Energy transient growth in curved channel flow
Cheng Chen, Bo-Fu Wang, Zhi-Wei Guo & De-jun Sun
Acta Mechanica ISSN 0001-5970 Volume 221 Combined 3-4 Acta Mech (2011) 221:341-351 DOI 10.1007/s00707-011-0513-z
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Author's personal copy Acta Mech 221, 341–351 (2011) DOI 10.1007/s00707-011-0513-z
Cheng Chen · Bo-Fu Wang · Zhi-Wei Guo · De-jun Sun
Energy transient growth in curved channel flow
Received: 15 January 2011 / Published online: 7 July 2011 © Springer-Verlag 2011
Abstract Transient growth of perturbations due to non-normality for the curved channel Poiseuille flow (CCPF) is presented. The study covers a wide range of the radii ratios η as well as axial and azimuthal modes, with the purpose of complementing the results of linear stability for this flow with a study of the optimal linear growth possible in the linearly stable parameter regions. For the wide-gap case, the transient growth of streamwise-azimuthal modes that grow most is of low level and suppressed by curvature. It is also found that as curved channel flow approaches the flow in a straight channel enough, both the normal and non-normal stability characteristics are almost identical to that of plane Poiseuille flow. The modulation of the basic circular Poiseuille flow by the presence of azimuthal streaks resulting from the significant growth of initial perturbations can be clearly visualized. For the transition region between the wide-gap case and the narrow-gap case, the sensitivity of eigenvalues is shown to be closely related to the magnitude of transient growth, which is tuned by curvature in a smooth way.
1 Introduction In the past decades, curved channel Poiseuille flow (CCPF) under a pressure gradient acting round the channel has been one of the most studied problems of fluid dynamics. It is important in yielding a better understanding of the physics of transition in general, and in other curved geometries in particular. Starting with the celebrated work of Dean [1], CCPF has been an experimental, theoretical, and numerical benchmark problem for bifurcation theory and hydrodynamic stability. Linear stability analysis shows that the primary instability occurs and steady Dean vortices develop, if the so-called Dean number De(De = Re d/ri∗ , where Re is the Reynolds number, d is the channel width, and ri∗ the radius of curvature of the inner wall of the channel) is above a critical value. At values of this parameter below the critical, curved channel flow is stable with respect to infinitesimal perturbations. Dean [1] first conducted this analysis in the narrow-gap approximation (i.e., radii ratio η ∼ 1) and obtained the critical Dean number (Decr ) as 35.92. For the narrow-gap case, Reid [2] and Hammerlin [3] considered this problem again and determined a part of the neutral stability curve. Walowit et al. [4] have also studied the finite-gap problem, confirming the value of Recr found experimentally by Brewster et al. [5]. For η ≥ 0.95, Gibson and Cook [6] examined the linear neutral stability of CCPF to infinitesimal non-axisymmetric disturbances. They found that when 0.95 ≤ η < (1 − 2.179 × 10−5 ), α = 0 modes are more unstable than β = 0 modes, but for η > (1 − 2.179 × 10−5 ), the reverse is true, where α and β are the wavenumbers in the streamwise and spanwise directions, respectively. Moreover, the linear stability analysis of CCPF has been extended to Taylor-Dean flow [7] and thread-annular flow [8], where the linear instability C. Chen · B.-F. Wang · Z.-W. Guo · D. Sun (B) Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, China E-mail:
[email protected] Tel.: +86-551-3606797
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boundaries are very different from the Dean problem. Both studies show that the instability is achieved not with the lowest mode. As De is increased further, the primary instability may be susceptible to several new types of instabilities. Finlay, Keller, and Ferziger [9] made the first numerical simulation of the stability of curved channel flow and found two different types of secondary instability of traveling wave type, one with a long (many channel heights) and one with a short wave length (of the order of the channel height). These were denoted undulating and twisting vortices, respectively. Various experiments on CCPF stability have also been performed and are comprehensively reviewed by Matsson et al. [10,11]. They reported from hot-wire measurements that two types of secondary instability, with distinctly different frequencies are at hand simultaneously, and their spatial distribution and growth are determined. It is believed that the secondary instability described is of relevance for the breakdown to turbulence in flows with Dean vortices. Recently, the researchers have found that subcritical transition from laminar to turbulent flow in wallbounded or unbounded shear flows (Poiseuille, Couette, boundary layer flows, etc) is closely