RMI Zabuskey 1999

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Annu. Rev. Fluid Mech. 1999. 31:495536

Copyright c 1999 by Annual Reviews. All rights reserved

VORTEX PARADIGM FOR

ACCELERATED INHOMOGENEOUS

FLOWS: Visiometrics for the

Rayleigh-Taylor and Richtmyer-Meshkov

EnvironmentsNorman J. ZabuskyLaboratory for Visiometrics and Modeling, Department of Mechanical and Aerospace

Engineering, and Computer Aids for Industrial Productivity (CAIP) Center, Rutgers

University, Piscataway, New Jersey 08855; e-mail: [email protected]

KEY WORDS: accelerated flows, vortex dynamics

ABSTRACTWe illustrate how cogent visiometrics can provide peak insights that lead to path-

ways for discovery through computer simulation. This process includes visualiz-

ing, quantifying, and tracking evolving coherent structure morphologies. We use

the vortex paradigm (Hawley & Zabusky 1989) to guide, interpret, and model phe-

nomena arising in numerical simulations of accelerated inhomogeneous flows,

e.g. Richtmyer-Meshkov shock-interface and shock-bubble environments and

Rayleigh-Taylor environments. Much of this work is available on the Internet at

the sites of my collaborators, A Kotelnikov, J Ray, and R Samtaney, at our Vizlab

URL, http://vizlab.rutgers.edu/vizlab.html.

The next El Dorado lies in the mathematical understanding of coherent structures

in weakly dissipative fluids: the formation, evolution, and interaction of metastable,

vortex-like solutions of nonlinear partial differential equations. By juxtaposing ex-

perimental results with computer simulations, I hope to focus on essential analytical

directions.

(ZABUSKY 1977)

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1. INTRODUCTION

Nonlinear processes are usually unstable, chaotic, and turbulent. Often they

are unpredictable beyond early time eras, and laminar (or smooth) features

evolve into a multiplicity of space scales (sometimes with complex topologies)

and singular and near-singular space-time events. To acquire a deeper under-standing of special phenomena, those that are considered by the community of

scientists to be the contemporary primary focus of attention, we find it useful

to formulate low-degree-of-freedom models. These models are often derived

from insights gained by juxtaposing many-degree-of-freedom direct numerical

simulations (dns) with laboratory and field experiments. If the latter experi-

ments are not available, then care must be taken to elucidate the accuracy and

content of the dns massive data sets, e.g. the convergence of the discretized and

regularized numerical simulations to solutions of the continuum formulation.

A practical goal is to obtain efficient numerical models that aid the prediction

and design of natural and man-made processes.

I will relate a few of my experiences with simulations of the Fermi-Pasta-

Ulam nonlinear lattice and the Korteweg de Vries (KdV) partial differential

equation in one space dimension that led to the discovery of the soliton (Zabusky

& Kruskal 1965; Zabusky 1981, 1984; Weissert 1997; Heyerhoff 1998). These

will illustrate the visiometric approach, namely how the use of well-chosen

visualizations and quantifications aided in posing and answering key questions.

I will show the robustness of this modus operandi by reviewing recent work that

emphasizes the vortex dynamics of accelerated inhomogeneous flows. In par-

ticular, I discuss progress on applying the vortex paradigm (Hawley & Zabusky

1989) to the Richtmyer-Meshkov (RM) and the Rayleigh-Taylor (RT) insta-

bility environments where inhomogeneous density flows are accelerated. The

vortex-related insights have led us recently to conjecture that some astrophys-

ical objects, called knots, that are seen in the vicinity of astrophysical jets and

supernova remnants may be vortex projectiles (Zabusky et al 1997; Zabusky

et al, submitted), as described below.

2. VISIOMETRICS AND REDUCED MODELING

2.1 VisiometricsBy visiometrics, I mean the visualization, quantification, and juxtaposition of

data sets (Bitz & Zabusky 1990). In quantification, I imply the extraction

and tracking of coherent structures above a low-lying and more random (less

coherent) background. These data sets are obtained from laboratory and field

experiments, remote observations of natural phenomena, and computer simu-

lations. In this paper, we emphasize direct numerical simulations (dns) of thecompressible Euler equations of fluid motion.



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VORTEX PARADIGM AND VISIOMETRICS FOR RT & RM 497

By juxtaposition, I mean visualized and quantified comparisons among

overlaid data sets. These comparisons are of similar or different functions

at the same or different times from simulations of the same or linked math-

ematical models in a relevant domain of parameters. An example is an in-

compressible configuration that models a corresponding low-Mach-numberRM configuration. If one is tracking coherent structures, it is often useful to

be in a translating and/or rotating frame with only a subset (zoomed) of the

total space (Winkler et al 1987; Zabusky et al 1992, 1993; Silver & Zabusky

1993).

For nonlinear phenomena, one must be aware, particularly beyond early time

epochs in a simulation, of the effects due to specific forcing functions and noise

and the explicit and code-implicit dissipative and dispersive regularizations (e.g.

for very small scales the turbulence models, de-aliasing and filtering functions,

etc). This information willallow a validation of candidate mathematical models.In essence, visiometrics is the way to mathematical metaphors, the mother lode

referred to in the opening quotation (Zabusky 1977).

2.1.1 VISUALIZATION By visualization, I mean the conversion of numerical

data into a variety of images. First, we preprocess or filter the data (to reduce

noise or size) and categorize it by choosing appropriate color maps, opacity

parameters, contour levels, etc. It can then be transformed to an appropri-

ate geometric model for display. Standard visualization techniques include

continuous-tone contour maps in two dimensions; volume rendering a functionin three dimensions [as it would appear when radiating or reflecting light from

various sources (Upson & Keeler 1988), etc]; displaying isosurfaces (contours)

from connected polygons that bound regions of functions extracted by thresh-

olding (Lorensen & Cline 1987) or wavelets (Siegel & Weiss 1997, Farge et al

1998); drawing icons or hedgehogs (arrows) for vector fields and drawing lines

for streamlines, vector field lines, etc.

A visualization community has arisen from diverse workers in computer

graphics, computer vision, computational geometry, image processing, mor-

phological mathematics, and applications people in every discipline (Hes-selink 1988, Jahne 1991, Patrikalakis 1991). Many new journals, special jour-

nal issues, and conference proceedings report on environments such as AVS,

EXPLORER, IDL, etc. For example, see the proceedings of the ninth IEEE

Visualization Conference, which took place in October of 1998. Many of

these are now listed on Internet homepages. An excellent place to start is:

http://www.nas.nasa.gov/RNR/Visualization/annotatedURLs.html.

2.1.2 QUANTIFICATION Visualization alone is not enough. Intuition to guide

the development of new models requires the quantification of evolving mor-phologies or structures. This process begins with structure (or region) isolation,



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Table 1 Glossary for interacting fluid dynamical coherent structures

Interact

Structure Move Come together Come apart

bubble mushroom advect accrete bifurcate

blast wave packet convect aggregate breakup

blob patch diffuse agglomerate burst

cloud point disperse align disassemble

critical pt. ring flow bind disrupt

depletion roll hop collapse finger

dipole separatrix intensify condense fission

dislocation sheet migrate entangle phase-shift

eddy shock wave percolate fission plow

filament soliton propagate focus reconnect

finger spike spike fold reflect

gradient spiral stream fuse split

gyre stem swirl merge striate

hairpin striation transport pair strip

helix surface wind roll-up

hole tendril wind-about

interface triple-point

jet tube

knot vortex

layer whorl

extraction, and tracking. Each structure is in a data base with its diagnosed

properties as described below.

In Table 1, a list of some fluid dynamical structures (or features or objects),

actions, and interactions is presented. The move and interactcategories describe

how the objects may evolve. (Some of these verbs may serve as nouns in the

object column.)

Some diagnostic operations are:

1. Global quantification: measuring total quantities, some of which are invari-

ant in some region of parameter space. This includes energy, circulation,

enstrophy, palenstrophy, probability distribution functions (pdf), helicity,

drag force vector on an object, correlation functions, slopes and variances

of pdf and power density spectra, fractal dimension, moments of statistical

quantities (e.g. skewness, kurtosis, etc), etc.

2. Detailed quantification: Characterizing space-time topologies of coherent

structures, including evolution of critical and extremal points and surfaces

of scalar, vector, and tensor fields in appropriate moving frames (e.g. the rate

of strain tensor and its diagonal representation at every point in the field,



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or in projections onto a plane or interface, etc); space-time or parameter-

time projections of functions and their gradients, e.g. space-time diagrams

of transverse integrated vorticity (below); slopes, normals, curvatures, and

torsions of lines and surfaces; finite-time-interval particle trajectories and

Lyapounov exponents; low-order moment (e.g. ellipse and ellipsoidal rep-resentations) skeletal descriptions of elongated objects; distances between

extrema (or centroids) and critical points; temporal variation of modal am-

plitudes and phases; wavelet extraction of coherent objects and a statistical

characterization of the remainder field.

