensitivity to generic base flow modifications (Eq. (19)), which will be used to estimate the eigenvalue drift following Eq. (23). The application of such equation requires the use of infinitesimal base flow modifications, in order to satisfy the assumption of a linear analysis. Therefore, the computation of local base flow ¯ induced by rotation of axisymmetric rotating modifications, δ u, bodies of different aspect ratios ℓ, will be done using small rotation increments, i.e. δ Ω = 0.05 (as presented in Section 3), although some computations for δ Ω = 0.01 are also performed for the sake of validation.

5.1. Sensitivity analysis of the LF mode We will first try to evaluate the influence of geometry on the stabilization or destabilization of the LF mode, by analyzing distributions of growth rate and frequency sensitivity, ∇u¯ ωi = R[∇u¯ ω] and ∇u¯ ωr = I [∇u¯ ω] respectively (Eq. (19)), for bodies with ℓ = 0 and 1.5. Figs. 11 and 12 display, for both bodies under study, the growth rate sensitivity to streamwise velocity modifications ∇u¯ x ωi , along with their advection ∇u¯ x ,A ωi and production ∇u¯ x ,P ωi components, and the frequency sensitivity to azimuthal velocity modifications ∇u¯ θ ωr (these contributions provide respectively with largest LF eigenvalue drifts), for Ω = 0 and respective Re ≃ Rec1 . In general, both bodies display similar sensitivity distributions, although the wake behind the ℓ = 1.5 body features larger magnitudes, in line with the structural sensitivity results. When the growth rate sensitivities are considered (Figs. 11a and 12a), distinct regions of interest for flow control are distinguished. First, for x ≥ 1, the growth rate may be decreased if the streamwise velocity is increased (resp. decreased) in the lower part (resp. upper part) of the shear layer. Second, the mode may be also stabilized by blowing just below the edge, preventing the boundary layer detachment. Although these two regions along the shear layer exhibit important values of sensitivity, the maxima are reached inside the recirculating zone. In fact, this third zone of high sensitivity suggests that the mode may be alternatively controlled either by decreasing the velocity at the base wall, e.g. through local base suction [28], or more efficiently, by increasing the velocity in the core of the recirculation around x = 0.2, e.g. through base blowing [11], where the wake is most receptive. The nature of such mechanisms can be further interpreted based on the decomposition into advection and production terms. As expected, the advection sensitivity component ∇u¯ x ,A ωi (Figs. 11c and 12c) shows that strategies of base suction or blowing act modifying the perturbation advection through the base flow. On the other hand, according to the production term ∇u¯ x ,P ωi (Figs. 11e and 12e), perturbation generation is mostly related to the velocity shear along the separation line. In any case, the advection mechanism seems to govern the mode instability, judging by the magnitude of sensitivity components. In addition, Figs. 11(g) and 12(g) display the frequency sensitivity to azimuthal velocity modifications, ∇u¯ θ ωr , which is the only contribution to frequency drift (recall that LF eigenfunctions uˆx and uˆr have only real parts at Ω = 0). Note that its magnitude is higher than that of growth rate sensitivities, reaching its maximum receptivity alongside the axis, in the near wake. Let us quantify now the eigenvalue drift δω induced by a slight rotation increment δ Ω = 0.05, by means of integrations over the computational domain of each component of ∇u¯ ω · ¯ as indicated in Eq. (23) (where δ u¯ are displayed in Fig. 5). δ u, The contribution to growth rate drift of streamwise velocity, δωi,¯ux , is comprised in Figs. 11(b) and 12(b), where the integrand ∇u¯ x ωi (x, r) · δ u¯ x (x, r) is depicted for the two bodies at hand. The wake behind the hemisphere presents a strong mode destabilization due to an increase of velocity shear (although there is also a weak velocity decrease above the separatrix), which is, as expected, purely related to the production of perturbations (see Fig. 11f). The latter is concurrent with a mode stabilization in the near wake caused by the weaker backflow induced by rotation and the resulting stronger advection (see Fig. 11d). Since both advection and production components display similar magnitudes, spatial integration over the domain to obtain δωi,¯ux is required to work out which is the dominant of these two effects (recall that according to Fig. 6a, destabilizing effects should prevail for the hemisphere). However, let us first continue the discussion on the qualitative differences between the different bodies. Fig. 12(b) displays the distribution of ∇u¯ x ωi (x, r) · δ u¯ x (x, r) corresponding

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Fig. 10. Low-frequency (LF) mode for ℓ = 0.5 body (a–c) and sphere (d–f), at respective thresholds of instability, i.e. (Ω , Re) = (0.6, 180) and (Ω , Re) = (0.75, 160): † (a, d) real part of the streamwise velocity, R(uˆ x ); (b, e) real part of the streamwise velocity of the adjoint mode, R(uˆ x ); and (c, e) structural sensitivity S(x). Solid lines indicate the separation line.

