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J. Fluid Mech. (2014), vol. 755, pp. 429–462.
c
Cambridge University Press 2014
doi:10.1017/jfm.2014.436
429
On the Richtmyer–Meshkov instability evolving
from a deterministic mult imode planar interface
V. K. Tritschler
1,2,
†, B. J. Olson
3
, S. K. Lele
2
, S. Hickel
1
, X. Y. Hu
1
and
N. A. Adams
1
1
Institute of Aerodynamics and Fluid Mechanics, Technische Universität München,
85747 Garching, Germany
2
Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305, USA
3
Lawrence Livermore National Laboratory, Livermore, CA 94550, USA
(Received 6 February 2014; revised 21 July 2014; accepted 23 July 2014)
We investigate the shock-induced turbulent mixing between a light and a heavy
gas, where a Richtmyer–Meshkov instability (RMI) is initiated by a shock wave
with Mach number Ma =1.5. The prescribed initial conditions define a deterministic
multimode interface perturbation between the gases, which can be imposed exactly for
different simulation codes and resolutions to allow for quantitative comparison. Well-
resolved large-eddy simulations are performed using two different and independently
developed numerical methods with the objective of assessing turbulence structures,
prediction uncertainties and convergence behaviour. The two numerical methods
differ fundamentally with respect to the employed subgrid-scale regularisation,
each representing state-of-the-art approaches to RMI. Unlike previous studies, the
focus of the present investigation is to quantify the uncertainties introduced by the
numerical method, as there is strong evidence that subgrid-scale regularisation and
truncation errors may have a significant effect on the linear and nonlinear stages
of the RMI evolution. Fourier diagnostics reveal that the larger energy-containing
scales converge rapidly with increasing mesh resolution and thus are in excellent
agreement for the two numerical methods. Spectra of gradient-dependent quantities,
such as enstrophy and scalar dissipation rate, show stronger dependences on the
small-scale flow field structures as a consequence of truncation error effects, which
for one numerical method are dominantly dissipative and for the other dominantly
dispersive. Additionally, the study reveals details of various stages of RMI, as the
flow transitions from large-scale nonlinear entrainment to fully developed turbulent
mixing. The growth rates of the mixing zone widths as obtained by the two numerical
methods are ∼t
7/12
before re-shock and ∼(t − t
0
)
2/7
long after re-shock. The decay
rate of turbulence kinetic energy is consistently ∼(t −t
0
)
−10/7
at late times, where the
molecular mixing fraction approaches an asymptotic limit Θ ≈ 0.85. The anisotropy
measure hai
xyz
approaches an asymptotic limit of ≈0.04, implying that no full recovery
of isotropy within the mixing zone is obtained, even after re-shock. Spectra of density,
turbulence kinetic energy, scalar dissipation rate and enstrophy are presented and show
excellent agreement for the resolved scales. The probability density function of the
heavy-gas mass fraction and vorticity reveal that the light–heavy gas composition
within the mixing zone is accurately predicted, whereas it is more difficult to capture
the long-term behaviour of the vorticity.
Key words: shock waves, turbulent mixing
† Email address for correspondence: volker.tritschler@aer.mw.tum.de
430 V. K. Tritschler and others
1. Introduction
The Richtmyer–Meshkov instability (Richtmyer 1960; Meshkov 1969) is a
hydrodynamic instability that occurs at the interface separating two fluids of different
densities. It shows similarities with the Rayleigh–Taylor instability (Rayleigh 1883;
Taylor 1950), where initial perturbations at the interface grow and eventually evolve
into a turbulent flow field through the transfer of potential to kinetic energy. In the
limit of an impulsive acceleration of the interface, e.g. by a shock wave, the instability
is referred to as a Richtmyer–Meshkov instability (RMI). In RMI, baroclinic vorticity
production at the interface is caused by the misalignment of the pressure gradient
(∇p) associated with the shock wave and the density gradient (∇ρ) of the material
interface. The baroclinic vorticity production term (∇ρ ×∇p)/ρ
2
is the initial driving
force of RMI. See Zabusky (1999) and Brouillette (2002) for comprehensive reviews.
RMI occurs on enormous scales in astrophysics (Arnett et al. 1989; Arnett
2000; Almgren et al. 2006), on intermediate scales in combustion (Yang, Kubota
& Zukoski 1993; Khokhlov, Oran & Thomas 1999) and on very small scales in
inertial confinement fusion (Lindl, McCrory & Campbell 1992; Taccetti et al. 2005;
Aglitskiy et al. 2010). Owing to the fast time scales associated with RMI, laboratory
experimental measurements have difficulties in characterising quantitatively initial
perturbations of the material interface and capturing the evolution of the mixing
zone. General insight into the flow physics of RMI relies to a considerable extent
on numerical investigations, where large-eddy simulations (LES) have become an
accepted tool during the past decade.
Hill, Pantano & Pullin (2006) performed a rigorous numerical investigation of
RMI with re-shock. The authors used an improved version of the tuned centred
difference–weighted essentially non-oscillatory (TCD-WENO) hybrid method of Hill
& Pullin (2004). The method employs a switch to blend explicitly between a TCD
stencil in smooth flow regions and a WENO shock-capturing stencil at discontinuities.
The TCD-WENO hybrid method is used together with the stretched-vortex model
(Pullin 2000; Kosovi
´
c, Pullin & Samtaney 2002) for explicitly modelling the
subgrid interaction terms. This approach was also used by Lombardini et al.
(2011) to study systematically the impact of the Atwood number for a canonical
three-dimensional numerical set-up, and for LES of single-shock (i.e. without
re-shock) RMI (Lombardini, Pullin & Meiron 2012).
Thornber et al. (2010) studied the influence of different three-dimensional broad-
and narrow-band multimode initial conditions on the growth rate of a turbulent
multicomponent mixing zone developing from RMI. In a later study (Thornber et al.
2011), the same authors presented a numerical study of a re-shocked turbulent mixing
zone, and extended the theory of Mikaelian and Youngs to predict the behaviour of a
multicomponent mixing zone before and after re-shock (cf. Mikaelian 1989; Thornber
et al. 2010). They used an implicit LES (Drikakis 2003; Thornber et al. 2008;
Drikakis et al. 2009) approach based on a finite-volume Godunov-type method to
solve the Euler equations with the same specific heat ratio for both fluids.
In a recent investigation, Weber, Cook & Bonazza (2013) derived a growth-rate
model for the single-shock RMI based on the net mass flux through the centre plane
of the mixing zone. Here, the compressible Navier–Stokes equations were solved
by a tenth-order compact difference scheme for spatial differentiation. Artificial
grid-dependent fluid properties, proposed by Cook (2007), were used for shock and
material-interface capturing as well as for subgrid-scale modelling.
Grid-resolution-independent statistical quantities of the single-shock RMI were
presented by Tritschler et al. (2013a). The kinetic energy spectra exhibit a Kolmogorov
On the Richtmyer–Meshkov instability 431
inertial range with k
−5/3
scaling. The spatial flux discretisation was performed in
characteristic space by an adaptive central-upwind sixth-order-accurate WENO scheme
(Hu, Wang & Adams 2010) in the low-dissipation version of Hu & Adams (2011).
LES relies on scale separation, where the energy-containing large scales are resolved
and the effect of non-resolved scales is modelled either explicitly or implicitly.
However, turbulent mixing initiated by RMI for typical LES mainly occurs at the
marginally resolved or non-resolved scales. The interaction of non-resolved small
scales with the resolved scales as well as the effect of the interaction of non-resolved
scales with themselves is modelled by the employed subgrid-scale model. Moreover,
discontinuities such as shock waves and material interfaces need to be captured by
the numerical scheme. Owing to the broad range of scales, coarse-grained numerical
simulations of RMI strongly rely on the resolution capabilities for the different
types of subgrid scales (turbulent small scales, shocks, interfaces) of the underlying
numerical scheme.
So far, research has mainly focused on the identification and quantification of
parameters that affect the evolution of Richtmyer–Meshkov unstable flows. The
influence of the Atwood number (Lombardini et al. 2011), the Mach number
(Lombardini et al. 2012) as well as the specific initial interface perturbations
(Schilling & Latini 2010; Thornber et al. 2010; Grinstein, Gowardhan & Wachtor
2011) on the temporal evolution of the instability have been investigated. Results
from numerical simulations have been compared to experiments (Hill et al. 2006;
Schilling & Latini 2010; Tritschler et al. 2013b) and theoretical models have
been derived (Thornber et al. 2011; Weber et al. 2013). These investigations have
assumed, based on standard arguments such as empirical resolution criteria, that the
marginally and non-resolved scales have a negligible effect on the resolved scales, and
therefore on the evolution of the instability. Uncertainties introduced by the numerical
method, i.e. the subgrid-scale regularisation and truncation errors, have not yet been
investigated systematically. There is, however, strong evidence that numerical model
uncertainty can significantly affect the linear and nonlinear stages of evolution, and in
particular the mixing measures. In fact, it is unclear how subgrid-scale regularisation
and dispersive or dissipative truncation errors can affect the resolved scales and
turbulent mixing measures.
In the present investigation, two independently developed and essentially different
numerical methods are employed to study the prediction uncertainties of RMI
simulations. The first method has a dominantly dissipative truncation error at the
non-resolved scales, whereas the second one exhibits a more dispersive behaviour.
At the marginally resolved scales, the numerical truncation error is not small and
the particular character of the truncation error is essential for the implicit modelling
capabilities of the method, and thus also affects the resolved scale solution. For the
purpose of investigating this effect, integral and spectral mixing metrics as well as
probability density functions are analysed on four computational grids with resolutions
ranging from 1562 to 195 µm. The simulations employing two different numerical
methods on a very fine grid resolution of 195 µm provide a data set with high
confidence in the results.
