Accademia delle Scienze, 13 Giugno 2007
Teoria dell’instabilita’ idrodinamica di OrrSommerfeld a cent’anni dalla sua prima
formulazione
Daniela Tordella
Politecnico di Torino
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Orr William M’Fadden
Arnold Sommerfeld
Mathematician
Physicist
1866 – 1934
1888 – 1951
Qeen’s Univeristy, Belfast
University of Gottingen
University College, Dublin
Aachen University
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University of Munich 5
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 Flusso base
 Dinamica non lineare
 Bassi numeri di Reynolds
 Fluido reale in tutto il dominio --- no decadimento
esponenziale
 Raccordo tra flussi interni ed esterni: vorticita’, gradiente di
pressione, velocita’ di entrainment
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0(k0, s0), r0(k0, s0).
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R = 35, x/D = 4.
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Frequency. Comparison between present solution (accuracy Δω = 0.05),
Zebib's numerical study (1987), Pier’s direct numerical simulations (2002),
Williamson's experimental results (1988) .
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Non-modal theory:
the initial-value problem
disturbance velocity
disturbance vorticity
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Initial and boundary conditions
Initial disturbances are periodic and bounded in the free stream:
asymmetric
or
symmetric
Velocity field bounded in the free stream
energy is finite.
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perturbation kinetic
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β0=1, Φ=0, y0=0. Present results (triangles: symmetric perturbation, circles: asymmetric
perturbation) and normal mode analysis by Tordella, Scarsoglio and Belan, 2006 (solid
lines). α=αr(x0) + iαi(x0) (where αr=k) is the most unstable wavenumber in any section of
the near-parallel wake (dominant saddle point in the local dispersion relation).
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r=0.0826
r=-0.0168
r=0.0038
(a)-(b): R=100, y0=0, x0=9, k=1.7, αi =-0.05, β0=1, symmetric initial condition,
(a) Φ=π/8, (b) Φ=(3/8)π. (c): R=100, y0=0, x0=11, k=0.6, αi=0.02, β0=1,
asymmetric initial condition, Φ=π/4.
where
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and
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Publications
 A synthetic perturbative hypothesis for the multiscale analysis of the
convective wake instability - D. Tordella, S. Scarsoglio and M. Belan - Phys.
Fluids, Vol. 18, No. 5 - (2006)
 22nd IFIP TC 7 Conference on System Modeling and Optimization - Analysis
of the convective instability of the two-dimensional wake (S. Scarsoglio, D.
Tordella, M. Belan) - 18/22 luglio 2005 – Torino
 6th Euromech Fluid Mechanics Conference
(EFMC6) - A synthetic
perturbative hypothesis for multiscale analysis of bluff-body wake instability
(D.Tordella, S. Scarsoglio, M. Belan) - June 26-30, 2006 - Stockholm, Sweden
 59th Annual Meeting Division of Fluid Dynamics (APS-DFD) - Initial-value
problem for the two-dimensional growing wake (S. Scarsoglio, D.Tordella and
W. O. Criminale) – November 19-21, 2006 - Tampa, Florida
 11th Advanced European Turbulence Conference - Temporal dynamics of
small perturbations for a two-dimensional growing wake (S. Scarsoglio,
D.Tordella and W. O. Criminale) - June 25-28, 2007 - Porto, Portugal
(submitted)
 Belan, M; Tordella, D
Convective instability in wake intermediate asymptotics
JOURNAL OF FLUID MECHANICS, 552 : 127-136 APR 10 2006.
 Tordella, D; Belan, M
On the domain of validity of the near-parallel combined stability analysis for
the 2D intermediate and far bluff body wake ZAMM, 85 (1): 51-65 JAN 2005
 Tordella, D; Belan, M
A new matched asymptotic expansion for the intermediate and far flow behind
a finite body PHYSICS OF FLUIDS, 15 (7): 1897-1906 JUL 2003
 Belan, M; Tordella, D
Asymptotic expansions for two dimensional symmetrical Laminar wakes
ZAMM, 82 (4): 219-234 2002
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Normal mode theory
Base flow is excited with small oscillations.
Perturbed system is described by Navier-Stokes model
u*(x,y,t) = U(x,y) + u(x,y,t)
v*(x,y,t) = V(x,y) + v(x,y,t)
p*(x,y,t) = P0 + p(x,y,t)
The linearized perturbative equation in term of stream function
is
Normal mode hypothesis
Perturbation is considered as sum of normal modes, which can be treated
separately since the system is linear.
complex eigenfunction,
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Physical problem
Steady, incompressible and viscous base flow described by continuity and
Navier-Stokes equations with dimensionless quantities U(x,y), V(x,y), P(x,y) and
 cost
R =UcD/
Boundary conditions:
symmetry
to
x,
uniformity at infinity
and field information
in the intermediate
wake.
