Shuang Zhang, Jian Chen and Norman J. Zabusky- Turbulent decay and mixing of accelerated inhomogeneous flows via a feature based analysis

8/3/2019 Shuang Zhang, Jian Chen and Norman J. Zabusky- Turbulent decay and mixing of accelerated inhomogeneous flow

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Zhang, Chen & Zabusky, Journal on Scientific Computing 1

Turbulent decay and mixing of accelerated inhomogeneous

flows via a feature based analysis

Shuang Zhang1, Jian Chen2, and Norman J. Zabusky1

Laboratory of Visiometrics and Modeling

1 Dept. of Mechanical and Aerospace Engineering

2 Dept. of Electrical and Computer Engineering

Rutgers University

Corresponding Author:

Shuang Zhang

Rutgers University, 64 Marvin LN, Piscataway, NJ, 08854.

Email: [email protected].

Tel: 732-445-5850. Fax: 732-445-3124

Abstract

In this paper, we focus on the late time turbulence of an accelerated inhomogeneous flow

environment, a generalization of Richtmyer-Meshkov environment. The numerical investigation is

based on two-dimensional (2D) compressible Euler simulation, which is initiated by a shock wave

hitting a gas layer (curtain). We observe excellent agreements with the previous decay analysis

on 2D viscous isotropic homogeneous turbulence at inertial range[2]. The baroclinic circulation in

this environment plays a major role on the mass transport and mixing. The mass transport

induced density gradient intensification, in turn, enhances the circulation baroclinically and


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provides the intrinsic forcing at intermediate to high frequency range. With assists of computer

graphics based feature extraction and tracking algorithm, we address quantitatively the spatial

and temporal diffusivity of the mixing zone. We study and compare both slow/fast/slow (a helium

curtain in air) and fast/slow/fast (a air curtain in helium) case to illustrate heuristically the

correlation of mass and momentum diffusivity.

Key words

Stratified turbulent mixing, Baroclinic acceleration, Diffusivity, Feature extraction

& tracking

1. Introduction

Flow environments with accelerations and inhomogeneities are unstable, due to

vorticity deposited baroclinically on the density interface, i.e., the misalignment of

pressure gradient and density gradient ( 0 p ). These flow environments

are usually referred to as Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM)

instability, when the initial acceleration is gravity and shock wave, respectively. In

[11], RT and RM instability environments are generalized as accelerated

inhomogeneous flows (AIF), and recent discovery of secondary baroclinic

circulation upon the vortex acceleration [5][15][18][19], further amplifies this

generalization. AIF are commonly seen in internal combustion, inertial

confinement (laser) fusion, and astrophysics such as supernova evolution

[11][17].


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The traditional research focus in the RT/RM community was on the so-called a-

dot problem: the growth rate of the initial small amplitude sinusoidal perturbation

under acceleration. However, many works were done experimentally and

numerically on evolutionary morphology of gas bubbles accelerated by shock

waves[14]. As we pointed out earlier, the primary physics in this

hydrodynamically unstable environment is the vorticity deposited baroclinicallyon

the gas interface. The nonlinearity of this instability is directly associated with the

initial geometry of the interface. Therefore, in the past 10 years, the inclined

planar interface has been advocated as more fundamental to the "vortex

paradigm" for RM/RT problems [4][7][8].

In all the aforementioned geometries, the baroclinic vorticity deposited by the

shock wave is mainly one sign in half wavelength. Most recently, studies were

extended to higher degree of complexity by introducing in addition to the initial

slow/fast interface (or the reverse), another fast/slow interface (or the reverse),

and make the inhomogeneity a slow/fast/slow (or fast/slow/fast) configuration.

This gas layer, referred to as a curtain, could be saw-tooth inclined [15], or

sinusoidal multi-modes [6]. This layer yields both signs of vorticity on the gas

interfaces upon the shock passage, and forms a vortex bilayer (VBL)

configuration, which accelerates the mixing and the transition to turbulence.