We emphasise that the purpose of this study is (i) to present RMI results with
a clear identification of the resolved scale range by systematic grid refinement, and
(ii) to assess the physical effects of numerical subgrid-scale regularisations on the
marginally resolved and on the non-resolved scale range. We do not intend to propose
or improve a certain subgrid-scale model or regularisation scheme.
The paper is structured as follows. The governing equations along with the
employed numerical models are described in § 2. Details about the computational
432 V. K. Tritschler and others
domain and the exact generic initial conditions are given in § 3. Results are presented
in § 4, and the key findings of the present study are discussed in § 5.
2. Numerical model
2.1. Governing equations
We solve the three-dimensional multicomponent Navier–Stokes equations:
∂ρ
∂t
+∇ · (ρu) =0, (2.1a)
∂(ρu)
∂t
+∇ · (ρuu +pδ − τ ) =0, (2.1b)
∂E
∂t
+∇ · [(E +p)u]−∇ · (τ · u −q
c
−q
d
) =0, (2.1c)
∂ρY
i
∂t
+∇ · (ρuY
i
) +∇ · J
i
=0. (2.1d)
In (2.1a), u is the velocity vector, p is the pressure, E is the total energy, ρ is the
mixture density, Y
i
is the mass fraction and J
i
is the diffusive mass flux of species
i =1, 2, . . . , K, with K the total number of species. The identity matrix is δ.
The viscous stress tensor τ for a Newtonian fluid is
τ =2 ¯µS +(β −
2
3
¯µ)δ(∇ · u), (2.2)
with the mixture viscosity ¯µ and the strain-rate tensor S. According to Fourier’s law,
we define the heat flux as
q
c
=−¯κ∇T (2.3)
and the inter-species diffusional heat flux (Cook 2009) as
q
d
=
K
X
i=1
h
i
J
i
, (2.4)
with
J
i
≈−ρ
D
i
∇Y
i
−Y
i
K
X
j=1
D
j
∇Y
j
!
. (2.5)
Here D
i
indicates the effective binary diffusion coefficient of species i, and h
i
is the
individual species enthalpy. The equations are closed with the equation of state for an
ideal gas,
p(ρe, Y
1
, Y
2
, . . . , Y
K
) =( ¯γ −1)ρe, (2.6)
where ¯γ is the ratio of specific heats of the mixture and e is the internal energy,
ρe =E −
1
2
ρu
2
. (2.7)
The multicomponent as well as the molecular mixing rules for ¯γ , ¯µ, D
i
and ¯κ are
given in appendices A and B.
2.2. Numerical methods
2.2.1. The Miranda simulation code
The Miranda simulation code has been used extensively for simulating turbulent
flows with high Reynolds numbers and multi-species mixing (Cook, Cabot & Miller
2004; Cabot & Cook 2006; Olson & Cook 2007; Olson et al. 2011; Weber et al.
2013). Miranda employs a tenth-order compact difference scheme (Lele 1992) for
On the Richtmyer–Meshkov instability 433
spatial differentiation and a five-stage fourth-order Runge–Kutta scheme (Kennedy,
Carpenter & Lewis 2000) for temporal integration of the compressible multicomponent
Navier–Stokes equations. Full details of the numerical method are given by Cook
(2007), which includes an eighth-order compact filter that is applied to the conserved
variables at each time step and smoothly removes the top 10 % of wavenumbers to
ensure numerical stability. For numerical regularisation of non-resolved steep flow
gradients, artificial fluid properties are used to damp locally structures that exist on
the length scales of the computational mesh. In this approach, artificial diffusion terms
are added to the physical ones that appear in equations (2.2), (2.3) and (2.5) as
µ = µ
f
+µ
∗
, (2.8)
β = β
f
+β
∗
, (2.9)
κ = κ
f
+κ
∗
, (2.10)
D
i
= D
f ,i
+D
∗
i
. (2.11)
This LES method employing artificial fluid properties was originally proposed by
Cook (2007), but has been altered by replacing the S (magnitude of the strain rate
tensor), with ∇ · u in the equation for β
∗
. Mani, Larsson & Moin (2009) showed that
this modification substantially decreases the dissipation error of the method. Here we
give the explicit formulation of the artificial terms on a Cartesian grid,
µ
∗
=C
µ
ρ|∇
r
S|∆
(r+2)
, (2.12)
β
∗
=C
β
ρ|∇
r
(
∇ · u
)
|∆
(r+2)
, (2.13)
κ
∗
=C
κ
ρc
s
T
|∇
r
e|∆
(r+1)
, (2.14)
D
∗
i
=C
D
|∇
r
Y
i
|
∆
(r+2)
1t
+C
Y
(|Y
i
|+|1 −Y
i
|−1)
∆
2
21t
, (2.15)
where S = (S : S)
1/2
is the magnitude of the strain-rate tensor, ∆ = (1x1y1z)
1/3
is the local grid spacing, c
s
is the sound speed and 1t is the time step size. The
polyharmonic operator, ∇
r
, denotes a series of Laplacians, e.g. r = 4 corresponds to
the biharmonic operator, ∇
4
= ∇
2
∇
2
. The overbar ( f ) denotes a truncated-Gaussian
filter applied along each grid direction as in Cook (2007) to smooth out sharp
cusps introduced by the absolute value operator. In LES of RMI, β
∗
acts as the
shock-capturing scheme. The µ
∗
is primarily used as a numerical stabilisation
mechanism rather than as a subgrid-scale model. The artificial shear viscosity is
found not to be needed to maintain numerical stability in the current calculations
and its inclusion has a small impact on the solution. The dissipation of the vortical
motion primarily depends on the eighth-order compact filter.
2.2.2. The INCA simulation code
The INCA simulation code is a multi-physics simulation method for single- and
multicomponent turbulent flows. With respect to the objective in this paper, it has
been tested and validated for shock-induced turbulent multi-species mixing problems
at finite Reynolds numbers (Tritschler et al. 2013a,b, 2014).
For all the simulations presented in this paper, we use a discretisation scheme that
employs for the hyperbolic part in (2.1a) a flux projection on local characteristics. The
Roe-averaged matrix required for the projection is calculated for the full multi-species
434 V. K. Tritschler and others
system (Roe 1981; Larouturou & Fezoui 1989; Fedkiw, Merriman & Osher 1997). The
numerical fluxes at the cell faces are reconstructed from cell averages by the adaptive
central-upwind sixth-order weighted essentially non-oscillatory (WENO-CU6) scheme
(Hu et al. 2010) in its scale separation formulation by Hu & Adams (2011).
The fundamental idea of the WENO-CU6 scheme is to use a non-dissipative sixth-
order central stencil in smooth flow regions and a nonlinear convex combination of
third-order stencils in regions with steep gradients. The reconstructed numerical flux
at the cell boundaries is computed from
ˆ
f
i+1/2
=
3
X
k=0
ω
k
ˆ
f
k,i+1/2
, (2.16)
where ω
k
is the weight assigned to stencil k with the second-degree reconstruction
polynomial approximation for
ˆ
f
k,i+1/2
. In the WENO-CU6 framework the weights ω
k
are given by
ω
k
=
α
k
3
X
k=0
α
k
, α
k
=d
k
C +
τ
6
β
k
+
q
, (2.17)
with being a small positive number = 10
−40
. The optimal weights d
k
are
defined such that the method recovers the sixth-order central scheme in smooth
flow regions. The constant parameters in (2.17) are set to C = 1000 and q = 4 (see
Hu & Adams 2011), τ
6
is a reference smoothness indicator that is calculated from a
linear combination of the other smoothness measures β
k
with
τ
6
=β
6
−
1
6
(β
0
+β
2
+4β
1
) (2.18)
and
β
k
=
2
X
j=1
1x
2j−1
Z
x
i+1/2
x−1/2
d
j
dx
j
ˆ
f
k
(x)
2
dx, (2.19)
and β
6
is also calculated from (2.19) but with the fifth-degree reconstruction
polynomial approximation of the flux, which gives the six-point stencil for the
sixth-order interpolation.
After reconstruction of the numerical fluxes at the cell boundaries, the fluxes are
projected back onto the physical field. A local switch to a Lax–Friedrichs flux is used
as entropy fix (see e.g. Toro 1999). A positivity-preserving flux limiter (Hu, Adams
& Shu 2013) is employed in regions with low pressure or density, maintaining the
overall accuracy of the sixth-order WENO scheme. It has been verified that the flux
limiter has negligible effect on the results, and avoids excessively small time step sizes.
Temporal integration is performed by a third-order total variation-diminishing Runge–
Kutta scheme (Gottlieb & Shu 1998).
3. Numerical set-up
3.1. Computational domain
We consider a shock tube with constant square cross-section. The fine-grid domain
extends in the y and z directions symmetrically from −L
yz
/2 to L
yz
/2 and in the x
direction from −L
x
/4 to L
x
. An inflow boundary condition is imposed far away from
On the Richtmyer–Meshkov instability 435
L
x
L
yz
L
yz
Shocked air
Unshocked air
Direction of shock
y
x
z
FIGURE 1. (Colour online) Schematic of the square shock tube and dimensions of the
computational domain for the simulations.
the fine-grid domain in order to avoid shock reflections. To reduce computational costs,
a hyperbolic mesh stretching is applied between the inflow boundary and −L
x
/4; L
x
is set to 0.4 m, and L
yz
= L
x
/4. At the boundaries normal to the y and z directions,
periodic boundary conditions are imposed and an adiabatic wall boundary at the end
of the shock tube at x =L
x
is used. A schematic of the computational domain is shown
in figure 1.
