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INSTABILITY OF LAMINAR SEPARATING FLOWS (SEPARATION BUBBLES) CONTENTS : 1. Fundamentals

Multimedia files - 6/ 13. INSTABILITY OF LAMINAR SEPARATING FLOWS (SEPARATION BUBBLES) CONTENTS : 1. Fundamentals ● Problem formulation ● Global response of separation bubbles to external forcing 2. Small-amplitude wavy disturbances: linear instability ● Waveform

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INSTABILITY OF LAMINAR SEPARATING FLOWS (SEPARATION BUBBLES) CONTENTS : 1. Fundamentals

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  1. Multimedia files - 6/13 INSTABILITY OF LAMINAR SEPARATING FLOWS (SEPARATION BUBBLES) CONTENTS: 1. Fundamentals ● Problem formulation ● Global response of separation bubbles to external forcing 2. Small-amplitude wavy disturbances: linear instability ● Waveform ● Growth rate ● Propagation velocity ● Effects of the axial symmetry ● Instability at separation of 3D boundary layers 3. Excitation of the instability waves in separation bubbles 4. Nonlinear phenomena ● Subharmonic resonance ● Coherent vortices ● Some other interactions

  2. 1. Fundamentals (problem formulation - 1/6) Separation bubbles are found in a variety of flow configurations a b c d notes

  3. 1. Fundamentals (problem formulation - 2/6) Transitional separation bubbles (Ward, 1963) (Brendel & Mueller, 1988) notes

  4. 1. Fundamentals (problem formulation - 3/6) Separation bubbles as: shear-layer transition noise amplifiers (convective instability, local stability analysis ) oscillators (self-sustained dynamics, absolute instability and global modes) e.g. large-scale vortices notes

  5. 1. Fundamentals (problem formulation - 4/6) In what follows, we are interested in such flows as noise amplifiers by focusing on the oscillations of the separated shear layer in terms of the classic stability theory rather than on the global dynamics of separation regions.

  6. 1. Fundamentals (problem formulation - 5/6) Natural velocity perturbations Laminar flow disturbances excited by environmental noise in a separation bubble on airfoil, Rec = 270 000 (Boiko et al., 1989) notes

  7. 1. Fundamentals (problem formulation - 6/6) Aspects of laminar-turbulent transition in separation regions notes

  8. 1. Fundamentals (global response - 1/5) Noteworthy is that excitation of instability waves modifies the entire separated flow pattern Mean-velocity profiles of the basic flow (1) and of the separation bubble perturbed by small-amplitude periodic oscillations (2) notes

  9. 1. Fundamentals (global response - 2/5) Global response of a transitional separation bubble on airfoil Maximum mean-flow variation (DU) vs. maximum amplitude of the harmonic shear-layer disturbances (u') generated locally upstream of separation on an airfoil at fc/U∞ = 10.4, contour levels are given as percentage of the oncoming-flow velocity U∞ (Gilev et al., 1984) notes

  10. 1. Fundamentals (global response - 3/5) Global response of a laminar separation bubble behind a 2D backward-facing step on flat plate Maximum amplitudes of 2D instability waves excited at fh/Uo = 0.029 (circles) and 0.034 (triangles) behind a backward-facing step (Boiko et al., 1991): in this case, the global response is found at the amplitude of oscillations in the region of reattachment as high as about 1% of the local external-flow velocity Uo (open symbols) notes

  11. 1. Fundamentals (global response - 4/5) Global response of a laminar separation bubble behind a 2D backward-facing step on flat plate Maximum mean-flow variation (DU) in the upstream part of separation bubble at x/h = 4.5 (top) and x/h = 13.6 (bottom) vs. maximum amplitude (u') of the controlled instability waves in downstream section x/h = 45, excitation frequency fh/Uo is 0.017 (□), 0.029(○) and 0.034 (∆) (Boiko et al., 1991)

  12. 1. Fundamentals (global response - 5/5) Effect of the instability waves on the mean-velocity profile Separated-flow profiles measured at 10 mm behind a 2D surface inflexion in natural conditions (●) and under controlled excitation of the instability wave (□) (Dovgal & Kozlov, 1983); experimental conditions: oncoming-flow velocity – 5.6 m/s, excitation frequency – 412 Hz, maximum local amplitude of the harmonic perturbation – 0.29% of Uo notes

