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Energy transfer in Rayleigh-Taylor instability
Andrew W. Cook and Ye Zhou
Lawrence Livermore National Laboratory, University of California, Livermore, California 94551
共Received 1 November 2001; revised manuscript received 8 February 2002; published 30 August 2002兲
The spatial structure and energy budget for Rayleigh-Taylor instability are examined using results from a
512⫻512⫻2040 point direct numerical simulation. The outer-scale Reynolds number of the flow follows a
rough t3power law and reaches a final value of about 5500. Taylor microscales and Reynolds numbers are
plotted to characterize anisotropy in the flow and document progress towards the mixing transition. A mixing
parameter is defined which characterizes the relative rates of entrainment and mixing in the flow. The spectrum
of each term in the kinetic energy equation is plotted, at regular time intervals, as a function of the inhomo-
geneous direction and the two-dimensional wave number for the homogeneous directions. The energy spectrum
manifests the beginning of an inertial range by the latter stages of the simulation. The production and dissi-
pation spectra become increasingly opposite and separate in wave space as the flow evolves. The transfer
spectrum depends strongly on the inhomogeneous direction, with the net transfer being from large to small
scales. Energy transfer at the bubble/spike fronts is strictly positive. Extensive cancellation occurs between the
pressure and advection terms. The dilatation term produces negligible energy transfer, but its overall effect is
to move energy from high to low density regions.
DOI: 10.1103/PhysRevE.66.026312 PACS number共s兲: 47.27.Ak, 47.27.Cn
I. INTRODUCTION
Rayleigh-Taylor instability 共RTI兲occurs at the interface
between two fluids of different densities whenever the
heavier fluid is decelerated by the lighter fluid 关1–4兴; i.e., if
density and pressure gradients are in opposite directions,
then vorticity, deposited at the interface through baroclinic
torque, will cause the fluids to interpenetrate and mix.
Richtmyer-Meshkov instability 共RMI兲corresponds to the
case of impulsive acceleration of an interface 关5,6兴, e.g.,
shock passage, and is sometimes considered a special case of
RTI 共with time-dependent acceleration兲. RTI presents a seri-
ous design challenge for inertial confinement fusion 共ICF兲
capsules, where high density shells are decelerated by low
density fuel. Depending on the acceleration history and the
ratio of shell radius to thickness, RTI may lead to break up of
the shell prior to ignition and/or significant mixing of the
fuel with the plastic ablator 关7兴. RTI also plays a prominent
role in supernovae, where ejecta are decelerated by circum-
stellar matter 关8,9兴. Furthermore, mixing from RTI alters
thermonuclear burn in supernovae in such a manner as to
affect the rates of formation of heavy elements; hence, the
relative abundance of elements in the universe, and the cor-
responding potential for life, are directly related to astro-
physical RTI mixing.
Most RTI research thus far has focused on predicting the
rate of growth of the turbulent mixing zone 关10–15兴. Mixing
zone amplitudes are routinely measured in high-energy laser
experiments conducted at very high Reynolds number 关16兴.
In its early stages, RTI growth is characterized by ‘‘spikes’’
of heavy fluid penetrating into light material and ‘‘bubbles’’
of light fluid rising into heavy material. In the strongly non-
linear stages, the bubbles and spikes merge to form larger
structures. If the only imposed length scale is from a constant
acceleration, then the mixing layer will grow quadratically in
time; e.g., a growth constant ‘‘
␣
’’ can be defined and mea-
sured 关17,18,10–13兴. However, if long wavelength perturba-
tions 共compared to domain size兲are present, the scaling
analysis is more complicated and growth may not be qua-
dratic 关19兴.
The range of scales participating in RTI dynamics con-
tinually grows as the flow evolves. Kelvin-Helmholtz insta-
bilities, occurring along the sides of the interpenetrating fin-
gers, along with vortex stretching and bending motions,
serve to broaden the energy spectrum. Eventually the flow
may become fully turbulent, while still remaining highly an-
isotropic. A complete description of the flow field requires
resolution down to the Kolmogorov scale 关20,21兴. Due to
limitations of diagnostics, laser experiments performed thus
far have not yielded much information on the internal struc-
ture of the mixing region. Larger-scale experiments 关22–24兴
have provided some information on mixing zone structure,
but lack the resolution necessary for a close investigation of
the energy budget.
