Nishihara

advertisement
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
ablation surfaceshock front ablation surface shock front
mass density
vorticity
International Conference on Turbulent Mixing and Beyond, Trieste, Italy, Aug.18-26, 2007
Introduction
of
Richtmyer-Meshkov instability
and
Laser Implosion
introduction (shocked interface)
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.
IS
I
dv0a
shocked interface
dv0b
vortex sheet
from linearized relation of the shock Rankin-Hugoniot

dvoa  ko 1 

where
ust
usi


u
vi , dvob  ko 1  sr
usi



vi  v1 

Matsuoka, Nishihara
Fukuda (PRE(03))
A=0.376, ξ0/λ=0.02
0 , k ; amplitude of the initial interface corrugation and its wave number,
usi , ust , usr ; incident, transmitted and reflected shock speeds, and
vi , v1; interface speed after the interaction and fluid velocity behind the incident shock.
introduction (accelerated interface)
Acceleration of different mass fluids also drives
velocity shear at the interface.
after Jacobs & Sheeley, PF (96)
http://infocenter.ccit.arizona.edu/~fluidlab/papers/paper4.p
df
two fluids with
surface perturbation
spring
0 
K
M
after the contact
http://scitation.aip.org/getpdf/servlet
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.
uy
ux
before the contact
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.
introduction
Better understanding of hydrodynamic instabilities
is essential for laser fusion.
laser
Instability of ablation surface
effective gravity
ablation surface
ablator
main fuel
shock
contact surface
vapor fuel
RMI
In this talk, we will mainly discuss
instabilities associated with
nonuniform vorticity deposited
at the interface.
RTI; Rayleigh-Taylor Instability
RMI; Richtmyer-Meshkov Instability
Richtmyer-Meshkov instability
(initial perturbation and wave equation)
rippled
corrugated
rippled
transmitted shock interface reflected shock
dv0a
y
x
dv0b
Perturbation of shocked interface, and
ripples on reflected and transmitted shock surfaces
linear RMI
IS
I
IS
I
IS
TS
t=t1
RS
t=0
t 0
time
transmitted shocks
TS
t=t2
0
r 0
reflected shocks
TS
t2
I
space
t1
incident shock trajectory
I
t=t2=0+
RS
RS
I ; interface
IS; incident shock
RS; reflected shock
TS; transmitted shock
initial amplitude of
rippled shocks (t=t2=0+)
ust
kt 0  k0 (1 
)
usi
usr
kr 0  k0 (1 
)
usi
linear RMI
Consider interaction between corrugated interface and rippled
shocks through sound wave and entropy wave between them.
Initial velocity shear (t=0+)
dvoa  ktovi
dvob  kro vi  v1 
Solve wave equations in the regions
between interface and shock fronts
for sound wave and entropy wave
with proper boundary conditions
dv0a
y
2
2
2
2 
dp  (kcs ) dp  cs 2 dp  0
2
t
x
dp exp(iky)
Boundary condition at shock front
normal velocity;
from linearized shock jump condition with respect
to ripple amplitude x  x  ust  as (t )eiky
tangential velocity;
continuous
x
dv0b
Propagation of a rippled shock
driven by a corrugated piston
(a)
P
S
(b)
P
S
y
x
Consider interaction
between corrugated piston (interface) and rippled shocks
through sound wave and entropy wave between them.
ripple shock
Solve wave equation for pressure perturbation between
shock and contact surface with proper boundary conditions
pressure perturbation
dp1  dp1 ( x, t ) exp(iky)
wave equation
2
2
2 
2 2
d
p

c
d
p

c
k dp1  0,
1
1
1
1
2
2
t 
x
change variables
where
( x, t )  (r , ) ,
where
2
1 
1 2
dp1 
dp1  dp1  2 2 dp1  0
2
r
r r
r 
x  x  v x1t , t   t
r cosh  kc1t, r sinh   kx
f
1
0
by introducing
Us
dp1  f  (r ) g ( )
d2
1 d
2
f 
f   (1  2 )f0   0,
2
dr
r dr
r
d2
2
g


g  0

2
d
piston
solution
shock
c
d
dp1   ( Da e   Db ecx)(cD
J
(
r
)