  13. References on Fundamentals: Boiko AV, Dovgal AV, Kozlov VV (1989). Soviet J. Appl. Phys., 3(2), 46–52. Boiko AV, Dovgal AV, Scherbakov VA (1991). Preprint ITAM 5–91. Brendel M, Mueller TJ (1988). J. Aircraft, 25(7), 612–617. Dovgal AV, Kozlov VV (1983). Dokl. Akad. Nauk, 270(6), 1356–1358 (translated in Phys. Dokl.). Gilev VM, Dovgal AV, Kozlov VV (1984). Preprint ITAM 6–84. Ward JW (1963). J. Royal Aeronaut. Soc., 67, 783–790. For basic features of transitional separation bubbles see: Allen T, Riley N (1995). Aeronaut. J., 99, 439–448. Eaton JK, Johnston JP (1981). AIAA J., 19(9), 1093–1100. Gaster M (1992). In M.Y. Hussaini, A. Kumar and C.L. Strett (Eds.), Instability, Transition, and Turbulence (pp. 212–215). Berlin Heidelberg New York: Springer. Horton HP (1967). Aeronaut. Research Council CP 1073. Kiya M (1989). In P. Germain, M. Piau and D. Caillerie (Eds.), Theoretical and Applied Mechanics (pp. 173–191). Elsevier. Mueller TJ, Batill SM (1982). AIAA J., 20(4), 457–463. Tani I (1964). Progr. Aeronaut. Sci., 5, 70-103. Van Ingen JL (1975). In AGARD-CP–168. Van Ingen JL (1977). In AGARD-CP–224.

  14. 2. Linear instability (waveform - 1/9) Normal-to-wall profiles of the instability waves in separation bubbles, natural disturbances Amplitude distributions u'(y) of the laminar flow perturbations amplifying at boundary-layer separation on an airfoil (Cousteix & Pailhas, 1979), notice the measurement sections upstream of reattachment marked in blue notes

  15. 2. Linear instability (waveform - 2/9) Normal-to-wall profiles of the instability waves in separation bubbles, natural disturbances Amplitude distributions u'(y) of the laminar flow perturbations amplifying at boundary-layer separation behind a 2D backward-facing step (Sinha et al., 1981) notes

  16. 2. Linear instability (waveform - 3/9) Normal-to-wall profiles of the instability waves in separation bubbles, controlled disturbances Among the first observations of controlled separated-flow perturbations were those by Gaster (1967) for boundary layer separation induced on a flat plate by streamwise pressure gradient notes

  17. 2. Linear instability (waveform - 4/9) Normal-to-wall profiles of the instability waves in separation bubbles, controlled disturbances Laminar flow oscillations generated upstream of a 2D hump on flat plate transform to the instability waves of the separation bubble and then turn back to those of the laminar reattached boundary layer: amplitude (top) and phase (bottom) of the perturbations excited at fh/Uo = 0.017 (Dovgal & Kozlov, 1990)

  18. 2. Linear instability (waveform - 5/9) Normal-to-wall profiles of the instability waves in separation bubbles, controlled disturbances Laminar flow oscillations generated in a separation bubble induced by streamwise pressure gradient on a flat plate: amplitudes vs. mean-flow profiles (left) and phase (right) of the perturbations (Häggmark et al., 2000)

  19. 2. Linear instability (waveform - 6/9) Normal-to-wall profiles of the instability waves in separation bubbles, LST Inviscid stability solution for a modified tanh-profile U(y) modeling the base flow at laminar separation: amplitude of the streamwise (u') and normal (v') velocities (Michalke, 1990) u' v' notes

  20. 2. Linear instability (waveform - 7/9) Normal-to-wall profiles of the instability waves in separation bubbles, LST 2D symmetric hump Step-by-step amplitude profiles of the streamwise disturbance velocity in a separation bubble behind a 2D hump on flat plate: stability analysis at a finite Reynolds number (Nayfeh et al., 1988) notes

  21. 2. Linear instability (waveform - 8/9) Normal-to-wall profiles of the instability waves in separation bubbles, DNS Amplitude distributions of the 2D instability wave propagating through a separation bubble induced by streamwise pressure gradient on a flat plate (Gruber et al., 1987) notes

  22. 2. Linear instability (waveform - 9/9) Normal-to-wall profiles of the instability waves in separation bubbles, DNS vs. LST Amplitude distribution of the 2D instability wave in a separation bubble induced by streamwise pressure gradient on a flat plate (Maucher et al., 1994) LST DNS notes