Over the past three decades, direct numerical simulation
共DNS兲has emerged as an accepted surrogate for experiment
when detailed information, not readily measured in the labo-
ratory, is needed. DNS is restricted to low Reynolds number
flows, due to the limited range of wave numbers that can be
supported on a computational mesh. Nevertheless, it has
proven capable of following the three phases of turbulent
mixing identified by Eckart 关25兴, i.e., entrainment, stirring,
and molecular mixing. It also provides a complete, three-
dimensional, time-dependent description of the flow field.
DNS data of RTI flow can be used to test and/or tune models
for the overall growth of the mixing region, and for devel-
oping subgrid-scale 共SGS兲models for large eddy simulation.
The latter is intimately connected with energy transfer to and
from unresolved scales. The primary goal of this paper is to
gain insight into the energy transfer processes in order to
guide future SGS model development.
The outline of this paper is as follows. In Sec. II, the
PHYSICAL REVIEW E 66, 026312 共2002兲
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governing equations and solution technique of the DNS are
described. In Sec. III, flow visualizations are presented,
along with statistical measures of flow structure, such as
growth rates and Reynolds numbers. In Sec. IV, a variable-
density formulation is proposed for energy transfer analyses
previously performed for isotropic 关26–31兴, anisotropic 关32兴,
and wall-bounded 关33兴flows. The analysis is carried out in
Sec. V, where the spectrum of each term in the kinetic energy
equation is computed from the DNS data. Finally, conclu-
sions are given in Sec. VI.
II. DIRECT NUMERICAL SIMULATION
A. Governing equations
The conservation laws governing the flow of two incom-
pressible fluids in a gravitational field with no surface ten-
sion are
Yl
t⫹
ujYl
xj
⫽
xj
冉
D
Yl
xj
冊
共l⫽1,2兲,共1兲
ui
t⫹
uiuj
xj
⫽⫺
p
xi
⫹
ij
xj
⫹
gi,共2兲
where
ij⫽2
冉
Sij⫺1
3
␦
ij
uk
xk
冊
,
Sij⫽1
2
冉
ui
xj
⫹
uj
xi
冊
.
Here
is the mixture density, Ylis the mass fraction of
species l,uiis the mass-averaged mixture velocity, pis the
pressure, Dis the Fickian diffusivity,
is the dynamic vis-
cosity, and gi⫽(0,0,⫺g) is the acceleration. The mass frac-
tions satisfy
Y1共x,t兲⫹Y2共x,t兲⫽1共3兲
and, defining
1and
2to be the constant densities of the
light and heavy fluids, respectively, the specific volume sat-
isfies
1
共x,t兲⫽Y1共x,t兲
1
⫹Y2共x,t兲
2.共4兲
Equations 共1兲,共3兲, and 共4兲can be used to derive the follow-
ing divergence relation for miscible fluids 关34兴:
uj
xj
⫽⫺
xj
冉
D
xj
冊
.共5兲
Hence, for incompressible mixing, a convenient equation for
is
t⫹uj
xj
⫽
xj
冉
D
xj
冊
.共6兲
B. Solution technique
The equations were solved in nondimensional form, with
length measured in units of box width L, time measured in
units of
冑
L/g, and density measured in units of
1. The
diffusivity was set to D⫽
/
1, with the viscosity being
⫽512⫺4/3 共in the units just described兲. The numerical
scheme for solving the governing equations is described in
detail in 关19兴. In summary, the code computes xand yde-
rivatives spectrally via fast Fourier transform. The zderiva-
tives are computed with an eighth-order compact scheme
关35兴. Periodic boundary conditions are applied in xand y,
with no-slip walls imposed in z. The zgrid spacing is set to
8
13 times the grid spacing in xand y, in order to account for
the difference in resolving power between the spectral and
compact methods. Time advancement is accomplished via a
pressure-projection algorithm with third-order, Adams-
Bashforth-Moulton integration.
The simulation was performed on a computational mesh
with 512⫻512⫻2040 grid points in x,y, and zdirections,
respectively. The bottom 共17
32兲of the domain was initialized
with
⫽
1⫽1 and the top 共15
32兲portion with
⫽
2⫽3. The
density interface between the fluids was specified in the same
manner as in 关19兴, i.e., as an error function in z共five grid
points thick兲with isotropic perturbations in xand y. The
perturbed interface was initially located at the z⫽0 plane.