Consider the coherent vortex structure (CVS), a localized region (sheet, layer

or tube) of fluid that persists for many periods of some relevant time interval.

We believe that in an optimal definition of a CVS, one will be able to track a

relatively compact object that contains a significant content (for vortices, thecirculation or enstrophy) for the longest time interval. For example, when a

CVS is distorting due to the presence of a strong strain field of nearby CVSs, one

should be able to track it until a complex interaction configuration among these

CVSs is formed and then afterward when another set of outgoing coherent

vortex states appears.

For a discussion of two-dimensional and three-dimensional feature identi-

fication, extraction, and tracking, see Carlbom et al (1992), Samtaney et al

(1994), and Silver & Wang (1996). An important aspect of segmentation is the

functional form of the variables available from a simulation that are used todefine the object (Buntine & Pullin 1989, Villasenor & Vincent 1992, Banks

& Singer 1994, Silver 1995, Jeong et al 1997). In many of these definitions,

the features are defined with some sort of connectibility criteria that enable the

method to partition the data set into important regions and background.

2.2 Reduced Modeling and the SolitonI propose a tentative definition: A reduced model is one in a hierarchy that

yields solutions to a cogent problem and minimizes an error-cost function.

The models in the hierarchy may be linked by means of mathematical asymp-totics and cogent means within a narrow but physically useful parameter range

(which includes the evolutionary time interval). The error-cost function could

be the product (solution error) (computational cost). The total error is acombination of several partial errors, including omission or inadequate rep-

resentation of realistic physical and chemical processes, roundoff, numerical

truncation, filtering (or regularization of small scales), etc. The computational

cost relates to algorithm (truncation accuracy and efficiency), parallelization

process, and storage and retrieval procedures (for massive data sets that are

post-processed, etc).



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For example, an explicit formula is usually good for steady, idealized pro-

cesses, e.g. the Kolmogoroff spectral indices for homogeneous, isotropic, forced,

incompressible turbulence. A set of linear simultaneous equations without and

with constraints may be the next level. This is followed in complexity by cou-

pled sets of linear ordinary, partial, or integro-differential equations (e.g. linearstability theory) and single or coupled sets of nonlinear ordinary, partial, or

integro-differential equations.

2.2.1 SOLITONS IN DISCRETE AND CONTINUOUS MEDIA A paradigm for visio-

metrics and asymptotical modeling is the discovery of the soliton by Zabusky

& Kruskal (1965). The soliton and its remarkable properties could have been

discovered earlier, because certain key experimental observations on small-

but finite-amplitude shallow water waves were made by Scott Russell in the

1840s and were known to Korteweg and de Vries (KdV) in 1895, who usingasymptotic methods, derived the single nonlinear partial differential equation

u

t+ up u

x+ 2p

3u

x 3= 0, (1)

with p = 1 to model observed phenomena (Korteweg & de Vries 1895). Theyfound stationary isolated and periodic solutions. This (1 + 1) nonlinear partialdifferential equation models many situations, including the one-dimensional

cubic nonlinear lattice, as derived by M Kruskal (Zabusky 1995, Kruskal 1995)

to model the FPU simulations. Direct numerical simulations of the KdV equa-tion were performed with Gary Deem in 1965 at Bell Labs in Whippany, New

Jersey, and the near recurrence of the solution to its initial state was visualized

cinematically (Deem et al 1975).

We studied the waveforms and space-time diagrams of the solution, u(x , t),

and realized that the initial condition disassembled itself into localized coherent

structures that looked like the solitary wave solutions when they were well sep-

arated; we dubbed them solitons. What we didnt realize was that this system

is mathematically integrable and the solitons are exact mathematical entities

whose number is determined by the parameters of the problem. This exact-ness was realized when the key breakthrough was made by Gardner, Greene,

Kruskal, and Miura in 1967 (Gardner et al 1967), who transformed the nonlinear

evolution equation to the linear Schrodinger equation in one dimension.

The application of the soliton concept to a potentially successful commercial

venture, namely the issue of productivity of computational science, has taken

place recently. Ingenious experimentalists, materials scientists, theoreticians,

mathematicians, and simulation specialists built on the early ideas for optical

soliton properties of Hasegawa & Tappert (1973) [Hasegawa (1995), Hasimoto

& Kodama (1995), and Akhmediev & Ankiewicz (1997)], who both were at



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Bell Labs in the early soliton days. Optical solitons, manifest as light pulses

of picosecond width, were first observed in glass fibers by Linn Mollenauer,

Roger Stolen, and James Gordon at Bell Laboratories (Stolen et al 1980) in

1980. However, in nature, solitons do decay, and many technical problems

had to be overcome. Recently, 20-gigabit/sec soliton signals were transmittederror-free in a loop over a distance of 14,000 km.

To review,

1. We realized that a continuum model could characterize the nonlinear dy-

namics of a lattice.

2. Kruskal derived the KdV equation from a dispersively regularized nonlinear

wave equation after making a Galilean transformation to a reference frame

moving with the linear speed of the lattice.

3. We made simulations of the KdV with parameters corresponding to the non-

linear lattice with progressive wave initial condition and obtained agreement

in the near-recurrence time.

4. We made simulations with two solitary waves isolated in a periodic domain

and observed nearly nondispersive passage with recurrence to the initial

waveforms and with a phase-shift.

5. We observed similar numerical phenomena for the modified KdV equation(p = 2 above).

Aspects from mathematics and other fields are in Bullough & Caudrey 1995,

who review mathematical developments, Weissert (1997), who presents a thor-

ough history of FPU and nonlinear lattices and other low-dimension chaotic

nonlinear systems (from 1954 to 1974); and Heyerhoff (1998), who presents a

broad history of (1+1) nonlinear wave and related partial differential equations(from 1834 to 1967).

3. ACCELERATED INHOMOGENEOUS FLOWSAND THE VORTEX PARADIGM

3.1 Application DomainsThere are many density-stratified classical and contemporary high-energy fluid

environments where accelerated flows are paramount, including combustion

(Curran et al 1996); inertial confinement (laser) fusion, or ICF (Lindl 1995);

and astrophysics, particularly supernova remnants (Wang & Robertson 1985,

Muller et al 1991, Chevalier et al 1992, Klein et al 1994, Stone & Norman 1994,



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502 ZABUSKY

Chevalier & Blondin 1995, Xu & Stone 1995, Dohm-Palmer & Jones 1996).

The latter two domains were brought into focus at a March 1998 symposium

in Tucson, Arizona (Remington & Arnett 1999).

In ICF, one desires to confine an amount of nuclear reactive gaseous mixture,

such as D-T, until it is heated to a temperature where the nuclear reactionreleases enormous amounts of energy. One procedure advocated at present

is to illuminate with lasers a small spherical capsule containing gaseous DT

surrounded by layers of solid DT and a solid ablator. The short-duration laser

pulses deposit energy within a narrow layer, causing an ablative ejection of

material in the direction of the incoming radiation, while an intense shock

wave is driven toward the center of the pellet.

If everything were ideal, the process would be one-dimensional and the shock

would be transmitted and reflected at the interfaces. However, despite the care

taken to obtain perfect radial symmetry during fabrication and illumination, theinterface between the two media is perturbed, breaking the one-dimensional

symmetry, and the dynamical processes become three-dimensional. Hence, in

this domain the goal is to understand and control the interfacial density per-

turbations that are the locations of vortex roll-up and subsequent turbulence

and mixing activity, a process that inhibits the elevation of temperature of the

volume fraction of the ignitable nuclear mixture.

3.2 Overview of Rayleigh-Taylor (RT)

and Richtmyer-Meshkov (RM) EnvironmentsThe contemporary study of accelerated incompressible flows at a density-

stratified interface began with the linear stability analysis of a harmonically

perturbed interface by Taylor (1950) and the corresponding experiments of

Lewis (1950) (undoubtedly brought on by wartime considerations). This con-

stant acceleration environment is called the Rayleigh-Taylor (RT) instability.

Lewis concluded that the agreement of his experiments with Taylors exponen-

tial growth held until the initial small sinusoidal amplitude grew to 0.4 , where

is the initial wavelength. The RT linear instability, including viscosity and

magnetic fields, was elaborated in the 1950s and was comprehensively sum-marized (Chandrasekhar 1961). The effects of diffusion were also examined

experimentally (Duff et al 1962).

