POD analysis of the unsteady behavior of a laminar separation bubble

Accepted Manuscript POD analysis of the unsteady behavior of a laminar separation bubble Davide Lengani, Daniele Simoni, Marina Ubaldi, Pietro Zunino PII: DOI: Reference:

S0894-1777(14)00153-8 http://dx.doi.org/10.1016/j.expthermflusci.2014.06.012 ETF 8248

To appear in:

Experimental Thermal and Fluid Science

Received Date: Revised Date: Accepted Date:

15 January 2014 30 April 2014 11 June 2014

Please cite this article as: D. Lengani, D. Simoni, M. Ubaldi, P. Zunino, POD analysis of the unsteady behavior of a laminar separation bubble, Experimental Thermal and Fluid Science (2014), doi: http://dx.doi.org/10.1016/ j.expthermflusci.2014.06.012

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POD analysis of the unsteady behavior of a laminar separation bubble Davide Lengania,∗, Daniele Simonia , Marina Ubaldia , Pietro Zuninoa a

DIME Universit´a di Genova, Via Montallegro 1, I-16145 Genoa, Italy

Abstract Particle Image Velocimetry (PIV) measurements have been performed in order to analyze the unsteady flow field developing along the separated flow region of a laminar separation bubble. Data have been post-processed by means of Proper Orthogonal Decomposition (POD) to improve the understanding of the physics of this complex phenomenon. The paper shows that the first two POD modes of the normal to the wall velocity component are coupled. Thus, they are representative of a vortex shedding phenomenon which is identified to be induced by Kelvin-Helmholtz instability. The POD allows the phase identification of each PIV image within the vortex shedding cycle. The computed eigenvectors are used to sort the experimental snapshots and then reconstruct a phase-averaged velocity field which highlighted the motion of vortices shed close to the bubble maximum displacement. Moreover, other sources of deterministic fluctuations characterized by frequencies which are different from the one induced by the Kelvin-Helmholtz instability are also revealed. Indeed, the most energetic POD mode of the streamwise velocity component is not related to the shedding frequency, while it describes large velocity fluctuations in the shear layer region upstream of the bubble maximum displacement, where the turbulent activity is not yet present. The POD decomposition presented here identifies the large scale structures within the flow, thus separately accounts for both coherent and stochastic contributions to the overall energy of the velocity fluctuations. Keywords: Separated-flow transition, Kelvin-Helmholtz instability, POD, Ultra-High-Lift turbine blade profile

1. Introduction The formation of a laminar separated flow region over airfoils, as well as on turbomachinery bladings, is frequently observed as a consequence of high aerodynamic loading (inducing strong adverse pressure gradients), low Reynolds number conditions and high angles of attack. In the case of boundary layer laminar separation, the inherently unstable separated shear layer typically rolls up inducing a vortex shedding phenomenon responsible for high unsteadiness and energy losses within the flow (e.g. Refs. [1–5]). A close inspection of the dynamics of the vortex shedding phenomenon can be obtained experimentally by means of fast response PIV systems [6–8], as well as through phase-locked acquisition procedures [9]. Anyway, in this latter

∗ Corresponding

author Email address: [email protected] (Davide Lengani)

Preprint submitted to Elsevier

June 17, 2014

case, the phase-jitter induced by slight cycle to cycle variations of the phenomenon (which is typically not perfectly repeatable) may introduce statistical errors in computing phase-averaged quantities, as discussed in Perrin et al. [10]. To this purpose, the work of van Oudheusden et al. [11] clearly shows that Proper Orthogonal Decomposition (POD) can be successfully adopted to phase-lock and sort the unrelated flow maps obtained by means of not time-resolved PIV instrumentation. The POD technique was first introduced for revealing coherent structures in turbulent flows by Lumley [12], while the snapshot POD method was successively derived by Sirovich [13] taking care of ergodic theorem in order to reduce the computational effort required to obtain POD modes and their temporal coefficients. POD has been widely applied in recent years to analyze the unsteady characteristics of the wakes shed by bluff bodies [10, 14, 15], coherent structures embedded within free-jets [16] and highly swirling flows [17]. The spatial distribution of the first two POD modes has been found to be typically coupled in pairs when a well established convective vortex shedding phenomenon takes place, as described in the theoretical formulation of Legrand et al. [18, 19]. Moreover, when the flow is dominated by large-scale convective structures the first pair of modes represents the orthogonal components of the harmonics of the vortex-shedding process, and they can be adopted to construct a low order model able to describe the coherent pattern of the phenomenon, as shown in the works of Wee et al. [20] and Ben Chiekh et al. [14]. To this end, Konstantinidis and Balabani [21] recommended the use of only modes which exhibit a sufficient degree of symmetry or anti-symmetry to indicate a repetitive coherent modal behavior, while less energetic POD modes represent the small-scale turbulent structures. However, wall proximity effects may alter the spatial distribution of the most energetic pair of modes as shown in Shi et al. [15], where these effects on the time-varying characteristics of the wake shed by a rectangular square section cylinder have been surveyed. POD mode distributions are poorly investigated in the case of laminar separation bubbles. For example POD has been applied on phase-averaged data obtained in the case of a laminar separation bubble excited by means of small amplitude waves (e.g. [22]). Similarly, it has been used for the investigation of periodically excited separation bubble (e.g. under the influence of periodic convective wakes as in the work of Sarkar [23]). In both cases there is a loss of information: fluctuations that occur at a frequency different from the exciting source could not be resolved in this way since they are smeared out by the ensemble averaging procedure. In the present work, the vortex shedding phenomenon induced by the Kelvin-Helmholtz instability process has been analyzed in detail. The combination of PIV measurements and POD analysis would shed more light on the dynamics of the vortex shedding in the rear part of the laminar separation bubble, as well as in the mechanisms driving the turbulence production. The analysis of the POD modes helps to identify the convective (or not) nature of the different velocity fluctuations growing in the separated flow region. To this end, a phase identification method, similar to that described in van Oudheusden et al. [11], has been employed to sort the PIV instantaneous vector maps in order to reconstruct the time-dependent shedding phenomenon induced by the Kelvin-Helmholtz instability process. Particular care has been paid in the choice of the velocity component (streamwise, normal to the wall and the velocity vector) adopted to accomplish this procedure, since for laminar separation bubble the first two POD modes are not 2

