Teoria dell’instabilita’ idrodinamica di Orr-Sommerfeld a cent’anni dalla sua prima...

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13 June 2007 1 Teoria dell'instabilità idrod inamica Teoria dell’instabilita’ Teoria dell’instabilita’ idrodinamica di Orr-Sommerfeld a idrodinamica di Orr-Sommerfeld a cent’anni dalla sua prima cent’anni dalla sua prima formulazione formulazione Daniela Tordella Daniela Tordella Politecnico di Torino Politecnico di Torino Accademia delle Scienze, 13 Accademia delle Scienze, 13 Giugno 2007 Giugno 2007

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Teoria dell’instabilita’ idrodinamica di Orr-Sommerfeld a cent’anni dalla sua prima formulazione. Accademia delle Scienze, 13 Giugno 2007. Daniela Tordella Politecnico di Torino. Orr William M’Fadden. Arnold Sommerfeld. Mathematician 1866 – 1934 Qeen’s Univeristy, Belfast - PowerPoint PPT Presentation

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13 June 2007 1Teoria dell'instabilità idrodinamica

Teoria dell’instabilita’ Teoria dell’instabilita’ idrodinamica di Orr-Sommerfeld a idrodinamica di Orr-Sommerfeld a

cent’anni dalla sua primacent’anni dalla sua primaformulazioneformulazione

Daniela TordellaDaniela TordellaPolitecnico di TorinoPolitecnico di Torino

Accademia delle Scienze, Accademia delle Scienze, 13 Giugno 200713 Giugno 2007

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Orr William M’Fadden

Mathematician1866 – 1934Qeen’s Univeristy, BelfastUniversity College, Dublin

Arnold SommerfeldPhysicist

1888 – 1951University of GottingenAachen UniversityUniversity of Munich

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Flusso baseFlusso base

Dinamica non lineareDinamica non lineare

Bassi numeri di ReynoldsBassi numeri di Reynolds

Fluido reale in tutto il dominio --- Fluido reale in tutto il dominio --- no no decadimento esponenzialedecadimento esponenziale

Raccordo tra flussi interni ed esterni: Raccordo tra flussi interni ed esterni: vorticita’, gradiente di pressione, velocita’ di vorticita’, gradiente di pressione, velocita’ di entrainmententrainment

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00(k(k00, s, s00), r), r00(k(k00, s, s00). R = 35, x/D ). R = 35, x/D = 4. = 4.

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Frequency. Comparison between present solution (accuracy Frequency. Comparison between present solution (accuracy ΔωΔω = 0.05), Zebib's numerical study (1987), Pier’s direct = 0.05), Zebib's numerical study (1987), Pier’s direct numerical simulations (2002), Williamson's experimental numerical simulations (2002), Williamson's experimental results (1988) . results (1988) .

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Non-modal theory: Non-modal theory: the initial-value problemthe initial-value problem

disturbance disturbance velocityvelocitydisturbance disturbance vorticityvorticity

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Initial and boundary conditionsInitial and boundary conditions Initial disturbances are periodic and bounded in Initial disturbances are periodic and bounded in the free stream:the free stream:

asymmetricasymmetric oror symmetricsymmetricVelocity field bounded in the free stream Velocity field bounded in the free stream perturbation kinetic perturbation kinetic energy is finite.energy is finite.

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ββ00=1, =1, ΦΦ=0, y=0, y00=0. Present results (triangles: symmetric =0. Present results (triangles: symmetric perturbation, circles: asymmetric perturbation) and normal mode perturbation, circles: asymmetric perturbation) and normal mode analysis by Tordella, Scarsoglio and Belan, 2006 (solid lines). analysis by Tordella, Scarsoglio and Belan, 2006 (solid lines). αα==ααrr(x(x00) + i) + iααii(x(x00) (where ) (where ααrr=k) is the most unstable wavenumber in =k) is the most unstable wavenumber in any section of the near-parallel wake (dominant saddle point in the any section of the near-parallel wake (dominant saddle point in the local dispersion relation).local dispersion relation).

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(a)-(b): R=100, y(a)-(b): R=100, y00=0, x=0, x00=9, k=1.7, =9, k=1.7, ααii =-0.05, =-0.05, ββ00=1, =1, symmetric initial condition, (a) symmetric initial condition, (a) ΦΦ==ππ/8, (b) /8, (b) ΦΦ=(3/8)=(3/8)ππ. (c): . (c): R=100, yR=100, y00=0, x=0, x00=11, k=0.6, =11, k=0.6, ααii=0.02, =0.02, ββ00=1, asymmetric =1, asymmetric initial condition, initial condition, ΦΦ==ππ/4./4.

where where andand

r=0.0826 r=-0.0168

r=0.0038

utente
case 3, ic: v=0.5*exp(-y^2)*sin(y)
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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 - Analysisof 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)

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Belan, M; Tordella, D Convective instability in wake intermediate asymptotics JOURNAL OF FLUID MECHANICS, 552 : 127-136 APR 10 2006.

