Initial conditions for turbulent mixing simulations
In the context of the classical Rayleigh-Taylor hydrodynamical instability, we examine the much debated question of models for initial conditions and the possible influence of unrecorded long wave length contributions to the instability growth rate α.
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irk-123456789-321252012-04-10T12:15:53Z Initial conditions for turbulent mixing simulations Kaman, T. Glimm, J. Sharp, D.H. In the context of the classical Rayleigh-Taylor hydrodynamical instability, we examine the much debated question of models for initial conditions and the possible influence of unrecorded long wave length contributions to the instability growth rate α. У контексті класичної гідродинамічної нестійкості Релея - Тейлора вивчено дискусійне питання моделей для початкових умов і можливий вплив незафіксованих далекосяжних вкладів на швидкість росту нестійкості альфа. 2010 Article Initial conditions for turbulent mixing simulations / T. Kaman, J. Glimm, D.H. Sharp // Condensed Matter Physics. — 2010. — Т. 13, № 4. — С. 43401:1-7. — Бібліогр.: 15 назв. — англ. 1607-324X PACS: 47.27.wj, 47.27.tb, 47.27-i http://dspace.nbuv.gov.ua/handle/123456789/32125 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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In the context of the classical Rayleigh-Taylor hydrodynamical instability, we examine the much debated question of models for initial conditions and the possible influence of unrecorded long wave length contributions to the instability growth rate α. |
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Kaman, T. Glimm, J. Sharp, D.H. |
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Kaman, T. Glimm, J. Sharp, D.H. Initial conditions for turbulent mixing simulations Condensed Matter Physics |
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Kaman, T. Glimm, J. Sharp, D.H. |
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Initial conditions for turbulent mixing simulations |
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Initial conditions for turbulent mixing simulations |
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Initial conditions for turbulent mixing simulations |
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Initial conditions for turbulent mixing simulations |
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initial conditions for turbulent mixing simulations |
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Initial conditions for turbulent mixing simulations / T. Kaman, J. Glimm, D.H. Sharp // Condensed Matter Physics. — 2010. — Т. 13, № 4. — С. 43401:1-7. — Бібліогр.: 15 назв. — англ. |
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Condensed Matter Physics |
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AT kamant initialconditionsforturbulentmixingsimulations AT glimmj initialconditionsforturbulentmixingsimulations AT sharpdh initialconditionsforturbulentmixingsimulations |
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Condensed Matter Physics 2010, Vol. 13, No 4, 43401: 1–7
http://www.icmp.lviv.ua/journal
Initial conditions for turbulent mixing simulations
T. Kaman1, J. Glimm1,2, D.H. Sharp3
1 Department of Applied Mathematics and Statistics, Stony Brook University,
Stony Brook, NY 11794–3600, USA
2 Computational Science Center, Brookhaven National Laboratory, Upton, NY 11793–6000, USA
3 Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Received June 23, 2010
In the context of the classical Rayleigh-Taylor hydrodynamical instability, we examine the much debated ques-
tion of models for initial conditions and the possible influence of unrecorded long wave length contributions to
the instability growth rate α.
Key words: turbulent mixing layers, turbulent diffusion
PACS: 47.27.wj, 47.27.tb, 47.27-i
1. Introduction
Turbulent mixing has been a challenge for sixty years, attracting the talents of leading physicists
and mathematicians. Recently we have proposed a new LES algorithm, [1, 2] based on Front
Tracking [3, 4] (to control unphysical numerical diffusion of species mixture) and subgrid scale
(SGS) models to regularize the effects of the unresolved scales, as they influence the resolved ones
[5]. We have success with matching simulation with experiment according to the following criteria:
• simulation agreement with 14 different experiments [2, 6–8],
• agreement with sufficient accuracy (often) to distinguish between distinct experiments [8],
• agreement for both immiscible and miscible experiments, the latter for high, moderate and
low Schmidt number [8, 9],
• agreement both where initial conditions were recorded (and used to initialize the simulation)
and where they were not [8],
• agreement with other simulations in one of the few cases where simulation-experiment agree-
ment was achieved by others [8].