Diffusion velocities were initialized as in 关19兴in order to
satisfy the divergence relation 共5兲.
The spectral/compact algorithm was chosen to ensure that
numerical dissipation did not enter into the calculation. No
filtering or artificial diffusion of any kind was applied in the
simulation, i.e., the viscous and diffusive terms in Eqs. 共2兲
FIG. 1. 共Color兲Initial density perturbations on z⫽0 plane.
ANDREW W. COOK AND YE ZHOU PHYSICAL REVIEW E 66, 026312 共2002兲
026312-2
and 共6兲were solely responsible for energy dissipation. Con-
sequently, the simulation eventually became unstable 共for
Re⬎5500兲once significant energy reached the Nyquist wave
number. A prominent symptom of under-resolution 共and con-
sequent aliasing errors兲beyond Re⫽5500 is a curling up of
the energy spectrum at the highest wave numbers. The data
presented herein were selected at times prior to those where
significant aliasing errors occurred.
FIG. 2. 共Color兲Snapshots of density field from DNS of Rayleigh-Taylor instability. Images, proceeding from upper left to lower right,
were taken at t⫽1, 2, 3, 4, 5, and 6. The heavy fluid is red (
⫽3), the light fluid is blue (
⫽1), and the mixed fluid is green (
⫽2).
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III. FLOW STRUCTURE
A. Visualization
The initial perturbations are depicted in Fig. 1, which pro-
vides a top-down view of density on the z⫽0 plane at t
⫽0. The flow was seeded with fine scale perturbations to
minimize the influence of the periodic boundary conditions.
Figure 2 displays a time sequence of the density field. The
images, taken at unit time intervals, illustrate the evolution of
the fluid interface 共defined as the
⫽2 isosurface兲. The early
evolution is weakly nonlinear and is characterized by the
formation of bubbles of light fluid rising upwards and spikes
of heavy fluid penetrating downward. Later on, the bubbles
and spikes begin to merge and the flow becomes strongly
nonlinear.
The kinetic energy,
⬅
uiui/2, on the back, bottom, and
side planes of the flow domain is shown in Fig. 3 for t⫽3, 4,
5, and 6. It grows rapidly and appears to be fairly evenly
distributed across the mixing layer. The gravitational poten-
tial provides the source for kinetic energy production. Figure
4 displays the kinetic energy on the z⫽0 plane at the same
times. Like the density field, the kinetic energy is homoge-
neous and isotropic in xand y. Also, large values of
appear
to become concentrated in localized regions of the flow.
The source of vorticity 共and consequently energy genera-
tion兲is the baroclinic torque term in the vorticity equation,
which is nonzero only in the mixing region. The magnitude
of vorticity,
储
“⫻u
储
, and the magnitude of baroclinic torque,
储
“
⫻“p
储
/
,att⫽6, are plotted in Fig. 5. A few spikelike
and bubblelike structures are discernible in the fields; how-
FIG. 3. 共Color兲Kinetic energy 共
兲on side
boundary planes of DNS domain. Images were
taken at t⫽3, 共upper left兲,4共upper right兲,5
共lower left兲, and 6 共lower right兲.
ANDREW W. COOK AND YE ZHOU PHYSICAL REVIEW E 66, 026312 共2002兲
026312-4
ever, the fields are quite chaotic. Early on, when the bubbles
and spikes are growing more or less independently, and
hence are readily identified, vorticity 共and consequently en-
ergy兲is primarily generated along the sides of the structures.
However, by late time, the flow has become weakly turbulent
and vorticity generation occurs throughout the mixing zone.
The complexity of the baroclinic torque field increases as
centrifugal forces pull pressure gradients out of alignment
with the gravity vector.
B. Statistics
The penetration lengths of the bubbles and spikes, hb(t)
and hs(t), respectively, are defined as in 关19兴, i.e., by aver-
aging the heavy fluid mole fraction 关X⫽(
⫺
1)/(
2⫺
1)兴
in xand y, and measuring the distance from z⫽0 for which
具
X
典
xy⭐0.99 and
具
X
典
xy⭓0.01 共
具典
xy denoting horizontal aver-
age兲. The bubble and spike penetrations are plotted in Fig. 6.
The change in slope around t⫽2 occurs as modal growth
overtakes diffusive growth.