The fine-grid domain is discretised by four different homogeneous Cartesian grids
with 64, 128, 256 and 512 cells in the y and z directions and 320, 640, 1280 and 2560
cells in the x direction, resulting in cubic cells of size 1562 µm . ∆
xyz
. 195 µm.
The total number of cells in the fine-grid domain amounts to ≈ 1.3 × 10
6
for the
coarsest resolution and to ≈670 ×10
6
for the finest resolution.
3.2. Initial conditions
We consider air as a mixture of nitrogen (N
2
) and oxygen (O
2
) with (in terms of
volume fraction) X
N
2
= 0.79 and X
O
2
= 0.21. The equivalent mass fractions on the
air side give Y
N
2
= 0.767 and Y
O
2
= 0.233, i.e. Y
air
= Y
N
2
+ Y
O
2
. The heavy gas is
modelled as a mixture of SF
6
and acetone (Ac) with mass fractions Y
SF
6
= 0.8 and
Y
Ac
= 0.2, i.e. Y
HG
= Y
SF
6
+ Y
Ac
. The material interface between light (air) and heavy
gas is accelerated by a shock wave with Mach number Ma =1.5 that is initialised at
x = −L
x
/8 propagating in the positive x direction. The pre-shock state is defined by
the stagnation condition p
0
=23 000 Pa and T
0
=298 K. The corresponding post-shock
436 V. K. Tritschler and others
Quantity Post-shock Pre-shock Pre-shock
light-gas side heavy-gas side
ρ (kg m
−3
) 0.498 69 0.267 84 1.040 57
U (m s
−1
) 240.795 0 0
p (Pa) 56 541.7 23 000 23 000
T (K) 393.424 298 298
D
N
2
(m
2
s
−1
) 5.919 ×10
−5
8.981 ×10
−5
—
D
O
2
(m
2
s
−1
) 5.919 ×10
−5
8.981 ×10
−5
—
D
SF
6
(m
2
s
−1
) — — 1.846 ×10
−5
D
Ac
(m
2
s
−1
) — — 1.846 ×10
−5
¯µ (Pa s) 2.234 ×10
−5
1.826 ×10
−5
1.328 ×10
−5
c
p
(J kg
−1
K
−1
) 1008.35 1008.35 815.89
TABLE 1. Initial values of the post-shock state and the pre-shock states of the light- and
heavy-gas sides.
thermodynamic state is obtained from the Rankine–Hugoniot conditions,
ρ
0
air
= ρ
air
(γ
air
+1)Ma
2
2 +(γ
air
−1)Ma
2
, (3.1a)
u
0
air
= Ma c
air
1 −
ρ
air
ρ
0
air
, (3.1b)
p
0
air
= p
0
1 +2
γ
air
γ
air
+1
(Ma
2
−1)
, (3.1c)
with c
air
=
√
γ
air
p
0
/ρ
air
. The initial data of the post-shock state of the light gas as well
as the pre-shock state of the light and heavy gases are given in table 1.
Tritschler et al. (2013a) introduced a generic initial perturbation of the material
interface that resembles a stochastic random perturbation but being, however,
deterministic and thus exactly reproducible for different simulation runs. This
multimode perturbation is given by the function
η(y, z) =a
1
sin(k
0
y) sin(k
0
z) +a
2
13
X
n=1
15
X
m=3
a
n,m
sin(k
n
y +φ
n
) sin(k
m
z +χ
m
) (3.2)
with the constant amplitudes a
1
=−0.0025 m and a
2
=0.000 25 m and wavenumbers
k
0
=10π/L
yz
, k
n
=2πn/L
yz
and k
m
=2πm/L
yz
. The amplitudes a
n,m
and the phase shifts
φ
n
and χ
m
are given by
a
n,m
= sin(nm)/2, (3.3a)
φ
n
= tan(n), (3.3b)
χ
m
= tan(m). (3.3c)
To facilitate a grid sensitivity study, we impose an initial length scale by prescribing
a finite initial interface thickness in the mass fraction field as
ψ(x, y, z) =
1
2
1 +tanh
x −η(y, z)
L
ρ
(3.4)
On the Richtmyer–Meshkov instability 437
10
–10
10
–5
10
0
10
1
10
2
FIGURE 2. (Colour online) Initial power spectra of density from Miranda (dark grey; blue
online) and INCA (light grey; red online). The different resolutions are represented as
dotted line (64), dashed line (128), solid line (256) and solid line with open squares for
Miranda and open diamonds for INCA (512).
with L
ρ
=0.01 m being the characteristic initial thickness. The individual species mass
fractions are set as
Y
SF
6
=0.8ψ, Y
Ac
=0.2ψ, (3.5a,b)
Y
N
2
=0.767(1 − ψ), Y
O
2
=0.233(1 − ψ). (3.6a,b)
The material interface is initialised at x − η(y, z) = 0 m. Combined with the
multicomponent and molecular mixing rules given in appendices A and B, the
flow field is fully defined at t =0.
Figure 2 shows the initial condition in terms of the power spectrum of density
for Miranda and INCA at all grid resolutions. The initial perturbation given in (3.2)
and shown in figure 2 has been designed with the objective to obtain a reproducible
and representative data set. Nevertheless, we cannot exclude the possibility that some
of the observations presented in this paper do not apply to very different initial
perturbations.
4. Results
To explore the effect of the finite truncation error arising from grid resolution
and numerical method, four meshes were used to compute the temporal evolution
of RMI with both Miranda and INCA. The simulation reaches t = 6.0 ms, which
is well beyond the occurrence of re-shock at t ≈ 2 ms. At this stage, the effects of
reflected shock waves and expansion waves on the shock location have become small,
as the shock wave is attenuated with each subsequent reflection. The space–time (x–t)
diagram shown in figure 3 depicts the propagation of the shock wave and interface
during the simulation.
The initial conditions described in the previous section are entirely deterministic and,
owing to their band-limited representations, are identically imposed at the different
grid resolutions and for the two numerical methods. Therefore, the obtained results
438 V. K. Tritschler and others
0
1
2
3
4
5
6
7
–5 5 101520253035
Adiabatic wall
Interface
Shock wave
First shock
Re-shock
x (cm)
t (ms)
FIGURE 3. (Colour online) Space–time (x–t) diagram depicting the propagation of the
shock wave and interface during the simulation. The effect of the shock wave on the
interface location is attenuated with each subsequent reflection.
exhibit uncertainties due only to the numerical method and to grid resolution, but
exclude initial-data uncertainties.
For illustration we show the three-dimensional contour plots of species mass fraction
of the heavy gas Y
HG
obtained with Miranda and INCA, respectively, in figure 4.
Similarities at the large scales are clearly visible after re-shock, but also differences
exist at the fine scales, more clearly visible from the inset.
4.1. Integral quantities
Integral measures of the mixing zone are presented here for both numerical models
and all resolutions. Often, these time-dependent integral measures are the only metrics
available for comparison with experiment and are therefore of primary importance for
validation.
Figure 5 shows the transition process predicted by the reference grid with a
resolution of 512 cells in the transverse directions. The numerical challenge, prior
to re-shock, is to predict the large-scale nonlinear entrainment and the associated
interface steepening. The interface eventually becomes under-resolved when its
thickness reaches the resolution limit of the numerical scheme and further steepening
is prevented by numerical diffusion. The equilibrium between interface steepening
and numerical diffusion occurs later in time as the grid is refined. The accurate
prediction of the interface steepening phenomenon is one of the main challenges in
modelling pre-transitional RMI where large-scale flow structures are still regular. This
is because the numerical model largely determines the time when mixing transition
occurs. In nature, mixing transition is due to the presence of small-scale perturbations,
whereas in numerical simulation, the transition is triggered by backscatter from the
under-resolved scales as predicted by the particular numerical model. Hence, details
of mixing transition of the material interface evolve differently for the two codes.
On the Richtmyer–Meshkov instability 439
(a)(b)
FIGURE 4. (Colour online) Three-dimensional contour plots of species mass fraction of
the heavy gas from (a) Miranda and (b) INCA data. Data are from the finest grid at
t =2.5 ms that show contours of the heavy-gas mass fraction Y
HG
from 0.1 (at the bottom;
blue online) to 0.9 (at the top; red online). Note that, although some large-scale features
remain consistent between codes, small and intermediate scales are quite different at this
stage.
0.5 ms 2 ms 2.5 ms 6 ms
Re-shock
(a)
(b)
FIGURE 5. (Colour online) Colour-coded plots of species mass fraction of SF
6
gas from
(a) Miranda and (b) INCA at various times where data are taken from the finest grid. The
contours range from 0.05 (white) to 0.75 (dark grey; blue online).
440 V. K. Tritschler and others
0 1 2 3 4 5 6
0.02
0.04
0.06
0.08
0.10
0.001 0.002
0.004 0.006
10
–2
10
–1
t (s) t (s)
(a)(b)
FIGURE 6. (Colour online) Time evolution of the mixing zone width from Miranda
(dark grey; blue online) and INCA (light grey; red online). The different resolutions are
represented as dotted line (64), dashed line (128), solid line (256) and solid line with open
squares for Miranda and open diamonds for INCA (512).
Nevertheless, similarities before re-shock are striking and large-scale similarities
in the resolved wavenumber range even persist throughout the entire simulation
time. Following re-shock, the large interfacial scales break down into smaller scales
and develop a turbulent mixing zone as can be seen in figures 4 and 5. By visual
inspection of figure 5, one finds that the post-re-shock turbulent structures are very
similar, whereas the long-term evolution of the small scales appears to be different
between the codes. Differences in the observed flow field at t = 6 ms may indicate
slightly different effective Reynolds numbers for the two numerical methods and
therefore they also exhibit different decay rates of enstrophy (Dimotakis 2000;
Lombardini et al. 2012), as can be seen in figure 8 after re-shock.
