generation of nonuniform vorticity at interface and its linear and nonlinear growth...
TRANSCRIPT
Generation of Nonuniform Vorticity at Interface and
Its Linear and Nonlinear Growth(Richtmyer-Meshkov and RM-like Instabilities)
K. Nishihara, S. Abarzhi, R. Ishizaki, C. Matsuoka, G. Wouchuk and V. Zhakhovskii
Institute of Laser Engineering, Osaka University
Incident laser light
shock frontablation surface ablation surface shock front mass density vorticity
International Conference on Turbulent Mixing and Beyond, Trieste, Italy, Aug.18-26, 2007
Introductionof
Richtmyer-Meshkov instability and
Laser Implosion
After an incident shock hits a corrugated interface, ripples on reflected and transmitted shocks are induced and RM instability is driven by velocity shear left by the rippled shocks at the interface.
ISIshocked interface
vortex sheet
;,, srstsi uuu
v0a v0
b
,1 isi
sto
ao v
u
ukv
11 vv
u
ukv i
si
sro
bo
from linearized relation of the shock Rankin-Hugoniot
;,0 k
;, 1vvi
incident, transmitted and reflected shock speeds, and
amplitude of the initial interface corrugation and its wave number, where
interface speed after the interaction and fluid velocity behind the incident shock.
introduction (shocked interface)
Matsuoka, NishiharaFukuda (PRE(03))A=0.376, ξ0/λ=0.02
introduction (accelerated interface)
Acceleration of different mass fluids also drives velocity shear at the interface. after Jacobs & Sheeley, PF (96)
http://scitation.aip.org/getpdf/servlet
spring
two fluids with surface perturbation
M
K0
yu
before the contact
after the contactxu
http://info-center.ccit.arizona.edu/~fluidlab/papers/paper4.pdf
During the contact of a container with a spring,phase inversion of the corrugated interface occurs and velocity shear is induced due to the acceleration.
outline of talk
We first show that the RMI is driven essentially by nonuniform velocity shear induced at an interface, instead of impulsive acceleration.
In early stage of the growth, we show the importance of the interaction between the corrugated interface and rippled shocks through sound wave and entropy wave.
There exist similar instabilities caused by the interaction, such as rippled shock interaction with uniform interface, and instability of the laser ablation surface.
Nonlinear evolution of the instability is analyzed, treating the interface as a vortex sheet with finite density ratio for incompressible fluids.
Nonlinear evolution of the instability in cylindrical geometry is investigated both analytically and with the use of molecular dynamic simulations.
ablation surface
shock contact surface
ablator
main fuel
vapor fuel
laser
effective gravity
RTI; Rayleigh-Taylor InstabilityRMI; Richtmyer-Meshkov Instability
Instability of ablation surface
RMI
In this talk, we will mainly discussinstabilities associated with nonuniform vorticity deposited at the interface.
Better understanding of hydrodynamic instabilitiesis essential for laser fusion.
introduction
Richtmyer-Meshkov instability (initial perturbation and wave equation)
v0a v0
b
x
y
rippled reflected shock
corrugated interface
rippled transmitted shock
t2
t1
time
space
incident shock trajectory
reflected shockstransmitted shocks
Perturbation of shocked interface, and ripples on reflected and transmitted shock surfaces
ISI
t=t2
TS RSI
t=t1
ISI
t=0
RSTS
IS
linear RMI
)1(00si
srr u
ukk
)1(00si
stt u
ukk
initial amplitude of rippled shocks (t=t2=0+)
t=t2=0+
TS RSI
00t
0r
I ; interfaceIS; incident shockRS; reflected shockTS; transmitted shock
Consider interaction between corrugated interface and rippled shocks through sound wave and entropy wave between them.
linear RMI
1vvkv irobo
Initial velocity shear (t=0+)
itoao vkv
0)(2
222
2
2
px
cpkcpt ss
)exp(ikyp
Solve wave equations in the regions between interface and shock frontsfor sound wave and entropy wave with proper boundary conditions
v0a v0
b
x
y
ikyss etatuxx )(
Boundary condition at shock front normal velocity; from linearized shock jump condition with respect to ripple amplitude tangential velocity; continuous
Propagation of a rippled shockdriven by a corrugated piston
P S(a) (b)
x
y
P S
Consider interaction between corrugated piston (interface) and rippled shocks through sound wave and entropy wave between them.
