Analysis of single-mode Richtmyer–Meshkov instability using high-order incompressible vorticity—streamfunction and shock-capturing simulations

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RESEARCH ARTICLE

FEBRUARY 13 2024

Analysis of single-mode Richtmyer–Meshkov instability

using high-order incompressible vorticity—streamfunction

and shock-capturing simulations



Special Collection:

Shock W

aves

Marco Latini

;

Oleg Schilling



;

Daniel I. Meiron

Physics of Fluids

36, 0241

15 (2024)

https://doi.org/10.1063/5.0179157

28 August 2024 21:59:52

Analysis of single-mode Richtmyer

–

Meshkov

instability using high-order incompressible

vorticity

—

streamfunction and shock-capturing

simulations

Cite as: Phys. Fluids

, 024115 (2024);

doi: 10.1063/5.0179157

Submitted: 29 September 2023

Accepted: 12 January 2024

Published Online: 13 February 2024

Marco

Latini,

1,a)

Oleg

Schilling,

2,b)

and Daniel I.

Meiron

AFFILIATIONS

Applied and Computational Mathematics, California Institute of Technology, Pasadena, California 91125, USA

Lawrence Livermore National Laboratory, Livermore, California 94550, USA

Graduate Aerospace Laboratories and Applied and Computational Mathematics, California Institute of Technology, Pasadena,

California 91125, USA

Note:

This paper is part of the special topic, Shock Waves.

Present address:

Northrop Grumman Aeronautics Systems, Palmdale, California 93550, USA.

Author to whom correspondence should be addressed:

schilling1@llnl.gov

ABSTRACT

Two- and three-dimensional simulation results obtained using a new high-order incompressible, variable-density vorticity

–

streamfunction

(VS) method and data from previous ninth-order weighted essentially nonoscillatory (WENO) shock-capturing simulations [M. Latini and

O. Schilling,

“

A comparison of two- and three-dimensional single-mode reshocked Richtmyer-Meshkov instability growth,

”

Physica D

401

132201 (2020)] are used to investigate the nonlinear dynamics of single-mode Richtmyer

–

Meshkov instability using a model of a Mach 1.3

air(acetone)/SF

shock tube experiment [J. W. Jacobs and V. V. Krivets,

“

Experiments on the late-time development of single-mode

Richtmyer

–

Meshkov instability,

”

Phys. Fluids

, 034105 (2005)]. A comparison of the density fields from both simulations with the experi-

mental images demonstrates very good agreement in the large-scale structure with both methods but differences in the small-scale structure.

The WENO method captures the small-scale disordered structure observed in the experiment, while the VS method partially captures such

structure and yields a strong rotating core. The perturbation amplitude growth from the simulations generally agrees well with the experi-

ment. The simulation bubble and spike amplitudes agree well at early times. At later times, the WENO bubble amplitude is smaller than the

VS amplitude and vice versa for the spike amplitude. The predictions of nonlinear single-mode instability growth models are shown to agree

with the simulation amplitudes at early-to-intermediate times but underpredict the amplitudes at later times in the nonlinear regime.

Visualizations of the mass fraction and enstrophy isosurfaces, velocity and vorticity fields, and baroclinic vorticity production and vortex

stretching terms from the three-dimensional simulations indicate that, with the exception of the small-scale structure within the rollups, the

VS and WENO results are in good agreement.

Published under an exclusive license by AIP Publishing.

https://doi.org/10.1063/5.0179157

I. INTRODUCTION

The classical Richtmyer

–

Meshkov instability

1,2

is an important

fluid dynamics instability with many applications in science and tech-

nology. When this instability is induced by a shock wave interacting

with a perturbed interface separating two fluids, the subsequent growth

and development of the instability and mixing between the fluids are

challenging to study theoretically, experimentally, and numerically. A

large number of experiments and numerical simulations of the

Richtmyer

–

Meshkov instability have been performed over the last

several decades, which have increased the understanding of the insta-

bility mechanisms and their consequences for applications.

3,4

Although multimode perturbations and reshock of evolving mixing

layers occur in nature and technological applications, the singly-

shocked, single-mode Richtmyer

–

Meshkov instability in planar geom-

etry remains widely studied numerically, experimentally, and theoreti-

cally because of its fundamental as well as relatively simple nature.

