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Global Instabilities in Walljets

Global Instabilities in Walljets. Gayathri Swaminathan 1 A Sameen 2 & Rama Govindarajan 1 1 JNCASR, Bangalore. 2 IITM, Chennai. In the context of 'Transient Growth', we define a new non-dimensional number, to be used Ѭ = ___________ = ___________ << 1

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Global Instabilities in Walljets

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  1. Global Instabilities in Walljets Gayathri Swaminathan1 A Sameen2 & Rama Govindarajan1 1JNCASR, Bangalore. 2IITM, Chennai.

  2. In the context of 'Transient Growth', we define a new non-dimensional number, to be used Ѭ= ___________ = ___________ << 1 This talk: 0 < Ѭ < 1e-5 only in this talk, Study we DID on transient growth Past tense Future tense Study we WILL DO on transient growth

  3. y M.B.Glauert 1956 x Re=Umaxδ/υ U ~ x-1/2 δ ~ x3/4 Re ~ x1/4 Umax δ(x)

  4. Related previous work • 1967, Chun et al. studied the linear stability of Glauert's similar profile using Orr-Sommerfeld equation. • 1970, Bajura et al, confirmed the existence of self-similar solutions experimentally. 1975, they reported the 'dominance' of the outer region in the instability mechanism. • 2005, Levin et al defined the developing region of a Blasius walljet with boundary layer approximations (Blasius boundary layer combined with a free shear layer), and studied its stability using the PSE. • Recrit ~ 57; αcrit ~ 1.18

  5. Strong Non-parallel effects • Wave-like assumption • Wave-like is not good here • Wave-like is valid here • Non-parallelism is very high • Non-parallelism is less Very strongly non-parallel analysis • Global Stability Analysis • ψ(x,y,t) = φ(x,y) e -iωt

  6. Locally global stability of walljet Less strongly non-parallel analysis • Following relations hold for a walljet: • Umax = 0.498 (F/xυ) ½ • .yumax = 3.23 (υ 3x3/F) ¼ .x/δ = Re/C Re = Umax δ /υυ Re ~ x1/2 Periodic boundary conditions δ x x1 xn 2π/α

  7. Locally global stability of walljet

  8. Locally global stability of walljet Normal disturbance velocity Re=300 α=0.45

  9. Global stability of walljets Strongly non-parallel analysis • Consider a long domain • Neumann boundary conditions on the derivatives of the velocity perturbations. • Results are presented for the following case: • Restart = 200; Reend = 254; domain length = 63δ; grid size=121x41. • Chebyshev spectral discretization in both x and y, with suitable stretching.

  10. Restart = 200

  11. Restart = 200 ω = (0.7415444, -0.00158584) ω = (0.7323704, -0.0289041) Restart = 200

  12. Restart = 200 ω = (0.29521599, -0.03928137)

  13. Superposition of global modes To talk about Transient Growth • 1997 Chomaz, 'a suitable superposition of the non-normal global modes produces a wave-packet, which initially grows in time and moves in space. • 2005 Ehrenstein et al, in a flat plate boundary layer, convective instability is captured by superposition of global modes. • 2007 Henningson et al, separated boundary layer, sum of global modes gives a localized disturbance.

  14. Restart = 40 Mere superposition of few modes. Not the optimal growth! G

  15. Restart = 200 Mere superposition of few modes. Not the optimal growth!

  16. Local and global stability of walljet • Study on the Glauert’s similarity profile does not reveal a region of absolute instability, YET. (Not surprising ). • Global stability will be performed on the 3D mean flow.(DNS under development). ? !

  17. Future Work • Understand the effect of non-parallelism by studying the global modes. • Study the stability of the developing region of the wall jet using global analysis • To study the transient growth dynamics from global modes.

  18. Thank You

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