Instability and topology bifurcations on a hemisphere-cylinder at high angle of attack

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(1)UNIVERSIDAD POLITÉCNICA DE MADRID ESCUELA TÉCNICA SUPERIOR DE INGENIEROS AERONÁUTICOS. INSTABILITY AND TOPOLOGY BIFURCATIONS ON A HEMISPHERE-CYLINDER AT HIGH ANGLE OF ATTACK. Doctoral Thesis By. Soledad Le Clainche Martı́nez Mechanical Engineer. Madrid, December 2013.

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(3) UNIVERSIDAD POLITÉCNICA DE MADRID. INSTABILITY AND TOPOLOGY BIFURCATIONS ON A HEMISPHERE-CYLINDER AT HIGH ANGLE OF ATTACK Doctoral Thesis. by Soledad Le Clainche Martı́nez Mechanical Engineer supervised by. V. Theofilis and D. Rodrı́guez Ph.D. in Aeronautical Engineering. Escuela Técnica Superior de Ingenieros Aeronáuticos Dpto. Motopropulsión y Termofluidodinámica. Madrid, December 2013.

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(7) Tribunal nombrado por el Sr. Rector Magfco. de la Universidad Politécnica de Madrid, el día...............de.............................de 20.... Presidente: Vocal: Vocal: Vocal: Secretario: Suplente: Suplente:. Realizado el acto de defensa y lectura de la Tesis el día..........de........................de 20 ... en la E.T.S.I. /Facultad.................................................... Calificación .................................................. EL PRESIDENTE. LOS VOCALES. EL SECRETARIO.

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(9) ”Let’s not pretend that things will change if we keep doing the same things.”. A. Einstein.

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(11) Abstract The hemisphere-cylinder may be considered as a simplified model for several geometries found in industrial applications such as aircrafts’ fuselages or submarines. Understanding the complex flow phenomena that surrounds this particular geometry is therefore of major industrial interest. This thesis presents an investigation of the origin and evolution of the complex flow pattern; i.e. separation bubbles, horn vortices and leeward vortices, around the hemisphere-cylinder under separated flow conditions. To this aim, threedimensional Direct Numerical Simulations (DNS) and experimental tests, using Particle Image Velocimetry (PIV) techniques, have been performed for a variety of Reynolds numbers (Re) and angles of attack (AoA). Critical point theory has been applied to the numerical simulations to provide, for the first time for this geometry, a bifurcation diagram that classifies the different flow topology regimes as a function of the Reynolds number and the angle of attack. A complete characterization about the origin and evolution of the complex structural patterns of this geometry has been put in evidence. Surface critical points and surface and volume streamlines were able to describe the main flow structures and their strong dependence with the flow conditions up to reach the structurally stable state. This state was associated with the pattern of the horn vortices, found on ranges from low to high Reynolds numbers and from incompressible to compressible regimes. In addition, different structural analysis techniques have been employed: Proper Orthogonal Decomposition (POD), Dynamic Mode Decomposition (DMD) and Fourier analysis. These techniques have been applied to the experimental and numerical data to extract flow structure information (i.e. modes and frequencies). Experimental and numerical modes are shown to be in good agreement. A dominant frequency associated. v.

(12) with an instability of the leeward vortices has been identified in both, experimental and numerical results..

(13) Resumen La configuración de un cilindro acoplado a una semi-esfera, conocida como ’hemispherecylinder’, se considera como un modelo simplificado para numerosas aplicaciones industriales tales como fuselaje de aviones o submarinos. Por tanto, el estudio y entendimiento de los fenómenos fluidos que ocurren alrededor de dicha geometrı́a presenta gran interés. En esta tesis se muestra la investigación del origen y evolución de los, ya conocidos, patrones de flujo (burbuja de separación, vórtices ’horn’ y vórtices ’leeward’) que se dan en esta geometrı́a bajo condiciones de flujo separado. Para ello se han llevado a cabo simulaciones numéricas (DNS) y ensayos experimentales usando la técnica de Particle Image Velocimetry (PIV), para una variedad de números de Reynolds (Re) y ángulos de ataque (AoA). Se ha aplicado sobre los resultados numéricos la teorı́a de puntos crı́ticos obteniendo, por primera vez para esta geometrı́a, un diagrama de bifurcaciones que clasifica los diferentes regı́menes topológicos en función del número de Reynolds y del ángulo de ataque. Se ha llevado a cabo una caracterización completa sobre el origen y la evolución de los patrones estructurales caracterı́sticos del cuerpo estudiado. Puntos crı́ticos de superficie y lı́neas de corriente tridimensionales han ayudado a describir el origen y la evolución de las principales estructuras presentes en el flujo hasta alcanzar un estado de estabilidad desde el punto de vista topológico. Este estado se asocia con el patrón de los vórtices ’horn’, definido por una topologı́a caracterı́stica que se encuentra en un rango de números de Reynolds muy amplio y en regı́menes compresibles e incompresibles. Por otro lado, con el objeto de determinar las estructuras presentes en el flujo y sus frecuencias asociadas, se han usado distintas técnicas de análisis: Proper Orthogonal Decomposition (POD), Dynamic Mode Decomposition (DMD) y análisis de Fourier. Dichas técnicas se han aplicado sobre los datos experimentales y numéricos, demostrándose la vii.

(14) buena concordancia entre ambos resultados. Finalmente, se ha encontrado en ambos casos, una frecuencia dominante asociada con una inestabilidad de los vórtices ’leeward’. —————————————————————-.

(15) Agradecimientos A mi hermana y a mi madre En primer lugar quiero agradecer a mi director de tesis, Prof. Vassilis Theofilis, el haberme dado la oportunidad de introducirme en el mundo cientı́fico y que me enseñara que la ciencia es algo que va más allá de los libros. En segundo lugar, quiero agradecer a mi co-director de tesis, Daniel Rodrı́guez, toda su dedicación y que me diera una visión madura y útil sobre la manera de afrontar la ideas y aplicarlas al concimiento. También quiero agradecer a mis compañeros de grupo de investigación, no solo el concimiento compartido durante este tiempo, sino también la maravillosa compañı́a durante los viajes por el mundo (conferencias y estancias) que hemos tenido la suerte de disfrutar: Hawaii, Cambridge, Melbourne, São Paulo... Por orden cronológico de entrada al grupo, gracias a José Miguel Pérez, Elmer Gennaro, Pedro Paredes, Paco Gómez, Juan Ángel Tendero, Mamta Jotkar, Qiong Liu y Wei He. Agradezco a los profesores que me han acogido durante las estancias académicas que realicé gracias al proyecto Marie Curie Grant PIRSES-GA-2009-247651 ”FP7-PEOPLEIRSES: ICO-MASEF, Julio Soria en la Universidad de Monash y Julio Meneghini en la Universidad de São Paulo. Agradezco también a Ivy Li su ayuda con los experimentos que forman parte de esta tesis. Agradezco el respaldo del Multi-modal Australian ScienceS Imaging and Visualisation Environment (MASSIVE) (www.massive.org.au), en donde realicé las simulaciones numéricas de la tesis. Quiero dar gracias a todos aquellos que estuvieron conmigo en algunos de los momentos áridos de mi doctorado, porque sin ellos todo hubiera sido mucho más difı́cil. Especialmente, gracias a Fernando por su incondicional apoyo, a Laura y Assal por su compresión, a Marı́a, Miguelin, por los buenos momentos juntos y a Guillem por su soporte ix.

(16) técnico. Y por supuesto sin olvidar a los de siempre, gracias a Laurita, Guille, Suu, Elena, Edu, Ruth, Bea y Paco III. Por supuesto, quiero agradecer también todo el apoyo de mi familia, quienes con su ejemplo me enseñaron a no rendirme nunca y con su cariño me ayudaron a seguir adelante. Gracias a Lucı́a, a mis padres, a mi abuela y a todos. Finalmente quiero agradecer a Esteban el enorme apoyo que me ha dado, imposible de expresar con palabras. Gracias por tu ayuda, cariño, paciencia, comprensión... porque sin ti, nada hubiera sido igual..