–

Most direct, large-eddy, and implicit large-eddy simulations of the

Richtmyer

–

Meshkov instability are performed using shock-capturing

Phys. Fluids

, 024115 (2024); doi: 10.1063/5.0179157

, 024115-1

Published under an exclusive license by AIP Publishing

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methods, which treat shock propagation explicitly and account for com-

pressibility effects. A wide range of numerical algorithms are used with

varying spatial and temporal accuracy and resolving capabilities. These

simulation methods are computationally expensive, even on modern

supercomputing platforms. Another class of numerical methods that

has had success in describing the interfacial hydrodynamic instability

evolution induced by Rayleigh

–

Taylor, Richtmyer

–

Meshkov, and

Kelvin

–

Helmholtzunstableflowsarevortexmethods.Whilethese

methods are typically incompressible, they can describe many of the

essential variable-density features of such flows and are appealing

because they are directly related to the main physical mechanism that

initially drives these instabilities

—

baroclinic vorticity production

through the deposition of vorticity by a shock on a perturbed interface.

These methods are computationally efficient and can provide useful

insights into the physics of these complex hydrodynamic flows.

Vortex methods have been applied to the Richtmyer

–

Meshkov

instability, as well as to Rayleigh

–

Taylor and Kelvin

–

Helmholtz insta-

bilities. A semi-Lagrangian vortex-in-cell method with a fourth-order

Runge

–

Kutta time-evolution scheme, a Lagrangian vortex blob

method, and an Eulerian second-order Godunov method were used to

perform two-dimensional simulations of incompressible Richtmyer

–

Meshkov experiments

11,12

including the second shock (or reaccelera-

tion).

13,14

A vortex-in-cell method and the piecewise-parabolic method

were used to simulate the incompressible and compressible, respec-

tively, single-mode Richtmyer

–

Meshkov instabilities in two dimen-

sions with a finite interfacial transition layer.

The incompressible

Richtmyer

–

Meshkov instability was simulated for different Atwood

numbers using a vortex sheet method applied to the Birkhoff

–

Rott

equation.

Inviscid and viscous point vortex models were demon-

strated for the Richtmyer

–

Meshkov instability.

The motion of vortex

sheets in three-dimensional inviscid and incompressible Richtmyer

–

Meshkov and Rayleigh

–

Taylor instabilities with axial symmetry was

investigated analytically and numerically using the vortex blob

method.

The nonlinear interaction between bulk point vortices and

an unstable interface with nonuniform velocity shear such as the

Richtmyer

–

Meshkov instability was investigated using the vortex

method.

The vortex sheet model was used to study linear and nonlin-

ear interactions between an interface and bulk vortices in the

Richtmyer

–

Meshkov instability.

A multiscale model based on an

asymptotic Birkhoff

–

Rott equation was developed for Rayleigh

–

Taylor

and Richtmyer

–

Meshkov instabilities in two-dimensional, inviscid,

compressible vortical flows.

Here, two-dimensional (2D) and three-dimensional (3D) simula-

tions and analysis of single-mode Richtmyer

–

Meshkov instability are

performed using a high-order vorticity

–

streamfunction (VS) method

with a comparison to results

obtained using the formally ninth-order

weighted essentially nonoscillatory (WENO) shock-capturing method.

The WENO simulation results are considered as reference results

expected to be accurate for shock-induced flows. As for all other

numerical methods, the realized computational accuracy diminishes to

at best first-order near the shock. Although the global order of the

solution is reduced near a shock, formally higher-order methods better

resolve complex flow features at long evolution times than traditional

second- and third-order methods (such as Godunov methods using

limiters). Higher order schemes are more computationally efficient

than lower order schemes for the same accuracy. In particular, WENO

methods of sufficiently high-order are well-suited for the simulation of

complex, compressible evolving flows containing shocks and structures

with a wide range of scales. Following previous investigations,

–

the

ninth-order WENO method was used

for the present study: the sim-

ulations were performed using mix initial conditions, an initial sinusoi-

dal perturbation, and a grid resolution corresponding to 512 points

per initial perturbation wavelength

In addition to the comprehensive study of the dynamics of the

single-mode Richtmyer

–

Meshkov instability realized in the Jacobs

–

Krivets experiment (extended to include reshock) presented recently,

the same experimental configuration was previously simulated.

27,28

The predictions of the second-order MUSCL and fifth- and ninth-

order WENO methods were compared.

The 3D compressible, multi-

component Euler equations were solved, with an emphasis on the

effects of grid resolution. A sharp interface was used in these simula-

tions. The predictions of the amplitudes obtained from all three meth-

ods were shown to be in very good agreement with the experimental

data points. This experiment was later simulated in two dimensions

using an adaptive central-upwind, sixth-order WENO method and the

fifth-order WENO method.