We have presented numerical evidence that the mixing rate α is nonuniversal, in that it depends
on at least six nondimensional groups [9]. It depends on long wave contributions to the initial
interface perturbation, as is generally acknowledged, and has been shown by others. We show that
α also depends on two dimensionless characterizations of the short wave initial conditions, namely
the ratios of the maximally unstable wave length λ = λc and of the thickness of the initial diffusion
layer to the dominant short wave length perturbation. We also show that it depends on the fluid
transport properties (Schmidt, Prandtl and Grashof numbers) [8, 9]. Here the maximally unstable
wave length λc comes from the theory of dispersion relations. Dispersion theory is a theoretical
solution of the Rayleigh-Taylor instability problem, linearized for small perturbation amplitudes;
its solution is a growth rate as a function of wave length. The mixing rate α is defined in terms of
the penetration distance h of the light fluid into the heavy fluid (“bubbles”), via the formula
h = αAgt2, (1)
c© T. Kaman, J. Glimm, D.H. Sharp 43401-1
http://www.icmp.lviv.ua/journal
T. Kaman, J. Glimm, D.H. Sharp
where g denotes the acceleration of a fluid discontinuity and the Atwood number is a dimensionless
measure of the density contrast between the two fluids, A = (ρ2 − ρ1)/(ρ2 + ρ1).
The ongoing debate over α centers on initial conditions and can perhaps be summarized as
follows:
(i) Fourier spectral amplitudes A(k)2 ∼ k−2 and k−3 have been proposed for the initial pertur-
bations to yield the growth law (1) based on superposition of modes, nonlinear single mode
RT growth rates of the Fourier modes, and saturation of this growth at larger amplitudes
[10, 11]. The difference between the two proposed exponents for the spectral amplitudes has
to do with the lumping of modes for adjacent frequencies k into a single effective mode.
(ii) Given that long wave length perturbations do influence the growth rate α, how can valid
experimental-simulation comparisons be achieved for experiments for which these initial con-
ditions were not recorded?
(iii) In the high Re limit, dependence on nonuniversal fluid parameters such as the Schmidt
number should disappear, and then α becomes universal, other than its dependence on initial
conditions.
(iv) Many simulation codes achieve growth rates with an α as in (1) half or less than the experi-
mentally observed values when initialized without long wave length perturbations. Consensus
of simulation codes has been claimed as an indication of scientific truth.
The first goal of this paper is to examine the k−3 or k−2 hypothesis of (i) in the context of
experimental data [12, 13]. We find evidence for a negative exponent, A2 ∼ k−a, a ≈ 4 at the initial
time. We note a strong time dependence for the relative spectral amplitudes, with A2 ∼ k0 at the
early time (third plate) plot, and the fraction of long wave spectral energy dropping from 50% to
80% initially to about 2% at the time of the third plate. k−2 and k−3/2 A2 initial spectra have
been observed in the machined surfaces present in plasma experiments [14]. The second purpose
of this paper is to develop a plan for use of experimental data in simulation initialization.
Concerning (iii), our validation-simulation tests [8] show a 20% decrease in α as Sc changes
from 560 to 1 at Re = 40, 000, thus documenting a nonuniversal and Sc dependent α well above
the transition value Re = 10, 000 sometimes cited as a universal Reynolds number for the rapid
transition to hard turbulence.
Item (iv) was previously discussed [8, 9]. Briefly, most RT simulations neglect the influence of
numerical mass diffusion and of the six dimensionless groups which have been shown to influence
the value of α, beyond the fact that the argument in (iv) is prima facie weak, and generally not
acceptable as a basis for scientific conclusions.
