Multi-scale character of the nonlinear
dynamics of the Rayleigh-Taylor and
Richtmyer-Meshkov instability
Snezhana I. Abarzhi
Many thanks to the co-authors:
M. Herrmann (Stanford, Arizona State), J. Glimm (SUNY)
P. Moin (Stanford), K. Nishihara (ILE), A. Oparin (ICAD),
R. Rosner (ANL)
International Conference Turbulent Mixing and Beyond, 18-26 August 2007, ICTP, Trieste, Italy
Preamble
Ptolemeus, 100 AD
new methods of measurements
and calculations;
Geocentric model with adjustable
parameters, describing epicycles.
The curve fit was perfect for 1500 years.
Copernicus 1543:
Heliocentric model provided
worse agreement with observations
A model, based on a wrong idea and used
adjustable parameters, may agree with
observations
Supernova 1572: Brahe, Kepler, Newton
Chandra pictures, 2003
Rayleigh-Taylor / Richtmyer-Meshkov instability
Fluids of different densities are accelerated against
the density gradient.
A turbulent mixing of the fluids ensues with time.
RT/RM turbulent mixing controls
 fusion, plasmas, laser-matter interaction
 supernovae explosions, thermonuclear flashes, photo-evaporated clouds
 flames and fires, geophysics, impact dynamics, spray formation,…
RT/RM flows are non-local, inhomogeneous and anisotropic.
Grasping essentials of the mixing process is
a fundamental problem in fluid dynamics.
One of the primary issues is the dynamics of
the large-scale coherent structure of bubbles and spikes.
This structure appears in the nonlinear regime of RTI/RMI.
The nonlinear regime bridges a gap between
the initial and turbulent (perhaps, self-similar) stages of RTI/RMI.
Rayleigh-Taylor /Richtmyer-Meshkov instability
laboratory experiments in fluids and gases

h
h
g
l
nonlinear RMI, Jacobs et al2004
h l  3,  ~ 6cm, t ~ 10ms
RT mixing, Ramaprabhu & Andrews2004
h  l  h ~ 103 ,  ~ 1cm, t ~ 1s
Rayleigh-Taylor / Richtmyer-Meshkov evolution
•
 ~ h  l  g h  l ,
•
h ~ h0 exp t  
linear regime
 ~  max
h ~ t 
nonlinear regime
light (heavy) fluid penetrates
heavy (light) fluid in bubbles (spikes)
•
h ~ gt 2  h  l   h
turbulent mixing
Richtmyer-Meshkov (g=0)

hb s  ~ t b  s 

h
h
l
g
RT/RM flow is
characterized by:
 large-scale structure
 small-scale structures
 energy transfers to
large and small scales
Large-scale coherent dynamics
Conservation laws
    v  0
v h  n  0  v l  n  0
v  v  v   p  ...  0
ph 0  pl 0
no mass flux
momentum
initial conditions
~  max 
v h z    v l
z  
0
no mass sources
 v0
symmetry
Singular and non-local aspects of the interface evolution cause significant
difficulties for theoretical and numerical of studies of RMI.
scale separation
group theory
Abarzhi1990s, 2001
v h l    h l     Ψh l 
~
 
passive
active
Dynamical system
• Fourier expansion
• Local expansion
conservation laws
surface variables ζ ij
~
 l   m 
 h   m 
x, y ~ 0
z  z0 

*

2i 2 j



t
x
y
 ij
N 1 i  j  N
local dynamical system ODEs
moments
~
M n t , M n (t ) 

