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Lagrangian investigations on velocity

gradients in compressible turbulence:

Examination of viscous process and ﬂow-ﬁeld

topology

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)

We perform Lagrangian investigations (following ﬂuid particles) of the dynamics of

velocity gradients in compressible decaying turbulence. Speciﬁcally, we examine (i) the

evolution of the viscous process in the governing equation of the velocity gradient

tensor, and (ii) evolution of the invariants of the velocity gradient tensor. Well-resolved

direct numerical simulations over a range of Mach and Reynolds number along with a

Lagrangian particle tracker are employed for our investigations. We ﬁnd that an increase

in the initial turbulent Mach number tends to intensify the viscous process following ﬂuid

particles. We provide evidence and explain that this intensiﬁcation is attributable to the

development of a large disparity in the magnitude of the velocity gradients associated

with contracting and expanding ﬂuid particles combined with the overall preferences

of the these contracting and expanding ﬂuid particles to change their dilatation rate.

Further, we examine the performance of the so-called linear Lagrangian diﬀusion model

(LLDM) of the viscous process. Subsequent to identifying the shortcomings of the model,

we propose some suggestions for model improvement as well. In the second part of the

study, we employ our DNS results to examine the trajectories of ﬂuid particles in the

the space of the invariants of the velocity gradient tensor. Such an examination allows us

to accurately measure the lifetimes of major topologies of compressible turbulence and

provide explanation why some selective topologies tend to exit longer than the others.

1. Introduction

Gradients of the small-scale velocity ﬁeld and its dynamics in a turbulent ﬂow inﬂuence

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 the subject of interest employing experimental

measurements (L¨uthi et al. 2005), direct numerical simulations (DNS) (Ashurst et al.

1987b), as well as simple autonomous dynamical models (ordinary diﬀerential equations)

(Vieillefosse 1982; Cantwell 1992) of velocity gradients. The pioneering work done by

the cited 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 (Soria et al. 1994; Pirozzoli & Grasso

†Email address for correspondence: nishantparashar14@gmail.com

2N. Parashar, S. S. Sinha and B. Srinivasan

2004; Suman & Girimaji 2009, 2010b, 2012; Wang & Lu 2012; Chu & Lu 2013; Vagheﬁ

& Madnia 2015; Danish et al. 2016a; Bechlars & Sandberg 2017; 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 per-

formed using the Eulerian approach. However, it is desirable to investigate various

ﬂow physics following individual ﬂuid particles (the Lagrangian tracking) as well. Such

an investigation is especially required from the point of view of developing/improving

simple dynamical models of the velocity gradients 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 of an initial reference time in order to cause intense

vortex stretching, and the alignment tendency as shown by the Eulerian one-time ﬁeld

(with the instantaneous intermediate eigenvector) was merely a transient and incidental

picture.

In incompressible ﬂows, Lagrangian studies using the direct numerical simulation

of decaying turbulence have earlier been performed by Yeung & Pope (1989) and ?.

(see ref. 48 in 2008chevillard). Yeung and Pope Yeung & Pope (1989), focused on

Lagrangian statistics of velocity, acceleration, and dissipation. ?examined the evolution

of material elements in incompressible decaying turbulence. Recently, Xu et al. (2011)

have complemented their experimental observations of vorticity alignment with the

Lagrangian data extracted from DNS of forced isotropic turbulence as well. 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 re-

cently 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 behavior in terms of the dynamics of the inertia tensor of ﬂuid

particles and conservation of angular momentum of tetrads representing 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 viscous process and

the role of the deformation gradient tensor in it, and (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 modeled the

viscous process using the gradient-diﬀusion hypothesis, wherein diﬀusion is allowed to

Lagrangian statistics in compressible turbulence 3

be ampliﬁed as a 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 some 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 ﬁrst examine

the inﬂuence of compressibility on the exact viscous process in the Lagrangian dynamics

of velocity gradients, and subsequently subject the LLD modeling approach to a direct

scrutiny by comparing its Lagrangian evolution history against that of the exact process

it represents. 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. The inﬂuence of compressibility on the exact viscous process

is parameterized in terms of Mach number and dilatation rate. Finally, we also propose a

new modiﬁed model of the viscous process to address the identiﬁed shortcomings of the

original LLDM of Jeong & Girimaji (2003).