The outer-scale Reynolds number is plotted, versus time,
in Fig. 7. The Reynolds number is based on the vertical
extent of the mixing region, h⫽hb⫺hs, and its rate of
growth, h
˙, i.e.,
Re⫽共
1⫹
2兲h
˙h
2
.共7兲
The Reynolds number grows roughly like t3, which is ex-
pected if h⬃t2and h
˙⬃t.共Reasons why hmay depart from
quadratic growth are discussed in 关19兴.兲The terminal outer-
scale Reynolds number of 5500 is about a third of the Rey-
nolds number 关(1–2)⫻104] suggested by Dimotakis 关36兴as
the critical value for reaching the mixing transition and
achieving fully developed turbulence. It should be noted,
however, that the Reynolds number for mixing transition has
yet to be documented for RTI flow. If data above the mixing
transition could be obtained for RTI flow, it would be inter-
FIG. 4. 共Color兲Kinetic energy 共
兲on z⫽0 plane at t⫽3, 共upper left兲,4共upper right兲,5共lower left兲, and 6 共lower right兲.
ENERGY TRANSFER IN RAYLEIGH-TAYLOR INSTABILITY PHYSICAL REVIEW E 66, 026312 共2002兲
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esting to see if the ‘‘hot spots’’ in kinetic energy would per-
sist or if the energy would become more evenly distributed.
It is possible to define Taylor microscales and Reynolds
numbers for this flow in a manner that accommodates the
anisotropic forcing. A microscale in the idirection can be
defined as 关37,38兴
i⫽
冋
具
ui
2
典
xy
具
共
ui/
xi兲2
典
xy
册
1/2
共no sum on i兲,共8兲
with statistics computed in the (z⫽0) plane. With statistical
isotropy in the (z⫽0) plane, the xand ymicroscales are very
close and can be averaged to define a single horizontal mi-
croscale,
xy⫽x⫹y
2.共9兲
Figure 8 depicts the temporal growth of the vertical and
horizontal Taylor microscales in the (z⫽0) plane. The ver-
tical and horizontal scales both grow as the bubbles increase
in size, broadening the velocity correlation functions. The
difference between the vertical and horizontal scales gives a
FIG. 5. 共Color兲Magnitude of vorticity 共left兲and magnitude of baroclinic torque 共right兲at t⫽6.
FIG. 6. Amplitude of bubbles (hb) and spikes (hs) in the mixing
region. The total width of the layer is h⫽hb⫺hs.FIG. 7. Outer-scale Reynolds number, based on extent and rate
of growth of mixing region.
ANDREW W. COOK AND YE ZHOU PHYSICAL REVIEW E 66, 026312 共2002兲
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direct measure of anisotropy in the flow. The ratio z/xy
starts out near unity, increases during the diffusive growth
stage to a maximum value of about 3.7, and then appears to
asymptote to a value around 1.4 in the far nonlinear regime.
Figure 9 depicts the temporal evolution of the horizontal
and vertical Taylor Reynolds numbers on the (z⫽0) plane.
These are defined as
Re,i⫽
具
典
xyi关
具
ui
2
典
xy兴1/2
共no sum on i兲,共10兲
again, with spatial averages computed in the (z⫽0) plane.
As with the microscales, horizontal isotropy permits a hori-
zontal Taylor Reynolds number to be defined as the average
of Re,xand Re,y, i.e.,
Re,xy⫽Re,x⫹Re,y
2.共11兲
The anisotropy in microscales 共Fig. 8兲is also manifest in the
Taylor Reynolds numbers. A Taylor Reynolds number of
roughly 100 is required to cross the mixing transition 关36兴.
Judging from the proximity of the outer-scale Reynolds num-
ber to the critical value of 10000–20000, it appears that the
transition criterion would probably apply best to the horizon-
tal Taylor Reynolds number, rather than the vertical.
In order to quantify the degree of mixing within the layer,
a parameter, analogous to the Youngs 关13兴‘‘molecular mix-
ing fraction,’’ 共⌰兲 is defined. Assuming a passive, equilib-
rium chemical reaction between fluids, the chemical product
is
Xp⫽
再
X/Xsif X⭐Xs
共1⫺X兲/共1⫺Xs兲if X⬎Xs,共12兲
FIG. 8. Vertical 共labeled z兲and horizontal 共labeled xy兲Taylor
microscales on the z⫽0 plane.