The shock-induced impulsive acceleration of an interface with sinusoidal

perturbations was studied analytically by Richtmyer (1960), who used a linear

representation after shock passage and derived a simple formula that captures

the early time motion of the interface for a restricted range of Mach numbers

and density ratios; and experimentally by Meshkov (1969), whose results at

early times were in qualitative agreement with Richtmyers formula. For these

reasons, the shock-accelerated interface is called the Richtmyer-Meshkov (RM)



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environment. A correspondingly important experimental study of a shock-

spherical bubble was made (Rudinger & Somers 1960), and attention was paid

to the emerging vortex rings. Figure 1 shows three primary focus geometries,

where (a) is half a wavelength of the sinusoidal perturbation, (b) is the planar

inclined interface, and (c) is representative of a symmetrical cylinder in twodimensions or an axisymmetric sphere in three dimensions. The last assumes

symmetry about the lower boundary and thus represents the model used in our

Euler dns.

Sharp (1984) reviewed the linear RT instability in various geometries. He

briefly mentions compressibility and the RM environment. He includes exam-

ples of numerical simulations of the evolving sinusoidal interface (somewhat

beyond the linear phase) and ad-hoc theories of bubble dynamics. Bubble is

a term associated with the smoother part of the undulating (e.g. sinusoidal or

sawtooth) interface that is evolving, as discussed below. He did not assess therole of vorticity in either RT or RM environments.

3.2.1 LABORATORY EXPERIMENTS ON SHOCK-ACCELERATED INTERFACES

Since the early 1980s, many careful and innovative experiments were made by

Sturtevant, Haas, Meshkov, Zaitsev, and their collaborators (Haas & Sturtevant

1987, Brouillette & Sturtevant 1989, Aleshin et al 1993). For shock tubes, there

had been concern about the fabrication and properties during shock interaction

of the interfacial polymer membrane. Sturtevant (1987) gave a review of the

RM environment and showed many pioneering experimental results, includ-ing Meshkovs 1969 experiment; a continuation to a more turbulent phase by

Andronov et al (1976); a reshock experiment by Zaitsev et al (1985) (where

the second shock arises from the reflection of the incident shock from the

downstream wall of the shock tube); the shock-sphere (helium in air); shock in-

clined planar interface and double inclined interface (recently called a curtain).

Sturtevants experiments were carried to intermediate times, and he commented

about visual manifestations of vorticityin particular, interface roll-up.

3.2.2 NUMERICAL SIMULATIONS OF SHOCK-ACCELERATED INTERFACES: VOR-TICITY DEPOSITION AND MODELING Although vorticity was displayed by

Chalmers et al (1988) and Picone & Boris (1988), and divergence by the for-

mer, the first serious quantification of vorticity was given by Hawley & Zabusky

(1989). Their focus of attention was the single planar inclined interface geom-

etry; they showed the space-time integrated vorticity for positive and negative

Atwood numbers and introduced the vortex paradigm as a guide to understand-

ing and modeling. Comparisons of juxtaposed images, amplitudes, and spike

and bubble velocities by Yang et al (1992) with Sturtevants experiments on pla-

nar interfaces showed excellent agreement at early times. Quantitative estimates



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Figure 1 Schematic of primary-focus two-dimensionalphysical domains and parameters in shock-

accelerated density-stratified interfaces prior to shock interaction. (a) Sinusoidal interface. (b)

Planar interface. (c) Circular or spherical-axisymmetric interface.



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for circulation deposition on interfaces for a wide variety of parameters was

given by Samtaney & Zabusky (1992, 1994). Studies of sinusoidal interfaces

(Holmes et al 1995) also showed qualitative agreement with experiments.

At the 18th (International Symposium on Shock Waves) ISSW in 1991, the

RM environment was discussed. Rupert (1992) summarized research for planarinterfaces parallel to the shock front (i.e. not inclined) at various Mach numbers

up to 3.4. Several color images of vorticity and divergence were shown. She

remarked that the numerical results seem to differ from the observations by a

significant amount. Zabusky et al (1992) summarized research for the inclined

interface and the shock cylinder at M = 4. Many new effects were revealed thatare not present at lower Mach number. Furthermore, there was emphasis on

novel visiometrics, including quantifying total positive and negative circula-

tion in the domain and visualizing vorticity and integrated vorticity space-time

diagrams.Since 1991, there has been a large increase in the number of RM papers in

the fluid and plasma communities. This is particularly manifest in the quality

and quantity of the papers at the recent 1997 Marseilles meeting (Jourdan &

Houas 1997) of the sixth International Workshop on the Physics of Compress-

ible Turbulent Mixing (IWPCTM), the focus group for RM research. At these

MIX workshops, groups from around the world present analytical, experi-

mental, and computational results, e.g. a history of the work at the Russian

VNIIEF center was outlined (IIkaev et al 1997).

3.2.3 PRIMARY FOCUS Many papers have been written on the linear theory

and early time simulations of the RM environment and are available in the

reviews by Sharp (1984) and Rupert (1992). In this paper, our primary focus

will be on linear and nonlinear models and vortex-related phenomena up to

early-intermediate times, as discussed below. For example, in recent years,

it has been found convenient to quantify, at early times, the rate of change

of amplitude of the sinusoidally perturbed interface extrema, i.e. the spike

and bubble velocities and their difference,

a(t). However, this low-order in-

formation, although essential for making low-order experiment-simulation anddifferent-code comparisons, provides little insight into the physics of vortex

deposition and its localization (or roll-up) phenomena and the emergence of

small-scale structures. Furthermore, the curvature, nearness, and magnitude of

the large gradients of the rolling density interface are important for deposition

of vorticity in re-acceleration and reshock experiments, as described below.

Thus we will not discuss the multimode-statistical (Ofer et al 1995) and tur-

bulent mixing aspects of interfacial region growth, a vital area of late-time three-

dimensional research in this field, as discussed in Jourdan & Houas (1997).



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4. EQUATIONS, NUMERICAL METHODS,PARAMETERS, GEOMETRIES,AND TIME EPOCHS

4.1 Euler Equations of EvolutionThe Euler equation set that we solved in two dimensions is written in conser-

vation form as

Ut + F(U)x + G(U)y = 0, (2)where U = {, u, v, E, }T,F(U) = {u, u2 + p, uv, (E+ p)u, u}T,and G(U) = {v ,uv ,v2 +p, (E+p)v,v}T. E is the total energy relatedto the pressure, p, by p = ( 1)[E 1

2(u2 + v2)]; and is the interface

tracking function. The boundary conditions are reflecting in the y-direction and

inflow/outflow in the x-direction.We have used the conservative level set formulation (Osher et al 1988) in

which a function (x, t) defines the interfacial domain everywhere in the do-

main. We initialize (x, 0) = +1(1) in the incident (transmitted) gas. Thelast PDE in the system of equations (2) governs (x, t) and is coupled to the

other equations through the variable (x, t), which is determined by linear

interpolation for a particular cell. Note this system is not strictly hyperbolic.

4.2 Numerical Methods

Many finite-difference numerical methods with fixed and adaptive meshes havebeen used. Comparisons among them for convergence, accuracy, etc, usually

involve the primitive variables, e.g. density and temperature, and almost never

anything related to rate of strain, vorticity, or divergence. Some examples in-

clude Kang et al (1994), Don & Quillen (1995), Quirk & Karni (1996), Lee

& Giltrud (1996), Ofgeneim et al (1995), and Holmes et al (1998). Investiga-

tors use, often without mention, various regularization and filtering techniques

to make the codes more robust. This includes the introduction of numerical

viscosities near strong shocks and shear layers as well as artificial steepening

of contact discontinuities, etc. In the future it would be desirable to developstandards for comparative tests so that we may enhance the believability of

simulation predictions.

Our results are from two numerical methods. Both are second-order accurate

in space and time and include interface tracking. For the first, we assume that:

the flow is inviscid; the gases are perfect and in thermodynamic equilibrium; and

there are no chemical reactions between the gases. The method is generalization

of Godunovs method, and is similar to the Eulerian MUSCL algorithm (Colella

1985), which is suitable for flows involving nonlinear wave interactions. The

solution at each time step is interpolated to give a piecewise linear distribution in



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each grid zone. Van Leers monotonicity constraints (van Leer 1977) are used to

mitigate short-wavelength numerical oscillations. The linearized characteristic

equations are solved to get the solution at an intermediate time step. The flux

terms are then obtained at each cell boundary by solving the full nonlinear

one-dimensional Riemann problem (Smoller 1982).In our implementation, we use (x, t) to determine the volume fraction of the

gases in each cell where | (x, t)| < 1. We use alternating sweeps in the x- andy-directions, which formally yields second-order accuracy (Strang 1968). We

do not employ artificial viscosities or gradient augmenters for contact discon-

tinuities. The details of the numerical method are given in Samtaney (1993).

