harish dixit and rama govindarajan with anubhab roy and ganesh subramanian
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Instabilities in variable-property flows and the continuous spectrum An aggressive ‘passive’ scalar. Harish Dixit and Rama Govindarajan With Anubhab Roy and Ganesh Subramanian Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore September 2008. Re=3000, unstratified. - PowerPoint PPT PresentationTRANSCRIPT
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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