(17) Contents List of Figures. xv. List of Tables. xxi. 1 Introduction 1.1 Novelty and overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Journal Papers (JCR Peer-Reviewed) . . . . . . . . . . . . . 1.2.2 Conference Papers (Peer-reviewed with ISBN and/or ISSN). . . . .. . . . .. . . . .. 1 4 6 6 7. 2 Theory 2.1 Linear stability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Local instability: one-dimensional LNSE . . . . . . . . . . . . . . . 2.1.2 BiGlobal instability analysis: two-dimensional LNSE . . . . . . . . 2.1.3 TriGlobal instability analysis: three-dimensional LNSE . . . . . . . 2.2 Flow structures and global stability analysis . . . . . . . . . . . . . . . . . 2.2.1 Proper Orthogonal Decomposition . . . . . . . . . . . . . . . . . . 2.2.1.1 Algorithm of Snapshots Method . . . . . . . . . . . . . . 2.2.1.2 Proper Orthogonal Decomposition and Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Koopman modes and Dynamic Mode Decomposition . . . . . . . . 2.2.2.1 Algorithm of DMD . . . . . . . . . . . . . . . . . . . . . 2.2.3 Residual Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 On the relation between Residual Algorithm and Dynamic Mode Decomposition in linear stability analysis and saturated flow regimes 2.3 Flow topology and critical points theory . . . . . . . . . . . . . . . . . . . 2.4 Critical points theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 On the relation between structural stability and linear stability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9 13 15 16 17 17 18 19. 3 Implementation and Validation 3.1 Flow structures and global stability analysis . . . . . . . . . . . . . . . . . 3.1.1 Validation test: BiGlobal linear stability analysis on a lid driven cavity with alternative matrix-free numerical methods . . . . . . . 3.2 High order numerical methods for global stability analysis . . . . . . . . . 3.2.1 Dispersion-Relation-Preserving finite difference scheme . . . . . . . 3.2.2 Compact finite difference schemes . . . . . . . . . . . . . . . . . . 3.2.3 Summation by Parts Operators for finite difference approximations. 35 35. xi. 21 22 23 25 26 27 27 33. 36 40 41 42 42.

(18) Contents 3.2.4 3.2.5. xii Stable high-order finite-difference methods based on non-uniform grid points distributions . . . . . . . . . . . . . . . . . . . . . . . Derivative matrix based on Chebyshev polynomial and GaussLobatto grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Validation test: local stability analysis with FD-q . . . . . . . . . 3.2.6.1 Case 1: Eigenspectrum of plane Poiseuille flow . . . . . 3.2.6.2 Case 2: Pseudospectrum of plane Poiseuille flow . . . . 3.2.6.3 Case 3: Eigenspectrum of the Blasius boundary layer . 3.2.6.4 Case 4: Pseudospectrum of the Blasius boundary layer. . 43 . . . . . .. 44 44 45 47 48 49. 4 Hemisphere-Cylinder 4.1 Model description and parameters . . . . . . . . . . . . . . . . . . . . . . 4.2 Direct Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Numerical code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Computational domain and boundary conditions . . . . . . . . . . 4.2.3 Computational resources . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Study of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Particle Image Velocimetry and experimental facility . . . . . . . . . . . . 4.4 Classification of the different regimes obtained in the numerical simulations. 51 52 52 53 53 54 55 57 57 59. 5 Topology of 3D Separated Flow 5.1 Surface critical points and streamlines . . . . . . . . . . . . . . . . . . . . 5.1.1 Topological variations with Reynolds number fixing the angle of attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1.1 Attached flow . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1.2 Separated flow . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1.3 On the identification of the separation bubble . . . . . . 5.1.2 Topological variations with angle of attack maintaining the Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Three-dimensional flow topology and surface streamlines . . . . . . . . . . 5.2.1 On the formation of the horn vortices . . . . . . . . . . . . . . . . 5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 65 65. 3.2.6. 6 Flow Structures 6.1 Base flow and coherent structures . . . . . . . . . . . . . . . . . . . . . . 6.2 Sampling theory and Fourier analysis . . . . . . . . . . . . . . . . . . . . 6.2.1 Fourier analysis of the experimental data . . . . . . . . . . . . . 6.2.2 Fourier analysis in the numerical simulations . . . . . . . . . . . 6.3 Computational domain in POD and DMD analyses of the numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Proper Orthogonal Decomposition . . . . . . . . . . . . . . . . . . . . . 6.4.1 POD in saturated flow . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1.1 Case 1: Re= 350, AoA= 20◦ . . . . . . . . . . . . . . . 6.4.1.2 Case 2: Re= 400, AoA= 20◦ . . . . . . . . . . . . . . . 6.4.1.3 Case 3: Re= 1000, AoA= 20◦ . . . . . . . . . . . . . . . 6.5 Dynamic Mode Decomposition . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 DMD in saturated flow . . . . . . . . . . . . . . . . . . . . . . .. 68 69 72 74 80 83 92 98. 101 . 101 . 102 . 104 . 107 . . . . . . . .. 108 111 111 112 116 117 121 125.

(19) Contents. 6.6. 6.5.1.1 6.5.1.2 6.5.1.3 Conclusions . .. xiii Case Case Case . . .. 1: Re= 350, AoA= 20◦ . . . 2: Re= 1000, AoA= 20◦ . . . 4: Experiments at Re= 1000, . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . AoA= 20◦ . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 125 129 130 136. 7 Final Remarks and Recommendations for Further Work. 139. A Dispersion-Relation-Preserving Finite Difference Schemes. 143. B Compact Finite Difference Schemes. 151. C Chebyshev Polynomial and Gauss Lobatto Grid. 155. D Local Stability Analysis D.1 Characterization of the separation bubble at the symmetry plane D.1.1 Local stability analysis within the separation bubble . . . D.2 Characterization of the wake at the symmetry plane . . . . . . . D.2.1 Local satability analysis within the wake . . . . . . . . . .. Bibliography. . . . .. . . . .. . . . .. . . . .. . . . .. 157 158 164 165 167. 173.

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(21) List of Figures 1.1. 2.1 2.2 2.3 2.4. 3.1. 3.2. 3.3. 3.4. Sketch of topology patterns on a hemisphere-cylinder at high AoA. Extracted from [39] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SVD mathematical interpretation [55, 82]. . . . . . . . . . . . . . . . . . . Critical points classification on the PQ-Chart [87]. . . . . . . . . . . . . . Classification of critical points on a two-dimensional case. . . . . . . . . . Critical points classification on the QR-Chart. Nomenclature: SF/S ⇒ stable foci/ stretching; UF/C ⇒ unstable focus/ contracting; SN/S/S ⇒ stable node/saddle/saddle; USN/S/S ⇒ unstable node/saddle/saddle [17]. Top: Leading eigenmode of the regularized LDC obtained using BiGlobal Instability Analysis (BG), Time-stepping (TS), Koopman analysis (DMD) and residual algorithm (RA) and first Proper Orthogonal Decomposition (POD) mode at Re= 2000. Bottom: Second eigenmode of the regularized LDC obtained using BiGlobal Instability Analysis (BG), Timestepping (TS), Koopman analysis (DMD) and residual algorithm (RA) at Re= 2000. Eigenvectors are normalized with u bmax and vbmax . Dashed lines mean negatives values. 21 equidistant isolines from u b = −1 to u b=1 and from vb = −1 to vb = 1. Line-thickness agreement is obtained between the results of all algorithms employed. . . . . . . . . . . . . . . . . . . . Top: Dependence of damping ratio σ with time showing the exponential decay of two travelling modes (ωr = 0.958778, σ = 0.059983) and (ωr = 1.879050, σ = 0.065227), superimposed upon the steady mode (ωr ∼ 0, σ = 0.031703) at Re = 2000. Middle: Correspondence of the frequencies of the damped linear two-dimensional eigenmodes of the converged steady-states obtained from discrete Fourier transforms of the DNS signals at Re = 2000. Bottom: Frequency diagram of the first POD mode, (ωr = 0.000976, 0.012695, 0.041992) at Re = 2000. . . . . . . . . . Relative error for the amplification rate of the leading eigenmode of plane Poiseuille flow at Re = 10000, α = 1 [50, 80], obtained by (black) spectral collocation using CGL and (blue) high-order finite-difference methods of order 8: STD, Padé, DRP, SBP, as well as (red) FD-q with q = 8 and q = 16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eigenspectrum and pseudospectrum of plane Poiseuille flow at Re = 104 , α = 1 [80], obtained by spectral collocation using CGL and highorder finite-difference method FD-q. Solid lines and empty circles: CGL, Dashed lines and solid circles: FD-q16., both of them with N = 128. Levels from inner to outer isoline, 10−7 , 10−6 , 10−5 , 10−4 , 10−3 , 10−2 . Note that c = ω/α refers to phase velocity. . . . . . . . . . . . . . . . . . . . .. xv. 2 22 30 31. 32. . 38. . 39. . 46. . 48.

