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1

Lagrangian statistics in compressible
turbulence: Examination of viscous process
and invariant dynamics of velocity gradients
Nishant Parashar1 †, Sawan Suman Sinha
1

1

and Balaji Srinivasan2

Department of Applied Mechanics, Indian Institute of Technology Delhi, New Delhi 110016,
India
2
Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai
600036, India
(Received xx; revised xx; accepted xx)

1. Introduction
The gradients of the small-scale velocity field and its dynamics in a turbulent flow
hold the key to understanding many important nonlinear turbulence processes like
cascade, mixing, intermittency and material element deformation. Thus, examination
of the velocity gradient tensor in canonical turbulent flow fields have been pursued
using experimental measurements (Lüthi et al. (2005)), direct numerical simulations
(DNS, Ashurst et al. (1987b)), as well as by employing simpler autonomous dynamical
models (ordinary differential equations, Vieillefosse (1982); Cantwell (1992) ) of velocity
gradients. The pioneering work done by these authors have been further followed up
extensively by several researchers for both incompressible (Ashurst et al. (1987b,a);
Girimaji (1991); Girimaji & Speziale (1995); Ohkitani (1993); Pumir (1994); da Silva
& Pereira (2008); O’Neill & Soria (2005); Chevillard & Meneveau (2006, 2011)) and
compressible turbulence (Pirozzoli & Grasso (2004); Suman & Girimaji (2009, 2010b,
2012); Danish et al. (2016a); Parashar et al. (2017a)). These efforts have led to an
improved understanding of small-scale turbulence.
Most DNS or experiment-based studies of fluid mechanics have so far been performed
using one-time Eulerian flow field. However, it is desirable to investigate the flow statistics
following individual fluid particles (the Lagrangian tracking). Such an investigation is
especially required from the point of view of developing/improving simple dynamical
models of the velocity gradient dynamics like the restricted Euler equation (REE)
(Cantwell (1992); Girimaji & Speziale (1995); Meneveau (2011)) and the enhanced
homogenized Euler equation model of Suman & Girimaji (2009). Such simple models, in
turn, can be used for closure of Lagrangian PDF method of turbulence (Pope (2002)). An
apt example of how Lagrangian statistics can reveal more profound insights into velocity
gradient dynamics is the recent experimental study of Xu et al. (2011), wherein the
authors provided evidence of the so-called “Pirouette effect”. Even though the vorticity
vector had always been expected to align with the largest strain-rate eigenvector, Eulerian
investigations invariably revealed a counterintuitive picture of vorticity aligning most
strongly with the intermediate eigenvector of the instantaneous local strain-rate tensor.
Xu et al. (2011), with their experimental Lagrangian investigations, provided first-hand
evidence that indeed the vorticity vector dynamically attempts to align with the largest
strain-rate eigenvector at an initial reference time in order to cause intense vortex
† Email address for correspondence: nishantparashar14@gmail.com

2

N. Parashar, S. S. Sinha and B. Srinivasan

stretching, and the alignment tendency as shown by the Eulerian one time field (with
the instantaneous intermediate eigenvector) was merely a transient incidental event.
In incompressible flows, Lagrangian studies using the direct numerical simulation of
decaying turbulence have earlier been performed by Yeung & Pope (1989), though the
authors’ focus was on Lagrangian statistics of velocity, acceleration, and dissipation.
Recently, Xu et al. (2011) have complemented their experimental observations of vorticity
alignment with the Lagrangian data extracted from DNS of forced isotropic turbulence.
Chevillard & Meneveau (2011) evaluated the Lagrangian model for velocity gradient
tensor in its capability to predict vorticity alignment using Lagrangian data obtained
from DNS of forced isotropic turbulent flow. Bhatnagar et al. (2016) quantified the
persistence time of fluid particles in vorticity-dominated and strain-dominated topologies
using Lagrangian data obtained from DNS of isotropic forced incompressible turbulence.
In compressible turbulence, Lagrangian statistics of velocity gradients have been recently
studied by Danish et al. (2016a) and Parashar et al. (2017a). While Danish et al.
(2016a) provided the first glimpse of compressibility effects on the alignment tendencies
of the vorticity vector, Parashar et al. (2017a) followed it up and made attempts at
explaining the observed behaviour in terms of the dynamics of the inertia tensor of
fluid particles and conservation of angular momentum of tetrads representing the fluid
particles. In continuation of our effort to develop deeper insight into the dynamics of
small-scale turbulence from a Lagrangian perspective, in this work, we focus on another
two important aspects of velocity gradient dynamics: (i) evolution of the deformation
gradient tensor, (ii) dynamics of flow field topology in compressible turbulence.
Our primary motivation behind investigating the dynamics of the deformation gradient
tensor is that this quantity has been used in modeling the viscous processes in both
restricted Euler equation (REE) by Jeong & Girimaji (2003). The authors modelled
the viscous process using gradient-deformation hypothesis-like assumption, wherein the
diffusion is allowed to be amplified as function of the deformation gradient tensor. This
model was called the linear Lagrangian diffusion model (LLDM). Later the same model
was used by Suman & Girimaji (2009) in their enhanced homogenized Euler equation
(EHEE) model. While the REE is the simple dynamical representation of velocity gradient dynamics in incompressible flows, the EHEE model is the counterpart for compressible
flows. Even though the EHEE model employing the LLDM approach does capture various
Mach number and Prandtl number effects, further improvements are indeed desirable
(Danish et al. (2014)). From this point of view, in the first part of this work, we subject
the LLDM modeling approach to a direct scrutiny by comparing its Lagrangian evolution
history against that of the exact process it represents−an examination that has not been
previously attempted neither in incompressible nor in compressible turbulence. Direct
numerical simulation data of decaying compressible turbulence over a wide range of
Mach number along with a well-validated Lagrangian particle tracker is employed for
the purpose. Further, the influence of compressibility−parameterized in terms of Mach
number, dilatation rate, and topology is also investigated.
In the second part of this work, we examine the evolution of topology itself in
compressible turbulence following the Lagrangian trajectories of the invariants of the
velocity gradient tensors. The local topology of a compressible flow field depends on
the local state of the velocity gradient tensor. Topology can also be visualized as
the local streamline pattern as observed with respect to a reference frame which is
translating with the centre of mass of a local fluid particle (Chong et al. (1990)). Topology
actually depends on the nature of eigenvalues of the velocity gradient tensor, and can
also be readily determined by knowing the three invariants of the velocity gradient
tensor (Cantwell & Coles 1983; Chong et al. 1990). Topology can not only be used for

Lagrangian statistics in compressible turbulence

3

visualization of a flow field; it has been observed to reveal deeper insights into various
nonlinear turbulence processes as well (Cantwell (1993); Soria et al. (1994)). Recently,
Danish et al. (2016b) have also attempted developing models for scalar mixing using
topology as conditioning parameter.
Traditionally, due to the prohibitive demand of computational resources, dynamics
of topology have been studied employing an approximate surrogate method called the
conditional mean trajectory (CMT). The idea of CMT was proposed by Martı́n et al.
(1998). The authors employed merely one-time velocity gradient data of the entire flow
field and computed bin-averaged rates-of-change of second and third invariants using
the right-hand-side of evolution equations of the invariants. These bin-averaged rates of
change conditioned upon their locations were subsequently used to plot trajectories in
the Q-R space. The authors called these trajectories as conditional mean trajectories
(CMT) and used them as a surrogate approach to study invariant dynamics. Several
authors have employed the CMTs to investigate various aspects of dynamic of topology
both for incompressible (Ooi et al. (1999); Meneveau (2011); Atkinson et al. (2012)) and
compressible flows (Chu & Lu (2013); Bechlars & Sandberg (2017)). Indeed the work done
by previous researchers employing the approximate approach of CMTs have improved our
understanding of the distribution and dynamics of topology in compressible turbulence.
Even though CMTs provide useful information about dynamics of invariants, they are
after all an approximation and merely a surrogate approach in the absence of adequate
computational resources (Martı́n et al. (1998)). An investigation of the Lagrangian
dynamics in compressible turbulence must be performed if adequate computational
resources are available. Indeed such an investigation of invariants using Lagrangian
trajectories have been recently performed by Bhatnagar et al. (2016) for incompressible
turbulence. Thus, we identify the following objectives for the second part of this work: (i)
identifying and understanding the differences, if any, between CMT and the Lagrangian
trajectory (LT) in compressible turbulence, and (ii) employing the LTs to investigate
lifetime of topologies and their interconversion processes.
To address the identified objectives of both parts of this paper, we employ direct
numerical simulations of decaying isotropic compressible turbulence and over a wide
range of turbulent Mach number (0.6, 1.5) and a moderate range of Reynolds number
(70, 350). The Lagrangian dynamics are obtained using an almost time continuous set
of flow field along with spline-aided Lagrangian particle tracker (more details in §4).
This paper is organized into seven sections. In §2 we present the governing equations.
In §3 we provide details of our direct numerical simulations and the Lagrangian particle
tracker. In §4 we explain our study plan. In §5 we evaluate the LLDM model of Jeong
& Girimaji (2003) in terms of its ability to mimic the exact viscous diffusion process.
In §6 we study the dynamics of topology, compare CMT and LT and quantify lifetime
of various flow-topologies existing in compressible turbulence. Section 7 concludes the
paper with a summary.