Tordella, D; Belan, MOn 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, MA 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, DAsymptotic expansions for two dimensional symmetrical Laminar wakes ZAMM, 82 (4): 219-234 2002

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Normal mode theoryNormal mode theoryBase flow is excited with small oscillations.Base flow is excited with small oscillations.

Perturbed system is described by Navier-Stokes modelPerturbed system is described by Navier-Stokes model

The linearized perturbative equation in term of stream The linearized perturbative equation in term of stream function is function is

Normal mode hypothesisNormal mode hypothesis Perturbation is considered as sum of normal modes, which Perturbation is considered as sum of normal modes, which can be treated separately since the system is linear.can be treated separately since the system is linear. complex eigenfunction, complex eigenfunction,

u*(x,y,t) = U(x,y) + u*(x,y,t) = U(x,y) + u(x,y,t) v*(x,y,t) = u(x,y,t) v*(x,y,t) = V(x,y) + v(x,y,t) V(x,y) + v(x,y,t) p*(x,y,t) = Pp*(x,y,t) = P00 + + p(x,y,t)p(x,y,t)

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Physical problemPhysical problemSteady, incompressible and viscous base flow described by Steady, incompressible and viscous base flow described by continuity and Navier-Stokes equations with dimensionless continuity and Navier-Stokes equations with dimensionless quantitiesquantities U(x,y), V(x,y), P(x,y) U(x,y), V(x,y), P(x,y) and and costcost

Boundary Boundary conditions:conditions:symmetry to symmetry to xx, , uniformity at uniformity at infinity and field infinity and field information in information in the intermediate the intermediate wake. wake. The physical The physical domain is domain is divided into two divided into two regions both regions both described by described by Navier-Stokes Navier-Stokes modelmodel..

R =R =UUccD/D/

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Inner flowInner flow -> ->

Outer flowOuter flow -> -> Physical quantities involved in matching criteria are the Physical quantities involved in matching criteria are the pressure longitudinal gradientpressure longitudinal gradient, the , the vorticityvorticity and and transverse velocitytransverse velocity. The composite expansion, f. The composite expansion, fcncn = f = finin + + ffonon – (f – (fonon))inin, is continuous and differentiable over the whole , is continuous and differentiable over the whole domain (Belan & Tordella, 2002; Tordella & Belan, 2003).domain (Belan & Tordella, 2002; Tordella & Belan, 2003).

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R = 60

Normal mode analysisNormal mode analysisBase flow: inner expansion (both longitudinal and Base flow: inner expansion (both longitudinal and transversal velocity components) up to the third order.transversal velocity components) up to the third order.Initial-value problemInitial-value problemBase flow: inner expansion (only the longitudinal velocity Base flow: inner expansion (only the longitudinal velocity component) up to the second order (component) up to the second order (xx parameter). parameter).

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Non-modal theory: Non-modal theory: the initial-value problemthe initial-value problem

Linear, three-dimensional perturbative equations Linear, three-dimensional perturbative equations (non dimensional (non dimensional quantities with respect to the base flow and quantities with respect to the base flow and spatial scales);spatial scales); Steady, incompressible and viscous base flow;Steady, incompressible and viscous base flow; Base flowBase flow:: 2D asymptotic Navier-Stokes expansion 2D asymptotic Navier-Stokes expansion (Belan & (Belan & Tordella, 2003) parametric in Tordella, 2003) parametric in xx

disturbance disturbance velocityvelocitydisturbance disturbance vorticityvorticity

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FormulationFormulation Moving coordinate transform Moving coordinate transform ξξ = x – U = x – U00t t (Criminale & Drazin, (Criminale & Drazin, 1990), U1990), U00=U=Uyy

Fourier transform in Fourier transform in ξξ andand z z directions: directions:

ααrr = k cos( = k cos(ΦΦ)) wavenumber in wavenumber in ξξ-direction -direction γγ = k sin( = k sin(ΦΦ)) wavenumber in z-direction wavenumber in z-direction

ΦΦ = tan = tan-1-1((γγ//ααrr)) angle of obliquity angle of obliquity k = (k = (ααrr22 + + γγ22))1/21/2 polar polar

wavenumberwavenumber

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Order zero theoryOrder zero theory Homogeneous Orr-Sommerfeld Homogeneous Orr-Sommerfeld equation (parametric in x)equation (parametric in x)

eigenfunctions and a discrete set of eigenvalues eigenfunctions and a discrete set of eigenvalues 0n0n First order theoryFirst order theory Non homogeneous Orr-Sommerfeld Non homogeneous Orr-Sommerfeld equation (x parameter)equation (x parameter)

Stability analysis through multiscale Stability analysis through multiscale approachapproach

Slow spatial and temporal evolution of the system: xSlow spatial and temporal evolution of the system: x11 = = x, x, tt1 1 = = x, x, = 1/R. = 1/R.

Hypothesis: and are expansions in term Hypothesis: and are expansions in term of of ..