2. Scaling laws for initial perturbation amplitudes
The essence of our approach for determining experimental initial perturbations is to transfer
information from the experimentally observed early time RT experimental photographs to time
t = t0 = 0. We start by measuring the locations of all bubble minima in the third experimental
plate. As is standard in the analysis of these experiments, the edge (corner) bubbles are not included
as they have an exceptional amplitude. For confirmation purposes, we also record the bubble
minima in the second experimental plate where they are clear enough to be reliable (experiment
#105 [13]), or in the fourth plate when the second plate is not clear (#56 and #63 [12] and
#104, #114 [13]). The bubble minima heights are Fourier analyzed, giving Fourier amplitudes as a
function of wave number n. The mode n = 0 represents the mean bubble position, and it is clearly
the dominant short wave length datum. The other modes can be regarded as long and short wave
length perturbations of this dominant signal. These other modes are generally of small amplitude
and (especially if the methodology of (i) is to be accepted), have a time evolution governed by
superposition of the propagation of single mode solutions for the RT fluid equations. The linear
propagation is based on the theory of dispersion relations, which defines an exponential growth
43401-2
Initial conditions for turbulent mixing simulations
n
A
(n
)
0 5 10 15
0.2
0.4
0.6
0.8
1
1.2
#56 alpha=0.058
#63 alpha=0.069
#104 alpha=0.068
#105 alpha=0.072
#114 alpha=0.060
n
A
(n
)
0 5 10 15
0
0.002
0.004
0.006
0.008
0.01
0.012
#56 alpha=0.058
#63 alpha=0.069
#104 alpha=0.068
#105 alpha=0.072
#114 alpha=0.060
Figure 1. Left: measured values for the spectral amplitude at t = t3. Right: t = t0 inferred
spectral amplitudes A(n) (cm) for the bubble minima for several experiments [12, 13] vs. n.
Here and below, wave numbers are cut off at the Nyquist value nmax/2.
rate. Using this theory, we transfer time tj mode amplitudes back to time t0, where tj denotes
the time of the jth plate, for j > 1. The criteria for choosing linear vs. nonlinear single mode
propagation is based on the ratio of wave length to amplitude, and as this is ratio not too large,
the simpler linear theory is used.
The mode n = 0 is assigned the most rapid growth rate, as it represents the dominant bubble
size. Since this mode is never small at the time of the third experimental plate, we proceed differ-
ently in this case. The maximum bubble location has been recorded experimentally, and we start
from the earliest of these recorded values for the backward in time propagation of the n = 0 mode.
There is a difference between the mean bubble height, referred to here, and the maximum bubble
height, as recorded experimentally. In previous analysis of the same data as considered here [15],
we observed a time independent ratio for the maximum to mean bubble height. Using this ratio,
we infer the mean bubble height at the earliest recorded time. Using this mean bubble minima at
this time, we find that the height to wave length ratio allows propagation back to time t = t0 by
linear RT single mode analysis. See figure 1.
To split the t = t0 or t = tj modes into long and short wave lengths, we combine the n = 0 and
the collective short wave length signal into a combined short wave length mode and we combine
the rest into a long wave length signal. The wave length interval [ 2
3
λc,
4
3
λc] comprises the short
wave length spectrum. Using the relation n = L/λ where L = 15 for all experiments considered
here, the short wave number n interval is [3L/4λc, 3L/2λc]. Larger n values, if any, are included
in the short wave length spectrum, as is the n = 0 value. The smaller n values (other than n = 0)
describe the long wave length perturbation, often referred to as “long wave length noise”. The long
wave length spectrum comprises most of figure 1, in the region to the left, other than the n = 0
value.