2. Governing Equations
The governing equations of compressible flow field of a perfect gas are the continuity,
momentum, energy and state equations:
∂ρ
∂ρ
+ Vk
=
∂t
∂xk
∂Vi
∂Vi
+ Vk
=
∂t
∂xk

−
−

∂Vk
;
∂xk
1 ∂p
1 ∂σik
+
,
ρ ∂xi
ρ ∂xk

ρ

(2.1)
(2.2)

4

N. Parashar, S. S. Sinha and B. Srinivasan
∂T
∂T
∂Vi
n − 1 ∂qk
+ Vk
= − T (n − 1)
−
∂t
∂xk
∂xi
ρR ∂xk
n−1 ∂
+
(Vi σji ) ,
ρR ∂xj
p = ρRT,

(2.3)
(2.4)

where Vi , xi , ρ, p, T, R, σik , qk , n denote velocity, position, density, pressure,
temperature, gas constant, stress tensor, heat flux and ratio of specific heat values,
respectively. The quantities σij and qk obey the following constitutive relationships:


∂Vj
∂Vk
∂Vi
σij = µ
+
+ δij λ
;
(2.5)
∂xj
∂xi
∂xk
∂T
qk = −K
,
(2.6)
∂xk
where δij is the Kronecker delta, K represents the thermal conductivity, and µ and λ
denote the first and second coefficients of viscosity respectively (λ = − 2µ
3 ) .
The velocity gradient tensor is defined as:
Aij ≡

∂Vi
.
∂xj

The evolution equation of Aij can be obtained by taking the gradient of momentum
equation 2.2, as


∂
1 ∂p
DAij
= −Aik Akj −
Dt
∂xj ρ ∂xi
{z
}
|
Pij

+

∂
∂xj
|

D
Dt (≡



 

1 ∂
∂Vi
∂Vk
2 ∂Vp
,
µ
+
−
δik
ρ ∂xk
∂xk
∂xi
3 ∂xp
{z
}

(2.7)

Υij

∂
∂t

Vk ∂x∂ k )

where, the operator
+
stands for the substantial derivative, which
represents the rate of change following a fluid particle. In equation 2.7, the first term on
its right-hand side (RHS) represents the self-deformation process of velocity-gradients.
The term Pij is called the pressure Hessian tensor, whereas Υij represents the action of
viscosity on the evolution of the velocity gradient tensor.

3. Direct numerical simulations and particle tracking
In this work dynamics of invariants of the velocity gradient tensor (VGT) are studied
using the direct numerical simulation (DNS) of decaying turbulent flows. Our simulations
are performed using the gas kinetic method (GKM). The gas kinetic method (GKM) was
originally developed by Xu et al. (1996) has been shown to be quite robust in terms of
numerical stability and has the ability to capture shock without numerical oscillations for
simulating compressible turbulence (Kerimo & Girimaji 2007; Liao et al. 2009; Kumar
et al. 2013; Parashar et al. 2017b). Our computational domain is of size 2π with a uniform
grid and periodic boundary conditions imposed on opposite sides of the domain.
The initial velocity field is generated at random with zero mean and having the
following energy spectrum E(κ):


E(κ) = A0 κ4 exp −2κ2 /κ20 ,
(3.1)

Lagrangian statistics in compressible turbulence
1

0
-0.5

A
B
C
D
E
F
G
H
I

k/k0

0.6
0.4

A
B
C
D
E
F
G
H
I

-2

Su

0.8

5

-4

0.2
0
0

2

4

6

8

0

2

4

6

8

t/τ

t/τ

(a)

(b)

Figure 1. Evolution of (a) normalized turbulent kinetic energy kk0 and (b) Velocity derivative
skewness Su , in Simulations A-G: (Table 1).

where κ is wavenumber. Values for spectrum constants A0 and κ0 are provided in Table 1
for various simulations employed in this work. The relevant Reynolds number for isotropic
turbulence is the one based on Taylor micro-scale (Reλ ):
r
20
Reλ =
k,
(3.2)
3ν
where k, , and ν represent turbulent kinetic energy, its dissipation-rate, and kinematic
viscosity. For compressible isotropic turbulence, the relevant Mach number is the turbulent Mach number (Mt ):
r
2k
Mt =
,
(3.3)
nRT
where T represents mean temperature. Following the work of Kumar et al. (2013), we
have used 4th order accurate weighted-essentially-non-oscillatory (WENO) method for
interpolation of flow variables, in-order to simulate high Mach number compressible
turbulent flows. Our solver has been extensively validated with established DNS results
of compressible turbulent flows (Danish et al. (2016a)). In total, this study employs seven
different simulations (Simulations A-G). Descriptions of these simulations are presented
in Table 1.
In Figure 1(a) we present evolution of turbulent kinetic energy (k) observed in Simulations A-G. In Figure 1(b), we present the evolution of skewness of the velocity derivative
(SV ) defined as:
k=

1
Vi Vi ;
2
 3

SVi = 


∂Vi
∂xi

∂Vi
∂xi

2 3/2

(3.4)

,

(3.5)

SV1 + SV2 + SV3
.
(3.6)
3
Note that the time has been normalized using τ , which represents eddy turnover time
(Yeung & Pope 1989; Elghobashi & Truesdell 1992; Samtaney et al. 2001; Martı́n et al.
SV =

6

N. Parashar, S. S. Sinha and B. Srinivasan

Simulation

Reλ

Mt

Grid size

A0

A.
B.
C.
D.
E.
F.
G.

70
350
150
100
70
70
70

0.075
0.6
1.0
1.0
1.0
1.25
1.5

1283
10243
5123
5123
2563
2563
2563

0.000023
0.0015
0.0042
0.0042
0.0042
——
0.0094

κ0
4
4
4
4
4
4
4

Table 1. Initial parameters of DNS simulations.

2006):
τ=

λ0
;
u00

(3.7)

where u00 and λ0 are the root mean square (rms) velocity and integral-length-scale of the
initial flow field (at time, t = 0).
To extract Lagrangian statistics, a Lagrangian particle tracker (LPT) is used to extract
the full time-history of tagged fluid particles. Our LPT obtains the trajectory (X+ (y, t))
of a fluid particle by solving the following equation of motion:


∂X+ (t, y)
= V X+ (t, y) , t ,
∂t

(3.8)

where the superscript “+” represents a Lagrangian flow variable, and y indicates the
label/identifier assigned to the fluid particle at a reference time (tref ). The initial value
of X+ at a reference time is chosen at random. Using this initial condition, we then
integrate Equation 3.8 by employing second order Runge-Kutta method. However, upon
integration, the position of the fluid particle at a subsequent time instant may not fall
exactly on one of the grid points of computational domain used in the parent DNS.
Therefore, an interpolation method is required to find relevant flow quantities at the
particle’s subsequent locations. Following the work of Yeung & Pope (1988), we choose
cubic spline interpolation for this purpose. Like our DNS solvers, our LPT algorithm and
implementation have been adequately validated. Details are available in Danish et al.
(2016a).

4. Plan of study
In this section we present our plan of study and also explain the quantities that are
employed to perform the desired investigations. In §4.1 we present the study plan for the
first part of the work, where we examine the Lagrangian dynamics of the deformation
gradient tensor. In §4.2 we explain our study plan for the second part of this work, which
involves comparing CMT and LT and consequently using these trajectories to investigate
interconversion processes of topologies existing in compressible turbulence.

Lagrangian statistics in compressible turbulence

7

4.1. Study I
In the first part of this work we focus on the viscous process Υ in the evolution equation
of the velocity gradient (Equation 2.7):


∂Aij
∂Akk
ν ∂ρ ∂Aik
1 ∂Akk
Υij = ν
+ν
−
+
.
(4.1)
∂xk ∂xk
∂xi ∂xj ρ ∂xj ∂xk
3 ∂xi
| {z
} | {z
} |
{z
}
ΥIij

ΥIIij

ΥIIIij

In Equation 4.1, ΥIij and ΥIIij are essentially the diffusion terms and ΥIIIij is an
interaction between density gradient and viscous process of the Aij tensor. From the
point of view of a dynamical equation of Aij (like REE of Vieillefosse (1982) and HEE
of Suman & Girimaji (2009)), each of the three viscous terms (ΥIij , ΥIIij and ΥIIIij )
represents a non-local, unclosed process. Jeong & Girimaji (2003) proposed a model for
the first term ΥIij . This model is called the linear Lagrangian diffusion model (LLDM).
The LLDM model approximates the viscous diffusion term ΥIij as:
ν

C −1
∂Aij
≈ kk Aij ,
∂xk ∂xk
3τν

(4.2)

where, Cij represents the right Cauchy Green tensor, which is derived from the deformation gradient tensor Dij :
∂xi
Dij ≡
;
(4.3)
∂Xj
Cij ≡ Dki Dkj .

