Time-resolved Particle Image Velocimetry and ...

AIAA 2012-3275

42nd AIAA Fluid Dynamics Conference and Exhibit 25 - 28 June 2012, New Orleans, Louisiana

Time-resolved Particle Image Velocimetry and structural analysis on a hemisphere-cylinder at low Reynolds numbers and large angle of incidence ∗

†

Downloaded by MONASH UNIVERSITY on June 22, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2012-3275

Soledad Le Clainche1 , Jingyun I. Li 2 , Vassilis Theofilis1 and Julio Soria

1

2,3

‡

School of Aeronautics, Universidad Politécnica de Madrid, Pza. Cardenal Cisneros 3, E-28040 Madrid, Spain

2

Laboratory for Turbulence Research in Aerospace and Combustion, Monash University Melbourne Wellington Road, Clayton, Victoria 3800, Australia 3

Department of Aeronautical Engineering, King Abdulaziz University Jeddah, Kingdom of Saudi Arabia

Time-resolved planar Particle Image Velocimetry (PIV) experiments have been carried out in a water tunnel over a hemisphere-cylinder body at Reynolds number 1000, 2000 and 3000 and angle of incidence of 20. Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) analysis have been performed over such experimental data in order to identify flow patterns and provide a quantitative description of fluid motion and behavior. Results are in good agreement with Power Spectral Density (PSD) predictions. Three dominant DMD modes reproducing three-dimensional separation bubble and shear layer instabilities were detected and compared with POD results.

I.

Introduction

The effect on stability and control of flow separation on axisymmetric bodies of revolution is interesting from both an academic and an applications point of view. Flow around a hemisphere-cylinder configuration ∗ Correspondence

to: [email protected] Professor of Fluid Mechanics, Associate Fellow AIAA ‡ Research Professor, Associate Fellow AIAA † Research

1 of 22 Copyright © 2012 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

American Institute of Aeronautics and Astronautics

at non-zero angle of attack has serverd as a model and has been a subject of several experiments and computations whose main goal is to understand flow patterns in the vicinity of the nose of axisymmetric bluff bodies ranging from commercial airliners forebodies to submarines. From an academic point of view, separation around the nose is a prototype of essentially three-dimensional laminar flow separation. At high angle of atack the stability of the pair of the so-called ”horn vortices” conditions vehicle stability, owing to the unsteady nature of this phenomenon. Fairlie1 studied prolate-spheroid and hemisphere-cylinder geometries and was the first to observe the appearance of the pair of horn vortices mentioned before in both geometries at angle of attack AoA > 17.5 in the hemisphere-cylinder. Surface flow visualizations revealed two spiral nodes symmetrically placed about


the leeward plane of symmetry that gives rise to a quite strong vortex pair shed downstream. Flow topology in terms of describing skin friction lines and their relationship to the flow patterns motivated another sequence of investigations above the hemisphere-cylinder model. In 1982 this problem was studied by Tobak and Peake.2, 3 Several authors followed to contribute such research4–8 based on the critical point topological analysis, described in detail, e.g. by Perry and Chong.9 In the 90’s , a sequence of experiments and numerical computations were performed, in order to study the influence of Reynolds number and incidence on this flow. In their experimental investigations, Meade et al.10 performed pressure distribution experiments, confirming what Tobak and Peake2 postulated namely that minimum local pressure distributions where related with node separation (focus) and the maximum with node attachment. Hoang et al.11, 12 performed flow visualizations at low and moderate Reynolds numbers, finding that a separation bubble forms a ring around the nose at low incidences, while its width decreases as Reynolds number increases. Related computations Hsieh et al.7 found different kinds of separations and vortices. He posulated that connections of primary and nose separation takes place at high incidence. Pantelatos et al.13 confirmed the lack of asymmetries in this kind of bodies and the earlier appearence of separation when incidence increases. In both, experimental and numberical investigations, the Reynolds number employed was up to 10000. However, Gross et al.14 recently compared high Reynolds number numerical simulations with experiments at very low Reynolds number (2000 and 5000). Shear layer instabilities where found on the symmetry plane at low incidence (AoA = 10◦ ) which was transformed on a linear instability at higher incidence (AoA = 30◦ ). On the other hand, linear stability theory, in part motivated by research into laminar-turbulent flow transition, has occupied a substantial part of fluid mechanics research for over a century. Numerical experience has confined the bulk of the efforts into analyzing one-dimensional shear flows. The classic linear stability theory of Tollmien15 is mainly concerned with individual sinusoidal waves propagating in the boundary layer parallel to the wall. The first quantitative connection between flow topology and global (2D Global, modal) linear instability has been made by Rodr´ıguez and Theofilis,16, 17 via the construction of the Jacobian matrix mirroring the fundamental decomposition of linear theory. These authors were capable of connecting linear amplification of the leading stationary global mode of separation bubbles with well-known topological patterns of separation, such as U-separation on an APG boundary layer and Stall Cells on an airfoil at high

