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Lagrangian statistics in compressible

turbulence: Examination of viscous process

and invariant dynamics of velocity gradients

Nishant Parashar1†, Sawan Suman Sinha 1and Balaji Srinivasan2

1Department of Applied Mechanics, Indian Institute of Technology Delhi, New Delhi 110016,

India

2Department 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 (L¨uthi 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

2N. 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 gradi-

ent 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 Mart´ın 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 (Mart´ın 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

∂t +Vk

∂ρ

∂xk

=−ρ∂Vk

∂xk

; (2.1)

∂Vi

∂t +Vk

∂Vi

∂xk

=−1

∂p

∂xi

∂σik

∂xk

,(2.2)

4N. Parashar, S. S. Sinha and B. Srinivasan

∂T

∂t +Vk

∂T

∂xk

=−T(n−1)∂Vi

∂xi−n−1

ρR

∂qk

∂xk

+n−1

ρR

∂

∂xj

(Viσji),(2.3)

p=ρRT, (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 qkobey the following constitutive relationships:

σij =µ∂Vi

∂xj

+∂Vj

∂xi+δij λ∂Vk

∂xk

; (2.5)

qk=−K∂T

∂xk

,(2.6)

where δij is the Kronecker delta, Krepresents the thermal conductivity, and µand 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

DAij

Dt =−AikAkj −∂

∂xj1

∂p

∂xi

| {z }

Pij

+∂

∂xj1

∂

∂xkµ∂Vi

∂xk

+∂Vk

∂xi−2

∂Vp

∂xp

δik

| {z }

Υij

,(2.7)

where, the operator D

Dt (≡∂

∂t +Vk∂

∂xk) 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 2πwith 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) = A0κ4exp −2κ2/κ2

0,(3.1)

Lagrangian statistics in compressible turbulence 5

02468

t/τ

0.2

0.4

0.6

0.8

k/k0

(a)

02468

t/τ

-4

-2

-0.5

(b)

Figure 1. Evolution of (a) normalized turbulent kinetic energy k

k0and (b) Velocity derivative

skewness Su, in Simulations A-G: (Table 1).

where κis wavenumber. Values for spectrum constants A0and κ0are 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):

Reλ=r20

3ν k, (3.2)

where k,, and νrepresent turbulent kinetic energy, its dissipation-rate, and kinematic

viscosity. For compressible isotropic turbulence, the relevant Mach number is the turbu-

lent Mach number (Mt):

Mt=r2k

nRT ,(3.3)

where Trepresents 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 Simula-

tions A-G. In Figure 1(b), we present the evolution of skewness of the velocity derivative

(SV) deﬁned as:

k=1

2ViVi; (3.4)

SVi=∂Vi

∂xi3

∂Vi

∂xi23/2,(3.5)

SV=SV1+SV2+SV3

3.(3.6)

Note that the time has been normalized using τ, which represents eddy turnover time

(Yeung & Pope 1989; Elghobashi & Truesdell 1992; Samtaney et al. 2001; Mart´ın et al.

6N. Parashar, S. S. Sinha and B. Srinivasan

Simulation ReλMtGrid size A0κ0

A. 70 0.075 12830.000023 4

B. 350 0.6 102430.0015 4

C. 150 1.0 51230.0042 4

D. 100 1.0 51230.0042 4

E. 70 1.0 25630.0042 4

F. 70 1.25 2563—— 4

G. 70 1.5 25630.0094 4

Table 1. Initial parameters of DNS simulations.

2006):

τ=λ0

; (3.7)

where u0

0and λ0are 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)

∂t =VX+(t, y), t,(3.8)

where the superscript “+” represents a Lagrangian ﬂow variable, and yindicates 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 Υin the evolution equation

of the velocity gradient (Equation 2.7):

Υij =ν∂Aij

∂xk∂xk

| {z }

ΥIij

+ν∂Akk

∂xi∂xj

| {z }

ΥIIij

−ν

∂ρ

∂xj∂Aik

∂xk

∂Akk

∂xi

| {z }

ΥIIIij

.(4.1)

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∂Aij

∂xk∂xk≈C−1

3τν

Aij ,(4.2)

where, Cij represents the right Cauchy Green tensor, which is derived from the deforma-

tion gradient tensor Dij :

Dij ≡∂xi

∂Xj

; (4.3)

Cij ≡DkiDkj .(4.4)

Here, xiis the Eulerian position vector of the particle at time t, initially located at

position vector Xjat time tref .τνis 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 Υundergoes 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:

r(t, tref ) = pΥij (t)Υij (t)

pΥij (tref )Υij (tref ),(4.5)

where, Υij (t) and Υij (tref ) are values of the quantity Υij associated with an identiﬁed ﬂuid

particle at an arbitrary time tand 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 La-

grangian 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)iconditioned upon selected particles with a speciﬁed Mt, or aii or Tat tref .

