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

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

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5. OrrWilliam M’Fadden Arnold Sommerfeld Mathematician 1866 – 1934 Qeen’s Univeristy, Belfast University College, Dublin Physicist 1888 – 1951 University of Gottingen Aachen University University of Munich Teoria dell'instabilità idrodinamica

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8. 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 Teoria dell'instabilità idrodinamica

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13. 0(k0, s0), r0(k0, s0). R = 35, x/D = 4. Teoria dell'instabilità idrodinamica

14. 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) . Teoria dell'instabilità idrodinamica

15. Non-modal theory: the initial-value problem disturbance velocity disturbance vorticity Teoria dell'instabilità idrodinamica

16. Initial and boundary conditions • Initial disturbances are periodic and bounded in the free stream: • asymmetric • or • symmetric • Velocity field bounded in the free stream perturbation kinetic energy is finite. Teoria dell'instabilità idrodinamica

17. 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). Teoria dell'instabilità idrodinamica

18. 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 and Teoria dell'instabilità idrodinamica

19. 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)

20. 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 Teoria dell'instabilità idrodinamica

21. Normal mode theory Base flow is excited with small oscillations. Perturbed system is described by Navier-Stokes model 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, 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) Teoria dell'instabilità idrodinamica

22. 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 Navier-Stokes model. Teoria dell'instabilità idrodinamica

23. 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). Teoria dell'instabilità idrodinamica

24. 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). Teoria dell'instabilità idrodinamica

25. 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 Teoria dell'instabilità idrodinamica

26. Formulation • Moving coordinate transform ξ = x – U0t (Criminale & Drazin, 1990), U0=Uy • Fourier transform in ξand 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 Teoria dell'instabilità idrodinamica

27. 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) Teoria dell'instabilità idrodinamica

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29. 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 Teoria dell'instabilità idrodinamica

30. … 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) Teoria dell'instabilità idrodinamica

31. (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. Teoria dell'instabilità idrodinamica

32. (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. Teoria dell'instabilità idrodinamica

33. (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. Teoria dell'instabilità idrodinamica

34. 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 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 Conclusions Teoria dell'instabilità idrodinamica