FIG. 9. Vertical 共zlabel兲and horizontal 共xy label兲Taylor Rey-
nolds numbers on the z⫽0 plane.
FIG. 10. Mixing parameter, indicating the ratio of actual chemi-
cal product 共for a hypothetical infinite-rate reaction兲to the product
that would be formed if the fluid inside the mixing region were
completely mixed 共no xy variation兲.
FIG. 11. Evolution of two-dimensional energy spectrum 共E兲at
z⫽0.
ENERGY TRANSFER IN RAYLEIGH-TAYLOR INSTABILITY PHYSICAL REVIEW E 66, 026312 共2002兲
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where Xsis the 共heavy-fluid兲mole fraction for a stoichio-
metric mixture, here taken as X⫽1
2. A mixing parameter, ⌶,
is defined as the ratio of mixed to entrained fluid, i.e.,
⌶⫽
冕
H
具
Xp共X兲
典
xydz
冕
HXp共
具
X
典
xy兲dz,共13兲
where His the height of the flow domain. Thus, ⌶⫽1 indi-
cates fluids are completely mixed within the mixing zone;
whereas, ⌶⫽0 corresponds to fully segregated fluids 共im-
miscible case兲. Note that with no perturbations (X⫽
具
X
典
xy),
for an increasingly sharp interface, the numerator and de-
nominator go to zero at the same rate; hence, ⌶⫽1 for a
Heaviside function. The mixing parameter ⌶is plotted in
Fig. 10 as a function of time. Initially, the layer is diffuse
with small amplitude perturbations; hence, ⌶starts out near
unity. As the perturbations grow, they entrain fluid at a rate
proportional to their wavelengths 共longer wavelengths result
in bigger ‘‘gulps’’ of pure fluid兲. At early times, the rate of
entrainment exceeds the rate of mixing and ⌶decreases.
This is a consequence of the fact that, early on, the surface
area across which the fluids can diffuse is relatively small.
However, later on the interface begins to wrinkle due to
baroclinic vorticity 共mushroom caps兲and Kelvin-Helmholtz
instabilities in the shearing regions along the mushroom
necks; the interfacial surface area then rapidly increases, the
mixing rate overtakes the entrainment rate, and the curve
reverses direction. The curve appears to asymptote to a value
somewhere around 0.8; however, mixing and entrainment
rates have not come into balance within the time span of the
simulation. Furthermore, this curve is likely to rise after the
mixing transition occurs.
IV. ENERGY BUDGET
In order to extend the methodology of constant-density
energetics to the variable-density case, a new variable is in-
troduced, i.e.,
vi⬅
1/2ui,共14兲
such that, the kinetic energy may be written as
⫽vivi/2.
This variable has been used for similar purposes by various
authors 关39–41兴. The left-hand side of the Navier-Stokes
equation can then be written as
ui
t⫹
uiuj
xj
⫽
1/2
vi
t⫹1
2
⫺1/2vi
t
⫹
1/2
viuj
xj
⫹1
2
⫺1/2viuj
xj
⫽
1/2
冉
vi
t⫹
viuj
xj
⫺1
2vi
uk
xk
冊
,
such that the transport equation for vibecomes
vi
t⫽
1/2gi⫺
viuj
xj
⫺
⫺1/2
p
xi
⫹1
2vi
uk
xk
⫹2
⫺1/2
xj
冋
冉
Sij⫺1
3
␦
ij
uk
xk
冊
册
.共15兲
For simplicity, Eq. 共15兲is rewritten as
vi
t⫽Fi⫹Ni⫹Di,共16兲
where the buoyant forcing is
Fi⫽
1/2gi,共17兲
the nonlinear 共quadratic, pressure, and dilatation兲contribu-
tion is
Ni⫽⫺
viuj
xj
⫺
⫺1/2
p
xi
⫹1
2vi
uk
xk,共18兲
and the viscous diffusion is
Di⫽2
⫺1/2
xj
冋
冉
Sij⫺1
3
␦
ij
uk
xk
冊
册
.共19兲
FIG. 12. 共Color兲Energy spectrum 共E兲versus z共vertical兲and
log10(k)共horizontal兲at t⫽3共top left兲,4共top right兲,5共lower left兲,
and 6 共lower right兲. Blue⫽0 and red⫽5⫻10⫺4.
