instability of laminar separating flows (separation bubbles) contents : 1. fundamentals

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 bubbles4. Nonlinear phenomena

● Subharmonic resonance● Coherent vortices● Some other interactions

Multimedia files - 6/13

1. Fundamentals (problem formulation - 1/6)

Separation bubbles are found in a varietyof flow configurations

a b

c d

notes

notes

Transitional separation bubbles

(Ward, 1963)

(Brendel & Mueller, 1988)


oscillators(self-sustained dynamics,absolute instabilityand global modes)

noise amplifiers(convective instability,local stability analysis )

shear-layertransition

e.g. large-scale vortices

Separation bubbles as:

notes



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.

Natural velocity perturbations

notes

Laminar flow disturbances excitedby environmental noise in a separation

bubble on airfoil, Rec = 270 000(Boiko et al., 1989)


notes

Aspects of laminar-turbulent transition in separation regions


1. Fundamentals (global response - 1/5)

Noteworthy is that excitation of instability waves modifies the entire separated flow pattern

notes

Mean-velocity profiles of the basic flow (1) and of the separation bubble perturbed by small-amplitude periodic oscillations (2)


Global response of a transitional separation bubble on airfoil

notes

Maximum mean-flow variation (U) 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

Global response of a laminar separation bubblebehind a 2D backward-facing step on flat plate

Maximum amplitudes of 2Dinstability waves excited

at fh/Uo = 0.029 (circles)and 0.034 (triangles) behinda backward-facing step(Boiko et al., 1991):in this case, the globalresponse is found at theamplitude of oscillationsin the region of reattachment as high as about 1% of the

local external-flow velocity Uo

(open symbols)


Global response of a laminar separation bubblebehind a 2D backward-facing step on flat plate

Maximum mean-flow variation(U) in the upstream partof separation bubble at x/h = 4.5 (top)and x/h = 13.6 (bottom)vs. maximum amplitude (u')of the controlled instability wavesin downstream section x/h = 45,

excitation frequency fh/Uo

is 0.017 (□), 0.029(○) and 0.034 (∆)(Boiko et al., 1991)


Effect of the instability waves on the mean-velocity profile

notes

Separated-flow profiles measuredat 10 mm behind a 2D surface inflexionin natural conditions (●) and undercontrolled excitation of the instabilitywave (□) (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

For basic features of transitional separation bubbles see:

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.

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.

2. Linear instability (waveform - 1/9)

Normal-to-wall profiles of the instability wavesin separation bubbles, natural disturbances

notes

Amplitude distributions u'(y)of the laminar flow perturbationsamplifying at boundary-layerseparation on an airfoil(Cousteix & Pailhas, 1979),notice the measurement sections upstreamof reattachment markedin blue


Normal-to-wall profiles of the instability wavesin separation bubbles, natural disturbances

notes

Amplitude distributionsu'(y) of the laminar flowperturbations amplifyingat boundary-layerseparation behinda 2D backward-facing step(Sinha et al., 1981)


Normal-to-wall profiles of the instability wavesin separation bubbles, controlled disturbances

notes

Among the first observationsof controlled separated-flowperturbations were thoseby Gaster (1967) for boundarylayer separation inducedon a flat plate by streamwisepressure gradient


Laminar flow oscillations generatedupstream of a 2D hump on flat platetransform to the instability wavesof the separation bubble and thenturn back to those of the laminarreattached boundary layer:amplitude (top) and phase (bottom)of the perturbations excited

at fh/Uo = 0.017(Dovgal & Kozlov, 1990)




Laminar flow oscillations generatedin a separation bubble inducedby streamwise pressure gradient on a flat plate: amplitudes vs. mean-flow profiles (left) and phase (right) of the perturbations(Häggmark et al., 2000)


notes

Normal-to-wall profiles of the instability wavesin separation bubbles, LST

Inviscid stability solutionfor a modified tanh-profileU(y) modelingthe base flow at laminarseparation: amplitudeof the streamwise (u')and normal (v') velocities(Michalke, 1990)

u'

v'


notes

Normal-to-wall profiles of the instability wavesin separation bubbles, LST

Step-by-step amplitude profiles of the streamwise disturbance velocityin a separation bubble behind a 2D hump on flat plate: stability analysisat a finite Reynolds number (Nayfeh et al., 1988)