Fig. 11. LF mode sensitivities for ℓ = 0 (Ω = 0, Re = 170): (a) growth rate sensitivity to streamwise velocity variations ∇u¯ x ωi , with its (c) advection ∇u¯ x ,A ωi and (e) production ∇u¯ x ,P ωi components, and (g) frequency sensitivity to azimuthal velocity variations ∇u¯ θ ωr . Corresponding distributions of local contributions to the growth rate drift δωi,¯ux and frequency drift δωr ,¯uθ for δ Ω = 0.05 − 0 are also depicted: (b) ∇u¯ x ωi (x, r) · δ u¯ x (x, r), (d) ∇u¯ x ,A ωi (x, r) · δ u¯ x (x, r), (f) ∇u¯ x ,P ωi (x, r) · δ u¯ x (x, r), and (h) ∇u¯ θ ωr (x, r) · δ u¯ θ (x, r). Dashed lines indicate the separation line.

Table 3 LF mode growth rate drift δωi and frequency drift δωr , obtained for base flow modifications induced when δ Ω = Ω2 − Ω1 = 0.05 − 0 and δ Ω = Ω2 − Ω1 = 0.01 − 0, in the wake behind the ℓ = 0 and ℓ = 1.5 bodies (Re = 170 and 330 respectively). δωi,u¯ corresponds to growth rate drift due to the base flow modifications, decomposed and computed according to Eq. (23), into its components: δωi,¯ux , δωi,¯ur and δωi,¯uθ ; whereas δωi is computed as the difference between eigenvalue growth rates: δωi = ωi,Ω1 − ωi,Ω2 .

ℓ=0 δ Ω δωi,¯ux ,A

δωi,¯ux ,P

δωi,¯ux

δωi,¯ur

δωi

δωr ,¯uθ

0.05 0.00007 0.01 0.283 × 10−5

0.00006 0.249 × 10−5

0.00013 0.532 × 10−5

−0.00002 0.00011 −0.087 × 10−5 0.445 × 10−5

δωi,u¯

0.00014 0.457 × 10−5

0.03433 0.03433 0.02391 0.0052 0.0052 0.0049

δωr ,u¯

δωr

δωi,¯ux ,P

δωi,¯ux

δωi,¯ur

δωi

δωr ,¯uθ

ℓ = 1.5 δ Ω δωi,¯ux ,A

0.05 −0.00179 −0.00030 −0.00209 0.00014 0.01 −4.934 × 10−5 −1.183 × 10−5 −6.117 × 10−5 0.572 × 10−5

δωi,u¯

δωr ,u¯

δωr

−0.00205 −0.00589 0.08157 0.08157 0.05290 −5.544 × 10−5 −6.501 × 10−5 0.01546 0.01546 0.01355

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Fig. 12. LF mode sensitivities for ℓ = 1.5 (Ω = 0, Re = 330): (a) growth rate sensitivity to streamwise velocity variations ∇u¯ x ωi , with its (c) advection ∇u¯ x ,A ωi and (e) production ∇u¯ x ,P ωi components, and (g) frequency sensitivity to azimuthal velocity variations ∇u¯ θ ωr . Corresponding distributions of local contributions to the growth rate drift δωi,¯ux and frequency drift δωr ,¯uθ for δ Ω = 0.05 − 0 are also depicted: (b) ∇u¯ x ωi (x, r) · δ u¯ x (x, r), (d) ∇u¯ x ,A ωi (x, r) · δ u¯ x (x, r), (f) ∇u¯ x ,P ωi (x, r) · δ u¯ x (x, r), and (h) ∇u¯ θ ωr (x, r) · δ u¯ θ (x, r). Dashed lines indicate the separation line.

Fig. 13. Wake behind the sphere at Re = 215: (a) base flow streamwise velocity modifications, δ u¯ x , and (b) corresponding local contribution to growth rate drift of the LF mode, δωi,¯ux , induced by rotation when δ Ω = 0.05 − 0. Dashed lines indicate the separation line.