The compressible Euler equations, aug-

mented by an equation for the heavy fluid volume fraction, were

solved. Simulations using a single value of the adiabatic exponent for

the gas mixture were compared with simulations using the different

adiabatic exponents for the gases. A diffuse interface was used, with

256 cells per wavelength. It was shown that the sixth-order method

captured the small-scale secondary instabilities within the rollups,

while the fifth-order method did not. The effect of varying the width of

the initial diffusion layer was also studied. The amplitude growth from

these simulations was not compared to the experimental data points or

to the PLIF images.

A hybrid compressible

–

incompressible solver that removes the

low-Mach-number limitations of compressible solvers, thus allowing

numerical simulations of late-time mixing, was developed and used to

investigate late-time, multimode Richtmyer

–

Meshkov mixing with

Atwood number 0.5.

This method is motivated by the observation

that the instability is compressible at early times after the passage of

the shock through the interface and nearly incompressible at later

times. More recently, 2D, single-mode Richtmyer

–

Meshkov instability

was simulated for a wide range of Atwood numbers, Mach numbers,

and perturbation amplitudes.

The perturbation amplitudes were

compared to analytic growth models, and a modification to them was

proposed to describe the growth reduction found at large initial pertur-

bation amplitudes and large Mach numbers.

Jacobs and Krivets

modified the vertical shock tube previously

used in the experiments of Jones and Jacobs

and Collins and Jacobs

to include a longer driver section, allowing a stronger shock with

Mach number Ma

3 to be generated. The test section had an 8

cm square cross section and a length of 75 cm. A membraneless initial

condition was created as follows: a mixture of air and acetone vapor

[denoted air(acetone) hereafter] and sulfur hexafluoride (SF

)gas

flowed toward each other, exiting from small slits located at the

entrance of the test section, generating a stable surface at the test sec-

tion entrance. An oscillation imposed on the shock tube formed stand-

ing sinusoidal waves. This technique created a well-defined, slightly

diffused, 2D initial perturbation amplitude. By contrast, the use of

membranes gives sharp initial conditions, but the effects of their frag-

mentation on the development of the instability remain partially

understood.

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Previously, using well-defined, membraneless sinusoidal initial

conditions and shocks with Ma

1 and 1.2, Collins and Jacobs

reported excellent agreement between their experimental measure-

ments of the amplitude growth and the predictions of linear and non-

linear models. However, the late-time development of the instability

was limited by the arrival of the transmitted shock during the reshock

phase. Using a different configuration of the shock tube, the larger

Mach number Ma

3 achieved in the subsequent Jacobs

–

Krivets

experiment

allowed the investigation of

“

late-time

”

effects, as the

instability developed more rapidly and for a longer evolution time for

the larger Mach number. Scaling time by

;

(1)

where

(2)

is the initial Richtmyer

–

Meshkov perturbation velocity,

SF6

air

SF6

air

(3)

is the post-shock Atwood number,

is the perturbation wave-

number,

is the post-shock amplitude, and

is the velocity jump

at the interface following shock passage; the instability development

attained larger values of

as a result of the larger

. In addition to

the

9 cm experiment, they also considered a perturbation with

6 cm, resulting in a larger wavenumber

, allowing the investiga-

tion of even later-time effects. Only the

9 cm experiment is con-

sidered here.

Vorticity

–

streamfunction simulations using the same initial con-

ditions were performed for the current study, and the results are com-

pared to those from the WENO simulations. The main objective is to

investigate the dynamics of the single-mode instability in two and

three dimensions and to compare the amplitude growth to the experi-

mental results of Jacobs and Krivets,

as well as to the predictions of a

variety of nonlinear amplitude growth models. The extension of the

Jacobs

–

Krivets initial condition to three dimensions is regarded here

as a hypothetical case of a 3D sinusoidal initial condition. This study

also demonstrates the utility of the VS method, indicating similarities

and differences with the WENO method.

This paper is organized as follows: in Sec.

, overviews of the

incompressible VS and compressible WENO method are provided.

The initial conditions used in the simulations for both methods are

described in Sec.

III

. In Sec.

, visualizations of the 2D instability

evolution, including the dynamics of the velocity and vorticity, are

presented. The structures observed within the spike roll-ups are

compared between the two methods and to the experimental PLIF

images. A comparison between the bubble and spike amplitudes

and velocities obtained from the VS and WENO simulations and

from linear and nonlinear single-mode perturbation amplitude

model predictions is presented. In Sec.