2D symmetric hump


Normal-to-wall profiles of the instability wavesin separation bubbles, DNS

Amplitude distributions of the 2Dinstability wave propagatingthrough a separation bubbleinduced by streamwise pressuregradient on a flat plate(Gruber et al., 1987)

notes


Normal-to-wall profiles of the instability wavesin separation bubbles, DNS vs. LST

Amplitude distributionof the 2D instabilitywave in a separation bubble induced by streamwise pressure gradient on a flat plate(Maucher et al., 1994)

notes

DNS

LST

2. Linear instability (growth rate - 1/9)

Streamwise growth of the small-amplitude disturbances,controlled oscillations

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 Uo are: 0.03 (□), 0.08 (○), 0.15 (∆) (left) and 0.02 (□), 0.07 (○), 0.13 (∆) (right)

notes

Amplification of the instability wavesbehind a 2D surface inflexionat different initial amplitudesof the oscillations (Dovgal & Kozlov, 1983)



Linearity of the separatedlayer disturbances excited

at fh/Uo = 0.017 behinda 2D hump on flat plate,maximum disturbanceamplitudes u' in the mostupstream section (x/h = 6.1)

as percentage of Uo are:0.014 (□), 0.026 (○), 0.047 (∆)and 0.080 (●)(Dovgal & Kozlov, 1990)


Streamwise growth of the small-amplitude disturbances,controlled oscillations vs. LST

notes

Wind-tunnel data on thegrowth rates of 2D instabilitywaves behind a backward-facing step on flat plate (symbols) (Boiko et al., 1990) as compared to inviscid (solid line) and finiteRe-number (dotted line)stability solutions obtainedfor the experimentalmean-velocity profiles(Michalke, 1991)

LST, inviscid

LST, finite Reexperiment


Streamwise growth of the small-amplitude disturbances,controlled oscillations vs. LST

notes

Amplification curves of theinstability waves behind2D steps on a flat plate calculatedby Masad & Nayfeh (1993)for the experimental conditionsof (Dovgal & Kozlov, 1990):stability solutions (lines)and hot-wire data (symbols)

a b

c d

fh/Uo = 0.009 0.014

0.022 0.022

experiment

LST


Streamwise growth of the small-amplitude disturbances,controlled oscillations vs. LST and DNS

Amplification of the2D instability wavepropagating througha separation bubbleinduced by streamwisepressure gradienton a flat plate:experiment (symbols)and calculations (lines)(Häggmark et al., 2001)

experiment

DNS

LST


Streamwise growth of the small-amplitude disturbances,DNS vs. LST

Amplification rate (top)and the wave number (bottom) of a 2D instability wave propagating througha separation bubble behind a smoothbackward-facing stepon flat plate:DNS (solid lines) and LST(dashed lines)(Bestek et al., 1993)

DNS

DNS

LST

DNS

LST

wall contour



Growth rates of 2D( = 0) and 3D ( > 0)instability wavespropagating througha separation bubbleinduced by streamwisepressure gradienton a flat plate(Rist & Maucher, 1994)

DNS

LST



Stability solutions (symbols)by Theofilis et al. (2000) comparing to DNS results(lines) by Rist & Maucher (1994), see the previousfigure