In our previous curtain study[15][18][19], we emphasized on the vortex dynamics

at early and intermediate time, such as the initial deposition and evolution of

VBL, the innovative discovery of secondary baroclinic circulation and the

qualitative understanding of the formation and evolution of vortex projectiles


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(VP), i.e., coherent vortex structures with different sign of vorticity binding. In this

study, we extend our investigation to late time and focus on the mixing and decay

of the turbulent regime.

The inhomogeneity in our flow field motivates us to develop and apply a feature

extraction and tracking scheme [8] to quantify the time evolution of coherently

evolving vortex and mass structures. For a systematic comparison, two curtains

are investigated in this paper: helium curtain in air (slow/fast/slow, denoted S/F/S

hereafter, with density ratio =2/1=0.14) and air curtain in helium

(fast/slow/fast, F/S/F, =7.0). We note the different correlation of mass and

momentum evolution. In S/F/S case, the primary vortex structures are entrained

inside the helium bubbles; while in F/S/F case, the vorticity rolls up in ambient

helium and tends to elongate the air bubble. The comparison of these two

configurations provides insight understandings on mass (through density field)

and momentum (through vorticity field) diffusivity.

Sections 2 discusses briefly the simulation setups and numerical methods.

Section 3 introduces our visiometrics mode of working and the impact of feature

based analysis. Section 4 and 5 discuss the phenomena and some global

quantification, with emphasis on vortex activity and turbulent decay. Section 6

focuses on the application of feature based analysis on transport, mixing, and

diffusivity of the momentum and mass fields.

2. Geometry, initial condition, and numerical scheme


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Fig. 1 shows the schematic and parameter space of the 2D computational

domain. The boundary conditions and initial conditions are also shown. The

incident shock has the strength M= 2, where Mstands for Mach number. The

domain is resolved by 2048 256 cells with ?x=?y=3.910-3. Note the units of

the simulation are normalized: the size of the shock tube is normalize by the

width and hence gives H=1; all the simulations start as p1=1.0, 1=1.0. A frame

velocity Uf is imposed to window the simulation to late time with a relatively

smaller domain.

We introduce a interfacial transition layer (ITL) between two gases, which

assumes an error function profile defined as:

=.5(1+2)(1+Aerf((x-x0)/)),

where A is the Atwood number; x0 is the central position of the profile, 1.5 for the

upstream interface and 1.927 for the downstream interface, which gives the same

curtain thickness as in [15]; is the maximum slope thickness. Note the actual

thickness of the transition layer for the truncatederror function must be at least

2.5~3, to minimize gradient effects at the truncation points.

The ITL is very important to simulate the gas-diffusion absent in the Euler

simulation and make the solution well-posed by introducing vortex layers instead

of vortex sheets upon shock deposition. In [13], a careful study is made of

spreading of 1D various thickness ITLs.

We solve the following set of partial differential equations [11]:

0)()( =++ yxt GF UUU

where


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U={, u, v, E}T

F(U)= ={u, u2+p, uv, (E+p)u}T

G(U)= ={v, uv, v2+p, (E+p)v}T

Eis the total energy per unit mass. The subscript denotes derivatives. The above

equation set is solvedwith the Piecewise Parabolic Method (PPM) algorithm [2],

which gives second order accurate solutions in both space and time.

Table 1 summarizes the parameters under investigation. We focus on two sets of

simulation data. The only difference of these two runs is the reversed initial

density ratio. The Mach number and Atwood number scaling laws of the curtain

geometry were studied with more detail in [15][18][19] and hence omitted here.

Here we focus on the hydrodynamic effect and assume the specific heat ratio to

be the same for different gases [11].

3. Visiometrics via feature based analysis

By visiometrics, we mean the systematic process of visualizing, juxtaposing and

quantifying data from numerical simulations [1]. As a comprehensive post-

processing pipeline, visiometrics is being appreciated more and more as a way to

mathematical metaphors in nonlinear physics. For example, data projections to

lower dimensions and space-time diagram have proven to be a powerful tool for

interpreting multi-dimensional evolutionary phenomena [15].

The quantities we visualize and juxtapose are: density , vorticity = u,

baroclinic source2/ p and dilatation u.