2 of 22 American Institute of Aeronautics and Astronautics

AoA values. To-date topological ideas are yet to be applied to results of compressible 2D or 3D Global Instability analysis. In this paper, flow stability is analyzed employing two alternative methodologies of structures detection that project the flow as a expansion of modes, namely Dynamic Mode Decomposition (DMD)18 and Proper Orthogonal Decomposition (POD).19 Time-resolved Particle Image Velocimetry (PIV) experiments have been carried out on the symmetry plane of a hemisphere-cylinder at different Reynolds number values (Re= 1000, 2000 and Re= 3000) all at a single AoA=20o . The main goal of this paper is to understand the flow topology in the symmetry plane on a hemisphere-cylinder detecting coherent structures using DMD and POD analysis. Flow instabilities and U −separation patterns as function of Reynolds number are detected with these techniques. Shear layers


instabilities were found on the symmetry plane and were compared with the literature.

In Section II the experimental facility and techniques are presented. In Section III flow topology and structural analysis techniques are briefly explained. In Section IV topologial analysis description, DMD analysis and POD analysis over the experimental data on a hemisphere-cylinder is presented.

II.

Model Description and Experimental Facility

In the present investigation, planar PIV experiments have been carried out in the horizontal water tunnel experimental facility of Laboratory for Turbulence Research in Aerospace and Combustion (LTRAC) of Monash University. The turbulent pattern flow around an hemisphere-cylinder of diameter d = 25mm and length l = 200mm have been analyzed at angle of attack of 20 and Reynolds number 1000, 2000 and 3000 (Figure 1). The plane of measurements is the symmetry plane, parallel to the incoming flow. Reynolds number is defined in the following way: Re =

U∞ d ν

Figure 2 presents the setup in the water tunnel where the experiments were carried out. The test section of the water tunnel has a cross-section area of 500mm × 500mm and is approximately 3m in length. The

measurement is carried out downstream, away from the contraction in order to achieve optimum water quality. The free stream turbulent intensity is estimated to be less than 1%. The tunnel is seeded with 11 micro Potter’s glass sphere which has the density of 1.01kg/m3 through the experiment. High speed laser and camera are employed in the experiment to take time resolved planar PIV measurement. The camera used is a PCO. Dimax camera equipped with 100mm Zeiss Macro Lens. The Quantronix laser which has up to 400mJ in power has the frequency set to between 250 − 550Hz adjusting to the change in Reynolds

number. Optical lenses are set up to divert and trim the laser into a sheet of 2mm in thickness. The field of view of this experiment is 90mm × 90mm. The magnification is estimated to be 0.045mm/pixel. The experimental parameters are listed in Table 1.

The sampling frequency of the PIV measurement is the same as the laser frequency, which increases with the Reynolds number. This makes the Nyquist frequency to be 125Hz, 200Hz and 275Hz respectively. 4000


images have been taken for each condition. The time spans of the measurement are 16s, 10s and 7.2s with respect to different Reynolds number. Each data set has a size of 30GB. Each set of image has been preprocessed. The images are first subtracted from their averaged value and then being thresholded. These steps helps to remove the background noise of the image. The data is then being processed by programs developed in house. The details of multigrid cross-correlation can be found in Soria et al.20

III.

Theoretical analysis concepts

Flow topology is a suitable tool in order to study and understand the features of the fluid motion. Downloaded by MONASH UNIVERSITY on June 22, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2012-3275

Flow behaviour in the neighborhood of critial points is described in a quantitative way. Several criteria to define coherent structures exists in the literature. This paper follows the idea to undertand flow patterns on a hemisphere cylinder and to conceptualize turbulent strucutres as an expansion of modes performing Dynamic Mode Decomposition18 and Proper Orthogonal Decomposition19 over planar-PIV experimental data. A.