Solve wave equation for pressure perturbation between shock and contact surface with proper boundary conditions
)exp(),(11 ikytxpp pressure perturbation
wave equation
tvxx x1 tt 0122
112
22112
2
pkcpx
cpt
,where,
Us
0
0
piston shock
1
f
sx ucc 01 cos
011
12
2
21112
2
pr
pprr
pr
tkcr 1cosh xkr sinh,),(),( rtx change variables where,
0)1(1
2
2
2
2
fr
fdr
d
rf
dr
d02
2
2
ggd
d,
)()(1 grfp by introducing
))()()((1 rNDrJDeDeDp dcba
solution
where )(),( rNrJ daD
are Bessel functions are coefficients,
ripple shock
-0.01
-0.005
0
0.005
0.01
0 1 2 3 4 5 6 7
Normalized time vst /
Sho
ck fr
ont r
ippl
ea
s /
Amplitude of shock ripple decays with t-1/2
solid line; analytical solutioncircles; simulation resultdotted line; CCW approximation solution
)()12(1
)12(12)(
)(222
222
00
sss
ss
ss rJM
MrJ
a
ra
2
1 1 ss tkcr
ssn MJ ,,
shock front ripple
, where ,
are Bessel function, shock Mach number ahead and behind the shock
ripple shock
Richtmyer-Meshkov instability (linear theory)
asymptotic growth rateeffects of compressibility
0)(2
222
2
2
px
cpkcpt ss
)exp(ikyp
Solve wave equations in the regions between interface and shock frontsfor sound wave and entropy wave with proper boundary conditions
v0a v0
b
x
y
Both tangential velocity and normal velocity reach asymptotic values
linear RMI
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0 0.5 1 1.5 2 2.5 3
dim
en
sio
nle
ss
ve
loc
ity
xt/
tangential velocity at the contact surface in fluid "a"
a=1.8,
a=1.1, R
0 = 3
0
0.02
0.04
0.06
0.08
0.1
0 0.5 1 1.5 2 2.5 3
dim
en
sio
nle
ss
ve
loc
ity
xt/
normal perturbation velocity at the contact surface
a = 1.8,
b = 1.1, R
0 = 3
J. G. Wouchuk and K. Nishihara, Phys. Plasmas 4, 1028 (1997),J. G. Wouchuk, Phys. Rev. E 63, 056303 (2001), Phys. Plasmas 8, 2890 (2001).
time evolution of tangential velocity time evolution of normal velocity
Asymptotic growth rates depend on the whole compressible evolution:
linear RMI
iy pk
dt
vd
)()( 00ybybbfyayaaf vvvv
x
p
dt
vd ix
Integrate equation of motion from 0+ to
From pressure continuity at the interface, we have for tangential velocity
which is valid for any value of the initial parameters:shock intensity, fluid density and fluid compressibility.
It should be noted that the F terms are proportional to a spatial average of the vorticity field left by the rippled shock fronts.
bfa , : density at t=0+
By defining the difference between normal and tangential velocities at each side of the interface,
we get an exact expression for the asymptotic linear growth rate:
biyb Fvv ayai Fvv
afbf
aafbbf
afbf
yaafybbfi
FFvvv
00
In a weak shock limit.the F-terms can be neglected.
Efects of the compressibility: Freez-out of the growth asympotically occurs due to the compressibility
linear RMI
-0,04
-0,03
-0,02
-0,01
0
0,01
0,02
-2 -1,5 -1 -0,5 0 0,5 1 1,5 2
vo
rtic
ity
(a
.u.)
x/
As the shocks separate away, their ripples will change in time, generating at the same time sound waves and vorticity/entropy.
A typical spatial vorticity/entropy profile:
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.5 1 1.5 2 2.5 3
dim
en
sio
nle
ss
ve
loc
ity
xt/
normal perturbation velocity at the contact surface
a = 1.8,
b = 1.1, M
i = 5
R0 = 1.1579271...