necessarily coupled, as it will be shown in the paper. Later, a low order model limited to a finite number of POD modes has been constructed to highlight the dynamics of generation and propagation of the coherent structures shed as a consequence of the shear layer roll-up, as well as to identify the spatial/temporal distribution of other velocity fluctuations not directly associated with the shedding phenomenon. Thus, both coherent and stochastic contributions to the overall energy of the velocity fluctuations can be separately analyzed, highlighting the regions of the separation bubble affected by deterministic (Kelvin-Helmholtz induced) or stochastic (turbulence induced) velocity fluctuations.

Nomenclature

3

C

cross-correlation matrix

Fu

vector of POD filtered velocity u

fu

element of vector Fu

L

flat plate length

KH

Kelvin-Helmholtz

N

number of instantaneous velocity fields

T

period

t

time

U0

freestream velocity at x/L=0.3

u

streamwise velocity component

v

normal to the wall velocity component

x

axial coordinate

y

coordinate normal to the wall

ϑ

phase of the vortex shedding phenomenon

λ (k)

POD eigenvalue of mode k

χ (k)

POD eigenvectors of mode k

X

Matrix of POD eigenvectors

φ

POD mode

Φ

vector of POD modes

<>

deterministic periodic component

Subscripts MAX

maximum value in the measurement domain

RMS

root mean square

Superscripts −

time averaged properties

∼

phase-averaged properties

ˆ

instantaneous slip velocity component

′

stochastic fluctuating component

2. Experimental apparatus and data reduction 2.1. Test facility and instrumentation The experiment has been performed in the open-circuit low-speed wind tunnel of the Aerodynamics and Turbomachinery Laboratory of the University of Genova. As shown in Fig. 1, the test article consists of a thick flat plate with 4

x/L=1 x/L=0 PIV fields of view

3.6

p -p Cp= pt0-p

Cp

3.2

t0

2.8



2

2.4 2 0.4

0.6

0.8

1

x/L Figure 1: Top: Sketch of the test section and PIV interrogation areas: the first one extends from x/L = 0.315 up to x/L = 0.5 (red) while the second one from x/L = 0.49 to x/L = 0.71 (blue). Bottom: pressure coefficient distribution [25].

(8:1) elliptic leading edge and a sharp trailing edge. The flat part of the plate including the leading edge is 200 mm long and 300 mm wide. The plate has been installed between two contoured walls producing the prescribed adverse pressure gradient, typical of Ultra-High-Lift turbine profiles. The endwall geometry is scaled from the test case of Lou and Hourmouziadis [24]. A trip wire has been used to force transition of the endwall boundary layers in order to overcome separation. This ensure that the pressure gradient (bottom of Fig. 1) induces a large laminar separation bubble on the plate for low Reynolds numbers, as documented in Simoni et al [25]. The inlet sidewall boundary layers were removed by means of lateral gaps in order to avoid inlet blockage effects. In detail, the Reynolds number of the present experiment is 70000, the inlet turbulence intensity is 1.5% and the momentum thickness Reynolds number at separation is 242. The boundary layer developing along the rear part of the plate was surveyed by means of Particle Image Velocimetry (PIV). The region from x/L = 0.315 to x/L = 0.71 (the test section throat is located at x/L = 0.285) has been investigated using two interrogation areas: the former extends from x/L = 0.315 up to x/L = 0.5 while the latter from x/L = 0.49 to x/L = 0.71. Measurements were performed at midspan of the plate, hence three dimensional effects that may occur along the spanwise direction, such as streaks, G¨ortler vortices as well as the 3D turbulent breakdown of Kelvin-Helmholtz structures as revealed in the work of Lang et al. [26], could not be analyzed. For each investigation area 200 instantaneous velocity fields have been computed for statistical moments evaluation as well as to collect a satisfactory number of snapshots for the POD analysis (e.g., Ben Chiekh et al. [14], Lacarelle et al. [27]). To this end, Lacarelle et al. [27] stated that the shape of the modes obtained with 500 snapshots is already recognizable in the modes obtained with four times less snapshots. The instrumentation is constituted by a double-cavity Nd: Yag pulsed laser BLUESKY-QUANTEL CFR200 (energy 2x100 mJ per pulse at 532 nm, pulse duration 8 ns, repetition rate 10 Hz). The optical system forms a light 5