  m km n
m 0
 Layzer-type approach
regular asymptotic solutions are absent
• non-local dynamics
interplay of harmonics
• continuous family
• family parameters
• physically significant is
singularities
regular asymptotic solutions
symmetry, 2D/3D
the fastest stable solution
Nonlinear dynamics of RT bubble
RT evolution is characterized by two length scales, period  and position h
time
curvature
,
t  
 , v ~ expt 
,
t  
   A, RT
 1
velocity
Agk
v  v A, RT
A 1
 A, RT  k 81  1  A 8
v A, RT  g k 1  31  A 16 
A0
 A, RT  3k 16A1 3
v A, RT  3 2
 Universal dynamics for all A
32
v
2
A, RT
3
Ag k
 A, RT ~ g4
Velocity and curvature vs Atwood, RTI
 1 /k
v /(g/k)1/2
 A, RM = 0
vA, RT
A
L
vD, RT
vL, RT
 A, RT
A
v A, RT
v L, RT  Ag k
v D, RT  2 Ag 1  Ak
D
 A, RT
 L   Ak 8
D   k 8
non-local theory
empiric model
drag model
 empirical / drag / Layzer-type solutions violate the conservation of mass
 velocity is not a sensitive diagnostic parameter
 curvature is sensitive parameter, which tracks the conservation of mass
Nonlinear dynamics of RM bubble
RM evolution is characterized by two length scales, period  and position h
time
tv0 k  1
curvature
velocity
  k t 
v  v0   Av0 t 
tv0 k ~ 1
  k
t v0 k  
 0
v0

C 
bi 

v~
1   Bi tv0 k  

Akt  i 1

Rebi   0
• flat bubbles move faster and experience stronger deceleration and drag
• for smaller values of A, bubbles move faster
t 
A0
v ~ ta
1  a  0
• for velocity, for a finite sequence of data points and short dynamic range
the exponent -1 and coefficient C may be hard to distinguish
Empirical drag models and single-mode solutions
Drag model assumes
Shvarts et al 1995, 2001
• RM evolution has a single-scale character, h = h (
• curvature  is uniquely determined by period 
    3
Drag model suggests an empirical formula
1  kv0t
vD  v0
2
1  1  Akv0t  E kv0t 
1 A
E 3
3 A
 To calculate the drag model solution in a single-mode Layzer-type
approximation, Goncharov 2002 introduced
a time-dependent inhomogeneous mass source of the light fluid.
 Experiments do not have any mass source yet report a reasonable
agreement with the drag model for the position h (velocity v).
• The formula is a curve fit with free parameters (exponents, coefficients).
• To prove the formula as an empirical law,
huge statistics and large dynamic range are required.
Statistics standards: 30 data points per each parameter, ~3011 (~1.7x1016).
Nonlinear dynamics: power-laws
Velocity of RM bubble given by the non-local theory with account for
next-order correction in time (solid) and the single-scale drag-model

C 
bi 
vA ~
1   Bi tv0 k  
Akt  i 1

vD 
1 A/ 3
1  Akt
Reliable quantitative
statements are very
hard to make.
Flattening of the front
of RM bubble
is a qualitative effect.
Flow AixCo
Marcus Herrmann, Stanford and Arizona State University
(JFM)
• Solves fully compressible Navier-Stokes equations (two-dimensional)
• Operator splitting
• Convective terms solved by explicit 2nd order Godunov type scheme
• Diffusive terms solved by explicit 2nd order Runge-Kutta scheme
• Hybrid tracking-capturing scheme
• Interface is tracked by level set approach
• 5TH order WENO
• Shocks and waves are treated by a standrad capturing scheme
• In-cell-reconstruction scheme ensures the interface remains a
discontinuity
 High Atwood numbers and strong shocks can be modeled
Computational setup
•
•
Mach = 1.2
four cases: A=0.55, 0.663, 0.78, A=0.9
•
•
•
speed of sound of the light gas cl = 347.2 m/s
viscosity of air for the Navier-Stokes equations
several test runs of the Euler equations
•
Initial perturbation a(t) = a0 cos(2 x / 
 = 3.75 cm,
a0 = 0.064 
•
•
•
•
Computational domain [- 40.667  , +1.333  ] x [-0.5 , 0.5 ]
grid 5376 x 128 (256); equidistant grid cells
outflow conditions in the z-direction
symmetry conditions in the x-direction
•
•
Simulations stop as the reflected shock hits the interface.
The run time is longer than in most observations of the nonlinear RMI.
Interface evolution induced by RMI
A=0.55
A = 0.663
A=0.78
A=0.9
v ~ 0.1cl
• RMI develops relatively to a background motion with velocity v
at which the interface would move if it would be ideally planar.
length scale 
time scale   
v
v ~ 0.1cl
• The bubble (spike) dynamics is quantified in the frame of references
moving with velocity v
Validation
Linear theory of Wouchuk 2001
adequate value of the growth-rate
Experiments of Jacobs 1997
A=0.663; Mach = 1.1
A=0.9
• Oscillations are caused by reverberations of sound waves
• Oscillations do not result in significant pressure fluctuations
• The oscillations induce an error ~10% in theoretical value and
up to ~30-40% in experimental value of RMI growth-rate
• Experimental data sampling do not capture the velocity oscillations
Nonlinear velocity
The bubble is accelerated “impulsively” and then decelerates.
Other earlier observations
- set time-scale using the value of the growth-rate v0
- calculated the bubble (spike) position relative to the “middle line”
- did not capture the high frequency components
Nevertheless
- The velocity time-dependence was evaluated quantitatively (pre-factor)
Diagnostics of the velocity
Log-log plot of velocity vs time
v
- Time-scale is set by the velocity
- Bubble (spike) position calculated in the moving frame of references.
- High frequency components are captured.
v0
Only asymptotic value of the velocity can be evaluated
Accurate quantitative estimate of the velocity time-dependence (exponent)
is prevented due to
- oscillations caused by reverberations of sound waves
- unknown contribution of higher order terms and short dynamic range
Diagnostics of the interface
Bubble curvature  and its rms deviation ‘ vs time
curvature is
calculated via a
least square fit
circle, |x/| < 4/64
solid line is 
(left scale),
dashing lines is ‘/
(right scale),
|x/| < 4/64
• RM bubble flatten asymptotically with time
• The flattening process is slower for smaller values of A
Multi-scale character of the dynamics
velocity v(t) vs curvature (t) with time t
white square is
our non-local
solution,
black square is
the drag model
solution
• velocity v=dh/dt and curvature  mutually depend on one another
dh dt  v f   
• deceleration d2 h / dt2 and flattening d () /dt are interrelated processes
• they indicate a multi-scale character of RM evolution
Multi-scale dynamics in RTI/RMI