By substituting in the linearized perturbative equation, one By substituting in the linearized perturbative equation, one hashas

(ODE dependent on ) + (ODE dependent on ) + (ODE dependent on , ) (ODE dependent on , ) + + O O ((22))

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Transient dynamics …Transient dynamics … Total kinetic energy Total kinetic energy EE and the kinetic energy density and the kinetic energy density ee of the perturbation of the perturbation are defined (Blossey et al., submitted 2006) as:are defined (Blossey et al., submitted 2006) as:

The growth function The growth function GG defined in terms of the defined in terms of the normalized energy densitynormalized energy density

can effectively measure the growth of the energy at time can effectively measure the growth of the energy at time tt, for a given , for a given initial condition at initial condition at tt = 0. = 0. For asymptotically stable cases:For asymptotically stable cases:

• if G>1 for some time t>0 algebraically if G>1 for some time t>0 algebraically unstable flowunstable flow• if G=1 for all time algebraically if G=1 for all time algebraically neutral flowneutral flow• if G<1 for all time algebraically if G<1 for all time algebraically stable flowstable flow

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… … and asymptotic behavior of the and asymptotic behavior of the perturbationsperturbations

Considering that the amplitude of the disturbance is Considering that the amplitude of the disturbance is proportional to ,proportional to , the temporal growth rate the temporal growth rate rr can be defined (Lasseigne et can be defined (Lasseigne et al., 1999) asal., 1999) as

Computations to evaluate the long time asymptotics are Computations to evaluate the long time asymptotics are made by made by integrating the equations forward in time beyond the integrating the equations forward in time beyond the transient until the transient until the growth rate growth rate rr asymptotes to a constant value (for asymptotes to a constant value (for example example dr/dtdr/dt < < εε ~ 10 ~ 10-4-4).).

The angular frequency The angular frequency ff can be defined by taking the can be defined by taking the phase of the phase of the complex wave at a fixed transversal station and then complex wave at a fixed transversal station and then considering its time considering its time derivative (Whitham, 1974)derivative (Whitham, 1974)

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(a)-(b): (a)-(b): R=100, R=100, yy00=0, k=1.2, =0, k=1.2, ααii=-0.1, =-0.1, ββ00=1, =1, xx00=10.15, =10.15, symmetric symmetric initial initial condition, condition, ΦΦ=0, =0, ππ/8, /8, ππ/4, /4, (3/8)(3/8)ππ, , ππ/2. /2.

(c)-(d): (c)-(d): R=50, yR=50, y00=0, =0, k=0.9, k=0.9, ααii=0.15, =0.15, ΦΦ=0, x=0, x00=14, =14, asymmetric asymmetric initial initial condition, condition, ββ00=1, 3, 5, 7.=1, 3, 5, 7.

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(a)-(b): (a)-(b): R=100, yR=100, y00=0, =0, ααi i =-0.01, =-0.01, ββ00=1, =1, ΦΦ==ππ/2, /2, xx00=7.40, =7.40, symmetric symmetric initial initial condition, condition, k=0.5, 1, 1.5, k=0.5, 1, 1.5, 2, 2.5.2, 2.5.

(c)-(d): R=50, (c)-(d): R=50, yy00=0, k=0.3, =0, k=0.3, ββ00=1, =1, ΦΦ=0, =0, xx00=5.20, =5.20, symmetric symmetric initial initial condition, condition, ααii =-0.1, 0, =-0.1, 0, 0.1.0.1.

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(a)-(b): R=50, (a)-(b): R=50, k=1.8, k=1.8, ααi i

=0.05, =0.05, ββ00=1, =1, ΦΦ==ππ/2, x/2, x00=7, =7, asymmetric asymmetric initial initial condition, condition, yy00=0, 2, 4, 6.=0, 2, 4, 6.

(c)-(d): (c)-(d): R=100, R=100, k=1.2, k=1.2, ααii =- =-0.01, 0.01, ββ00=1, =1, ΦΦ==ππ/8, x/8, x00=12, =12, symmetric symmetric initial initial condition, condition, yy00=0, 2, 4, 6.=0, 2, 4, 6.

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ConclusionsConclusionsNormal mode theoryNormal mode theory Accurate analytical description Accurate analytical description

of the base flow;of the base flow; Non-parallel effects, multiple Non-parallel effects, multiple

spatial and temporal scales;spatial and temporal scales; Synthetic perturbative Synthetic perturbative

hypothesis (saddle point hypothesis (saddle point sequence); sequence);

Good agreement with numerical Good agreement with numerical and experimental results; and experimental results;

Ordinary differential equations;Ordinary differential equations;• No information on the early time No information on the early time

history of the perturbation; history of the perturbation; • Two-dimensional disturbancesTwo-dimensional disturbances

Initial-value Initial-value problemproblem

Early transient and asymptotic Early transient and asymptotic behavior of the disturbance; behavior of the disturbance;

Three-dimensional (symmetrical Three-dimensional (symmetrical and asymmetrical) arbitrary and asymmetrical) arbitrary initial disturbances imposed for initial disturbances imposed for different configurations; different configurations;

Good agreement with normal Good agreement with normal mode theory; mode theory;

• SSimplified description of the implified description of the spatial evolution of the system spatial evolution of the system (base flow parametric in x); (base flow parametric in x);

• Partial differential equations in Partial differential equations in time and spacetime and space