The assessment (i) comes from plotting the long wave length data only, with log-log scaled
coordinates. In the log-log plot, see figure 2, the exponent of n is the slope of the data. There is
over a decade of long wave length n values, sufficient for the approximate determination of the
slope, but the noise in the data only allows an approximate assessment of the spectral exponent,
A(n)2 ∼ n−a, a ≈ 4. The power law with amplitudes A(n)2 ∼ n−a, a = 2 or a = 3 has been
mentioned especially, as, if substituted into a dispersion relation growth rate, with a cut off in
growth at a saturation amplitude A ≈ λ, a Rayleigh-Taylor growth rate h ∼ Agt2 can be obtained
through a purely theoretical argument. The exponent a is, however, strongly time dependent, and
for most of the experimentally self similar mixing regime, it is close to zero. See table 1. Thus
the theory that the observed self similar RT growth rates can be determined from the long wave
length perturbations in the initial data is inconsistent with the experimental data. The alternate
43401-3
T. Kaman, J. Glimm, D.H. Sharp
n
A
(n
)
100 101 10210-4
10-3
10-2
10-1
100
#56 alpha=0.058
#63 alpha=0.069
#104 alpha=0.068
#105 alpha=0.072
#114 alpha=0.060
n
A
(n
)
100 101 10210-6
10-5
10-4
10-3
10-2
10-1
#56 alpha=0.058
#63 alpha=0.069
#104 alpha=0.068
#105 alpha=0.072
#114 alpha=0.060
Figure 2. Log-log plot of amplitudes vs. wave number, excluding n = 0, inferred at t = t0. This
data satisfies a power law A(n)2 ∼ n−a, a ≈ 4. Left: measured data from t = t3. Right: inferred
data for t = t0.
n
A
(n
)
100 101 10210-6
10-5
10-4
10-3
10-2
10-1
#104 t3->t0
#104 t4->t0
#104 t5->t0
#104 t6->t0
n
A
(n
)
100 101 10210-4
10-3
10-2
10-1
100
#104 t3
#104 t4->t3
#104 t5->t3
#104 t6->t3
Figure 3. Log-log plot of amplitudes vs. wave number, excluding n = 0, from experiment #104.
Left: predictions of t = t0 data starting from different times t = tj . Right: Prediction of the
measured t = t3 data starting from different t = tj .
explanation of RT growth as due to bubble merger [15] remains the only theoretical explanation for
the observed RT growth rates that is consistent with experimental data. To assess the accuracy of
this time transfer of spectral amplitudes, we use the method to predict the results with known data,
and we use it to predict the same unknown data, but from different starting times, see figure 3.
3. Initialization of RT simulations
The second purpose of this paper is to construct initial conditions for new RT simulations that
reflect experimental initial conditions. Thereby, we address item (ii).
Carrying out the transfer of spectral amplitudes from t = t3 to t = t0, we have spectral
amplitudes and relative amplitudes for the long wave length perturbation, at the time of the
third experimental plate (directly observed), and at an experimental or simulation initial time. We
introduce the L2 norms squared (spectral energies) L2 = A2
long and S2 = A2
short of the long and
short wave length portions of the Fourier spectrum. Also we define T and R: T 2 = L2 + S2 and
R2 = L2/T 2. To specify the normalization, we define L2 = A2
long =
∑
k,long A(k)
2.
43401-4
Initial conditions for turbulent mixing simulations
Table 1. Spectral exponents for A2 at experimental initial and recorded data times.
Exp. #56 [12] #63 [12] #104 [13] #105 [13] #114 [13]
t3,0 –4.5 –4.1 –4.3 –3.0 –3.8
t4,0 –3.9 –3.6 –4.3 –3.6 –3.6
t5,0 –4.0 –3.2 –4.0 –3.6 –3.0
t3,1 –2.5 –2.3 –3.6 –2.8 –2.7
t3,2 –1.4 –1.3 –1.7 –0.9 –1.6
t3 –0.4 –0.2 –0.4 –0.1 –0.4
t4 –0.3 –0.4 –0.2 –0.2 –0.1
t5 –0.5 –0.1 –0.2 –0.3 –0.4
Table 2. Long and short wave length spectral energies, observed directly at t = t3 and t2 or
t4, and inferred/confirmed from this data for t = t0. Also shown is the relative difference ∆
in these two inferences for t = t0 data. Experiments #56 and #104 had an apparent small
amplitude long wave length perturbation from visual observation of the data, while #63 was
vibrated mechanically to induce a long wave length perturbation. Experiments #105 and #114
had no experimentally noted long wave length nor was any detectable from examination of the
early time plates.