These conditional statistics are symbolically represented as hr(t, tref )|Mti,hr(t, tref )|aiii

8N. Parashar, S. S. Sinha and B. Srinivasan

and hr(t, tref )|T i respectively. Direct numerical simulations with a range of Mtand 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:

aij =Aij /pAmnAmn.(4.6)

The normalized dilatation-rate of a ﬂuid particle (henceforth, referred to as just “dilata-

tion”) represents the normalized rate of change in density of a local ﬂuid particle:

dρ

dt0=−aii (4.7)

where dt0=dtpAij Aij represents time normalized with the local magnitude of the

velocity gradient tensor itself. A positive value of aii implies an expanding ﬂuid ele-

ment, 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 Mtand Reλare 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 ):

rm(t, tref ) = qC−1

kk (t)pAij (t)Aij (t)

qC−1

kk (tref )pAij (tref )Aij (tref )

,(4.8)

where, qC−1

kk (t)pAij (t)Aij (t) and qC−1

kk (tref )pAij (tref )Aij (tref ) represents quanti-

ties associated with an identiﬁed ﬂuid particle at tand 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:

hrm(t, tref )i=*qC−1

kk (t)pAij Aij (t)

qC−1

kk (tref )pAij Aij (tref )+.(4.9)

Note that in the LLDM model of Jeong & Girimaji (2003) τνrepresents 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 )iis examined and careful comparisons of this quantity is

made against the behaviour of hr(t, tref )iin §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 < 0p > 0 Eigenvalues of aij

SFS r < 0r < 0 & S2>0r < 0 complex

UFC r > 0r > 0r > 0 & S2<0 complex

UNSS r > 0 & q < 0r > 0r > 0 & q < 0 real

SNSS r < 0 & q < 0r < 0 & q < 0r < 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−rspace, 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-n-

ode/unstable-node/unstable-node (UN/UN/UN), stable-focus-compressing (SFC), stable-n-

ode/stable-node/stable-node (SN/SN/SN).

velocity gradient tensor P, Q, R:

P=−Aii, Q =1

2P2−Aij Aji, and

R=1

3−P3+ 3P Q −Aij Ajk Aki.(4.10)

Correspondingly, the normalized invariants (p,q,r) of the local velocity gradient tensor

(aij ) are deﬁned as:

p=−aii, q =1

2p2−aij aji, and

r=1

3−p3+ 3pq −aij ajkaki.(4.11)

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 deter-

mines 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)):

dt =P2−2Q−Sii,

dt =QP −2P

3Sii −3R−Aij S∗

ji,

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

UFC

S1bS1a

q-axis

SFS

(a)

S1b

S1a

q-axis

UFC

SFC

SFS

(b)

UFC

S1b

S1a

q-axis

UFS

SFS

(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).)

dt =−Q

3Sii +P R −P Aij S∗

ji)−AikAkj S∗

ji (4.12)

where, Sij is the source term in the evolution equation of velocity-gradient-tensor

(Equation 2.7) and S∗

ij is the traceless part of Sij tensor deﬁned as:

Sij =−Pij +Υij , and

S∗

ij =Sij −Skk

3δij .(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

pAij Aij

, q =Q

Aij Aij

, and r =R

(Aij Aij )3/2(4.14)

Subsequently, using equations (4.12 & 4.14), the evolution equation of normalized

invariants (p,q,r) can be derived:

dt =d

dtP

pAij Aij =1

pAij Aij

dt −P

(Aij Aij )3/2Aij

dAij

dt ,

dt =d

dtQ

Aij Aij =1

Aij Aij

dt −2Q

(Aij Aij )2Aij

dAij

dt ,

Lagrangian statistics in compressible turbulence 11

(a) (b)

(e) (f)

(g) (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).)