In the second part of this work, we examine the evolution of ﬂow-ﬁeld topology in

compressible turbulence following the Lagrangian trajectories of the invariants of the

velocity gradient tensors. Topology can also be visualized as the local streamline pattern

as observed with respect to a reference frame which is translating with the center of mass

of a local ﬂuid particle (Chong et al. 1990). Topology actually depends on the nature of

eigenvalues of the velocity gradient tensor (VGT), 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 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 a 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), who employed merely one-time velocity gradient data of the entire ﬂow ﬁeld

and computed bin-averaged rates-of-change of second and third invariants of VGT 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 space of VGT invariants. The authors called these trajectories as conditional

mean trajectories (CMT) and used them as a surrogate approach to study invariant

dynamics. Subsequently, several authors have employed the CMTs to investigate various

aspects of topology dynamics 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 some 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 exact Lagrangian dynamics in compressible turbulence must be per-

formed, 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.Based on such a motivation, we identify the

following objectives for the second part of this work: (i) highlighting the diﬀerences,

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

between CMT and the exact 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 (0.01, 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 §??).

This paper is organized into seven sections. In §2 we present the governing equa-

tions. In §3 we provide details of our direct numerical simulations and the Lagrangian

particle tracker. In §?? we explain our study plan. In §4 we examine the inﬂuence of

compressibility on the viscous processes of the velocity gradient dynamics, evaluate the

LLD model and propose modiﬁcations in the model to address its shortcomings. In §5 we

study the dynamics of topologies, compare CMT and LT and quantify lifetime of various

ﬂow-topologies existing in compressible turbulence. Section 6 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)

∂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

Lagrangian statistics in compressible turbulence 5

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

Our direct numerical simulations of nearly incompressible and compressible decaying

turbulence are performed using the gas kinetic method (GKM). The gas kinetic method

(GKM) was originally developed by Xu et al. (1996) and has been shown to be quite

robust in terms of numerical stability. Further, GKM has the ability to capture shocks

without numerical oscillations. Several workers have employed GKM for simulating

compressible decaying 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)

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 a 4th order accurate weighted-

essentially-non-oscillatory (WENO) method for interpolation of ﬂow variables. Our solver

has been extensively validated with established DNS results of compressible turbulent

ﬂows (Danish et al. 2016a). In total, this study employs eight diﬀerent simulations

(Simulations A-H). Descriptions of these simulations are presented in Table 1.

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

0 2 4 6 8

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

In Figure 1(a) we present evolution of turbulent kinetic energy (k) observed in Simula-

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

(SV) deﬁned as (your expressions are not meaningful - what does the index i represent–

please be careful- such things annoy reviewer and the reader alike. Please pay attention

to such details..if we want our work to be published we will have to pay attention to such

details):

SVi=∂V 0

∂xi3

∂V 0

∂xi23/2,(3.4)

SV=SV1+SV2+SV3

3.(3.5)

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.

2006):

τ=λ0

; (3.6)

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,(3.7)

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 3.7 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.

Lagrangian statistics in compressible turbulence 7

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 25630.0065 4

G. 70 1.5 25630.0094 4

Table 1. Initial parameters of DNS simulations.

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. Study I: Lagrangian investigations on the viscous process of

velocity gradients

In this section we ﬁrst examine the inﬂuence of compressibility on the exact process of

velocity gradient dynamics (Section 4.1). Following a similar methodology, in Section 4.2

we examine the performance of the LLD model of Jeong & Girimaji (2003). Subsequently,

in Section ?? we propose modiﬁcation in the LLD model to address its shortcomings.

4.1. DNS-based examination of viscous process

The viscous process Υin the evolution equation of the velocity gradient tensor (Equa-

tion 2.7) is:

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 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/or HEE of Suman

& Girimaji (2009)), each of the three viscous terms (ΥIij ,ΥIIij and ΥIIIij ) represents a

non-local, unclosed process.

To gauge the relative importance of these three constituent viscous processes, 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

.(4.2)

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

Any of these fractions approaching unity can be used as an evidence of the corresponding

process to be the dominant one. Using the ﬂow-ﬁeld at peak dissipation time from several

simulations (A-H, 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 synonymous

with Υij .

Our interest is to examine how the viscous process ΥIij undergoes change 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 ):

r(t, tref ) = pΥIij (t)ΥIij (t)

pΥIij (tref )ΥIij (tref ),(4.3)

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

ﬂuid particle at an arbitrary time tand at the reference time tref , respectively. Since

an individual particle represents just one realization, we obtain 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.