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In AIF environment, localized coherent structures are the dominant phenomena,

either in terms of density structures such as bubbles/spikes or in terms of vortex

structures such as vortex projectiles and dipoles. The analysis of the latter is also

referred to as vortex paradigm. These structures are defined as features. The

mass, momentum and energy transport associated with the interactions of

features with different scales, e.g., generation, merging, splitting, dissipation, etc,

are of crucial importance in turbulent decay and mixing. Access to these features

is crucial to quantify their evolution. Schemes to extract and track these localized

features are the only way to accomplish this goal.

Fig. 2 shows a feature analysis pipeline. The basic steps are: 1. Obtain data from

experiments, simulations or observations, with appropriate preprocessing (e.g.,

transformation and juxtaposition); 2. Visualize (in traditional way) the simulation

data, quantify fields globally and identify main feature of interests (vortices and

mass bubbles in this case); 3. Extract these features [9] and quantify them to get

abstract description; 4. Track the time evolution of these features [9]; 5. Isolate

individual features (interactively) to obtain evolution quantification.

The feature based analysis pipeline is a useful tool in both enhanced

visualization and reduced mathematical modeling. In this paper, we use the

information output to analyze turbulent decay and mixing. In the following

sections, well discuss these steps in more detail and the physical insight we

obtained accordingly.

4. Phenomena morphologies


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To identify the main physical process, we first show in Fig. 3 the visualization of

the simulation data. Here we summarize the main phenomena and emphasize

the juxtaposition of two runs with reversed density ratio. A more detail description

of the shock curtain interaction process could be found in [15].

At t=0, the shock front arrives at the curtain interface. Fig. 3a, 3b, and 3e are

three time steps for S/F/S curtain and Fig. 3c, 3d, and 3f are three time steps for

F/S/F curtain. At each time, vorticity is shown above and density below. Note the

two runs have different time scales and we picked the time when the evolutions

display similar physical behavior.

We see at early time Fig 3a and 3c, the shock wave passes the upstream

interface and deposits a strong vortex layer, positive for S/F/S and negative for

F/S/F, respectively. In density images at this time, we see the transmitted and

reflected wave front, denoted as T and R respectively. Density interface is

denoted as D. The intensification of the curtain density by shock wave

compression is also seen clearly.

Later, the shock wave passes the whole curtain and deposits another vortex

layer with opposite sign (negative for S/F/S and positive for F/S/F) on the

downstream curtain interface. This gives a vortex bilayer (VBL) configuration, as

already seen in Fig. 3a, an important phenomenon in AIF. The opposite signed

vorticity drives these two layers approaching each other, bending at lower

boundary for S/F/S case (Fig. 3b) and at upper boundary for F/S/F case (Fig. 3d)

respectively, and penetrating into the curtain. During the penetration, VBLs tip


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rolled-up as a dipolar VP structure. The spanwise velocity associated with this

VP, denoted as VP1, makes the curtain deform and accelerates its topological

change and the associated mixing. VP1 finally hits the wall and split. In S/F/S

curtain case, VP1 evolves into two streamwise leading VPs, binds with their

mirror images, shoots out downstream (VP3) and upstream (VP2) respectively.

In the F/S/F curtain case, the free ends of the VBL at lower boundary roll up into

VP3 and VP2 before VP1 hits the boundary.

The subsequent times in the simulation show more active vortex interactions,

e.g., merging, splitting and dissipating. Fig. 3e and 3f show late time visualization

images. It is obvious that the F/S/F run (Fig. 3f) is more turbulent than the S/F/S

run (Fig. 3e). The black arrows illustrate the direction of the translation velocity of

some major VPs. Note the correlation of the vortices with bubbles in S/F/S case,

i.e., each VP corresponds to a helium gas bubble as numbered accordingly,

which is absent in F/S/F case.

In Fig. 4, we show the circulation evolution for these two runs, which are defined

as:

+(t) =?? +(t) dxdy,

-(t) =?? -(t)dxdy,

(t)=??(t)dxdy

where ?+

(?-

) denotes positive (negative) vorticity. The solid and dash-dotted

curves are circulations for F/S/F and S/F/S, respectively. Time is normalized by

tn, the time for an M=2.0 incident shock traversing through the curtain in 1D,


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which is 0.47 for S/F/S run and 1.2 for F/S/F run. The circulation for S/F/S run is

multiplied by 1 to compare with the F/S/F case.