sheet of 1 mm thickness. The light scattered by the seeding particles (mineral oil droplets with a mean diameter of 1.5µ m) is recorded on a high sensitive digital camera with a cooled CCD matrix of 1280 x 1024 pixels (with single pixel dimension of 6.7 x 6.7 µ m2 ). The camera maximum frame rate in the double frame mode is 4.5 Hz, and the minimum frame interval is 200 ns. This image acquisition rate does not allow a time-resolved sampling of the shedding phenomenon occurring in the rear part of the laminar separation bubble. A similar phase identification method to that proposed by van Oudheusden et al. [11] is adopted to sort the PIV images allowing the reconstruction of the phase-dependent vortex shedding phenomenon. The magnification factor for the present experiments was set to 0.192 in order to obtain a high spatial resolution. The cross-correlation function has been calculated over a 16x16 pixels interrogation area with a 50% overlap. This corresponds to a spatial resolution of 0.28 x 0.28 mm2 . Considering a sub-pixel interpolation of 0.1 pixel, the experimental uncertainty for the instantaneous velocity has been estimated to be 3.0% of the maximum measurable velocity. 2.2. POD fundamentals The Proper Orthogonal Decomposition is nowadays a well established mathematical procedure (e.g., [13]) which has been largely used for the detection of coherent structures embedded within the flow. As recently shown by van Oudheusden et al. [11], and Legrand et al. [18, 19], this procedure may be also adopted to provide a temporal reconstruction of the flow from non-time resolved, statistically independent data. The POD is performed in the present work following the mathematical procedure described in Legrand et al. [18]. According to this procedure, given N statistical realizations (N PIV instantaneous flow fields), the problem is reduced in finding the eigenvalues λ (k) and the eigenvectors χ (k) of a cross-correlation matrix C [NxN]. The i, j element of the matrix C is defined as the surface integral of a scalar or a vector field. The streamwise u and the normal to the wall v velocity components, as well as the (u, v) vector field are concerned in the present investigation. The comparison of POD modes resulting from these three different definition of the cross-correlation matrix will allow to chose the best strategy to sort and lock the instantaneous PIV images, as it will be described later. In the following formula the i, j element of the cross-correlation matrix is written (for the sake of simplicity) for the scalar u: Ci, j = 1/N

Z Z

uˆi (x, y)uˆ j (x, y)dxdy

(1)

where the slip velocity component uˆi (x, y) is the streamwise velocity component of the ith PIV vector field at a datum (x, y) position minus its time-mean value (according to the Reynolds decomposition). In such way, C is a symmetric square matrix of size [NxN]. The eigenvalues of the matrix C are then real and non-negative and its eigenvectors are (k)

orthogonal. The kth POD mode of u, φu , at position (x, y) is then computed as: N

φu (x, y) = ∑ χi uˆi (x, y) (k)

(k)

(2)

i=1

(k)

where χi

is the ith element of the eigenvector χ (k) . The advantage of this procedure is that the eigenvalues and the

eigenvectors, and hence the POD modes, are ordered by their energy content: the POD modes with higher energy 6

are representative of the coherent structures within the flow field. At a fixed location (x, y), eq. 2 may be conveniently written in matricial form. The POD modes may be represented by a vector Φu of dimension [Nx1] defined as (1)

(2)

(N)

(φu , φu , ..., φu ). Rewriting eq. 2 as matrix product, it results: Φu = X ×U

(3)

where the vector U [Nx1] is constructed by means of the different velocity estimations (PIV snapshots) (uˆ(1) , uˆ(2) , ..., uˆ(N) ) at the fixed (x, y) position, and X is a matrix of dimension [NxN] defined as:   (1) (1) (1) χ1 χ2 ... χN    (2) (2) (2)   χ1 ... χN  χ2  X =    ... ... ... ...    (N) (N) (N) χ1 χ2 ... χN

(4)

Eq. 3 can be inverted in order to filter the low energy containing modes which are typically correlated to the stochastic contribution. Therefore, by setting the amplitude of the less energetic modes to zero, a new vector Fu of the “filtered POD” velocity component u may be computed to define a low order reconstructed flow field: Fu = X −1 × Φ∗u = X T × Φ∗u

(5)

where the identity is valid because X is an orthogonal matrix. The term Φ∗u represents the new series of POD modes, equivalent to Φu except for the higher POD modes (characterized by low energy content) which have been set to zero. This corresponds to the adoption of a limited number of modes to reconstruct the velocity distribution. In section 3.4 it will be shown how the operation of eq. 5 can be used to filter the instantaneous velocity fields removing the stochastic contribution from the instantaneous flow fields. Moreover, when only the first two modes of the normal to the wall velocity component v are considered, the temporal evolution of the vortex shedding in the rear part of the separation bubble can be correctly sorted in time by means of a phase identification technique, adopting a criterion similar to that presented in van Oudheusden et al. [11].

_ u/U0 -0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.04

POD zoom

y/L 0.02

0

0.35

0.4

0.45

0.5

0.55

0.6

0.65

x/L

0.7

Figure 2: Mean streamwise velocity component, the dotted line in the middle of the picture separates the two PIV interrogation areas, the following discussion is focused on the area marked as “POD zoom”.

7

3. Results 3.1. Time-mean flow field The time-mean flow field of the present experiment has been widely described in previous publications by Simoni et al. [25, 28, 29]. The time-mean streamwise velocity component obtained from the present PIV measurements is reported in Fig. 2, where the mean velocity is normalized by the free-stream velocity at the measuring domain inlet (U0 ). These results are consistent with the previously published works. The boundary layer separates at x/L =0.39 (± 0.01 x/L), where a null time-mean normal to the wall velocity gradient has been detected. The separation bubble reaches the maximum thickness at x/L =0.57 (± 0.01 x/L), while the reattachment process is completed at around x/L = 0.70. As observed in the LDV measurements [28], the maximum negative velocity is measured within the separated flow region at around x/L =0.60, and it is 8% of the local external free-stream velocity. This value suggests that the disturbance growth rate along the separated shear layer is governed by convective instability (e.g., [3, 4]). The hot-wire measurements of Simoni et al. [25] showed that velocity fluctuations are characterized by a large energy content in the Kelvin-Helmholtz frequency range around the bubble maximum displacement position (x/L ≈ 0.57). The authors supported this statement by means of instantaneous PIV velocity fields and identified in the region of the transition process a roll-up of the shear layer followed by vortex shedding and breakdown phenomena, within the range 0.57< x/L <0.70. The following sections are aimed at completing this investigation applying POD on data acquired in the window marked over Fig. 2 as “POD zoom” in order to analyze in depth the vortex shedding dynamics and propagation. 3.2. POD analysis As mentioned in section 2.2, POD has been applied to both velocity components and to the vector field, defining the covariance matrix C (eq. 1) with the streamwise u, with the normal to the wall v velocity components and with the velocity vector (u, v). Figure 3 shows the distribution of the normalized energy of the POD eigenvalues for the three cases. The highest energy is captured by the first eigenvalue of the decomposition concerning the v component. The modes of v show a classical distribution with similar energy of the 1st and 2nd modes, and also the following modes are paired as energy couples up to modes 7th and 8th . This mode pairing is made clear by the spatial distribution of (1)