D
N  (r ))
cos


u
1   0
s

where
J  (r ), N  (r )
are Bessel functions ,
Da d
are coefficients
ripple shock
Amplitude of shock ripple decays with t-1/2
solid line; analytical solution
circles; simulation result
dotted line; CCW approximation solution
0.01
Shock front ripple
as / 
0.005
0
-0.005
-0.01
0
1
2
3
4
Normalized time v
5
s
6
7
t/ 
shock front ripple
as (rs )
1  (2 22  1) M s2
 J 0 (rs )  2
J 2 (rs ) , where rs  kc1t 1   s2 ,
2
2
a0
1  (2 s  1) M s
J n , M s , s are Bessel function, shock Mach number ahead and behind the shock
Richtmyer-Meshkov instability
(linear theory)
asymptotic growth rate
effects of compressibility
dv0a
dv0b
Solve wave equations in the regions
between interface and shock fronts
for sound wave and entropy wave with
proper boundary conditions
2
2
2
2 
dp  (kcs ) dp  cs 2 dp  0
2
t
x
y
dp exp(iky)
x
linear RMI
Both tangential velocity and normal velocity
reach asymptotic values
time evolution of tangential velocity
time evolution of normal velocity
0.1
0
dimensionless velocity
-0.02
dimensionless velocity
tangential velocity at the
contact surface in fluid "a"
a =1.8, a =1.1, R 0 = 3
-0.01
-0.03
-0.04
-0.05
-0.06
0.08
0.06
normal perturbation velocity
at the contact surface
a = 1.8, b = 1.1, R 0 = 3
0.04
0.02
-0.07
-0.08
0
0.5
1
1.5
xt /
2
2.5
3
0
0
0.5
1
1.5
2
2.5
3
xt /
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).
linear RMI
Asymptotic growth rates depend on
the whole compressible evolution:
Integrate equation of motion from 0+ to
ddv y

ddv x
dpi

dt
x
dt
From pressure continuity at the interface, we have for tangential velocity

 kdpi

af (dvya  dv0ya )  bf (dvyb  dv0yb )
a,bf : density at t=0+
By defining the difference between normal and tangential velocities
at each side of the interface,
dvi  dv ya  Fa
dv yb  dvi  Fb
we get an exact expression for the asymptotic linear growth rate:
0
0