In this work ﬂuid compressibility is parametrized based on the initial turbulent Mach

number (3.3) and the locally normalized dilatation rate. The normalized dilatation rate

(aii) is the trace of the locally normalized velocity gradient tensor, which is deﬁned as:

aij =Aij /pAmnAmn.(4.4)

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 (4.5)

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 element,

a negative value of aii implies a contracting ﬂuid element. A ﬂuid particle with instan-

taneous aii = 0 implies a volume preserving ﬂuid element. While the turbulent Mach

number is a global/statistical indicator of compressibility, the normalized dilatation is a

local parameter and thus it aptly represents the inﬂuence of compressibility on a speciﬁc

local ﬂuid particle. Note that aii is algebraically bounded between ±√3.

To understand the inﬂuence of initial Mt, in Figure 2 we present Lagrangian mean

hr(t, tref )ifrom simulations E-G. These simulations have identical initial Reynolds num-

ber (70) but diﬀerent initial Mach numbers (1.00, 1.25, 1.50) In each of these simulations,

the exact viscous process shows a two-stage evolution. In the ﬁrst stage, hr(t, tref )i

increases and reaches a peak value. In the second stage, it decays a monotonic decay.

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 almost match

(tpeak−dissipation ≈0.8τ). The ampliﬁcation in the ﬁrst stage can be attributed to the

steepening of gradients due to the rapid spread of the spectrum. The decay in the second

Lagrangian statistics in compressible turbulence 9

0 1 2 3 4 5 6 7

t/τ

hr(t, tref )i

Figure 2. Mach number dependence on evolution of exact viscous term.

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 compressibility parameter like Mt, the same statistics when

conditioned upon appropriate local compressibility parameters like normalized dilatation

(aii) reveal, signiﬁcant variations and insightful physics. Examining the inﬂuence of local

parameters like aii 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. Thus to gain further insight, into the viscous diﬀusion process (ΥIij ), we subject

hr(t, tref )ito conditioned averaging on discrete values of aii at tref .

In Figure 3 we present conditional Lagrangian mean of hr(t, tref )ifrom Simulation

G, with aii as the conditioning parameter. The reference time is chosen to be

xx eddy-turnover time. In Figure 3a, separate curves are presented for aii =

−1.0,−0.75,−0.50,−0.25 and 0, whereas in Figure 3b 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, the conditional mean,hr(t, tref )|aiii, 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 )ihappens (Figure 3) seems to be strongly

aﬀected by the value of aii that a ﬂuid particle has at t=tref . In Figure 3a 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 )|aiii

reduces. On the other hand, in Figure 3b we observe the opposite trend. For ﬂuid

particles with high positive dilatation (fast expanding ﬂuid particles) the peak value of

hr(t, tref )|aiiitends to increase.

In our attempt to understand and explain the behavior observed in Figure 3a, 3b in

Figure 4a and 4b 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 3. We measure this ampliﬁcation as:

hrA(t, tref )i=Aij (t)Aij (t)

Aij (tref )Aij (tref ).(4.6)

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

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

0 1 2 3 4 5 6 7

(t−tref )/τ

+r(t, tref )-

-aii,

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

+r(t, tref )-

-aii,

aii = +1.00

aii = +0.75

aii = +0.50

aii = +0.25

aii = 0.00

(b)

Figure 3. Dependence of dilatation rate on evolution of exact viscous term for simulation G

at tref = 0.5τ

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 dilatation tends to lower the peak of hrA(t, tref )|aiii. This similarity in the trends

shown by hr(t, tref )|aiiiin Figure 3 and that shown by hrA(t, tref )|aiiiin Figure 4 is not

completely unexpected, and it indeed substantiates a gradient diﬀusion like hypothesis

which assumes ∆A ∝A(Mart´ın et al. 1998; Jeong & Girimaji 2003). However, in the

light of Figure 4a, 4b our primary curiosity that why the peaks of hr(t, tref )i(in Figure

3a, 3b) rise for expanding ﬂuid particles and reduce for contracting ﬂuid particles can

now possibly be explained based on the behavior of Aij itself. In Figure 5 we show

pAij (t)Aij (t)aiiat three time instants, tref ,tref +τ/4 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 almost 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 behavior. Extremely fast contracting particles (very high

negative aii) have a very large pAij Aij . As dilatation becomes 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 5). In Figure

6a we present averaged rate of change in dilatation (ψ) of tagged ﬂuid particles over one

Kolmogorov time:

ψ= (aii(τκ+tref )−aii(tref )) /τκ,(4.7)

where, τκis the Kolmogorov time at tref .