(2)

the POD modes 1 and 2 of the v component (φv , φv ), which are shown in Fig. 3. For clarity, the shear layer may be identified by means of the isocontour lines of the mean velocity which are superimposed to the plot. The POD modes appear very similar, sorted by their energy, but characterized by a spatial shift of 1/4 of the spatial wavelength (λ identified over the plot between two maxima of the POD mode 1). This indicates that the first two POD modes exhibit spatial correlation, i.e., one appears as a sine wave and the other as a cosine one. This distribution of the first two modes, with regions of similar dimensions at negative and positive values shifted by 1/4 of the wavelength, is typical of a vortex shedding phenomenon (e.g., [14, 20]). The characteristic wavelength λ of the 8

Energy captured [%]

16

u v (u,v)

12

8

Integral up to Mode 16: 55% of energy (u) 60% of energy (v)

4

0 0

10 20 Mode number

30

Figure 3: Relative energy of POD eigenvalues for the two velocity components u, v and for the velocity vector (u, v)

Normalized φv 0.04

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 -1.0 0.0

0.4 0.6 0.8

1 1.0

Mode 1

λ

y/L 0.02

Control points 0 0.55

0.6

0.65

Mode 2

0.04

y/L 0.02

0 0.55

0.6

0.65

x/L

Figure 4: First two POD modes of the normal to the wall velocity component v. Isocontour lines of u/U ¯ 0 are superimposed on the plots, the contour levels from bottom to top of the frame are 0.0, 0.2, 0.4, 0.6, 0.8. The control points (x1 , y1 ) and (x2 , y2 ) are marked in the top and bottom plot, respectively.

first 2 POD modes of the v velocity component corresponds to that of the Kelvin-Helmholtz instability found in the same condition by Simoni et al. [25]. The plot clearly indicates where the shedding begins: the first POD mode shows a local minimum in the shear layer for 0.57< x/L <0.59. Similarly, the second POD mode shows a local maximum at around x/L =0.57. This position agrees well with the bubble maximum displacement position, where the growth rate of the oscillations in the Kelvin-Helmholtz frequency range starts to be saturated, as shown in Simoni et al. [25] and in Burgmann and Schr¨oder [7]. Hence, in the range 0.57< x/L <0.59, the KH instability starts to induce the shedding of large scale vortical structures as a consequence of the shear layer roll-up, and this is clearly highlighted by the beginning of relevant values in the spatial distributions of the first two POD modes of the normal to the wall velocity component, whose wavelength corresponds to that of the KH instability. A completely different scenario characterizes the POD modes distribution of the streamwise velocity component 9

u and of the vector field (u, v). The energy content of the first POD mode of u (see Figure 3) appears sensibly higher than that characterizing the 2nd mode. The first two modes appear consequently uncoupled, while mode pairing seems to be re-established for the 2nd and the 3rd modes. The first three POD modes of the vector field (u, v) have similar energy, and hence they do not form a typical pair of energetically coupled modes. These distributions of the POD modes of u and (u, v) can be explained considering the work of Shi et al [15]: they found that the most energetic POD modes, in case of vortex shedding under the influence of endwalls, do not appear as energetically coupled pairs. This is true, in the present analysis, for the POD modes of the vector field (u, v) and of the streamwise velocity component (u). In contrast, the first two POD modes of the wall-normal velocity (v) are clearly energetically paired. Thus, it can be concluded that the first two POD modes of the v velocity component should be considered the best choice to sort the PIV instantaneous velocity fields in order to phase-average the vortex shedding phenomenon. (k)

As observed by Legrand et al. [18], the eigenvector χi

is the contribution of the ith instantaneous field to the

kth POD mode. The authors proved that for pseudo periodic convective flows (such as the shedding of large scale (1)

vortical structures), the first two eigenvectors χi Considering eq. 5, for the v component, the

ith

(2)

and χi

can be adopted to define a “temporal sorting coefficient”.

instantaneous velocity field filtered using only the 1st POD mode

(1)

(1)

( f vi (x, y)) is obtained as follows (the vector Φ∗v is defined in this case by only one not null element φv (1)

(1) (1)

f vi (x, y) = χi φv (x, y)

and zeros): (6)

This quantity, that is equivalent to the low order model of the velocity field described in Perrin et al. [10], might be (1)

even considered a temporal sorting coefficient, since φv (x, y) does not depend on time. In the present investigation (1)

(2)

f vi (x, y) and f vi (x, y) have been used as sorting coefficients. Particularly, two control points have been identified (as marked over Fig. 4). The employment of two distinct control points avoids the intrinsic ambiguity in the determination of the direction of the convective flow associated with the phase identification (see Legrand et al. [18]). The phase of the instantaneous image (ϑi ) has been then derived as (see also van Oudheusden et al. [11] and Perrin et al. [10]):

ϑi = arctan(

(1)

(2)

(2)

(1)

f vi (x1 , y1 ) f vMAX f vi (x2 , y2 ) f vMAX

)

(7)

Where, the terms f v(1,2) have been normalized by their maximum values f vMAX . The two points (x1 , y1 ) and (x2 , y2 ) are the coordinates of the control points which have to be properly chosen, as previously mentioned, to obtain the correct direction of convection. The right convection direction of the phenomenon results to be well established if the two control points are chosen within area of negative and positive POD mode values for the 1th and the 2th modes, respectively. Once the phase ϑi of each instantaneous velocity field is determined, the velocity maps have been correctly sorted within the shedding cycle. The sorted slip velocity (vˆi ) is represented in Fig. 5 for the control point (x1 , y1 ). According to Legrand et al. [19], the next post-processing step is the reconstruction of the flow fields by means of phase averaging. In the present contribution, a further reduction is applied to the original PIV vector fields before 10