The mixing width δ
x
is a length scale that approximates the large-scale temporal
evolution of the turbulent mixing zone. It is defined as an integral measure by
δ
x
(t) =
Z
∞
−∞
4φ(1 −φ)dx, with φ(x, t) =hY
SF
6
+Y
Ac
i
yz
, (4.1)
where h·i
yz
denotes the ensemble average in the cross-stream yz plane. For a quantity
ϕ it is defined by
hϕi
yz
(x, t) =
1
A
ZZ
ϕ(x, y, z, t) dy dz, with A =
ZZ
dy dz. (4.2)
The mixing width plotted in figure 6(a) shows that data from both numerical methods
converge to a single solution throughout the entire simulation time. Furthermore, it is
observed that even with very-high-order models a minimum resolution of ∼400 µm
appears to be necessary for an accurate prediction of the mixing zone width. As will
be shown later, coarser grids tend to overpredict not only the growth of the mixing
zone but also molecular mixing.
Figure 6(b) shows the mixing zone width time evolution on a log–log scale. The
(bubble) growth-rate model of Zhou (2001) predicts accurately the pre-re-shock
mixing zone growth rate that is consistently recovered by both numerical methods as
∼t
7/12
. However, this is, according to Zhou (2001), the growth rate that is associated
with turbulence of Batchelor type (Batchelor & Proudman 1956) with E (k) ∼ k
4
On the Richtmyer–Meshkov instability 441
as k →0. The kinetic energy spectra in the present investigation are of Saffman type
(Saffman 1967a,b) with E(k) ∼k
2
as k →0 (Tritschler et al. 2013a), for which Zhou
(2001) predicts a growth that scales with ∼t
5/8
. The present growth rates are also in
good agreement with the experimental and numerical results of Dimonte, Frerking &
Schneider (1995) with ∼t
β
and β =0.6 ±0.1 and their model predictions ∼(t −t
i
)
1/2
,
where t
i
accounts for the time the shock needs to traverse the interface. As the mixing
zone has not yet reached self-similar evolution, the initial growth rate depends on the
specific initial conditions.
Llor (2006) found that the self-similar growth rate of the energy-containing
eddies, i.e. the integral length scale, for incompressible RMI at vanishing Atwood
number, should scale as δ
x
∼ t
1−n/2
with 2/7 6 1 − n/2 6 1/3, if the turbulence
kinetic energy (TKE) decays as ∼t
−n
. These growth rates slightly differ from the
growth-rate prediction for homogeneous isotropic turbulence, 1/3 6 1 − n/2 6 2/5,
by the same author. The predictions of Llor (2006), however, are at odds with
Kolmogorov’s classical decay law (Kolmogorov 1941) for TKE ∼t
−10/7
and more
recent investigations of decaying isotropic turbulence by Ishida, Davidson & Kaneda
(2006) and Wilczek, Daitche & Friedrich (2011), which found Kolmogorov’s decay
law to hold if the Loitsyansky integral is constant and if the Taylor-scale Reynolds
number exceeds Re
λ
> 100. Based on Rayleigh–Taylor experiments driven by either
sustained or impulsive acceleration at various Atwood numbers, Dimonte & Schneider
(2000) found scaling laws for the bubble and spike growth rate. For the present
density ratio, the exponents become 1 − n
B
/2 ≈ 0.25 ± 0.05 for the former and
0.256 1−n
S
/26 0.43 for the latter. The late-time mixing zone growth rate is therefore
expected to correlate with the spike growth rate. The late-time growth-rate prediction
of the present work is ∼(t − t
0
)
2/7
, i.e. 1 − n/2 = 2/7, once the turbulent mixing
zone is fully established. Time t
0
is a virtual time origin set to t
0
= 2 ms. This is
consistent with the mixing zone width growth-rate predictions of Llor (2006) and the
late-time growth-rate predictions of Dimonte & Schneider (2000), but underestimates
the predictions of Zhou (2001), with a scaling of t
1−n/2
with 0.35 6 1 − n/2 6 0.45
long after re-shock, once the nonlinear time scale has become the dominant time
scale. In the numerical investigation of Lombardini et al. (2012), the authors found
the mixing zone width to grow as 0.2 6 1 −n/2 6 0.33. Before re-shock the infrared
part of the kinetic energy spectrum (see figure 14) exhibits a k
2
range, for which a
post-re-shock growth rate of ∼t
2/7
is predicted by the model of Youngs (2004), which
is in good agreement with the present data.
The definition of the molecular mixing fraction Θ (Youngs 1991, 1994) is given as
Θ(t) =
Z
∞
−∞
hX
air
X
HG
i
yz
dx
Z
∞
−∞
hX
air
i
yz
hX
HG
i
yz
dx
(4.3)
and quantifies the amount of mixed fluid within the mixing zone. It can be interpreted
as the ratio of molecular mixing to large-scale entrainment by convective motion.
As bubbles of light air and spikes of heavy gas begin to interfuse, the initially
mixed interface between the fluids steepens and the fluids become more segregated
on the molecular level (see figure 7a). The molecular mixing fraction reaches its
minimum at t ≈ 1.3 ms before Kelvin–Helmholtz instabilities lead to an increase
of molecular mixing. The onset of secondary instabilities is very sensitive to the
numerical method, as the numerical scheme determines how sharp the material
interface can be represented or whether numerical diffusion or dispersion effects lead
to an early mixing transition.
442 V. K. Tritschler and others
0 1 2 3 4 5 6
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 1 2 3 4 5 6
0.5
1.0
1.5
2.0
2.5
t (s) t (s)
(a)(b)
FIGURE 7. (Colour online) Molecular mixing fraction Θ and scalar dissipation rate χ
from Miranda (dark grey; blue online) and INCA (light grey; red online). The different
resolutions are represented as dotted line (64), dashed line (128), solid line (256) and solid
line with open squares for Miranda and open diamonds for INCA (512).
After re-shock molecular mixing is strongly enhanced and reaches its maximum of
Θ ≈ 0.85 by the end of the simulation. This finding is consistent with Lombardini
et al. (2012), who also found an asymptotic late-time mixing behaviour with Θ ≈0.85
independent of the shock Mach number but without re-shock. The asymptotic limit is
already accurately calculated on grid resolutions of ∼400 µm. As the second shock
wave compresses the mixing zone, the instability becomes less entrained yet equally
diffused (at least in the y and z directions) and therefore causes a steep rise in Θ.
A gradual increase of the mixing fraction after the steep rise occurs as the mixing
zone becomes more homogeneously distributed (Thornber et al. 2011) due to turbulent
motion.
The temporal evolution of the scalar dissipation rate is plotted in figure 7(b) and
is derived from the advection–diffusion equation for a scalar. The instantaneous scalar
dissipation rate of the three-dimensional RMI is estimated from the SF
6
concentration
field as
χ(t) =
Z
∞
−∞
D
SF6
∇Y
SF6
· ∇Y
SF6
dx dy dz, (4.4)
which quantifies the rate at which mixing occurs. For consistency of post-processing, a
second-order central difference scheme has been used for the calculation of the spatial
derivatives in (4.4) and (4.6) for all simulation data sets. Note that the order of the
finite difference scheme with which the gradients in (4.4) and (4.6) are approximated
affect their results.
The variation of the scalar dissipation rate with grid resolution before re-shock is
largely due to the under-resolved material interface and the onset of mixing transition.
Mixing is strongly enhanced after the second shock–interface interaction, but the
mixing zone is also confined to a much smaller region, which results in a decrease
of the integral χ. Also, χ only represents the resolved part of the dissipation rate
and therefore certainly underestimates the true value.
The TKE and the enstrophy (ε) are integrated over cross-flow planes in the mixing
zone that satisfy
4φ[1 −φ]> 0.9. (4.5)
This region is referred to as the inner mixing zone (IMZ) in the following.
On the Richtmyer–Meshkov instability 443
0
0
1
1
1
2
2
012
2
2
0.2
0.4
0.005
0.010
0.6
0.8
1.0
3
3
4
4
4
56
0 123456
t (s) t (s)
(a)(b)
TKE
FIGURE 8. (Colour online) Enstrophy ε and turbulence kinetic energy (TKE) from
Miranda (dark grey; blue online) and INCA (light grey; red online). The different
resolutions are represented as dotted line (64), dashed line (128), solid line (256) and
solid line with open squares for Miranda and open diamonds for INCA (512).
Baroclinic vorticity is deposited at the material interface during shock passage. The
amount of generated vorticity scales directly with the pressure gradient of the shock
wave and the density gradient of the material interface. The enstrophy is calculated by
ε(t) =
Z
IMZ
ρ(ω
i
ω
i
) dx dy dz, (4.6)
where ω
i
is the vorticity.
As can be seen from figure 8, the enstrophy also exhibits a strong grid dependence.
Fully grid-converged results are only obtained for times up to t ≈ 0.7 ms. As
the interface steepens due to strain and shear, the effective interface thickness is
determined by numerical diffusion, which appears to occur at t ≈ 0.7 ms. This is
consistent with the evolution of Θ shown in figure 7(a). Following Youngs (2007)
and Hahn et al. (2011), integration of enstrophy with a theoretical scaling of k
1/3
up to the cut-off wavenumber yields a proportionality between enstrophy and grid
resolution as ε ∝ ∆
−4/3
xyz
. From this follows an increase of enstrophy by a factor
of approximately 2.5 from one grid resolution to the next finer, which is in good
agreement with the present data.
