Linear stability analysis

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60th Annual Meeting Division of Fluid Dynamics
A multiscale approach to study the stability
of long waves in near-parallel flows S.
Scarsoglio, D.Tordella and W. O. Criminale
Dipartimento di Ingegneria Aeronautica e
Spaziale, Politecnico di Torino, Torino,
Italy Department of Applied Mathematics,
University of Washington, Seattle,
Washington, Usa EFMC7, Manchester, September
14-18, 2008
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Outline

• Physical problem

• Initial-value problem

• Multiscale analysis for the stability of long
  waves

• Conclusions

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Physical Problem

• Flow behind a circular cylinder steady,
  incompressible and viscous

• Approximation of 2D asymptotic Navier-Stokes
  expansions (Belan Tordella, 2003), 20Re100.
  

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Initial-value problem

• Linear, three-dimensional perturbative equations
  in terms of vorticity and velocity (Criminale
  Drazin, 1990)

• Base flow parametric in x and Re U(y
  x0, Re)

• Laplace-Fourier transform in x and z directions
  for perturbation quantities

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ar k cos(F) wavenumber in x-direction ?
k sin(F) wavenumber in z-direction F
tan-1(?/ar ) angle of obliquity k
(ar2 ?2)1/2 polar wavenumber
ai 0
spatial damping rate
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• Periodic initial conditions for

symmetric
asymmetric
and

• Velocity field vanishing in the free stream.

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Early transient and asymptotic behaviour of
perturbations

• The growth function G is the normalized kinetic
  energy density
• and measures the growth of the perturbation
  energy at time t.

• The temporal growth rate r (Lasseigne et al.,
  1999) and the angular frequency ? (Whitham, 1974)

perturbation phase
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Exploratory analysis of the transient dynamics
(a) R100, y00, x06.50, ai0.01, n01,
asymmetric, F3/8p, k0.5,1,1.5,2,2.5.
(b) R50, y00, x07, k0.5, F0, asymmetric,
n01, ai 0,0.01,0.05,0.1.
(c) Wave spatial evolution in the x direction
for kar0.5, ai 0,0.01,0.05,0.1.
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(d) R100, y00, x011.50, k0.7, asymmetric,
ai0.02, n01, F0, p/2.
(e) R100 x012, k1.2, ai0.01, symmetric,
n01, F p/2, y00,2,4,6.
(f) R50 x014, k0.9, ai0.15, asymmetric,
y00, F p/2, n01,3,5,7.
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..
(a)-(b)-(c)-(d) R100, y00, k0.6, ai0.02,
n01, Fp/4, x011 and 50, symmetric and
asymmetric.
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gtgt
(a) R100, y00, x09, k1.7, ai 0.05, n01,
symmetric, Fp/8.
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Asymptotic fate and comparison with modal analysis

• Asymptotic state the temporal growth rate r
  asymptotes to a constant value (dr/dt lt e 10-4).

(a)-(b) Re50, ai0.05, ?0, x011, n01, y00.
Initial-value problem (triangles symmetric,
circles asymmetric), normal mode analysis (black
curves), experimental data (Williamson 1989, red
symbols).
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Multiscale analysis for the stability of long
waves

• Different scales in the stability analysis
• Slow scales (base flow evolution)
• Fast scales (disturbance dynamics)

• In some flow configurations, long waves can be
  destabilizing (for example Blasius boundary layer
  and 3D cross flow boundary layer)

• In such instances the perturbation wavenumber of
  the unstable wave is much less than O(1).

Small parameter is the polar wavenumber of the
perturbation kltlt1
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Full linear system
base flow (U(x,yRe), V(x,yRe))
Multiple scales hypothesis

• Regular perturbation scheme, kltlt1

• Temporal scales

• Spatial scales

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Order O(1)
Order O(k)
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Comparison with the full linear problem
(a)-(b) Re100, k0.01, ?p/4, x010, n01,
y00. Full linear problem (black circles
symmetric, black triangles asymmetric),
multiscale O(1) (red circles symmetric, red
triangles asymmetric).
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(a) R50, y00, k0.03, n01, x012, Fp/4,
asymmetric, ai0.04, 0.4.
-gt lt-
(b) R100, y00, n01, x027, F0, symmetric,
ai0.2, k0.1, 0.01, 0.001.
(c) R100, y00, k0.02, x013.50, n01, Fp/2,
ai0.08, sym and asym.
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(a)-(b) R50, y00, k0.04, n01, x012, Fp/2,
asymmetric, ai0.005, 0.01, 0.05 (multiscale
O(1)), ai0 (full problem).
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Conclusions

• Synthetic perturbation hypothesis (saddle point
  sequence)

• Absolute instability pockets (Re50,100) found in
  the intermediate wake

• Good agreement, in terms of frequency, with
  numerical and experimental data

• No information on the early time history of the
  perturbation

• Different transient growths of energy

• Asymptotic good agreement with modal analysis and
  with experimental data (in terms of frequency and
  wavelength)

• Multiscaling O(1) for long waves well
  approximates full linear problem.

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where
and