(4.4)

Here, xi is the Eulerian position vector of the particle at time t, initially located at
position vector Xj at time tref . τν is a molecular viscous relaxation time scale (Jeong &
Girimaji 2003) .
Suman & Girimaji (2009) have employed the same LLDM model for the closure of their
enhanced homogenized Euler equation model (EHEE). Even though the LLDM achieves
a mathematically closed form, in this work, we intend to perform direct scrutiny of this
model using our DNS results. Such an investigation is required for a deeper understanding
of the model and may lead to further improvement in its performance.
Our interest is to examine how the viscous process Υ undergoes changes in comparison
to its state at a reference time following a fluid particle. For monitoring this change we
define an amplification ratio r(t, tref , which is defined as:
p
Υij (t)Υij (t)
,
(4.5)
r(t, tref ) = p
Υij (tref )Υij (tref )
where, Υij (t) and Υij (tref ) are values of the quantity Υij associated with an identified fluid
particle at an arbitrary time t and at a reference time tref respectively. Since an individual
particle represents just one realization, we obtain a relevant statistics by calculating
the mean of r(t, tref ) over several identified fluid particles of a homogeneous flow field.
The resulting quantity is referred as hr(t, tref )i, and is truly a two-time Lagrangian
correlation.
Direct numerical simulation of compressible decaying turbulence along with our Lagrangian particle tracker (LPT) are employed to access hr(t, tref )i. A set of 1,000,000
particles are identified at tref for the purpose. Further, to identify the role of turbulent
Mach number (Mt ), normalized dilatation (aii ) and topology (T ), we also calculate
hr(t, tref )i conditioned upon selected particles with a specified Mt , or aii or T at tref .
These conditional statistics are symbolically represented as hr(t, tref )|Mt i, hr(t, tref )|aii i

8

N. Parashar, S. S. Sinha and B. Srinivasan

and hr(t, tref )|T i respectively. Direct numerical simulations with a range of Mt and Reλ
are employed for our work. The relevant results and discussion are presented in §5.
The normalized dilatation rate, which is used as a conditioning parameter, is the trace
of the normalized velocity gradient tensor, defined as:
p
(4.6)
aij = Aij / Amn Amn .
The normalized dilatation-rate of a fluid particle (henceforth, referred to as just “dilatation”) represents the normalized rate of change in density of a local fluid particle:
1 dρ
= −aii
ρ dt0

(4.7)

p
where dt0 = dt Aij Aij represents time normalized with the local magnitude of the
velocity gradient tensor itself. A positive value of aii implies an expanding fluid element, a negative value of aii implies a contracting fluid element. A fluid particle with
instantaneous aii = 0 implies a volume preserving fluid element.
Direct numerical simulations with a range of initial Mt and Reλ are employed for
studying the behaviour of hr(t, tref )i, we examine the behaviour of viscous process
modelled using the LLDM model.
????????????????????????? ???????????????????????
????????????????????????????????????????????????
?????????????????????????????????????????????????
We define another ratio rm (t, tref ):
q
p
−1
Ckk
(t) Aij (t)Aij (t)
rm (t, tref ) = q
,
(4.8)
p
−1
Ckk
(tref ) Aij (tref )Aij (tref )
q
q
p
p
−1
−1
where, Ckk
(t) Aij (t)Aij (t) and Ckk
(tref ) Aij (tref )Aij (tref ) represents quantities associated with an identified fluid particle at t and tref respectively. Similar to
r(t, tref ), rm (t, tref ) is also accessible uing our DNS database and the Lagrangian particle
tracker. The corresponding mean calculated using the same set of 1,000,000 particles as
before, leads to the Lagrangian statistics:
* q −1 p
+
Ckk (t) Aij Aij (t)
hrm (t, tref )i = q
.
(4.9)
p
−1
Ckk
(tref ) Aij Aij (tref )
Note that in the LLDM model of Jeong & Girimaji (2003) τν represents a laminar viscous
time scale. Even though we do not have a direct estimate of this quantity for our DNS
set-up of compressible decaying turbulence, it has no net contribution to hrm (t, tref )i.
The evolution of hrm (t, tref )i is examined and careful comparisons of this quantity is
made against the behaviour of hr(t, tref )i in §5.2.
4.2. Study II
In this second part of this study, we focus on the dynamics of the topology of
compressible turbulence.
The topology of a fluid particle is the local streamline pattern as observed with respect
to a reference frame which is translating with the center of mass of the fluid particle. The
topology of a fluid particle depends on the nature of eigenvalues of the velocity gradient
tensor. However, it can also be inferred with the knowledge of the three invariants of the

Lagrangian statistics in compressible turbulence

9

Acronyms

p=0

p<0

p>0

Eigenvalues of aij

SFS

r<0

r < 0 & S2 > 0

r<0

complex

UFC

r>0

r>0

r > 0 & S2 < 0

complex

UNSS

r>0&q<0

r>0

r>0&q<0

real

SNSS

r<0&q<0

r<0&q<0

r<0

real

UFS

—

r < 0 & S2 < 0

—

complex

UN/UN/UN

—

r<0&q>0

—

real

SFC

—

—

r > 0 & S2 > 0

complex

SN/SN/SN

—

—

q>0&r>0

real

Table 2. Zones of various topologies on p − q − r space, where acronyms are:
stable-focus-stretching (SFS), unstable-focus-compressing (UFC), unstable-node/saddle/saddle
(UNSS), stable-node/saddle/saddle (SNSS), unstable-focus-stretching (UFS), unstable-node/unstable-node/unstable-node (UN/UN/UN), stable-focus-compressing (SFC), stable-node/stable-node/stable-node (SN/SN/SN).

velocity gradient tensor P, Q, R:


1
P 2 − Aij Aji , and
2


1
R=
−P 3 + 3P Q − Aij Ajk Aki .
3

P = −Aii , Q =

(4.10)

Correspondingly, the normalized invariants (p,q,r) of the local velocity gradient tensor
(aij ) are defined as:


1 2
p − aij aji , and
2


1
3
r=
−p + 3pq − aij ajk aki .
(4.11)
3
Determination of the topology of a fluid particle can also be done using the invariants
of the normalized velocity gradient tensor (aij ) as well. Chen et al. (1989) categories
topological patterns (Table 2) that can be observed in an incompressible field into UNSS,
SNSS, SFS, UFC. In compressible flows additional four more major topologies can exist:
SFS and SNSNSN in contracting fluid particles and UFS and UNUNUN in expanding
fluid particles. Figure 2 shows different topologies existing in different p-planes in p-q-r
space. Figure 3 present schematics of these topological patterns. The reader is referred
to Chong et al. (1990) and Suman & Girimaji (2010b) for further details on topology.
Since the value of the three invariants of the velocity gradient tensor uniquely determines the topology associated with a local fluid particle, the dynamics of topology can be
studied in terms of the dynamics of invariants themselves. Using the evolution equation
of the velocity-gradient-tensor (Equation 2.7), the time-evolution of invariants (P,Q,R)
of the velocity-gradient-tensor Aij can be found out (Bechlars & Sandberg (2017)):
p = −aii , q =

dP
= P 2 − 2Q − Sii ,
dt
dQ
2P
∗
= QP −
Sii − 3R − Aij Sji
,
dt
3

10

N. Parashar, S. S. Sinha and B. Srinivasan
SFS

UFC

q-axis

q-axis

SFS
S1a

S2

q 0
S1a

SFC
S2

q 0

S1b

S1b
0

0 r
(a)

r

UFC

(b)

q-axis
UFC
UFS

S1a
S2

q 0
S1b
SFS

0 r

(c)
Figure 2. Flow topologies represented in different p-planes: a) p = 0, b) p > 0 and c) p < 0.
(Figures to be reproduced with permission from Suman & Girimaji (2010a).)

dR
Q
∗
∗
= − Sii + P R − P Aij Sji
) − Aik Akj Sji
dt
3

(4.12)

where, Sij is the source term in the evolution equation of velocity-gradient-tensor
∗
(Equation 2.7) and Sij
is the traceless part of Sij tensor defined as:
Sij = −Pij + Υij , and
Skk
∗
Sij
= Sij −
δij .
3

(4.13)

Here Pij is the pressure hessian tensor and Υij represents the contribution of viscosity
in the evolution equation of the velocity-gradient tensor (Aij ), as shown in Equation
2.7. The relation between non-normalized invariants (P,Q,R) and normalized invariants
(p,q,r) is shown in equation 4.14:
p= p

P
Q
R
,q =
, and r =
Aij Aij
(Aij Aij )3/2
Aij Aij

(4.14)

Subsequently, using equations (4.12 & 4.14), the evolution equation of normalized
invariants (p,q,r) can be derived:


dp
d
P
1
dP
P
dAij
p
=
=p
−
Aij
,
dt
dt
dt
(Aij Aij )3/2
Aij Aij
Aij Aij dt


dq
d
Q
1 dQ
2Q
dAij
=
=
−
Aij
,
2
dt
dt Aij Aij
Aij Aij dt
(Aij Aij )
dt

Lagrangian statistics in compressible turbulence

(a)

(b)

(c)

(d)

(e)

(f)

(g)

11

(h)

Figure 3. Flow patterns corresponding to different flow topologies: a) UNSS, b) SNSS, c)
SFS, d) UFC, e) UFS, f) SFC, g) SNSNSN and h)UNUNUN. (Figures to be reproduced with
permission from Suman & Girimaji (2010a).)



d
R
1
dQ
3R
dAij
dr
=
=
−
Aij
.
dt
dt (Aij Aij )3/2
dt
(Aij Aij )3/2 dt
(Aij Aij )5/2

(4.15)

While following a fluid particle in physical space and tracking its invariants information,
we can track the movement of the fluid particle in the p-q-r space. We refer to such a
trajectory of the fluid particle in p-q-r space as the Lagrangian trajectory (LT). Mean
Lagrangian trajectory (MLT) can be obtained by calculating the mean position of a
selected number of particles originating from the same location in p-q-r space (within a
specified tolerance) at some reference time (tref ). Note that this procedure involves no
approximation.
As mentioned in the Introduction, many researchers have adopted an alternate though
approximate procedure of examining trajectories in the p-q-r space. This alternative
method does not track the individual fluid particles, but uses the averaged value of
the RHS of Equation 4.15 conditioned on a chosen set of p,q,r. In this method, merely
a one-time Eulerian dataset of the flow field is used. The statistics thus obtained are
essentially the conditional averages of the rate of change of the invariants with the