In Figure 6b 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 initial

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

Lagrangian statistics in compressible turbulence 11

0 1 2 3 4 5 6 7

(t−tref )/τ

0.2

0.4

0.6

0.8

1.2

1.4

+rA(t, tref )-

-aii,

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

+rA(t, tref )-

-aii,

aii = +1.00

aii = +0.75

aii = +0.50

aii = +0.25

aii = 0.00

(b)

Figure 4. Dependence of dilatation rate on evolution of pAij Aij for simulation G at

tref = 0.5τ.

dilatation than that at lower positive dilatation. Indeed this behavior is observed in

Figure 3a. For contracting particles the dynamics seem to be more complicated because

of non-monotonic and highly non-linear distribution of pAij Aij |aii(Figure 5). At

low negative dilatation (say aii ≈0.25), the dominant tendency of particles is to

move towards zero dilatation (Figure 5). The association of higher pAij Aij at zero

dilatation compare to that at low negative dilatation still allows hrA(t, tref )|aiiito

show a substantial magniﬁcation at early times (Figure 4a). 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 5) 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 4b to the curve of aii =−0.25 in Figure 4a).

For faster contracting particles (say those with initial aii ≈ −0.75), Figure 5 shows

that (like other contracting particles) they begin their journey towards zero dilatation.

However, Figure 5 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 4a.

In the light of the foregoing discussion we now make an attempt to explain the

inﬂuence of initial turbulent Mach number observed in Figure 2. As the initial turbulent

Mach number is increased (Simulations xx, yy and zz), more particles tend to have

non-zero dilatation, and also, larger magnitudes of normalized dilatation are generated.

In Figure xx we show pdf of dilatation at the peak dissipation time in Simulation xx,

yy and zz.We observe that (i) the population of fast expanding particles and the fast

contracting particles increase as the initial turbulent Mach number is increased, and

(ii) the population distribution is almost symmetric on the positive and the negative aii

sides. In Figure 3 we have already observed that that expanding particles tend to undergo

much more ampliﬁcation os r(t, tref) than the contracting particles (for example compare

the peak of hr(tref ))|aii = +0.75iin Figure 3a and the peak of hr(tref ))|aii =−0.75iin

Figure 3b). Thus at a higher Mach number, the higher ampliﬁcation achieved by the

particles with high positive dilatation seem to oﬀset the lower ampliﬁcation of the viscous

process in the contracting particles resulting into a net increase in the peak value of the

overall hr(tref )ias evident in Figure 2.

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

-1 -0.5 0 0.5 1

aii

1000

3000

5000

7000

pAij Aij

t=tref

t=tref +τ/4

t=tref +τ/2

Figure 5. Variation in magnitude of Aij tensor with dilatation aii for simulation G

(tref = 0.5τ).

-1 -0.5 0 0.5 1

aii

-0.5

0.5

(a)

-1 -0.5 0 0.5 1

aii

0.2

0.4

0.6

0.8

(b)

Figure 6. Variation of rate of change of dilatation ψwith dilatation over one Kolmogorov time

(at tref = 0.5τ) expressed as: a) average rate of change of dilatation b) Fraction of particles

moving towards +ve aii (symbol +) and -ve aii (symbol o).

4.2. Evaluation of the LLD model

Having examined the behavior of the exact process ΥIij following ﬂuid particles, now

we examine the performance of the LLD model of Jeong & Girimaji (2003), which intends

to capture the essential physics of this exact process. The primary motivation of Jeong

& Girimaji (2003) to develop the LLD model was to ensure that the viscous process is

large enough to eliminate the ﬁnite time singularity problem seen earlier in the restricted

Euler dynamics Cantwell & Coles (1983). While the LLD model was found to achieve

this requirement by quickly amplifying the modeled expression of the viscous action using

the trace of the Cauchy-Green tensor, the exact nature of its evolution and other time-

dependent aspects remain questionable - especially its anticipated exponential growth

at late times (Chevillard et al. 2003). In this work we pursue a detailed examination of

the behavior of the LLD model following ﬂuid particles and employing the exact DNS

ﬂow ﬁeld in the background. For this examination, we use the results of two simulations:

xx and yy. Both these simulations have the same initial Reynolds number (150) but

Lagrangian statistics in compressible turbulence 13

diﬀerent Mach numbers. Simulation xx with initial Mt≈0.01 can be treated as almost

incompressible, while Simulation yy has considerable compressibility with initial Mach

number being xx.