Exp. #56 [12] #63 [12] #104 [13] #105 [13] #114 [13]
L2
t3 0.022 0.031 0.010 0.0058 0.0054
S2
t3 1.152 0.977 0.551 0.334 0.422
T 2
t3 1.174 1.008 0.561 0.339 0.427
R2
t3 0.019 0.031 0.018 0.017 0.0125
R2
t4 0.067 0.058 0.0224 0.009 0.0128
R2
3,0 0.998 0.9999 0.9997 0.947 0.998
R2
4,0 0.999 0.9999 0.9996 0.939 0.995
∆R2
0 0.001 0.0000 0.0001 0.008 0.003
In table 2, we present the related L2 norms squared (spectral energies) for the rocket rig data.
We define L2
tj as the long wave length L2 norm squared as measured and Fourier analyzed for
t = tj , j = 3, 4. We infer t = t0 spectral energies from those at t = t3. All data obtained from
t = t3 is confirmed by an independent analysis using t = t4 data and t = t2 data where possible.
These inferred spectral energies are labeled L2
j,0 = L2
tj→t0 . We define similar short wave length
and total spectral energies S2 and T 2 and ratios R2. As a justification for these constructions, we
compare the differences in the R2
j,0, defining ∆R2
0 = |R2
3,0 − R2
4,0|, or a similar expression using
t = t2 data if available. It is interesting to note the striking decrease in the long wave relative long
wave length spectral energy R2 from t = 0 to t = t3.
Conventional initialization of Rayleigh-Taylor mixing is to use uniform amplitudes over the
short wave length interval with random phases. This can be referred to as “ideal random initial
conditions”. At the time t = t3 of the third experimental plate, the observed wave length for the
experiments #104, #105, #114 is approximately equal to λc, indicating that no mode doubling
has occurred at this time. The observed wave length at the time t = t3 for experiments #56 and
#63 is about two times λc, indicating one round of mode doubling has occurred at the time t = t3.
The Fourier modes (especially the long wave length ones in the above sense) can be used
to provide an improved initialization. We propose to use the inferred long wave length Fourier
amplitudes and phases at a tsim = 0, with tsim defined through limiting the fastest growing mode to
small amplitude. These amplitudes will be combined with an ideal short wave length perturbation
as initialization, with the relative amplitudes of the two governed by the ratios computed as in
table 2. This construction yields data for only one of the required two dimensions for initialization.
43401-5
T. Kaman, J. Glimm, D.H. Sharp
Table 3. Simulation-experiment comparisons for various experiments. The wave length in column
3 (units cm) comes from dispersion relations in which all physical transport parameters are used.
Ref. # λc (cm) αexp αsim
[12] #56 0.27 0.058 0.070
[12] #63 0.27 0.069 0.070
[13] #104 0.43 0.068 0.070
[13] #105 0.46 0.072 0.070
[13] #114 0.44 0.060 0.070
The other horizontal direction, not visible in the experimental plates, has one third the size of
the visible direction, 5 cm vs. 15 cm. Thus the experiment does not allow for long wave length
perturbations in this direction, and only marginally allows medium length perturbations outside
the range of the ideal short wave length perturbations. We propose to choose an ideal short wave
length initialization in this direction.
The simulation results in the last column of table 3 were performed with ideal random initial
conditions. The most unstable wave length λc, which determines these initial conditions, was set
by dispersion relations using surface tension alone, i.e. without corrections due to viscosity. It is
for this reason that one simulation [6] is used to model all the experiments in table 3.