associated with transient energy growth of small disturbances [12–14]. Namely, the perturbation energy can grow up to a very high level over short times, and this phenomenon leads to the elongated streamwise streak structures that trigger transition to turbulence ultimately. Mathematically, the amplification of perturbations is due to the effect of non-normality in the linearized evolution operator and non-orthogonality of eigenmodes derived from the linear stability analysis. Physically, a lift-up mechanism has been widely accepted to explain this short-term growth in asymptotically stable flows, which states that spanwise vorticity in basic flow transfers energy to streamwise velocity under the action of normal velocity perturbation [15]. Note that as η → 1, CCPF approaches plane Poiseuille flow infinitely. Besides, plane Poiseuille flow exhibits substantial transient growth as reported in an early investigation by Reddy and Henningson [14]. Thus, it is natural to guess that near the limit, i.e., in the narrow-gap case, the transient growth characteristic of CCPF might be similar to that of plane Poiseuille flow. Moreover, it is also natural to enquire what is the transient growth characteristic for finite-gap cases, and these questions motivate the present study. In this paper, we investigate the effect of so-called transient growth mechanisms on CCPF, with the purpose of complementing the results of linear stability analysis for this flow in asymptotically stable flow regimes. In addition to the single non-dimensional parameter—the Reynolds number—of plane Poiseuille flow, CCPF is characterized by the other parameter denoting curvature. Though there also have been many studies on transient growth in confined flows with remarkable curvature recently [16–18], detailed discussions about the effect of curvature are rare. In the following, our research is devoted to approaching the non-normal instability of plane Poiseuille flow via CCPF, aiming to investigate the role of curvature in tuning the non-normality of the system. The remainder of this paper is organized as follows. In Sect. 2, we present the governing equations and boundary conditions and define the energy norm that measures the transient growth of perturbations. In Sect. 3, the problem is solved numerically, and we provide a comprehensive exploration of the optimal transient growth for different azimuthal and axial modes. Finally in Sect. 4, main conclusions are drawn. 2 Mathematical formulation Consider the CCPF of an incompressible fluid of kinematic viscosity γ and density ρ confined in a curved channel, as sketched in Fig. 1, where the inner and outer walls of the channel have radii of curvature ri∗ and ro∗ , respectively. We work in cylindrical polar coordinates (r, θ, z) with the z-axis coinciding with the common axis of the walls. It is assumed that the flow is driven by a constant azimuthal pressure gradient ∂ p/∂θ . In the absence of perturbation, the basic azimuthal shearing velocity profile is given by {(ro∗2 log ro∗ − ri∗2 log ri∗ )r − ri∗2 ro∗2 log(ro∗ /ri∗ )/r } ∂p 1 r ln r − V (r ) = . (1) 2ργ ∂θ ro∗2 − ri∗2 By writing d=
ro∗
− ri∗ , η
r∗ d2 = i∗ , Vm = ro 2ργ ri∗
∂p Vm d , Re = , ∂θ γ
(2)
all variables will be rendered dimensionless using d, d/Vm , ρVm2 as units for space, time, pressure, respectively. The flow is thus confined between ri = η/(1 − η) and ro = 1/(1 − η). The dimensionless form of (1) is:
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Fig. 1 Definition of flow geometry in curved channel flow
η E V (r ) = r ln r + Cr + , ri ≤ r ≤ r o 1−η r
(3)
where C =−
ro2 ln ro − ri2 ln ri ro2
− ri2
,
E=
ri2 ro2 ro2
− ri2
ln
ro . ri
For our analysis, the mean flow is perturbed by a velocity disturbance (u, v, w)eimθ +iβz and pressure disturbance peimθ +iβz . We take m and β to be real wavenumbers, and the complex amplitudes u, v, w and p are functions of time t and radius r . The linearized Navier-Stokes equations become ∂u m2 + 1 imV u 2V v 2imv −1 (r u ) 2 + − = − p + Re − β + u− 2 , (4) ∂t r r r r2 r m2 + 1 imp (r v ) 2imu ∂v imV v (V r ) u + + =− + Re−1 − β2 + v+ 2 , (5) 2 ∂t r r r r r r m2 (r w ) ∂w imV w + = −iβp + Re−1 − β2 + 2 w , (6) ∂t r r r imv (r u) + + iβw, (7) 0= r r where a prime denotes differentiation with respect to r . The boundary conditions on the disturbance velocity are no slip on the walls, i.e., u = v = w = 0 at r = ri , ro . Introduce a vector q = (u (r, t), v (r, t), w (r, t), p (r, t))T , where T means transpose. The full evolution equation is written symbolically as M
∂q ∂q = Lq or = M−1 Lq, ∂t ∂t
(8)
where M and L are operator matrices that can be easily derived from the governing equations. Suppose the solution to Eq. (8) is of the form q = q˜ exp (−iωt), where ω is the complex eigenvalue, and q˜ = (u˜ (r ), v˜ (r ), w˜ (r ), p˜ (r ))T is the eigenvector consisting of eigenfunctions. Thus, the evolution equation is transformed into a generalized eigenvalue problem as follows: ˜ − iωq˜ = M−1 Lq.