to the ℓ = 1.5 body, where a different scenario is evidenced. In that case, rotation clearly stabilizes the wake by fostering a stronger advection through the weakening of backflow in the near wake (Fig. 12d), while the mode destabilization through shear production is negligible (see magnitudes in Fig. 12f). Finally, the analysis of frequency sensitivity shows that, for both wakes, ωi increases (as observed in Fig. 6) because of the growth in azimuthal velocity in the near wake (see maps of ∇u¯ θ ωr (x, r) · δ u¯ θ (x, r) in Figs. 11h and 12h). Interestingly, these streamwise locations nearly overlap the regions identified using the structural sensitivity formalism from [15] (see Figs. 5d and 6d), which may be then consider wavemakers in terms of global mode frequency selection. Values of total growth rate and frequency drifts δωi,u¯ and δωr ,u¯ , computed according to Eq. (23), are listed in Table 3 (note that ∇u¯ θ ωi , ∇u¯ x ωr and ∇u¯ r ωr are nill for Ω = 0), where respective positive and negative values of δωi,u¯ indicate the LF mode destabilization for a ℓ = 0 body and the stabilization for ℓ = 1.5 body, which are mainly due to variations of streamwise base flow velocity. In fact, the wake behind the ℓ = 1.5 is efficiently stabilized due to the reduction in the velocity recirculation and the consequent stronger advection, as the value of δωi,¯ux ,A evidences. However, the destabilization mechanism for the hemisphere wake is a bit more subtle, since it is not only controlled by the increase of shear in the rear of the recirculation region (positive production term δωi,¯ux ,P ), but also by an

overall weakening of advection inside the recirculation region (positive advection term δωi,¯ux ,A ), which does not exist for the ℓ = 1.5 body (see Figs. 11d and 12d). To validate quantitatively the sensitivity analysis, we compare values of δωi,u¯ and the actual growth rate drift δωi , computed directly by subtracting the growth rates of the leading eigenvalues for Ω = 0.05 and 0 (Section 4), obtaining similar values for the hemisphere case (20% error), but big discrepancies for the ℓ = 1.5 body. This may be explained considering that the linear sensitivity analysis requires δ u¯ to be small in order to be accurate; and that a rotation of δ Ω = 0.05 gives rise to values of δ u¯ which are one order of magnitude larger for the ℓ = 1.5 body (Fig. 5). When a smaller δ Ω is considered for the analysis, e.g. 0.01, the relative error between δωi,u¯ and δωi reduces drastically for both bodies, proving that a sufficiently small forcing δ Ω allows accurate prediction of the stabilizing or destabilizing effect of rotation. Moreover, the frequency growth (positive δωr ) is also properly captured by the analysis of sensitivity, with relative errors similar to those associated with δωi . Finally, to complete the discussion, we also analyze the sensitivity of the LF mode in the wake behind the sphere. As Fig. 13(b) shows for Re = 215 ≃ Rec1 (Ω = 0), the instability mechanism of this mode resembles that described for the hemisphere. In fact, the overall reduction in base flow streamwise velocity caused by rotation (Fig. 13a), entails a fostering of the velocity shear

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Fig. 14. HF mode sensitivities for ℓ = 1.5 (Ω = 0.2 and Re = 410): (a) growth rate sensitivity to streamwise velocity variations ∇u¯ x ωi , (c) growth rate sensitivity to azimuthal velocity variations ∇u¯ θ ωi and (e) frequency sensitivity to azimuthal velocity variations ∇u¯ θ ωr . Corresponding spatial distributions of local contributions to the growth rate drift δωi and frequency drift δωr are also depicted: (b) ∇u¯ x ωi (x, r) · δ u¯ x (x, r), (d) ∇u¯ θ ωi (x, r) · δ u¯ θ (x, r), and (f) ∇u¯ θ ωr (x, r) · δ u¯ θ (x, r). Eigenvalue drifts are computed for base flow modifications induced when δ Ω = 0.25 − 0.2. Dashed lines indicate the separation line.

below the separation line (see Fig. 13b), and the weakening of the perturbation advection inside the recirculation bubble, which contributes increasing the growth rate. As earlier, this drift is ¯ , r), being quantified through spatial integration of ∇u¯ ωi (x, r) ·δ u(x δωi,u¯ = 0.0058 the value predicted by the sensitivity analysis for δ Ω = 0.05 − 0. The largest contribution is again that related to streamwise velocity variations, δωi,¯ux = 0.0066, with important destabilizing effects of both advection and production terms. The actual growth rate drift, obtained by means of direct stability analysis, is slightly higher, i.e. δωi,u¯ = 0.0071, but again, the consideration of smaller δ Ω provides with lower relative errors and more accurate predictions. 5.2. Sensitivity analysis of the HF mode The latter sensitivity analysis may be now applied to understand the HF mode destabilization as rotation increases, focusing on the ℓ = 1.5 body, where the instability is more pronounced (Fig. 6). Note that, since the HF eigenvalue is not the leading one at low values of Ω , the assumption of axisymmetric base flow no longer holds and therefore sensitivity and eigenvalue drift analyses might lead to slightly misleading quantitative results [11]. However, these shortcomings will be progressively less relevant as Ω grows, since the HF mode eventually becomes the leading mode (Fig. 6). We therefore focus on the unstable HF mode in the wake for Ω = 0.2 and Re = 410 ≃ Rec1 , whose values of growth rate drift δωi,u¯ and frequency drifts δωr ,u¯ , computed according to Eq. (23) and using δ Ω = 0.25 − 0.20 = 0.05, are listed in Table 4. There, it is seen that main contributions to mode destabilization and frequency growth are those related to azimuthal base flow velocity variations, i.e. δωi,¯uθ = 0.0306 and δωr ,¯uθ = 0.0401, while axial and radial velocities act stabilizing the mode (as mentioned, the quantitative errors arising from comparison with actual values of δωi and δωr may be reduced using a smaller δ Ω ). The spatial contribution to such eigenvalue drift is illustrated in Fig. 14. In particular, Fig. 14(a) displays the axial component of growth rate sensitivity, ∇u¯ x ωi , where most sensitive regions are located, as expected for a shear-type instability, alongside the shear layer (note the similarity with the structural sensitivity map of Fig. 9c). The stabilizing contribution of axial velocity modifications (see Fig. 14b) is shown to stem from a stronger advection in the near base and a weaker shear. In addition, Fig. 14(c) and (d)