For the second method, we use the equilibrium flux method of Pullin (1980).

Because of its diffusive nature, we can simulate strong shock phenomena. The

code has been extended by Samtaney & Meiron (1997) to include an ideal

dissociating model of a gas.

4.3 ParametersThe classical geometries for the RT and RM are those amenable to a linear

stability analysis, namely a discontinuous density-stratified interface, perpen-

dicular to an external acceleration or the normal vector of a planar shock wave,

respectively. The interface is perturbed with a small-amplitude single harmonic

ka0 1, where k = 2/ and a0 is the initial amplitude prior to shock arrival,as shown in Figure 1a for RM (where only a half wavelength is shown).

The density ratio across the interface prior to shock arrival is = 1/0, oralternatively, the Atwood number is A = (1)/(+1). At present, we do notinclude a model for the interface, e.g. the thin polymer membrane that separates

the two gases in a shock tube. In fact, the initial density jump is usually over

three zones and causes grid-scale perturbations on the front, as discussed below.

Recent experiments with a sinusoidal interface include Zaytsev et al (1993) and

Aleshin et al (1993, 1996).

The excitations are the acceleration g (for RT) which may be a function

of time, or the shock (for RM) as measured by the normalized pressure jump,

S (pbefore pafter)/pafter, or the Mach number, M, M2 = 1 + ( + 1)/(2 ))[S/(1 S)]. For compressible media, we have an ideal equation of state wherethe specific heats in each medium are 0 and 1.

4.4 GeometriesA more fundamental configuration is the inclined planar interface in Figure 1b,

because it allows an exact calculation of circulation deposition in many cases

(Henderson 1966, Sturtevant 1987, Henderson 1989, Samtaney & Zabusky,

1994) and because a self-similar solution can be found that includes vortex

roll-up and other fundamental vortex configurations in the vicinity where the



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508 ZABUSKY

incident shock approaches the interface (Samtaney & Pullin 1996). Another

basic configuration, shown in Figure 1c, is the circular cylinder, or axisymmet-

rical spherical bubble, discussed below (Haas Sturtevant 1987, Jacobs 1992,

Bourne & Field 1992, Yamada et al 1996, Zabusky & Zeng 1998, Klein 1998).

Numerical simulations and laboratory experiments are also being conductedin other configurations: the vertical interface with multiple small-amplitude si-

nusoidal modes or multiple piecewise linear sections (Budzinski et al 1994); the

planar layer of identically inclined interfaces or a sawtooth curtain (Sturtevant

1987, Zabusky & Yang 1989, Yang et al 1990); the sinusoidal curtain, or a

vertical layer where each interface is a single small-amplitude sinusoidal mode

(Rightley et al 1997); and the elliptical cylindrical bubble (Zabusky et al 1999).

Of course, one could exaggerate aspects of all the above geometries and ob-

tain a multitude of mushrooming interfaces. For example, one could increase

the amplitude or change the wavelength or phase of the various harmonics orsawtooth segments (Aleshin et al 1993), ka0 1.0. For example, for large-amplitude single harmonic interfaces, one sees secondary pressure waves arise

as the shock traverses the interface, waves that were not present for very small

amplitudes.

In addition to geometrical complexity, one could consider the complexity

of excitation, e.g. multiple planar shocks arriving from both directions of the

shock tube (the reshock problem) as well as consider cylindrical or spherically

imploding shocks in two and three dimensions. Finally, all the above can be stud-

ied in three dimensions. If the initial condition is controlled, then vortex-relatedinsights gathered from two-dimensional dynamics will provide pathways to un-

derstand three-dimensional effects up to intermediate times. However, at later

times three-dimensional effects are essential for the turbulence and mixing that

arise.

4.5 Equations of Vorticity EvolutionIf one takes the curl of the momentum equation for a compressible Navier-

Stokes fluid, one obtains the vorticity evolution equation

t= u + u ( u) + 1

2( p)

+ 2

( ) 432

[ ( u)] +

2

+

1

2

3( u)() + 2(u) () + ()

,

(3)



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where we have assumed that the viscosity, , is variable. The last two lines show

that viscous effects may be important (Hewett & Madnia 1998), particularly

the third row at high-gradient interfaces.

In all discussions below, we omit viscosity. The remaining terms on the

right of Equation 3 are the usual advection term, which when combined withthe left side gives the material derivative; the vortex stretching term, which is

essential for discussions of three-dimensional turbulence and mixing (We will

ignore this term below, as all discussions are of two-dimensional and non-swirl

axisymmetric flows); the vortex dilatation term, which is important only for

highly compressible fluids; and the baroclinic term, which is most essential in

our discussions.

Other important sources of vorticity that are not immediately evident from

this equation arise from curvature in shock waves, as discussed recently by

Kevlahan (Kevlahan 1997) and shock triple-point phenomena that arise fromMach reflections (Ben Dor et al 1980, Hornung 1985). Here, slip-lines, or vortex

sheets (in modern parlance), arise and are unstable and the source of turbulence.

In this context, the triple-point phenomenon was discussed by Colella et al

(1986) for the blast wave surface interaction and by Winkler et al (1987) and

Zabusky & Zeng (1998) for the shockaxisymmetric bubble interaction.

4.6 Time EpochsFor RT and RM, the evolution of a small-amplitude sinusoidal interface driven

by an external acceleration or the passage of a single shock wave, respectively,is sketched in Figure 2 in the left and right columns, respectively. The config-

urations, i.e. the direction of g or direction of incident shock, M or M,and medium through which it moves (heavy = slow and light=fast) empha-sizes the deposition of vorticity. The sign of the vorticity on the interface is

shown as ( + + ) to the right or ( ) to the left in allpanels and is obtained from the directions of p and g, as in-dicated. The right-hand rule for motion induced is implied. Also indicated

is whether the panel shows phase reversal (lower panels) or not (upper pan-

els). The lower left RT panel has two branches, one for a constant accelerationg = constant (which produces stable oscillations or gravity waves) and onefor g = 0 (at right), which illustrates the roll-up following the impulsive RM-like case. Note in all panels that the roll-up in the downward-moving spike

region is indicative of the localization of vorticity for a finite Atwood num-

ber, smaller than one. Also, the single roll-up domain (rather than many roll-

ups along the interface) is indicative of a smallbut finiteinterfacial layer

thickness.

For the RM environment, we split the evolution into the four following time

epochs:



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510 ZABUSKY



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1. Very Early Time (VET). The interval when the incident shock wave com-

pletely traverses the interface, a distance of 2a0, for the first time. For a

very small amplitude perturbation or a very small interface angle, the linear

theory applies.

2. Early Time (ET). The interval when secondary reflected and transmitted

pressure or shock waves pass over the interface (e.g. less than four transit

times of a linear wave across L T, the width of the shock tube). Nonlinear

effects arise early in this interval due to the growing amplitude of the inter-

facial perturbation, and the magnitude of the growth rate has achieved its

maximum. During part of this interval, the interfacial evolution may be mod-

eled by nonlinear theories, as discussed below, or with an evolving vortex

sheet on a nearly incompressible, density-stratified interface.

3. Intermediate Time (IT). The interval following ET when the vortex sheet

or layer on the interface localizes or collapses into one or more CVSs. For

sinusoidal perturbations, one may conveniently begin this interval when

the contact discontinuity becomes a multivalued function of the transverse

distance y, as discussed below.

4. Late Time (LT). The very large interval beginning when the flow is driven by

an array of compact vortices on the density-stratified interface and continu-

ing to late in the phase when a turbulent-like collection of vortex domainsmerges, binds, splits, and mixes the interface.

The time of transition between these intervals depends specifically on details

of the run, e.g. the strength of the shock or amplitude of the harmonic (or

alternatively, the inclination of the interface).

Figure 2 A schematic of the very-early and early time behavior for the classical RT and RMconfigurations, illustrated in the left and right columns, respectively. The configurations, namely

the direction of g or direction of incident shock, IS, moving in the direction of M and medium

through which it moves (heavy= slow and light = fast) emphasizes the deposition of vorticity.We see positive to the right and negative to the left (obtained from the directions of ,p, and g)and a downward-moving spike in all panels. The right-hand rule, for motion induced, is implied.