We can see three main time epochs in Fig. 4 for both runs, as summarized in

Table 2:

Time epoch I (eI), the primary baroclinic deposition epoch by the incident

shock wave. The circulation increases with a nearly linear rate.

Time epoch II (eII), the secondary baroclinic deposition epoch. The

circulation increases with a non-linear rate and reached maximum.

Time epoch III (eIII), the dissipation epoch.

We can see that the S/F/S run has a relatively short period of secondary

baroclinic process (4.2 time units), compare to 10.4 time units in F/S/F curtain

case. This difference is due to the different mixing processes explained in the

following sections. Fig. 3a and 3b show stronger vorticity maximum/minimum in

S/F/S than in F/S/F. However, in terms of global circulation in Fig 4, F/S/F curtain

reaches a bigger value than S/F/S. It is the long secondary baroclinic deposition

epoch is contributing to the stronger circulation enhancement in F/S/F case.

5. Decaying stratified turbulence

In this section, we present the decay analysis of our simulation data, and address

the different turbulent state in the two runs. Note in Euler simulation, it is

numerical dissipation we are studying.

Fig. 5 shows the enstrophy scaling of the two runs, defined as:

=?? dxdy


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We observe an oscillatory plateau region of the enstrophy (t* =1.0~3.0 for S/F/S

and t* =2.0~8.0 for F/S/F). In the enstrophy evolution of 3D compressible

turbulence [10], there is not such a plateau. By referring to Fig. 4 and Table 2, we

conclude that this plateau corresponds to the secondary baroclinic deposition

time epoch, eII. The enstrophy decays at late time as t-0.4, which is expectedly

slower than the previous study at lower resolution [18].

Fig. 6 shows the late time evolution of specific kinetic energy:

KE=?(u + v)dxdy

which approaches a constant after the shock passage. This is consistent with all

our previous runs [18] and 2D homogeneous incompressible turbulence study

[2].

In our previous papers[18], weve shown that after the shock passage, the

energy spectrum behaves in the same asymptotic mannar for different Mach

numbers and different density ratios greater than 1. Here in Fig. 7. we show a

juxtaposition of power spectrum for the two runs in this paper at the latest time,

when the circulation and enstrophy decay. Note power spectrum at an addtional

time (t=10.4) for F/S/F curtain is also plotted for comparison, when the two

streamwise VPs (VP2 and VP3) has roughly the same distance with its S/F/S

correspondence (at t=8.5), however, the circulation is still increasing for F/S/F run

at this time.

The attatched table shows quantitatively the power-laws at different wave

number range. Noticeble enough, in the middle range, all simulation data decay

universally with coefficients between 3 and 4.


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Fig. 8 shows density gradient (||) distribution at selected times, and explains the secondary circulation

growth during eII in Fig. 4. Note although initially both S/F/S and F/S/F curtains have the same interface

profile and density jump, the distribution of density gradient, denoted by dashed curves, are different. This

is because the gradient distribution depends on the absolute value of the jump.

For both runs, the gradient is intensified from initial condition. We see large gradient distribution difference

between F/S/F (cyan and black curve-triplet) and S/F/S (red dash-dot curve), which explains the different

behavior of the secondary baroclinic circulation enhancement and decay for these two runs in Fig. 4. We

plot F/S/F curves at three different times, and indicate the trends of gradient distribution by the black dotted

arrows. At high gradient range, the dissipation dominants the mixing zone after the shock compression. At

intermediate range (20 < || < 200), we see the gradient is intensified over a large number of grid points

(note the log scale), and contribute to the secondary baroclinic circulation enhancement in turn. We

measure the slopes of the latest gradient distribution curve at intermediate range for both runs, and get a

similar power law 0.8. We believe this slope is associated with the saturation of the gradient

intensification and important in future effort of modeling this process.

6. Mixing: mass transport and vortex projectiles

In this section, we study the mass transport and mixing associated with the

evolution of VPs based on feature analysis.

