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