, a comparison of 2D and

3D simulation results using both methods is presented. Bubble and

spike amplitudes and velocities from the simulations are compared

to the predictions of models applicable to 3D instabilities.

Visualizations of a number of fields obtained from the simulations

are compared and discussed. A summary of the findings and con-

clusions is given in Sec.

II. NUMERICAL METHODS

In the present study, simulation results from consistently initial-

ized incompressible VS and compressible WENO methods are com-

pared and contrasted, as well as compared to available experimental

data and to the predictions of many nonlinear amplitude growth mod-

els. Comparisons of the bubble and spike amplitudes between the sim-

ulations are also used to estimate how important compressibility

effects are for simulating the Ma

3Jacobs

–

Krivets experiment. In

the discussion of the numerical methods and simulation initial

conditions, it will be assumed that the shock propagates along the

direction (i.e., streamwise direction) and that the

-direction (and the

-direction in three dimensions) (i.e., spanwise direction) is periodic.

A. Incompressible vorticity

–

streamfunction (VS)

method

Vortex methods are useful because the vorticity deposition

–

evo-

lution formulation results in improved physical insight into the insta-

bility evolution

34,35

and are also numerically advantageous

when

compared with more expensive compressible simulations.

Compressible shock-capturing simulations typically require large com-

putational domains with large numbers of grid points, while vortex

methods require much smaller computational domains localized to

regions with nonzero vorticity. The vorticity and vortex dynamics par-

adigm

37,38

further recognize that the principal physical mechanism

driving the classical Richtmyer

–

Meshkov instability is the deposition

of localized vorticity on the interface during the shock refraction pro-

cess through baroclinic vorticity production. Following shock refrac-

tion, a transmitted shock enters the second fluid and a reflected wave

returns back into the first fluid.

A second mechanism of vorticity deposition is the interaction of

the interface with the pressure perturbations from stable perturbed

shock fronts,

39,40

including the reflected and transmitted shocks but

not a reflected rarefaction. Typically, the pressure perturbations from

the stable shock front slow down the growth rate, causing in some

cases full stabilization or

“

freeze out.

”

This second mechanism of

vorticity generation is not included in the present incompressible sim-

ulations. However, the results of the present study suggest that such a

contribution is not significant for the classical Richtmyer

–

Meshkov

instability at relatively small Mach numbers.

“

Richtmyer

–

Meshkov-

”

instabilities, including

“

anti-collisions,

”

where such a mechanism

becomes relevant, have been discussed in the context of shock

–

piston

flows relevant to the modeling of laser experiment targets.

Local vor-

ticity deposition from curved shocks can also significantly affect the

late-time growth of single-mode Richtmyer

–

Meshkov instability, par-

ticularly in the case of an initial light-to-heavy gas transition.

The use of vortex methods also provides insight into later stages

of the instability development not readily accessible using analytical

treatments. For example, current analytical methods are limited to

weakly nonlinear analysis

43,44

or Layzer-type expansions

–

extend-

ing earlier work

pertaining to the Rayleigh

–

Taylor instability. Such

weakly nonlinear treatments are valid up to a dimensionless time

4[seeEq.

(1)

]. By contrast, vortex methods offer an alternative

description of the interface dynamics from the linear to the weakly-

nonlinear and to the fully-nonlinear stages. In the fully-nonlinear

stage, the spike rolls up to a spiral due to the nonuniform vorticity dis-

tribution on the interface.

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Vortex methods are fundamentally based on a discretization of

the vorticity equation on a uniform grid. The thin vortex sheet of a

purely Lagrangian and of a hybrid Lagrangian

–

Eulerian method is

replaced by a thickened vortex sheet. For flows with variable density,

the vorticity equation is augmented by the density equation to com-

pute the baroclinic vorticity production term and describe the instabil-

ity evolution. The Eulerian method used here was developed in two

dimensions and extended to three dimensions.

The third-order

semi-implicit Adams

–

Bashforth backward differentiation method

(AB/BDI3) applied here includes mass diffusivity and viscosity in the

density and the vorticity equations (which also regularizes the numeri-

cal algorithm). The solution of a pressure Poisson equation using

Neumannboundaryconditionsisrequired.

The incompressible variable-density vorticity equation,

$

$

$

$

;

(4)

is obtained by taking the curl of the incompressible momentum

equation

$

$

;

(5)

where

is the (constant) kinematic viscosity (

is the dynamic

viscosity) and

is the Laplacian.