(22) List of Figures 3.5. 3.6. 4.1 4.2 4.3 4.4 4.5 4.6. 4.7. 5.1. 5.2. 5.3. 5.4. 5.5. 5.6 5.7. xvi. Eigenspectrum of Blasius flow at Reδ∗ = 580 and α = 0.179 [64], obtained with spectral collocation based on mapped CGL and two high-order finitedifference methods FD-q of order 12 and 24 with N = 100. Note that c = ω/α refers to phase velocity. . . . . . . . . . . . . . . . . . . . . . . . 49 Eigenspectrum and pseudospectrum of the Blasius boundary layer at Reδ∗ = 580 and α = 0.179 [64], obtained by spectral collocation using CGL and FD-q. Solid lines and empty circles: CGL; Dashed lines and solid circles: FD-q16, both of them with N = 128. Levels from inner to outer isoline, 10−7 , 10−6 , 10−5 , 10−4 , 10−3 , 10−2 . . . . . . . . . . . . . . . . 50 (a) Hemisphere-cylinder dimensions (side view) (b) Azimuthal angle into the hemisphere-cylinder (Front view). . . . . . . . . . . . . . . . . . . . . Mesh M2 containing ∼ 2.8 × 106 tetrahedral cells. . . . . . . . . . . . . . OpenFoam 2.0 scalability test in m2. Hemisphere-cylinder in mesh M 1 at Re= 1000, AoA= 20◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Planar-PIV setup in the horizontal LTRAC water tunnel . . . . . . . . . . Planar-PIV measurements in the horizontal LTRAC water tunnel . . . . . (a) Phases Diagram of the numerical simulations as function of the Reynolds number and the angle of attack on the hemisphere-cylinder with AR= 8. Red circles represent experimental measurements. . . . . . . . . . . . . . . Critical points and surface streamlines on the hemisphere-cylinder with AR= 16 at Re= 1000 and AoA= 20◦ . . . . . . . . . . . . . . . . . . . . . . Surface stream lines and critical points on the hemisphere-cylinder on the plane XZ (parallel to separation bubble) at AoA= 15◦ , Re= 1.9 × 106 and Mach number Ma= 0.7, obtained from experiments performed in [5]. . (a) Surface stream lines and critical points on the hemisphere-cylinder on the plane XZ (parallel to separation bubble). (b) Surface stream lines and critical points on the hemisphere-cylinder on the plane XY (parallel to symmetry plane). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Critical points and surface streamlines at Re= 10, AoA= 20◦ ; (b) Critical points and surface streamlines at Re= 50, AoA= 20◦ ; (c) Critical points and surface streamlines at Re= 100, AoA= 20◦ . Top left: body surface. Top right: back of the body surface. Bottom: critical points on the back surface of the body. . . . . . . . . . . . . . . . . . . . . . . . . (a) Critical points and surface streamlines at Re= 150, AoA= 20◦ ; (b) Critical points and surface streamlines at Re= 200, AoA= 20◦ ; (c) Critical points and surface streamlines at Re= 200, AoA= 20◦ . In (a) and (b): top left corresponds to the body surface, top right corresponds to the back of the body, bottom corresponds to the back surface of the body. In (c): top corresponds to the separation bubble, bottom corresponds to the origin of the leeward vortices (saddle S). . . . . . . . . . . . . . . . . . . (a) Critical points and surface streamlines at Re= 350, AoA= 20◦ ; (b) Critical points and surface streamlines at Re= 400, AoA= 20◦ ; (c) Critical points and surface streamlines at Re= 500, AoA= 20◦ . . . . . . . . . . (a) Critical points and surface streamlines at Re= 500, AoA= 20◦ ; (b) Critical points and surface streamlines at Re= 1000, AoA= 20◦ . . . . . Pressure coefficient distribution along streamwise axis X/D at AoA= 20◦ , 100 <Re< 1000 and Φ = 0◦ . . . . . . . . . . . . . . . . . . . . . . . . . .. 52 54 55 59 60. 62 63. . 66. . 67. . 70. . 73. . 75 . 76 . 78.

(23) List of Figures 5.8. 5.9. 5.10. 5.11 5.12 5.13. 5.14. 5.15. 5.16. 5.17. 5.18 5.19. 5.20. 5.21. 5.22. Pressure coefficient distribution in the area surrounding the separation bubble along streamwise axis X/D at AoA= 20◦ , 100 <Re< 1000 and Φ = 0◦ . Blue square: point of separation. Green square: point of attachment. . Contours of surface distribution of Cp and surface streamlines on the hemisphere-cylinder AR= 8 at AoA= 20◦ . 11 iso-contours equidistributed from −1 to 1. Blue colors corresponds to Cp < 0, green with Cp = 0 and red with Cp > 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface distribution of pressure coefficient along streamwise axis X/D in 0 < Φ < 180. Blue square: point of separation. Green square: point of attachment. Yellow square: NI or F . Red square: SI . OUT: region out of the bubble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Critical points and surface streamlines at Re= 500, AoA= 20◦ ; (b) Critical points and surface streamlines at Re= 1000, AoA= 10◦ . . . . . . . Surface stream lines and critical points on the hemisphere-cylinder at Re= 1000 and AoA= 30◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . Top: Surface stream lines and critical points on the hemisphere-cylinder on the plane XZ (parallel to separation bubble) at Re= 1000 and AoA= 30◦ . Bottom: Surface stream lines and critical points on the hemispherecylinder on the plane XY (parallel to symmetry plane) at Re= 1000 and AoA= 30◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Streamlines on a set of YZ planes along X axis (streamwise direction) in X/D= 1.2, 2, 4, 8 , considering X/D= 0 the junction point between the semi-sphere and the cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . Three-dimensional flow topology of the hemisphere-cylinder at Re= 500, AoA= 20◦ . Surface and volume streamlines. (b) and (c) include Iso2 = 500. . . . . . . . . . . . . . . . . . . . . surfaces of instantaneous Q/U∞ Three-dimensional flow topology of the hemisphere-cylinder at Re= 500, AoA= 20◦ . Surface and volume streamlines. General view. Iso-surfaces 2 = 500. . . . . . . . . . . . . . . . . . . . . . . . . of instantaneous Q/U∞ Three-dimensional flow topology of the hemisphere-cylinder at Re= 1000, AoA= 20◦ . Surface and volume streamlines. (b) and (c) include Iso2 = 1300. . . . . . . . . . . . . . . . . . . . surfaces of instantaneous Q/U∞ Three-dimensional flow topology of the hemisphere-cylinder at Re= 1000, 2 = 1300. AoA= 20◦ . Surface and volume streamlines. Iso-surfaces of Q/U∞ Three-dimensional flow topology of the hemisphere-cylinder at Re= 1000, AoA= 20◦ . Surface and volume streamlines. General view. Iso-surfaces 2 = 1300. . . . . . . . . . . . . . . . . . . . . . . . . of instantaneous Q/U∞ Three-dimensional flow topology of the hemisphere-cylinder at Re= 1000, AoA= 30◦ . Surface and volume streamlines. (b) include Iso-surfaces of 2 = 4000. . . . . . . . . . . . . . . . . . . . . . . . . . instantaneous Q/U∞ Three-dimensional flow topology of the hemisphere-cylinder at Re= 1000, AoA= 30◦ . Surface and volume streamlines. General view. Iso-surfaces 2 = 4000. . . . . . . . . . . . . . . . . . . . . . . . . of instantaneous Q/U∞ Global separation. Scheme of the critical points evolution on the separation bubble corresponding with the critical points presented in Table 5.2.1. Top: surface critical points sketch represented on the surface of the bubble. Middle: separation bubble on the symmetry plane. The critical point corresponds to the foci FS . Bottom: eigenvalues representing the evolution from NI to F in the QR chart. . . . . . . . . . . . . . . . . . . .. xvii. 79. 80. 81 82 83. 84. 85. 88. 89. 91 92. 93. 94. 95. 99.