4

instantaneous value Bin average Polynomial fit

v^ i [m/s]

2

0

-2

-4 0

90

180


[°]

270

360

i

Figure 5: Instantaneous slip velocity vˆi sorted with POD modes 1 and 2 and phase averaging procedures, at control point (x1 , y1 ).

phase averaging: the instantaneous values have been filtered with POD (see eq. 5). This procedure has been adopted to remove spurious contributions due to the higher order POD modes associated with stochastic motion and turbulence. Particularly, the modes have been analyzed and after the 16th mode they present very high spatial frequencies without symmetric or anti-symmetric pairing characteristics. It has to be noted that, according to Fig. 3, the reconstruction of the data accounting for up to the 16th mode takes into account more than 55% of the energy for both velocity components. As an example of velocity fields reconstructed with the first 16 modes, Fig. 6 shows the comparison between an original instantaneous velocity field (top of the figure) and the filtered one (bottom of the figure). The vectors of the instantaneous fluctuating velocities (uˆi ,vˆi ) are depicted on the picture. The filtered velocity field has lower velocities than the original one. However, the filtered map maintains and makes more clearly visible the characteristic large scale vortices and it removes a large number of spurious vectors. In order to obtain a correct phase-averaged velocity field, once POD filtering is applied, two different averaging procedures have been taken into account. One of the procedure considered is a classical bin-to-bin phase averaging [19] (black dots of Fig. 5), while the other one consist in a polynomial fitting (red line of Fig. 5) which solves a 5th order polynomial fitting of a least square problem. Since the number of acquired images was limited to 200, the procedure based on bin-to-bin averaging (16 bins in the example of the picture) is characterized by a quite large statistical error, as suggested by the scattering of the data. The phase variation of instantaneous images within the same bin (due to the finite dimension of the bin), provokes a scattering of the computed phase-averaged results. Otherwise, the statistical error of the polynomial interpolation is independent from the number of time steps used to reconstruct the phase-averaged distribution (it of course depends on the number of acquired images). This is why, in Fig. 5, the polynomial interpolation has been displayed as a continuous line. The results shown in the following section are finally obtained using this second procedure. Thus, the time evolution of the velocity components has been obtained, for each measuring point, from the values estimated by the polynomial fitting for the different time steps chosen to analyze the vortex shedding phenomenon. 11

To conclude this section, the post-processing procedure can be summarized in the following steps: 1) The PIV data, concerning scalar quantities u, v and vector field (u, v), were POD analyzed. 2) The energetically coupled first modes of v were identified as the best solution for the characterization of the vortex shedding. 3) Data have been reordered in the KH vortex shedding period by means of a phase identification technique applied to the first two POD modes of v. 4) The PIV snapshots were POD-filtered (retaining 16 modes) and sorted according to the phase. 5) Phase averaging, based on polynomial fitting, was applied at any (x, y) to the sorted and filtered velocity fields.

Original Vector map 0.04

y/L 0.02

0 0.55

0.6

0.65

x/L

POD filtered (modes 1-16) Vector map 0.04

y/L 0.02

0 0.55

0.6

0.65

x/L

Figure 6: Instantaneous slip velocity vector map (top frame), and POD filtered vector map using the first 16 POD modes (bottom frame)

3.3. Time-resolved phase-averaged flow field The phase averaged experimental data are analyzed according to the triple decomposition of Hussain and Reynolds [30]: u(t) = u+ < u(t) > +u′ (t)

(8)

where, the term < u(t) > represents the deterministic contribution associated, in this case, with the Kelvin-Helmotz frequency, and the term u′ (t) is the random fluctuation associated mainly with turbulence. The contributions of the time-mean and the purely periodic parts give the phase-averaged velocity u(t): ˜ u(t) ˜ = u+ < u(t) >

(9)

The periodic components < u > and < v > are obtained from this equation as the difference between the phaseaveraged velocities (u, ˜ v) ˜ and their time-mean values (u, ¯ v). ¯ 12