depict respective growth rate sensitivity and local contribution of azimuthal velocity variations, where it is seen that the mode is mainly destabilized because body rotation increases the swirl just below the separatrix line, inside the core of recirculation (although important local stabilization is also achieved close to the axis). Furthermore, values in Table 4 also show that the HF mode destabilization mainly occurs through the weakening of perturbation advection by the base flow azimuthal velocity, since δωr ,¯uθ = 0.0292 ≃ δωr ,¯uθ ,A = 0.0295. It must be noted that the mode destabilization shown by the increase of azimuthal velocity below the shear layer in Fig. 14(d), is mainly related to the advection component, ∇u¯ θ ,A ωi (x, r) · δ u¯ θ (x, r). Finally, the frequency growth is also controlled by the azimuthal base flow velocity term (see Table 4), and the most sensitive regions concentrate again inside the recirculation bubble (Fig. 14e). It is interesting to highlight that the zone that contributes the most in increasing the frequency, according to the base flow modifications considered in Fig. 14(f), practically coincides with that of maximum perturbation production (not shown), suggesting a connection of the sensitivity production term to the concept of wavemaker in the frequency selection sense [13]. 5.3. Sensitivity analysis of the MF mode After evaluation of neutral diagrams in Section 4 (Fig. 6), it can be inferred that the onset of instability for the MF mode requires a critical angular momentum in the near wake, which is only provided when the body is large enough, as the differences in base flows for ℓ = 0 and ℓ = 1.5 bodies indicate (see Fig. 5). That said, we will apply the previous sensitivity analysis to the MF mode for Ω = 0.45 and Re = 330, in the vicinity of the instability threshold (see Fig. 6), to investigate the instability mechanisms activated by the base flow modifications at moderate values of Ω . Fig. 15 shows growth rate and frequency sensitivities to base flow modifications and their respective local effect on the eigenvalue drift, computed using δ Ω = 0.5 − 0.45 = 0.05. Note that such base flow modifications (not shown here for the sake of conciseness) are qualitatively and quantitatively similar to those depicted in Fig. 5 for ℓ = 1.5, which may be taken as a reference for the discussion. Again, axial and, mainly, azimuthal velocities variations have the largest impact on the growth rate, whereas azimuthal variations account for frequency increase, as shown in Table 5, where components of eigenvalue drift δω (Eq. (23)) are

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J.I. Jiménez-González, C. Manglano-Villamarín and W. Coenen / European Journal of Mechanics / B Fluids 76 (2019) 205–222 Table 4 HF mode growth rate drift δωi and frequency drift δωr , obtained for base flow modifications induced when δ Ω = Ω2 − Ω1 = 0.25 − 0.20, in the wake behind the ℓ = 1.5 bodies (Re = 410). δωi,u¯ corresponds to growth rate drift due to the base flow modifications, computed and decomposed according to Eq. (23), into its components: δωi,¯ux , δωi,¯ur and δωi,¯uθ ; whereas δωi is computed as the difference between eigenvalues growth rates: δωi = ωi,Ω1 − ωi,Ω2 .

ℓ = 1.5 δΩ

δωi,¯ux

δωi,¯ur

δωi,¯uθ ,A

δωi,¯uθ ,P

δωi,¯uθ

δωi,u¯

δωi

δωr ,¯ux

δωr ,¯ur

δωr ,¯uθ

δωr ,u¯

δωr

0.05–0

−0.0013

−0.0002

0.0373

−0.0067

0.0306

0.0291

0.0393

0.0012

0.0018

0.0401

0.0432

0.0413

Table 5 MF mode Growth rate drift δωi and frequency drift δωr , obtained for base flow modifications induced when δ Ω = Ω2 − Ω1 = 0.5 − 0.45 in the wake behind the ℓ = 1.5 body (Re = 330). δωi,u¯ corresponds to growth rate drift due to the base flow modifications, computed and decomposed according to Eq. (23), into its components: δωi,¯ux , δωi,¯ur and δωi,¯uθ ; whereas δωi is computed as the difference between eigenvalue growth rates: δωi = ωi,Ω1 − ωi,Ω2 .