  23. 2. Linear instability (growth rate - 1/9) Streamwise growth of the small-amplitude disturbances, controlled oscillations Amplification of the instability waves behind a 2D surface inflexion at different initial amplitudes of the oscillations (Dovgal & Kozlov, 1983) Experimental conditions: oncoming-flow velocity – 5.6 m/s, excitation frequency – 311 Hz (left) and 412 Hz (right), maximum disturbance amplitudes u' in the most upstream section (x = 12 mm) as percentage of Uoare: 0.03 (□), 0.08 (○), 0.15 (∆) (left) and 0.02 (□), 0.07 (○), 0.13 (∆) (right) notes

  24. 2. Linear instability (growth rate - 2/9) Streamwise growth of the small-amplitude disturbances, controlled oscillations Linearity of the separated layer disturbances excited at fh/Uo = 0.017 behind a 2D hump on flat plate, maximum disturbance amplitudes u' in the most upstream section (x/h = 6.1) as percentage of Uoare: 0.014 (□), 0.026 (○), 0.047 (∆) and 0.080 (●) (Dovgal & Kozlov, 1990)

  25. 2. Linear instability (growth rate - 3/9) Streamwise growth of the small-amplitude disturbances, controlled oscillations vs. LST Wind-tunnel data on the growth rates of 2D instability waves behind a backward-facing step on flat plate (symbols) (Boiko et al., 1990) as compared to inviscid (solid line) and finite Re-number (dotted line) stability solutions obtained for the experimental mean-velocity profiles (Michalke, 1991) LST, inviscid LST, finite Re experiment notes

  26. 2. Linear instability (growth rate - 4/9) Streamwise growth of the small-amplitude disturbances, controlled oscillations vs. LST a b experiment LST Amplification curves of the instability waves behind 2D steps on a flat plate calculated by Masad & Nayfeh (1993) for the experimental conditions of (Dovgal & Kozlov, 1990): stability solutions (lines) and hot-wire data (symbols) fh/Uo = 0.009 0.014 c d 0.022 0.022 notes

  27. 2. Linear instability (growth rate - 5/9) Streamwise growth of the small-amplitude disturbances, controlled oscillations vs. LST and DNS Amplification of the 2D instability wave propagating through a separation bubble induced by streamwise pressure gradient on a flat plate: experiment (symbols) and calculations (lines) (Häggmark et al., 2001) experiment DNS LST

  28. 2. Linear instability (growth rate - 6/9) Streamwise growth of the small-amplitude disturbances, DNS vs. LST LST wall contour DNS Amplification rate (top) and the wave number (bottom) of a 2D instability wave propagating through a separation bubble behind a smooth backward-facing step on flat plate: DNS (solid lines) and LST (dashed lines) (Bestek et al., 1993) DNS DNS LST

  29. 2. Linear instability (growth rate - 7/9) Streamwise growth of the small-amplitude disturbances, DNS vs. LST DNS Growth rates of 2D (g = 0) and 3D (g > 0) instability waves propagating through a separation bubble induced by streamwise pressure gradient on a flat plate (Rist & Maucher, 1994) LST

  30. 2. Linear instability (growth rate - 8/9) Streamwise growth of the small-amplitude disturbances, DNS vs. LST DNS Stability solutions (symbols) by Theofilis et al. (2000) comparing to DNS results (lines) by Rist & Maucher (1994), see the previous figure LST, non-local (parabolized stability equations) LST, local

  31. 2. Linear instability (growth rate - 9/9) Streamwise growth of the small-amplitude disturbances, controlled oscillations free shear layer Amplification rates of the controlled 2D instability waves determined through wind-tunnel testing of the separation bubbles in different configurations (Dovgal & Kozlov, 1983, 1984, 1990; Boiko et al., 1990) comparing to the instabilities of Blasius boundary layer (Levchenko et al., 1975) and free shear layer (Monkewitz & Huerre, 1982); H = d*/q is the shape factor averaged over the region of the exponential growth, where d * and q are the displacement and momentum thickness flat-plate boundary layer at Red* = 1320 notes

  32. 2. Linear instability (propagation velocity - 1/4) Phase velocity of the 2D waves, controlled oscillations free shear layer Wind tunnel data on the 2D perturbations of separation bubbles behind a 2D hump on flat plate (□) (Boiko & Dovgal, 1992) and on an airfoil (●) (Boiko et al., 1989) as compared to the dispersion curve for free shear layer (Monkewitz & Huerre, 1982) notes