Fig. 9 shows the feature extraction result corresponds to Fig. 3e and 3f, and table

3 summarizes the main parameters for feature extraction. Note both density field

and vorticity field (absolute value) are extracted.

6.1 Evolution of mixing and role of vorticity: temporal mass and momentum

diffusivity

As observed again in Fig. 9a, the vortex objects are always associated with certain bubble mass mixed with

the ambient for S/F/S run. F/S/F run, on the contrary, shows mismatches: the VPs are formed outside the


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heavy gas bubble (note obj 272 in vorticity extraction and the big hole at the same location in density

extraction). Depending on the gas bubble heavier or lighter than the ambient, the mixing process are totally

different, an important statement well amplify in the following.

Because of the strong correlation between the vortex and mass structures, in S/F/S run, the vortex objects

tend to refrain the mass from diffusing into the ambient. Consequently, the stronger the vorticity is, the

weaker the mass mixing becomes. While in F/S/F run, the VPs are formed away from the bubble. They

elongated the bubble and rolled in some of the bubble mass. This means that the stronger the vorticity is,

the stronger the mass mixing becomes. This is evident in Fig. 10, the quantifications of the mass objects

evolution, i.e., feature analysis on density field.

In Fig. 10a, we show the mass object number evolutions, which are increase versus time for both runs. This

increase is associated with the two-phase circulation deposition addressed in section 5. It continues for

F/S/F curtain till the end of the simulation; while for S/F/S curtain, it tends to saturate when the circulation

enhancement stops (refer to Fig.4). The data between time 3.5-6.5, when both runs are within secondary

circulation enhancement time epoch, shows that the object number increases with the same power-law t1.5

.

Figure 10b shows the area of the density bubbles integrated over the computational domain, normalized by

initial curtain areaAGL, which is the same for both runs. The mass objects area spreads with a power law t0.6

in F/S/F run,while it approaches a constant for S/F/S run. The vorticity is dragging the heavy gas in F/S/F

curtain into the ambient, and causes the area more spread; in S/F/S curtain run, the vorticity is withholding

the light gas from mixing with the ambient, which prevents the area from increasing.

Figure 10c shows the total mass (normalized by initial curtain mass MGL) of all the bubbles extracted,

which is conserved after the shock wave flew out and before the mass dissipation dominates the flow. Note

although initially, the F/S/F curtain has much larger mass, the normalization makes the initial mass unity

for both cases. Consequently, because of higher ratio of shock compression, the mass plot here shows

larger value in S/F/S run.

We further juxtapose the quantification of Fig. 10b and 10c in Fig. 10d. In this plot, the mixing in S/F/S run

is indicated by an increase of averaged density avg =M/A, with a power law t0.23

, while the mixing of

F/S/F curtain is indicated by a decrease of avg , with a power law t-0.55

. In addition, the rate of the increase


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/ decrease reflects the mass mixing rate, which is much larger in the F/S/F case (0.55) than the S/F/S case

(0.23).

Figure 11 shows the similar feature analysis for vorticity field. We note in Fig. 11a, the number of vortex

objects decreases at intermediate time for S/F/S run, while in F/S/F case it is still increase, but obviously

with a much slower rate than the mass objects in Figure 10a. This tells that the vorticity (momentum) is

more sensitive to dissipation than the mass.

Fig. 11b shows the area of vortex objects versus time. At late time, vortex objects area decreases for S/F/S

run and increases with a much slow rate for F/S/F run.

Fig. 11c, shows the sum of absolute value of the positive and negative circulation of the extracted vortex

objects, which is consistent with enstrophy evolution in Fig. 5.

In Fig. 11d, we defined an average vorticity: avg=/A, for all vortex objects. Although the light curtain

has stronger averaged vorticity, it diffuses faster, which means the vorticity driving the mixing process is

getting weaker at a higher rate in the S/F/S case than the F/S/F case.