12

N. Parashar, S. S. Sinha and B. Srinivasan

conditional parameters being the local values of p,q and r. The trajectories thus obtained
are basically instantaneous streamlines in p-q-r space. Such trajectories are referred to
as the conditional mean trajectories (CMT) (Martı́n et al. 1998).
CMTs are approximate, and their use in past can only be justified as a surrogate
tool in the absence of adequate computational resources (Martı́n et al. 1998). In §6.1,
with enough resources to calculate MLTs, we compare CMTs against MLTs and evaluate
the extent to which CMTs can still be useful as a tool to study topology dynamics
in compressible turbulence. Subsequently in §6.2 we employ LTs to investigate mean
lifetimes of various topologies, and the influence of initial Mt , Reλ and dilatation (aii )
on them. To identify the lifetime of topologies, we track a large number of initially
identified fluid particles using our DNS datasets and the Lagrangian particle tracker.
Particle tracking is performed over a long duration (≈ 4τ ) and the residence time of
each particle in various topological regions in the p-q-r space is recorded. Subsequent
averaging of these residence times (one corresponding to each topology) over a number
of fluid particles is thus examined as the mean life of that particular topology. Further
details are provided along with results and discussion in §6.2.

5. Lagrangian investigations on the viscous process of velocity
gradients
We first analyze the evolution of the exact viscous term in §5.1. After understanding
the evolution characteristics of the exact viscous term, we then compare the performance
of the LLDM model term in approximating the exact viscous process in §5.2.
5.1. DNS based examination of viscous process
In Equation 4.1 we decompose the exact viscous process Υij into ΥIij , ΥIIij and
ΥIIIij : these processes represent diffusion of velocity gradient tensor, viscous Hessian of
dilatation and interaction of density gradient with velocity gradient components. At this
point it is pertinent to examine the relative importance of the three constituent process
in a compressible turbulent flow field. For this examination we define three fractions (fI ,
fII , fIII ):
p
ΥIij ΥIij
p
p
;
fI = p
ΥIij ΥIij + ΥIIij ΥIIij + ΥIIIij ΥIIIij
p
ΥIIij ΥIIij
p
p
fII = p
,
ΥIij ΥIij + ΥIIij ΥIIij + ΥIIIij ΥIIIij
p
ΥIIIij ΥIIIij
p
p
p
fIII =
.
(5.1)
ΥIij ΥIij + ΥIIij ΥIIij + ΥIIIij ΥIIIij
Clearly, 0 6 fI 6 fII 6 fIII . Any of these fraction approaching unity can be used as an
evidence of the corresponding process to be the dominant process. Using the flow-field
at peak dissipation time from several simulations (B-G, Table 1), we have computed the
volume-averaged values of fI , fII and fIII . We find that in all these simulations fI ≈ 0.9,
fII ≈ 0.09 and fIII ≈ 0.01. Based on these findings, we conclude that for the range of
Mach number and Reynolds number considered in this study, the exact process Υij is
almost solely represented by ΥIij , itself. Thus in the rest of this study we focus only on
ΥIij and assume it to be almost synonymous with Υij . Accordingly, the ratio hr(t, tref )i

Lagrangian statistics in compressible turbulence
defined earlier in Equation 4.5 is redefined as:
p
ΥIij (t)ΥIij (t)
r(t, tref ) = p
.
ΥIij (tref )ΥIij (tref )

13

(5.2)

To understand the influence of initial Mt , in Figure 4 we present Lagrangian mean
of hr(t, tref )i from simulations E-G. In both case, tref = 0.5τ . In both simulations
this time instant is smaller than the time of peak dissipation event. In each of these
simulations, the exact viscous process shows a two-stage evolution. In the first stage
of evolution, hr(t, tref )i increases and reaches a peak value. In the second stage, it
decays in magnitude. This evolution pattern is reminiscent of the evolution of dissipation
itself. Indeed the time instant of the peak of dissipation and that of the viscous process
match (tpeak−dissipation = 2τ ). The amplification in the first stage can be attributed to
steepening of gradients due to the rapid spread of the spectrum. The decay in the second
stage of evolution can be attributed mainly to the decay in kinetic energy. We observe
that as initial Mt increases, the peak value of hr(t, tref )i increases.
In several previous studies like Suman & Girimaji (2010b) and Parashar et al. (2017a),
it has been demonstrated that even when the unconditioned Eulerian or Lagrangian
statistics obtained using DNS data of homogeneous turbulence do not show any discernible influence of a global compressiblity parameter like Mt , the same statistics
when conditioned upon appropriate local compressibility parameters like normalized
dilatation (aii ) and local topology (T ) reveal, significant variations and insightful physics.
Examining the influence of local local parameters like aii and T on turbulence processes
is especially useful from the point of view of Lagrangian based statistical closure of
turbulence (Lagrangian PDF methods, Pope (2002))., wherein dynamical equations are
typically cast in terms of local flow variables directly. Thsus to gain further insight,
into the viscous diffusion process (ΥIij ), we subject hr(t, tref )i to conditioned averaging
on discreate values of aii at tref . In Figure 5 we present conditional Lagrangian mean
of hr(t, tref )i from Simulation G, with aii as the conditioning parameter. In Figure 5a,
separate curves are presented for aii = −1.0, −0.75, −0.50, −0.25 and 0, whereas in Figure
5b separate curves are presented for aii = 1.0, 0.75, 0.50, 0.25 and 0. In both figures we
observe profound influence of both the magnitude and the sign of aii on the evolution
of hr(t, tref )i. Even though almost at all dilatation levels, hr(t, tref )|aii i seem to retain
the two-stage evolution patterns as shown by the unconditioned statistics hr(t, tref )i,
the extent to which magnification of hr(t, tref )i happens (Figure 5) seems to be strongly
affected by the value of aii that a fluid particle has at t = tref . In Figure 5a we observe
that as the dilatation level change from being zero (volume preserving fluid particles) to
being more negative (contracting fluid particles), the peak value of hr(t, tref )|aii i reduces.
On the other hand, in Figure 5b we observe the opposite trend. For fluid particles with
high poistive dilatation (expanding fluid particles) the peak value of hr(t, tref )|aii i tends
to increase.
In our attempt to understand and explain the behaviour observed in Figure 5a, 5b in
Figure 7a and 7b we present the mean value of the amplification of the magnitude of
the velocity gradient tensor Aij itself, following the same set of fluid particles as used in
Figures 5. We measure this amplification as:


Aij (t)Aij (t)
.
(5.3)
hrA (t, tref )i =
Aij (tref )Aij (tref )
We observe that over almost the entire range of dilatation considered in this work, the
trend shown by hrA (t, tref )|aii i is similar to hr(t, tref )|aii i. Like hr(t, tref )|aii i, a more
positive dilatation tends to move the peak of hrA (t, tref )|aii i higher and a more negative

14

N. Parashar, S. S. Sinha and B. Srinivasan

dilatation tends to lower the peak of hrA (t, tref )|aii i. One to one correspondence between
the trends shown by hr(t, tref )|aii i in Figure 5 and that shown by hrA (t, tref )|aii i in
Figure 7 is not completely unexpected, and it indeed substantiates a gradient diffusion
like hypothesis which assumes ∆A ∝ A. However, in the light of Figure 7a, 7b our
primary curiosity that why the peaks of hr(t, tref )i (in Figure 5a, 5b) rise for expanding
fluid particles and reduce for contracting fluid particles can now

 possibly be explained
p
Aij (t)Aij (t) aii at
based on the behaviour of Aij itself. In Figure 9a we show
two
 time instants, tref and tref + τ /2. We observe that the one time statistics of
p
Aij (t)Aij (t) aii does not show significant variations with time.
p
The dependence of Aij Aij on aii is monotonic and alomost linear for aii > 0. A
p
faster expanding fluid particle (large positive aii ) is associated with smaller Aij Aij
than a slower expanding fluid particle (small positive aii ). On the other hand contracting
particles show a more complex behaviour.
Extremely fast contracting particles (very
p
p high
negative aii ) have a very large Aij Aij . As dilatation decomes less negative Aij Aij
drops
p first (till aii ≈ 0.25) and then again tends to be larger. Thus the dependence
of Aij Aij on aii is non-monotonic as well as (apparently) non-linear. (Figure 9a) In
Figure ?xxxxxxxxxx?a we present averaged rate of change in dilatation (ψ) of tagged
fluid particles over one Kolmogorov time:
ψ = (aii (τκ + tref ) − aii (tref )) /τκ ,