Flow Topology

Critical point theory is used to describe the topological features of flow patterns. It considers fluid motions that are describable by the leading terms of a Taylor series expansion for the velocity field in terms of space coordinates. Therefore, in a linear vector field the local connection of velocity components and the corresponding position vector can be described via the Jacobian matrix. 

or

  x ˙  1   j11     x˙  =  j  2   21    x˙3 j31

j12 j22 j32

  j13   x1      j23    x2    j33 x3

x˙ = Jx

(1)

(2)

The eigenvalues of the Jacobian matrix satisfy the equation: λ3 + P λ2 + Qλ + R = 0

(3)

1 2 (P − tr[A2 ]) and R = −det[A] (in incompressible flow P = 0). 2 Depending on the nature of the eigenvalues, Dallman? and Perry & Chong21 clasify critical points as nodes where P = −tr[J] , Q = ∆[J] =

(stable, unstable), saddle, focus (stable, unstable) and a sequence of borderline cases. Streamlines are defined by the permissible connections between critical points, while the limit x2 → 0 defines wall-streamlines.


B.

Dynamic Mode Decomposition

Koopman modes22 has recently been introduced for the description of fluid flow structures as a particular class of technique for nonlinear systems analysis and reduction by Rowley et al.23 and Schmid.18 Linear or nonlinear flow coherent structures are described, since the linear operator corresponding to DMD analysis is defined for any nonlinear dynamical system. When DMD is applied to the linearized Navier-Stokes equations, Koopman modes reduce to linear global modes and when DMD is applied to time-periodic flows, Koopman modes reduce to Fourier modes. DMD algorithms open different possibilities of structural analysis with the same tool, spatial and temporal decomposition. However, in this paper, attention is focused in the temporal analysis. The DMD algorithm


presented by Schmid et al.24 is briefly described. Considering a data sequence of N + 1 snapshots, two different snapshots matrices can be constructed: V1N = {v(x, t1 ), v(x, t2 ), ..., v(x, tN )} (from the first snapshot to the N th snapshot) and

V2N +1 = {v(x, t2 ), v(x, t3 ), ..., v(x, tN +1 )} (from the second snapshot to the (N + 1)th snapshot).

Assuming a linear mapping constant over the snapshot sequence A that connects the flow field vi with the subsequent flow field vi+1 (vi+1 = Avi ), it will be able to formulate the sequence of flow field as a Krylov sequence25 V1N = {v1 , Av1 , A2 v1 , . . . , AN −1 v1 }

(4)

V2N +1 = AV1N .

(5)

Thus,

For a long data snapshots sequence got from experiments, it is possible to express any further snapshots sequence by linear combination of the previous one. Therefore, selecting a companion matrix S (constructed from the snapshots data) which can be thought of as a projection of A onto the snapshot basis V1N Eq. (5) can be approximated by V2N +1 � V1N S.

(6)

The idea behind DMD is to describe the dynamical process defined by A (and approximated by S). This process will be defined through numerical solution of the eigenvalue problem Sµ = λµ,

�

j=1

(7)

where the dynamical modes Φ are the projection of the eigenvectors µ on the snapshot basis V1N , Φ = µj (V1N )j , and the eigenvalues follow the next transformation ω = log(λ)/∆t, where ∆t is the time interval

between snapshots.


C.

Proper Orthogonal Analysis

Proper Orthogonal Decomposition (POD) is a technique able to represent a least-order snapshots ensemble. Data analysis using the POD is often conducted to extract ’mode shapes’ or basis functions, form experimental data or detailed simulations of high-dimensional systems, for subsequent use in Galerkin projections that yield low-dimensional dynamical models. The idea behind POD analysis is to find a function basis φi (x) that most faithfully represents a random vector function, u(x, t) using the form:

uk (x, t) =

N �

aki (t)φi (x)

(8)


i=1

with ψ = [φ1 φ2 . . . φN ], such that describes u(x, t) better than any other representation of the same dimension using any other function basis. Each time-dependent expansion coefficient, aki , is termed chronos (time), while the space-dependent coefficient φi is called topos (space).26 Given a set of data (snapshots) {uk ∈ H | k = 1, ..., N } the aim is to determine a subspace S of dimensions

M < N , such that the error E(�u − PS u�) is minimized, (it is employed the induced norm of the Hilbert

space H �.�, with inner product �., .�, where Ps denotes the orthogonal projection onto S and E(.) is an average operator over k). This minimization problem leads to an eigenvalue problem.27

There are two different methods to perform POD, the classical method28 and the snapshot POD.19 In this paper attention is focused to snapshot method.