(24) List of Figures 6.1 6.2. 6.3. 6.4. 6.5 6.6. 6.7 6.8. 6.9. 6.10 6.11. 6.12 6.13. Iso-surfaces of spanwise velocity Uz in the hemisphere-cylinder at AoA= 20◦ . (a) Re= 350, Uz = ±0.03. (b) Re= 1000, Uz = ±0.08. . . . . . . . . PIV experimental analysis in the symmetry plane. Streamwise velocity component non-dimensionalized with maximum value. (a) Field of view of the PIV experiments. (b) spatial coordinates corresponding to the experimental field of view. The point X/D= 0 corresponds to the junction between the semi-sphere and the cylinder. The point Y/D= 0 corresponds to the wall of the hemisphere-cylinder. . . . . . . . . . . . . . . . . . . . PIV experimental results in the symmetry plane. (a) PSD in a probe got into the shear layer. (b) Exponential growth rate curves associated with the dominant frequencies in the shear layer. a/a0 is the amplification of the frequency in each point non-dimensionalized with initial amplification a0 . (X−X0 )/(XL −X0 ): X is streamwise position, X0 is streamwise initial position and XL is streamwise final position. . . . . . . . . . . . . . . . PSD in planes YZ along X axis. PSD applied in points: Pt1=(X/D,0.004,90), Pt2=(X/D,0.06,-64.5), Pt3=(X/D,0.06,-25.4), Pt4=(X/D,0.7,0) and Pt5=(X/D,0.004,0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PSD in (a) Pt2, (b) Pt3 and (c) Pt5 in plane X/D= 2. . . . . . . . . . . Computational domain for global analyses. Hemisphere-cylinder in red and computational domain in grey: (a) Computational domain D1 . (b) Computational domain D2 . (c) Computational domain D3 . . . . . . . . POD analysis in domains D1 and D2 at Re= 350 and AoA= 20◦ . Kinetic Energy distribution (left) and PSD of POD coefficients (right). . . . . . POD analysis in domain D1 at Re= 350 and AoA= 20◦ . POD eigenmodes. Red color: positive value. Blue color: negative value. From left to right: streamwise velocity Ux , wall normal velocity Uy and spanwise velocity Uz . (a) POD mode 1 Re= 350, AoA= 20◦ , computational domain D1 , Ux ,Uy ,Uz = ±0.003. (b) POD mode 3 Re= 350, AoA= 20◦ , computational domain D1 , Ux ,Uy ,Uz = ±0.5. . . . . . . . . . . . . . . . POD analysis in domain D2 at Re= 350 and AoA= 20◦ . POD eigenmodes. Red color: positive value. Blue color: negative value. From left to right: streamwise velocity Ux , wall normal velocity Uy and spanwise velocity Uz . (a) POD mode 1: Ux ,Uy ,Uz = ±0.03. (b) POD mode 3: Ux ,Uy ,Uz = ±0.03 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . POD analysis in domain D1 at Re= 400 and AoA= 20◦ . Kinetic Energy distribution (left) and PSD of POD coefficients (right). . . . . . . . . . . POD analysis in domain D1 at Re= 400 and AoA= 20◦ . POD eigenmodes. Red color: positive value. Blue color: negative value. From left to right: streamwise velocity Ux , wall normal velocity Uy and spanwise velocity Uz . (a) POD mode 1: Ux ,Uy ,Uz = ±0.003. (b) POD mode 3: Ux ,Uy ,Uz = ±0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . POD analysis in domains D1 , D2 and D3 at Re= 1000 and AoA= 20◦ . Kinetic Energy distribution (left) and PSD of POD coefficients (right). . POD analysis in domain D1 at Re= 1000 and AoA= 20◦ . POD eigenmodes. Red color: positive value. Blue color: negative value. From left to right: streamwise velocity Ux , wall normal velocity Uy and spanwise velocity Uz . (a) POD mode 1: Ux ,Uy ,Uz = ±0.5. (b) Left: POD mode 3: Uz = ±0.5. (b) Right: POD mode 5: Uz = ±0.05 . . . . . . . . . . . . .. xviii. . 102. . 105. . 106. . 107 . 109. . 110 . 114. . 115. . 116 . 117. . 118 . 119. . 122.

(25) List of Figures 6.14 POD analysis in domain D2 at Re= 1000 and AoA= 20◦ . POD eigenmodes. Red color: positive value. Blue color: negative value. From left to right: streamwise velocity Ux , wall normal velocity Uy and spanwise velocity Uz . (a) POD mode 1: Ux ,Uy ,Uz = ±0.5. (b) POD mode 3: Ux ,Uy ,Uz = ±0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.15 POD analysis in domain D3 at Re= 1000 and AoA= 20◦ . POD eigenmodes. Red color: positive value. Blue color: negative value. From left to right: streamwise velocity Ux , wall normal velocity Uy and spanwise velocity Uz . (a) POD mode 1: Ux ,Uy ,Uz = ±0.03. (b) POD mode 3: Ux ,Uy ,Uz = ±0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.16 DMD analysis in domain D1 and D2 at Re= 350 and AoA= 20◦ . (a)Ritz diagram in D1 (b) Frequency spectrum in D1 (c) Frequency spectrum in D2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.17 DMD analysis in domain D1 at Re= 350 and AoA= 20◦ . DMD modes. Red color: positive value. Blue color: negative value. From left to right: streamwise velocity Ux , wall normal velocity Uy and spanwise velocity Uz . (a) DMD Steady mode, Uz = ±5. (b) DMD mode St' 0.065, Ux ,Uy ,Uz = ±0.0003. (c) DMD mode St' 0.11, Ux ,Uy ,Uz = ±0.0003. . . . . . . . . . 6.18 DMD analysis in domain D1 at Re= 350 and AoA= 20◦ . DMD modes. Red color: positive value. Blue color: negative value. From left to right: streamwise velocity Ux , wall normal velocity Uy and spanwise velocity Uz . (a) DMD Steady mode, Uz = ±0.0001. (b) DMD mode St' 0.065, Ux ,Uy ,Uz = ±0.0003. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.19 DMD analysis in domain D1 , D2 and D3 at Re= 1000 and AoA= 20◦ . (a) Frequency spectrum in D1 ; (b) Frequency spectrum in D2 ; (c) Frequency spectrum in D3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.20 DMD analysis in domain D1 at Re= 1000 and AoA= 20◦ . DMD modes. Red color: positive value. Blue color: negative value. From left to right: streamwise velocity Ux , wall normal velocity Uy and spanwise velocity Uz . (a) DMD Steady mode, Uz = ±1. (b) DMD mode St' 0.09, Ux ,Uy ,Uz = ±0.003. (c) DMD mode St' 0.16, Ux ,Uy ,Uz = ±0.003. . . . . . . . . . . 6.21 DMD analysis in domain D2 at Re= 1000 and AoA= 20◦ . DMD modes. Red color: positive value. Blue color: negative value. From left to right: streamwise velocity Ux , wall normal velocity Uy and spanwise velocity Uz . (a) DMD Steady mode, Uz = ±1.6. (b) DMD mode St' 0.12, Ux ,Uy ,Uz = ±0.01. (c) DMD mode St' 0.24, Ux ,Uy ,Uz = ±0.001. . . . . 6.22 DMD analysis in domain D3 at Re= 1000 and AoA= 20◦ . DMD modes. Red color: positive value. Blue color: negative value. From left to right: streamwise velocity Ux , wall normal velocity Uy and spanwise velocity Uz . (a) DMD Steady mode, Uz = ±2.5. (b) DMD mode St' 0.12, Ux ,Uy ,Uz = ±0.003. (c) DMD mode St' 0.24, Ux ,Uy ,Uz = ±0.003. . . . 6.23 Computational domain for the DMD calculations in the experiments. Streamwise mean velocity Ux iso-contours. (a) PIV computational domain. (b) Domain captured from the numerical simulations. . . . . . . . 6.24 DMD with experiments: frequency diagram at Re= 1000, AoA= 20◦ . . .. xix. . 123. . 124. . 127. . 128. . 129. . 131. . 132. . 133. . 134. . 135 . 136.

(26) List of Figures. xx. 6.25 DMD performed on the symmetry plane of the hemisphere-cylinder. The black area represents the hemisphere-cylinder. Numerical simulations, St' 0.12: (a) DNS, streamwise velocity Ux . (b) DNS, wall normal velocity Uy . Experiments, St' 0.11: (c) PIV, streamwise velocity Ux . (d) PIV, wall normal velocity Uy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 D.1 Separation bubble. Symmetry plane of the hemisphere-cylinder. Streamwise velocity contours non-dimensionalized with maximum streamwise velocity. Numerical simulations: (a) Re= 1000, AoA= 20◦ and (b) Re= 1000, AoA= 30◦ . Experiments: (c) Re= 1000, AoA= 20◦ and (d) Re= 3000, AoA= 20◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 D.2 Velocity profiles along the streamwise componend X/D within the separation bubble of the hemisphere-cylinder extracted from the DNS data. . 161 D.3 Velocity profiles along the streamwise componend X/D within the separation bubble of the hemisphere-cylinder extracted from the PIV data. . . 162 D.4 Boundary layer characterization of the separation bubble in the symmetry plane. Left: boundary layer displacement thickness δ, momentum thickness θ, inflexion point Yinf . Right: level of non-parallelism P AR. . . 163 D.5 Boundary layer characterization of the separation bubble in the symmetry plane. Left: boundary layer displacement thickness δ, momentum thickness θ, inflexion point Yinf . Right: level of non-parallelism P AR. . . 164 D.6 OSE in the DNS data at Re= 1000, AoA= 20◦ . Streamwise wave length α, Strouhal number St = ωr · sin(AoA) · D/U , frequency ωr , growth rate ωi .166 D.7 OSE in the experimetal PIV data at Re= 1000, AoA= 20◦ . Streamwise wave length α, Strouhal number St = ωr · sin(AoA) · D/U , frequency ωr , growth rate ωi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 D.8 OSE in the DNS data at Re= 1000, AoA= 30◦ . Streamwise wave length α, Strouhal number St = ωr · sin(AoA) · D/U , frequency ωr , growth rate ωi .168 D.9 OSE in the experimetal PIV data at Re= 3000, AoA= 20◦ . Streamwise wave length α, Strouhal number St = ωr · sin(AoA) · D/U , frequency ωr , growth rate ωi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 D.10 Characterization of the wake of the hemisphere-cylinder at Re= 1000, AoA= 20◦ . Wake profiles and definition of velocity deficit and wake half-width. The ponit X/D= 8 corresponds to the end of the hemispherecylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 D.11 Characterization of the wake of the hemisphere-cylinder at Re= 1000, AoA= 20◦ . Evolution of velocity deficit and wake half-width with the distance from the hemisphere-cylinder. The ponit X/D= 8 corresponds to the end of the hemisphere-cylinder. . . . . . . . . . . . . . . . . . . . . 171 D.12 OSE in the DNS data at Re= 1000, AoA= 20◦ . Streamwise wave length α, Strouhal number St = ωr · sin(AoA) · D/U , frequency ωr , growth rate ωi .172.

