1. Global Instabilities in Walljets Gayathri Swaminathan 1 A Sameen 2 & Rama Govindarajan 1 1 JNCASR, Bangalore. 2 IITM, 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 Past tense Future tense Study we DID on transient growth Study we WILL DO on transient growth only in this talk,

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

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.  Re crit ~ 57; α crit ~ 1.18

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

6. Following relations hold for a walljet: U max = 0.498 (F/x υ ) ½.y umax = 3.23 ( υ 3 x 3 /F) ¼ Locally global stability of walljet x1x1 xnxn 2π/α2π/α x.x/ δ = Re/C Re = U max δ / υυ Re ~ x 1/2 δ Periodic boundary conditions Less strongly non-parallel analysis

7. Locally global stability of walljet

8. Re=300 α =0.45 Normal disturbance velocity

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

10. Re start = 200

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

12. Re start = 200 ω = (0.29521599, -0.03928137)

13. 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. Superposition of global modes To talk about Transient Growth

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

15. Re start = 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