IV. A.

Results

Flow Topology

PIV measurements have been performed on the symmetry plane of a hemisphere-cylinder and topology patterns have been studied and compared with the literature. Three different Reynolds numbers (1000, 2000 and 3000) have been analyzed at AoA= 20. Figure 3 shows longitudinal and transversal (U and V respectively) mean velocity values of the hemispherecylinder at different Reynolds numbers on the symmetry plane, where the flow is statistically stationary. Flow separation is evident in all the cases in the U mean velocity contour, where there exists an area surrounding the body in which the velocity is very small, negative or equal to zero. On the other hand, as Reynolds number increases, the presence of maximum positive V velocity is closer to the body surface and the negative V velocity is smoother, thus lift increases with Reynolds number. Streamlines of symmetry plane of the hemisphere-cylinder at Reynolds number 1000, 2000 and 3000 are shown in Figure 4. These results can be completed with the Tobak & Peake29 topological analysis in order to visualize the three-dimensional separation phenomenon that occurs in the area close to the nose of the body. For Reynolds number 2000 and 3000, the presence of an unstable flocus on the symmetry plane explains the appearance of a vortex perpendicular to this surface (symmetry plane). Following the surface flow pattern of


Tobak & Peake29 the flow topology for the hemisphere-cylinder in these cases of study is shown in Figure 5, where surface and fild vortex are schematically presented. A pair of horn vortex appear perpendicular to the body surface from the stable focus F2 , at the same time a vortex reemerge from the unstable focus F1 located in the symmetry plane. The latter is perpendicular to the horn vortex. The vortex system is conected by the saddle point S and the nodal point N . Such vortex joins the two horn vortex and defines the region of the U −separation. As Reynolds number increases, vortex center is displaced to the nose of the body and separation bubble width decreases. Figure 6 (right) shows the three-dimensional vortex reconstruction of the flow pattern described before, presented by Dallmann.30 Similar structural patterns can be found in the literature in other laminar separated flows.11, 14


On the other hand, Reynolds number 1000 is still too low in order to get the flow pattern already described. However, there exist separation and the appearance of a laminar separation bubble. Rodriguez and Theofilis17 studied the appearance of this topology patterns of U −separation on a steady laminar separated flow. Figure

6 (right) presents laminar separation bubble topological patterns and vortex shedding mechanism described by Theofilis et al.31 This sedding pattern is thought to be followed by the hemisphere-cylinder ant very low Reynolds number (smaller than 1000). Further investigations are need in order to further examine this hypothesis. B.

Structural analysis: Power Spectral Density and Dynamic Mode Decomposition

DMD is a suitable technique in order to detect and analyze dominant structures on a flow. Dominant frequency of the flow motion for the three different cases of study is obtained performing a Power Spectral Density (PSD) analysis over the PIV experimental data. In order to find the global mode frequency, the points selected to perform the PSD analysis are those which constitute a shear layer, calculated with transversal velocity fluctuations (averaged in time) < v � >, since v � is symmetric for linear instability modes (v � has a maximum and is symmetric). In Figure 4 this shear layer is shown (blue dots). Following the topological pattern described in Section A, two different shear layer shapes can be found, depending on the Reynolds number. At Reynolds number 2000 and 3000 a single layer goes through the symmetry plane where a vortex center emerges, while at Reynolds number 1000, the shear layer is divided in two parts, one of them limiting the separation bubble. Some of the points composing the shear layer have been selected in order to perform Fourier analysis (PSD) and to find the dominant frequency for each Reynolds number, where the PSD distribution over the entire frequency range is shown (Figure 7, left). The slope of each line are compared (Figure 7, right). Dominant frequencies identified at these conditions are: ωi = 0.4885Hz for Re= 1000, ωi = 1.5656Hz for Re= 2000 and ωi = 1.0763Hz for Re= 3000. Once that the dominant frequencies of the three different cases of study are determined, a DMD analysis is performed over the cases with Re= 1000 and Re= 3000 in order to better understanding the flow motion behaviour on the symmetry plane of the hemisphere-cylinder. The domain set is reduced in the x direction for the case Re= 3000 in order to increase accuracy without the need of increasing the number of snapshots,32


thus (in pixels) for Re= 1000: x0 = 0px and xL = 1983px and for Re= 3000: x0 = 0px and xL = 1535px. The number of snapshots required to converge each case is proportional to its dominant frequency and ∆T , since in Fourier analysis ∆f = 1/(N · ∆T ). The value of the time interval is ∆T = (1/250)s for Re= 1000

and ∆T = (1/550)s for Re= 3000. The number of snapshots set in DMD work as a windowing process in