(9)
In order to obtain the eigenvalues and eigenfunctions numerically, we discretize the spatial radial coordinate using a Chebyshev spectral collocation method. First, the radial coordinate r is mapped onto the computational space y ∗ ∈ [−1, 1] through an appropriate linear transformation r=
y ∗ + ri + r o . 2
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Table 1 The leading eigenvalues as a function of the number of collocation points, η = 0.999999, De = 65, α = 3 and β = 6 N
ωr
ωi
50 60 80 100 120 150
−0.7480 6059 085 −0.7480 3882 580 −0.7480 3808 687 −0.7480 3809 011 −0.7480 3809 012 −0.7480 3809 012
−0.001993 1906 769 −0.002083 2095 776 −0.002076 5492 435 −0.002076 5444 140 −0.002076 5444 149 −0.002076 5444 146
Second, using N collocation points for this discretization, Eq. (9) is transformed into a matrix equation. Finally, the eigenvalue problem is solved using a MATLAB code. In order to measure the growth of arbitrary initial disturbance q, we define the energy norm by means of the inner product ||q||2E
1 = (q, q) E = 2
ro
q∗ · qr dr,
(10)
ri
where * stands for complex conjugation. We consider the matrix of inner products between the eigenvectors Mi j = (q˜ i , q˜ j ) E ,
(11)
where q˜ i and q˜ j denote the ith and jth eigenfunctions. This matrix is positive definite, and its elements represent the non-orthogonality of two eigenfunctions. The maximal energy growth for q evolving according to (8) is defined by maximizing the ratio between the energy norm of the perturbation at time t and its initial norm, G (t) = sup
q(0) =0
q (t) 2E
q (0) 2E
,
(12)
which is calculated by the methodology described in Refs. [15,19–21]. The optimal energy growth in time is defined as G o = G (to ) = supt≥0 G (t), where to is the corresponding optimal time. In order to compare with plane Poiseuille flow, we introduce α = m/ri as the streamwise wavenumber. Then, the peak value of G o is defined in the wavenumber plane as G P = G o (α P , β P ) = supα,β G o (α, β), where α P and β P are the corresponding peak wavenumbers. Furthermore, we define the Dean number as De = Re
η(1 − η2 )2 − 4η3 (ln η)2 , 4(1 − η)5/2 (1 − η2 )
the form of which is derived to be consistent with that defined by Walowit et al. [4]. The convergence of the numerical method is reflected in Table 1, where the real and imaginary parts of the leading eigenvalues are presented as functions of the number of collocation points for the case η = 0.999999, De = 65, α = 3 and β = 6 being the maximum values of the four parameters used in our computation. The number of modes used for computation was N = 80 in most cases of this work. While for some special cases like the one mentioned earlier, we have that N = 120. Before proceeding to the study of transient growth in curved channel flow, we examine the critical values of Re. The critical parameter has been computed by solving the eigenvalue problem (9) and imposing the condition that the imaginary part of the uppermost eigenvalue be zero. As mentioned earlier, the most unstable disturbance changes from α = 0 mode to β = 0 mode at the transition point ηt = (1 − 2.179 × 10−5 ). From now on, they are referred to as streamwise-azimuthal mode and spanwise-axial mode, respectively. Figure 2a plots critical Reynolds numbers for both modes, which agreed to at least five significant digits with the published results of Gibson and Cook. Figure 2b depicts the computed results of G(t) for stable and unstable plane Poiseuille flow with α = 1, β = 0. The stable case corresponds to Re = 5000 and the unstable to Re = 8000. The curves herein are in very good agreement with the results of Reddy and Henningson, testifying the validity of our numerical code.