Also indicated is amplitude phase reversal (lower panels) or not (upper panels). The RT panel

(b) has two branches: g = constant (which produces stable oscillations or gravity waves), andg = 0 at right, which illustrates the roll-up following an impulse, as in the experiment of Jacobs& Niederhaus (1997). The roll-up in the spike region is indicative of an Atwood number smaller

than one.



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512 ZABUSKY

5. VISIOMETRIC METHODS AND CONVERGENCE

5.1 Visiometric Methods for RM EnvironmentsAs described in Bitz & Zabusky (1990), we visualize and quantify density;

dilatation, (div u) (to highlight wavefronts and shock waves); vorticity, =ez u; numerical shadowgraph, (to juxtapose contact discontinuitiesand shock fronts); and baroclinic generation, 2( p). The Lapla-cian (second derivative) of may oscillate and so is reserved for early time

analysis.

To obtain a global view of the flow, we project various functions to lower

dimensions and often present space-time diagrams. For example, we present

integrated positive, negative, and total vorticities (or circulations per unit length)

for planar or axisymmetric flows. For example, the projections

(x, t) =

(x, r, t) dr, (4)

where = (1/2)( ||), are either the positive or negative values of thez or components of vorticity, respectively. Similar diagrams are given for

integrations of vorticity along the shock direction, as seen in Figure 6, and also

the pressure on axis, p(x , 0, t).

Figure 3 (see color section) (Figure 4.6 of Samtaney 1993) shows the inte-

grated total vorticity up to the late IT phase. The run is for a shock (M = 1.5)interacting with a planar layer (Air/R22, = 3) at an angle of = 60. (SeeYang et al 1992 at early time for Air/R22 and M = 1.2 and = 30). Twoframes of reference are used: laboratory frame for 0 < t < 200; and a frame

translating uniformly with the velocity of the post-shock fluid. (The latter

frame is chosen so the page is filled optimally with data.) The dominant fea-

tures are the smooth deposition of circulation at VET (the negative sloping line

over the black region); the localization or roll-up of vorticity at the lower wall

(at left), a dominant feature, particularly when examined in color; the array of

vortices that emerge from the initially deposited smooth layer as a result of aKelvin-Helmholtz instability, as indicated by the multitude of positively sloping

thin domains emerging at 300 < t< 500. Here, perturbations to the vortex layer

arise mainly from the mesh-related initialization of the density. The merger of

these vortices (VM) with the wall vortex and with each other is a characteris-

tic process in two-dimensional turbulence. Also seen are white regions around

the increasing stratified vortex at lower left. This is due to the entrainment

of the shear layer generated after shock transmission and the baroclinic gen-

eration of vorticity on the rolling interface, an important process yet to be

modeled.



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Figure 3 y-integrated vorticity for a shock (M = 1.5) interacting with a planar layer (Air/R22, = 3) at an angle of = 60. Two frames of reference are used: laboratory frame for 0 < t 200.

The dominant features are the deposition of a layer of vorticity ( uniform black triangular region)

and emergence of lower wall vortex at left; array of vortices emerging from the layer as a result

of the Kelvin-Helmholtz instability (positive sloping, thin, dark regions) and the merger of these

vortices (VM) with the wall vortex and with the neighboring vortex, a characteristic process in

two-dimensional turbulence. Also seen are white regions around the increasing stratified vortex,

mainly at lower left. This is due to the entrainment of the shear layer generated after shock

transmission and the further baroclinic generation of vorticity.



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514 ZABUSKY

At any instant, the total circulation on the interface is calculated (Samtaney

& Zabusky 1994) as

(t) =D

(i, j, t)xy, (5)

where we use discretized representations

(i, j, t) = v(i + 1, j, t) v(i 1, j, t)2x

u(i, j + 1, t) u(i, j 1, t)2y

(6)

and where D = {(i, j, t) such that | (i, j, t) < 1 t h|}, and th is a chosenthreshold.

The domain of integration is chosen to highlight special phenomena. For

example, to examine the negative circulation on the interface and within thebubble, we accumulate only those cells with < 0 that coincide with thelocation of1, the bubble fluid.

5.2 Visiometrics of Circulation on Interfacesand Within Elliptical Domains

We now examine several runs in detail and compare models for them in sections

below. We consider mostly the two-dimensional domain with a sinusoidal inter-

face, ka0

=0.1 separating air and R22 where

=3.0, 0

=1.4, 1

=1.172.

Panel c2 of Figure 2 provides heuristic sketches. The undisturbed incidentgas (0) is initialized with unit pressure and density, and the incident shock

of M = 1.5 is initialized by density and pressure jumps in accord with theRankine-Hugoniot conditions. The simulations were made with our Godunov

code, the domain was discretized with three meshes (800 40, 1600 80, and3200 160), and a0 corresponds to 4, 8, and 16 zones, respectively. It takes4.5, 9.0, and 18.0 time units, respectively, for the M = 1.5 shock to cross theinterface, 2a0. At t = 0, the shock strikes the interface.

5.2.1 VISIOMETRICS AND CONVERGENCE OF INTERFACIAL CIRCULATION InFigure 4 we see images of vorticity with the one mid-height density contour

superimposed, for an Air/R-22 interface with M = 1.5, ka0 = 0.1 , and reso-lution 1600 80, shown at these times: (a) t = 113.2 (end of the VET phase);Figure 4 Images of vorticity with the one mid-height density contour superimposed. For an

Air/R-22 interface with M = 1.5, = 3.0, ka0 = 0.1 , and resolution 1600 80. The time forthe shock to cross the interface, 2a0, or 16 grid points, is nine units. Shown at times (a) t = 113.2;(b) t

=484.7, and (c) t

=981.3. Note in (a) the initial position of the incident shock with respect

to the interface (to scale) is shown in the box inserted at left and the present position of the slightlycurved transmitted shock is at right.



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516 ZABUSKY

(b) t = 484.7 (end of the ET phase); and (c) t = 981.3 (early in the IT phase).The actual shapes and relative positions of the initial condition (shock with

respect to the interface) is shown in the insertion at left in (a), and t = 0 occurswhen the shock first strikes the interface. The present position of the slightly

curved transmitted shock is at right. In (b) the density contour first becomesmultivalued (kinks), and in (c) the vorticity has localized and rolled up and

the density contour clearly shows the formation of a spike at left and a bubble

at right. Note the localization of the vorticity.

In Figure 5, we see temporal variations of circulation, obtained as described

above. The simulations were made with the Godunov code at three resolutions,

as mentioned above. Curves 2 and 3 are sufficiently close for us to conclude

Figure 5 Convergence of interfacial circulation for an Air/R-22 interface with parameters as given

in Figure 4. Simulations with the Godunov code at resolutions (1) 800 40, (2) 1600 80, and(3) 3200 160.



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that the compressible vortex dynamics is captured with sufficient accuracy to

allow modeling during this time interval. The first minimum of the negative

circulation corresponds to the time when the shock has traversed the interface,

2a0. The early undulations are due to reflected shocks passing over the interface

(depositing first positive and then negative vorticity, respectively.) The last twocrossing times are larger than the initial transit time because these reflected

shock waves are weaker than the original shock and are moving across the

lateral extent of the shock tube.

5.2.2 QUANTIFICATION OF INTERFACIAL CIRCULATION In Figure 6, we see the

corresponding x-integrated vorticity plotted as a function of y at seven times.

The rolling and localization of the interfacial vorticity is manifest in that the

curve is narrowest in the interval between 4 and 5, 465 < t < 616, where the

interface become multivalued.

Figure 6 x-integrated vorticity across the interface as a function of y at seven times for an Air/R-

22 interface with parameters as given in Figure 4. The shock strikes the interface at t = 0 andtraverses it in nine units.



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518 ZABUSKY

In Figures 7 and 8, we see results of quantifying the interfacial vorticity

with a quadratic form, ellipses fitted to CVS at 30% of maximum vorticity.

In Figure 7 we see the evolution of aspect ratio and normalized major axis.

Curve 1 at M = 1.5 corresponds to the previous quantifications. Curve 2 atM = 3.0 shows that the larger the deposited circulation, the more rapidly doesthe circulation localize, as with an incompressible layer (Baker & Shelly 1990).Also, later in time, curve 2 grows slowly, possibly due to diffusion, and has a

superimposed quasiperiodic behavior due to the rotation of the localized vortex

domain. Figure 8 shows the evolution of normalized y-centroid position of

these ellipses toward the tip of the spike, a result seen in Figure 4c. To obtain

good models of interface dynamics we must elucidate and model the temporal

approach to one or more asymptotical states of centroid, major axis, and aspect

ratio, etc, as illustrated in these last figures.