ℓ = 1.5 δΩ

δωi,¯ux = δωi,¯ux ,A + δωi,¯ux ,P

δωi,¯uθ = δωi,¯uθ ,A + δωi,¯uθ ,P

δωi,u¯

δωi

δωr ,¯ux

δωr ,¯uθ

δωr ,u¯

δωr

0.5–0.45

0.0159 = 0.0169 − 0.0010

0.0362 = 0.0269 + 0.0080

0.0518

0.0367

−0.0095

0.0349

0.0270

0.0271

Fig. 15. Sensitivities to base flow modifications of the MF mode for ℓ = 1.5 at Ω = 0.45 and Re = 330: growth rate sensitivity to (a) axial velocity variations ∇u¯ x ωi , (c) azimuthal velocity variations ∇u¯ θ ωi and (e) frequency sensitivity to azimuthal velocity variations ∇u¯ θ ωr . Corresponding spatial distributions of local contributions to the growth rate drift δωi,¯ux , δωi,¯uθ and frequency drift δωr ,¯uθ are also depicted: (b) ∇u¯ x ωi (x, r) · δ u¯ x (x, r), (d) ∇u¯ θ ωi (x, r) · δ u¯ θ (x, r) and (f) ∇u¯ θ ωr (x, r) · δ u¯ θ (x, r). Eigenvalue drifts are computed for base flow modifications induced when δ Ω = 0.5 − 0.45. Dashed lines indicate the separation line.

included. The growth rate sensitivity to axial base flow velocity modifications, ∇u¯ x ωi (Fig. 15a), shows that the eigenvalue is very sensitive to any modification of axial velocity along the axis inside the recirculation region and the zone just downstream of its rear stagnation point. When the magnitude ∇u¯ x ωi (x, r) · δ u¯ x (x, r) is analyzed (Fig. 15b), the effect of axial velocity modifications is twofold: first, the reduction in the backflow induced by swirl at the near wake acts increasing the growth rate value; and second, the decrease in axial velocity originated at the rear end of the recirculation bubble contributes destabilizing the mode by increasing the shear below the separatrix, and stabilizing beyond it. These two opposing effects compose a complex scenario for which the weakening of advection of perturbations dominates over a slight reduction of perturbation production (see Table 5). On the other hand, the regions most sensitive to azimuthal base flow velocity modifications (Fig. 15c, e) are located near the axis, around the rear stagnation point of the recirculation bubble. In fact, a velocity increase promotes a local eigenvalue destabilization (Fig. 15d) along the separatrix and shortly downstream of its rear edge. There is also an important stabilizing contribution below the separatrix, although the total growth rate drift δωi,¯uθ , obtained after integration of ∇u¯ θ ωi (x, r) · δ u¯ θ (x, r), is positive, as listed in Table 5. As for the previous cases, the corresponding growth rate rise is mainly due to a weaker advection of perturbations. Similarly, the total eigenvalue frequency increase, δωr , is

clearly related to a strong positive contribution of ∇u¯ θ ωr (x, r) · δ u¯ θ (x, r) around the rear stagnation point, as shown in Fig. 15(f). Again, a strong resemblance can be observed between the structural sensitivity (Fig. 9f) and the frequency sensitivity to base flow modifications and the corresponding frequency eigenvalue drift in Fig. 15, which display the largest magnitudes near the rear stagnation point of the recirculation bubble. The nature of the mode instability and the role of the stagnation point will be further discussed hereafter in terms of the structural sensitivity. 6. Discussion on the mechanism of instability at large rotation values From the point of view of control, the destabilization of the MF mode is clearly dominated by modifications of azimuthal velocity induced by rotation, as the values included in Table 5 show. However, unlike in the case of the LF and HF modes, which has been proven to stem from modifications by rotation of the well known SS and RSP modes in non-rotating wakes [9], the MF mode nature is still unclear. In that sense, it is seen that sensitivity to azimuthal velocity modifications ∇u¯ θ ω is almost one order of magnitude larger than the corresponding to axial velocity modifications, ∇u¯ x ω. The former involves only terms related to azimuthal and radial components of perturbations (see Eq. (22)), so it might be worth analyzing in depth the MF mode in terms of