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+
+ +
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1/(1-η)×10
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t
Fig. 2 Comparison between the numerical results of Gibson and Cook [6], Reddy and Henningson [14] and the present work. a The critical Reynolds number as a function of 1/(1 − η). b G(t) for stable and unstable plane Poiseuille flow, α = 1, β = 0 105
104
104 3
10
102
2
Recr
GP
103 10
wide-gap
transition region
101 100 100
102
104
narrow -gap
101
100 106
1/(1-η) Fig. 3 The variation of the peak value G P and the critical Reynolds number Recr for the most unstable mode as a function of η. The solid line corresponds to G P , and the dashed one to Recr
3 Results and discussion Since the normal-mode stability of CCPF is dominanted by the streamwise-azimuthal mode or spanwise-axial mode in accordance with the degree of channel curvature, it is natural to investigate the transient growth of these two modes. Figure 3 plots the variation of the peak value G P as a function of η. The critical Reynolds number, which corresponds to the most unstable mode, is also shown. It is observed that G P increases initially at relatively small η, attains a maximum near ηt , and then decreases. Thus, we suggest that CCPF can be classified into three cases, namely the wide-gap (η < 0.999), narrow-gap (0.999 ≤ η < 0.99999), and the transition region (η ≥ 0.99999). In the following, we discuss results obtained by transient growth analysis combined with eigenvalue analysis, focusing on the transient and long-term behavior for the three cases.
3.1 The wide-gap case For the wide-gap case, the results of our exploration for streamwise-azimuthal modes are summarized in Fig. 4, as the disturbances uniform in the azimuthal direction are proved to experience the maximum transient growth among all azimuthal wavenumbers. The shaded zone represents the region of the (η, De) plane where CCPF is linearly unstable, i.e., G o → ∞. This region has a lower boundary that is the critical Dean number Decr (η) where the first linear instability appears. Below this critical boundary, contours of G o are shown for the purpose of exploring the effect of curvature with the axial wavenumber maximizing G o . Different features can be pointed out. First, energy growth is of very low level for the radius ratio in the range 0.1 ≤ η ≤ 0.98. The fact implies that curvature plays an important role in suppressing energy amplification, and such a suppression is so strong that G o < 10 even with relatively small curvature (say, η = 0.98). Second, it is clearly visualized that G o at a fixed Dean number initially decreases as η is increased, attains a minimum, and then increases until the maximum η considered. The non-monotonic change is due to the dependence of De on η. Actually, energy growth at fixed Re can be always enhanced by larger radii ratio. Finally, with increasing Re, especially for large-η cases, non-modal transient growth becomes progressively larger, as can be seen in Fig. 3.