dt =d

dtR

(Aij Aij )3/2=1

(Aij Aij )3/2

dt −3R

(Aij Aij )5/2Aij

dAij

dt .(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) (Mart´ın 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 (Mart´ın 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,Reλand 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 ):

fI=pΥIij ΥIij

pΥIij ΥIij +pΥIIij ΥIIij +pΥIIIij ΥIIIij

;

fII =pΥIIij ΥIIij

pΥIij ΥIij +pΥIIij ΥIIij +pΥIIIij ΥIIIij

fIII =pΥIIIij ΥIIIij

pΥIij ΥIij +pΥIIij ΥIIij +pΥIIIij ΥIIIij

.(5.1)

Clearly, 0 6fI6fII 6fIII . 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 13

deﬁned earlier in Equation 4.5 is redeﬁned as:

r(t, tref ) = pΥIij (t)ΥIij (t)

pΥIij (tref )ΥIij (tref ).(5.2)

To understand the inﬂuence of initial Mt, in Figure 4 we present Lagrangian mean

of hr(t, tref )ifrom 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 )iincreases 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 Mtincreases, the peak value of hr(t, tref )iincreases.

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 dis-

cernible 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 Ton 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 )ito conditioned averaging

on discreate values of aii at tref . In Figure 5 we present conditional Lagrangian mean

of hr(t, tref )ifrom 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 )|aiiiseem 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 )ihappens (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 )|aiiireduces.

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 )|aiiitends

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:

hrA(t, tref )i=Aij (t)Aij (t)

Aij (tref )Aij (tref ).(5.3)

We observe that over almost the entire range of dilatation considered in this work, the

trend shown by hrA(t, tref )|aiiiis similar to hr(t, tref )|aiii. Like hr(t, tref )|aiii, a more

positive dilatation tends to move the peak of hrA(t, tref )|aiiihigher and a more negative

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

dilatation tends to lower the peak of hrA(t, tref )|aiii. One to one correspondence between

the trends shown by hr(t, tref )|aiiiin Figure 5 and that shown by hrA(t, tref )|aiiiin

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

based on the behaviour of Aij itself. In Figure 9a we show pAij (t)Aij (t)aiiat

two time instants, tref and tref +τ /2. We observe that the one time statistics of

pAij (t)Aij (t)aiidoes not show signiﬁcant variations with time.

The dependence of pAij Aij on aii is monotonic and alomost linear for aii >0. A

faster expanding ﬂuid particle (large positive aii) is associated with smaller pAij 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 high

negative aii) have a very large pAij Aij . As dilatation decomes less negative pAij Aij

drops ﬁrst (till aii ≈0.25) and then again tends to be larger. Thus the dependence

of pAij 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, τκis 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 ψand 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 inital dilatation tend to acquire

lower positive dilatation levels and the associated pAij 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 )iwill 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 to be more complicated because of non-monotonic and highly

non-linear distribution of pAij Aij |aii(Figure 9a). At low negative dilatation (say

aii ≈0.25), the dominant tendency of particles is to move towards zero dilatation (Figure

9a). The association of higher pAij Aij at zero dilatation comare to that at low negative

dilatation still allows hrA(t, tref )|aiiito show a substantial magniﬁcation at early times

(Figure 7a). However, the relatively value of pAij 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 )|aiiiof particles with small negative

dilatation when compared to peak value of hrA(t, tref )|aiiiof particles with small positive

dilatations (compare the curve of aii=−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 towards zero dilatation.

However, Figure 9a suggests that as their dilatation reduces, pAij Aij severely drops

(almost exponential drop). This tendency, combined with the fact that initially also they

had a high value of pAij Aij , results into dramatic drop in hrA(t, tref )|aiiias observed

in Figure 7a. In summary, in this section we demonstrated that even though the viscous

Lagrangian statistics in compressible turbulence 15

02468

(t−tref )/τ

0.5

1.5

-ν∂2A

∂xk∂xk

-t

-ν∂2A

∂xk∂xk)-

-tref

Mt= 0.55

Mt= 0.40

Mt= 0.25

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

0 1 2 3 4 5 6 7

(t−tref )/τ

aii =−1.00

aii =−0.75

aii =−0.50

aii =−0.25

aii = 0.00

(a)

0 1 2 3 4 5 6 7

(t−tref )/τ

aii = +1.00

aii = +0.75

aii = +0.50

aii = +0.25

aii = 0.00

(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 in contracting and expanding particles is

attributable to the vastly diﬀerent values of pAij 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