(5.4)

where, τκ is the Kolmogorov time at tref . In Figure ?xxxxxxxxxx?b we present the
fraction of particles having increased/decreased their aii over one Kolmogorov time
relative to the dilatation the particles had at tref . We observe that expanding particles are
more associated with negative ψ and contracting particles are associated with positive
ψ. IN other words, both contracting and expanding fluid particles tend to reduce the
magnitude of their dilatation. Since particles with higher
p inital dilatation tend to acquire
lower positive dilatation levels and the associated
Aij Aij of particles with higher
positive dilatation is larger than those with lower positive dilatation, it is plausible to
expect that the peak of hr(t, tref )i will be more at higher positive dilatation than that at
lower positive dilatation. Indeed this behaviour is observed in Figure 5a. For contracting
particles the dynamics seem p
to be more complicated because of non-monotonic and highly
non-linear distribution of
Aij Aij |aii (Figure 9a). At low negative dilatation (say
aii ≈ 0.25), the dominant tendency
p of particles is to move towards zero dilatation (Figure
9a). The association of higher Aij Aij at zero dilatation comare to that at low negative
dilatation still allows hrA (t, tref )|aii i to show apsubstantial magnification at early times
(Figure 7a). However, the relatively value of Aij Aij at low negative dilatation (say
-0.25) as compared to that at low positive dilatation say aii = +0.25 (Figure 9a) seem
to somewhat restrict the peak value of hrA (t, tref )|aii i of particles with small negative
dilatation when compared to peak value of hrA (t, tref )|aii i of particles with small positive
dilatations (compare the curve of ai i = −0.25 in Figure 7b to the curve of aii = −0.25
in Figure 7a).
For faster contracting particles (say those with initial aii ≈ −0.75), Figure 9a shows
that (like other contracting particles) they begin their journey p
towards zero dilatation.
However, Figure 9a suggests that as their dilatation reduces, Aij Aij severely drops
(almost exponential p
drop). This tendency, combined with the fact that initially also they
had a high value of Aij Aij , results into dramatic drop in hrA (t, tref )|aii i as observed
in Figure 7a. In summary, in this section we demonstrated that even though the viscous

Lagrangian statistics in compressible turbulence
2

15

Mt = 0.55
Mt = 0.40

1.5
k

- 2 -ν ∂ A - ∂x2 k ∂xk- t
A
-ν ∂x∂ ∂x
)-

tref

Mt = 0.25

k

1

0.5

0
0

2

4

6

8

(t − tref )/τ

Figure 4. Mach number dependence on evolution of exact viscous term (tref = 0.5τ ).
14

5

aii
aii
aii
aii
aii

4

3

= −1.00
= −0.75
= −0.50
= −0.25
= 0.00

aii
aii
aii
aii
aii

12
10
8

= +1.00
= +0.75
= +0.50
= +0.25
= 0.00

6

2

4
1
2
0

0
0

1

2

3

4

(t − tref )/τ

(a)

5

6

7

0

1

2

3

4

5

6

7

(t − tref )/τ

(b)

Figure 5. Dependence of dilatation rate on evolution of exact viscous term (tref = 0.5τ ).

process ?xxxxxxxxxxxxx? influence of global parameters like Mt , it shows profound
influence of local nomralized dilatation rate. All expanding fluid particles when followed
tend to undergo magnification of the viscous process. On the other hand contracting
particles show reduced magnification tendency. Fast contracting particles seem to show
even less magnification thatn slow contracting fluid particles. Further, we showed that
this disparate behaviour of the viscous process
p in contracting and expanding particles is
attributable to the vastly different values of Aij Aij associated with these fluid particles
in compressible turbulence and the dominant tendency of these particles to move towards
zero dilatation.

16

N. Parashar, S. S. Sinha and B. Srinivasan
10

UNSS
SNSS

8

SFS
UFC
6

UFS
SFC

4

2

0
0

1

2

3

4

5

6

7

(t − tref )/τ

Figure 6. Dependence of topology on evolution of exact viscous term (tref = 0.5τ ).
1.4

aii
aii
aii
aii
aii

1.2
1

1.6

= −1.00
= −0.75
= −0.50
= −0.25
= 0.00

aii
aii
aii
aii
aii

1.4
1.2

0.8

1

0.6

0.8

0.4

0.6

0.2

0.4

0

= +1.00
= +0.75
= +0.50
= +0.25
= 0.00

0.2
0

1

2

3

4

5

6

7

0

1

2

3

(t − tref )/τ

4

5

6

(t − tref )/τ

(a)

(b)

Figure 7. Dependence of dilatation rate on evolution of A (tref = 0.5τ ).
1.4

UNSS
1.2

SNSS
SFS

1

UFC
0.8

UFS
SFC

0.6
0.4
0.2
0
0

1

2

3

4

5

6

7

(t − tref )/τ

Figure 8. Dependence of topology on evolution of A (tref = 0.5τ ).

7

Lagrangian statistics in compressible turbulence
2

7000

×10

17

7

Mt = 1.5

Mt = 1.5

Mt = 1.0

Mt = 1.0

1.5

1

p

p

Aij Aij

ΥIij ΥIij

5000

3000
0.5

1000
-1

-0.5

0

aii

(a)

0.5

1

0
-1

-0.5

0

0.5

1

aii

(b)

Figure 9. Variation in magnitude of a) Aij and b) ΥIij tensor with dilatation aii (tref = 0.5τ ).

18

N. Parashar, S. S. Sinha and B. Srinivasan

5.2. Evaluation of the LLDM model
Having examined the behavior of the exact process ΥIij following fluid particles, now we
examine the performance of the LLDM model of Jeong & Girimaji (2003), which intends
to capture the essential physics of this exact process. For this examination, we use the
results of Simulation G (same as what has been used in §5.1). The LLDM modelling
approach of Jeong & Girimaji (2003) uses Lagrangian-Eulerianchange in variables to
cast ΥIij as:


∂
∂Xm ∂Aij
∂ 2 Aij
=ν
,
(5.5)
ν
∂xk ∂xk
∂xk ∂xk ∂Xm
where, Xi and xi are Eulerian and Lagrangian spatial co-ordinates. Further expansion
of RHS of Equation 5.5 leads to:
ν

∂Aij ∂ 2 Xm
∂Xn ∂Xm ∂ 2 Aij
∂ 2 Aij
+ν
.
=ν
∂xk ∂xk
∂xk ∂xk ∂Xm ∂Xn
∂Xm ∂xk ∂xk
{z
} |
{z
}
|
A

(5.6)

B

(5.7)
Jeong & Girimaji (2003) neglects term B (first modelling assumption) on the RHS of
Equation 5.6 to arrive at the following equation:
ν

∂ 2 Aij
∂Xn ∂Xm ∂ 2 Aij
≈ν
.
∂xk ∂xk
∂xk ∂xk ∂Xm ∂Xn

(5.8)
(5.9)

Using the definition of the deformation gradient tensor Dij (Equation 4.3) and the right
Cauchy-Green tensor Cij (Equation 4.4), the RHS of Equation 5.8 can be expressed in
terms of Cij :
∂Xn ∂Xm
−1 −1
= Dkn
Dkm
∂xk ∂xk
∂Xn ∂Xm
−1
= (Dkm Dkn )
∂xk ∂xk
∂ 2 Aij
∂ 2 Aij
−1
≈ νCmn
ν
∂xk ∂xk
∂Xm ∂Xn

(5.10)
(5.11)
(5.12)
(5.13)

−1
Further, Jeong & Girimaji (2003) make the second modelling assumption wherein Cmn
is approximated as an isotropic tensor:

C −1
∂ 2 Aij
∂ 2 Aij
≈ ν kk δmn
∂xk ∂xk
3
∂Xm ∂Xn
−1
2
2
C
∂ Aij
∂ Aij
ν
≈ ν kk
∂xk ∂xk
3 ∂Xm ∂Xm
ν

(5.14)
(5.15)
(5.16)

2

∂ A

ij
Finally, the third approximation is made ( ∂Xm ∂X
=
m

Aij
δX 2 )

ν

C −1 Aij
∂ 2 Aij
≈ ν kk
∂xk ∂xk
3 (δX)2

(5.17)

ν

C −1
∂Aij
≈ kk Aij
∂xk ∂xk
3τν

(5.18)

Lagrangian statistics in compressible turbulence
10

×10 6

19

100

hrm (t, tref )i

hrm (t, tref )i
8

80

hr(t, tref )i

hr(t, tref )i
6

60

4

40

2

20

0

0
0

1

2

3

4

5

6

7

0

1

2

3

t/τ

4

5

6

7

t/τ

(a)

(b)

Figure 10. Comparison of LLDM model term and the exact viscous term: a) unscaled axis, b)
axis scaled to visualize the difference in growth rates of the two processes.
40

5

hr(t, tref )i

35

hr(t, tref )i
4

hrA (t, tref )i

30
25

hrA (t, tref )i

3

20
2

15
10

1
5
0

0
0

1

2

3

4

5

6

7

0

1

2

t/τ

3

4

5

6

7

t/τ

(a)

(b)

Figure 11. Evolution of magnitude of velocity gradient tensor |A|.

The evolution equation of the deformation gradient tensor (Dij ) is:
dDij
= Dik Akj .
dt
τν is the molecular viscous relaxation time scale, defined as:
τν = δX 2 /ν ≈ λ2τ /ν,

(5.19)

(5.20)

where, λτ is Taylor microscale.
In Figure 10, we compare the LLDM model with the exact viscous term. We observe
that unlike the evolution of the exact process, the LLDM model term shows monotonic
growth with time. At the early stages of evolution, the monotonic growth is at least
qualitatively the same as the exact process. However, at later stages (after the dissipation
peak event) this continued monotonic growth in is gross disagreement with the decaying
behavior of the exact process after reaching a peak value. In Figure 11 we present the
evolution of |A| with time. We observe that |A| does show a two-stage behavior and starts
decaying after the peak dissipation event. Comparing Figure 10 and Figure 11, it is clear
−1
that the coefficient Ckk
of the LLDM model grossly overestimates the influence of the
−1
C tensor.

