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
1. Fundamentals (problem formulation - 1/6)
Separation bubbles are found in a variety
of flow configurations
a
b
c
d
notes
1. Fundamentals (problem formulation - 2/6)
Transitional separation bubbles
(Ward, 1963)
(Brendel & Mueller, 1988)
notes
1. Fundamentals (problem formulation - 3/6)
Separation bubbles as:
noise amplifiers
(convective instability,
local stability analysis )
shear-layer
transition
oscillators
(self-sustained dynamics,
absolute instability
and global modes)
e.g. large-scale vortices
notes
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.
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
1. Fundamentals (problem formulation - 6/6)
Aspects of laminar-turbulent transition in separation regions
notes
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
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 oncomingflow velocity U∞
(Gilev et al., 1984)
notes
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
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)
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
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.
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
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
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
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)
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)
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
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
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
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
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 Uo are: 0.03 (□),
0.08 (○), 0.15 (∆) (left) and 0.02 (□),
0.07 (○), 0.13 (∆) (right)
notes
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 Uo are:
0.014 (□), 0.026 (○), 0.047 (∆)
and 0.080 (●)
(Dovgal & Kozlov, 1990)
2. Linear instability (growth rate - 3/9)
Streamwise growth of the small-amplitude disturbances,
controlled oscillations vs. LST
LST, inviscid
LST, finite Re
experiment
notes
Wind-tunnel data on the
growth rates of 2D instability
waves behind a backwardfacing 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)
2. Linear instability (growth rate - 4/9)
Streamwise growth of the small-amplitude disturbances,
controlled oscillations vs. LST
a
b
experiment
LST
fh/Uo = 0.009
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)
0.014
c
d
0.022
0.022
notes
2. Linear instability (growth rate - 5/9)
Streamwise growth of the small-amplitude disturbances,
controlled oscillations vs. LST and DNS
experiment
DNS
LST
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)
2. Linear instability (growth rate - 6/9)
Streamwise growth of the small-amplitude disturbances,
DNS vs. LST
LST
wall contour
DNS
DNS
DNS
LST
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)
2. Linear instability (growth rate - 7/9)
Streamwise growth of the small-amplitude disturbances,
DNS vs. LST
DNS
LST
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)
2. Linear instability (growth rate - 8/9)
Streamwise growth of the small-amplitude disturbances,
DNS vs. LST
DNS
LST, non-local
(parabolized stability
LST, local
equations)
Stability solutions (symbols)
by Theofilis et al. (2000)
comparing to DNS results
(lines) by Rist & Maucher
(1994), see the previous
figure
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
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
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
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
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
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
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
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
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
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
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
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
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.
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.
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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 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
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
3. Excitation of the instability waves in separation bubbles (3/5)
Separated flow disturbances coming from the pre-separated
boundary layer, controlled oscillations
vibrating ribbon
at x/c = 0.14
minimum static pressure
at x/c = 0.35 – 0.40
notes
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)
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
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
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.
For other details see:
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 perturbations
of a separation bubble
on airfoil under
periodic forcing
at fc/U∞ = 8.6 (left)
and 14.7 (right),
Rec = 270 000
(Boiko et al., 1989)
notes
4. Nonlinear phenomena (subharmonic disturbances - 2/5)
Subharmonic generation, controlled oscillations
fundamental wave
subharmonic,
at resonance
subharmonic,
linear instability
notes
Resonant interaction
of the 2D fundamental (f)
and 2D subharmonic (f/2)
disturbances both excited
in a controlled manner
at flow separation on an airfoil,
fc/U∞ = 14.7, Rec = 270 000
(Boiko et al., 1989)
4. Nonlinear phenomena (subharmonic disturbances - 3/5)
Subharmonic generation, controlled oscillations
Spanwise amplitude (○, ●) and phase
(∆) distributions of the controlled 2D
subharmonic disturbance in the resonance
region on airfoil, fc/U∞ = 14.7, x/c = 0.693,
Rec = 270 000 (Boiko et al., 1989)
at resonance
at resonance
linear instability
notes
4. Nonlinear phenomena (subharmonic disturbances - 4/5)
Subharmonic generation, controlled oscillations
fundamental
wave
subharmonic,
at resonance
subharmonic,
linear instability
notes
Resonant interaction of the 2D
fundamental (f) and oblique
subharmonic (f/2) disturbances
behind a 2D hump on flat plate
at fh/Uo = 0.032: g = 0° (plane
wave) (a), 20° (b)
and 37° (c) where g
is the subharmonic
wave angle
(Boiko et al., 1991)
4. Nonlinear phenomena (subharmonic disturbances - 5/5)
Subharmonic instability in calculations
hump
pairing mode
Blasius
boundary
layer
notes
Subharmonic growth
rate over the spanwise
wave-number spectrum
behind a 2D hump
on flat plate:
3D perturbations
at increasing the hump
height as shown
by arrow (left) and
2D oscillations (right)
(Nayfeh & Ragab,1987)
4. Nonlinear phenomena (coherent vortices - 1/3)
Effect of the initial spectrum on the perturbed flow pattern, controlled
oscillations
Spectra of velocity perturbations in the aft part
of separation bubble on an airfoil (at x/c = 0.729)
for different levels of external periodic forcing
where u'exc is the amplitude of excited instability
wave 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
4. Nonlinear phenomena (coherent vortices - 2/3)
Effect of the initial spectrum on the perturbed flow pattern, controlled
oscillations
Spectra and oscilloscope traces
of velocity perturbations
in a separation bubble on airfoil
under 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
4. Nonlinear phenomena (coherent vortices - 3/3)
Suppression of the background disturbances
by controlled oscillations
u'/Uo = 0.22%
excited
natural
natural
excited
notes
Velocity fluctuations
in the upstream part
of separation bubble
behind a 2D hump on flat
plate under the excitation
of instability wave
at fh/Uo = 0.023 with
the local maximum
amplitude u'/Uo = 0.22%:
normal-to-wall distributions
of the low-frequency random
perturbations (a) and spectral
data (b) at x/h = 5
(Boiko et al., 1991)
4. Nonlinear phenomena (some other interactions - 1/3)
Wave combinations, controlled disturbances
Multiplication of spectral components
in a separation bubble on airfoil
at the interaction of two instability waves
fc/U∞ = 13.4 and 16.8 with
the amplitudes in the upstream section
x/c = 0.643 as high as 1.87 and 1.60
as percentage of U∞
notes
4. Nonlinear phenomena (some other interactions - 2/3)
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
4. Nonlinear phenomena (some other interactions - 3/3)
Oblique breakdown, controlled disturbances
Amplitude contours of velocity
perturbations generated behind
a 2D hump on flat plate
at the excitation of a pair
of oblique waves at fh/Uo = 0.036
and the wave angles ± 45°:
z – t planes at x/h = 6 (top)
and 31 (bottom)
(Ablaev et al., 1998)
notes
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.
For other details see:
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.