(27) List of Tables 2.1. Critical points clasification on the QR-Chart. Nomenclature: P1, P2, P3 ⇒ 2D Planes; SN ⇒ stable node; S ⇒ saddle; USN ⇒ unstable node; SF ⇒ stable foci; ST ⇒ stretching; UF ⇒ unstable focus; C ⇒ contracting [17]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31. 3.1. Damping rate of the leading stationary eigenmode obtained by solution of the global instability eigenvalue problem and DMD analysis of transient DNS data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40. 4.1 4.2 4.3. DNS cases on the hemisphere-cylinder with AR= 8. . . . . . . . . . . . . DNS cases on the hemisphere-cylinder with AR= 2 and AR= 16. . . . . OpenFoam 2.0, scalability test in m2. Speed up as function of the number of processors. Hemisphere-cylinder in mesh M 1 at Re= 1000 and AoA= 20◦ . Grid convergence study in the numerical simulations of the hemispherecylinder at Re= 1000, AoA= 20◦ with AR= 8. . . . . . . . . . . . . . . . . DNS case on the hemisphere-cylinder with AR= 2 at Re= 300, AoA= 0◦ . Planar PIV experimental setup . . . . . . . . . . . . . . . . . . . . . . . .. 4.4 4.5 4.6. 53 53 55 57 57 58. 5.1. Global spearation phenomenon. Critical points on the separation bubble of the hemisphere-cilinder. Correspondence: S saddle, N node, F foci, X/L spanwise direction non-dimensionalized with the lenght of the hemisphere-cylinder, Φ azimutal angle measured clockwise, Y /D wall normal component non-dimensionalized with the diameter of the body. . . . 96. 6.1. PSD in PIV shear layer probes at Re= 1000, AoA= 20◦ . Half-segment overlapping PSD. Total number of samples N, sampling interval ∆t, number of samples contained in each segment Nsamp , sampling distance nondimensionalized with Strouhal number ∆ St and number of segments Nens .105 PSD in numerical simulations at Re= 1000, AoA= 20◦ . Half-segment overlapping PSD. Total number of samples N, sampling interval ∆t, number of samples contained in each segment Nsamp , sampling distance nondimensionalized with Strouhal number ∆ St and number of segments Nens . Probes extracted in Pt1, Pt2, Pt3, Pt4 and Pt5 Figure 6.4. . . . . . 107 POD and DMD analyses computational domain. . . . . . . . . . . . . . . 110 POD convergence. Re= 350, AoA= 20◦ , domain D1. Kinetic energy E(%) contained in modes M1, M2, M3 and M4 as function of the number of snapshots N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 POD convergence. Re= 350, AoA= 20◦ , domain D2. Kinetic energy E(%) contained in modes M1, M2, M3 and M4 as function of the number of snapshots N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. 6.2. 6.3 6.4. 6.5. xxi.

(28) List of Tables POD convergence. Re= 400, AoA= 20◦ , domain D1. Kinetic energy E(%) contained in modes M1, M2, M3 and M4 as functions of the number of snapshots N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 POD convergence. Re= 1000, AoA= 20◦ , domain D1. Kinetic energy E(%) contained in modes M1, M2, M3 and M4 as function of the number of snapshots N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 POD convergence. Re= 1000, AoA= 20◦ , domain D2. Kinetic energy E(%) contained in modes M1, M2, M3 and M4 as function of the number of snapshots N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 POD convergence. Re= 1000, AoA= 20◦ , domain D3. Kinetic energy E(%) contained in modes M1, M2, M3 and M4 as function of the number of snapshots N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Snapshots number (N) and time interval between snapshots ∆t employed to solve the POD problem at Reynolds number Re in the computational domain D. PSD half-segment overlapping: Sampling distance nondimensionalized with Strouhal number ∆St, number of samples Nsamp and number of segments Nens . . . . . . . . . . . . . . . . . . . . . . . . 6.11 Snapshots number (N) and time interval between snapshots ∆t employed to solve the DMD problem at Reynolds number Re in the computational domain D. Sampling distance non-dimensionalized with Strouhal number ∆St. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xxii. 6.6. . 112. . 112. . 112. . 112. . 113. . 125. D.1 Boundary layer characterizaation on the separation bubble. Symmetry plane. Numerical (DNS) and experimental (PIV) results. . . . . . . . . . 163.

(29) Chapter 1. Introduction Flow separation over inclined axisymmetric bodies affects their stability and control. Due to its several applications in the industry, such as aircrafts’ fuselages and submarines, the study of the flow around a hemisphere-cylinder model, at both zero and non-zero angle of attack (AoA), has been the subject of several experimental and computational investigations. The main goal has been to understand flow separation effects on aerodynamic properties, and particulary in forces and moments. The importance of the prediction and control of 3D separation relies on the strong undesirable effect that it may cause over the aerodynamic and structural characteristics of the object. Massive flow separation leads to stall conditions on wings of high aspect ratio and low velocity, or to the interaction and destabilization of vortex motion downstream of control surfaces. In addition, separation increases local heat transfer on reattachment zones causing problems in high speed flights (e.g. hypersonic Space Shuttle ’Columbia’ and slender-wing supersonic transport aircraft ’Concorde’) [84]. Flow separation around the nose of the hemisphere-cylinder is a prototype case of essentially three-dimensional laminar flow separation. At certain flow conditions, depending on Reynolds number (Re), Mach number and angle of attack, the qualitative physical picture emerging when massive flow separation occurs on a hemisphere-cylinder, is a laminar separation bubble which at high incidence of the body results in a pair of counter-rotating vortices originating in the separation bubble, commonly referred to as ’Horn’ vortices. These vortices emerge from the nose of the body and condition vehicle stability, owing to their unsteady nature as the Reynolds number increases. In addition, the transversal pressure gradient on the inclined circular cylinder causes the boundary layer to separate on the lee sides and roll up generating the so-called ’Leeward’ vortices (See Figure 1.1).. 1.

(30) 2. Figure 1.1: Sketch of topology patterns on a hemisphere-cylinder at high AoA. Extracted from [39]. Motivated by the improvements in capabilities of air-to-air missiles, intensive research was conducted in order to improve the existing predictive methods for aerodynamic performances. The high angles of attack attained by these vehicles lead to massive 3D separation, and hence the methods used traditionally in aircraft industry, mainly based on potential flow, are not adequate. It is then, when the flow topology laws, introduced sequentially by Legendre [58], Lighthill [60] and Perry and Fairlie [85], become useful to analyze flow structures in 3D separated flows to deduce mean flow characteristics and mechanisms. In addition, these laws provide a theory for guiding the analysis of the flow patterns in situations where lack of resolution (both experimental and numerical) may cause inaccuracies in diagnosis of these kind of complex flows. Hsieh et al. [43, 44, 45] were the first to study the flow around a hemisphere cylinder at zero incidence in low supersonic flow. As function of Reynolds number, Mach number and AoA, the authors were able to identify three different separation patterns:. I The crossflow separation, originated by the transverse pressure gradient (direct consequence of incidence), that causes a pair of leeside vortex sheets that wind over themselves forming the leeward vortices. II The nose separation bubble, caused by the meridional pressure gradient. III The secondary separation line defining secondary leeward vortices, that appears when the transverse pressure gradient is intense enough.. On the other hand, Fairlie [27] studied prolate-spheroid and hemisphere-cylinder geometries in incompressible flow at Reynolds number 1.5x106 and high AoA. He found, for the first time in the hemisphere-cylinder, that when the AoA was higher than 17.5◦ at this flow conditions, the horn vortices were to emerge from the body surface: two spiral nodes, symmetrically located about the leeward plane of symmetry, give rise to a pair of counter-rotating vortices that are shed downstream..