Comparing to Fig. 10d, we note that the vorticity and mass diffuse differently for the two curtains. We

know that vorticity tends to diffuse to light medium. In F/S/F case, it carries the bubble gas with it. This

makes the mass more distributed and hence contributes to the circulation enhancement through the

intensified density gradient. However, in S/F/S case, it tends to confine the light bubble gas from mass

diffusion. This competition process is at the cost of stronger circulation diffusion, i.e., the mass is less

distributed, hence it has weaker gradient enhancement at late time.

6.2 Late time mixing state and role of vortex projectiles: spatial mass and

momentum diffusivity

In both simulations, VPs are playing crucial roles in the mixing process, in terms

of formation of dipolar jets or interactions such as merging, splitting, and local

leap-frogs[18]. In Fig. 12 we track and quantify the mass fractions of the leading

VP objects for the S/F/S curtain case. Two major objects numbered consistently

with Fig. 9a are shown. Again, initial curtain mass is used for normalization. We

can see that the upstream leading VP obj 23 (VP2 in Fig. 3) is isolated at very


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early stage of the simulation. It evolves with nearly constant mass. While obj 1,

the downstream VP (VP3 in Fig. 3), oscillates due to the shedding of small vortex

filaments. These filaments always roll up together with some mass from original

obj 1 and explains the periodic sudden drop in the curve. Also for obj 1, before

the shedding, there is always an gradual increase in mass, because of the mixing

with the ambient gas. These two large objects, together with the largest object in

the center turbulent region Obj 6 (VP6 in Fig. 3), have 52.9% of the total mass.

Fig. 12b shows the density minimum normalized by the initial gas density of obj 1

and 23. Regardless the different shape and the evolution procedure, the

minimum values of these helium bubbles behave similarly. Note the minimum

density value here is an indication of how well bubble got mixed, hence could

provide an important guidance whether the error function solution of diffusion

equation holds [16].

In Fig. 13, we look more closely to the evolution of leading objects, by

juxtaposing mass and vorticity information for F/S/F and S/F/S runs. Note that in

the F/S/F case, the mass objects are not associated with the vortex objects. In

Figure 13a and 13b, we show the local circulation and vorticity maximum for two

leading objects in S/F/S run and one in F/S/F run. As is obvious in the plot, vortex

objects in S/F/S run is much stronger than F/S/F run. The circulation is nearly

constant, because the diffusion effect is not significant for large scale structures.

If we look at maximum of vorticity magnitude, Fig. 13b, we obtain the same

statement as from Fig. 11d: VPs in S/F/S curtain diffuse at a higher rate. Fig. 13c


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shows the invariant quantity correlating the vorticity and density field, and we see

almost identical evolution for both objects in S/F/S run.

We discussed in previous sections the temporal evolution of the diffusivity associated with this

stratified turbulent environment. In Fig. 14 we show the spatial diffusivity issues by plotting the

late time feature quantifications vs. area for both runs.

Fig. 14a shows mass versus area, which gives nearly linear behavior, and is invariant at late time.

Figure 14b shows the density extrema versus area, which gives the spatial mixing rates. Larger

object corresponds to higher (lower) density in F/S/F (S/F/S) case. The S/F/S case has a larger

mixing rate over space because smaller objects are easier to diffuse due to the weaker vorticity

entraining it. However, due to the absence of this vorticity and mass correlation, in S/F/S case,

smaller object doesnt have to diffuse faster all the time because the vorticity somewhere else

might be intensifying them.

This lack of correlation is also shown in Fig. 14c and 14d. In Fig. 14c, the vorticity maximum

distribution, the smaller area of objects doesnt necessarily mean weaker vorticity in F/S/F run.

Actually there is a range of objects area, which has roughly the same vorticity (note the plateau

for A > 0.01). While S/F/S run displays a nearly linear distribution over the area: larger objects

correspond to stronger vortices.

Figure 14d shows the circulation versus area. The structures follow different turbulent mixing law

for larger objects and smaller objects in S/F/S run. However, most of F/S/F objects at this time

are mixed similarly (with the power law fit) regardless their size.