DNS

LST, local

LST, non-local(parabolized stabilityequations)


notes

Amplification rates of the controlled 2Dinstability waves determined throughwind-tunnel testing of the separationbubbles in different configurations(Dovgal & Kozlov, 1983, 1984, 1990;Boiko et al., 1990) comparing to theinstabilities of Blasius boundary layer(Levchenko et al., 1975) and free shearlayer (Monkewitz & Huerre, 1982);H = */ is the shape factor averagedover the region of the exponentialgrowth, where * and are thedisplacement and momentum thickness


free shear layer

flat-plate boundary layer

at Re* = 1320

2. Linear instability (propagation velocity - 1/4)

Phase velocity of the 2D waves, controlled oscillations

Wind tunnel data on the 2Dperturbations of separationbubbles behind a 2D humpon flat plate (□)(Boiko & Dovgal, 1992)and on an airfoil (●)(Boiko et al., 1989)as compared to the dispersioncurve for free shear layer(Monkewitz & Huerre, 1982)

notes

free shear layer


Phase velocity of the 2D waves, LST

notes

Inviscid stability solutionsfor modified tanh-profilesU(y) modeling the basicflow at laminar separation:phase velocity of theinstability waves dependingon the distance betweenthe separated shear layerand the wall, d/ (Michalke, 1990)


Phase velocity of the 2D waves, controlled oscillations vs. LST

Wind tunnel data on the 2D perturbations of a separation bubble behind a 2D humpon flat plate (□) (Boiko & Dovgal, 1992)comparing to LST results (lines) from

LST

LST

notes

the previous figure,two dispersioncurves calculated by Michalke (1990)for the mean velocity profiles most closeto the experimental U(y) – distributions are taken for comparison


Streamwise propagation of the plane 3D waves,controlled oscillations

Variation of the streamwise wavenumber of 3D waves () normalizedby that of the 2D disturbances (2d)with the wave angle () in theseparation 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 comparedto the dispersion curve in Blasius

boundary layer at F = 139 . 10-6

(dashed line) (Kachanov, 1985)

notes

2. Linear instability (effects of the axial symmetry - 1/4)

Instability of axisymmetric separation bubbles,controlled oscillations vs. LST

notes

Growth rates of the axisymmetricdisturbances 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

Instability of axisymmetric separation bubbles,controlled oscillations

Wind-tunnel data on the growthrates of the axisymmetricdisturbances behind circularbackward-facing steps whereH = */ is the shape factoraveraged over the regionof the exponential amplificationof the instability waves(Dovgal et al., 1995)


notes

Instability of axisymmetric separation bubbles, LST

Effect of the flow curvatureon the separated layerinstability: amplificationrates of the axisymmetricdisturbances calculatedby Michalke et al. (1995)for modified tanh-profilesU(y) at two Reynolds numbers (solid and dashed lines), the curvature parameter grows as shown by arrows


notes

Instability of separation bubbles in axisymmetric and plane configurations, controlled oscillations

Maximum growth ratesof the controlledinstability waves:plane vs. axisymmetricseparation bubbles(Dovgal et al., 1995)

2. Linear instability (separation of 3D boundary layers - 1/3)

Instability of separation bubbles in swept configurations,controlled oscillations

notes

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∞



Phase contours of the wave packets on the straight (left) and swept (right) wings from the previous plot


notes


Growth rates of the instability waves behind a 2D surface inflection and in thesame configuration at the 30° sweep angle (left) measured in close mean flowconditions (right) (Dovgal & Kozlov, 1983; Dovgal et al., 1988b)

2D flow3D flow

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.

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.

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.

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.


3. Excitation of the instability waves in separation bubbles (1/5)

Two paths of the separated flow receptivityto the external flow perturbations

notes

Instability waves can beexcited in a separationbubble by the oscillationsof the pre-separatedboundary layer (top)or/and induced by theexternal disturbancesin the vicinity of separationpoint (bottom)


