Harish Dixit and Rama Govindarajan With Anubhab Roy and Ganesh SubramanianJawaharlal Nehru Centre for Advanced Scientific Research, BangaloreSeptember 2008

Instabilities in variable-property flows and the continuous spectrum

An aggressive ‘passive’ scalar

Re=3000, unstratified

Building block for inverse cascade

`Perpendicular’ density stratification: baroclinic torque

(+ centrifugal + other non-Boussinesq effects)

Heavy

Light1 2

ρ(y)

y

ρ

Brandt and Nomura, JFM (2007): stratification upto Fr=2, Boussinesq

Stratification aids merger at Re > 2000

At lower diffusivities, larger stratifications?

Re=3000, Pe=30000, Fr (pair) = 1

Large scale overturning: a separate story

Why does the breakdown happen?

Consider one vortex in a (sharp) density gradient

In 2D, no gravityDt

uD

Dt

D

1

A

rl

N

rUFr d

2

//

dl

Pe

2

A

1

1dl

Heavy

Light

point vortex

Initial condition: Point vortex at a density jump

Homogenised within the yellow patch, if Pe finite

A single vortex and a density interfaceInviscid

The locus seen is not a streamline!

Scaling tUr

t

rrr nn

3

1Density is homogenised for

tld

dh lPer 3/1most at is

ds lPer 2/1

e.g. Rhines and Young (1983)Flohr and Vassilicos (1997) (different from Moore & Saffman 1975)

When Pe >>> 1, many density jumps between rh and rs

Consider one such jump, assume circular

22

m

rji

2j

r r

m

Linearly unstable when heavy inside light, Rayleigh-Taylor

Vortex sheet of strength

Rotates at m times angular velocity of mean flow

Point vortex, circular density jump

Ar

gm

ji

)](exp[ˆ tmiuu rr

Radial gravity Non-Boussinesq, centrifugal

riuu 2

Non-Boussinesq: e.g. Turner, 1957, Sipp et al., Joly et al., JFM 2005

m = 2

Vortex sheet at rj

)(2 4cr rOiuu

In unstratified case: a continuous spectrum of `non-Kelvin’ modes

Rankine vortex with density jumps at rjs spaced at r3

r

2

20

j

c

r

r0r

u

cr

1r

12

0

0

Kelvin (1880): neutral modes at r=a for a Rankine vortex

r

r

urmrr

Drmgrm

dr

dumrDmrDDrm

)2'(1

)(

113)(

2222

2222

Vorticity and density: Heaviside functions

).....](),(),[( 321 rrrrrrdr

ρd

)(23 0 crrrD

For j jumps: 2j+2 boundary conditions

ur and pressure continuous at jumps and rc

Green’s function, integrating across jumpsFor non-Boussinesq case:

22

02

21242

0

122

02

3

21

220

2

2

)(

)()(

cjc

cjm

jcj

rmrrm

rmrArrmrA

11

1c1

31

2

c1

1

:

r r :

r :

r r r

r rArA

r rA

um

mm

m

r

For one density jump

m = 5

Multiple (7) jumps

rj = 2 rc, =0.1

Single jumpStep vs smoothdensity change

Single jump: radial gravity (blue), non-Boussinesq (red)

m = 2, = 0.01

(circular jump: pressure balances, but) Lituus spiral

t

r

2tan

2

Dt

uD

Dt

D

1

)(2

3

jrrt

r

t

Dominant effect, small non-dimensional)

)log(/ tAUU

KH instability at positive and negative jump

growing faster than exponentially

In the basic flow

ttu ˆ

Simulations: spectral, interfaces thin tanh, up to 15362 periodic b.c.

Heavy

Light

Non-Boussinesq equations

)( 6 zgupuut

u

6

ut

0 u

t=0

3.18

6.4

t=1.59

Boussinesq, g=0, density is a passive scalar

9.5

t=12.7

time=0

time=12.7

Vorticity

1.6

3.82

Non-BoussinesqA=0.2

t=4.5

t=5.1

5.73

t=3.2Notice vorticity contours

t=4.5

5.1

5.73

A=0.12, t = 7.5Г/rc2

λ ~ 2.5ld (λstab ~ 4ld)

Viscous simulations: same instability

Re = 8000, Pe = 80000, rho1 = 0.9, rho2 = 1.1 (tanh interface), Circulation=0.8, thickness of the interface = 0.02, rc = 0.1, time = 2.5, N=1024 points

Initial condition: Gaussian vortex at a tanh interface

Conclusions:

Co-existing instabilities: `forward cascade’unstable wins

Beware of Boussinesq, even at small A

What does this do to 2D turbulence?