  33. 2. Linear instability (propagation velocity - 2/4) Phase velocity of the 2D waves, LST Inviscid stability solutions for modified tanh-profiles U(y) modeling the basic flow at laminar separation: phase velocity of the instability waves depending on the distance between the separated shear layer and the wall, d/q (Michalke, 1990) notes

  34. 2. Linear instability (propagation velocity - 3/4) Phase velocity of the 2D waves, controlled oscillations vs. LST Wind tunnel data on the 2D perturbations of a separation bubble behind a 2D hump on flat plate (□) (Boiko & Dovgal, 1992) comparing to LST results (lines) from the previous figure, two dispersion curves calculated by Michalke (1990) for the mean velocity profiles most close to the experimental U(y) – distributions are taken for comparison LST LST notes

  35. 2. Linear instability (propagation velocity - 4/4) Streamwise propagation of the plane 3D waves, controlled oscillations Variation of the streamwise wave number of 3D waves (a) normalized by that of the 2D disturbances (a2d) with the wave angle (g) in the separation bubbles behind a 2D hump on flat plate at fh/Uo = 0.019 (○) (Boiko et al., 1991) and on an airfoil at fc/U∞ = 10.4 (solid line) (Gilev et al., 1988)as compared to the dispersion curve in Blasius boundary layer at F = 139 . 10-6 (dashed line) (Kachanov, 1985) notes

  36. 2. Linear instability (effects of the axial symmetry - 1/4) Instability of axisymmetric separation bubbles, controlled oscillations vs. LST Growth rates of the axisymmetric disturbances behind a circular backward-facing step: wind-tunnel data (○) and LST results for the mean-velocity profiles in the upstream (solid line) and downstream (dotted line) sections of the experimental domain where the amplification rates were determined (Dovgal et al., 1995) notes

  37. 2. Linear instability (effects of the axial symmetry - 2/4) Instability of axisymmetric separation bubbles, controlled oscillations Wind-tunnel data on the growth rates of the axisymmetric disturbances behind circular backward-facing steps where H = d*/q is the shape factor averaged over the region of the exponential amplification of the instability waves (Dovgal et al., 1995) notes

  38. 2. Linear instability (effects of the axial symmetry - 3/4) Instability of axisymmetric separation bubbles, LST Effect of the flow curvature on the separated layer instability: amplification rates of the axisymmetric disturbances calculated by Michalke et al. (1995) for modified tanh-profiles U(y) at two Reynolds numbers (solid and dashed lines), the curvature parameter grows as shown by arrows notes

  39. 2. Linear instability (effects of the axial symmetry - 4/4) Instability of separation bubbles in axisymmetric and plane configurations, controlled oscillations Maximum growth rates of the controlled instability waves: plane vs. axisymmetric separation bubbles (Dovgal et al., 1995) notes

  40. 2. Linear instability (separation of 3D boundary layers - 1/3) Instability of separation bubbles in swept configurations, controlled oscillations Wave packets of the harmonic shear-layer disturbances generated upstream of separation on a straight wing at fc/U∞ = 10.4 (left) (Gilev et al., 1984) and on the same model at the 30° sweep angle at fc/U∞ = 9.9 (right) (Dovgal et al., 1988a), contour levels (u') are given as percentage of the oncoming-flow velocity U∞ notes

  41. 2. Linear instability (separation of 3D boundary layers - 2/3) Instability of separation bubbles in swept configurations, controlled oscillations Phase contours of the wave packets on the straight (left) and swept (right) wings from the previous plot

  42. 2. Linear instability (separation of 3D boundary layers - 3/3) Instability of separation bubbles in swept configurations, controlled oscillations 2D: U(y), x = 10 mm, Uo = 8.5 m/s; 3D: Ux(y), x' = 10 mm, Uo = 8.7 m/s. 2D: U(y), x = 30 mm, Uo = 8.5 m/s; 3D: Ux(y), x' = 30 mm, Uo = 8.7 m/s. 2D flow 3D flow Growth rates of the instability waves behind a 2D surface inflection and in the same configuration at the 30° sweep angle (left) measured in close mean flow conditions (right) (Dovgal & Kozlov, 1983; Dovgal et al., 1988b) notes

