1. 11th European Turbulence Conference Temporal dynamics of small perturbations for a two-dimensional growing wake 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 Porto, Portugal June 25-28, 2007

2. 2 Physical problem  Flow behind a circular cylinder steady, incompressible and viscous;  Approximation of 2D asymptotic Navier-Stokes expansions (Belan & Tordella, 2003) parametric in x

3. 3 Formulation  Linear, three-dimensional perturbative equations in terms of vorticity: disturbance velocity disturbance vorticity  Moving coordinate transform ξ = x – U 0 t (Criminale & Drazin, 1990), U 0 =U y 

4. 4  Fourier transform in ξ and z directions: α r = k cos( Φ ) wavenumber in x -direction γ = k sin( Φ ) wavenumber in z-direction Φ = tan -1 ( γ/α r ) angle of obliquity k = ( α r 2 + γ 2 ) 1/2 polar wavenumber    k r z x y cylinder axis

5. 5  Initial disturbances periodic and bounded in the free stream: asymmetric or symmetric  Velocity field bounded in the free stream perturbation kinetic energy is finite.

6. 6  Total kinetic energy E and kinetic energy density e of the perturbation Early transient and asymptotic behaviour  The growth function G measures the growth of the perturbation energy at time t.

7. 7  The temporal growth rate r (Lasseigne et al., 1999) is perturbation phase  The angular frequency f (Whitham, 1974) is

8. 8  Asymptotic behaviour the temporal growth rate r asymptotes to a constant value (dr/dt < ε ~ 10 -4 ). Comparison with normal mode theory

9. (a): R=50, y 0 =0, k=0.9, α i =-0.15, Φ=0, x 0 =14, asymmetric initial condition, β 0 =1, 3, 5, 7. Results (b): R=100, y 0 =0, k=1.2, α i =0.1, β 0 =1, x 0 =10.15, symmetric initial condition, Φ=0, π/8, π/4, (3/8)π, π/2.

10. 10 (d): R=100, y 0 =0, α i =0.01, β 0 =1, Φ=π/2, x 0 =7.40, symmetric initial condition, k=0.5, 1, 1.5, 2, 2.5. (c): R=50, y 0 =0, k=0.3, β 0 =1, Φ=0, x 0 =5.20, symmetric initial condition, α i =-0.1, 0, 0.1.

11. β 0 =1, Φ=0, y 0 =0. Present results (triangles: symmetric perturbation, circles: asymmetric perturbation) and normal mode analysis by Tordella, Scarsoglio and Belan, 2006 (solid lines). Complex wavenumber  Real wavenumber 

12. 12 where and (a): R=100, y 0 =0, x 0 =9, k=1.7, α i =0.05, β 0 =1, symmetric initial condition, Φ=π/8. (b): R=100, y 0 =0, x 0 =11, k=0.6, α i =-0.02, β 0 =1, asymmetric initial condition, Φ=π/4. r=0.0826 r=0.0038

13. 13 Conclusions R=100, y 0 =0, x 0 =9, k=1.7, α i =0.05, β 0 =1, symmetric initial condition, Φ=(3/8)π.  Transients can last up to hundred time scales;  Complex wavenumber in x-direction;  Different transient configurations;  Asymptotically good agreement with normal mode theory. r=-0.0168

14. 14 Outline  Physical problem  Formulation  Early transient and asymptotic behavior  Results  Conclusions

15. 15 (a)-(b): R=50, k=1.8, α i =-0.05, β 0 =1, Φ=π/2, x 0 =7, asymmetric initial condition, y 0 =0, 2, 4, 6. (c)-(d): R=100, k=1.2, α i =0.01, β 0 =1, Φ=π/8, x 0 =12, symmetric initial condition, y 0 =0, 2, 4, 6.