6. ANALYSIS AND SIMULATION

6.1 Linear StabilityBecause of mathematical simplicity, the growth of the small-amplitude single

harmonic perturbation of a two-dimensional interface, ka0 1,hasbeenamajorfocus of attention in the 1990s. Richtmyer analyzed the temporal problem and

presented a simple scaling formula for the terminal or asymptotic growth rate

aftershock passage,

aI ka0 AU, (7)often called the impulse formula or impulse model. Here, k is the wave num-

ber of the perturbation, a0 is related to the initial amplitude of the harmonicperturbation to the interface or contact surface (as elaborated below), A =( 1)/( + 1) is the post-shock Atwood number (i.e. = 1 /0 ), andU is the speed of an interface after shock passage from medium zero to one

in a corresponding one dimensional interaction (if the interface is at rest prior

to the arrival of the shock). Richtmyer suggested using post-shock amplitudes

above, or a0 = a0+ for the case when the reflected wave is a shock. Notethat we designate the pre-shock amplitude or initial interface perturbation as

a(0) a0. Recently, several investigators have clarified and elaborated on thelinear theory (Fraley 1986, Mikaelian 1994a,b, Yang et al 1994, Wouchuk &

Nishihara 1997).

The work of Yang et al (1994) presented a comprehensive study of the linear

(or VET) RM environment for sinusoidal interfaces that separate regions of

different density and . [Note in their paper that the shock moves from region

2 to region 1 and that their interface growth rate is designated as

a0(t) while

ours is a(t).] They introduced a new self-similar partial differential equation



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Figure 7 Evolution of aspect ratio (solid) and normalized major axis ( ) of ellipses

fitted to CVS at 30% of maximum vorticity. For an Air/R-22 interface with parameters as given in

Figure 4, and (1) M= 1.5 and (2) M= 3.0. The times for the shock to cross the interface are 9.0

and 4.5 units, respectively.

Figure 8 Evolution of normalized y-centroid position. For an Air/R-22 interface with parameters

as given in Figure 4, and (1) M = 1.

5 and (2) M = 3.

0. The times for the shock to cross the

interface are 9.0 and 4.5 units, respectively.



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520 ZABUSKY

for the pressure as function of the variables (t, ) = (t,x /t) (different fromthe one used by Richtmyer) and solved it by consistent second-order accurate

numerical methods.

Yang et als Figures 11 and 12 show the linear terminal growth rate, a0()vs incident shock strength, S = (pbefore pafter)/pafter, for reflected shocks(fast/slow configurations) and reflected rarefactions (slow/fast configurations),respectively. As expected, for small S and , the growth rate magnitudes increase

with increasing S and . However, they note without explanation that almost all

the curves are non-monotonic as S increases. Yang et al agreed with Richtmyer

that post-shock densities (on both sides of the interface) are to be used in the

impulsive formula. They also agreed with the consensus of investigators that

in the formula for aI : a0 a0+, for a reflected shock, a0 12 (a0 + a0+),

for a reflected rarefaction, where the last had been suggested by Meyer &

Blewett (1972). Careful comparisons of the terminal growth rate were made tothe impulsive model aI over a range of parameters for cases where the wave

reflected from the interface was a rarefaction or a shock. These results are given

in Tables I and II and Figures 16 and 17 and show strong disagreement for

S > 0.8.

Yang et al conclude with the following comments:

1. The impulsive formula with parameters chosen as above agree well in the

weak incident shock limit.

2. The agreement between the impulsive model and the linear theory improvesas the adiabatic exponent s of the two fluids increase while remaining

approximately equal. This is especially true for the reflected rarefaction

case.

3. For strong incident shocks, large discrepancies between the linear theory and

impulsive model appear, especially when the s are substantially different

(see their Table 1).

6.2 Nonlinear TheoryRecently Velikovich & Dimonte (1996) (for A = 1) Zhang & Sohn (1997) andBerning & Rubenchik (1998) presented formal perturbation calculations (fol-

lowing the work of Haan 1991) for the growth of a very small amplitude, single

harmonic perturbation to a vertical interface beyond early times, a nonlinear

theory. The latter two assumed incompressibility and irrotationality off the

interface and derive a Taylor series of four terms whose highest-order term in-

cludes the fourth harmonic multiplied by terms O(a0A3), etc. It is obvious from

the multivalued character of the evolving interface (due to voricity localization)

that this description is inadequate after a very short time.



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The authors (Zhang & Sohn 1997) recognized the divergent nature of their

series, and because the overall growth rate decays at large times, they introduced

Pade approximants and a heuristic inner-outer matching recipe. . . proposed

by Prandtl to obtain growth rates for the bubble and spike (their Equations 58

and 59) and total amplitude asab = ali n/D1 + ali nA/D2, (8)

as = +ali n /D1 + ali nA/D2, (9)

a da/dt = (1/2)(as ab) = ali n/D1, (10)where

D1 = 1 + a0+k + max0, a20+k2 A2 +1

2

2

, and

D2 = 1 + 2a0+k + (4/3)

3a20+k2 A2 + 12.

(11)

Here, A is the Atwood number based on post-shock densities, = kali n t, andali n a0 is the time-varying solution obtained from their linear analysis. Notethat in the Di s we have used the post-shock amplitude, a0+, as described abovefor the case of a reflected shock.

Their growth rates are a product of the time-dependent linear growth rate,

ali n, multiplied with an approximant that has an asymptotic property of varying

like t1. However, the true asymptotic velocity is a constant, +0 (t1), becausea vortex dipole forms in the spike region. Thus, the Pade process converts the

divergent Taylor series into a function that may be close to the comparable a

from the dns, over the ET range. However, at later times, particularly in the

spike region, the discrepancies are larger. Furthermore, their figures of smooth

waveforms of the interface do not show the small-scale structures that truly arise

on very high-gradient interfacial layers. For additional comments on nonlinear

theories at ET, see Neuvazhayev & Parshukov (1997).

7. VORTEX PARADIGM MODELS

7.1 OverviewThe deposition of vorticity in a two-dimensional inviscid fluid is based on the

equation

D/Dt = p2

u, (12)

=

[

Du/Dt

+g]

u, (13)



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522 ZABUSKY

which is obtained from Equation 3, if the viscous terms and the vortex stretching

term are omitted. The first (baroclinic term) is dominant in our discussions. For

RT, we replaced p/ by terms in the momentum equation to make explicitthe forcing by the external acceleration, which helps elucidate Figure 2.

Most RM models do not recognize the importance of the vortex view-point. This situation is changing. Following Hawley & Zabusky (1989),

Samtaney & Zabusky (1994) derived formulas for circulation per unit length

deposited on discontinuous planar and curved interfaces. Samtaneys thesis

(1993) contains a multitude of simulations with visualizations and quantifica-

tions. The early time vortex models discussed in Section 7 were first presented

there. Jacobs et al (1995), Jacobs & Sheely (1996), and Rightley et al (1997)

interpreted experimental data from compressible shock layers (curtains) and im-

pulsively accelerated, falling tanks, respectively, with von Karmanlike point

vortex models of a nonstratified fluid. Despite these simplifications, they es-timated sufficient parameters to achieve collapse of the data over a short time

interval. A more systematic approach up to intermediate times was made by

Ray et al (1996) using both incompressible vortex sheet and point models, and

is given below.

7.2 Caveats on Interfaces and Models in Twoand Three Dimensions

The problem of a two-dimensional vortex sheet in a homogeneous inviscid

medium (the Kelvin-Helmholtz instability) is ill-posed and gives rise to theMoore singularity in a finite time (Hou 1995). This singular behavior is the

source of rapidly growing small-scale perturbations that roll up in numerical

computations. A comparison of a Navier-Stokes simulation with a regularized

single sheet of vorticity shows the limited capability of such single-sheet models

beyond a time of two rolls (Tryggvason et al 1991). With surface tension

this singularity is suppressed, but at large Weber number, other singularities

may arise (Hou et al 1997). Similarly, for an incompressible discontinuous-

density RT environment, vorticity is deposited on a sheet and the problem is ill-

posed (Baker et al 1993). Samtaney & Pullin (1996) and Samtaney (1997) havereexamined the RM situation by comparing computer-generated self-similar

solutions with highly resolved evolutions and have found small-scale roll-up

phenomena in the latter, comparable to those shown in Figure 3. They note that

the small-scale structures appear sooner as the resolution increases, a result

to be expected, because the layer thickness is smaller ( x) and hence thevorticity is larger, since circulation per unit length is conserved. In all further

discussions, we assume that the numerical solutions are regularized by pro-

cesses associated with grid-related truncation errors. For long-time turbulent

mixing calculations, the interfacial layer should be adequately resolved anddescribed by appropriate physical processes.