As we prepare to repeat the simulations of [6], we propose to:
• determine λc from complete physics (use of distinct viscosities in the two fluids) to contribute
to the setting of the length scale for the ideal random short wave length initial conditions,
• use the inferred long wave length perturbations to set amplitude and phases for the long
wave length perturbations in the observed direction combined with ideal short wave length
perturbations in both directions,
• use laminar and turbulent viscosity of the experimental fluids with an LES simulation with
dynamic subgrid models.
In conclusion, we have shown that the power law A(n)2 ∼ n−a, a ≈ 4, is consistent with
experimental data. We have shown a striking decrease in the long wave length portions of the
spectral energies from 99% initially to about 1% after the first observable times and a decrease
in the negative spectral power law exponent from a ≈ −4 to a ≈ −0.3. Thereby, the long wave
length initial spectra does not play an important role in the experimentally observed self similar
scaling law for α. We have developed a plan for a UQ study of α related to uncertainty in initial
conditions.
Acknowledgements
It is a pleasure to thank D. Youngs for helpful comments and for sharing unpublished experi-
mental data. This research was supported in part by the U.S. Department of Energy grants DE–
FC02–06–ER25770,DE–FG07–07ID14889, DE–FC52–08NA28614 and DE–AC07–05ID14517. This
work is supported by the Army Research Organization grant W911NF0910306. This manuscript
has been co-authored by Brookhaven Science Associates, LLC, under Contract No. DE–AC02–
98CH10886 with the U.S. Department of Energy. The United States Government retains, and the
publisher, by accepting this article for publication, acknowledges, a world-wide license to publish
or reproduce the published form of this manuscript, or allow others to do so, for the United States
Government purposes. This work has a Los Alamos Laboratory preprint number LA–UR 10–03424
and a Stony Brook University preprint number SUNYSB–AMS–10–03.
43401-6
Initial conditions for turbulent mixing simulations
References
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Stony Brook University: SUNYSB–AMS–09–02; Preprint of the Los Alamos National Laboratory:
LA–UR–09–01364.
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SUNYSB–AMS–08–07; Preprint of the Los Alamos National Laboratory: LA–UR–08–07725.
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8. Lim H., Iwerks J., Glimm J., Sharp D.H., PNAS, 2010; Preprint of the Stony Brook University:
SUNYSB–AMS–09–05; Preprint of the Los Alamos National Laboratory: LA–UR–09–06333.
9. Lim H. et al., Physica Scripta, 2009 (in press); Preprint of the Stony Brook University: SUNYSB–
AMS–09–07; Preprint of the Los Alamos National Laboratory: LA–UR–09–07240.
10. Dimonte G., Phys. Rev. E, 2004, 69, 056305.
11. Youngs D.L., American Institute of Aeronautics and Astronautics, Technical report 4102, 16th AIAA
Computational Fluid Dynamics Conference, 2003.
12. Burrows K.D., Smeeton V.S., Youngs D.L., AWE report number 0 22/84, 1984.
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Початковi умови для моделювання турбулентного
змiшування
T. Каман1, Дж. Глiмн1,2, Д.Г. Шарп3
1 Факультет прикладної математики i статистики, Унiверситет Стонi Брук, Стонi Брук, США
2 Науковий обчислювальний центр, Нацiональна лабораторiя Брукхейвена, Уптон, США
3 Лос Аламоська нацiональна лабораторiя, Лос Аламос, США
У контекстi класичної гiдродинамiчної нестiйкостi Релея-Тейлора ми вивчаємо дуже дискусiйне
питання моделей для початкових умов i можливий вплив незафiксованих далекосяжних вкладiв на
швидкiсть росту нестiйкостi α.
Ключовi слова: шари турбулентного змiшування, турбулентна дифузiя
43401-7
Introduction
Scaling laws for initial perturbation amplitudes
Initialization of RT simulations
|