K. O. Mikaelian, Phys. Fluids 6, 356 (1994), Wouchuk and Nishihara, Phys. Rev. E 70, 026305 (2004)
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0 0.2 0.4 0.6 0.8 1
shock intensity
CO2- Air
Xe - Ar
SF6- Air
no
rma
l v
elo
cit
y
Efects of the compressibility: At high incident shock intensitythe asymptotic growth rate decreases, which agrees well with simulations by Yang et al.
linear RMI
a rarefaction is reflected back
different pairs of gases
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8 1
as
ym
pto
tic
ve
loc
ity
shock intensity
VMG
R-M
our model andYang et al simulations
Air - SF6
asym
ptot
ic v
eloc
ity
a shock is reflected back
Y. Yang et al, Phys. Fluids, 6, 1856 (1994), J. Wouchuk, Phys. Rev. E63, 056303 (2001), andPhys. Plasmas, 8, 2890 (2003).
Exact linear formula also agrees well with laser experimentswith solid target at high Mach number of 10 and 15 (rarefaction was reflected)
linear RMI
-12
-10
-8
-6
-4
-2
0
-12 -10 -8 -6 -4 -2 0
mo
de
l p
red
icti
on
Nova experiment ( m/ns)
LP100/14
HP100/14
HF100/4
LF100/4
LF150/10
LF100/10
HF150/10
LF100/14
HF100/10
LF150/10
J. Wouchuk, Phys. Plasmas, 8, 2890 (2001). G. Dimonte et al., Phys. Plasmas 3, 614 (1996); R. L. Holmes et al., J. Fluid Mech. 389, 55 (1999).
solve wave equations in regions 1, 2 and 3with proper boundary conditions.
RMI-like Instability (1)
Instability induced when a ripple shock hits uniform interface
phase 1 phase 2 phase 3
time derivative of ripple
shock front ripple
sa
sa
Since shock front ripple oscillates, phase of oscillation at the interaction changes dynamics of interface after
instability due to rippled shock
Growth rate of contact surface ripple depends on the phase of the incident ripple shock at the incident
gro
wth
rat
e o
f co
nta
ct s
urf
ace
phase 1
phase 2
phase 3
dotted line; instantaneous valuecircles; simulation
solid line; time integrated value
instability due to rippled shock
R. Ishizaki et al., Phys. Rev E53, R5592 (1996).
Analytical solutions agree with simulations
Incident laser light
shock frontablation surface ablation surface shock front
(a) nonuniform target surface (b) nonuniform laser irradiation
RMI-like Instability (2)
Instabilities associated with laser ablation (nonuniform target or nonuniform laser)
trajectory of shock,ablation surface,and sonic point
Chapman-Jouguet condition at sonic point density profile
Energy deposited at heat wave front induces ablation pressure, and laser ablation drives a shock wave ahead (like a piston)
Energy deposition at heat wave front corresponds to combustion in rocket engine
heat wave
eTtemperature
eT heat flux
eT divergence of heat flux
ablationsurface instability
distance
tim
eflow diagram
ablation surface
shock front
dash-dot line; ablation surface deformationsolid line; ripple shock driven laser ablationdotted line; ripple shock driven rigid piston
-1
-0.5
0
0.5
1
0
2
4
6
8
10
0 10 20 30 40
0 10 20 30 40 50
Sho
ck fr
ont r
ippl
e
as /
a0
Abl
atio
n su
rfac
e de
form
atio
n
aa /
a0
Normalized time rs
Normalized time ra
Ablation deformation monotonically increases, andamplitude of shock ripple is small compared with a case of a rigid piston
ablationsurface instability
R. Ishizaki and K. Nishihara, Phys. Rev. Lett., 78, 1920 (1997).
This instabilitty now called ablative RMI afterV. N. Goncharov, Phys. Rev. Lett., 82, 2091 (1999).