(31) 3 Another series of investigations were carried out to study the three-dimensional complex topology patterns developed above the body surface, based on critical-point theory. This theory was introduced by Dallmann [22], Perry and Chong [87], Hornung and Perry [42] and Perry and Hornung [86] to describe flow separation in several applications, including blunt bodies at incompressible regimes. The latter authors clasified the existing surface streamlines and critical points for different bodies and flow regimes and described their possible 3D topological connections. On the other hand, Tobak and Peake [83, 121, 122] were the first to identify and classify critical points and separation lines on the hemisphere-cylinder body surface in compressible flows. They pointed out the existence of two unstable foci, associated with the horn vortices emergence, a saddle and a node point, related to a separation bubble, as well as a reattachment line and primary and secondary lines of separation, related to the leeward vortices. Additionally, these authors found that compressibility alters quantitative the flow characteristics but that the topology patterns on the surface of the hemisphere-cylinder were equivalent between compressible and incompressible regimes. Supplementary to this hypothesis, Bippes and Turk [5] studied the influence of Reynolds number, Mach number and angle of attack over the same geometry and also found that the structural flow patterns in stable laminar flow were very similar to the ones found in turbulent separated flow. A sequence of experiments performed in the early 800 s using surface hot-film sensors [54, 69] and laser-Doppler velocimetry [19, 20] studied more in detail the pair of vortices on the leeward region, far from the body nose, previously detected by Hsieh et al. [43] (leeward vortices). In particular, Costis and Telionis [20] and Costis et al. [19] studied the origin of such vortices and the effects of transition on their development. It was postulated that the effects of boundary layer transition cause the development of vortex sheets to form this pair of leeward vortices, which are not able to remain laminar for large Reynolds numbers. This effect is consequence of the contribution of substantial crossflow and heavily inflectional profiles of boundary layer on the leeside. In addition, Hsieh and Wang [46] provided numerical evidence in support of this flow pattern. Several authors followed up to contribute in such research. Meade and Schiff [68] studied surface pressure distributions and their variations in the separation bubble region in supersonic flows. Ying et al. [132] suggested that the horn vortices may appear in an asymmetric fashion, but Hoang et al. [39, 40], who studied experimentally the influence of Reynolds number and angle of attack on these vortex pair in incompressible flows, did not find evidence of such asymmetry. In addition, the latter authors found a lack of connectivity between the nose structures and the separation lines, which give rise to the vortical structures over the aft part of the body..

(32) 4 More recently, Gross et al. [35] performed direct numerical simulations and experiments at moderate Reynolds numbers (Re= 2000 and Re= 5000) and moderate-to-high incidence. They determined the absence of a separation bubble for very low angles of attack. Additionally, they pointed out the existence of a shear layer instability on the symmetry plane at low incidence (AoA= 10◦ ). They suggested that the dominant mechanisms in flow separation may be an inviscid (linear) shear layer instability. On the other hand, Bohorquez et al. [7] and Sanmiguel-Rojas et al. [104] performed direct numerical simulations of the same geometry at AoA= 0◦ finding bifurcations of the linear global stability modes on the wake of the hemisphere-cylinder as a function of Reynolds number and aspect ratio of the body (length/diameter). A global mode was found to be associated with the instability of the counter-rotating vortex pair forming the three-dimensional wake, with associated non-dimensional frequency St' 0.12. Similar frequencies were found by many authors that studied flows past bluff bodies [105, 124]. As found by Gross et al., variations in the flow behavior driven by changes in angle of attack and Reynolds number can be associated with linear instability phenomena. Three dimensional separated flows are dominated by inflectional instability and become unstable at relatively small Reynolds number. However, the three-dimensional nature of the flow around a hemisphere-cylinder prevents, in principle, the use of traditional linear stability analysis based on 1D velocity profiles. Methodologies for global instability analysis need to be employed. On the other hand, large scale coherent-structures in turbulent flows are often reminiscent of flow structures generated by linear instabilities, specially in mixing layers flows (e.g. Crow and Champagne [21], Gaster et al. [31], Reynolds and Hussain [93], etc).. 1.1. Novelty and overview. In the past, topology patterns have been analyzed over different configurations on a hemisphere-cylinder at high angle of attack in compressible and incompressible regimes employing both, experimental and numerical techniques, at high Reynolds number. However, the creation of the topological patterns in laminar regimes starting from very low values of the Reynolds number up to reaching the more classical configuration presented in the literature [122] is still an open topic. This thesis includes the discussion of the origin and evolution of the critical points on the surface of a hemisphere-cylinder, and details the analysis of 3D topological patterns that preceed the formation of the three well known topology patterns found on this geometry: the separation bubble, the horn vortices and the leeward vortices. For the first time, a bifurcation diagram that classifies the different flow topology regimes as function of.

(33) 5 Re and AoA is presented for this geometry. In addition, the 3D formation, evolution and interaction between the three different topology patterns, described previously, as function of Reynolds number and AoA are presented. The flow structures and their associated frequencies have been found, for the first time for this geometry, using Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD). This thesis combines Direct Numerical Simulations (DNS) and time-resolved Particle Image Velocimetry (PIV) experiments to contribute in the study of flow separation and unsteadiness in the hemisphere-cylinder. Direct Numerical Simulations have been carried out for a range of Reynolds number between 10 to 1000 at angle of attack of 20. Different angles of attack (from 0 to 30) whilst maintaining the Reynolds number fixed to 1000 have been studied, in order to contrast and understand 3D topology patterns that are consequence of flow separation. The selected aspect ratio (AR) of the geometry analyzed on this thesis is L/D= 8 (L and D are the length and diameter of the body respectively). However, the same geometry with double aspect ratio L/D= 16 has also been considered in some selected cases in order to determine the influence of the length of the body on 3D flow separation. Frequencies and structures have been detected employing Power Spectral Density analysis (PSD), Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD). In addition, PIV experiments have been performed over the same geometry (AR= 8) at one of the most unstable flow conditions of the latter cases (Re= 1000, AoA= 20◦ ) and at higher Reynolds number (Re= 3000, AoA= 20◦ ). Numerical and experimental results were found to be in good agreement. The present thesis is divided into eight chapters, each one addressing a differentiated objective of the present research work.. • Chapter 2 details the theory and numerical methods employed to the analysis of the DNS results. In this Chapter, linear stability theory and the theory behind the different structural analysis techniques used to find flow structures and their associated frequencies are presented: POD, DMD and Residual Algorithm (RA). • Chapter 3 presents the implementation and the validation of the numerical codes used in this thesis. In this Chapter different high-order matrix-forming numerical methods to solve linear instability problems are compared with matrix-free methods. A new high-order finite difference method, called FD-q, is found to be much more efficient in terms of accuracy and computational cost than the traditional numerical methods employed to solve stability analysis, and hence is used in this thesis to solve the local stability analysis presented in Appendix D. In addition, a validation of the flow structures techniques (POD, DMD and RA) is presented in.

(34) 6 this Chapter. The last two methods have been proposed as alternative matrix-free methods to solve global instability analysis problem. • Chapter 4 details the hemisphere-cylinder geometry, summarizes the cases studied and briefly discusses the parameters employed to solve the Navier-Stokes equations. The cases studied are classified as a function of their topology patterns and are introduced in a bifurcation diagram. Different flow regimes were found when varying the Reynolds number and angle of attack. Additionally, the Chapter presents the experimental facility and flow conditions for the experiments, which are compared to the numerical results. • Chapter 5 shows critical points and surface streamlines for different flow regimes on the hemisphere-cylinder and presents a detailed description of 3D flow patterns. In addition, a complete description of the formation of the 3D separation bubble and the link between this pattern and the two other possible flow patterns for this geometry is detailed in this Chapter. • Chapter 6 studies the flow structures present on the hemisphere-cylinder at different flow regimes. POD, DMD and Fourier analyses have been performed on saturated flow regime to detect the dominant flow frequencies and their associated flow structures. Both, experimental and numerical results were found to be in good agreement. • Chapter 7 summarizes the final conclusions and suggests future lines of research.. 1.2. Publications. Most of the material in this thesis has been presented in the following publications:. 1.2.1. Journal Papers (JCR Peer-Reviewed). • Gómez, F., Le Clainche, S., Paredes, P., Hermanns, M., Theofilis, V. ”Four Decades of Studying Global Linear Instability: Progress and Challenges”, AIAA Journal, Vol. 50, No 12, 2012, pp 2731-2743. • Paredes P., Hermanns, M., Le Clainche, S., Theofilis, V. ”Order 104 speed up in global linear instability analysis using matrix formation”. Computer Methods in Applied Mechanics and Engineering, Vol. 253, 2013, pp 287 304. • Le Clainche, S., Li, J. I. , Theofilis, V., Soria, J. ”Experimental and Numerical analysis of the flow around a hemisphere-cylinder at high angle of attack and moderate Reynolds number”, Aerospace Sciences and Technology, (To appear )..