20

N. Parashar, S. S. Sinha and B. Srinivasan
1

Rγ

0.8

Rβ
Rα

0.6

0.4

0.2

0
0

0.5

1

1.5

2

(t − tref )/τ

Figure 12. Evolution of locally normalized eigenvalues of C −1

To better understand the reason for the failure of the LLDM model, we revisit the
modelling assumptions. One of the assumptions used in the model is that the tensor C −1
is isotropic. To scrutinize whether this assumption is contributing to the model failure,
we examine the eigenvalues of the C −1 tensor. The C −1 tensor is always symmetric (since
C = DDT is symmetric), hence the eigenvalues of C −1 are always real. To examine the
validity of the isotropic assumption, we plot the mean evolution of the three eigenvalues
(α > β > γ). In Figure 12 we present the Lagrangian statistics of Rα , Rβ and Rγ as a
function of time. We observe that the eigenvalues begin to depart from each other. Rα
increases appreciably during the evolution phase, while Rβ and Rγ decreases to negligible
values. This indicates that the C −1 tensor is strongly biased towards α−eigenvector.
Thus, it is clear that the assumption of the isotropy of the tensor is incorrect.
As shown in §4, (Equation 5.17 and 5.18) viscous term in LLDM model can also be
∂ 2 Aij
recovered by assuming the 4th order tensor ∂Xm ∂X
to be isotropic. However, it is not
n
possible to find the eigen-values of this tensor directly. By lagrangian change of variables,
∂ 2 Aij
∂Xm ∂Xn can be approximated as:
∂ 2 Aij
xm xn ∂ 2 Aij
≈
∂Xm ∂Xn
Xm Xn ∂xm ∂xn

(5.21)

Since a product of an anisotropic tensor with an isotropic tensor is always anisotropic,
∂ 2 Aij
we rather evaluate the eigenvalues of ∂xm ∂x
tensor. Since, this tensor is of order 4, we
n
dissociate the tensor into 9 second-order tensors and analyze their eigenvalues. Since,
∂ 2 Aij
∂xm ∂xn is not symmetric, it is not guaranteed to have real eigenvalues. Indeed, the
eigenvalues of the tensor are not purely real with mean ratio of the imaginary part to
the real part for α, β and γ eigenvalues to be 0.13, 0.77 and 0.17 respectively . However,
inorder to measure the extent of anisotropy, we plot the ratio of real part of the eigenvalues
∂ 2 Aij
tensor is anisotropic with
in terms of Rα , Rβ and Rγ in Figure 13. Clearly the ∂xm ∂x
n
ratio of the eigenvalues as: α : β : γ :: 1.9 : 1.0 : 1.1. Hence, it can be concluded that the
∂ 2 Aij
∂Xm ∂Xn tensor is also anisotropic.
In summary, our investigations reveal that the performance of the LLDM model is
unrealistic in the later stage of evolution of decaying turbulence. In the first stage of
evolution, even though qualitatively LLDM captures the right behavior, it tends to
overestimate the value. Our investigations reveal that the isotropy assumption of the
∂ 2 Aij
C −1 and ∂Xm ∂X
tensor is one of the major cause of the unrealistic behavior of the
n
LLDM model term as compared to the exact viscous term.

Lagrangian statistics in compressible turbulence

21

1

0.8

0.6

0.4

0.2

0
0

0.5

1

1.5

2

2.5

3

3.5

(t − tref )/τ
∂2A

ij
. Three
Figure 13. Evolution of locally normalized real part of eigenvalues of ∂xm ∂x
n
different colors of the line plot viz. black, light gray, dark gray represents Rα , Rβ and Rγ
respectively. Different markers represents eigenvalues corresponding to different 2nd order tensors
(components) of the original 4th order tensor. Rij represents ratio of normalized eigenvalue for

∂ 2 Aij
.
∂Xm ∂Xn

Different marker identifiers are −→ +: R11 , o: R12 , ∗: R13 , <: R21 , >: R22 , x: R23 , :
R31 , 7: R32 , D: R33

6. Comparisons of Eulerian and Lagrangian investigations of flow
field topology
In §6.1 we present a comparative study between the CMT and MLT and highlight
some important differences between the two. Further in the §6.2 we study the life of
topology using LTs and examine the influence of compressibility on it.
6.1. CMT vs. MLT
CMT has been extensively used as a method to predict particle trajectories in pq-r space. CMTs are basically instantaneous streamlines of particles in p-q-r space.
Several researchers have drawn conclusions based on the time-integrated behavior of
CMT as a complete substitute of MLT. However, using CMT for predicting long-term
behaviour−such as finding the life of topology may not be as accurate as MLT. We present
a discussion here highlighting the shortcomings of CMT over MLT. As most of the CMT
based studies have been performed for incompressible flows (Ooi et al. (1999); Meneveau
(2011); Lozano-Durán et al. (2015)), we use nearly incompressible simulation (case A 1),
to demonstrate the difference between CMT and MLT. We further condition the data-set
at very small dilatation value (|aii | < 0.01) to assert very weak compressibility effects.
To highlight the difference, we show CMT and MLT emerging from a small region in
q-r plane (p = 0 ± 0.01) in Figure 14(a-d) (for all four topologies that exist in this plane).
It is evident from Figure 14 that the instantaneous CMT does not coincide with the
MLT of fluid particles in q-r plane. There is no directional preference of fluid particles
to rotate in spiral order and converge to the origin of q-r plane, as inferred by CMT
(Figure 14(a-d)). In-fact the mean trajectory (MLT) converging directly to origin with
no tendency to rotate around the origin, explains no directional preference and a clear
tendency to randomly move in the q-r space. This argument is further supported by
Figure 15(a), where the root-mean-squared value of q and r is plotted with time.
Using the CMT approach, it can be shown that in around 3 eddy-turnover time, a fluid
particle completes one complete rotation around the origin (Ooi et al. (1999)). However, it
can be clearly seen in Figure 15(a), that in just 1 eddy-turnover time, the rms approaches

N. Parashar, S. S. Sinha and B. Srinivasan
0.4

0.4

0.2

0.2

0

0

q

q

22

-0.2

-0.2

-0.4

-0.4
-0.1

-0.05

0

0.05

0.1

-0.1

-0.05

0

r

0.4

0.4

0.2

0.2

0

0

-0.2

-0.2

-0.4

-0.4
-0.05

0.1

0.05

0.1

(b)

q

q

(a)

-0.1

0.05

r

0

0.05

0.1

-0.1

-0.05

0

r

r

(c)

(d)

Figure 14. Comparison of instantaneous CMTs (dotted line) and actual mean Lagrangian
trajectory MLT (solid line) of fluid particles with bin dimensions r ∈ r ± 0.01 and q ∈ q ± 0.025
for (a)UNSS, (b)SNSS, (c)SFS and (d)UFC topology. Dashed lines represent surfaces: S1a, S1b
and S2.
0.25

0.4
q
r

q

RMS

0.2

0

-0.2
0.05

0

high concentration
(Vieillefosse tail)

-0.4
0

1

2

3

4

5

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

r

(a)

(b)

Figure 15. (a) Evolution of root mean squared value of invariants q and r starting from a
bounded region r (-0.05 ± 0.01) and q (0.3 ± 0.025) (b) Instantaneous CMT (solid line) and final
spread of Lagrangian particles after 1 eddy-turnover time starting from the bounded region.
Sample size of conditioned particles in the bounded region ≈ 5000.

its maximum value (qrms ≈ 0.24 and rrms ≈ 0.05), indicating maximum spread of the
fluid particles in q-r plane. In-fact these RMS values are identical to the RMS of q and
r location of global unconditioned sample. To further understand this difference, we plot
the location of various fluid particles starting from a small region in q-r plane after 1
eddy-turnover time (Figure 15(b)). It can be observed that the population spread after
1 eddy-turnover time, is in-fact similar to the global spread of particles in q-r plane with
major concentration along the Vieillefosse tail.
Table 3 shows the percentage topology composition after three eddy-turnover times

Lagrangian statistics in compressible turbulence

UNSS
tref
100
tref + 3τ
27

23

100% UNSS
100% SNSS
100% SFS
100% UFC
SNSS
SFS UFC UNSS
SNSS
SFS UFC UNSS
SNSS
SFS UFC UNSS
SNSS
SFS UFC
0
100
0
0
0
0
100
0
0
0
0
100
0
0
0
8
38
26
27
8
39
26
23
7
41
28
26
8
39
27

Table 3. Percentage topology composition for particles after 3 eddy-turnover time starting
with 100% UNSS, SNSS, SFS and UFC sample respectively.