The reconstructed phase-averaged velocity field is represented as a plot sequence in Fig. 7. The phase-averaged evolution of the normalized streamwise velocity (u/U ˜ 0 ) is depicted on the left column of the plot. The vector maps of the phase-averaged periodic velocity components < u > and < v > are depicted on the right column. The isocontour lines (on the left) and the isocountour colour plot (on the right) of the normalized time-mean streamwise velocity u/U ¯ 0 are superimposed on the plots in order to compare the time-mean flow field with the phase-averaged one. A time increment of t/T =0.125 between subsequent frames have been adopted to give a dynamic view of the bubble motion and of the vortical structures shed as a consequence of the Kelvin-Helmholtz instability process. The shape of the separation bubble in the first time step (left plot for t/T =0.0) is rather similar to the time-mean one. In the shear layer region, up to the bubble maximum displacement (0.55< x/L <0.57), the isocontour lines of the time-mean streamwise velocity u/U ¯ 0 are almost superimposed on the contour distribution of the phase-averaged value u/U ˜ 0 . The time-resolved flow starts to sensibly deviate from the time-mean flow field immediately downstream of the bubble maximum displacement in the upper part of the shear layer. Whereas, for this time step, in the bottom part of the shear layer the phase-averaged velocity is similar to the time-mean: up to x/L =0.62, the bottom isocontourline, which corresponds to u/U ¯ 0 =0.0, is almost coincident with the countour level of u/U ˜ 0 =0.0. Furthermore, the minimum velocity is measured between x/L=0.60 and x/L = 0.61, this is slightly downstream of the time-mean minimum value. The largest differences between the time-mean and the time-resolved velocities are observed further downstream (0.62< x/L <0.69), where the contours of u/U ˜ 0 assume a “wavy” shape while the isocontour lines of u/U ¯ 0 are rather straight. In the front part of the pictures, up to the bubble maximum displacement (0.55< x/L <0.57), all the time steps present a similar trend: there are small differences between the time-mean and the time-resolved flow within the separated shear layer. This implies that the separated shear layer prior to the bubble maximum displacement position is not affected by velocity fluctuations at the shedding frequency. However, the occurrence of velocity fluctuations associated with different frequencies, such as the low frequency fluctuations of the shear layer mentioned in Hain et al. [8], cannot be revealed in these ensemble-averaged plots since their effects are smeared out by the phase-averaging process. The time-dependent reconstructed flow field of Fig. 7 is in agreement with the dynamics of the vortex shedding described in literature (e.g. [4]), as well as with the previously published work [25]. In this latter the spectral analysis of hot wire measurements showed the occurrence of only low frequency oscillations in the front part of the bubble. Otherwise, Kelvin-Helmholtz oscillations have been revealed far downstream of the detachment position and are amplified as far as the bubble maximum displacement, where saturation (finite amplitude oscillations) has been observed. Coherently, the time-resolved flow field reported in Fig. 7 starts to sensibly deviate from the time-mean flow only downstream of x/L =0.58. The most clear feature induced by the KH instability may be observed within the separated area. Indeed, the aft region of the bubble fluctuates along the axial direction: in the first time step, u/U ˜ 0 is equal to 0 at the wall for x/L =0.63, after half a period the same condition is obtained for 0.64< x/L <0.65, and later at t/T =0.875 this condition is satisfied for x/L =0.62. This agrees with what shown in the work of Burgmann and W. Schr¨oder [7], clearly 13

_ u/U0

~ u/U0 -0.1 0 0.04

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

-0.1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

t/T=0

y/L 0.02

0 0.04

t/T=0.125

y/L 0.02

0 0.04

t/T=0.25

y/L 0.02

0 0.04

t/T=0.375

y/L 0.02

0 0.04

t/T=0.5

y/L 0.02

0 0.04

t/T=0.625

y/L 0.02

0 0.04

t/T=0.75

y/L 0.02

0 0.04

y/L

t/T=0.875

0.02

0 0.55

0.6

0.65

x/L

0.55

0.6

0.65

x/L

Figure 7: Time resolved streamwise velocity (left) and vector map (right) of the phase-averaged velocity field. Isocontour lines of the mean streamwise velocity are superimposed on the plot on the left (the contour14 levels are reported in the caption of Fig. 4). The contour plot of the mean streamwise velocity is depicted in each plot on the right.

highlighting a periodic variation of the bubble length induced by the shedding phenomenon. A similar fluctuations may be identified following the time evolution of the position of minimum velocity within the bubble: at t/T =0.375 the minimum velocity is measured for 0.57< x/L <0.58 while, at the previous time instant (t/T =0.25) the minimum of u/U ˜ 0 occurs at 0.61< x/L <0.62. The time-space distribution of the phase-averaged vector maps (right column of Fig. 7) provides further details on the unsteady behavior of the separation bubble. It has to be mentioned that these vector maps represent the perturbation to the time-mean shear layer. Since the time-mean vorticity induced by the shear layer is orientated in the clockwise direction, the clockwise vortical structures are vortical structures shed by the shear layer, the counter-clockwise structures represent a state of lower vorticity within the time resolved flow field. A series of alternating vortical perturbations are identified in the first time step (t/T =0.0). By a closer view of this frame, for 0.55< x/L <0.57, the generation of a first clockwise rotating vortex could be identified. This vortex has the same sign vorticity of the shear layer, and it is consequently related to the shear layer roll up induced by the KH instability. A series of three vortices can be identified at the first time step at different abscissa. The vortex train moves downstream increasing its propagating velocity. Inclined lines superimposed on the plots help to identify the vortex propagation, black and white lines follow the counter-and clockwise-rotating vortical perturbation, respectively. The propagation line of the first vortex (0.55< x/L <0.57 in the first frame) is dotted up to the time snapshot t/T =0.5 where this clockwise rotating vortex assumes a more defined shape. The low definition of the vortex in this flow portion agrees well with the fact that vortex shedding is typically revealed only behind the bubble maximum displacement (e.g. [2]). Because of the periodicity of the phenomenon, the clockwise rotating vortices may be easily followed during their propagation (consider the white propagation lines): the vortex originating at t/T =0.0 between 0.55< x/L <0.57 moves up to x/L =0.6 in the last time snapshot, then for periodicity the vortex continues its movement at t/T =0.0 (0.6< x/L <0.61) and ends up its trajectory at x/L =0.66 for t/T =0.875. Along this motion the vortex lifts up: it originates at the bottom of the separated shear layer and it moves upward after that flow reattachment is attained. Moving upwards it slightly accelerates (as suggested by the skew change of the two white lines) before breaking up to turbulence. This latter condition is not clearly identified on the phase-averaged field. However, the amplitude of the purely deterministic components of velocity is considerably reduced in the range 0.67< x/L <0.69, hence, a purely turbulent flow may develop just afterwards. Simoni et al. [25], analyzing a set of three instantaneous images, identified such condition for x/L >0.65 from where the large scale organized structures were no more visible (see also Fig. 6). The POD analysis corrects this statement and reveals instead that this condition occurs just after x/L=0.69 since organized structures are observed up to this location. The phase-averaged time-resolved evolution presents further differences with respect to the instantaneous image of Fig. 6. In this last picture the clockwise rotating vortex appearing between 0.575< x/L <0.595 is located at a higher y coordinate than the corresponding one in the phase-resolved flow (time snapshot t/T = 0.625). Similar considerations may be done comparing the phase-resolved evolution reported in Fig. 7 with the set of instantaneous images 15

previously published in Simoni et al. [25]. This effect is due to the presence of deterministic velocity fluctuations that occur at a different frequency as compared with the shedding one, which provokes a low frequency motion of the separated shear layer similarly to that described in [8], as it will be discussed in the next section.