The amount of TKE created by the impulsive acceleration of the interface is
calculated as
TKE(t) =
Z
IMZ
K dx dy dz, with K(x, y, z, t) =
ρ
2
u
00
i
u
00
i
. (4.7)
The fluctuating part ϕ
00
of a quantity ϕ is calculated from
ϕ
00
=ϕ − ¯ϕ, with ¯ϕ =hρϕi
yz
/hρi
yz
, (4.8)
where ¯ϕ is the Favre average of ϕ.
Grid-converged TKE is obtained on grids with a minimum resolution of ∼400 µm
(see figure 8). This is consistent with the convergence rate of the mixing zone
width. The total TKE deposited in the IMZ by the first shock–interface interaction
can be seen in the inset of figure 8. The re-shock occurring at t ≈ 2 ms deposits
approximately 40 times more TKE than the initial shock wave. Hill et al. (2006)
444 V. K. Tritschler and others
0.002 0.004 0.006
10
–3
10
–2
10
–1
10
0
t (s)
TKE
FIGURE 9. (Colour online) Log–log representation of TKE from Miranda (squares; blue
online) and INCA (diamonds; red online) taken from the finest grid (512).
found a similar relative increase by the re-shock at the same shock Mach number.
A significant decay in energy occurs immediately following re-shock. The material
interface interacts with the first expansion fan (see figure 3) and results in a further
increase in TKE between 3 and 3.5 ms. The amount of energy deposited by the first
expansion wave, however, is much weaker than that deposited by the reflected shock
wave. Hill et al. (2006) and Grinstein et al. (2011) found the amplification of TKE
by the first rarefaction to be much stronger than for our data. Such differences are
not surprising, because Grinstein et al. (2011) reported a strong dependence of energy
deposition on the respective initial interface perturbations. After the first expansion
wave has interacted with the interface, TKE decays slowly and the pressure gradients
associated with the subsequent rarefactions are too shallow to generate any further
noticeable increase in TKE.
Lombardini et al. (2012) found the decay rate of TKE to be larger than ∼t
−6/5
,
approaching ∼t
−10/7
. In our data, the late-time TKE decay is also approximately
∼(t − t
0
)
−10/7
, with t
0
= 2 ms being the virtual time origin (see figure 9). This
scaling would be characteristic for Batchelor-type turbulence (Batchelor & Proudman
1956) with a constant Loitsyansky integral (Kolmogorov 1941; Ishida et al. 2006) in
contrast to ∼t
−6/5
typical for turbulence of Saffman type (Saffman 1967a,b).
In the limit of a self-similar quasi-isotropic state, the temporal evolution of the
integral length scale δ
x
is related to the evolution of TKE in the mixing zone. From
TKE ∝ t
−n
, the growth rate of the integral scale follows as δ
x
∝ t
1−n/2
. Llor (2006)
derived a maximum decay rate of TKE ∼t
−10/7
that corresponds to a growth-rate
scaling of the energy-containing eddies of δ
x
∼t
2/7
. These predictions are in excellent
agreement with the growth-rate predictions of the mixing zone width of the present
investigation (see figure 6) and the decay rate of TKE (see figure 9).
The scalings indicated for the growth rate of the mixing zone and the decay rate
of TKE in figures 6 and 9 were not fitted in a strict sense. They merely serve as
reference for comparison with incompressible isotropic decaying turbulence. The
narrow data range of only ≈2 ms after re-shock for which the flow exhibits a
self-similar regime precludes any precise estimates for decay and growth-rate laws.
On the Richtmyer–Meshkov instability 445
0 1 2 3 4 5 6
–0.5
0
0.5
0
0 1 2 3 4 5 6
–0.5
0
0.5
0 1 2 3 4 5 6
0
(a)
(b)
(c)
t (s)
FIGURE 10. Anisotropy hai
yz
as a function of the dimensionless mixing zone coordinate
ξ and time from (a) Miranda and (b) INCA. (c) The volume-averaged anisotropy hai
xyz
of
the inner mixing zone from Miranda (squares) and INCA (diamonds). All data are taken
from the finest grid (512).
4.2. Anisotropy and inhomogeneity of the mixing zone
In the following, the anisotropy in the mixing zone is investigated. We define the local
anisotropy as
a(x, y, z, t) =
|u
00
|
|u
00
|+|v
00
|+|w
00
|
−
1
3
, (4.9)
where a = 2/3 corresponds to having all TKE in the streamwise velocity component
u
00
, whereas a = −1/3 corresponds to having no energy in the streamwise u
00
component. In figure 10(a,b) we show the yz plane averaged anisotropy hai
yz
as
a function of the dimensionless mixing zone coordinate ξ and time from Miranda
and INCA. The dimensionless mixing zone coordinate ξ is defined as
ξ =
x −x
∗
(t)
δ
x
(t)
, (4.10)
with x
∗
(t) being the x location where 4(1 −φ(x, t))φ(x, t) is maximal.
The light-gas side of the mixing zone remains more anisotropic than the heavy-gas
side but with a homogeneous anisotropy distribution after re-shock on either side. The
volume-averaged anisotropy in the inner mixing zone hai
xyz
is shown in figure 10(c).
No full recovery of isotropy of the mixing zone is achieved, and the re-shock
does not significantly contribute in the sense of the volume-averaged quantity hai
xyz
,
but leads to a stratified anisotropy distribution around the centre of the mixing
zone. After t ≈ 4.5 ms an asymptotic limit of hai
xyz
≈ 0.04 is reached, which
446 V. K. Tritschler and others
temporally coincides with the onset of the self-similar decay of TKE (see figures 9
and 10). The positive value of hai
xyz
implies that the streamwise component u
00
remains, despite re-shock, the dominant velocity component throughout the simulation
time. Lombardini et al. (2012) also found a temporal asymptotic limit of the
isotropisation process in their simulations. Grinstein et al. (2011) observed that
the velocity fluctuations in the mixing zone are more isotropic when the initial
interface perturbations also include short wavelengths, in which case the authors
nearly recovered full isotropy. When Grinstein et al. (2011) used long-wavelength
perturbations, the mixing zone remained anisotropic except for a narrow range on the
heavy-gas side.
In order to quantify the homogeneity of mixing, we calculate the density-specific
volume correlation (Besnard et al. 1992)
hbi
yz
(ξ, t) =
−
1
ρ
00
ρ
00
yz
=
1
ρ
yz
h
ρ
i
yz
−1, (4.11)
which is non-negative. The value hbi
yz
= 0 corresponds to homogeneously mixed
fluids with constant pressure and temperature. Large values indicate spatial inhomo-
geneities in the respective yz plane. The density-specific volume correlation has
gained some attention in recent years and was the subject of several experimental
investigations of the RMI – see Balakumar et al. (2012), Balasubramanian et al.
(2012), Balasubramanian, Orlicz & Prestridge (2013), Orlicz, Balasubramanian &
Prestridge (2013), Tomkins et al. (2013) and Weber et al. (2014).
Figure 11(a,b) shows the density-specific volume correlation normalised by the
maximal value at time t,
f
hbi
yz
=hbi
yz
(ξ, t)/max(hbi
yz
)(t), (4.12)
as a function of the dimensionless mixing zone coordinate ξ and time from Miranda
and INCA. The largest values of
f
hbi
yz
are found around the centre of the mixing
zone slightly shifted towards the heavy-gas side. The value of
f
hbi
yz
peaks around the
region where mixing between light and heavy gas occurs and tends to zero outside
the mixing region, towards the respective pure-gas side. Weber et al. (2014) observed
in their experiment that the peak of the density-specific volume correlation is initially
shifted towards the light-gas side, but moves towards the centre of the mixing zone
with increasing time.
In contrast to the anisotropy, where the re-shock does not contribute to the
isotropisation and which levels out after t ≈ 4.5 ms, the mixing zone becomes
significantly more homogeneous after re-shock, as can be observed from the temporal
evolution of the volume average of the density-specific volume correlation in the inner
mixing zone hbi
xyz
. Following re-shock the fluids become more and more mixed (see
figure 11c), with a value of hbi
xyz
≈0.13 at the latest time. The measured values of the
density-specific volume correlation in the single shock–interface interaction experiment
of Weber et al. (2014) at Ma =2.2 are in good agreement with our simulated values
at late times O(0.1), whereas at the lower Mach number Ma =1.6 Weber et al. (2014)
observed a more inhomogeneous mixing zone O(0.2). These values are significantly
larger than those measured for instance in the shock-gas-curtain experiments of Orlicz
et al. (2013) and Tomkins et al. (2013).
4.3. Spectral quantities
From homogeneous isotropic turbulence, it is well known that vorticity exhibits
coherent worm-like structures with diameter of the order of the Kolmogorov length
On the Richtmyer–Meshkov instability 447
0 1 2 3 4 5 6
–0.5
0
0.5
0 0.2 0.4 0.6 0.8 1.0
0 1 2 3 4 5 6
–0.5
0
0.5
0 1 2 3 4 5 6
0.1
0.2
0.3
0.4
t (s)
(a)
(b)
(c)
FIGURE 11. (Colour online) Normalised density-specific volume correlation
f
hbi
yz
as a
function of the dimensionless mixing zone coordinate ξ and time from (a) Miranda and
(b) INCA. (c) The volume-averaged density-specific volume correlation hbi
xyz
of the inner
mixing zone from Miranda (squares) and INCA (diamonds). All data are taken from the
finest grid (512).
scale and of a length that scales with the integral scale of the flow. The work of
Jiménez et al. (1993) suggests that these structures are especially intense features of
the background vorticity and independent of any particular forcing that generates the
vorticity. In contrast to forced homogeneous isotropic turbulence, where self-similar
stationary statistics are achieved, shock-induced turbulent mixing is an inhomogeneous
anisotropic unsteady decay phenomenon. Nevertheless, homogeneous isotropic
turbulence is used as theoretical framework for most of the numerical analysis
of RMI. However, it is unclear at what time and at what locations the mixing zone
exhibits the appropriate features and if homogeneous isotropic turbulence is achieved
at all. A fully isotropic mixing zone is never obtained, as the anisotropy, even though
decreasing with time, reaches an asymptotic limit at t ≈4.5 ms.