J.I. Jiménez-González, C. Manglano-Villamarín and W. Coenen / European Journal of Mechanics / B Fluids 76 (2019) 205–222

the structural sensitivity S(x), in order to evaluate local feedback between these components of perturbation velocity uˆθ and uˆr . Wakes behind rotating axisymmetric bodies keep some resemblance with other swirling flows, such as swirling jets, so that an evaluation of their intrinsic dynamics could help understanding instability nature of the MF mode. In that sense, it has been shown that both swirling jets and wakes might undergo vortex breakdown if the level of swirl is sufficiently high [29]. Typically a two-parameter Grabowski profile [30] is used, which allows to define the axial and azimuthal velocity components piecewise for the regions inside and outside a characteristic radius (the radial velocity component is nil). For swirling jets (which in the limit exhibits a radially uniform axial velocity profile), the base flow bifurcates towards a steady axisymmetric bubble vortex breakdown at moderate values of swirl. This state is unstable to helical (m = −1) perturbations, giving rise to a spiral vortex breakdown [31]. On the contrary, the wake-like profile (featuring radial changes of axial velocity inside the characteristic radius) is first destabilized by a helical breakdown for moderate swirl and Reynolds numbers, resembling the scenario found in the present work for Re < 400. In any case, the mechanism of spiral vortex breakdown seems to be equivalent for both type of swirling flows. More precisely, by comparing the changes in the feedback between components of the perturbation velocity, it has been concluded that the onset of spiral instability is related to the conservation of angular momentum [32]. Following this approach, we present here the structural sensitivity complex-valued † tensor Sij = uˆ i (uˆ j )∗ , which quantifies eigenvalue sensitivity to changes in the feedback between perturbation velocity compoˆ Fig. 16 depicts the absolute value of nine components nents u. of feedback between the perturbation velocity vector uˆ and the components of the momentum equations (see Appendix), for Ω = 0.5 and Re = 330. While real and imaginary parts allow phase identification of feedbacks, absolute values represent the maximum possible coupling between perturbation velocity components. It can be observed that the strongest coupling is produced between radial and azimuthal sensitivities, since the † † highest sensitivities are reached for the components uˆ r uˆ r , uˆ r uˆ θ , † † uˆ θ uˆ r and uˆ θ uˆ θ (Fig. 16e, f, g, i). This is a typical feedback mechanism involving conservation of angular momentum, and is also mainly responsible for the spiral vortex breakdown [see Figs. 6 and 8 in 32]. To a lesser extent, spiral breakdown in swirling jets is also the outcome of the Kelvin–Helmholtz mechanism, although it has a weaker impact on the MF eigenvalue. To show that, we include in Fig. 17 the rate-of-strain tensor of the base ¯ T ], where a quick look at the off-diagonal flow, ϵ = 21 [∇ u¯ + (∇ u) frames allows to identify zones of strong axial shear in the x − r components and azimuthal shear in the r − θ and x − θ components, which are related to the solid body rotation. However, the most sensitive regions involving feedback between the radial and azimuthal perturbation components (Fig. 16e, f, h, i) do not correspond to the areas of high shear of Fig. 17, and feedback involving the axial perturbation velocity and the axial momentum equation (left column and top row of Fig. 16) features a very weak sensitivity. The equivalence between instability scenarios for the spiral vortex breakdown [29,32] and the MF mode studied here is striking. Consequently, we conclude that the same mechanism due to conservation of angular momentum triggers both spiral unstable modes, and that herefore, the intrinsic dynamics of the MF mode is controlled by an efficient feedback between the radial and azimuthal perturbation velocity component. According to the wavemaker interpretation of the sensitivity, where the largest magnitude of S(x) may be understood as the absolutely unstable location that controls the global mode dynamics, the instability emerges from the rear stagnation point of the recirculation bubble. However, the region of large sensitivity