0 1 2 3 4 5 6 7

(t−tref )/τ

UNSS

SNSS

SFS

UFC

UFS

SFC

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

0 1 2 3 4 5 6 7

(t−tref )/τ

0.2

0.4

0.6

0.8

1.2

1.4 aii =−1.00

aii =−0.75

aii =−0.50

aii =−0.25

aii = 0.00

(a)

0 1 2 3 4 5 6 7

(t−tref )/τ

0.2

0.4

0.6

0.8

1.2

1.4

1.6 aii = +1.00

aii = +0.75

aii = +0.50

aii = +0.25

aii = 0.00

(b)

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

0 1 2 3 4 5 6 7

(t−tref )/τ

0.2

0.4

0.6

0.8

1.2

1.4

UNSS

SNSS

SFS

UFC

UFS

SFC

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

Lagrangian statistics in compressible turbulence 17

-1 -0.5 0 0.5 1

aii

1000

3000

5000

7000

pAij Aij

Mt= 1.5

Mt= 1.0

(a)

-1 -0.5 0 0.5 1

aii

0.5

1.5

pΥIij ΥIij

×10 7

Mt= 1.5

Mt= 1.0

(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:

ν∂2Aij

∂xk∂xk

=ν∂

∂xk∂Xm

∂xk

∂Aij

∂Xm,(5.5)

where, Xiand xiare Eulerian and Lagrangian spatial co-ordinates. Further expansion

of RHS of Equation 5.5 leads to:

ν∂2Aij

∂xk∂xk

=ν∂Xn

∂xk

∂Xm

∂xk

∂2Aij

∂Xm∂Xn

| {z }

+ν∂Aij

∂Xm

∂2Xm

∂xk∂xk

| {z }

.(5.6)

(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∂2Aij

∂xk∂xk≈ν∂Xn

∂xk

∂Xm

∂xk

∂2Aij

∂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

∂xk

∂Xm

∂xk

=D−1

kn D−1

km (5.10)

∂Xn

∂xk

∂Xm

∂xk

= (DkmDkn)−1(5.11)

ν∂2Aij

∂xk∂xk≈νC−1

∂2Aij

∂Xm∂Xn

(5.12)

(5.13)

Further, Jeong & Girimaji (2003) make the second modelling assumption wherein C−1

is approximated as an isotropic tensor:

ν∂2Aij

∂xk∂xk≈νC−1

3δmn

∂2Aij

∂Xm∂Xn

(5.14)

ν∂2Aij

∂xk∂xk≈νC−1

∂2Aij

∂Xm∂Xm

(5.15)

(5.16)

Finally, the third approximation is made ( ∂2Aij

∂Xm∂Xm=Aij

δX2)

ν∂2Aij

∂xk∂xk≈νC−1

Aij

(δX)2(5.17)

ν∂Aij

∂xk∂xk≈C−1

3τν

Aij (5.18)

Lagrangian statistics in compressible turbulence 19

01234567

t/τ

10 ×10 6

hrm(t, tref )i

hr(t, tref )i

(a)

01234567

t/τ

100

hrm(t, tref )i

hr(t, tref )i

(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.

0 1 2 3 4 5 6 7

t/τ

hr(t, tref )i

hrA(t, tref )i

(a)

0 1 2 3 4 5 6 7

t/τ

hr(t, tref )i

hrA(t, tref )i

(b)

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

The evolution equation of the deformation gradient tensor (Dij ) is:

dDij

dt =DikAkj .(5.19)

τνis the molecular viscous relaxation time scale, deﬁned as:

τν=δX2/ν ≈λ2

τ/ν, (5.20)

where, λτis 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

that the coeﬃcient C−1

kk of the LLDM model grossly overestimates the inﬂuence of the

C−1tensor.