The temporal evolution of the initial perturbation is depicted in figure 12. Before
re-shock the dominant modes of the initial perturbation slowly break down. After re-
shock, however, the additional vorticity deposited during the second shock–interface
interaction rapidly destroys structures generated by the initial perturbation and initial
shock, leading to a self-similar decay after t ≈4 ms.
Thornber et al. (2010, 2012) found, formally in the limit of infinite Reynolds
numbers, a persistent k
−3/2
scaling of the TKE spectrum as well as a k
−3/2
spectrum
with a k
−5/3
spectrum at high wavenumbers that covers more and more of the
spectrum as time proceeds. Furthermore, the same authors (Thornber et al. 2011)
found (depending on the initial conditions) a k
−5/3
or a k
−2
scaling range after
448 V. K. Tritschler and others
0 1 2 3 4 5 6
10
0
10
1
10
2
10
0
10
1
10
2
–5 –4 –3 –2 –1
0 1 2 3 4 5 6
(a)
(b)
t (s)
FIGURE 12. (Colour online) Power spectra of density from (a) Miranda and (b) INCA
as a function of wavenumber k(L
yz
/2π) and time. The data are taken from the finest
grid (512).
re-shock. Long after re-shock, however, these scalings return to a k
−3/2
scaling at
intermediate scales and to a k
−5/3
scaling at high wavenumbers, close to the cut-off
wavenumber. The authors evaluated the radial spectra either in the centre of the
mixing zone or averaged over a fixed number of yz planes within the mixing zone. A
different scaling behaviour was observed by Hill et al. (2006) and Lombardini et al.
(2012), who found in their multicomponent LES at finite Reynolds numbers a k
−5/3
scaling in the centre of the mixing zone, whereas Cohen et al. (2002) found a k
−6/5
scaling range for the single-shock RMI averaged over four transverse slices within the
mixing zone. In a recent experimental investigation of a shock-accelerated shear layer,
Weber et al. (2012) showed a k
−5/3
inertial range followed by an exponential decay in
the dissipation range of the scalar spectrum. This result was numerically reproduced
by Tritschler et al. (2013a). Here, the authors averaged over a predefined IMZ.
All spectra shown in this section are radial spectra with a radial wavenumber that
is defined as k = (k
2
y
+ k
2
z
)
1/2
. The radial spectra are averaged over all yz planes
within the IMZ in the x direction that satisfy the condition in (4.5).
The radial power spectra of density are plotted in figure 13, where 13(a,b) show
the spectra before and 13(c,d) after re-shock. The power spectra of density and mass
fraction concentration (not shown) show a close correlation, even though they are not
directly related, as the mass fractions are constrained to be between zero and one.
Before re-shock, the dominant initial modes slowly break down and redistribute
energy to smaller scales. Re-shock causes additional baroclinic vorticity production
with inverse sign that results in a destruction process of the pre-shock structures (see
also figure 12). This process in conjunction with a vorticity deposition that is one
order of magnitude larger than the pre-shock deposition leads to rapid formation of
complex disordered structures, which eliminates most of the memory of the initial
interface perturbation, as can be seen in figures 12–14. Schilling, Latini & Don (2007)
reported that during re-shock vorticity production is strongly enhanced along the
interface where density gradients and misalignment of pressure and density gradients
is largest. The vorticity deposited by the re-shock transforms bubbles into spikes
and vice versa, which subsequently results in more complex and highly disordered
structures.
On the Richtmyer–Meshkov instability 449
k
–3/2
k
–3/2
k
–3/2
10
0
10
0
10
1
10
2
10
0
10
1
10
2
10
0
10
1
10
2
10
0
10
1
10
2
10
–8
10
–6
10
–4
10
–2
10
0
10
–8
10
–6
10
–4
10
–2
10
0
10
–8
10
–6
10
–4
10
–2
10
0
10
–8
10
–6
10
–4
10
–2
(c)
(a)
(d)
(b)
FIGURE 13. (Colour online) Power spectra of density from Miranda (dark grey; blue
online) and INCA (light grey; red online) before re-shock at (a) t =0.5 ms and (b) t =
2 ms and after re-shock at (c) t =2.5 ms and (d) t =6 ms. The different resolutions are
represented as dotted line (64), dashed line (128), solid line (256) and solid line with open
squares for Miranda and open diamonds for INCA (512).
At late times the power spectra of density appear to be more shallow than k
−3/2
,
and rather approach k
−6/5
, as was found by Cohen et al. (2002). The smallest length
scale in scalar turbulence is the Batchelor scale. For isotropic turbulence and Schmidt
numbers of order unity, it has the same order of magnitude as the Kolmogorov
microscale λ
B
≈ η. Therefore, the TKE spectra are closely correlated with the scalar
power spectra. Figure 14 shows the spectra of TKE before and after re-shock.
The significant increase in TKE is mainly due to the interaction of the enhanced
small-scale structures with comparatively steep density gradients and the reflected
shock wave. The re-shock at t ≈ 2 ms leads to a self-similar lifting of the spectrum
(see figure 14). The destruction process of the vortical structures initiated by the
re-shock leads to the formation of small scales, which rapidly remove the memory
of the initial condition. The intense fluctuating velocity gradients past re-shock are
rapidly smoothed out by viscous stresses. This results in a fast decay of the TKE
following the first ≈0.5 ms after re-shock (see figures 8 and 14c,d).
The sharp drop-off of the spectral energy in figures 13 and 14 in the Miranda
data at high wavenumbers is due to the filtering operator of the numerical method.
Opposite behaviour, that is, an increase of spectral energy at the highest wavenumbers,
is observed for the less dissipative INCA code, where the spurious behaviour at the
non-resolved scales is mainly dispersive.
450 V. K. Tritschler and others
10
0
10
1
10
2
10
0
10
1
10
2
10
0
10
1
10
2
10
0
10
1
10
2
10
–4
10
–2
10
2
10
0
10
–4
10
–2
10
2
10
0
10
–4
10
–2
10
2
10
0
10
–4
10
–2
10
2
10
0
(c)
(a)
(d)
(b)
k
–3/2
k
–3/2
k
–3/2
FIGURE 14. (Colour online) Spectra of TKE from Miranda (dark grey; blue online) and
INCA (light grey; red online) before re-shock at (a) t =0.5 ms and (b) t =2 ms and after
re-shock at (c) t =2.5 ms and (d) t =6 ms. The different resolutions are represented as
dotted line (64), dashed line (128), solid line (256) and solid line with open squares for
Miranda and open diamonds for INCA (512).
The scaled TKE spectra kE
kin
(k) represent the effective energy contributed by each
mode. Artifacts of the initial conditions still exist immediately before re-shock at
t =2.0 ms, as can be seen in figure 15(a), where most energy is contained at mode
k(L
yz
/2π) = 7. At re-shock, baroclinic vorticity is deposited at the interface and the
energy-containing wavenumber range immediately widens as vortex stretching and
tangling introduce new scales and higher vorticity. This broader profile is plotted in
figure 15(b), which clearly shows that the relative difference between the imposed
initial length scale k(L
yz
/2π) = 7 and the remaining length scales (both larger and
smaller) is vanishing. Indeed, as the mixing layer fully transitions to turbulence, the
flow reaches a self-similar state where the memory of initial perturbations is lost.
The spectra of the scalar dissipation rate χ in figure 16 quickly build up in the
cut-off wavenumber range after the initial shock impact (see figure 16b). After
re-shock and at late time (see figure 16c,d), the inertial subrange broadens to
wavenumbers where numerical dissipation damps out structures. The inertial range is
observed to scale with k
1/2
after re-shock, which is consistent with the k
−3/2
scaling
observed for E
ρ
and E
kin
. For the resolved wavenumbers, there is good agreement
between both codes at the finest two resolutions. Differences observed in figure 16(b)
are also reflected in figures 7 and 11. A sharper material interface and the associated
segregation of the fluids lead to a higher scalar dissipation rate (χ ), whereas at late
On the Richtmyer–Meshkov instability 451
10
0
10
1
10
2
10
0
10
1
10
2
0
500
1000
1500
0
500
1000
1500
(a)
(b)
FIGURE 15. (Colour online) Scaled spectra of TKE from Miranda (dark grey; blue online)
and INCA (light grey; red online) before re-shock at (a) t =2 ms and (b) t =2.5 ms. The
different resolutions are represented as dotted line (64), dashed line (128), solid line (256)
and solid line with open squares for Miranda and open diamonds for INCA (512).
10
0
10
1
10
2
10
0
10
1
10
2
10
0
10
1
10
2
10
0
10
1
10
2
10
–4
10
–5
10
–2
10
–1
10
–3
10
0
10
–4
10
–5
10
–2
10
–1
10
–3
10
0
10
–4
10
–5
10
–2
10
–1
10
–3
10
0
10
–4
10
–5
10
–2
10
–1
10
–3
10
0
k
1/2
k
1/2
(c)
(a)
(d)
(b)
FIGURE 16. (Colour online) Spectra of scalar dissipation rate from Miranda (dark grey;
blue online) and INCA (light grey; red online) before re-shock at (a) t = 0.5 ms and
(b) t =2 ms and after re-shock at (c) t =2.5 ms and (d) t =6 ms. The different resolutions
are represented as dotted line (64), dashed line (128), solid line (256) and solid line with
open squares for Miranda and open diamonds for INCA (512).