219

is typically somewhat spread out around that rear stagnation point, which may be due to a viscous effect stemming from the laminar nature of the flow under investigation. Therefore, to complement the study on the nature of the instability, we perform an additional analysis of the sensitivity distribution, whereby the perturbation viscous term is penalized by using a damping coefficient β ∈ [0, 1], to evaluate the influence of viscosity on the instability at the perturbation level (see Appendix). Note that the stability analysis for low viscosity, e.g. with low values of β , is equivalent to that performed by Citro et al. [33] to identify the core of the inviscid mechanism of instability in an open cavity flow, within the inviscid structural sensitivity framework. Thus, results for the MF mode at Ω = 0.5 and Re = 330 are shown in Fig. 18, where the sensitivity maps (Fig. 18a) and the eigenvalue drift (Fig. 18b) corresponding to different values of the damping coefficient β are included (note that mesh #3, defined in Table 1, has been used to compute cases of low β in order to ensure good convergence). As the viscous term is increasingly damped, the growth rate ωi grows exponentially, while the frequency ωr decreases, tending towards an inviscid constant value. Interestingly, the magnitude of the structural sensitivity grows with a decreasing value of β (Fig. 18a), and the core of sensitivity concentrates precisely on the stagnation point of the recirculation bubble, as clearly observed for β = 0.1. Thus, the MF mode destabilization is shown to be driven by an inviscid instability mechanism which acts at the rear saddle point. A remaining question still needs to be addressed, which concerns the similarity between the neutral curves of Fig. 6 at low values of Re, regardless of the unstable mode governing the flow bifurcation. In particular, as shown in Section 4, the structural sensitivity of the LF mode, for bodies with ℓ < ℓc3 and large values of Ω (see Fig. 9), also displays a moderate amplitude close to the rear stagnation point, as in the case of the unstable MF mode. Hence, to get a deeper insight into the nature of the unstable LF mode at large rotation rates, we perform a similar inviscid analysis as the previous one for the MF mode. Fig. 19 presents the structural sensitivity and corresponding eigenvalue drift for different values of β , for the LF mode of the ℓ = 0.5 body at Ω = 0.6 and Re = 180. As expected, as the perturbation viscous term is damped, the growth rate increases, and the structural sensitivity is strongly modified. More precisely, the sensitivity located initially for the reference case (β = 1) at the core of the recirculation bubble, vanishes with decreasing β , thus highlighting the viscous nature of the instability mechanism emanating from the SS mode of the non-rotating body. Conversely, the rear region clearly concentrates at the rear stagnation point, as depicted for β = 0.1. The remarkable similarity between the β = 0.1 sensitivity maps from Figs. 18(a) and 19(a), highlights the common inviscid nature of the instability mechanisms of the LF and MF modes at large values of Ω , that leads to coincident profiles of neutral stability for low Reynolds number, as displayed in Fig. 6. 7. Conclusions Global stability and sensitivity analyses have been performed for wakes behind axisymmetric bodies, including the sphere and blunt-based bodies that feature a hemispherical nose and a cylindrical trailing edge of growing aspect ratio, ℓ, for moderate values of the rotation parameter, Ω < 1, and the Reynolds number, Re < 500. The study reveals important differences in the stability of the swirling wakes as the aspect ratio ℓ grows (Fig. 6). In particular, the wake behind short bodies, i.e. the sphere or the hemisphere (ℓ = 0), breaks its axisymmetry through the destabilization of a helical LF mode, stemming from the SS mode at Ω = 0 [9] for the whole range Ω < 1, and this transition takes place at

220

J.I. Jiménez-González, C. Manglano-Villamarín and W. Coenen / European Journal of Mechanics / B Fluids 76 (2019) 205–222

†

ˆ i (uˆ j )∗ for the MF mode at Ω = 0.5 and Re = 330 (ℓ = 1.5 body). Sensitivity magnitude Fig. 16. Absolute value of the components of the sensitivity tensor Sij = u ranges from 0 to 35. Dashed lines indicate the separation line.

¯ T ] for the base flow at Ω = 0.5 and Re = 330 (ℓ = 1.5 body). Contours levels range from −2 Fig. 17. Components of the rate of strain tensor ϵ = 21 [∇ u¯ + (∇ u) (blue) to 2 (red). Dashed lines indicate the separation line . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

lower Re as Ω grows. However, when the aspect ratio grows, this destabilizing effect of rotation at low Ω is hindered, until eventually, from a critical value ℓc1 ≃ 0.25, the axisymmetry breaking transition takes place at higher values of Re as Ω rises, achieving an important stabilization of the wake with rotation. The enlargement of the trailing edge also involves a slight destabilization of the helical HF mode, which stems from the unsteady RSP mode at Ω = 0 [9], which, for ℓ > ℓc2 ≃ 0.5, governs the unsteady bifurcation at moderate values of Ω . Moreover, there is a third critical value of ℓ, 0.5 < ℓc3 < 1, from which a new unstable helical MF mode emerges at high values of Ω . Consequently, it is proven that the stability scenario found in [7]

for bullet-like bodies stems from the rotation of a sufficiently large trailing cylinder, and not from the blunt base. Next, the use of the sensitivity to base flow modifications has allowed to study the physical mechanisms that give rise to the distinct stability properties for bodies of different ℓ. Regarding the LF mode, it has been found that, at low values of ℓ, base flow modifications induced by rotation act by increasing the axial shear along the separation line (Fig. 11), so that the destabilization of this mode is therefore related to a perturbation production mechanism, which is also endorsed by an overall decrease in the perturbation advection inside the recirculation bubble. This mechanism has been also identified in the wake behind the swirling sphere. Conversely, for larger ℓ, base flow modifications

J.I. Jiménez-González, C. Manglano-Villamarín and W. Coenen / European Journal of Mechanics / B Fluids 76 (2019) 205–222

221

Fig. 18. (a) Structural sensitivity of the MF mode for the ℓ = 1.5 body (Ω = 0.5 and Re = 330) at different values of β , and (b) and corresponding eigenvalue drift. Dashed lines in (a) indicate the separation line, while arrow in (b) represents the decreasing sense of β = 1, 0.9, 0.8, . . . , 0.1.