3.4. Identification of velocity fluctuation unrelated with the KH frequency In order to closely analyze the separated shear layer motion induced by deterministic structures characterized by different frequencies than the phase-resolved one (hence not observable in the time sequence of Fig. 7), the POD modes computed for the streamwise velocity component u, before and after phase averaging, are depicted in Fig. 8. The first five modes of u at the top of Fig. 8 are obtained from the instantaneous images, while the first POD mode computed from the phase-averaged value of u is shown on the bottom plot. Whereas, the POD modes after phase averaging of v are not shown since they match the POD modes of Fig. 4 obtained from the instantaneous velocity field. The spatial distributions of the first two POD modes of Fig. 8 suggest that they are uncoupled, as also previously observed by the normalized energy distribution of the eigenvalues reported in Fig. 3. Therefore, the first POD mode of u is associated with a non convective phenomenon, since this latter condition requires POD modes energetically paired as described in Legrand et al. [18]. In detail, this mode identifies a region (large negative values of the mode) which oscillates but does not convect in the measurement plane. This motion characterizes the former part of the separated shear layer and part of the shedding region. On the other hand, the following POD modes of u depicted in Fig. 8 show (negative or positive) large values just in the shedding region. Modes 2 and 3, as well as 4 and 5, seem now coupled (see also Fig. 3 to support this), hence they represent a convective phenomenon which is associated with the KH vortex shedding. To further support this statement, Fig. 9 reports the oscillations at the KH frequency of the first eigenvectors of v and u within one cycle of the shedding period. These oscillations are extrapolated by means of FFT analysis of the sorted POD eigenvectors, the time base of which is the non-dimensional time t/T computed from the phase θi of eq. 7. The picture is a schematic representation of the temporal correlation between POD modes of the different variables.The eigenvectors of mode 1 and 2 of v have the same amplitude and are shifted by a quarter of phase. Similarly, modes 2 and 3 of u have the same amplitude, lower than that of the eigenvectors of v, and they have a similar relation of the phase. The eigenvector of mode 1 of u is not related to them: its amplitude at this frequency is almost null. This is the reason why the first POD mode computed after the phase-averaging (last plot of Fig. 8) differs from the first POD mode of u computed from the instantaneous PIV images. Indeed, since the most energetic deterministic fluctuation of u (associated with the first mode of the instantaneous PIV images) occur at a frequency different from that characterizing the KH instability (see also Fig. 9 for instance), its effects on the plot sequence of Fig. 7 are smeared-out by the phase-averaging procedure. It also explain why the POD mode of u after phase-averaging (bottom of Fig. 8) does not show negative contribution in the former part of the separated shear layer. Otherwise, a local 16

Normalized φu

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 -1.0 0.0

0.4 0.6 0.8

1 1.0

Obtained from instantaneous data

Mode 1

0.04

y/L 0.02

0 0.55

0.6

0.65

Mode 2

0.04

y/L 0.02

0 0.55

0.6

0.65

Mode 3

0.04

y/L 0.02

0 0.55

0.6

0.65

Mode 4

0.04

y/L 0.02

0 0.55

0.6

0.65

Mode 5

0.04

y/L 0.02

0 0.55

0.6

0.65

Obtained from phase averaged data

x/L Mode 1

0.04

y/L 0.02

0 0.55

0.6

0.65

x/L

Figure 8: POD modes of velocity component u: 1st five modes obtained from the instantaneous images (top) and POD mode 1 obtained from the phase-averaged velocity field (bottom). Isocontour lines of u/U ¯ 0 are superimposed on the plots (the contour levels are mentioned in the caption of Fig. 4)

maximum of the mode can be observed within the shear layer between 0.55< x/L <0.57, which is the position where the smallest clockwise rotating vortex starts to roll up in the first time step of Fig. 7. This vortex, identified with the white dotted line in Fig. 7, is triggered by a local maximum of u˜ in the shear layer where the vectors of the purely deterministic components point in the streamwise direction. 17

The presence of POD modes unrelated with the KH shedding is supported by the model described in Simoni et al. [25]: i.e., the KH instability starts to exponentially grow in the shear layer after that the small frequency fluctuations (inherently present in the flow) are amplified within the shear layer. Similarly, the spatial distribution of the first POD mode of u, which does not occur at the KH frequency, characterizes the whole shear layer whereas the other POD modes, related to the KH shedding characterize the aft-part of the bubble. 3.5. Coherent and stochastic contributions to the overall energy of the velocity fluctuations Once phase-locked data are computed, the triple decomposition (Hussain and Reynolds [30]) can be used to filter the contribution to the overall energy of the velocity fluctuations due to the coherent motion. Thus, the fluctuating velocity uˆ can be decomposed as: uˆ =< u > +u′

(10)

The integral over time of the square of this expression gives: uˆ2RMS =< u >2RMS +u′2 RMS

(11)

Where RMS stands for root mean square, and the integral over time of < u > u′ does not appear in the equation because it is null. The RMS of the terms u′ and < u > (Hussain-Reynolds [30] decomposition) and of uˆ (Reynolds decomposition), and the analogous ones for the other velocity component (v′ , < v > and v), ˆ has been computed. Figure q ′2 10 shows the composition of these terms made non-dimensional by the reference velocity U0 : u′2 RMS + vRMS /U0 is q q depicted in the plot on top, < u >2RMS + < v >2RMS /U0 in the middle and uˆ2RMS + vˆ2RMS /U0 on bottom. The first term presented (top plot of the figure) is proportional to the merely turbulent kinetic energy of the flow. Thus, it identifies the region where stochastic fluctuations occur. Its maximum is measured downstream of the bubble maximum displacement at x/L = 0.625. However, high RMS values persist also at the end of the measurement plane, where the reattached turbulent boundary layer is developing. For what concern the deterministic fluctuations (