Fourier transformations and its dependence with the windows size. Thus when the number of snapshots increases, the ability of DMD applied to experimental data of capturing low frequencies increases. In Figure 8, most relevant eigenvalues of DMD in terms of energy23 are shown at Re= 1000 and Re= 3000. For longer observations, eigenvalues will approach to the neutral line.24 Therefore, DMD convergence at Re= 3000 is slowerr than at Re= 1000, but it is sufficient to accurately capture the dominant frequency. An


eigenvalue close to the origin without frequency represents the mean flow for each case (blue). On the other hand, the mode that captures the dominant frequency is unstable in both cases (red). Figures 9 and 10, present the real part of DMD modes asociated to the eigenvalues presented earlier at Re= 1000 and Re= 3000 respectively. These modes are presented from up to down in ascending order of frequency at Re= 1000. The idea is to study frequency and amplification evolution for the same eigenmode as function of Reynolds number. Three relevant eigenmodes present prominent structures: 2nd mode (unstable), associated with the separation region (bubble or vortex), 5th mode (unstable), associated with the dominant frequency and representing shear layer effects, and 6th mode (stable), also representing shear layer effects. Attention is focused first on Re= 1000 (Figure 9). The 5th mode (unstable) presents the dominant frequency also captured with PSD. U vector fields reveals the appearance of reverse flow surrounding the separation region, while in V antisymmetric vortical structures are present. This mode is associated with a shear layer instability. Relevant structures can be found for higher frequency (6th mode, stable), where antisymmetric vortex appear in pairs around the U-separation region. This mode also captures the shear layer of the flow. Intensity of positive vortex is higher than intensity of negative vortex, therefore such vortical structures are not perpendicular to the symmetry plane. On the other hand, the 2nd mode is associated with the separation bubble. This mode is unstable and presents a flaping vortex that controls the layer of the separation region. Finally, when frequency is lower than the dominant frequency, structures appear closer to the nose of the model, at the beginning of the separation region. Antysimmetric vortex are dominant at these frequencies. As Reynolds number increases, the separation bubble breaks up and a vortical structure appear. Structures start to be translated downstream the separation region and get closer to the model. They appear in the adherence area (see Figure 10). At Re= 3000, eigenmodes frequency and amplification is higher than for Re= 1000. In particular, the 2nd mode (U-separation mode) presents a relative frequency (in the eigenspetrum) higher than in Re= 1000. Further analysis of these data is underway.


C.

Proper Orthogonal Decomposition

Following the structural analysis related with the topology on a hemisphere-cylinder, POD is a suitable technique in order to deretmine the most energetic flow structures in terms on kinetic energy. The study of these structures provide complementary understanding of the flow motion. POD is able to detect spatial changes, but not changes in temporal response dynamics since POD modes are composed by a mixture of frequencies. Figure 11 presents the energy contained on the 15 first POD modes at Reynolds number 1000 and 3000 and AoA= 20. In all the cases, almost 80% of kinetic energy is contained in the six first modes. As in the DMD study, computational domain is reduced to x0 = 0px, xL = 1535px for Re= 3000 in order to increase


accuracy. POD modes present a wave-like behaviour. Similar to DMD modes, POD modes are focused around the U-separation area and as Reynolds number increases they are traslated downstream the U-separation region and get closer to the model. POD captures the most energetic structures, but unlike DMD, each POD mode can be composed by a mixture of frequencies. At Re= 1000 the DMD dominant freqency mode (5th mode) can be considered similar to the sum of the first and second most energetic POD mode, thus almost 50% of the energy contained in the flow motion is related with the dominant frequency predicted by PSD and is related to shear layer effects. Finally, at Re= 3000, as in DMD, POD only identify one vortex dowmstream the separation bubble in most of the cases. Shear layer effects are present in the 4th , 5th and 6th modes. Experimental frame should be longer in order to study this phenomenon more in depth. For this reason, additional experiments are currently underway.

V.

Conclusions

Separation effects are studied on a flow around an hemisphere-cylinder at different Reynolds number (1000, 2000 and 3000) and incidence (AoA= 20). Time-resolved PIV measurements were performed in the symmetry plane of such model. Topological patterns were studied and compared with the literature. Additional, structures analysis was carried out employing DMD and POD techniques. Dominant frequencies were detected performing a PSD analysis and eigenmodes associated with such frequencyare independently confirmed by DMD analysis. At Re= 1000 a separation bubble appear. As the Reynolds number increases, this bubble breaks up and a vortex appear in the same region. The origin of this vortex is translated closer to the nose of the model when Reynolds number increases. DMD captures three relevant structures on the flow: an unstable structure associated with the shear layer pattern, which possesses the dominant frequency, a stable structure associated with the shear layer pattern and an unstable structure related to the separation bubble or the vortex of the separation area. Frequency and amplification rates increases with Reynolds number. In particular, the separation bubble presents a higher relative frequency. In addition, structures start to be translated downstream the separation region and get closer to the model when the Reynolds


number increases. Finally POD analysis concludes that 80% of kinetic energy is contained in the six first modes. At Re= 1000, the sum of the two most energetic POD modes reproduce the dominant frequency DMD mode.