Additional equations for the velocity

¼ð

;

, pressure

and density

are needed to supplement Eq.

(4)

.Notethatinacom-

pressible flow, Eq.

(4)

would include the vorticity compression term

$

on the right side.

The density is obtained by solving the continuity equation

$

ðÞ

(6)

Diffusion is modeled as Fickian so that the mass diffusion flux of the

heavy gas is

,where

is the (constant) mass diffusivity,

and

is the heavy gas mass fraction. When combined with the heavy

mass fraction equation

$

$

;

(7)

this gives the density diffusion equation

54,55

consistent with binary

fluid mixing

$

$

$

;

(8)

where the kinematic viscosity and mass diffusivity are chosen here

so that the Schmidt number is Sc

1. In this case,

$

$

is zero only if

0orif

;

Þ¼

constant

However, for relatively small Mach numbers, the velocity divergence is

small after the passage of the shock through the perturbed interface, justify-

ing the incompressible approximation

$

0usedfortheVSmethod.

The velocity field is obtained from the vorticity

–

streamfunction

Poisson equation with

$

0 as follows:

;

(9)

$

;

(10)

where

¼ð

;

is the streamfunction vector (also called the

vector potential). Taking the divergence of the momentum equation

(5)

gives the pressure Poisson equation

$

$

ðÞ

$

$

(11)

for incompressible flow. The appearance of the mean pressure gradient

on the right side of this equation necessitates the following two-step

algorithm. First, the equation with only the first term on the right side

is solved at the time level

. Then, the equation including both terms is

solved using

substituted into the second term on the right side to

obtain an updated value of the pressure. Note that the flow field is

locally Rayleigh

–

Taylor unstable where

$

$

To determine the boundary conditions for Eqs.

(9)

and

(11)

,con-

sider a periodic domain in the

-and

-directions with rigid walls at

the left and right (

-direction)

;

. The boundary

value problem for the streamfunction is given by Eq.

(9)

with

;

Þ¼

;

Þ¼

;

(12)

which represents three separate boundary value problems for each

component

. The boundary value problem for the pressure is given

by Eq.

(11)

with

;

Þ¼

;

Þ¼

;

Þ¼

;

Þ¼

(13)

Equations

(4)

and

(8)

are supplemented by the boundary value prob-

lems for the streamfunction and pressure [Eqs.

(12)

and

(13)

]andfor

the velocity from the streamfunction [Eq.

(10)

] constitute a complete

VS method in two or three dimensions.

In the vorticity and density equations

(4)

and

(8)

, the spatial

operator is written as a sum of a linear diffusion and a nonlinear trans-

port term as follows:

;

Þþ

;

(14)

;

Þ¼

$

$

$

$

;

(15)

;

Þ¼

;

(16)

;

Þþ

;

(17)

;

Þ¼

$

$

$

;

(18)

;

Þ¼

(19)

The baroclinic vorticity production vector is

$

$

,and

the vorticity stretching vector is

$

(zero in two dimensions).

The governing equations above are discretized using a semi-

implicit scheme in which the linear part

is treated implicitly

and the nonlinear part

is treated explicitly. This overcomes the

time step limitations of a purely-explicit formulation and the need

for a nonlinear solver for a fully-implicit formulation. The third-

order Adams

–

Bashforth backward differentiation (AB/BDI3)

scheme is adopted for the time-evolution, which uses multiple

time-levels for both the temporal and spatial operators.

56,57

Writing

fg¼

;

, the resulting spatiotemporal dis-

cretization (

is the time level) is

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ðf

gÞþ

ðf

gÞþ

ðf

gÞ

(20)

in three dimensions (with the 2D case given by

0and

0). This equation is solved by following the same procedure as in

the derivation of the modified nine-point method for the Poisson

equation.

This results in an overall method that is third-order accu-

rate in time and fourth-order accurate in space (see Ref.

for addi-

tional details).

B. Compressible weighted essentially nonoscillatory

(WENO) method

The numerical simulations

of the reshocked Richtmyer

–

Meshkov instability modeled after the Jacobs

–

Krivets experiment were

performed using the characteristic projection-based, finite-difference

WENO shock-capturing method using ninth-order flux reconstruc-

tion.

59,60

That study

compared 2D and 3D simulation results, com-

pared the density fields and bubble and spike (and total) amplitude

from the simulations with experimental data

and with the predic-

tions of linear and nonlinear amplitude growth models, and consid-

ered the physics and modeling of reshock (which was not present in

the original experiment). Note that the long streamwise computational

domains (and large number of associated grid points) for these

WENO simulations were used to allow reshocked mixing layer simula-

tions, i.e., the domain length along the shock propagation direction

matched the experimental test section length.