Separated flow disturbances coming from the pre-separated boundary layer, controlled oscillations

notes

Excitation of the instabilitywave at laminar flowseparation on an airfoil

by external sound, fc/U∞ = 19.2:

amplitude (○) and phase (●)distributions of the hot-wiresignal - a superpositionof the acoustic forcingand the generated vorticaldisturbances, streamwisegrowth of the instability waveextracted from the totalsignal (solid line)(Dovgal & Kozlov, 1983)

minimum static pressureat x/c = 0.35 – 0.40


Separated flow disturbances coming from the pre-separated boundary layer, controlled oscillations

notes

Amplification curvesof the 2D instabilitywaves excitedon an airfoil by externalsound (open symbols)and by a vibratingribbon (filled symbols)

at fc/U∞ = 14.2 (□, ■),

19.2 (○, ●) and 21.6 (∆, ▲)(Dovgal & Kozlov, 1983)

minimum static pressureat x/c = 0.35 – 0.40

vibrating ribbonat x/c = 0.14


Generation of the instability waves close to separation,controlled oscillations

notes

Excitation of the instabilitywave at laminar flowseparation behind a 2D stepon flat plate by external sound,

fh/Uo = 0.034: streamwiseamplitude distributionof the hot-wire signal –a superposition of theacoustic forcing andthe generated vorticaldisturbances(Boiko et al., 1990), see also(Dovgal & Kozlov, 1990)

step


Generation of the instability waves close to separation,controlled oscillations

notes

Streamwise growth of the instabilitywaves excited at a 2D step on flatplate by external sound (line)and by a vibrating ribbon (□)

at fh/Uo = 0.034 (Boiko et al., 1990),see also (Dovgal & Kozlov, 1990)

upper bound of the vortical perturbations before the stepat their acoustic excitation

vibrating ribbonat x/h = –90

step

Excitation of the instability waves - references:


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.).Dovgal AV, Kozlov VV (1983). Fluid Dyn., 18(2), 205–209.Dovgal AV, Kozlov VV (1990). In D. Arnal and R. Michel (Eds.), Laminar-Turbulent Transition (pp. 523–531). Berlin Heidelberg New York: Springer.

Asai M, Kaneko M (1998). In Proc. Third Internat. Conf. Fluid Mechanics (pp. 231–237). Beijing: Beijing Institute of Technology.Bodonyi RJ, Welch WJC, Duck PW, Tadjfar M (1989). J. Fluid Mech., 209, 285–308.Dovgal AV, Kozlov VV, Michalke A (1996). European J. Mech. B Fluids, 15(4), 651–664.Goldstein ME (1984). J. Fluid Mech., 145, 71–94.Goldstein ME, Leib SJ, Cowley SJ (1987). J. Fluid Mech., 181, 485–517.Michalke A (1993). European J. Mech. B Fluids, 12(4), 421–445.Michalke A (1995). European J. Mech. B Fluids, 14(4), 373–393.Michalke A (1997). European J. Mech. B Fluids, 16(1), 17–37.Michalke A, Al-Maaitah AA (1992). European J. Mech. B Fluids, 11(5), 521–542.Ruban AI (1985). Fluid Dyn., 19(5), 709–716.

4. Nonlinear phenomena (subharmonic disturbances - 1/5)

Subharmonic generation, controlled oscillations

Velocity perturbationsof a separation bubbleon airfoil under periodic forcing

at fc/U∞ = 8.6 (left)

and 14.7 (right),

Rec = 270 000(Boiko et al., 1989)

notes


notes

Resonant interactionof the 2D fundamental (f)and 2D subharmonic (f/2)disturbances both excitedin a controlled mannerat flow separation on an airfoil,

fc/U∞ = 14.7, Rec = 270 000

(Boiko et al., 1989)


fundamental wave

subharmonic,linear instability

subharmonic,at resonance


notes


Spanwise amplitude (○, ●) and phase(∆) distributions of the controlled 2Dsubharmonic disturbance in the resonance

region on airfoil, fc/U∞ = 14.7, x/c = 0.693,

Rec = 270 000 (Boiko et al., 1989)

linear instability

at resonance

at resonance


notes


Resonant interaction of the 2Dfundamental (f) and obliquesubharmonic (f/2) disturbancesbehind a 2D hump on flat plate

at fh/Uo = 0.032: = 0° (plane wave) (a), 20° (b) and 37° (c) where is the subharmonic wave angle (Boiko et al., 1991)

fundamentalwave

subharmonic,at resonance

subharmonic,linear instability


Subharmonic instability in calculations

notes

Subharmonic growthrate over the spanwisewave-number spectrumbehind a 2D humpon flat plate:3D perturbationsat increasing the humpheight as shownby arrow (left) and2D oscillations (right)(Nayfeh & Ragab,1987)