Acknowledgments Support of the Marie Curie Grant PIRSES-GA-2009-247651 ”FP7-PEOPLE-IRSES: ICOMASEF – Instability and Control of Massively Separated Flows” is gratefully acknowledged. Support of the Spanish Ministry of Science and Innovation through Grant MICINN-TRA2009-13648: ”Metodologias computacionales para la predicci´ on de inestabilidades globales hidrodin´ amicas y aeroac´ usticas de flujos Downloaded by MONASH UNIVERSITY on June 22, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2012-3275

complejos” is gratefully acknowledged. Support of the Defence Science and Technology Organisation of Australia is gratefully acknowledged.

References 1 Fairlie

B.D., “Flow Separation on bodies of revolution at incidence,” 7th Australian Hydraulics and Fluid Mechanics

Conference, Brisbane, 18-22 August, 1980. 2 Tobak

M. and Peak D.J., “Topology of two-dimensional separated flows.” Annual Review of Fluid Mech., Vol. 14, 1992,

pp. 61–85. 3 Peak

D.J. and Tobak M., “Three-dimensional flows about simply components at angle of attack.” NASA Technical

Memorandum., Vol. 84226, 1992. 4 Meade

A.J. and Schiff L.B., “Experimental study of three-dimensional separated flow surrounding a hemisphere-cylinder

at incidence,” AIAA Paper , 1987, pp. 87–2492. 5 Ying,

S. X., Schiff, L. B. and Steger, J. L., “A numerical study of three-dimensional separated flow past a hemisphere

cylinder.” AIAA Paper , 1987, pp. 87–1207. 6 Wang

K.C. and Hsieh T., “Separation patterns and flow structures about a hemisphere-cylinder at high incidence.” AIAA

Paper , 1992, pp. 92–2712. 7 Hsieh

T. and Wang K.C., “Three-dimensional separated flow structure over a hemisphere-cylinder.” J. of Fluid Mech.,

Vol. 324, 1996, pp. 83–108. 8 Hoang

N.T., Rediniotis O.K. and Telionis D.P., “Symmetric and asymmetric separation patterns over a hemisphere

cylinder at low Reynolds number and high incidences,” J. of Fluids and Structures, Vol. 11, 1997, pp. 793–817. 9 Perry

A.E. and Chong M.S., “A description of eddying motions and flow patterns using critical-point concepts,” Ann.

Rev. Fluid Mech., Vol. 19, 1987, pp. 125–155. 10 Meade

J., S. L., “Experimental study of three-dimensional separated flow surrounding a hemisphere-cylinder at incidence.”

NASA report, 1995. 11 Hoang

N.T., Rediniotis O.K, T. D., “Symmetric and asymmetric separation patterns over a hemisphere cylinder at low

Reynolds numbers and high incidences.” J. of Fluid and Struct., Vol. 11, 1997, pp. 793–817. 12 Hoang

N.T., Rediniotis O.K, T. D., “Hemisphere Cylinder at Incidence at Intermediate to High Reynolds Numbers.” J.

of Fluid and Struct., Vol. 37, 1997, pp. 1240–1250. 13 Pantelatos

D. K., M. D. S., “Experimentalflowstudyoverablunt-nosedaxisymmetricbody at incidence,” J. of Fluid and

Struct., Vol. 19, 2004, pp. 1103–1115. 14 Gross

A., Jagadeesh C., F. H., “Numerical investigation of Three-Dimensional Separation on Axisymmetric Bodies at

Angle of Attack.” AIAA paper , Vol. 2012-0097, 2012.


15 Tollmien,

W., Uber die Entstehung der Turbulenz. 1. Mitteilung, Nachr. Ges. Wiss. Goettingen, Math. Phys. Klasse,

1929. 16 Rodr´ ıguez

D. and Theofilis V., “Structural changes of laminar separation bubbles induced by global linear instability,”

J. of Fluid Mech., Vol. 655, 2010, pp. 280–305. 17 Rodriguez

D., Theofilis V., “On the birth of stall cells on airfoils.” Theor. and Comp. Fluid Dynamics, Vol. 21, 2011,

pp. 105–117. 18 Schmid

P. J, “Dynamic mode decomposition of numerical and experimental data.” J. Fluid Mech., Vol. 656, 2010,

pp. 5–28. 19 Sirovich 20 Soria

L., “Turbulence and the dynamics of coherent structures, Parts I-III.” Q. Appl. Math., Vol. 45, 1987, pp. 561.