7. Conclusion

In this paper, we show the simulation results of fast/slow/fast and slow/fast/slow

accelerated inhomogeneous flow environments. A careful comparison of these

two configurations and associated turbulent mixing processes are investigated


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with the assists of feature extraction and tracking. We identify vorticity deposition

and dissipation as the most fundamental mechanism of the turbulence, and study

the baroclinic acceleration of the circulation, especially beyond the initial shock

deposition phase, i.e., the secondary baroclinic enhancement. Detailed

quantifications are obtained on the local quantities and are associated, the first

time, with the turbulent mixing and decay. The correlation of mass and

momentum diffusivity is addressed.

We are aware the lack of the physical viscosity in our simulation. However, our

first step on the diffusivity issues associated with this stratified turbulence

analysis uncovers a whole new set of questions and points the direction of future

approaches.

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submitted. 2003.

[17] S. Zhang, N. J. Zabusky, & K. Nishihara, "Vortex Structures and Turbulence Emerging in a

Supernova 1987A Configuration: Interactions of "Complex" Blast Waves and Cylindrical/Spherical

Bubbles ", Laser and Particle Beams. To appear 2003.

[18] S. Zhang, N.J. Zabusky, & G-Z Peng, "A study on Shock - SF6/Helium curtain interaction:

secondary baroclinic instability and late time inhomogeneous turbulence", Journal of Fluid

Mechanics, Submitted, 2002.


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[19] S. Zhang and N.J. Zabusky, " Shockplanar curtain interactions: Strong secondary baroclinic

deposition and emergence of vortex projectiles (VPs) and decaying inhomogeneous turbulence ".

Laser and Particle Beams. To appear 2003.


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Figure 1. Schematic and parameter space of the computational domain,

boundary conditions and initial conditions.

At t=0 Upper Reflecting Boundary

Lower Reflecting Boundary

1,1,1, a1

x

H

WGL

a1

2, 2,2, a2

shock

M

OutflowInflow

Moving Frame Uf

1,1,1, a1

a2

Transition layer with error function profile,

=.5(1+2)(1+Aerf((x-x0)/))


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Parameters S/F/S(Slow/Fast/Slow)

(Helium curtain in Air)

FSF(Fast/Slow/Fast)

(Air curtain in Helium)

Mach Number: M 2.0 2.0

Atwood #:A=(1-2)/ (1+2) 0.75 -0.75

Background Density:1 Air: 1.0 Helium: 1.0

2 Helium: 0.14 Air: 7.14

Specific heat ratio: 1=2 1.4 1.4

Speed of sound: a1 331m/s(air, 0C) 965m/s (Helium, 0C)

a2 965m/s (Helium, 0C) 331m/s(air, 0C)

Inclined angle: a1 = a1 30 30

Initial effective curtain

thickness: WGL

0.427H

(H: Shock tube width)

0.427H

Table 1. Summary of run-time parameters.


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Figure 2. Feature based analysis in turbulence study

Data transformation,

juxtaposition &

visualization

Feature identification

Feature Extraction

Feature Tracking

(Color coding)

Feature Isolation

Evolution

quantification

Decay study:

Power law

Spectrum

Mixing study:

Events query

Correlation

Scales

Density

Vorticity u=

t1 t2

t1 t2

I

II

I

II

I

II

t1 t2

Feature DescriptionEnhanced

Visualization


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(e). S/F/S, t=8.5

VP2 VP3VP5

VP6

25

3

6

0.35 3.02

2.3 16.5

-125.4 165.0

-57.3 57.3

(f). F/S/F, t=10.4

(c). F/S/F

t=0.3

(d). F/S/F

t=2.0

(a). S/F/S

t=0.3(b). S/F/S

t=0.9

Figure 3. Simulation visualization, vorticity above and density below

VBL

T

VP1

VP1VP2

VP3

VP2VP3

Density

Vorticity

Density

Vorticity

Density

Vorticity

Density

Vorticity

Density

Vorticity

Density

Vorticity

R

T

R T

R

D D D

D

D


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Figure 4. Circulation scaling. Time is normalized by tn, the time for an M=2.0

incident shock traversing through the curtain medium. Red curves are +, blue

curves are -and green curves are . Because the deposition sequence of the

positive and negative circulation is different for the two runs, circulation of SFS

run is multiplied by 1 for easier comparison. Please refer to Table 2 for

classifications of different time epochs for the two runs.