  43. Linear instability waves - references: Bestek H, Gruber K, Fasel H (1993). In K. Gersten (Ed.), Physics of Separated Flows – Numerical, Experimental, and Theoretical Aspects (Vol. 40, pp. 73–80). Braunschweig: Vieweg. Boiko AV, Dovgal AV (1992). Sib. Fiz.-Techn. Zh., 34(3), 19–24 (In Russian). Boiko AV, Dovgal AV, Kozlov VV (1989). Soviet J. Appl. Phys., 3(2), 46–52. Boiko AV, Dovgal AV, Kozlov VV, Scherbakov VA (1990). Izv. Sib. Otd. Akad. Nauk SSSR, Ser. Tekhn. Nauk, 1, 50–56 (translated in Siberian Phys. Techn. J.). Boiko AV, Dovgal AV, Simonov OA, Scherbakov VA (1991). In V.V. Kozlov and A.V. Dovgal (Eds.), Separated Flows and Jets (pp. 565–572). Berlin Heidelberg New York: Springer. Cousteix J, Pailhas G (1979). Rech. Aérospat., 1979(3), 213–218. Dovgal AV, Kozlov VV (1983). Dokl. Akad. Nauk, 270(6), 1356–1358 (translated in Phys. Dokl.). Dovgal AV, Kozlov VV (1984). Fluid Mech. – Soviet Res., 13(1), 137–143. Dovgal AV, Kozlov VV (1990). In D. Arnal and R. Michel (Eds.), Laminar-Turbulent Transition (pp. 523-531). Berlin Heidelberg New York: Springer. Dovgal AV, Kozlov VV, Michalke A (1995). Eur. J. Mech. B Fluids, 14(3), 351–365. Dovgal AV, Kozlov VV, Simonov OA (1988a). Izv. Sib. Otd. Akad. Nauk SSSR, Ser. Tekhn. Nauk, 3(11), 43–47 (translated in Siberian Phys. Techn. J.). Dovgal AV, Kozlov VV, Simonov OA (1988b). Soviet J. Appl. Phys., 2(4), 18–24. Gaster M (1967). ARC R&M 3595. Gilev VM, Dovgal AV, Kachanov YuS, Kozlov VV (1988). Fluid Dyn., 23(3), 393–399. Gilev VM, Dovgal AV, Kozlov VV (1984). Preprint ITAM 6-84. Gruber K, Bestek H, Fasel H (1987). AIAA Paper 87–1256. Häggmark CP, Bakchinov AA, Alfredsson PH (2000). Philos. Trans. R. Soc. Lond. A, 358, 3193–3205. Häggmark CP, Hildings C, Henningson DS (2001). Aerosp. Sci. Technol., 5(5), 317–328. Kachanov YuS (1985). In V.V. Kozlov (Ed.), Laminar-Turbulent Transition (pp. 115–123). Berlin Heidelberg New York: Springer.

  44. Linear instability waves - references: Levchenko VY, Volodin AG, Gaponov SA (1975). Stability characteristics of boundary layers. Novosibirsk: Nauka. (In Russian.) Masad JA, Nayfeh AH (1993). In D. E. Ashpis, T. B. Gatski and R. Hirsh (Eds.), Instabilities and Turbulence in Engineering Flows (pp. 65–82). Dordrecht: Kluwer. Maucher U, Rist U, Wagner S (1994). In Computational fluid dynamics ’94 (pp. 471–477). New York: Wiley. Michalke A (1990). Z. Flugwiss. Weltraumforsch., 14, 24–31. Michalke A (1991). In V.V. Kozlov and A.V. Dovgal (Eds.), Separated Flows and Jets (pp. 557–564). Berlin Heidelberg New York: Springer. Michalke A, Kozlov VV, Dovgal AV (1995). Eur. J. Mech. B Fluids, 14(3), 333–350. Monkewitz PA, Huerre P (1982). Phys. Fluids, 25(7), 1137–1143. Nayfeh AH, Ragab SA., Al-Maaitah AA (1988). Phys. Fluids, 31(4), 796–806. Rist U, Maucher U (1994). In AGARD-CP–551 (pp. 36.1–36.7). Sinha SN, Gupta AK, Oberai MM (1981). AIAA J., 19(12), 1527–1530. Theofilis V, Hein S, Dallmann U (2000). Philos. Trans. R. Soc. Lond. A, 358, 3229-3246. For other details see: Al-Maaitah AA, Nayfeh AH, Ragab SA (1990). Phys. Fluids A, 2(3), 381–389. Al-Maaitah AA, Nayfeh AH, Ragab SA (1990). AIAA J., 28(11), 1916–1924. Bestek H, Gruber K, Fasel H (1989). In The Prediction and Exploitation of Separated Flow (pp. 14.1–14.16). London: R. Aeronaut. Soc. Hein S. (2000). In H.F. Fasel and W.S. Saric (Eds.), Laminar-Turbulent Transition (pp. 681–686). Berlin Heidelberg New York: Springer. Hein S, Theofilis V, Dallmann U. (1998). DLR-IB 223–98 A 39. Goettingen.