J., New T.H., Lim T.T., Parker K., “Multigrid CCDPIV Measurements of Accelerated Flow Past an Airfoil at an

Angle of Attack of 30◦ .” Experimental Thermal and Fluid Science, Vol. 27, 2003, pp. 667–676. 21 Perry

A.E., Chong M.S., “A description of eddyng motions and flow patterns using critical-point concepts.” Ann. Rev.


Fluid Mech., Vol. 191, 1987, pp. 125–155. 22 Koopman

B.O., “Hamiltonian systems and transformations in Hilbert space.” Proc. Natl. Acad. Sci. USA, Vol. 17, 1931,

pp. 315–318. 23 C.

W. Rowley and I. Mezi´ c and S. Bagheri and P. Schlatter and D. S. Henningson, “Spectral analysis of nonlinear flows.”

J. of Fluid Mech., Vol. 641, 2009, pp. 115–127. 24 Schmid

P. J., Li L., Juniper M., Pust O., “Applications of the dynamic mode decomposition.” Theor. Comp. Fluid

Dynamics, Vol. 25, 2011, pp. 249–259. 25 Trefethen 26 Aubry

L.N., Bau D., “Numerical Linear Algebra.” SIAM , 1997.

N., Guyonnet R., Lima R., “Spatiotemporal analysis of complex signals: theory and applications.” J. Stat. Phys.,

Vol. 64, 1987, pp. 683–739. 27 Holmes

P., Lumley J., Berkooz G., “Turbulence, Coherent Structures, Symmetry and Dynamical Systems.” Cambridge

University Press, 1996. 28 Lumley

J., “The structure of inhomogeneous turbulent flows. Atmospheric Turbulence and Radio Wave Propagation.”

A. Yaglom and V. Tatarski, 1967. 29 Tobak

M., Peake D.J., “Topology of two-dimensional and three-dimensional separated flows.” AIAA paper , Vol. 79, 1979,

pp. 1480. 30 Dallman

U., “Three-dimensional vortex structures and vorticity topology.” Fluid Dynamic Research, Vol. 3, 1988,

pp. 183–189. 31 Theofilis,

V., “Advances in global linear instability of nonparallel and three-dimensional flows,” Prog. Aero. Sciences,

Vol. 39 (4), 2003, pp. 249–315. 32 Duke

D., Soria J., Honnery D., “An error analysis of the Dynamic Mode Decompositions.” Experiments in Fluids, Vol. 52,

2011, pp. 529–542.


Table 1. Planar PIV experimental setup

Camera

PCO Dimax (2016pixel×2016pixel)

Flow Seeding

11µs potter glass spheres, ρ = 1.01Kg/m3

Laser

Nd: YAG up to 400 mJ

Laser sheet thickness

2mm

Laser frequency

250Hz for Re = 1000, 400Hz for Re = 2000, 550Hz for Re = 3000

Magnification

0.0453mm/pixel


Field of View (image space)

90 × 90mm2

Maximum Displacement

12pixel

Lens Aperture

11

Figure 1. Hemisphere-cylinder, diameter d = 25mm and length l = 200mm.



Figure 2. Planar-PIV setup in the horizontal LTRAC water tunnel

Figure 3. Mean velocity field at AoA= 20 and Re= 1000, 2000, 3000 from up to down. Left: U mean, Right: V mean. Velocity field is normalized with Umax and Vmax . 21 equidistant isolines



Figure 4. Streamlines of flow motion on hemisphere-cylinder at AoA= 20 and Re= 1000, 2000, 3000 from up to down. Blue dots represents shear layer calculated with < v � >.



Figure 5. Topological structure for a flow over a hemisphere-cylinder at AoA= 20 and Re= 2000, 3000. Streamlines and critical points.

Figure 6.

Topology patterns on a hemisphere-cylinder.

Right: Schematic representation of a mechanism

for vortex shedding from a laminar separation bubble, on account of global instability, Hemisphere-cylinder at Re= 1000 and AoA= 20 (Theofilis, 2003). Left: three-dimensional vortex reconstruction of topology on a hemisphere-cylinder at AoA= 20 and Re= 2000, 3000(Dallmann, 1988).


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Figure 7. Upper Left:Power Spectral Density in shear layer of velocity fluctuations v � at AoA= 20 and Re= 1000 along x in x1 = 383, x2 = 399, x3 = 415, x4 = 431. Right: Energy spectrum comparing amplification of the dominant frequencies in the shear layer: f 1 = 0.6106Hz, f 2 = 0.4885Hz. Middle Left: Power Spectral Density in shear layer of velocity fluctuations v � at AoA= 20 and Re= 2000 along x in x1 = 943, x2 = 751, x3 = 655. Rigth: Energy spectrum comparing amplification of the dominant frequencies in the shear layer: f 1 = 0.7828Hz, f 2 = 1.5656Hz. Lower Left: Power Spectral Density in shear layer of velocity fluctuations v � at AoA= 20 and Re= 3000 along x in x1 = 687, x2 = 719, x3 = 799. Right: Energy spectrum comparing amplification of the dominant frequencies in the shear layer: f 1 = 177.59Hz, f 2 = 1.0763Hz.