The LLD modeling approach of Jeong & Girimaji (2003) uses Lagrangian-Eulerian

change in variables to cast ΥIij as:

ν∂2Aij

∂xk∂xk

=ν∂

∂xk∂Xm

∂xk

∂Aij

∂Xm,(4.8)

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

of rhs of Equation 4.8 leads to:

ν∂2Aij

∂xk∂xk

=ν∂Xn

∂xk

∂Xm

∂xk

∂2Aij

∂Xm∂Xn

| {z }

+ν∂Aij

∂Xm

∂2Xm

∂xk∂xk

| {z }

.(4.9)

(4.10)

Jeong & Girimaji (2003) neglects term B (ﬁrst modeling assumption) on the RHS of

Equation 4.9 to arrive at the following equation:

ν∂2Aij

∂xk∂xk≈ν∂Xn

∂xk

∂Xm

∂xk

∂2Aij

∂Xm∂Xn

.(4.11)

(4.12)

Using the deﬁnition of the deformation gradient tensor Dij =∂xi

∂Xiand the right Cauchy-

Green tensor Cij =DkmDkn (Equation ??), the RHS of Equation 4.11 can be expressed

in terms of the tensor C:

∂Xn

∂xk

∂Xm

∂xk

=D−1

kn D−1

km (4.13)

∂Xn

∂xk

∂Xm

∂xk

= (DkmDkn)−1(4.14)

ν∂2Aij

∂xk∂xk≈νC−1

∂2Aij

∂Xm∂Xn

(4.15)

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

is approximated as an isotropic tensor:

ν∂2Aij

∂xk∂xk≈νC−1

3δmn

∂2Aij

∂Xm∂Xn

(4.16)

ν∂2Aij

∂xk∂xk≈νC−1

∂2Aij

∂Xm∂Xm

(4.17)

(4.18)

Finally, the third approximation is made ( ∂2Aij

∂Xm∂Xm≈ −Aij

τL) leading to the following

equation:

ν∂2Aij

∂xk∂xk≈ − 1

τL

C−1

3Aij .(4.19)

(4.20)

Jeong & Girimaji (2003) consider the quantity τLto be a constant and interpret this

as a molecular viscous relaxation time scale. This model has been employed by Jeong

& Girimaji (2003) in the restricted Euler equation (REE) model. Recently, Suman &

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

Girimaji (2012), Danish et al. (2014) have employed it to capture the physics of viscous

diﬀusion process in the enhanced Homogenized Euler equation (EHEE) model, which is

the counterpart of REE for compressible ﬂows.

While introducing their recent ﬂuid formation closure hypothesis, ?employed almost

the same ﬁnal form for the viscous process as proposed by Jeong & Girimaji (2003),

however, they rationalized the model using a diﬀerent set of arguments. First, instead of

assuming the C−1tensor to be isotropic, the authors assumed the fourth-order tensor

∂2Aij

∂Xm∂Xn(Lagrangian Hessian of the tensor A) to be isotropic, and thereafter expressed

it in terms of the tensor Aij and a length scale δX associated with the ﬂow ﬁeld at tref :

∂2Aij

∂Xm∂Xm≈ −C−1

mnAij

(δX)2

δmn

3.(4.21)

Subsequently, equation 4.21 when combined with 4.15 led to the following form:

ν∂2Aij

∂xk∂xk≈ −νC−1

Aij

(δX)2,(4.22)

?interpret δX as the characteristic length scale that a ﬂuid particle traverses over

a Kolmogorov timescale, and thus δX ≈λ, where λis the Taylor-microscale of the

turbulent ﬂow ﬁeld at the reference time tref . Thus, the ﬁnal version of the LLD model

takes the form:

ν∂2Aij

∂xk∂xk≈ − ν

λ2

C−1

3Aij ,(4.23)