452 V. K. Tritschler and others
10
0
10
1
10
2
10
4
10
6
10
8
10
4
10
6
10
8
10
4
10
6
10
8
10
4
10
6
10
8
10
0
10
1
10
2
10
0
10
1
10
2
10
0
10
1
10
2
k
1/2
k
1/2
(c)
(a)
(d)
(b)
FIGURE 17. (Colour online) Spectra of enstrophy from Miranda (dark grey; blue online)
and INCA (light grey; red online) before re-shock at (a) t =0.5 ms and (b) t =2 ms and
after re-shock at (c) t =2.5 ms and (d) t =6 ms. The different resolutions are represented
as dotted line (64), dashed line (128), solid line (256) and solid line with open squares
for Miranda and open diamonds for INCA (512).
times the difference in the scalar dissipation rate does not significantly influence the
mixing measures hbi
xyz
and Θ.
Larger quantitative differences are observed in the power spectra of enstrophy
shown in figure 17. Immediately after either of the shock–interface interactions,
the quantitative agreement between the predicted enstrophy levels is excellent (see
figure 17a,c). The observed scalings of the inertial range following re-shock are
predicted consistently and agree with the inertial range scalings for the scalar
dissipation rate k
1/2
. However, the temporal decay of the small-scale enstrophy is
significantly different for either code, as can be seen immediately before re-shock
and long after re-shock in figures 17(b) and (d), respectively.
In isotropic homogeneous turbulence, the scaled spectra of the enstrophy (see
figure 18) has a single peak at the wavenumber where the dissipation range begins.
Therefore, under grid refinement this peak will shift to higher wavenumbers and
magnitudes as smaller scales are captured. The peak at k(L
yz
/2π) = 7 is associated
with the initial perturbation and disappears after re-shock as the flow becomes
turbulent (see figure 18). Good agreement for lower wavenumbers is observed between
codes and resolutions. Larger differences are observed at high wavenumbers, where
On the Richtmyer–Meshkov instability 453
10
0
10
1
10
2
0
2
4
6
8
10
10
0
10
1
10
2
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
(a)(b)
FIGURE 18. (Colour online) Scaled spectra of enstrophy from Miranda (dark grey; blue
online) and INCA (light grey; red online) after re-shock at (a) t =2.5 ms and (b) t =6 ms.
The different resolutions are represented as dotted line (64), dashed line (128), solid line
(256) and solid line with open squares for Miranda and open diamonds for INCA (512).
the dependence on numerical dissipation is greatest. At t = 2.5 ms the peak in the
scaled enstrophy spectra is at k(L
yz
/2π) ≈85 for both codes at the highest resolution.
Later, at t = 6.0 ms this peak has shifted to k(L
yz
/2π) ≈ 40 in INCA, whereas
in Miranda there is no apparent shift, although both have substantially decayed in
magnitude.
As RMI is a pure decay process after re-shock, differences in the numerical
approach become most apparent at late times. The numerical models of this study
predict different turbulence decay rates, as is evident from differences in the enstrophy
spectrum (figures 17 and 18) and in TKE (figure 9). The differences in enstrophy
(ε) and scalar dissipation rate (χ) have a qualitative effect that becomes apparent in
the fine-scale structures of figure 5 at t =6 ms. Although INCA resolved less scales
with smaller enstrophy levels, it does resolve steeper mass fraction gradients, which
is reflected in the higher χ and higher levels of E
χ
. Although it is unclear which
dissipation rate (scalar or kinetic) has most effect on the mixing process, both are
important (Dimotakis 2000).
4.4. Probability density functions
The bin size for computing the discrete probability density function (p.d.f.) is defined
as 1ϕ = [ϕ
max
− ϕ
min
]/N
b
for a quantity ϕ(x, y, z, t). The number of bins for all
quantities and all grid resolutions is N
b
= 64. Each discrete value of ϕ is distributed
into the bins, yielding a frequency N
k
for each bin. The p.d.f. is then defined by
P
k
(ϕ, t) =
N
k
1ϕN
, (4.13)
such that
P
N
k
k=1
P
k
1ϕ = 1, with N as the total number of cells in the IMZ that fall
within the range ϕ
min
6 ϕ 6 ϕ
max
. The limits ϕ
max
and ϕ
min
are held constant for all
resolutions and times.
The p.d.f. of the heavy-gas mass fraction is constrained to be 0.1 6 Y
HG
6 0.9.
Figure 19 shows the p.d.f. at times before re-shock (t = 0.5 ms, t = 2 ms) and
454 V. K. Tritschler and others
Y
HG
Y
HG
0.1 0.3 0.5 0.7 0.9
0.1
0.3 0.5 0.7
0.9
0.1 0.3 0.5 0.7 0.9
0.1 0.3 0.5 0.7 0.9
0
0.5
1.0
1.5
2.0
2.5
3.0
0
0.5
1.0
1.5
2.0
2.5
3.0
0
0.5
1.0
1.5
2.0
2.5
3.0
0
0.5
1.0
1.5
2.0
2.5
3.0
p.d.f.p.d.f.
(a)
(c)
(b)
(d)
FIGURE 19. (Colour online) Probability density function of Y
HG
from Miranda (dark grey;
blue online) and INCA (light grey; red online) before re-shock at (a) t = 0.5 ms and
(b) t =2 ms and after re-shock at (c) t =2.5 ms and (d) t =6 ms. The different resolutions
are represented as dotted line (64), dashed line (128), solid line (256) and solid line with
open squares for Miranda and open diamonds for INCA (512).
following re-shock (t = 2.5 ms, t = 6 ms). From figure 19(a) it is evident that
at early times following the initial shock–interface interaction the IMZ consists
mostly of segregated fluid, as the large peaks at the p.d.f. bounds indicate. Before
re-shock, inter-species mixing is largely dominated by the inviscid linear and nonlinear
entrainment. Molecular diffusion processes have not yet had enough time to act (see
figure 19b). Following re-shock, a fundamental change in the p.d.f. of Y
HG
(P(Y
HG
))
is observed (see figure 19c,d). The additional vorticity deposited by the re-shock
leads to rapid formation of small and very intense vortical structures that lead to very
effective mixing and destruction of the initial interface perturbation. The p.d.f. takes
a unimodal form at t =2.5 ms, as also reported by Hill et al. (2006). The peak value,
however, is not as well correlated with the average value of the mixture mass fraction
as was reported by Hill et al. (2006). With our data the peak value is slightly shifted
towards the heavy-gas side centred around Y
HG
≈ 0.6. The degree of convergence
between codes and resolutions is reassuring at t =2.5 ms. Note that P(Y
HG
) is a very
sensitive measure of the light–heavy gas mixing.
The rarefaction wave at t ≈ 3.2 ms does not significantly contribute to the mixing,
as it is not as pronounced as found in comparable investigations (Hill et al. 2006;
Grinstein et al. 2011). Long after re-shock the mixing process continues, which is
reflected in narrower tails of P(Y
HG
). The peak value of Y
HG
predicted by Miranda now
On the Richtmyer–Meshkov instability 455
0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6
0.8
0 0.2 0.4 0.6 0.8
0.5
2
4
6
8
10
12
2
4
6
8
10
12
2
4
6
8
10
12
1.0
1.5
2.0
2.5
p.d.f.p.d.f.
(a)
(c)
(b)
(d)
FIGURE 20. (Colour online) Probability density function of ω(λ
L
/v
s
) from Miranda (dark
grey; blue online) and INCA (light grey; red online) before re-shock at (a) t =0.5 ms and
(b) t =2 ms and after re-shock at (c) t =2.5 ms and (d) t =6 ms. The different resolutions
are represented as dotted line (64), dashed line (128), solid line (256) and solid line with
open squares for Miranda and open diamonds for INCA (512).
coincides with the average value of the mixture mass fraction. In the INCA results,
this value remains slightly shifted towards the heavy-gas side. However, the bimodal
character of P(Y
HG
) reported by Hill et al. (2006), who used air–SF
6
as light–heavy
gases, is not observed on the finest grid. Despite the strong mixing past re-shock the
turbulent mixing zone remains inhomogeneous until the end of the simulation time,
which makes the observed p.d.f. very sensitive to their location of evaluation within
the mixing layer.
The p.d.f. of the normalised vorticity is constrained between 0 6 ˜ω 6 0.8 with
˜ω = ω(λ
L
/v
s
), where v
s
is the initial shock velocity and λ
L
is a characteristic length
scale of the perturbations taken as λ
L
=L
yz
/
˜
k
max
, where L
yz
is the width of the domain
in the transverse direction and
˜
k
max
=k
max
(L
yz
/2π) =16.
Figure 20 shows the p.d.f. of the normalised vorticity P( ˜ω). Before re-shock,
mixing is driven by weak large-scale vortices (see figure 20a,b). Following re-shock,
however, structures with very intense vorticity develop with a dual-mode shape in
P( ˜ω) at t =2.5 ms on the finest grid. The early times after the second shock–interface
interaction are again consistently predicted by both codes (figure 20c). Nevertheless,
we observe larger differences for P( ˜ω) at t =6 ms. The peak and distribution of P( ˜ω)
from Miranda are shifted towards larger values of ˜ω as compared to INCA. This
supports the previous observation that the vorticity decay is affected by the numerical
456 V. K. Tritschler and others
approach. The difference in the vorticity intensity observed in figure 20, however,
does not lead to noticeable differences for the integral mixing measures shown in
figures 7(a) and 11 or for the integral length scale in figure 6.