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

0 0.5 1 1.5 2

(t−tref )/τ

0.2

0.4

0.6

0.8

Rγ

Rβ

Rα

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−1tensor. The C−1tensor is always symmetric (since

C=DDTis symmetric), hence the eigenvalues of C−1are 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,Rβand Rγas a

function of time. We observe that the eigenvalues begin to depart from each other. Rα

increases appreciably during the evolution phase, while Rβand Rγdecreases to negligible

values. This indicates that the C−1tensor 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

recovered by assuming the 4th order tensor ∂2Aij

∂Xm∂Xnto be isotropic. However, it is not

possible to ﬁnd the eigen-values of this tensor directly. By lagrangian change of variables,

∂2Aij

∂Xm∂Xncan be approximated as:

∂2Aij

∂Xm∂Xn≈xm

∂2Aij

∂xm∂xn

(5.21)

Since a product of an anisotropic tensor with an isotropic tensor is always anisotropic,

we rather evaluate the eigenvalues of ∂2Aij

∂xm∂xntensor. Since, this tensor is of order 4, we

dissociate the tensor into 9 second-order tensors and analyze their eigenvalues. Since,

∂2Aij

∂xm∂xnis 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 γeigenvalues 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

in terms of Rα, Rβand Rγin Figure 13. Clearly the ∂2Aij

∂xm∂xntensor is anisotropic with

ratio of the eigenvalues as: α:β:γ:: 1.9 : 1.0 : 1.1. Hence, it can be concluded that the

∂2Aij

∂Xm∂Xntensor 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

C−1and ∂2Aij

∂Xm∂Xntensor is one of the major cause of the unrealistic behavior of the

LLDM model term as compared to the exact viscous term.

Lagrangian statistics in compressible turbulence 21

0 0.5 1 1.5 2 2.5 3 3.5

(t−tref )/τ

0.2

0.4

0.6

0.8

Figure 13. Evolution of locally normalized real part of eigenvalues of ∂2Aij

∂xm∂xn. Three

diﬀerent colors of the line plot viz. black, light gray, dark gray represents Rα, Rβand 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

∂2Aij

∂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 p-

q-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-Dur´an 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

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

-0.1 -0.05 0 0.05 0.1

-0.4

-0.2

0.2

0.4

(a)

-0.1 -0.05 0 0.05 0.1

-0.4

-0.2

0.2

0.4

(b)

-0.1 -0.05 0 0.05 0.1

-0.4

-0.2

0.2

0.4

(c)

-0.1 -0.05 0 0.05 0.1

-0.4

-0.2

0.2

0.4

(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.

012345





0.05

0.25

RMS

(a)

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

-0.4

-0.2

0.2

0.4

high concentration

(Vieillefosse tail)

(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 23

100% UNSS 100% SNSS 100% SFS 100% UFC

UNSS SNSS SFS UFC UNSS SNSS SFS UFC UNSS SNSS SFS UFC UNSS SNSS SFS UFC

tref 100 0 0 0 0 100 0 0 0 0 100 0 0 0 0 100

tref + 3τ27 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.

UNSS SNSS SFS UFC

% anticlockwise 32 51 33 33

% clockwise 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 eddy-

turnover 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

0123

t−tref

Topology %

UNSS

SNSS

SFS

UFC

UFS

SFC

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 Lκfor 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=

i=1

lκi

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 −→ UF C −→ UN SS −→ SN SS.

Table 6 shows the average velocity (|Upqr |=|∂p

∂t ˆp+∂q

∂t ˆq+∂r

∂t ˆr|) 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 25

UNSS SNSS SFS UFC

average velocity |Uqr |0.23 0.28 0.23 0.26

(units: s−1)

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

∂t ˆp+∂q

∂t ˆq+∂r

∂t ˆr) for simulation

case A.

-0.1 -0.05 0 0.05 0.1

-0.4

-0.2

0.2

0.4

SNSS UNSS

Large particle concentration

(Vieillefosse tail)

UFC

SFS

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

-2 -1 0 1 2

aii

10 -4

10 -2

10 0

10 2

PDF

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 −→ UNSS −→ UF C −→

SNSS. 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 compressibil-

ity are present in the ﬂow ﬁeld. However, statistically, by looking at the probability-

density-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 −→ UNSS −→ UF C −→ SNSS.

Further, at very high Mach numbers the order stability based on life-time of existence

is found to be as follows:

SF S −→ UNSS −→ UF C −→ UF S −→ SNSS −→ SF C −→ UNUNUN −→ SNSNSN.