% anticlockwise
% clockwise

UNSS SNSS SFS UFC
32
51
33
33
68
49
67
67

Table 4. Clockwise and anti-clockwise movement of particles in q-r plane. [Note: This table
shows percentage transfer of particles. The topology composition remains invariant with time
i.e. Total particles moving in and out of a particular topology is same.]

of particles initially belonging to a distinct topology. It is evident that after three eddyturnover times the particles gets distributed throughout the q-r plane with the final
composition identical to global unconditioned sample. In fact, the composition after
three eddy-turnover times is approximately equal to the global topology composition for
isotropic incompressible flow as reported by Suman & Girimaji (2009). This observation
challenges the CMT approach that approximates particle motion in the q-r plane using
instantaneous Eulerian flow field. Table 4 shows anticlockwise and clockwise movement
of particles starting from a particular topology using LT approach. It can be seen that a
significant fraction of particles moves anti-clockwise (1/2 for SNSS topology and 1/3 for
UNSS, SFS and UFC topology). Therefore it can be concluded that the CMT approach
inaccurately predicts cyclic rotation of particles around the origin.
Hence, from the above analysis, we conclude that CMT does not represent actual
motion of fluid particles in q-r plane for incompressible flow. In general, the fluid
particles move around randomly in q-r plane, such that at every time instant the overall
distribution of particles is identical, with major concentration along the Vieillefosse tail.
Although, the above analysis is performed for incompressible flow, we expect even for
compressible flow, the motion of fluid particles to be such that their topology composition
approach global topology composition with time. To prove this hypothesis, we show
time-evolution of percentage composition of particles originating from a discrete p-plane
(p = +0.5 ± 0.05) for compressible simulation E (Table 1) in Figure 16. It can be seen
in Figure 16 that despite a significant variation in initial composition of topology as
compared to global composition, the particles moves around randomly in p-q-r space
such that their percentage composition tends toward the global composition.
6.2. Life of topology
In this section, we quantify the life-time of existence of particles in different topologies.
Starting with 10,00,000 particles, we tag the particles based on their topology at tref and
track them until they lose their initial topology. The life-time of each particle (lκ ) in a
particular topology is measured as a fraction of Kolmogorov time, τκ (measured at tref ),
calculated by recording the time from tref to the instant the particle loses it’s initial
topology. Further, we calculate the life-time of topology (Lκ ) as the mean life-time of all

24

N. Parashar, S. S. Sinha and B. Srinivasan
UNSS
SNSS
SFS
UFC
UFS
SFC

40

Topology %

30

20

10

0

0

1

2

t − tref
τ

3

Figure 16. Evolution of topology-composition for particles initially conditioned at
p = +0.5 ± 0.05 for 6 major topologies viz. UNSS, SNSS, SFS, UFC, UFS, SFS (Simulation E).
Initial composition at tref = 4τ : UNSS = 11.6%, SNSS = 28.9%, SFS = 25.6%, UFC = 16.7%,
UFS = 0%, SFC = 17.0%, SNSNSN = 0.2%, UNUNUN = 0%. Lines without markers represents
topology composition of global unconditioned sample.

UNSS SNSS SFS UFC
Sample %
25.2
5.4
43.5 25.9
Life of topology (κτ ) 1.80 0.53 3.32 2.08
Life %
23.32 6.86 42.95 26.91
Table 5. Life of topology Lκ for nearly incompressible flow (case A). The sample is further
conditioned on dilatation (|aii | < 0.01) to ensure strong incompressibility. Sample size ≈ 1,25,000
particles (conditioned).

the particles in a particular topology (in terms of τκ ):
Lκ =

N
X
lκ

i

i=1

N

(6.1)

6.2.1. Incompressible flow
We first show life of topology for nearly incompressible simulation (case A) in Table
5. To assert very mild compressibility, we further condition our sample on dilatation
(|aii | = |p| < 0.01). It can be observed that life of topology is proportional to the
percentage composition of topology. SFS topology is the most stable with average lifetime
of 3.3κτ , next is the UFC topology with average lifetime of 2κτ , next most stable is UNSS
with a lifetime of 1.8κτ . SNSS topology is found to be least stable with average lifespan
of 0.5κτ . Hence, for incompressible flow, the order of stability of topology is as follows:
SF S −→ U F C −→ U N SS −→ SN SS.
∂q
∂r
Table 6 shows the average velocity (|Upqr | = | ∂p
∂t p̂ + ∂t q̂ + ∂t r̂|) of particles in different
topologies in p-q-r space. It is interesting to observe that although the average velocity

Lagrangian statistics in compressible turbulence

average velocity |Uqr |
(units: s−1 )

UNSS
0.23

SNSS
0.28

SFS
0.23

Table 6. Average velocity of particles in p-q-r space (Upqr =
case A.

∂p
p̂
∂t

25

UFC
0.26

+

∂q
q̂
∂t

+

∂r
r̂)
∂t

for simulation

0.4
SFS

UFC

0.2

q

Large particle concentration
(Vieillefosse tail)
0

-0.2

UNSS

SNSS

-0.4
-0.1

-0.05

0

0.05

0.1

r

Figure 17. Region of high concentration of particles in q-r space.

for different topologies is comparable in magnitude, there is a significant difference in
their lifetimes (Lκ ). It can also be seen in Table 5, that the proportion of life of different
topologies is identical to their percentage composition. To explain the variation in Lκ
for different topologies, we focus on the distribution of the population in various zones
of topology rather than just percentage composition. Figure 17 shows region of high
concentration of particles in q-r plane. This region of high particle concentration is also
termed “Vieillofosse tail”. It can be seen in Figure 17 that, while SFS and UFC region
have an equal area in the q-r plane, their population distribution is not alike. In UFC, the
population has a spread closer to the surfaces of unlike topologies, than for SFS topology.
Closer proximity to the nearby surfaces of unlike topologies explains the likelihood of
UFC to be more prone to change than SFS topology, having known that their average
speeds in p-q-r space are comparable in magnitude (Table 6). Similarly, UNSS and SNSS
have equal area. Still, UNSS is found to be more stable than SNSS topology. This is
because for SNSS topology the bulk of the population is found closer to the origin where
nearby surfaces separating different topologies are closer leading to higher probability of
particles crossing the zone of SNSS topology into other topologies. However, in UNSS
topology, the population although highly concentrated near the origin, has significant
population spread away from the origin, where nearby surfaces for interconversion are
not very close. Hence, despite having comparable average velocities, different topologies
have different lifetimes (Lκ ).
6.2.2. Compressible flow
We show average life-time of topology for compressible simulations (case E-I) in Table
7. It can be seen that the life-time of topology is again a strong function of particle
concentration. In-fact the percentage life of different topologies is almost identical to the
composition of their populations (Table 7). However, the order of stability seems to be

26

N. Parashar, S. S. Sinha and B. Srinivasan
10 2
I
H
G
F
E
A

10 0

PDF

Mt

10 -2

10 -4

-2

-1

0

1

2

aii

Figure 18. PDF of noramilized dilatation aii for different simulations (Table 1).

influenced by Mt (Table 7). For weakly compressible flow (simulation A), the order of
stability based on lifetime for four major topologies is SF S −→ U N SS −→ U F C −→
SN SS. However, global turbulent Mach number of the flow seems to affect the order of
stability.
To explain this variation in the order of stability with Mt , we now focus on the
localized origin of compressibility. As shown by Suman & Girimaji (2010a),
√ the extent
√ of
compressibility can be determined solely by the strength of dilatation (− 3 < aii < 3).
Compressibility is a localized phenomenon, i.e., regions of weak and strong compressibility are present in the flow field. However, statistically, by looking at the probabilitydensity-function (PDF) of dilatation one can determine the extent of compressibility. A
larger spread of the dilatation PDF, represent a high strength of compressibility. For
incompressible flow (aii = −p ≈ 0), only 4 flow topologies exist viz. UNSS, SNSS,
SFS and UFC topology. But compressibility gives rise to new flow-topologies (Figure 2),
existing in p-q-r space in planes of non-zero dilatation (|p| > 0). The population of these
topologies depends upon the spread of dilatation. Weak compressibility, accompanied
by low dilatation spread leads to a low population of topologies existing in non-zero
p-planes and vice-versa for highly compressible flow. In Figure 18, we show the PDF of
dilatation for different simulations. It can be seen that the dilatation spread increases
with turbulent Mach number. However, there seems to be little to no dependence on
Reynolds number as evident from simulations F-H (Figure 18).
From the above discussion, it can be concluded that there is no general order of stability
of topology for compressible flows. However, for major 4 topologies (UNSS, SNSS, SFS
and UFC), there is a particular order of stability:
SF S −→ U N SS −→ U F C −→ SN SS.
Further, at very high Mach numbers the order stability based on life-time of existence
is found to be as follows:
SF S −→ U N SS −→ U F C −→ U F S −→ SN SS −→ SF C −→ U N U N U N −→ SN SN SN.
In order to explain the variation of lifetime of topology with Mt , we report the

Lagrangian statistics in compressible turbulence

Simulation

UNSS

SNSS

SFS

UFC

UFS

SFC

27

SNSNSN UNUNUN

case A
M=0.075
R = 70

Sample %
Life of topology
Life %

25.2
1.80
23.32

5.4
0.53
6.86

43.5
3.32
42.95

25.9
2.08
26.91

0
0
0

0
0
0

0
0
0

0
0
0

case E
M=0.6
Re=350

Sample %
Life of topology (κτ )
Life %

24.68
1.31
24.62

8.35
0.44
8.27

33.78
1.80
33.83

21.67
1.15
21.62

5.17
0.28
5.26

4.53
0.24
4.51

0.09
0.05
0.01

0.09
0.05
0.01

case F
M=1
Re=150

Sample %
Life of topology (κτ )
Life %

27.81
1.04
25.43

10.26
0.31
7.58

28.42
1.13
27.63

20.88
0.85
20.78

8.44
0.35
8.56

3.89
0.24
5.87

0.14
0.06
1.47

0.15
0.11
2.69

case G
M=1
Re=100

Sample %
Life of topology (κτ )
Life %

26.61
1.01
24.51

9.82
0.31
7.52

26.26
1.18
28.64

21.50
0.86
20.87

8.39
0.35
8.50

4.14
0.25
6.07

0.14
0.06
1.46

0.14
0.10
2.43

case H
M=1
Re=70

Sample %
Life of topology (κτ )
Life %

26.52
1.00
24.63

9.45
0.29
7.14

29.96
1.20
29.56

21.52
0.86
21.18

8.08
0.33
8.13

4.23
0.24
5.91

0.13
0.06
1.48

0.10
0.08
1.97

case I
M=1.5
Re=70

Sample %
Life of topology (κτ )
Life %

26.07
0.87
24.51

10.10
0.26
7.32

27.44
0.94
26.48

21.16
0.74
20.85

10.49
0.35
9.86

4.32
0.23
6.48

0.23
0.07
1.97

0.21
0.09
2.54

Table 7. Life of topology for compressible flows (case A, E-I). Sample size = 10,00,000
particles.