5. Conclusion
We have investigated the shock-induced turbulent mixing between a light (N
2
, O
2
)
and heavy (SF
6
, acetone) gas in highly resolved numerical simulations. The mixing
was initiated by the interaction of a Ma = 1.5 shock wave with a deterministic
multimode interface. After the initial baroclinic vorticity deposition, the shock wave
is reflected at the opposite adiabatic wall boundary. The reflected shock wave impacts
the interface (re-shock) and deposits additional vorticity with enstrophy that is more
than two orders of magnitude larger than that of the initial vorticity deposition. The
transformation of spike structures into bubbles and vice versa in conjunction with
a large increase in vorticity results in the formation of disordered structures which
eliminate most of the memory of the initial interface perturbation.
A proposed standardised initial condition for simulating the RMI has been assessed
by two different numerical approaches, Miranda and INCA, over a range of grid
resolutions. The deterministic interface definition allows for spectrally identical initial
conditions for different numerical models and grid resolutions. A direct comparison
shows that larger energy-containing scales are in excellent agreement. Different
subgrid-scale regularisations affect marginally resolved flow scales, but allow for a
clear identification of a resolved scale range that is unaffected by the subgrid-scale
regularisation.
Mixing widths are nearly identical between the two approaches at the highest
resolution. At lower resolutions, the solutions differ, and we found a minimum
resolution of ∼400 µm to be necessary in order to produce reasonable late-time
results. The initial mixing zone growth rate scaled with δ
x
∼t
7/12
, whereas long after
re-shock the predicted growth rate was ∼t
2/7
. The decay of TKE was also found
to be consistent and in good agreement between the approaches. The decay scaled
with TKE ∼ t
−10/7
at late times, which corresponds to a growth-rate scaling of the
energy-containing eddies of ∼t
2/7
. The agreement in the large scales of the solution
between the two approaches is striking and has not been observed before.
Previous work on three-dimensional LES of RMI has examined numerical
dependence only indirectly. For example, Thornber et al. (2010) performed a code
comparison of single-shock RMI. However, the initialisation was different for the two
codes and the purpose was not to quantify the effect of different numerical methods.
Accordingly, the comparison of results at different resolutions for mean, spectral and
gradient-based quantities was limited. With the current work, we have presented for
the first time a comprehensive quantitative analysis of numerical effects on RMI.
Our results conclusively show that the large scales are in excellent agreement
for the two methods. Differences are observed in the representation of the material
interface. We conclude that the numerical challenge, prior to re-shock, is to predict
the large-scale nonlinear entrainment and the associated interface sharpening. Under
shear and strain, the interface steepens and eventually becomes under-resolved with
a thickness defined by the resolution limit of the numerical scheme. Therefore,
the saturation of the interface thickness by the numerical method occurs later as
the grid is refined. The molecular mixing fraction reached an asymptotic limit as
Θ ≈ 0.85 after re-shock, which was already correctly calculated on grid resolutions
of ∼400 µm. The yz plane averaged anisotropy hai
yz
revealed that the mixing zone
On the Richtmyer–Meshkov instability 457
exhibits a stratified anisotropy distribution with lower anisotropy on the heavy-gas
side and higher anisotropy on the light-gas side. Moreover, the volume-averaged
anisotropy hai
xyz
approached an asymptotic limit of hai
xyz
≈ 0.04, implying that
the fluctuating velocity component u
00
remains the dominant component even after
re-shock and that no full recovery of isotropy of the mixing zone is obtained. The
density self-correlation has been investigated in order to better understand the mixing
inhomogeneity in the mixing zone. The volume-averaged density-specific volume
correlation hbi
xyz
showed that the re-shock significantly increases mixing homogeneity
approaching a value of hbi
xyz
≈0.13 at the latest time.
The spectra demonstrate a broad range of resolved scales, which are in very
good agreement. Data also show that differences exist in the small-scale range. The
frequency dependence of the velocity and density fluctuations shows the existence
of an inertial subrange and that the two approaches agree at lower frequencies. The
observed spectral scalings were consistent among the methods with k
−3/2
.
Quantities that are gradient-dependent and therefore more sensitive to small scales,
such as the scalar dissipation rate and enstrophy, exhibit stronger dependence on
numerical method and grid resolution. The flow field shows visual differences for the
fine-scale structures at late times. The tenth-order compact scheme and the explicit
filtering and artificial fluid properties used in Miranda resolved more small scales
in TKE and enstrophy, whereas the sixth-order WENO-based scheme used in INCA
resolves more of the small-scale scalar flow features as observed in the spectra of
density and scalar dissipation rate. This result is somewhat intuitive given the numerics
of the two codes. High-order compact methods are capable of resolving higher
modes than explicit finite difference methods (Lele 1992). Given that the artificial
shear viscosity in Miranda has only a small effect on the solution compared to the
effect of the eighth-order filter, the primary difference, we conclude, of the resolving
power between methods is due to the difference in order of accuracy and modified
wavenumber profiles between the schemes. The compact finite difference method with
high-order filtering appears to capture a broader range of dynamic scales at late times.
The p.d.f. statistics of heavy-gas mass fraction Y
HG
revealed that the IMZ remains
inhomogeneous until the end of the simulation and that the peak probability is centred
around Y
HG
≈0.6 and thus is slightly shifted towards the heavy-gas side. Although the
overall quantitative agreement was very good, the p.d.f. of the vorticity showed larger
differences once intense small-scale vortical structures exist. The decay of vorticity
differs accordingly between the numerical methods.
Acknowledgements
The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V.
(GCS, www.gauss-centre.eu) for providing computing time on the GCS Supercomputer
SuperMUC at Leibniz Supercomputing Centre (LRZ, www.lrz.de). This work was
performed under the auspices of the US Department of Energy by Lawrence
Livermore National Laboratory under contract number DE-AC52-07NA27344. V.K.T.
gratefully acknowledges the support of the TUM Graduate School. B.J.O. thanks
A. Cook and W. Cabot for valuable insight and for use of the Miranda code.
Appendix A. Multicomponent mixing rule s
The specific heat capacity of species i is found by
c
p,i
=
γ
i
γ
i
−1
R
i
, with R
i
=
R
univ
M
i
, (A 1)
458 V. K. Tritschler and others
where γ
i
is the ratio of specific heats. The ratio of specific heats of the mixture
follows as
¯γ =
c
p
c
p
−
¯
R
, with c
p
=
N
X
i
Y
i
c
p,i
, (A 2)
where Y
i
is the mass fraction of species i and
¯
R is the specific gas constant of the
mixture with
¯
R =R
univ
/
¯
M. The molar mass of the mixture is given by
¯
M =
N
X
i
Y
i
M
i
!
−1
. (A 3)
For the gas mixture, Dalton’s law p =
P
i
p
i
will be valid with p
i
= ρR
i
T. The
mixture viscosity ¯µ and the mixture thermal conductivity ¯κ are calculated from
(Reid, Pransuitz & Poling 1987)
¯µ =
N
X
i=1
µ
i
Y
i
/M
1/2
i
N
X
i=1
Y
i
/M
1/2
i
, ¯κ =
N
X
i=1
κ
i
Y
i
/M
1/2
i
N
X
i=1
Y
i
/M
1/2
i
. (A 4a,b)
The effective binary diffusion coefficients (diffusion of species i into all other species)
are approximated as (Ramshaw 1990)
D
i
=(1 − X
i
)
N
X
i6=j
X
j
D
ij
!
−1
, (A 5)
where X
i
is the mole fraction of species i. Equation (A 5) ensures that the inter-species
diffusion fluxes balance to zero.
Appendix B. Molecular mixing rules
The viscosity of a pure gas is calculated from the Chapman–Enskog model
(Chapman & Cowling 1990)
µ
i
=2.6693 × 10
−6
√
M
i
T
Ω
µ,i
σ
2
i
, (B 1)
where σ
i
is the collision diameter and Ω
µ,i
is the collision integral,
Ω
µ,i
=A(T
∗
i
)
B
+C exp(DT
∗
i
) +E exp(FT
∗
i
), (B 2)
with A = 1.161 45, B = −0.148 74, C = 0.524 87, D = −0.7732, E = 2.161 78 and
F = −2.437 87 and where T
∗
i
= T/(/k)
i
. Here (/k)
i
is the Lennard-Jones energy
parameter, with the minimum of the Lennard-Jones potential and k the Boltzmann
constant.
The thermal conductivity of species i is defined by
κ
i
=c
p,i
µ
i
Pr
i
, (B 3)
with Pr
i
the species-specific Prandtl number.
On the Richtmyer–Meshkov instability 459
Property Nitrogen Oxygen SF
6
Acetone
(/k)
i
(K) 82.0 102.6 212.0 458.0
σ
i
(Å) 3.738 3.48 5.199 4.599
M
i
(g mol
−1
) 28.0140 31.9990 146.0570 58.0805
γ
i
1.4 1.4 1.1 1.1
Pr
i
0.72 0.72 0.8 0.8
TABLE 2. Molecular properties of nitrogen, oxygen, SF
6
and acetone.
The mass diffusion coefficient of a binary mixture is calculated from the empirical
law (Reid et al. 1987)
D
ij
=
0.0266
Ω
D,ij
T
3/2
p
p
M
ij
σ
2
ij
, (B 4)
with the collision integral for diffusion
Ω
D,ij
=A(T
∗
ij
)
B
+C exp(DT
∗
ij
) +E exp(FT
∗
ij
) +G exp(HT
∗
ij
), (B 5)
where T
∗
ij
= T/T
ij
and A = 1.060 36, B = −0.1561, C = 0.193 00, D = −0.476 35,
E =1.035 87, F = −1.529 96, G =1.764 74, H =−3.894 11 and
M
ij
=
2
1
M
i
+
1
M
j
, (B 6a)
σ
ij
=
σ
i
+σ
j
2
, (B 6b)
T
ij
=
r
k
i
k
j
. (B 6c)
The molecular properties of all species in the present study are given in table 2.
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