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60
1.8 50
1.6
1. 5 1.4
40
1.2
1. 8
1.1
30
6 3 2
De
1.3
20
10 0.2
0.4
0.6
0.8
η Fig. 4 Contours of optimal growth for CCPF in the (η, De) plane. The dashed curve corresponds to G o (η, Re) = 1 0.8 M12 M23
inner product
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
η Fig. 5 The variation of the inner products M12 and M23 as functions of η
Following Trefethen et al., transient growth of flow perturbations that may evolve from certain initial conditions is thought to be an interpretation of non-normality [22]. Our computation shows increasing M12 and M23 with increasing η in Fig. 5. This strongly suggests that curvature does weaken the non-orthogonality of the first three eigenmodes and thus the non-normality of the linearized evolution operator in the wide-gap case. 3.2 The narrow-gap case Having examined optimal energy growth in the wide-gap case, we now turn to the narrow-gap case. As well known, plane Poiseuille flow exhibits substantial transient growth as reported in an early investigation by Reddy and Henningson [14]. It can be considered as the η → 1 limit of CCPF under an azimuthal pressure gradient. Therefore, it is natural to enquire what is the transient growth characteristic near this limit, and the question motivates the present study on the narrow-gap case. In Fig. 6, we have plotted G o and to in the (α, β) plane for a representative radii ratio of 0.99999. The Reynolds numbers are set to be Recr for spanwise-axial mode, since spanwise-axial mode herein is most unstable. For a general view, the level curves shown are nearly identical to those for plane Poiseuille flow, implying that transient behavior in the narrow-gap case is qualitatively consistent with that in a straight channel. Moreover, the minor quantitative difference is due to modifications for small, but nonzero, curvature. According to Fig. 6a, there are two peak values of G o , denoted by symbols A and B, corresponding to plane Poiseuille flow and the narrow-gap case, respectively. Both the peaks correspond to optimal streamwise-azimuthal disturbances having streamwise uniform wavenumber (α = 0), indicating that curvature does not alter the fact that the streamwise-azimuthal modes are optimal. Peak A lies at relatively low wavenumber β P = 4.10,
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(b)
B
4
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β
β
A
2
2
0
0 0
0.5
1
1.5
2
0
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1
α
1.5
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Fig. 6 Level curves of the optimal growth G o in the (α, β) plane (a). The curves, from left to right, correspond to G o (α, β, Recr ) = 6, 200, 5, 600, 5, 000, . . . , 800. Contours of the optimal time to (b). The curves, from left to right, correspond to to (α, β, Recr ) = 420, 260, 180, 140, 100, 80, 60. The dashed curves correspond to η = 0.99999, and the solid curve corresponds to plane Poiseuille flow
(a)
(b)
(c)
(d)
2π/β
2π/β
2π/β
2π/β
z
z
z
z
0
ri
r
ro
0
ri
r
ro
0
ri
r
ro
0
ri
r
ro
Fig. 7 Radial-axial components of the perturbation field for peak B at different times: (a) t = 0, (b) t = to /10, (c) t = to , and (d) t = 15to
the corresponding peak value G P = 6526.28; peak B lies at relatively large wavenumber β P = 4.15, and the corresponding peak value G P = 6737.14. These would imply that the axial wavelength of the optimal perturbation, which maximizes energy growth, would be decreased by a slight curvature. At the same time, the maximum energy amplification achieved is increased, as curvature leads to a larger critical Reynolds number Recr . In order to clarify the transient growth mechanism for the narrow-gap case, we have also studied the formation of azimuthal streaks resulting from the effects of streamwise-azimuthal perturbations, following Meseguer [16]. Figure 7 shows the time evolution of the radial-axial components of the velocity perturbation field in θ = const plane for peak B described earlier, and t = 0, to /10, to , 15to . The initial vector field at t = 0 is characterized by a pair of counter-rotating vortices (Fig. 7a). With time, we observe the perturbation velocity decays. For t = 15to , the energy of the radial-axial contribution has been transferred to the azimuthal direction by the lift-up mechanism. The modulation of the basic circular Poiseuille flow is clearly visualized in Fig. 8. For t = to (Fig. 8c), we observe the generation of azimuthal streaks near the inner and outer walls. After a long time, CCPF has almost recovered its initial state, as predicted by the modal analysis (Fig. 8d). In summary, CCPF in the narrow-gap case presents quite similar transient behaviors to that of plane Poiseuille flow, indicating that curvature tunes the non-normal stability of CCPF in a smooth fashion near the η → 1 limit. Note that CCPF in the narrow-gap case features spanwise-axial mode as the most unstable eigenmode.
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(a)
(b)
(c)
(d)
2π/β
2π/β
2π/β
2π/β
-0.0
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ri
r
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ro
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Fig. 8 Isovalues of the modulated azimuthal CCPF for peak B at different times: (a) t = 0, (b) t = to /10, (c) t = to , and (d) t = 15to
0
(b)
dominant
104 α=0 β=0
Go
ωimax ×10
3
10
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ηc
101
α=0 β=0
-2 0.999975
102
0
0.99998
η
0.999985
10 0.999975
0.99998
0.999985
η
Fig. 9 Plots of variation of the imaginary part of the most unstable eigenvalue (a) and the optimal energy growth G o (b) as a function of η
We conclude that in terms of temporal stability, either modal or transient, it is in good qualitative agreement with plane Poiseuille flow.