Simulation
A
E
F
G
H
I

UNSS
0.23
0.58
0.86
0.84
0.88
1.01

SNSS
0.28
0.88
1.39
1.35
1.37
1.57

SFS
0.23
0.60
0.95
0.92
0.97
1.13

UFC
0.26
0.66
1.00
0.99
1.03
1.18

UFS
0.62
0.82
0.80
0.87
0.90

SFC
0.96
1.55
1.49
1.46
1.76

SNSNSN
2.6
2.15
2.35
1.87
1.94

Table 8. Average velocity of particles in p-q-r space (|Upqr | = | ∂p
p̂ +
∂t
simulations A and E-I.

∂q
q̂
∂t

UNUNUN
1.80
1.33
1.21
1.37
1.27
+

∂r
r̂|)
∂t

for

∂q
∂r
magnitude of mean velocities (Upqr = ∂p
∂t p̂ + ∂t q̂ + ∂t r̂) of particles in p-q-r space in
Table 8. It can be seen that |Upqr | increases with Mt for all major topologies, while
for extreme topologies−UNUNUN and SNSNSN, the variation in |Upqr |, although most
likely opposite, seems less significant as compared to variation in rest of the 6 topologies.
For first 4 topologies (UNSS, SNSS, SFS, UFC), the decrease in life with increasing Mt
can be attributed to the increase in |Upqr | with increasing Mt . The rest of the topologies
come into existence only at high Mt , hence first their lifetime increases with Mt , but
further, with an increase in Mt their lifetime tend to remain constant, despite variation
in Upqr .
Further, as can be inferred from simulations F-H in Table 7, the Reynolds number
show negligible influence on the lifetime/stability of topology.

28

N. Parashar, S. S. Sinha and B. Srinivasan

6.2.3. Influence of initial dilatation
In Figure 19, we present the variation in Lκ for various topologies with initial dilatation
(aii ). To explain the variation in Lκ with dilatation we show joint-PDF (JPDF) of particle
population in different planes of discrete dilatation in Figure 20. For UNSS topology, peak
life-times (Lκ ) are observed at 0 dilatation (Figure 19(a)). Particles with UNSS topology
having initial positive dilatation have higher life as compared to those with initial negative
dilatation. This happens because for UNSS topology, the zone of existence in p-q-r space
shrinks along with decrease in population as dilatation decreases from high positive
dilatation to negative dilatation as shown in figure 20 (b-f). On the contrary, the region
of existence of SNSS topology widens while moving from high positive to high negative
population (20 (b-f)). This leads to lower population of SNSS topology at high positive
dilatations (Figure 19(b)).
Life-times (Lκ ) for SFS and UFC topology are shown in Figure 19(c) and 19(d)
respectively. It can be seen that both SFS and UFC topologies exhibit monotonic rise,
peaking in value for small positive dilatation, followed by monotonic fall while moving
from negative dilatation to positive dilatation. The variation is approximately symmetric,
slightly skewed towards positive values of dilatation. Variation ofLκ with initial dilatation
for UFS and SFC topology are shown in Figure 19(e) and 19(f) respectively. UFS and SFC
topologies have smaller Lκ and shows small variation with dilatation in their respective
regions of existence (aii > 0 for UFS and aii < 0 for SFC).
Hence, effect of initial dilatation is found to be maximum in UNSS and SFS topologies,
followed by UFC topology, while other topologies, with low global life-times seem to
exhibit mild variation in their life-times with varying initial dilatation.

Lagrangian statistics in compressible turbulence
2

2

1.5

1.5

1

1

0.5

0.5

-1

-0.5

0

0.5

1

-1

-0.5

(a)
2

1.5

1.5

1

1

0.5

0.5

-0.5

0

0.5

1

-1

(c)
2

1.5

1.5

1

1

0.5

0.5

0.5

(e)

0.5

1

-0.5

0

0.5

1

(d)

2

0

0

(b)

2

-1

29

1

-1

-0.5

0

(f)

Figure 19. Variation of life of topology Lκτ with initial dilatation aii (bin size: aii ± 0.05) for
6 major topologies: (a)UNSS, (b)SNSS, (c)SFS, (d)UFC, (e)UFS and (f)SFC. Symbol 4, , ∗,
×, and O represents life-time of topology for simulations E, F, G, H and I respectively.

30

N. Parashar, S. S. Sinha and B. Srinivasan
35

1

30

0.5

25

q

20

0

15
10

-0.5
-0.2

5

-0.1

0

0.1

0.2

r

(a)

(b)
35

1

35

1

30

0.5

30

0.5

q

20

0

25
20

q

25

0

15

15

10

-0.5
-0.2

10

-0.5
-0.2

5

-0.1

0

0.1

0.2

5

-0.1

0

r

r

(c)

(d)
35

1

0.1

0.2

35

1

30

0.5

30

25

0

15

20

q

q

20

25

0.5

15

0

10

-0.5
-0.2

5

-0.1

0

0.1

0.2

10

-0.5
-0.2

5

-0.1

0

r

r

(e)

(f)

0.1

0.2

Figure 20. Population spread of particles in p-q-r space shown as (a) surface plot of region of
high density (> 50% of maximum density) and Joint-PDF of population of particles at discrete
planes of dilatation: (b)aii = 0, (c)aii = −0.5, (d)aii = +0.5, (e)aii = −1, (f)aii = +1

Lagrangian statistics in compressible turbulence

31

7. Conclusions
We investigate the performance of the LLDM model of Jeong & Girimaji (2003) by
comparing the evolution of the LLDM model term with the exact viscous term. Further,
we compare the mean Lagrangian trajectory of fluid particles (MLT) with CMT and
investigate the lifetime of different topologies using Lagrangian trajectories (LTs). Wellresolved direct numerical simulations (up to 10243 ) of compressible decaying isotropic
turbulence with Reynolds number up-to 350 and Mach number up-to 1.5 are employed
to perform our studies. Along with this, a spline-interpolation based Lagrangian particle
tracker is used to track an identified set of fluid particles (at tref ) and extract their
Lagrangian statistics.
∂2A

ij
shows a twoWe found that the time evolution of the exact viscous term ∂xk ∂x
k
stage behavior. Its evolution is independent of turbulent Mach number. However, it’s
evolution is intensified at an elevated magnitude of dilatation. Further, we find that the
exact viscous process occurs at an amplified rate for rotation dominated topologies as
compared to strain-dominated topologies.
While comparing LLDM model term with the exact viscous term, we found that LLDM
model grossly overestimates the exact viscous process. LLDM model term undergoes an
exaggerated monotonic rise with time, failing to replicate the 2-stage evolution behavior
of the exact viscous term. We find that this anomaly in behavior can be attributed to
the LLDM modeling assumption of the isotropy of the inverse right Cauchy Green tensor
∂ 2 Aij
C −1 and the ∂Xm ∂X
tensor.
n
The actual motion of fluid particles in p-q-r space seems to show no particular tendency
to move in clockwise spiral orbit around the origin, as indicated by instantaneous CMTs.
In fact, there seems to be significant movement in the anticlockwise direction as well.
A group of chosen fluid particles is found to move randomly such that in very short
times (≈ 1 eddy-turnover time) the particles distribution mimics the global distribution,
which remains almost constant for fully developed turbulent flow. Computations for mean
life-time of topology reveals the following order of stability:
(i) Incompressible:

SF S −→ U F C −→ U N SS −→ SN SS
(ii) Mildly Compressible:
SF S −→ U N SS −→ U F C −→ SN SS −→ U F S −→ SF C −→ U N U N U N −→ SN SN SN.
(iii) Highly Compressible:
SF S −→ U N SS −→ U F C −→ U F S −→ SN SS −→ SF C −→ U N U N U N −→ SN SN SN.
The lifetime reduces with turbulent Mach number for topologies existing in the p = 0
plane (UNSS, SNSS, SFS, UFC). However, for topologies existing at high dilatation levels
viz. UFS, SFC, SNSNSN and UNUNUN, the lifetime first increases and later show little
variation with Mt . Reynolds number seems to have a negligible influence on the lifetime of
topology. Further, the lifetime of topology is found to decrease with increasing magnitude
of dilatation (|aii |).
The authors acknowledge the computational support provided by the HighPerformance Computing (HPC) Center of the Indian Institute of Technology Delhi, New
Delhi, India.

32

N. Parashar, S. S. Sinha and B. Srinivasan
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