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026312-8
Now let v
ˆi,F
ˆi,N
ˆi, and D
ˆibe horizontal Fourier transforms
of vi,Fi,Ni, and Di, respectively; e.g., v
ˆi(k,z,t)
⫽Fxy
兵
vi(x,t)
其
, where k⫽(kx,ky) is the horizontal wave
vector. If v
ˆi
*(k,z,t) denotes the complex conjugate of
v
ˆi(k,z,t), then multiplying the transform of Eq. 共16兲, and its
conjugate, by v
ˆi
*and v
ˆi, respectively, and adding the equa-
tions together yields
v
ˆi
*v
ˆi
t⫽v
ˆi
*F
ˆi⫹v
ˆiF
ˆi
*⫹v
ˆi
*N
ˆi⫹v
ˆiN
ˆi
*⫹v
ˆi
*D
ˆi⫹v
ˆiD
ˆi
*.
共20兲
Integrating Eq. 共20兲over Fourier annuli of radius k
⫽
冑
kx
2⫹ky
2leads to the energy budget equation
tE共k,z,t兲⫽⌸共k,z,t兲⫹T共k,z,t兲⫹E共k,z,t兲,共21兲
where the kinetic energy is
E⫽1
2
冖
共v
ˆi
*v
ˆi兲d
,共22兲
the production from gravity is
⌸⫽1
2
冖
共v
ˆi
*F
ˆi⫹v
ˆiF
ˆi
*兲d
,共23兲
the nonlinear transfer is
T⫽1
2
冖
共v
ˆi
*N
ˆi⫹v
ˆiN
ˆi
*兲d
,共24兲
and the viscous dissipation is
E⫽1
2
冖
共v
ˆi
*D
ˆi⫹v
ˆiD
ˆi
*兲d
,共25兲
with d
being a differential element of a wave space annu-
lus.
There are three contributions to the nonlinear energy
transfer. The first is from the quadratic term, which is respon-
sible for passive-vector advection, while the second and third
contributions are from pressure and dilatation effects. In or-
der to ascertain the relative importance of each process, the
total nonlinear transfer is subdivided into each individual
component, i.e.,
T共k,z,t兲⫽Tq共k,z,t兲⫹Tp共k,z,t兲⫹Td共k,z,t兲,共26兲
where
FIG. 13. 共Color兲Production 共⌸: left兲, transfer 共T: middle兲, and
dissipation 共E: right兲spectra versus z共vertical兲and log10(k)共hori-
zontal兲at t⫽3共top兲and 4 共bottom兲. Blue⫽⫺2⫻10⫺4, green⫽0,
and red⫽2⫻10⫺4.
FIG. 14. 共Color兲Production 共⌸: left兲, transfer 共T: middle兲, and
dissipation 共E: right兲spectra versus z共vertical兲and log10(k)共hori-
zontal兲at t⫽5共top兲and 6 共bottom兲. Blue⫽⫺2⫻10⫺4, green⫽0,
and red⫽2⫻10⫺4.
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Tm⫽1
2
冖
共v
ˆi
*N
ˆm,i⫹v
ˆiN
ˆm,i
*兲d
共m⫽q,p,d兲共27兲
and
Nq,i⫽⫺
viuj
xj,共28兲
Np,i⫽⫺
⫺1/2
p
xi,共29兲
Nd,i⫽1
2vi
uk
xk.共30兲
V. SPECTRA
The time evolution of the two-dimensional energy spec-
trum, E(k,z⫽0,t⫽0,2,4,6), is plotted in Fig. 11. The initial
diffusion velocities result in a nonzero spectrum at t⫽0. The
spectrum increases by several orders of magnitude as kinetic
energy is deposited into the flow. The peak of the spectrum
migrates toward lower wave numbers as bubbles and spikes
merge to form larger structures. The spectrum also fills out at
higher wave numbers as vortex stretching and bending mo-
tions transfer energy to smaller scales. Near the end of the
simulation it appears that an inertial range is just beginning
to form. In 关42兴, Zhou argues that the inertial range for the
RTI flow will follow a ⫺7
4power law. He reasons that the
spectrum will be modified from the classical Kolmogorov
共⫺5
3兲power law for isotropic turbulence as a result of the
external time scale introduced by gravity. Both power laws
are shown on the figure for comparison to the data. The
late-time spectra appear somewhat consistent with Zhou’s
theory; however, the statistical fluctuations in the spectra
共there are not many points in the Fourier annuli at lower
wavenumbers兲are larger than the difference in slope be-
tween k⫺7/4 and k⫺5/3. Furthermore, there is no clear begin-
ning to the dissipation range, which appears to extend well
into the lower wave numbers, thereby steepening the slope of
the spectrum. Due to the closeness of the power laws, much
higher Reynolds number data and improved statistics will be
needed in order to discriminate between the two.
