The spatial stability problem

1. The spatial stability problem André V. G. Cavalieri Peter Jordan (Visiting researcher, Ciência Sem Fronteiras)

2. In spatially developping flows, spatial stability seems a more appropriate description (M. Gaster 1962, 1965) Temporal x spatial stability

3. In spatially developping flows, spatial stability seems a more appropriate description (M. Gaster 1962, 1965) Temporal x spatial stability

4. In spatially developping flows, spatial stability seems a more appropriate description (M. Gaster 1962, 1965) Temporal x spatial stability Becker & Massaro JFM 1968

5. In spatially developping flows, spatial stability seems a more appropriate description (M. Gaster 1962, 1965) Temporal x spatial stability Brown & Roshko JFM 1974

6. In spatially developping flows, spatial stability seems a more appropriate description (M. Gaster 1962, 1965) Temporal x spatial stability Nishioka et al. JFM 1975

7. Equations for spatial stability The derivation did not specify temporal or spatial stability! Now: ω (real-valued) (=αc) is a parameter α (complex-valued) is the eigenvalue

8. Equations for spatial stability Multiply by α to obtain Now: ω (real-valued) (=αc) is a parameter α (complex-valued) is the eigenvalue The eigenvalue α appears nonlinearly!

9. Equations for spatial stability Now: ω (real-valued) (=αc) is a parameter α (complex-valued) is the eigenvalue The eigenvalue α appears nonlinearly! Solution: rewrite problem as: Exercise #6: obtain F0, F1, F2, F3, F4. Exercise #7: spatial stability of tanh profile. Compare with Michalke (1965)

10. Equations for spatial stability Exercise #6: obtain F0, F1, F2, F3, F4.

11. Spatial stability of a mixing layer Exercise #7: spatial stability of tanh profile. Compare with Michalke (1965) Note: Michalke (1965) solves the Rayleigh equation, we are solving O-S. The Kelvin-Helmholtz mode should be similar for high Re Growth rate of Kelvin-Helmholtz mode Results for Reθ=1000

12. Spatial stability of a mixing layer Exercise #7: spatial stability of tanh profile. Compare with Michalke (1965) Reθ=100

13. Spatial stability of a mixing layer Exercise #7: spatial stability of tanh profile. Compare with Michalke (1965) K-H αi = -0,1031 αr = 0,2254 Phase speed: Uc = ω/αr = 0,488

14. Spatial stability of a mixing layer Exercise #7: spatial stability of tanh profile. Compare with Michalke (1965) What about these modes? Are these all unstable?

15. Spatial stability of a mixing layer Exercise #7: spatial stability of tanh profile. Compare with Michalke (1965) ??? Besides Kelvin-Helmholtz, eigenspectrum has a continuous branch K-H

16. Spatial stability of a mixing layer Exercise #7: spatial stability of tanh profile. Compare with Michalke (1965) Besides Kelvin-Helmholtz, eigenspectrum has a continuous branch

17. Spatial stability of a mixing layer Let’s take a look at the eigenfunctions Kelvin-Helmholtz instability driven by the sheared, inflectional velocity profile

18. Spatial stability of a mixing layer Let’s take a look at the eigenfunctions A continuum of eigenfunctions “living” in the uniform flow

19. The continuous spectrum Grosch & Salwen JFM 1978 Fourier transform: eigenvalue problem (wave equation) A) Bounded domain: Solution: An infinite number of discrete eigenvalues and eigenfunctions (harmonics of a guitar string) B) Unbounded domain: Solution: A continuum ofeigenvalues and eigenfunctions (infinite guitar string, harmonics approach each other and become a continuum)

20. The continuous spectrum Orr-Sommerfeld equation, temporal stability Uniform flow: Tedious but straightforward algebra... Waves convected by the uniform flow and slowly damped by viscosity Spatial stability: similar results, slightly harder algebra

21. Spatial stability of a mixing layer Exercise #7: spatial stability of tanh profile. Compare with Michalke (1965) Continuous spectrum K-H

22. Spatial stability of a mixing layer Exercise #7: spatial stability of tanh profile. Compare with Michalke (1965) 1 Actually, four branches of continuous spectra: 1 – vorticity waves travelling downstream 2 – vorticity waves travelling upstream, strongly damped 3 – evanescent pressure waves travelling downstream 4 – evanescent pressure waves travelling upstream 3 4 2

23. Spatial stability of a mixing layer Exercise #7: spatial stability of tanh profile. Compare with Michalke (1965) Actually, four branches of continuous spectra: 3 – evanescent pressure waves travelling downstream 4 – evanescent pressure waves travelling upstream 3 and 4 depend on domain size

24. Spatial stability of a mixing layer Exercise #7: spatial stability of tanh profile. Compare with Michalke (1965) Actually, four branches of continuous spectra: 3 – evanescent pressure waves travelling downstream 4 – evanescent pressure waves travelling upstream 3 and 4 depend on domain size

25. Spatial stability of a mixing layer Exercise #7: spatial stability of tanh profile. Compare with Michalke (1965) 3 – evanescent pressure waves travelling downstream 4 – evanescent pressure waves travelling upstream Evanescent waves: exponential decay with x; behaviour as duct modes (see Aeroacoustics course) Need of larger domains for continuous representation of pressure waves

26. Eigenspectra of bounded flows

27. Eigenspectra of unbounded flows

28. Phase velocity Temporal stability: Spatial stability: Velocity for a constant phase, i.e. we are following the movement of a given phase of the wave (e.g. a wave crest)

29. Group velocity Waves should start and end somewhere. WAVEPACKET zero energy after this point zero energy before this point Group velocity: we follow the movement of the envelope Group velocity measures the speed at which energy is carried from a point to another

30. Calculation of group velocity A simple example

31. Calculation of group velocity A simple example Present notation:

32. Spatial stability of a mixing layer Waves with spatial growth or decay: Briggs’ criterion

33. Spatial stability of a mixing layer Waves with spatial growth or decay: Briggs’ criterion

34. Spatial stability of a mixing layer Waves with spatial growth or decay: Briggs’ criterion

35. Spatial stability of a mixing layer Waves with spatial growth or decay: Briggs’ criterion

36. Spatial stability of a mixing layer Waves with spatial growth or decay: Briggs’ criterion

37. Spatial stability of a mixing layer Waves with spatial growth or decay: Briggs’ criterion

38. Spatial stability of a mixing layer Waves with spatial growth or decay: Briggs’ criterion

39. Spatial stability of a mixing layer Waves with spatial growth or decay: Briggs’ criterion α+ modes (positive generalised group velocity) α- modes (negative generalised group velocity)

40. Spatial stability of a mixing layer Waves with spatial growth or decay: Briggs’ criterion

41. Spatial stability of a mixing layer Waves with spatial growth or decay: Briggs’ criterion

42. Spatial stability of a mixing layer Waves with spatial growth or decay: Briggs’ criterion

43. Spatial stability of a mixing layer Waves with spatial growth or decay: Briggs’ criterion

44. Spatial stability of a mixing layer Waves with spatial growth or decay: Briggs’ criterion K-H is an α+ mode, and grows in the direction of generalised group velocity All other modes decay in the direction of generalised group velocity!

45. Spatial stability of a mixing layer Exercise #8: We have seen methods for both temporal and spatial stability analysis. Which should be used to determine neutral curves and the critical Reynolds number?