Global Instabilities in Walljets Gayathri Swaminathan 1 A Sameen 2 & Rama Govindarajan 1 1 JNCASR, Bangalore. 2 IITM, Chennai.
Dec 30, 2015
Global Instabilities in Walljets
Gayathri Swaminathan1
A Sameen2
&
Rama Govindarajan1
1JNCASR, Bangalore.2IITM, Chennai.
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 tenseFuture tense
Study we DID on
transient growth
Study we WILL DO on
transient growth
only in this talk,
M.B.Glauert
1956
Umax
δ(x)
Re=Umax
δ/υ
U ~ x-1/2
δ ~ x3/4
Re ~ x1/4
x
y
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
Wave-like is valid hereWave-like is not good here
Strong Non-parallel effects
Global Stability Analysis ψ(x,y,t) = φ(x,y) e -iωt
Non-parallelism is very high Non-parallelism is less
Wave-like assumption
Very strongly non-parallel analysis
Following relations hold for a walljet:
Umax
= 0.498 (F/xυ) ½
.yumax
= 3.23 (υ 3x3/F) ¼
Locally global stability of walljet
x1 x
n2π/αx
.x/δ = Re/CRe = U
max δ /υυ
Re ~ x1/2
δPeriodic boundary conditions
Less strongly non-parallel analysis
Locally global stability of walljet
Locally global stability of walljet
Re=300
α=0.45
Normal disturbance velocity
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:
Restart
= 200; Reend
= 254; domain length = 63δ; grid
size=121x41.
Chebyshev spectral discretization in both x and y, with suitable
stretching.
Strongly non-parallel analysis
Restart
= 200
Restart
= 200ω = (0.7415444, -0.00158584)
Restart
= 200 ω = (0.7323704, -0.0289041)
Restart
= 200 ω = (0.29521599, -0.03928137)
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 modesTo talk about Transient Growth
Restart
= 40
Mere superposition of few modes. Not the optimal growth!
G
Restart
= 200
Mere superposition of few modes. Not the optimal growth!
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).
?!
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.
Thank You