The zdependence of the energy spectrum is graphically
portrayed in Fig. 12 at four different times. The plots are in a
semi-Fourier domain with log10(k) along the abscissa and z
along the ordinate. The bulk of the energy is initially depos-
ited at a moderate wave number, corresponding to the domi-
nant wavelength of the density perturbations. As time
progresses, E(k,z,t) expands in both kand zbut maintains
its maximum value close to z⫽0. The spectrum decreases
near the edges of the mixing zone, becoming negligible out-
side it.
The production 共⌸兲, transfer 共T兲, and dissipation 共E兲spec-
FIG. 15. 共Color兲Quadratic (Tq: left兲and pressure (Tp: right兲
contributions to transfer spectrum versus z共vertical兲and log10(k)
共horizontal兲at t⫽3共top兲and 4 共bottom兲. Blue⫽⫺5⫻10⫺4,
green⫽0, and red⫽5⫻10⫺4.
FIG. 16. 共Color兲Quadratic (Tq: left兲and pressure (Tp: right兲
contributions to transfer spectrum versus z共vertical兲and log10(k)
共horizontal兲at t⫽5共top兲, and 6 共bottom兲. Blue⫽⫺5⫻10⫺4,
green⫽0, and red⫽5⫻10⫺4.
ANDREW W. COOK AND YE ZHOU PHYSICAL REVIEW E 66, 026312 共2002兲
026312-10
tra are plotted in Figs. 13 and 14 on the same zversus
log10(k) domain for the same times. Early on, the peaks of
the production and dissipation spectra are close to one an-
other, but later the production moves toward lower wave
numbers while the peak of the dissipation spectrum stays
roughly fixed. By the end of the simulation, there is some k
separation between the two, i.e., as structures merge within
the mixing layer, energy is deposited at larger scales. The
increasing separation of peaks between the production and
dissipation spectra is a direct result of the increasing Rey-
nolds number for this transitional flow. As time progresses,
both spectra advance in zand k, becoming increasingly large
and opposite.
In contrast to the straightforward nature of ⌸and E, the
transfer spectrum behaves in a very complicated manner. It
exhibits an intricate web of positive and negative regions,
interspersed over a wide range in zand k. At higher wave
numbers Tis mostly positive, indicating a net cascade of
energy to smaller scales. It is also positive at the top and
bottom of the mixing zone, suggesting production of energy
at the bubble and spike fronts. Inside the mixing zone, back-
scatter appears approximately equal to forwardscatter. It fur-
ther appears that, at each instant in time, some zlocations
may be undergoing forward energy cascade, while neighbor-
ing regions are simultaneously experiencing inverse cascade.
In order to unravel the irregular patchwork that constitutes
T, each individual component 共Tq,Tp, and Td兲is plotted
separately. In Figs. 15 and 16, Tqand Tpare plotted on the
same semi-Fourier domain and at the same times as before.
The dilatation spectrum, Td, is plotted 共at all four times兲in
Fig. 17. The dilatation spectrum is roughly two orders of
magnitude smaller than Tqand Tp, and hence, makes negli-
gible contribution to T. Interestingly, the quadratic (Tq) and
pressure (Tp) components are near opposites of one another,
their net contribution to Tbeing the result of extensive can-
cellation between the two. The quadratic term is mostly
negative at lower wave numbers inside the mixing zone and
mostly positive at higher wave numbers and near the edges
of the mixing envelope. Regarding the pressure term, nearly
the opposite is true, except at higher wave numbers where
positive and negative regions appear to be roughly equally
distributed. The net positive transfer of energy to the higher
wave numbers 共usual cascade picture兲and to the bubble/
spike fronts is thus a result of quadratic interactions; with
pressure counterbalancing advection, for the most part.