(35) 7 • Le Clainche, S., Rodrı́guez, D. , Theofilis, V., Soria, J. ”On the formation of the three-dimensional separation bubble, the horn vortices and the leeward vortices in the hemisphere-cylinder”, Journal of Fluid Mechanics, (In preparation).. 1.2.2. Conference Papers (Peer-reviewed with ISBN and/or ISSN). • Le Clainche, S., Gómez, F., Li, I., Soria, J., Theofilis, V. ”Structural analysis on a hemisphere-cylinder at moderate Reynolds number at high angle of attack”. 51st AIAA Aerospace Sciences Meeting, 7 10 January 2013, AIAA paper 2013-0387. • Le Clainche, S., Li, J., Theofilis, V., Soria, J. ”Time-resolved Particle Image Velocimetry and structural analysis on a hemisphere-cylinder at low Reynolds numbers and large angle of incidence”.42nd AIAA Fluid Dynamics Conference and Exhibit, 25 28 June 2012, AIAA paper 2012-3275. • Theofilis, V., Le Clainche, S. ”Global Linear Instability at the Dawn of its 4th Decade: A List of Challenges (A Practical Guide on how to Contain the Euphoria and Avoid the Oversell)”. 6th AIAA Theoretical Fluid Mechanics Conference, 27 30 June 2011, AIAA paper 2011-3291..

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(37) Chapter 2. Theory Linear stability theory (LST), mainly motivated by research into laminar-turbulent flow transition, has occupied a substantial part of fluid mechanics research for over a century. The bulk of numerical efforts has been confined into analyzing one-dimensional shear flows. The classic linear stability theory of Tollmien [123] is mainly concerned with individual sinusoidal waves propagating in the boundary layer parallel to the wall. As far as the relation between linear global theory and flow structures is concerned, the first quantitative connection between flow topology and global (2D Global, modal) linear instability was made by Rodrı́guez and Theofilis[96, 97], 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 separated flow, such as U-separation on an adversure pressure gradient (APG) boundary layer and Stall Cells on an airfoil at high AoA values. To-date, topological ideas are yet to be applied to results of compressible 2D or 3D global instability analysis. With flow instability dominating the unsteady motion existing at even low Reynolds numbers, it is expected that analysis techniques based on empirical (either from experiment or numerical simulations) eigenfunctions are useful for understanding better the physics; i.e.: Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD). POD is a statistical technique introduced by Sirovich [108] suitable to detect flow structures on nonlinear saturated flows. The principal motivation of the application of the Karhunen-Loéve (K-L) theory / POD to fluid flow has been to provide a description of turbulence based on deterministic coherent structures, reconstructed using an empirical eigensystem [41]. Of no less significance has been the contribution of the theory to 9.

(38) 10 the ongoing efforts in constructing Reduced Order Models (ROM) in order to describe dynamic flow behavior efficiently, using a small number of empirical eigenmodes. From the point of view of LST (both local and global) a most interesting theoretical question concerns the relation between the empirical eigensystem and linear stability eigenmodes. The mathematical basis of POD is the spectral theory for compact, self-adjoint operators [41] and, as such, it delivers an orthogonal set of POD eigenfunctions. One might then expect a dichotomy between the non-self-adjoint operator describing linear stability (in both local and global contexts) and expansions based on POD eigenmodes. Indeed, Farrell and Ioannou [28] expressed misgivings regarding direct application of POD ideas to the linearized Navier-Stokes operator; quoting earlier work by North [78], they asserted that when the matrix discretizing the linearized operator is non-normal, the K-L decomposition of the correlation matrix does not identify the normal modes of that matrix. Then, the bi-orthogonal decomposition into the direct and adjoint LST eigensystems [36, 75] may be used in that situation. Breuer and Sirovich [9] were the first to apply POD analysis to a linear EVP governed by the Poisson operator (a normal operator, as opposed to linearized Navier-Stokes equations) and showed reproduction by POD of the analytically known eigenvalues of this operator in a rectangular two-dimensional dimension, before predicting via POD the eigenvalues and eigenfunctions of the same operator in a complex two-dimensional domain. By contrast, Rempfer and Fasel [91, 92] reconstructed coherent structures found in transitional data on the flat-plate boundary layer and classified them in pairs of nearidentical (but distinct) eigenstructures. Their work was the first to relate POD eigenfunctions with (one-dimensional) amplitude functions of Tollmien-Schlichting waves, the latter being solutions of the non-normal linearized Navier-Stokes operator. Interestingly, these authors did not invoke the (applicable to the flat-plate boundary layer instability problem) weakly-nonparallel flow assumption in their analysis. Rowley et al. [98, 100] extended the combined POD/LST analysis to two inhomogeneous spatial dimensions, studying DNS-obtained compressible flows over open cavity configurations. Certain analogies can be observed comparing the spatial structures of the leading POD modes with those of the local parallel LST in the two sets of data in the unstable shear-layer region at the open end of the cavity. In the same direction, recent research by Sengupta et al. [115] in the analysis of the flow past a circular cylinder has linked the POD modes and the instability modes of this flow by means of nonnonlinear interactions satisfying the Landau Stuart Eckhaus (LSE) equation, and also found qualitative analogies between LST and POD mode results. Merzari et al. [71], applied POD analysis to turbulent flows, extending their earlier global linear instability analysis of the same flows in the laminar regime [72]. They used the snapshots method,.

(39) 11 but instead of the generic decomposition, they introduced a decomposition which takes spatial homogeneity along one direction into account. These authors provided evidence by comparison of such POD eigenfunctions on the one hand, and results of their earlier BiGlobal instability analysis [72] that the leading POD eigenfunctions of turbulent flow exhibit strong spatial analogies with the amplitude functions of the leading BiGlobal eigenmodes. In addition, Merzari et al. [71] found that, within a reasonably wide range of Reynolds numbers the wavenumber of the leading POD eigenmodes does not depend on Reynolds number and is fixed by the geometry through the harmonic expansion along the homogeneous spatial direction considered. Finally, Oberleithner et al. [79] very recently presented direct comparison of empirical modes and linear stability eigenmodes of swirling-jet flow undergoing vortex breakdown. These authors present the first conclusive demonstration of excellent agreement between the leading POD and the leading eigenmode global flow. The successes of both Merzari et al. [71] and Oberleithner et al. [79] in comparing empirical and global turbulent flow eigenmodes in two and three inhomogeneous spatial directions, respectively, demonstrate that such analysis is feasible. It is certainly also desirable, in the sense that it paves the way to the description of the flow by simpler models and, ultimately, to its control. It can be speculated that the same successes may both be attributable to the predominance of a single Fourier harmonic in the flow dynamics: Tammisola et al. [114] compared on a viscous plane wake the frequency obtained using BiGlobal linear stability analysis on the linear regime with the frequency obtained in the nonlinear saturated regime and found that they were different. However, it is worth examining how the situation may be different in different kinds of flows and in (complex) flows in which additional frequencies are present. The rather old concept of Koopman modes [53] has been recently introduced to the analysis of fluid flow structures by Rowley et al. [101] as a particular class of techniques for nonlinear systems analysis and reduction discussed in the influential work of Mezic [74]. These authors introduced Koopman modes as a complement for POD modes. On the one hand, POD is a statistical technique that could be inaccurate if the principal directions in a set of data does not correspond with the dynamical important ones. On the other hand, DMD may provide associated mode frequencies and growth rates. Dynamic Mode Decomposition (DMD) is a technique introduced by Schmid [107] to compute approximations of Koopman modes. DMD is an Arnoldi-like algorithm that assumes a linear operator defined for any nonlinear dynamical system that describes linear or nonlinear flow coherent structures. Classical Arnoldi is a Krylov method in which each time step farfield, provided by the operator, is orthonormalized. However.

(40) 12 DMD is applied to a snapshots data sequence and it does not require explicit knowledge of the operator. Rowley et al. [99] introduced the possibility of applying DMD to different kind of flows. When DMD is applied to the linearized Navier-Stokes equations, Koopman modes reduce to linear global modes. On the other hand, when DMD is applied to time-periodic flows, Koopman modes reduce to temporal Fourier modes. Chen et al. [13] and Bagheri [3] applied DMD to study the flow around a circular cylinder in non-linear saturated flows. They showed that five dominant DMD modes (a steady and two pairs of complex conjugates), that are consequence of a Hopf bifurcation, may be identified on such case of study. The first mode vb0 is the steady mode and corresponds to the mean flow. A pair of complex conjugate eigenmodes vb1 corresponds to the flow field that oscillates with the fundamental vortex shedding frequency. Finally the second pair of eigenmodes vb2 is the second harmonic of the mode vb1 and oscillates with twice the fundamental frequency. This mode appears due to the interaction of the first mode with itself. The same eigenmodes are identified as the most energetic POD modes on this kind of flows by Noack et al. [76] who postulated that nonlinear stability assumes explicitly that the most energetic POD modes and instability modes are the same at least near the beginning of a supercritical Hopf bifurcation. However, for non-linear transient dymanics DMD may not be a suitable choice. This algorithm is not optimal to find eigenvalues and eigenvectors that describe the dynamics of the system. Duke et al. [25] proved that the algorithm can be sensitive to noise. In addition, Schmid [107] and Chen et al. [13] showed that, even though the algorithm is analytically correct, it may be ill-conditioned in practice . However, it is a reasonably inexpensive and effective method that has been proved to be effective to approximate linear eigenvalue problems (Schmid [107], Gomez et al. [34]), discrete Fourier transforms (Chen et al. [13]) and to capture modes on nonlinear systems (Bagheri [3]). Concerning the computational cost and linear stability analysis, nowadays there is still no evidence on the competitiveness of different LST algorithms compared with DMD. However, despite the very efficient high-order finite difference methods introduced by Paredes et al. [81] to solve global instability problems using matrix formation, the possibility of DMD to be applied as an alternative to time-stepping methods (Gomez et al. [34]), potentially allows to reduce computational cost. As far as Fourier analysis is concerned, it is known that DFT is cheaper than DMD in terms of memory and CPU time. However, the advantage of DMD against DFT is that, near-equilibrium and transient regimes, DFT modes energies have much slower decay due to the non periodicity of the flow. This implies the need of retaining a large number of samples to construct an accurate solution with DFT. The largest period of DFT that.