Blasiusboundarylayer

pairing mode

hump

4. Nonlinear phenomena (coherent vortices - 1/3)

notes

Effect of the initial spectrum on the perturbed flow pattern, controlled oscillations

Spectra of velocity perturbations in the aft partof separation bubble on an airfoil (at x/c = 0.729)for different levels of external periodic forcingwhere u'exc is the amplitude of excited instabilitywave measured close to the point of separation

(at x/c = 0.571) as percentage of U∞,

fc/U∞ = 13.8, Rec = 270 000 (Boiko et al., 1989)


notes

Effect of the initial spectrum on the perturbed flow pattern, controlled oscillations

Spectra and oscilloscope tracesof velocity perturbationsin a separation bubble on airfoilunder the excitation of instability

wave with the amplitude u'/U∞as high as 2.54% at x/c = 0.614,

fc/U∞ = 10.4, Rec = 270 000

(Boiko et al., 1989)


notes

Velocity fluctuationsin the upstream partof separation bubblebehind a 2D hump on flatplate under the excitationof instability wave

at fh/Uo = 0.023 withthe local maximum

amplitude u'/Uo = 0.22%:normal-to-wall distributionsof the low-frequency randomperturbations (a) and spectraldata (b) at x/h = 5(Boiko et al., 1991)

Suppression of the background disturbancesby controlled oscillations

u'/Uo = 0.22%

natural

excited

natural

excited

4. Nonlinear phenomena (some other interactions - 1/3)

notes

Wave combinations, controlled disturbances

Multiplication of spectral componentsin a separation bubble on airfoilat the interaction of two instability waves

fc/U∞ = 13.4 and 16.8 with

the amplitudes in the upstream sectionx/c = 0.643 as high as 1.87 and 1.60

as percentage of U∞


notes

Oblique breakdown, DNS

Instantaneous z-component of vorticity at the wall during oblique breakdown in a separation bubble induced by streamwise pressure gradient on a flat plate (Rist et al., 1996)


notes

Oblique breakdown, controlled disturbances

Amplitude contours of velocity perturbations generated behinda 2D hump on flat plateat the excitation of a pair

of oblique waves at fh/Uo = 0.036and the wave angles ± 45°:z – t planes at x/h = 6 (top)and 31 (bottom)(Ablaev et al., 1998)

Nonlinear phenomena - references:


Ablaev AR, Grek GR, Dovgal AV, Katasonov MM, Kozlov VV (1998). Preprint ITAM 7-98.Boiko AV, Dovgal AV, Kozlov VV (1989). Soviet J. Appl. Phys., 3(2), 46–52.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.Nayfeh AH, Ragab SA (1987). AIAA Paper 87–0045.Rist U, Maucher U, Wagner S (1996). In Computational Fluid Dynamics ’96 (pp. 319–325). New York: Wiley.

Dovgal AV, Boiko AV (2000). In H.F. Fasel and W.S. Saric (Eds.), Laminar-Turbulent Transition (pp. 675–680). Berlin Heidelberg New York: Springer.Masad JA, Nayfeh AH (1992). AIAA J., 30(7), 1731–1737.Maucher U, Rist U, Wagner S (2000). In H.F. Fasel and W.S. Saric (Eds.), Laminar-Turbulent Transition (pp. 657–662). Berlin Heidelberg New York: Springer.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 (1994). In AGARD-CP–551 (pp. 36.1–36.7).Smith FT (1987). In D.L. Dwoyer and M.Y. Hussaini (Eds.), Stability of Time Dependent and Spatially Varying Flows (pp. 104–147). Berlin Heidelberg New York: Springer.

instability of laminar separating flows (separation bubbles) contents : 1. fundamentals

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fundamentals global

separation regions1

separation bubbles4

flow variationdu

basic flow

notesglobal response

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