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7.3 Modeling VET Circulation Deposition with a LocalShock-Polar Analysis

7.3.1 INTRODUCTION The analysis assumes a frame of reference that is sta-

tionary with respect to the node where all the waves meet and both gases

are perfect and inviscid. The geometry and variables of interest are given inFigure 9. The notation is the same as that used by Samtaney & Zabusky (1994).

Here, mm is the interface; i, r, t are the incident, reflected, and transmitted

waves, respectively; s1 and s2 are the streamlines in the incident and transmit-

ted media; 0 and 1 are the deflections ofs1 due to i and r, respectively; and

b is the deflection ofs2 due to t. p0 and pb are the initial pressures in the inci-

dent and transmitted gases, respectively. Here p1, p2, and pt are (respectively)

the pressures behind the incident, reflected, and transmitted waves. The free-

stream Mach number in front (behind) the incident, reflected, and transmitted

shocks are M0(M1), M1(M2), and Mb(Mt), respectively. 0 and b are the ratioof specific heats in the incident and transmitted gases, respectively.

A shock polar is a graph of a deflection angle vs a pressure ratio. It represents

the locus of final states that can be connected to a given initial state for a

stationary oblique shock. For a planar interface, the angle between the incident

shock and the interface is constant. If is smaller than cr, all the three waves

are shocks and meet at a single node (called RRR). For very weak shock waves,

as is increased, the reflected wave becomes a Mach line and then a centered

rarefaction fan (called RRE). As is increased further, the reflected wave is not

a centered expansion fan and the refraction becomes anomalous. For stronger

Figure 9 Schematic of regular refraction, for three shocks at a fast-slow interface. Shown are

incident, i , reflected, r, and transmitted, t, shocks intersecting at a node on the interface, m. The

subscript notation (before and after) in this figure is from Samtaney & Zabusky (1994) and differsfrom that used in the present paper.



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524 ZABUSKY

shocks, the irregular refraction (called MRR) is typically characterized by the

appearance of a Mach stem at the interface.

Samtaney & Zabusky (1994) show (in their Figure 3) that for a large range

of > 1, cr decreases rapidly from 90 in 1 < M < 1.5 and then approaches

a near-constant small negative slope when 1.5 < M < 4.0. The larger , thelarger the decrease in cr. For example, for Air-R22 ( = 3.0, 0 = 1.4, b =1.172), cr = 54 at M = 4, and for He-Xe ( = 32.7, 0 = 1.667, b =1.667), cr = 36 at M = 4.1

7.3.2 EXACT CIRCULATION ON THE INTERFACE Samtaney & Zabusky (1994)

followed Henderson (1966) and obtained a 12th-order polynomial in p2. This

polynomial may have more than one real root or no real roots. We concern

ourselves with the physical root, which is the smallest real root.

They defined the circulation (vorticity deposition) per unit length of the

shocked interface as

d

ds vt v2, (14)

where vt and v2 are the velocities tangential to the shocked interface in the trans-

mitted gas and the twice-shocked incident gas, respectively. They renormalized

the circulation with respect to the original unshocked interface by multiplying

Equation 14 by the geometric factor ds/ds = [cos / cos( b)], which ac-counts for the instantaneous change in length of the interface due to the shock

passage

dds

= (vt v2)cos

cos( b)(15)

and obtained

= 12

0

sin

M2 + 2

b 1

1 (p2, b)

0/b

sin2

12

M2 + 20 1 (1 (p2/p1, 0) (p1, 0)) sin2 12 cos

cos( b) ,

(16)

where 2i = (i 1)/(i + 1) and (,) (1 + 2)( + 2). Note thatsin in Equation 16 has the same sign as p.

1For simplicity, we normalize by assuming pb = p0 = 0 1 and b = initially. Thus,the sound speed is

12

0 in the incident gas. In the sections that follow, we will normalize the

circulation on the interface by a length scale, which will be indicated. All the results will thus have

units of circulation per unit length, i.e. units of velocity. In order to convert to physical units, our

normalized circulation must be multiplied by the ratio of the physical sound speed to 12

0 .



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In the above equation, 12 represents physically the ratio of sound speed be-

hind and ahead of a shock. Equation 16 may only be used for RRR at a planar

interface, and we refer to it as the exact shock polar result for . Typically,for a slow-fast interface, the shock refraction is of the RRE type for small

(Puckett et al 1991). For RRE, one can derive a corresponding equation forcirculation by replacing the shock jump conditions across the reflected wave

with the Prandtl-Meyer expansion conditions. This is given in Samtaney et al

(1998).

Figure 10 shows the exact circulation, , in the (M1, 12 ) plane for =

1.4. The angle of the interface is = 15, 30, 45, and 60. The normalized

Figure 10 Circulation per unit length, normalized by in the M1, 1 plane. The angle

of the interfaces are (a)

=15, (b)

=30, (c)

=45, and (d)

=60. The light gray

(or black) regions show where is smaller (or larger) than 10% of. The white spaces showregions where the refraction is not RRR.



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526 ZABUSKY

circulation is set to zero where the shock refraction is not RRR (white space

indicated by IR). The scaling law holds exactly when the normalized circulation

is exactly equal to one. The 10% error boundaries (which bound the light grey

space) correspond to contour levels 1.1 and 0.9. For small , there are regions

where the scaling laws are within the specified error bounds up to very large Mat intermediate values of; and up to very large at low to moderate values of

M. For instance, for 30 , the error is less than 10% for 1.0 < M 1.32for all , and 5.8 32.6 for all M.7.3.3 APPROXIMATE MODEL FOR CIRCULATION ON THE INTERFACE When the

shock polar equations do not yield a physical root (an irregular interaction),

we have no exact formula for velocity jump. However, if we expand inEquation 16 as a series in sin about sin = 0, we obtain results close to thequantifications of our numerical simulations:

= 1 sin + 3 sin3 + O(sin5 ), (17)since the circulation on the interface is an odd function of sin . Here 1 =/ sin |sin =0 and 1 (the dominant term) and 3 are both real in 0 /2 and are given in Samtaney & Zabusky (1994).

7.3.4 SCALING LAW FOR CIRCULATION To simplify our discussions, we use

identical s, or = 0 = b. Hence in the limit of large M and small , weobtain

, where

=2

12

+ 1 (1 1

2 ) sin (1 + M1 + 2M2)(M 1). (18)

Thus, the circulation deposited per unit original length, , scales as

1. (1 + M1 + 2M2)(M 1) in Mach number. This leads to linear behaviorin M for large M.

2. (1 12 )for the density ratio. Note that for large , circulation is essen-tially independent of the density ratio. For , the circulation shouldcorrespond to the slip velocity at a solid surface.

3. (sin ) for the inclination of the interface with the shock.

4. 12 /( + 1) for the ratio of specific heats.

7.4 Reduced Vortex Models at Earlyand Intermediate Times

Ray et al (1996) present three reduced models that assume a Biot-Savart formu-

lation, where the vorticity drives the interface.Two of them, the stratification-



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modified Biot-Savart (SMBS) model and the line-vortex (LV) model, are pre-

sently limited in that they give accurate results over restricted time ranges.

Comparisons for a (the difference between the velocities of the x-extrema

[spike, y = 0, and bubble, y = /2] of the contact discontinuity) are given inFigure 11a and b among the impulse model (horizontal solid line); linear anal-ysis (described in Section 6.1); dns ( ); and NBS ();SMBS () and LV (), as described below. Note that thedoubling in M yields an approximate factor of two in time in the appearance

of all phenomena (which is associated in the almost factor of two in deposited

circulation).

7.4.1 NUMERICAL BIOT-SAVART (NBS) For the numerical Biot-Savart, or NBS,

the interfacial vorticity and location are extracted from the dns and substituted

into the Biot-Savart formula. The comparison with dns for two runs, M=

1.5

and M = 3.0 for = 3.0, is shown in Figure 11a and b. The excellent quality ofthe agreement at M = 1.5, after the waves have diminished, validates the codeand the idea that the evolution of vorticity is the cogent process for understanding

interface dynamics.

7.4.2 STRATIFICATION-MODIFIED BIOT-SAVART (SMBS) MODEL Here we assume

that only the original first harmonic is present and substitute into the equation

for x-component of velocity, u(x, y, t) to obtain

a = (t)k(2)4( + 1)

sin dk2a2(1 cos )2 + 2 , (19)

where = ky, (t) is a piecewise linear approximation to the extracted time-varying circulation, and where we have included (2

)/( + 1), a heuristic

factor obtained from linear theory (Chandrasekhar 1961), to account for the

stratified fluid. The comparison for two runs, M = 1.5 a n d M = 3.0 for = 3.0,is shown in Figures 11a and b by (). The model is valid in a small timeinterval where

a saturates, i.e. before the higher harmonics grow and couple to

each other.