< u >2RMS + < v >2RMS /U0 ), the maximum values are again

Mode 1 (u)

0.8

Normalized χΚΗ

q

Mode 2 (u) Mode 3 (u)

0.4 0 -0.4 Mode 1 (v)

-0.8

Mode 2 (v)

0

0.2

0.4

0.6

0.8

1

t/T

Figure 9: Amplitude of POD eigenvectors filtered at the shedding frequency

18

Hussain-Reynolds Decomposition (u’2RMS+v’2RMS)0.5/U0

0.01 0.03 0.05 0.07 0.09 0.11 0.13 0.15 0.17 0.19 0.21 0.11 0.01 0.21

0.04

y/L 0.02

0 0.55

0.6

2 RMS

(

2 0.5 RMS

+

) /U0

0.04

0.65

0.01 0.03 0.05 0.07 0.09 0.11 0.13 0.15 0.17 0.19 0.21 0.11 0.01 0.21

y/L 0.02

0 0.55

0.6

0.65

Reynolds Decomposition (u^RMS+v^RMS) /U0 2

2

0.04

0.5

0.01 0.03 0.05 0.07 0.09 0.11 0.13 0.15 0.17 0.19 0.21 0.11 0.01 0.21

y/L 0.02

0 0.55

0.6

0.65

x/L

Figure 10: RMS of stochastic and deterministic (KH frequency) velocity fluctuations decomposed according to Hussain and Reynolds [30], on top and RMS of slip velocities according to the Reynolds decomposition, on bottom. Isocontour lines of u/U ¯ 0 are superimposed on the plots (the contour levels are mentioned in the caption of Fig. 4)

observable behind the bubble maximum displacement position, where vortices due to the KH instability have been found to be shed. Otherwise, it appears sensibly reduced at the end of the measuring plate, where only the stochastic contribution tends to persist since KH vortices are still present but start to breakdown, as previously discussed. The term depicted in the bottom plot is obtained from the original data reduced according to the Reynolds decomposition. It represents the energy of velocity fluctuation without distinction between coherent and stochastic contributions, as classically observed from non-time resolved measurements. According to Eq. 11 the square of this term is equivalent to the sum of the square of the two terms depicted in the first two plots of Fig. 10. Comparing these plots, the deterministic contribution (< u >2RMS + < v >2RMS ) is about 30% of the total kinetic energy. It has ′2 to be further mentioned that part of the kinetic energy associated with the stochastic contribution (u′2 RMS + vRMS ) is

related with fluctuations that occur at a frequency different than the KH one, as identified from the first POD mode of the streamwise velocity in the previous section. Therefore, less than 70% of the energy of the overall fluctuating velocities (u, ˆ v) ˆ measured in the rear part of the laminar separation bubble is due to small scales structures associated with turbulence (higher POD modes), while the remaining contribution is due to the coherent fluctuations associated with KH instabilities or larger scale structures which do not occur at the KH frequency.

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4. Conclusion The unsteady behavior of a laminar separation bubble developing on a flat plate with a prescribed adverse pressure gradient has been investigated by means of PIV measurements. POD analysis has been carried out in order to identify the sources of unsteadiness that characterize this complex phenomenon. This analysis has been focused on the aftportion of the separation bubble, where the shear layer becomes unstable and sheds vortices. The POD mode distributions associated with the streamwise and the normal to the wall velocity components exhibit a completely different pattern. Indeed, the first two POD modes of the normal to the wall velocity component have been found to be coupled, hence they trace the propagation of convective phenomena within the measuring plane such as the vortices shed as a consequence of the KH instability. Moreover, the wavelength of the first two POD modes of the normal to the wall velocity component hes been found equal to the KH vortex wavelength. These two modes have been consequently used to sort the experimental PIV instantaneous images, in order to reconstruct phase-averaged data describing the time-resolved KH vortex shedding phenomenon. It is shown that a pair of counterrotating vortices originate just before the bubble maximum displacement. They cause the largest fluctuations of the wall-normal velocity close to the position of the time-mean reattachment, which has been found to sensibly move upward and downward in a vortex shedding cycle. The POD analysis identifies further sources of unsteadiness. The first POD mode of the streamwise velocity does not have any temporal or spatial relation with the other POD modes. The shear layer upstream of the bubble maximum displacement oscillates, then, at a frequency different from the KH frequency. This confirms that fluctuations of the streamwise velocity, which are inherently present in the flow, characterize the separated shear layer and they are not associated with convective phenomena propagating in the measuring plane. The triple decomposition of the velocity fluctuations allowed the quantification of the contribution due to the stochastic as well the coherent motions to the overall velocity fluctuations. It has been found that in the rear part of the bubble, where the maximum time-mean oscillations are revealed, only about 70% of the overall energy of the velocity fluctuations is due to the turbulence activity, while the remaining 30% is due to the coherent motion induced by the large scale vortical structure shed in this region as a consequence of the KH vortex shedding phenomenon.

5. Acknowledgments The authors gratefully acknowledge the financial support of the European Commission as part of the research project TATMo, “Turbulence And Transition Modeling for Special Turbomachinery Applications”. It is also acknowledged the financial support of the Italian Ministry for Instruction, University and Research (PRIN2007). [1] O. Marxen, M. Lang, U. Rist, S. Wagner, A combined experimental/numerical study of unsteady phenomena in a laminar separation bubble, Flow Turbul. and Combust. 71 (2003) 133–146. [2] B. McAuliffe, M. Yaras, Transition mechanisms in separation bubbles under low- and elevated-freestream turbulence, ASME J. of Turbomach. 132 (2010) 011004–10.

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