d
v


d
v
bf Fb   af Fa
bf
yb
af
ya

dvi 

bf   af
bf   af
In a weak shock limit.
the F-terms can be neglected.
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.
linear RMI
Efects of the compressibility: Freez-out of the growth
asympotically occurs due to the compressibility
As the shocks separate away,
their ripples will change in time,
generating at the same time
sound waves and vorticity/entropy.
dimensionless velocity
A typical spatial
vorticity/entropy profile:
0,02
vorticity (a.u.)
0,01
0
normal perturbation velocity
at the contact surface
a = 1.8, b = 1.1, M i = 5
0.02
R 0 = 1.1579271...
0.01
0
-0.01
-0.02
-0,01
-0,02
-0.03
-0,03
0
0.5
1
1.5
2
2.5
3
-0,04
-2
-1,5
-1
-0,5
0
x/
0,5
1
1,5
2
xt /
K. O. Mikaelian, Phys. Fluids 6, 356 (1994),
Wouchuk and Nishihara, Phys. Rev. E 70, 026305 (2004)
linear RMI
Efects of the compressibility: At high incident shock intensity
the asymptotic growth rate decreases,
which agrees well with simulations by Yang et al.
a shock is reflected back
a rarefaction is reflected back
different pairs of gases
0.3
0
VMG
-0.05
CO 2- Air
Air - SF 6
R-M
0.2
0.15
our model and
Yang et al simulations
0.1
normal velocity
asymptotic velocity
asymptotic velocity
0.25
-0.1
-0.15
Xe - Ar
-0.2
-0.25
0.05
-0.3
0
SF 6- Air
-0.35
0
0.2
0.4
0.6
shock intensity
0.8
1
0
0.2
0.4
0.6
0.8
shock intensity
Y. Yang et al, Phys. Fluids, 6, 1856 (1994),
J. Wouchuk, Phys. Rev. E63, 056303 (2001), and
Phys. Plasmas, 8, 2890 (2003).
1
Exact linear formula also agrees well with laser experiments
with solid target at high Mach number of 10 and 15
(rarefaction was reflected)
0
LP100/14
-2
LF100/4
HP100/14
mo del prediction
linear RMI
-4
HF100/4
LF150/10
LF100/10
-6
HF150/10
-8
LF100/14
HF100/10
-10
LF150/10
-12
-12
-10
-8
-6
-4
-2
0
Nova experiment ( m/ns)
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).
RMI-like Instability (1)
Instability induced when
a ripple shock hits uniform interface
solve wave equations in regions 1, 2 and 3
with proper boundary conditions.
instability due to
rippled shock
Since shock front ripple oscillates, phase of oscillation
at the interaction changes dynamics of interface after
time derivative of ripple
shock front ripple
phase 1
phase 2
phase 3
as
as
instability due to
rippled shock
Growth rate of contact surface ripple depends on
the phase of the incident ripple shock at the incident
phase 3
phase 2
phase 1
growth rate of contact surface
dotted line;
instantaneous value
circles; simulation
solid line;
time integrated value
Analytical solutions
agree with simulations
R. Ishizaki et al., Phys. Rev E53, R5592 (1996).
RMI-like Instability (2)
Instabilities associated with laser ablation
(nonuniform target or nonuniform laser)
Incident laser light
ablation surface shock front
(a) nonuniform target surface
ablation surface
shock front
(b) nonuniform laser irradiation
ablation
surface
instability
Energy deposited at heat wave front induces ablation pressure,
and laser ablation drives a shock wave ahead (like a piston)
trajectory of shock,
ablation surface,
and sonic point
heat wave
temperature
Te
flow diagram
 Te
time
heat flux
ablation surface
divergence
of heat flux
  Te
shock front
distance
Energy deposition at heat
wave front corresponds to
combustion in rocket engine
Chapman-Jouguet
condition at sonic point
density profile
Ablation deformation monotonically increases, and
amplitude of shock ripple is small compared with
a case of a rigid piston
Normalized time r
0
10
20
a
30
40
50
Shock front ripple
as / a 0
1
10
8
0.5
6
0
4
-0.5
2
-1
0
0
10
20
30
Normalized time r
dash-dot line; ablation surface deformation
solid line; ripple shock driven laser ablation
dotted line; ripple shock driven rigid piston
s
Ablation surface deformation
aa / a 0
ablation
surface
instability
40
R. Ishizaki and K. Nishihara,
Phys. Rev. Lett., 78, 1920 (1997).
This instabilitty now called ablative RMI after
V. N. Goncharov, Phys. Rev. Lett., 82, 2091 (1999).
ablation
surface
instability
Analytical solutions for both shock front ripple and
areal mass density perturbation agree well with laser experiments.
comparison with laser experiments (squares)
shock front ripple
(a)
areal mass density
(b)
Areal mass density
perturbation d l / ( d l) 0
Shock front ripple
as / a 0
1
0.5
0
-0.5
-1
5
4
3
2
1
0
0
0.5
1
Normalized time u
uniform laser
irradiation
1.5
s
2
t/ 
0
0.2
0.4
0.6
Normalized time u
0.8
s
t/ 
target surface
deformation
R. Ishizaki and K. Nishihara, Phys. Rev. Lett., 78, 1920 (1997).
T. Endo et al., Phys. Rev. Lett., 74, 3608 (1995).
1
ablation
surface
instability
Fairly good agreements were obtained between experiments
and theory, by assuming the ablative Rayleigh-Taylor growth
after rarefaction wave returns the ablation surface
after shock reach rare surface, exponential growth is assumed due to ablative RTI
10 -3
Areal mass density perturbation
( g / cm 2 )
ablative RTI
RMI-like
10 -4
10 -5
square and solid line: =100m, I0=0.4
circle and dotted line: =75m, I0=0.1
10 -6
0
nonuniform laser
irradiation
0.5
1
1.5
2
2.5
3
3.5
Time ( ns )
M. Nakai et al., Phys. Plasmas, 9, 1734 (2002).
H. Azechi et al., Phys. Plasmas, 5, 1945 (1998).
Richtmyer-Meshkov instability
(nonlinear theory)
(incompressible fluid approximation)
nonlinear RMI
Acceleration of different mass fluids drives
velocity shear at the interface.
http://infocenter.ccit.arizona.edu/~fluidlab/papers/paper4.p
df
two fluids with
surface perturbation
spring  0 
http://scitation.aip.org/getpdf/servlet
J. W. Jacobs and J. M. Sheely,
Phys. Fluids, 8, 405 (1996).
K
M
after the contact
uy
ux
before the contact
nonlinear RMI
We can obtain velocity shear induced at the interface due to
the acceleration during the contact between spring and container.
 coskx
Integrate equation for the amplitude of the interface perturbation
over the interval of the contact t f  (  2d ) / 0
d2
   Ag (t )k
2
dt
after the contact


v02 02
g (t )  g0 1  1  2 sin  0 t  d 
g0



g
K
d  arcsin 2 0 2 2
0 
,
 g v 
M
0 0
 0
where




uy
before the contact
K ; the spring constant, M ; mass of the container,
g0 ; the earth gravity and v0 ; initial velocity of the container
d
 uy ,
dt
u  0 ,
ux
(ux , uy )  ( sin kx, coskx)e  ky
nonlinear RMI
Nonlinear evolution of circulation at the interface
with finite density ratio: Bernoulli equation
Define interface velocity
by mass weighted velocity as
By introducing velocity potential
and circulation
u i   i
1u1   2 u 2
d
x u
1   2
dt
  1   2
which satisfies boundary condition
u1  n  u 2  n  u  n
Introducing vorticity
  u1  u 2
q
1
(u1  u 2 )
2
u becomes
u q
A
1   2
1   2
A