3.3 The transition region As mentioned earlier, CCPF in the narrow-gap case features large transient energy amplification as in plane Poiseuille flow. Such behavior contrasts with that of finite curvature, which leads to weak energy growth. Thereby, we now turn to the transition region, looking to bridge the gap between these two perspectives by considering the transient growth problem for the values of η between the wide-gap case and the narrow-gap case. It is noticeable that the change of curvature is equivalent to base flow variation. Accessing the modal instability of plane shear flows and pipe flows via base flow variation is of significant theoretical meaning and has inspired a number of investigations [23,24]. Our investigation proceeds somewhat in a different direction, focusing on the effect of base flow variation on non-modal stability and the relationship to that on modal instability in the transition region. In Fig. 9a, we plot the variation of the imaginary part of the most unstable eigenvalue with Reynolds number Re = 46, 100. As in the work of Gibson and Cook [6], we focus on twodimensional spanwise-axial and streamwise-azimuthal modes. For spanwise-axial modes, the most unstable eigenvalues are obtained by searching for the first local maximum of ωi max (α); for streamwise-azimuthal modes, they are obtained by searching for the first local maximum of ωi max (β). This is also the case for the optimal energy growth G o herein. When the basic flow is perturbed by larger curvature (η < ηc ), the streamwise-azimuthal modes are more unstable than spanwise-axial modes. As η is increased, the growth rate (i.e., ωi max ) of the streamwise-azimuthal mode decreases rapidly. This contrasts with the spanwise-axial mode
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Fig. 10 The evolution of kinetic energy in different directions for streamwise-azimuthal (a) and spanwise-axial (b) modes
case, in which the maximum growth rate remains almost unchanged. When curvature is reduced to a smaller level (η > ηc ), the spanwise-axial mode become the more unstable one. Thus, we conclude that the transition of the most unstable eigenmode is due to the discrepancy of eigenvalue sensitivity for different modes, e.g., spanwise-axial and streamwise-azimuthal modes. With the purpose of investigating the role of curvature in tuning the transient growth characteristics of CCPF in the transition region, Fig. 9b plots the variation of the optimal energy growth with radii ratio η. Nearly invariant optimal energy growth occurs even for streamwise-azimuthal modes, whose normal-mode stability is sensitive to base flow variation in terms of channel curvature. With decreasing η, the streamwiseazimuthal modes become eigenvalue unstable, i.e., G o → ∞ at η ≈ 0.9999782. Moreover, energy growth for the streamwise-azimuthal mode is about two orders of magnitude higher than that for the spanwise-axial mode. This observation is consistent with the arguments of Reddy et al. [25]. They have observed similar behaviors in parallel shear flows. Note that in contrast to two-dimensional spanwise-axial modes the streamwise-azimuthal modes are of the largest energy growth because of their slower viscous damping, as proposed in [26]. In order to provide a further demonstration of the effect of curvature, the evolution of kinetic energy in different directions, defined as ro E u (t) =
ro |u| r dr , 2
ri
E v (t) =
ro |v| r dr , 2
ri
E w (t) =
|w|2 r dr , ri
is shown in Fig. 10 for streamwise-azimuthal and spanwise-axial modes. For the streamwise-azimuthal mode, as seen in Fig. 10a, the energy in the radial and axial directions decreases with time, at all times shown. Further, the azimuthal velocity perturbation (not shown) grows rapidly, attains a maximum, and then decreases back to zero. The lift-up mechanism is sufficiently reflected by these facts. Note that the results show that the change of curvature has little influence on the evolution of the flow in the early stage, denoted as I, since the curves E u (t) and E w (t) corresponding to different values of η almost coincide. This would imply that for streamwise-azimuthal modes, transient behaviors seem to be insensitive to basic flow variation. Having passed the transition stage, denoted as II, the flow is dominanted in the stage III by the the least stable eigenvalue, the imaginary part of which is the energy change rate ωi . For each of the three values of η involved, both E u (t) and E w (t) curves have the same ωi , which changes observably with η. This would imply that curvature does have a significant effect on some of the eigenvalues. Considering the basic flow variation in terms of channel curvature as the operator perturbation, the observations about the sensitivity of eigenvalues and transient growth above could be explained by the discussions of Reddy et al. on the Orr-Sommerfeld operator [25]. They took a random matrix E as the operator perturbation and found that although perturbations have a significant effect on some of the eigenvalues, they have relatively little effect on the maximum transient growth. Figure 10b plots the kinetic energy in radial and azimuthal directions E u (t) and E v (t) for spanwise-axial modes. It is seen that during the whole period of time the energy curves coincide for different values of η. Consequently, there is a difference to streamwise-azimuthal modes. That is, after sufficiently long time, the growth rates corresponding to different curvature are yet nearly identical, implying that the eigenvalues herein seem not to be dependent on basic flow variation evidently. To illustrate the discrepancy of the sensitivity of eigenvalues between streamwise-azimuthal and spanwiseaxial modes, we associate transient growth with the condition number of an eigenvalue, which is a measure of
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C. Chen et al.