As regards the dilatation, Td, although its influence on the
energetics is likely negligible, it is interesting to observe that
it is strongly polarized, i.e., positive for z⬍0共spike region兲
and negative for z⬎0共bubble region兲. The velocity diver-
gence is related to diffusion through Eq. 共5兲. The net effect of
diffusion is to increase the density of fluid in the lower re-
gion and decrease the density of fluid in the upper region.
This results in a net transfer of energy in the ⫺zdirection
due to diffusion of heavy fluid into light fluid. At late times,
vigorous stirring at the center of the mixing zone causes the
positive and negative regions of Tdto overlap.
VI. CONCLUSIONS
We have examined the flow structure and energy budget
for Rayleigh-Taylor instability using the results of a high-
resolution direct numerical simulation. The outer-scale Rey-
nolds number was observed to follow a t3power law and
reached a final value of 5500, the highest Reynolds number
attained in a DNS or RTI flow to date. The curvature of
velocity correlation functions, as manifest in the Taylor mi-
croscales, exhibits strong anisotropy between the vertical and
horizontal directions, with a similar anisotropy observed in
the Taylor Reynolds numbers. This is due to the directed
forcing term in the governing equations. The energy spec-
trum, computed at the center of the mixing zone, appears to
manifest the beginning of an inertial range by the latter
stages of the simulation. Unfortunately, statistical fluctua-
tions in the spectrum make it difficult to establish whether
the inertial range follows a ⫺5
3Kolmogorov power law or
the ⫺7
4power law proposed by Zhou 关42兴.
A formulation of the kinetic energy equation was pro-
posed, which enables straightforward extension of method-
ologies commonly employed for constant-density, isotropic
turbulence. The spectrum of each term in the energy equation
was computed as a function of height, horizontal wave num-
ber, and time. The peak of the energy spectrum migrates to
lower wave numbers as structures merge inside the mixing
layer. The production spectrum also moves to lower wave
numbers as gravity acts on the larger structures. The dissipa-
tion spectrum expands in both zand kwhile its peak stays
roughly fixed in wave number space. The limited Reynolds
number of the DNS appears to inhibit movement of the peak
FIG. 17. 共Color兲Dilatation (Td) component of transfer spectrum
versus z共vertical兲and log10(k)共horizontal兲at t⫽3共top left兲,4共top
right兲,5共lower left兲,and6共lower right兲. Blue⫽⫺7⫻10⫺6, green
⫽0, and red⫽7⫻10⫺6.
ENERGY TRANSFER IN RAYLEIGH-TAYLOR INSTABILITY PHYSICAL REVIEW E 66, 026312 共2002兲
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dissipation to higher wave numbers.
The transfer spectrum depends strongly on the inhomoge-
neous direction z. While the net energy transfer is from large
to small scales, there is significant inverse cascade over a
wide range of z. Energy transfer at the bubble/spike fronts is
strictly positive. Examination of the individual contributions
to the transfer reveals this to be due to the quadratic 共advec-
tion兲term. Pressure acts to counterbalance advection, such
that the net transfer is substantially smaller than the transfer
from either single component. The dilatation term accounts
for energy transfer via diffusion of unequal-density fluids. It
is very small but serves to move energy from high to low
density regions.
The flow induced by Rayleigh-Taylor instability, as seen
here, has rather different character than that of homogeneous,
isotropic turbulence. The flow is highly anisotropic, even at
small scales, as evidenced by the Taylor microscales and
Reynolds numbers. Initial rates of entrainment and mixing
are determined by the initial conditions. Production rates al-
ways exceed dissipation rates; hence, the kinetic energy
grows rapidly in time. Furthermore, the evolution of the
spectra depends sensitively on initial conditions; since the
interfacial perturbations set the scale at which energy is ini-
tially injected into the flow. This has serious consequences
for large eddy simulations, where the initial perturbations
may be much smaller than the grid scale. In such cases,
subgrid-scale models must be capable of treating not only
backscatter, but also growth of spectra below the grid scale
and migration of the energy peak through the cutoff wave
number. The detailed analysis of the spectra, performed
herein, serves as a step toward developing subgrid-scale
models capable of treating RTI flows in ICF and astrophysics
applications.
ACKNOWLEDGMENTS
This work was performed under the auspices of the U.S.
Department of Energy by the University of California,
Lawrence Livermore National Laboratory, under Contract
No. W-7405-Eng-48. Additionally, we wish to thank Profes-
sor P. E. Dimotakis for many stimulating discussions on this
topic and for providing keen insights into this flow.
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