Harish Dixit and Rama Govindarajan With Anubhab Roy and Ganesh Subramanian
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Transcript of Harish Dixit and Rama Govindarajan With Anubhab Roy and Ganesh Subramanian
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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
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Re=3000, unstratified
Building block for inverse cascade
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`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?
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Re=3000, Pe=30000, Fr (pair) = 1
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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
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Heavy
Light
point vortex
Initial condition: Point vortex at a density jump
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Homogenised within the yellow patch, if Pe finite
A single vortex and a density interfaceInviscid
The locus seen is not a streamline!
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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
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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
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m = 2
Vortex sheet at rj
)(2 4cr rOiuu
In unstratified case: a continuous spectrum of `non-Kelvin’ modes
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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
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r
r
urmrr
Drmgrm
dr
dumrDmrDDrm
)2'(1
)(
113)(
2222
2222
Vorticity and density: Heaviside functions
).....](),(),[( 321 rrrrrrdr
ρd
)(23 0 crrrD
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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
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m = 5
Multiple (7) jumps
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rj = 2 rc, =0.1
Single jumpStep vs smoothdensity change
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Single jump: radial gravity (blue), non-Boussinesq (red)
m = 2, = 0.01
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(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
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Simulations: spectral, interfaces thin tanh, up to 15362 periodic b.c.
Heavy
Light
Non-Boussinesq equations
)( 6 zgupuut
u
6
ut
0 u
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t=0
3.18
6.4
t=1.59
Boussinesq, g=0, density is a passive scalar
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9.5
t=12.7
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time=0
time=12.7
Vorticity
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1.6
3.82
Non-BoussinesqA=0.2
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t=4.5
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t=5.1
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5.73
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t=3.2Notice vorticity contours
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t=4.5
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5.1
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5.73
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A=0.12, t = 7.5Г/rc2
λ ~ 2.5ld (λstab ~ 4ld)
Viscous simulations: same instability
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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
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Conclusions:
Co-existing instabilities: `forward cascade’unstable wins
Beware of Boussinesq, even at small A
What does this do to 2D turbulence?
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Single jump: Boussinesq (blue), non-Boussinesq (red)
m = 20, = 0.1
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Variation of ur eigenfunction with the jump location: rc = 0.1, m = 2
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Effect of large density differences
m = 2, = 1
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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
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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
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20,1 1 rrc
r
m
21r
mr
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i
20,1 1 rrc
m
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2,1 mrc
i 1r
mi
1r
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r
2,1 mrc
1r
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r
2,101 mrr c
cr
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Comparison: Boussinesq (blue), non-Boussinesq (red)
m = 2, = 0.1
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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:
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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
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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
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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
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gup
uut
u
2
0
2Dut
0 u
Density evolution
Continuity
Navier-Stokes: Boussinesq approximation , radial gravity
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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
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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
)(
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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