7.4.3 LINE VORTEX (LV) MODEL AT INTERMEDIATE AND LATE TIMES We assume

an array of point vortices of strength , placed for simplicity at y = (2n +1)/4, n = . . . 2, 1, 0, 1, 2, . . .. For the present, the circulation is thetotal circulation deposited after first shock passage, a quantity well estimated

at low M by the methods above. From this, we obtain

a = (3k/2 ) cosh(ak) or sinh ka = (3/2 )k(t ts ) + sinh ka(ts ).

(20)



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528 ZABUSKY



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The comparison for two runs, M = 1.5 and M = 3.0 for = 3.0, is shown inFigure 11a and b by (). The model improves in accuracy at later timeswhen the dominant vortex is well localized and a(t) is large, i.e. the interface

extrema are remote from the localized vortex centroid. See Jacobs 1995 for a

similar approach.This model can easily be improved when we relax the simplification of a

point vortex to an elliptical vortex domain and also place the vorticies at their

true centroid, (xc, yc) (from a model yet to be obtained or from the data, e.g. as

previously shown for yc), in Figure 8 so that we obtain an asymptotic a which

approaches a constant.

8. SHOCK BUBBLES AND VORTEXPROJECTILES (VPs)

The planar-shock bubble interaction is fundamental to many domains, but dif-

ficult to understand and model, particularly at higher Mach numbers and later

times (Klein et al 1994, Zabusky & Zeng 1998, Klein 1998, personal commu-

nication). The separation and fragmentation of the initial interfacial vorticity

layer introduces complex fluid dynamical processes.

We have introduced the concept of a translating vortex projectile (VP)

(Zabusky et al 1997, Zabusky & Zeng 1998) to synthesize a new understanding

of vortex dynamics in the vicinity of interacting nearby layers or structures of

opposite-signed vorticity, in particular, those deposited on contact surfaces orhigh-gradient regions when a shock wave interacts with a spherical or elliptical

bubble (Zabusky et al 1999) that is heavier than ambient or in reacceleration

and reshock environments for RT and RM, respectively.

The VPs are coherent vortex structures (CVSs) of positive and negative or

dipolar circulation in two-dimensions or oppositely-directed, nearby tube-like

regions (e.g. a vortex ring) in three dimensions. These VPs move under their

own self induction, i.e. they have a dominant translational motion in a frame of

reference in which the mean flow is zero. The simplest planar and axisymmet-

ric examples for homogeneous incompressible flow include, respectively, thedipole, a rectilinear pair of point vortices, (, ), separated by d (which trans-lates with speed /(2 d)) and the axisymmetric vortex ring [e.g. the limiting

form is Hills spherical vortex (Saffman 1992)]. Note that for three-dimensional

Figure 11 Evolution of circulation, 0.005 (dashed), a for the numerical simulation (dark solid)and the models: Impulse model, aI (horizontal line); Linear analysis (of Section 6.1) ;NBS (

); SMBS (

); and LV, (

). For Air/R-22,

=3.0, ka0

=0.1 ,

and resolution is 1600 80. (a) M = 1.5 and (b) M = 3.0.



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530 ZABUSKY

homogeneous, incompressible fluids, circular vortex rings are unstable (analysis

of Widnall and colleagues, 19711973, as described in Saffman 1992); ellipti-

cally deformed rings may break up into two ringlets in a finite time [Biot-Savart

simulations of Fernandez et al (1995)] and colliding coaxial rings will produce a

multitude of ringlets [experiments of Lim & Nickels (1992)]. The compressibleand stratified analogs of these and related configurations is yet to be investigated.

These two- and three-dimensional VPs may arise through instability of some

stationary or slowly evolving configurations of positive and negative circulation.

Couder and colleagues (Couder & Basdevant 1986), in pioneering experiments

and simulations on two-dimensional cylinder wakes, showed that perturbations

will induce strain fields that cause nearby opposite-signed vortices to link up

or bind and move together as a single dipolar VP, which is frequently manifest

as a mushroom or tadpole structure. They are an important component of

many geophysical flows (Fedorov & Ginsburg 1989). Note, in compressiblemedia, that isolated dipoles may move with variable speeds that depend upon

their Mach number and aspect ratio (Scheidegger 1997). Also, they have been

seen by Youngs (1994), who simulated a shock striking a randomly perturbed

two-dimensional interface and observed vortex pairs that propagate away from

the mixing zone.

The VP concept is generic in two and three dimensions and will play an

important role in interpreting reshock and reacceleration phenomena.

Also, in analogy with the geophysical environment and the CVS seen in

dns, I have conjectured that recent observations of explosion fragments(Aschenbach et al 1995) or knots (ODell & Handron 1996) in the vicinity of

supernova remnants and planetary nebula could be VPs. To validate the conjec-

ture, observationalists must quantify the temporal variation of the translational

velocity, radius, and internal rotational motion associated with VPs.

9. SUMMARY

We have shown how a vortex viewpoint, enhanced by visiometrics, will enrich

understanding of phenomena and the accuracy and modeling of acceleratedinhomogeneous flows, e.g. RM processes up to intermediate times. Similarly,

this modus will facilitate the understanding and statistical modeling of simula-

tions of turbulent-like phenomena in two and three dimensions, which many re-

searchers perform with inviscid codes. However, their results are useful only for

large-scale and qualitative aspects, i.e. their intermediate and small-scale struc-

tures are corrupted beyond intermediate times. This follows because most codes

are second-order accurate away from shocks and because the inviscid problem

is ill posed for vortex sheets. In the future we need codes of higher-order



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accuracy and interactive visiometric environments to handle the information

available from these massive data sets.

10. OMISSIONSThe author has tried to cite original papers, early reviews, and recent papers

in RT and RM, particularly those that emphasize the dynamics of vorticity.

Omissions from the list have been made because of space reasons or lack of

knowledge. I pay tribute to all, who in spite of valuable contributions, are not

mentioned in this brief paper. I hope they will send me their annotated works,

as I may have been unaware of them.

ACKNOWLEDGMENTS

The insights presented result from nearly a decade of work and collaborationswith many friends and colleagues. Some early insights in the shock interface

problem were obtained in joint work with K-H Winkler and P. Woodward in

1987 and John Hawley in 1989. Recent progress was made with X Yang, I-L

Chern, J Ray, S Zeng, A Kotelnikov, and particularly R Samtaney. I am grateful

to R Samtaney and W Rider for comments on the manuscript. J Ray assisted

in producing the manuscript. I am also grateful to other (at the time) graduate

students and staff in the VizGroup, including X Wang, M Gao, V Fernandez,

H-B Yao, P Mossip, W Townsend, and S Cooper.

In 1987 and 1989, collaborations on visiometrics began with F Bitz and

Prof. Deborah Silver, respectively. During the period 19891991, we had sup-

port from the National Science Foundation under Dr. Alvin Thaler (DMS-

89-01-900), and since 1993 our visiometric work has been supported by the

Department of Energy under Dr. Daniel Hitchcock and Dr. Fred Howes.

Computations were done on the Cray Y-MP and the CM-2 at the Pittsburgh

Supercomputer Center and on the CM-5 at NCSA. Recent computations were

done on the Cray T3D and T3E at the Pittsburgh Supercomputer Center, on the

IBM SP2 at the Cornell Theory Center, and the Origin2000 at NCSA. All these

centers are supported by the National Science Foundation. We also acknowl-

edge the support of the Center for Computer Aids for Industrial Productivity

(CAIP) at Rutgers University.

NOTE ADDED IN PROOF

There is a question of the asymptotic behavior of the growth rate for an ini-

tially small amplitude sinusoidal perturbation on an interface as discussed in

Section 6.2. The common belief of the community is that

a varies as (1/t).



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532 ZABUSKY

In the last paragraph of Section 6.2 we assert that it is = constant + 0 (t1).A Kotelnikov, J Ray, and I have recently completed a study of this asymp-

totical process which shows that for an incompressible situation, following an

implusive acceleration, the total circulation on the interface decreases because

of baroclinic effects and that the vorticity localizes at a centroid, Yc (A, M)/LTwhich approaches monotonically to a constant that is 0.4 for 0.3 < A < 0.9 and

M = 0 (incompressible) and decreases as M increases, as shown in Figure 8.It is observed that the decrease in YT is large due to finite M (compressibility)

at A = 0.5 and M = 3.0.

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