Fig. 19. (a) Structural sensitivity of the LF mode for the ℓ = 0.5 body (Ω = 0.6 and Re = 180) at different values of β , and (b) and corresponding eigenvalue drift. Dashed lines in (a) indicate the separation line, while arrow in (b) represents the decreasing sense of β = 1, 0.9, 0.8, . . . , 0.1.

strengthen the axial perturbation advection (Fig. 12), which becomes stronger as Ω grows, promoting therefore the stabilization of the mode. Moreover, the integration of local contributions to the growth rate and frequency drifts of the base flow modifications (Eq. (23)) provides satisfactory values and predictions of the destabilization (resp. stabilization) of the LF mode for small ℓ (resp. large ℓ) and of the increase in frequency, recovering the results of the direct global instability. As far as the HF mode is concerned, destabilization with increasing Ω at large ℓ is shown to be related to azimuthal base flow variations, and more precisely to the weakening of perturbation advection (Fig. 14). Similarly, the destabilization of the MF mode at larger values of Ω is also dominated by the azimuthal velocity modifications that rotation induces, although the role of axial velocity changes is not negligible. In general, the destabilization is driven by the weakening of perturbation advection, according to the predicted values of the growth rate drifts by means of a sensitivity analysis (Table 5). The most sensitive region is found to be located around the rear stagnation point of the recirculation region, which is also shown to control the frequency growth associated to the mode destabilization. Therefore, this location can be considered a wavemaker in the classical local stability sense. In fact, the analysis of the structural sensitivity distribution [15] identifies a similar receptive region at the rear stagnation point of the recirculation bubble (Fig. 9). To complete the discussion, the origin of the MF mode has been further investigated by means of the structural sensitivity tensor, where a typical feedback mechanism involving conservation of angular momentum is observed (Fig. 16), identified by means of the strong coupling between radial and azimuthal sensitivities. This mechanism is analogous to that described in [32] for spiral vortex breakdown in swirling jets, and as it happens for that type of flow, the instability requires a certain level of swirling motion to be efficiently activated. Additionally, the influence of viscosity at the perturbation level has been also evaluated, showing that the destabilization of the MF mode is mainly driven by

an inviscid mechanism that acts precisely at the rear stagnation point, in view of the sensitivity maps of Fig. 18. Furthermore, the nature of the LF mode destabilization at larger values of Ω has been also found to be related to a similar inviscid mechanism (Fig. 19). This common nature lies at the base of the similar neutral stability curves (Fig. 6) at low values of Re and large Ω for both the LF and MF modes. Acknowledgments This work has been supported by the Spanish Ministry of Science, Innovation and Universities (MICINN) and European Funds under Project DPI2017-89746-R. The authors are also grateful to Prof. Carlos Martínez-Bazán and Prof. Flavio Giannetti for valuable discussions. Appendix. Stability equations The generalized direct eigenvalue problem (Eq. (8)) is defined by the following equations: 1 ∂ (r uˆr ) imuˆθ ∂ uˆx + + , (A.1) ∂ x( r ∂ r r ) ∂ uˆx ∂ u¯ x ∂ uˆx ∂ u¯ x imu¯ θ uˆx − iωuˆx = − u¯ x + uˆx + u¯ r + uˆr + ∂x ∂x ∂r ∂r r ∂ pˆ 1 − + ∇ 2 uˆx , (A.2) ∂x Restb ( ∂ uˆr ∂ u¯ r ∂ uˆr ∂ u¯ r − iωuˆr = − u¯ x + uˆx + u¯ r + uˆr ∂x ∂x ∂r ∂r ) imu¯ θ uˆr 2u¯ θ uˆθ + − r r ( ) ∂ pˆ 1 uˆr 2imuˆθ − + ∇ 2 uˆr − 2 − , (A.3) ∂r Restb r r2 0=

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∂ uˆθ ∂ u¯ θ ∂ uˆθ ∂ u¯ θ imu¯ θ uˆθ + uˆx + u¯ r + uˆr + ∂x ∂x ∂r ∂r r ) u¯ r uˆθ u¯ θ uˆr + + r r ( ) impˆ uˆθ 1 2imuˆr − + ∇ 2 uˆθ − 2 + ; (A.4) 2

(

− iωuˆθ = − u¯ x

r

Restb

r

r

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