(41) 13 can be computed is the time span of the data set, however Chen et al. [13] proved that DMD is able to compute the exact solution with relatively small error (< 5 % ) with a set of data in time smaller than the period of the searched frequency. Finally, it is important to mention the discussion recently opened by Chen et al. [13] related to the DMD algorithm and the extraction of mean or base flow from the data set of instantaneous snapshots. These authors studied the flow past a circular cylinder and showed that when mean flow was subtracted from the raw data, DMD modes where exactly equivalent to DFT modes. On the other hand, they postulated the need to extract the base flow to the raw data in order to ensure that DMD modes will satisfy boundary conditions. However, these two variants applied to DMD are still to be proved on different kind of flows and to be studied their behavior. On nonlinear flows, close to a Hopf bifurcation where the flow contains a dominant structure and frequency, spatial mode recovered by DMD and POD or by LST and POD are nearly identical. However, the recovered frequencies associated with each DMD and POD or LST and POD modes present some small differences. Rowley et al. [99] applied DMD to a jet in crossflow and the results were compared to the POD analysis and LST global modes. The latter authors found that the LST and Koopman modes eigenfunctions were similar to the POD modes. However, they found that each POD mode contained by several frequencies, all different from the single frequency associated with the corresponding LST or Koopman mode. Gómez et al. compared DMD, POD and LST modes on a squared lid-driven cavity on the linear regime, finding that the spatial structure of the respective eigenfunctions was the same, but the frequency associated to the POD mode was different to the other frequencies. Finally, Le Clainche et al. [57] compared DMD and POD on the flow around a hemisphere-cylinder at high angle of attack. It was found that, on nonlinear saturated regimes, POD and DMD eigenfunctions are the same if the flow behavior is driven by a single dominant frequency and its harmonics. In addition, despite each DMD mode contains a single frequency and that each POD mode is composed by a mixture of frequencies, it was found that the mixture of frequencies associated with each POD mode was equivalent to the frequency associated with the corresponding DMD mode and its harmonics.. 2.1. Linear stability theory. The hydrodynamic linear stability theory studies the effect of small-amplitude perturbations over steady or time periodic laminar flows. The idea is to provide a better understanding of the processes involved in the transition from laminar flow to a turbulent regime..

(42) 14 The three-dimensional Navier-Stokes equations of a viscous, incompressible fluid in dimensionless form and Cartesian coordinates are ∇ · ũ = 0, 1 2 ∂ ũ + (ũ · ∇)ũ = −∇p̃ + ∇ ũ, ∂t Re. (2.1) (2.2). with the Reynolds number defined as: Re = U L/ν,. (2.3). being U and L the characteristic velocity and length scales, respectively, of the considered problem, and ν the kinematic viscosity. In the linear flow stability theory, the vector of fluid variables q̃ = [ũ, p̃]T is decomposed into a steady base flow Q = [U, P ]T and an unsteady small disturbance or perturbation ε q, with ε 1 and q = [u, p]T : q̃(x, t) = Q(x) + ε q(x, t).. (2.4). Introducing this decomposition of the perturbed flow into the Eq. (2.2), and subtracting the base flow (as it satisfies the Navier-Stokes and continuity equations itself) one arrives to the so-called perturbation equations or Linearized Navier Stokes Equations (LNSE). ∇ · u = 0, 1 2 ∂u + (U · ∇)u + (u · ∇)U = −∇p + ∇ u. ∂t Re. (2.5) (2.6). In these equations the O(ε) terms are retained, while the non-linear perturbation term (u∇u) is O(ε2 ) and have been neglected. Once this approximation is assumed, solutions to the initial-value-problem dq = L(Re, Q) q, dt. (2.7). are sought. Specific comments on the dependence of these quantities on the spatial coordinates, x, and time, t, will be made in what follows. The operator L is associated with the spatial discretization of the linearized Navier Stokes equations (LNSE) of motion and comprise the base state, Q(x), and its spatial derivatives. In case of steady base flows, the separability between time and space coordinates in (2.7) permits introducing.

(43) 15 a Fourier decomposition in time, q(x, t) = q̂(x) Θ(x, t),. Θ = θ(x) exp(−iωt). (2.8). with θ(x) a spatial phase function, which depends on the number of homogeneous directions of the problem. Substituting the Fourier decomposition into the linearized Navier-Stokes equations, leads to the following generalized matrix eigenvalue problem: A q̂ = ω B q̂.. (2.9). Here matrices A and B discretize the operator L, with B being singular due to the continuity equation. The sought complex eigenvalue is ω = ωr +iωi , the real part being a circular frequency, while the imaginary part being the temporal amplification/damping rate; and q̂(x; t) = (û, p̂)T is the vector comprising the amplitude functions of linear velocity-component and pressure perturbations.. 2.1.1. Local instability: one-dimensional LNSE. Throughout the largest part of last century, additional assumptions have been made to the base flow and the disturbances in order to make the problem solvable. The strongest of which was adopting the so-called parallel-flow assumption. The base flow is assumed to be homogeneous in two out of the three spatial directions, here x and z, and comprises components Q = [U, 0, W, P ]T (y),. (2.10). such that the coefficients of the resulting eigenvalue problem are x and z independent. Modal perturbations then get the form q(x, y, z, t) = q̂(y) exp[i(αx + βz − ωt)],. (2.11). where the periodicity lengths Lx = 2π/α and Lz = 2π/β are imposed to the disturbances’ shapes in the x and z directions respectively..

(44) 16 Upon substitution of Eq. (2.11) into the LNSE, the operators A and B defining (2.9) become:  A1D.    =  . Uy. 0. iα. 0. L1D. 0. 0. Wy. L1D. iα. Dy. iβ.  Dy   , iβ   0. where L1D = iαU + W iβ −. 1 Re (Dyy. . . L1D. B1D. i. 0 0 0. .    0 i 0 0    = ,  0 0 i 0    0 0 0 0. (2.12). − β 2 − α2 ) and Dy being the first derivative matrix. and Dyy the second derivative matrix respect to y direction. In addition, α, β ∈ R are wavenumber parameters, related with the periodicity length along the homogeneous spatial directions, x and z. The problem described by the system of four coupled equations can be reduced to a system of two coupled equations when W = 0, by eliminating the pressure variable in v−equation and introducing the normal vorticity η̂ = ∂ û/∂z − ∂ ŵ/∂x. The resulting set of equations, known as Orr-Sommerfeld and Squire equations, is 2 1 v̂ = 0, Dyy − k 2 (−iω + iαU ) Dyy − k 2 − iαUyy − Re 1 2 (−iω + iαU ) − η̂ = −iβUy v̂, Dyy − k Re. (2.13) (2.14). where the wavenumber k 2 = α2 + β 2 has been introduced. The phase velocity, is defined as c = ωr /k. This problem is usually complemented with the boundary conditions v̂ = dv̂/dy = η̂ = 0 at solid walls or in the far field. However, as for the BiGlobal problem, this choice of boundary conditions for the far field is not justified existing continuous spectrum, but is valid for the recovery of the discrete eigenmodes.. 2.1.2. BiGlobal instability analysis: two-dimensional LNSE. Assuming that the base flow is now dependent on two out of the three spatial coordinates Q = [U, V, W, P ]T (x, y),. (2.15). the coefficients of the LNSE are z independent, and modal perturbations now get the form q(x, y, z, t) = q̂(x, y) exp[i(βz − ωt)]. (2.16). The disturbances are still three-dimensional, but a sinusoidal dependence is assumed only in the homogeneous z direction, with the periodicity length Lz = 2π/β. Upon.