The nondissipative, nondiffusive compressible fluid dynamics

(i.e., Euler) equations (

,and

are the

density, velocity, total energy, internal energy, and pressure, respec-

tively) are augmented by an equation for mass fraction

conservation

of the heavier gas (with 1

the mass fraction of the lighter gas) as

follows:

ðÞ

;

ðÞ

;

ðÞ

;

ðÞ

(21)

with summation over repeated indices. WENO shock-capturing meth-

ods solving the nondissipative, nondiffusive Euler equations include

intrinsic dissipation and diffusion, which stabilize the numerical solu-

tion at small scales. See the Appendix of Ref.

for additional informa-

tion on the WENO method used.

As the WENO code is a single-

code, only a single value of the

adiabatic exponent can be specified. A multiple-species

formulation

introduces nonphysical pressure oscillations near the material interfa-

ces in conservative shock-capturing schemes for the multicomponent

fluid equations.

–

A five-equation, quasi-conservative model for

diffuse-interfaces can be used in compressible flows to address this

issue.

A methodology in which modified WENO weights were used

to solve the mass fraction equation in conservative form to prevent

temperature and species conservation errors was also developed.

Pressure errors are prevented by solving an additional transport equa-

tion for a particular function of

III. INITIAL CONDITIONS

The initial and boundary conditions used for the present simula-

tions are discussed here. The gases used in the experiment have differ-

ent adiabatic exponents. In a previous investigation,

upstream

conditions were matched so that the adiabatic exponent corresponding

to the air(acetone) mixture was used. However, mix initial conditions

were adopted for the simulations in Ref.

instead, where the adiabatic

exponent corresponding to a 50% mixture of air(acetone) and SF

volume was used. These initial conditions were used pre-shock for the

WENO method and post-shock for the VS method. The initial condi-

tions used for the WENO simulations are described in detail in Sec. 2.2

of Ref.

, and the mixture properties are derived in Sec. 2.3 of Ref.

(summarized in

Table I

). The single-mode sinusoidal perturbation in

two dimensions is

Þ¼

cos

ðÞ

;

(22)

with

the pre-shock perturbation amplitude,

the pertur-

bation wavenumber,

the perturbation wavelength, and

the pertur-

bation phase angle.

Comparing the mix and upstream initial conditions shows that

neither precisely captures the post-shock Atwood number At

(

5% difference). In addition,

is also different from the two-gas

initial conditions (

2% and

3%, respectively, compared to the exper-

iment). For mix initial conditions, the larger interface velocity and the

smaller post-shock Atwood numbers compensate, giving a perturba-

tion velocity

similar to that in the experiment. The reflected and

transmitted shock velocities are very close to the shock velocities in the

two-gas initial conditions.

Table II

includes the simulation parameters,

including the lengths of the physical domains

and

, the number of

grid points, and the corresponding grid sizes

and

based on a

resolution of 256 and 512 points per initial perturbation wavelength

for the VS and WENO simulations, respectively. The VS simulation

uses 1024 grid points along the shock propagation direction, compared

to 6726 grid points used in the WENO simulation.

A vortex method simulation of the Richtmyer

–

Meshkov instabil-

ity begins immediately after the passage of the shock through the inter-

face. As a result, a quantification of the circulation deposited on the

interface by the shock through the baroclinic production term is

needed to specify the initial circulation of the vortex markers. In Sec.

2.4 of Ref.

, it was shown that the circulation deposition on the inter-

face by the shock can be adequately modeled using the Samtaney

–

Zabusky model

or by linear instability theory.

Linear instability

theory provides results that are within 0.6% of the Samtaney

–

Zabusky

model prediction and within 4% of the WENO measured values

and

is used here because it has a more direct physical interpretation. Linear

instability theory was also used in the Lagrangian point-vortex simula-

tions of the single-mode Richtmyer

–

Meshkov instability.

The

vortex-dipole and the vortex sheet strength

;

Þ¼

cos

;

Þ¼

;

(23)

respectively, are specified, where

is a parameter associated with the

interface, and

is the initial instability velocity. The interface is

Physics of Fluids

ARTICLE

pubs.aip.org/aip/pof

Phys. Fluids

, 024115 (2024); doi: 10.1063/5.0179157

, 024115-5

Published under an exclusive license by AIP Publishing

28 August 2024 21:59:52