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Figure 8. Eigenvalues of four most important DMD modes in terms of amplification in hemisphere-cylinder at AoA= 20.

Upper: Re= 1000.

The eigenvalues are: (Mode I - blue) ω1 = −0.0001, (Mode II - green)

ω2 = 0.0013 ± 0.0783, (Mode III - purple) ω3 = −0.0026 ± 0.1632, (Mode IV - orange) ω4 = −0.0028 ± 0.3229, (Mode V - red) ω5 = 0.0009±0.4900, (Mode V I - black) ω6 = −0.0005±0.5689. Lower: Re= 3000. The eigenvalues are: (Mode I - blue) ω1 = −0.0002, (Mode II - green) ω2 = 0.0287 ± 0.6483, (Mode III - purple) ω3 = −0.0122 ± 0.1961, (Mode IV - orange) ω4 = −0.0260 ± 1.2806, (Mode V - red) ω5 = 0.0232 ± 1.0761, (Mode V I - black) ω6 = −0.0039 ± 1.5278.



(a) U-Mode I

(b) V-Mode I

(c) U-Mode II

(d) V-Mode II

(e) U-Mode III

(f) V-Mode III

(g) U-Mode IV

(h) V-Mode IV

(i) U-Mode V

(j) V-Mode V

(k) U-Mode V I

(l) U-Mode V I

Figure 9. Six most important DMD eigenmodes in terms of amplification in hemisphere-cylinder at AoA= 20 and Re= 1000. Left: Re(U ), Right: Re(V ). From up to down, the eigenvalue correspondence is: ω1 = −0.0001, ω2 = 0.0013 ± 0.0783, ω3 = −0.0026 ± 0.1632, ω4 = −0.0028 ± 0.3229, ω5 = 0.0009 ± 0.4900, ω6 = −0.0005 ± 0.5689. Eigenvectors are normalized with Umax and Vmax . 21 equidistant isolines



(a) U-Mode I

(b) V-Mode I

(c) U-Mode II

(d) V-Mode II

(e) U-Mode III

(f) V-Mode III

(g) U-Mode IV

(h) V-Mode IV

(i) U-Mode V

(j) V-Mode V

(k) U-Mode V I

(l) U-Mode V I

Figure 10. Six most important DMD eigenmodes in terms of amplification in hemisphere-cylinder at AoA= 20 and Re= 3000. Left: Re(U ), Right: Re(V ). From up to down, the eigenvalue correspondence is: ω1 = −0.0002, ω2 = 0.0287 ± 0.6483, ω3 = −0.0122 ± 0.1961, ω4 = −0.0260 ± 1.2806, ω5 = 0.0232 ± 1.0761, ω6 = −0.0039 ± 1.5278. Eigenvectors are normalized with Umax and Vmax . 21 equidistant isolines 19 of 22 American Institute of Aeronautics and Astronautics


Figure 11. POD energy spectrum of 15 first most energetic modes at AoA= 20 for Re= 1000 and 3000 (from up to down).



(a) U-Mode I

(b) V-Mode I

(c) U-Mode II

(d) V-Mode II

(e) U-Mode III

(f) V-Mode III

(g) U-Mode IV

(h) V-Mode IV

(i) U-Mode V

(j) V-Mode V

(k) U-Mode V I

(l) U-Mode V I

Figure 12. Six most energetic POD ”chrono-modes” (ordered from up to down) in hemisphere-cylinder at AoA= 20 and Re= 1000. Left: U , Right: V . Eigenvectors are normalized with Umax and Vmax . 21 equidistant isolines



(a) U-Mode I

(b) V-Mode I

(c) U-Mode II

(d) V-Mode II

(e) U-Mode III

(f) V-Mode III

(g) U-Mode IV

(h) V-Mode IV

(i) U-Mode V

(j) V-Mode V

(k) U-Mode V I

(l) U-Mode V I

Figure 13. Six most energetic POD ”chrono-modes” (ordered from up to down) in hemisphere-cylinder at AoA= 20 and Re= 3000. Left: U , Right: V . Eigenvectors are normalized with Umax and Vmax . 21 equidistant isolines