  45. For other details see: Hetsch T, Rist U (2009). European J. Mech. B Fluids, 28, 486–493. Hetsch T, Rist U (2009). European J. Mech. B Fluids, 28, 494–505. Kaltenbach H.-J, Janke G (2000). Phys. Fluids, 12, 2320–2337. Klebanoff PS, Tidstrom KD (1972). Phys. Fluids, 15(7), 1173–1188. Marxen O, Lang M, Rist U, Wagner S. (2003). In Flow, Turbulence, and Combustion (Vol. 71, pp. 133–146). Kluwer. Masad JA, Iyer V (1994). Phys. Fluids, 6(1), 313–327. Masad J.A, Malik MR (1994). AIAA Paper 94–2370. Masad JA, Nayfeh AH (1992). In Fourth Internat. Conf. Fluid Mechanics (Vol. 1, pp. 261–278). Alexandria. Nayfeh AH, Ragab SA, Masad JA (1990). Phys. Fluids A, 2(6), 937–948. Rist U (1994). In S.P. Lin, W.R.C. Phillips and D.T. Valentine (Eds.), Nonlinear Instability of Nonparallel Flows (pp. 324–333). Berlin Heidelberg New York: Springer. Rist U, Maucher U, Wagner S (1996). In Computational Fluid Dynamics ’96 (pp. 319–325). New York: Wiley. Smith FT, Bodonyi RJ (1985). J. R. Aeronaut. Soc., 89, 205–212. Stewart PA, Smith FT (1987). Proc. R. Soc. Lond. A, 409, 229–248. Taghavi H, Wazzan AR (1974). Phys. Fluids, 17(12), 2181–2183. Watmuff JH (1999). J. Fluid Mech., 397, 119–169.

  46. 3. Excitation of the instability waves in separation bubbles (1/5) Two paths of the separated flow receptivity to the external flow perturbations Instability waves can be excited in a separation bubble by the oscillations of the pre-separated boundary layer (top) or/and induced by the external disturbances in the vicinity of separation point (bottom) notes

  47. 3. Excitation of the instability waves in separation bubbles (2/5) Separated flow disturbances coming from the pre-separated boundary layer, controlled oscillations Excitation of the instability wave at laminar flow separation on an airfoil by external sound, fc/U∞ = 19.2: amplitude (○) and phase (●) distributions of the hot-wire signal - a superposition of the acoustic forcing and the generated vortical disturbances, streamwise growth of the instability wave extracted from the total signal (solid line) (Dovgal & Kozlov, 1983) minimum static pressure at x/c = 0.35 – 0.40 notes

  48. 3. Excitation of the instability waves in separation bubbles (3/5) Separated flow disturbances coming from the pre-separated boundary layer, controlled oscillations Amplification curves of the 2D instability waves excited on an airfoil by external sound (open symbols) and by a vibrating ribbon (filled symbols) at fc/U∞ = 14.2 (□, ■), 19.2 (○, ●) and 21.6 (∆, ▲) (Dovgal & Kozlov, 1983) vibratingribbon at x/c = 0.14 minimum static pressure at x/c = 0.35 – 0.40 notes

  49. 3. Excitation of the instability waves in separation bubbles (4/5) Generation of the instability waves close to separation, controlled oscillations Excitation of the instability wave at laminar flow separation behind a 2D step on flat plate by external sound, fh/Uo = 0.034: streamwise amplitude distribution of the hot-wire signal – a superposition of the acoustic forcing and the generated vortical disturbances (Boiko et al., 1990), see also (Dovgal & Kozlov, 1990) step notes

  50. 3. Excitation of the instability waves in separation bubbles (5/5) Generation of the instability waves close to separation, controlled oscillations step Streamwise growth of the instability waves excited at a 2D step on flat plate by external sound (line) and by a vibrating ribbon (□) at fh/Uo = 0.034 (Boiko et al., 1990), see also (Dovgal & Kozlov, 1990) vibrating ribbon at x/h = –90 upper bound of the vortical perturbations before the step at their acoustic excitation notes

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