Table 2 Comparison of the optimal energy growth to the sensitivity of the most unstable eigenvalue with Re = 46,100 η
Spanwise-axial mode
0.999975 0.999978 0.99998 0.999982 0.999984
Streamwise-azimuthal mode
ωm
Go
s(ωm )
0.2122159–0.0194089i 0.2122164–0.0194091i 0.2122168–0.0194092i 0.2122172–0.0194094i
51.25696 51.25711 51.25721 51.25729
145.8711 145.8715 145.8718 145.8721
ωm
Go
s(ωm )
−0.44405 × 10−4 i −0.90792 × 10−4 i −0.13907 × 10−3 i
7684.76 7357.97 7125.88
9.841 × 104 1.029 × 105 1.077 × 105
sensitivity to small perturbations. It is computed as s(ωk ) =
||yk ||||xk || , |ykH xk |
(13)
where xk and yk denote the right and left eigenvectors of the evolution operator corresponding to the eigenvalue ωk , H the conjugate transpose. The comparison of the optimal energy growth to the sensitivity of the most unstable eigenvalue is given in Table 2 for both spanwise-axial modes and streamwise-azimuthal modes, respectively, where ωm labels the most unstable eigenvalue. As is the case for the energy growth G o , the value of the sensitivity of ωm for streamwise-azimuthal modes is much larger than that for spanwise-axial modes, being consistent with the discrepancy mentioned earlier. Recovering the arguments of Schmid et al. [27], the sensitivity of the eigenvalues is closely related to the magnitude of transient growth, and they are both an effect of the non-normality in the linearized evolution operator. Streamwise-azimuthal modes feature the largest growth, and their normal-mode stability has strong sensitivity to perturbations, including basic flow variation; spanwise-axial modes experience much weaker growth, and their eigenvalue spectrum is nearly independent of perturbations. Moreover, we would point out that basic flow variation, tuning non-normality, may have a different effect on energy growth from that on the sensitivity of eigenvalues. We mention this because for streamwise-azimuthal modes, s(ωk ), increase with increasing η, while G o has an opposite trend. Nevertheless, for spanwise-axial modes, the increase of η leads to relatively small growth of both G o and s(ωm ). 4 Conclusions In this paper, we have investigated the transient behaviors of small disturbances in CCPF. The numerical results show that for wide-gap cases transient growth is strongly suppressed due to the effect of the curvature, which plays an important role in weakening the non-normality of the linearized evolution operator. For narrow-gap cases with η sufficiently close to 1, CCPF is found to be in good qualitative agreement with plane Poiseuille flow, viewing both modal and non-modal instability. Besides, the modifications due to curvature lead to the decrease of the axial wavelength of the optimal perturbation. When the transition region is considered, it is found that similar to the narrow-gap case curvature tunes the non-normal stability in a smooth way. This contrasts with the strong sensitivity of eigenvalues, which are thought to be closely related to the magnitude of transient growth. It is noticeable that basic flow variation, tuning non-normality, may have a different effect on energy growth from that on the sensitivity of eigenvalues. The present results can be regarded as an elementary exploration of the effect of basic flow variation on non-modal stability, and we believe this requires further investigation. Acknowledgments The authors are indebted to Professor Yin Xie-Yuan for valuable discussions that helped to improve the paper. The work was supported by the National Natural Science Foundation of China, Project No. 10772172.
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