h
l
h
 RTI, for all A
 RMI, for all A
g
The dynamics of RT/RM
large-scale coherent structure
is characterized by
 wavelength – initial conditions
 amplitude - small-scale
structures
( dh / dt )2  g ~ const
3
dh dt  v f   
• In RTI / RMI amplitude and wavelength contribute independently
to the nonlinear dynamics.
• The multi-scale character of the nonlinear evolution should be
accounted for in a description of the turbulent mixing process.
• The nonlinear dynamics is hard to quantify reliably (power-laws).
Coherent dynamics in RTI/RMI
• Any wave is characterized by wavelength, amplitude and phase.
• Coherence, symmetry and order are related to phase.
• Is it possible to create and maintain the order in RT flow?
• Group theory suggests: it is possible in principle as the flow should
maintain isotropy in the plane normal to the direction of acceleration
m
m
a1
pm
pm
2a1
2D flow: binary interaction, duplication of the wavelength
Coherent dynamics in RTI/RMI
p6mm
2a2
p2gm
p6mm
p3m1
a2
-3a1
a1
3(a1 + a2)
a2
a1
a1
3D flow: multi-pole interactions, growth of wavelength results in isotropy loss
RT mixing: between order and disorder ?
• Group theory suggests that RT/RM coherent structures with hexagonal
symmetry are the most stable and isotropic. Self-organization may occur.
• Our phenomenological model, which has found that the unsteady
turbulent mixing is more ordered compared to isotropic turbulence
vl v ~ l L 1 / 3
vl v ~ l L 1 / 2
Kolmogorov turbulence
unsteady turbulent flow
• How to impose the initial perturbation? Faraday waves may be a solution
Faraday waves
J.P. Gollub
Requirements on the precision and accuracy in the experiments are very high
Conclusions
The large-scale coherent dynamics in RTI and RMI
is studied theoretically and numerically.
Theory
• obeys the conservation laws
• has no adjustable parameters
• accounts for the higher-order correlations
• identifies the multi-scale character of the nonlinear dynamics
• suggests the disordered mixing may have coherence and order
Numerical solution in RMI (M. Herrmann)
• models weakly compressible and nearly inviscid fluids
• treats the interface as a discontinuity
• is applicable for fluids with very high values of the density ratio
The theory and the simulations
• validate each other, identify the reliable diagnostics of the interface dynamics
• indicate the non-local and multi-scale character of RMI
• show that the reliable quantification of RTI/RMI is a complex problem,
still open for a curious mind.