FSF: =7.14

SFS: =0.14

t*


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eI eII eIII

S/F/S 0-0.8 0.8-5.0 5.0-11.6

F/S/F 0-0.8 0.8-11.2 11.2-12

Table 2. Time epochs of the simulation according to physical process [10].


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t-0.42

t-0.43

t*

F/S/F

S/F/S

Figure 5. Enstrophy evolution and decay


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KE

t*

Figure 6. Kinetic energy evolution

F/S/F

S/F/S


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10-60 60-500 500-1024

F/S/F, t=14.0 k-2

k-3.47

k-1.93

F/S/F, t=10.4 k-1.9

k-3.8

k-2.8

S/F/S, t=8.5 k-1.12 k

-4.07 k-1.14

Power Spectrum

k

Figure 7. Energy power spectrum

F/S/F: t=14.0

S/F/S: t=8.5

F/S/F: t=10.4


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Figure 8. Density gradient distribution

S/F/S: t=8.5

F/S/F: t=10.4

F/S/F: t=14

F/S/F: t=4.

S/F/S: t=0

F/S/F: t=0

Primary baroclinic effect

Secondary baroclinic effect

||

8.0||

Shock

Compression


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Figure 9. Feature extreaction result, compare to Figure 3. Note the color isassigned randomly during extraction but is tracked in time.

Density

Vorticity

(a). S/F/S, t*=10.24,

Compare to Figure 3e

(b). F/S/F, t*=4.84

Compare to Figure 3f

obj 23

obj 6

obj 1

obj 1

obj 272

VP2 VP3

Density

Vorticity


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S/F/S F/S/F

Density threshold 3.67

Density Range 0.14-3.23 1.0-29.87

Vorticity threshold 0.02 0.02

Vorticity Range -1.11-0.93 -0.404-0.348

Initial Curtain Area 0.427 0.427

Initial Curtain Mass 0.05978 3.04878

Table 3 Feature extraction parameters


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Figure 10. Mass obejcts evolution (in density field)

time

(a). density object #, power fitting t=35-65 (b). density objects area (logscale)

(c). objects mass (logscale)

A/AGL

(M/A)

(d). object averaged density (logscale)

timetime

time

S/F/S

# of objects

F/S/F

S/F/S

S/F/S S/F/S

t0.23

F/S/F F/S/F

t-0.55

M/MGL

F/S/F

t1.5

t0.6


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Figure11. Vortex objects evolution (in vorticity field)

(a). Vortex object number

(d). circulation/area (logscale)(c). Vortex object circulation (logscale)

(b). Vortex object area (logscale)

F/S/F

t-0.23

(t=18~130)

F/S/F

S/F/S

F/S/F

time time

S/F/S

# of objects A

time

time

avg=/A

S/F/S

t-0.36

(t=5~50)

F/S/F

S/F/S


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Figure 12. Mass Fraction of leading objects, S/F/S run only

Total

obj 1

obj 23

a. Normalized Mass

min/0

b. Normalized density minimum

obj 23

obj 1


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Figure 13. Evolution of leading VPs

max

/

max/( min /GL)

S/F/S, obj 1

S/F/S, obj 23

F/S/F, obj 1

S/F/S, obj 1

S/F/S, obj 23

F/S/F, obj 1

a. Local Circulation b. Vorticity maximum

c. Correlation (S/F/S only)

S/F/S, obj 1

S/F/S, obj 23


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Figure14. Late time mass, circulation and their juxtaposed distribution.

S/F/S:A

0.93

F/S/F:A1.0

a. Mass vs. Area.

Invariant at late time

d. Circulation vs. Area

Circle: S/F/S

Left triangle: F/S/F

A1.14

b. Density extrema vs. Area.

S/F/S:A-0.208

F/S/F:A0.96

c. Vorticity maxima vs. Area.

Circle: S/F/S

Left triangle: F/S/F

Shuang Zhang, Jian Chen and Norman J. Zabusky- Turbulent decay and mixing of accelerated inhomogeneous flows via a feature based analysis

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