In order to explain the variation of lifetime of topology with Mt, we report the

Lagrangian statistics in compressible turbulence 27

Simulation UNSS SNSS SFS UFC UFS SFC SNSNSN UNUNUN

case A Sample % 25.2 5.4 43.5 25.9 0 0 0 0

M=0.075 Life of topology 1.80 0.53 3.32 2.08 0 0 0 0

R = 70 Life % 23.32 6.86 42.95 26.91 0 0 0 0

case E Sample % 24.68 8.35 33.78 21.67 5.17 4.53 0.09 0.09

M=0.6 Life of topology (κτ) 1.31 0.44 1.80 1.15 0.28 0.24 0.05 0.05

Re=350 Life % 24.62 8.27 33.83 21.62 5.26 4.51 0.01 0.01

case F Sample % 27.81 10.26 28.42 20.88 8.44 3.89 0.14 0.15

M=1 Life of topology (κτ) 1.04 0.31 1.13 0.85 0.35 0.24 0.06 0.11

Re=150 Life % 25.43 7.58 27.63 20.78 8.56 5.87 1.47 2.69

case G Sample % 26.61 9.82 26.26 21.50 8.39 4.14 0.14 0.14

M=1 Life of topology (κτ) 1.01 0.31 1.18 0.86 0.35 0.25 0.06 0.10

Re=100 Life % 24.51 7.52 28.64 20.87 8.50 6.07 1.46 2.43

case H Sample % 26.52 9.45 29.96 21.52 8.08 4.23 0.13 0.10

M=1 Life of topology (κτ) 1.00 0.29 1.20 0.86 0.33 0.24 0.06 0.08

Re=70 Life % 24.63 7.14 29.56 21.18 8.13 5.91 1.48 1.97

case I Sample % 26.07 10.10 27.44 21.16 10.49 4.32 0.23 0.21

M=1.5 Life of topology (κτ) 0.87 0.26 0.94 0.74 0.35 0.23 0.07 0.09

Re=70 Life % 24.51 7.32 26.48 20.85 9.86 6.48 1.97 2.54

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

particles.

Simulation UNSS SNSS SFS UFC UFS SFC SNSNSN UNUNUN

A 0.23 0.28 0.23 0.26 - - - -

E 0.58 0.88 0.60 0.66 0.62 0.96 2.6 1.80

F 0.86 1.39 0.95 1.00 0.82 1.55 2.15 1.33

G 0.84 1.35 0.92 0.99 0.80 1.49 2.35 1.21

H 0.88 1.37 0.97 1.03 0.87 1.46 1.87 1.37

I 1.01 1.57 1.13 1.18 0.90 1.76 1.94 1.27

Table 8. Average velocity of particles in p-q-r space (|Upqr |=|∂p

∂t ˆp+∂q

∂t ˆq+∂r

∂t ˆr|) for

simulations A and E-I.

magnitude of mean velocities (Upqr =∂p

∂t ˆp+∂q

∂t ˆq+∂r

∂t ˆr) of particles in p-q-r space in

Table 8. It can be seen that |Upqr|increases with Mtfor 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 Mttheir 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 Lκfor various topologies with initial dilatation

(aii). To explain the variation in Lκwith 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 ofLκwith initial dilatation

for UFS and SFC topology are shown in Figure 19(e) and 19(f) respectively. UFS and SFC

topologies have smaller Lκand 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 29

-1 -0.5 0 0.5 1



0.5

1.5



(a)

-1 -0.5 0 0.5 1



0.5

1.5



(b)

-1 -0.5 0 0.5 1



0.5

1.5



(c)

-1 -0.5 0 0.5 1



0.5

1.5



(d)

0 0.5 1



0.5

1.5



(e)

-1 -0.5 0



0.5

1.5



(f)

Figure 19. Variation of life of topology Lκτwith 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

(a)

-0.2 -0.1 0 0.1 0.2

-0.5

0.5

(b)

-0.2 -0.1 0 0.1 0.2

-0.5

0.5

(c)

-0.2 -0.1 0 0.1 0.2

-0.5

0.5

(d)

-0.2 -0.1 0 0.1 0.2

-0.5

0.5

(e)

-0.2 -0.1 0 0.1 0.2

-0.5

0.5

(f)

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). Well-

resolved 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.

We found that the time evolution of the exact viscous term 

∂2Aij

∂xk∂xkshows a two-

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

C−1and the ∂2Aij

∂Xm∂Xntensor.

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 −→ UF C −→ UN SS −→ SNSS

(ii) Mildly Compressible:

SF S −→ UNSS −→ UF C −→ SNSS −→ U F S −→ SF C −→ UNUNUN −→ SN SNSN.

(iii) Highly Compressible:

SF S −→ UNSS −→ UF C −→ UF S −→ SNSS −→ SF C −→ UNUNUN −→ SNSNSN.

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 High-

Performance 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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