2
  
1
1   2  q  
2
We obtain from Bernoulli equation

d
1
1
A
d

  2 A   q  q        q 
dt
2
8
2
 dt

d 
 u  
where
dt t
Circulation does not conserved
for a finite Atwood number A
nonlinear RMI
Interface dynamics with Lagrangian maker
Modified Birkhoff-Rott equation
Defining complex z from the interface position (x), y)
z   x   iy 
 : Lagrangian parameter
the interface trajectory is obtained from Modified Birkhoff-Rott equation
*
z
d *
A
z  u *  q*   
dt
2 s
s2  z z*
1
1

'


q 
d

'

'
s
cot
z
'

z




4i
2

Normalization
*
Bernoulli equation becomes
Finite Atwood number induces
locally stretching and shrinking
of the interface.
kvlint
Nonlocal.

 
d
A  d  
  2 Re z  q *   2 1  A 2 Re z  q*
dt
s
 dt  s 

kz
The similar equations
have been obtained
by Kotelnikov (PF(00))
but for different u.
Solve above coupled equations with initial conditions
x   
y   k0 cos( )
    2 sin 
    
Weakly nonlinear Theory of a Vortex Sheet : Expansion
Comparison with experiments
nonlinear RMI
accelerated interface
shocked interface
5
amplitude (cm)
4
spike
spike
3
bubble
2
1
bubble
0
-1
0
0.1
0.2
0.3
0.4
0.5
0.6
time (sec)
J. W. Jacobs and J. M. Sheely,
Phys. Fluids, 8, 405 (1996).
G. Dimonte et al.,,
Phys. Plasmas, 3, 614 (1996).
C. Matsuoka et al.,
Phys. Rev. E67, 036301 (2003).
expansion up to 3rd order
X    n X (n ) Y    n Y (n ) 1     i
n
n
n
n
(n )
e  nky cos nk x
nonlinear RMI
Dynamics of vortex sheet with density jump
nonlinear vortex generation, their self interaction
analytical model
K Vlin t = 0.80
K Vlin t = 0.05
double spiral shape of spike
and vorticity in simulation
K Vlin t = 6
K Vlin t = 12
Density jump at the interface introduces generation of vortex
and thus opposite sign of vortex appears,
which causes double spiral structure of spike
C. Matsuoka et al., Phys. Rev. E67, 036301 (2003).
nonlinear RMI
Fully nonlinear evolution: Double spiral structure is
observed as Jacobs & Sheeley experiment.
Color shows the vorticity
Parameters
A = 0.155
k0 = 0.2
kvlint = 0, 1, 2,,,,12
Jacobs
nonlinear RMI
Cylindrical vortex sheet in incompressible RMI.
spike
bubble
bubble
A=0.2, n=4
(inner: lighter fluid)
spike
A=-0.2, n=4
(inner: heavier fluid)
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
Details by Matsuoka
On Aug. 21
C. Matsuoka and K. Nishihara, Phys. Rev. E73, 055304 (2006),
Phys. Rev. E74, 066303 (2006).
Richtmyer-Meshkov instability
(Molecular Dynamic simulation)
(cylindrical geometry)
z
Fij
R
Potential barrier
as Piston
LJ
atoms
R
MD RMI
Nonlinear evolution of Richtmyer-Meshkov instability
in cylindrical geometry
mass density
mass density
vorticity
shock passing interface
mass density
vorticity
reflected shock hits interface
bubble
Mach stem appears
shock pass through interface
spike
shock reflected
anomalous mixing occurs
MD RMI
Molecular dynamics simulations show RM growth
driven by multiple shocks for different mode numbers.
Decay of nonlinear growth is mode dependent and higher mode decays slower,
which agrees with the model of cylindrical vortex sheet
trajectory
growth rate
160
0.4
8
1st
140
buble
5
3
80
5
60
spike
growth rate
3
100
40
8
0.2
0
shock
~ t 0.55
3rd
2nd 3rd
120
radius
2nd
~ t 0.7
1st
5
3
0
8
-0.2
20
-0.4
0
0
20
40
60
time
80
100
120
0
20
40
60
time
80
100
120
MD RMI
Whenever shocks pass through interface from heavy to light,
phase inversion occurs, which causes generation of higher harmonics
Richtmyer-Meshkov instability at shell surfaces
(light-heavy-light)
velocity
(radial)
density
initial
shock reaches
the center
reflected shock
reaches 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.
Download