-1
-0.5
0
0.5
1
0 0.5 1 1.5 2
Normalized time ust /
Sho
ck fr
ont r
ippl
ea
s /
a0
(a)
0
1
2
3
4
5
0 0.2 0.4 0.6 0.8 1
Normalized time ust /
Are
al m
ass
dens
ity
pert
urba
tion
l / (
l)
0
(b)
Analytical solutions for both shock front ripple and areal mass density perturbation agree well with laser experiments.
shock front ripple areal mass density
uniform laser irradiation
target surface deformation
comparison with laser experiments (squares)
ablationsurface instability
R. Ishizaki and K. Nishihara, Phys. Rev. Lett., 78, 1920 (1997).T. Endo et al., Phys. Rev. Lett., 74, 3608 (1995).
nonuniform laserirradiation
Fairly good agreements were obtained between experiments and theory, by assuming the ablative Rayleigh-Taylor growth after rarefaction wave returns the ablation surface
ablationsurface instability
M. Nakai et al., Phys. Plasmas, 9, 1734 (2002).H. Azechi et al., Phys. Plasmas, 5, 1945 (1998).
square and solid line: =100m, I0=0.4circle and dotted line: =75m, I0=0.1
after shock reach rare surface, exponential growth is assumed due to ablative RTI
10 -6
10 -5
10 -4
10 -3
0 0.5 1 1.5 2 2.5 3 3.5
Are
al m
ass
den
sity
pe
rtur
batio
n
( g
/ cm
2 )
Time ( ns )
RMI-like
ablative RTI
Richtmyer-Meshkov instability（ nonlinear theory ）
(incompressible fluid approximation)
Acceleration of different mass fluids drives velocity shear at the interface.
http://scitation.aip.org/getpdf/servlet
spring
two fluids with surface perturbation
M
K0
yu
before the contact
after the contactxu
http://info-center.ccit.arizona.edu/~fluidlab/papers/paper4.pdf
nonlinear RMI
J. W. Jacobs and J. M. Sheely, Phys. Fluids, 8, 405 (1996).
tg
vgtg 02
0
20
20
0 sin11)(
ktAgdt
d)(
2
2
20
20
20
0arcsin
vg
g,0 M
K
We can obtain velocity shear induced at the interface due to the acceleration during the contact between spring and container.
where ; the spring constant, ; mass of the container, ; the earth gravity and ; initial velocity of the container
K M0g 0v
kyyx ekxkxuu )cos,sin(),( yu
dt
d
, 0 u ,
0/)2( ftkxcosIntegrate equation for the amplitude of the interface perturbation
over the interval of the contact
yu
before the contact
after the contactxu
nonlinear RMI
Define interface velocity by mass weighted velocity as
dt
d
21
2211
uu
ux
which satisfies boundary condition
nununu 21
Introducing vorticity
21 uu
)( 212
1uuq
u becomes
qu2
A
21
21
A
iiu
21
212
1
By introducing velocity potentialand circulation
We obtain from Bernoulli equation
q
Aqq
dt
dA
dt
d
28
1
2
12
tdt
d uwhere
q
Circulation does not conserved for a finite Atwood number A
Nonlinear evolution of circulation at the interfacewith finite density ratio: Bernoulli equation
nonlinear RMI
Defining complex z from the interface position (x) y)
: Lagrangian parameter iyxz
*22
* Re1Re2
qzA
sq
dt
dz
s
A
dt
dBernoulli equation becomes Nonlocal. The similar equations
have been obtainedby Kotelnikov (PF(00))but for different u.
Interface dynamics with Lagrangian makerModified Birkhoff-Rott equation
nonlinear RMI
Solve above coupled equations with initial conditions
x )cos(0 ky sin2
s
zAquz
dt
d ****
2 *2
zzs
the interface trajectory is obtained from Modified Birkhoff-Rott equation
zzsd
iq '
2
1cot''
4
1 '*
Finite Atwood number induceslocally stretching and shrinkingof the interface.