Single jump: Boussinesq (blue), non-Boussinesq (red)

m = 20, = 0.1

Variation of ur eigenfunction with the jump location: rc = 0.1, m = 2

Effect of large density differences

m = 2, = 1

Reynolds number: Inertial / Viscous forces

For inviscid flow, no diffusion of density, Re, Pe infinite

2D simulations of Harish: Boussinesq approximation

2

Re

11 g

dt

d

21

Pedt

d

20Nb

Fr

Re

DPe

dy

dgN

2

Peclet number: Inertial / Diffusive

Froude number: Inertial / Buoyancy (1/Fr = TI N)

2, ,

20

00

bt

bUb v

Is the flow unstable?

rjr

r

r

urr

imgmf

dr

du

tD

D

gumrDmrDDrmf

ˆ )( ˆ)(

0ˆ

ˆˆ])}1(3){[(

2

222

Consider radially outward gravity

)exp()(ˆ Taking ftimruu rr

20,1 1 rrc

r

m

21r

mr

i

20,1 1 rrc

m

2,1 mrc

i 1r

mi

1r

r

2,1 mrc

1r

r

2,101 mrr c

cr

Comparison: Boussinesq (blue), non-Boussinesq (red)

m = 2, = 0.1

00

12

g

r

pu

u

t

u rr

p

rru

u

t

ur

0

12

01

u

rr

u

r

u rr

0

dr

du

t r

Governing stability PDE’s:

Component equations

gr

p

r

uu

r

u

r

uu

t

u rrr

r

0

2 1

p

rr

uuu

r

u

r

uu

t

u rr

0

1

01

u

rr

u

r

u rr

Continuity equations

Density evolution equations

0

r

u

ru

t r

Background literature:

Studying discontinuities of vorticity / densities or any passive scalar was initiated by Saffman who studies a random distribution of vortices as a model for 2D turbulence and predicted a k-4 spectrum

Bassom and Gilbert (JFM, 1988) studied spiral structures of vorticity and predicted that the spectrum lies between k-3 and k-4

Pullin, Buntine and Saffman (Phys. Fluid, 1994) verify the Lundgren’s model of turbulence based on vorticity spiral

Batchelor (JFM, 1956) argued that at very large Reynolds number, the vorticity field inside closed streamlines evolves towards a constant value.

Rhines and Young (JFM, 1983) showed that any sharp gradients of a passive scalar will be homogenized at Pe1/3

Bajer et al. (JFM, 2001) showed that the same holds true for the vorticity field, viz. thomo ~ Re1/3

Flohr and Vassilicos (JFM, 1997) showed that a spiral structure unique among the range of vorticity distribution. Closed spaced spiral lead to an accelerated diffusion

where Dk is the Kolmogorov capacity of the spiral )1(33/122 ~)()0( kDtPet

gup

uut

u

2

0

2Dut

0 u

Density evolution

Continuity

Navier-Stokes: Boussinesq approximation , radial gravity

Navier-Stokes: Non-Boussinesq equations.

For Boussinesq approxmiation, = 0

gupuut

u

2

2

Dut

0 u

Density evolution equation

Continuity equation: valid for very high D

Linear stability: mean + small perturbation, e.g.

),()( ),,()( rrrurru

timruu rr exp)(

00

12)(

g

dr

dpuumi r

pr

imruumi r

0

2)(

r

u

dr

du

m

iru rr

dr

dumi r

)(

Dtld ~

h

l)(ˆ tkxie

k

2

First: planar approximation, Rayleigh-Taylor instability

When U1 = U2, always unstable if ρheavy > ρlight

If D=0, growth rate

kg

i

~

Using kinematic conditions and continuity of pressure at the interface