The physical domain
is divided into two
regions
both
described by NavierStokes model.
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Inner flow ->
Outer flow ->
Physical quantities involved in matching criteria are the pressure longitudinal
gradient, the vorticity and transverse velocity. The composite expansion,
fcn = fin + fon – (fon)in, is continuous and differentiable over the whole domain
(Belan & Tordella, 2002; Tordella & Belan, 2003).
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R = 60
Normal mode analysis
Base flow: inner expansion (both longitudinal and transversal velocity
components) up to the third order.
Initial-value problem
Base flow: inner expansion (only the longitudinal velocity component) up to
the second order (x parameter).
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Non-modal theory:
the initial-value problem
 Linear, three-dimensional perturbative equations (non dimensional
quantities with respect to the base flow and spatial scales);
 Steady, incompressible and viscous base flow;
 Base flow: 2D asymptotic Navier-Stokes expansion (Belan &
Tordella, 2003) parametric in x
disturbance velocity
disturbance vorticity
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Formulation
 Moving coordinate transform ξ = x – U0t (Criminale & Drazin, 1990), U0=Uy
 Fourier transform in ξ and z directions:
αr = k cos(Φ) wavenumber in ξ-direction
γ = k sin(Φ) wavenumber in z-direction
Φ = tan-1(γ/αr) angle of obliquity
k = (αr2 + γ2)1/2 polar wavenumber
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Stability analysis through multiscale approach
Slow spatial and temporal evolution of the system: x1 = x, t1 = x,  = 1/R.
Hypothesis:
and
are expansions in term of .
By substituting in the linearized perturbative equation, one has
(ODE dependent on
) +  (ODE dependent on
,
) + O (2)
Order zero theory Homogeneous Orr-Sommerfeld equation (parametric in x)
eigenfunctions
and a discrete set of eigenvalues 0n
First order theory Non homogeneous Orr-Sommerfeld equation (x parameter)
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Transient dynamics …
 Total kinetic energy E and the kinetic energy density e of the perturbation
are defined (Blossey et al., submitted 2006) as:
 The growth function G defined in terms of the normalized energy density
can effectively measure the growth of the energy at time t, for a given
initial condition at t = 0.
For asymptotically stable cases:
• if G>1 for some time t>0
algebraically unstable flow
• if G=1 for all time
algebraically neutral flow
• if G<1 for all time
algebraically stable flow
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… and asymptotic behavior of the perturbations
 Considering that the amplitude of the disturbance is proportional to
the temporal growth rate r can be defined (Lasseigne et al., 1999) as
,
 Computations to evaluate the long time asymptotics are made by
integrating the equations forward in time beyond the transient until the
growth rate r asymptotes to a constant value (for example dr/dt < ε ~ 10-4).
 The angular frequency f can be defined by taking the phase of the
complex wave at a fixed transversal station and then considering its time
derivative (Whitham, 1974)
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(a)-(b): R=100,
y0=0, k=1.2,
αi=-0.1, β0=1,
x0=10.15,
symmetric initial
condition,
Φ=0, π/8, π/4,
(3/8)π, π/2.
(c)-(d): R=50,
y0=0, k=0.9,
αi=0.15, Φ=0,
x0=14,
asymmetric
initial condition,
β0=1, 3, 5, 7.
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(a)-(b): R=100,
y0=0, αi =-0.01,
β0=1, Φ=π/2,
x0=7.40,
symmetric initial
condition,
k=0.5, 1, 1.5, 2,
2.5.
(c)-(d): R=50,
y0=0, k=0.3, β0=1,
Φ=0, x0=5.20,
symmetric initial
condition,
αi =-0.1, 0, 0.1.
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(a)-(b): R=50,
k=1.8, αi =0.05,
β0=1, Φ=π/2,
x0=7, asymmetric
initial condition,
y0=0, 2, 4, 6.
(c)-(d): R=100,
k=1.2, αi =-0.01,
β0=1, Φ=π/8,
x0=12, symmetric
initial condition,
y0=0, 2, 4, 6.
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Conclusions
Normal mode theory





•
•
Accurate analytical description of
the base flow;
Non-parallel effects, multiple
spatial and temporal scales;
Synthetic perturbative hypothesis
(saddle point sequence);
Good agreement with numerical
and experimental results;
Ordinary differential equations;
No information on the early time
history of the perturbation;
Two-dimensional disturbances
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Initial-value problem



•
•
Early transient and asymptotic
behavior of the disturbance;
Three-dimensional (symmetrical
and asymmetrical) arbitrary
initial disturbances imposed for
different configurations;
Good agreement with normal
mode theory;
Simplified description of the spatial
evolution of the system (base flow
parametric in x);
Partial differential equations in time
and space
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