Normalization
tkvlin kz
Weakly nonlinear Theory of a Vortex Sheet : ExpansionComparison with experiments
nonlinear RMI
-1
0
1
2
3
4
5
0 0.1 0.2 0.3 0.4 0.5 0.6
amp
litu
de
(cm
)
time (sec)
spike
bubble
accelerated interface
J. W. Jacobs and J. M. Sheely, Phys. Fluids, 8, 405 (1996).
spike
bubble
shocked interface
G. Dimonte et al.,, Phys. Plasmas, 3, 614 (1996).C. Matsuoka et al.,Phys. Rev. E67, 036301 (2003).
)(n
n
nXX )(n
n
nYY nkxe nkyni
n
n cos)(1
expansion up to 3rd order
analytical model K Vlin t = 0.80 K Vlin t = 0.05
double spiral shape of spikeand vorticity in simulation
Dynamics of vortex sheet with density jumpnonlinear vortex generation, their self interaction
Density jump at the interface introduces generation of vortexand thus opposite sign of vortex appears, which causes double spiral structure of spike
K Vlin t = 6 K Vlin t = 12
nonlinear RMI
C. Matsuoka et al., Phys. Rev. E67, 036301 (2003).
Fully nonlinear evolution: Double spiral structure is observed as Jacobs & Sheeley experiment.
Color shows the vorticity Parameters A = 0.155 k0 = 0.2
kvlint = 0, 1, 2,,,,12
Jacobs
nonlinear RMI
A=0.2, n=4 (inner: lighter fluid)
A=-0.2, n=4 (inner: heavier fluid)
Cylindrical vortex sheet in incompressible RMI.nonlinear RMI
Features of cylindrical geometry, ・ two independent spatial scale, radius and wavelength nonlinear growth depends strongly on mode number ・ ingoing and outgoing of bubble and spike nonlinear growth depends inward and outward motion rather than spike and bubble
C. Matsuoka and K. Nishihara, Phys. Rev. E73, 055304 (2006),Phys. Rev. E74, 066303 (2006).
spike
bubble spike
bubble
Details by MatsuokaOn Aug. 21
Richtmyer-Meshkov instability （ Molecular Dynamic simulation ）
(cylindrical geometry)
Potential barrier Potential barrier
as Pistonas Piston
LJ atoLJ atomsmsFFijij
z
R R
Nonlinear evolution of Richtmyer-Meshkov instability in cylindrical geometry
MD RMI
mass density
shock passing interface
Mach stem appears
shock reflected
reflected shock hits interface
shock pass through interface
vorticitymass densityvorticitymass density
anomalous mixing occurs
bubble
spike
Molecular dynamics simulations show RM growth driven by multiple shocks for different mode numbers.
0 20 40 60 80 100 120
time
0
20
40
60
80
100
120
140
160
rad
ius
0
shock
buble
spike
8
8
5
3
5
3
trajectory
0 20 40 60 80 100 120
time
-0.4
-0.2
0
0.2
0.4
gro
wth
rat
e
8
5
3
~ t -0.55
~ t -0.7
growth rate
Decay of nonlinear growth is mode dependent and higher mode decays slower, which agrees with the model of cylindrical vortex sheet
1st
2nd 3rd
1st
2nd
3rd
MD RMI
Whenever shocks pass through interface from heavy to light, phase inversion occurs, which causes generation of higher harmonics
MD RMI
Richtmyer-Meshkov instability at shell surfaces(light-heavy-light)
densityvelocity(radial)
initial
shock reaches the center
reflected shockreaches shell
density
Conclusion
・ Both exact and asymptotic linear growth rates of the Richtmyer-Meshkov instability and RMI-like instabilities were obtained for compressible and incompressible fluids, which agrees with experiments.
・ By introducing mass weighted interface as a nonuniform vortex sheet between two fluids with finite density ratio, we have developed a fully nonlinear theory of the incompressible RM instability, which also agrees fairly well with experiments.
・ The theory is extended to a cylindrical geometry, in which nonlinear growth is determined from the inward and outward motion rather than bubble and spike, and it depends on mode number.
・ Molecular Dynamic simulation provides a new tool for a study of hydrodynamic instabilities, when CFD fails. We observed enhancement of the growth for sandwiched shell.
New features of such a system with density difference across interface, and nonuniform vorticity may provides a paradigm in vortex dynamics.