As mentioned in the Introduction, with access to DNS data of decaying turbulence

and a validated Lagrangian particle tracker, we intend to scrutinize the performance of

this model (4.23). For this study we select the same set of particles that we used in §4.1

to examine the exact ΥIij process. We follow the same particles and calculate C−1

kk ν

3λ2Aij

of each particle. Like the exact process (Equation ??), we deﬁne the ampliﬁcation ratio

of the viscous process modeled process as:

rm(t, tref ) = hν

λ2

C−1

3pAij Aij it

hν

λ2

C−1

3pAij Aij itref

(4.24)

At each time instant the ratio rm(t, tref ) is calculated, and subsequently the mean value

hrm(t, tref )iis computed by taking averages across the selected set of particles. A direct

comparison of hrm(t, tref )iis then performed against hr(t, tref )i. Note that to ﬁnd C−1

kk at

any arbitrary times we use the following exact evolution equation of deformation gradient

tensor Dij (Jeong & Girimaji 2003):

dDij

dt =DikAkj ,(4.25)

where, both Aand Dare calculated using DNS data ﬁelds at diﬀerent time instants and

the Lagrangian particle tracker. Thus, our evaluation procedure uses the exact states of

the Aand the Ctensors.

In Figure 7a, we present evolution of both hrm(t, tref )iand hr(t, tref )iusing results of

simulation A (nearly incompressible with initial Mtbeing 0.01 and Reλ=150). In Figure

7b, results are shown using DNS data from Simulation G (highly compressible with initial

Mtbeing 1.5 and Reλ=150). In each case tref = 0. We observe that in both simulations,

unlike the evolution of the exact process, the LLD model shows monotonic growth with

Lagrangian statistics in compressible turbulence 15

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 7. Comparison of LLD model term and the exact viscous term: a) unscaled axis, b)

axis scaled to visualize the diﬀerence in growth rates of the two processes.

time. At the early stages of evolution, this monotonic growth is at least qualitatively

the same as the exact process. However, at later stages (after the dissipation peak) the

continued monotonic growth of hrm(t, tref )iis in gross disagreement with hr(t, tref )i,

which shows a decaying behavior in the second stage. Our results clearly show that

even though the LLD model may eliminate the problem of ﬁnite time singularity of the

restricted Euler equation (Jeong & Girimaji 2003), its growth, especially at late times,

is unrealistic and severely overestimates the strength of the viscous process.

In an attempt to diagnose the problems of the LLD model (4.23), we revisit the

LLD modeling procedure of Jeong & Girimaji (2003) and ?. To begin with, we avoid

the assumption made by ?regarding the isotropic structure of the fourth-order tensor

∂2Aij

∂Xm∂Xn(or the assumption made by Jeong & Girimaji (2003) of C−1being isotropic).

Instead, we go back to (4.21) and approximate the left hand side (lhs) as:

∂2Aij

∂Xm∂Xn≈Aij

(δX)2Rmn.(4.26)

In (4.26) the symbol Rmn represents the (m−n)th component of a second-order tensor,

which is no more necessarily an isotropic tensor as assumed by ?. Now combining (4.26)

with (refeq:lldm6mod), we arrive at:

ν∂2Aij

∂xk∂xk≈νC−1

∂2Aij

∂Xm∂Xn≈ −νC−1

Aij

(δX)2Rmn (4.27)

Equation 4.27 has so far been written in terms of the components of various tensors using

a coordinate system ﬁxed to the laboratory. The equation can be readily expressed in its

full tensor form as:

ν∇2A≈ − ν

(δX)2(C:R)A(4.28)

If we now express C−1and Rtensors in the eigen-system of the instantaneous C−1

tensor, then (4.28) can be expressed as:

ν∇2A≈ − ν

(δX)2(αRα+βRβ+γRγ)A(4.29)

where, α,βand γare the three eignevalues of the instantaneous C−1sorted as α>

16 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

hRγi

hRβi

hRαi

Figure 8. Evolution of locally normalized eigenvalues of C−1

β>γ, and these symbols when sub-scripted, imply the component of a tensor along the

corresponding eigenvector.

Our DNS results from both incompressible and compressible cases show that C−1

tensor evolves to a highly skewed state with its largest eigenvalue being overwhelmingly

dominant over the other two - Figure 8 show the instantaneously normalized eigenvalues

of C−1from Simulation G. The plotted quantities Tα,Tβand Tγare deﬁned as:

Tα=α2

α2+β2+γ2;

Tβ=β2

α2+β2+γ2;

Tγ=γ2

α2+β2+γ2; (4.30)

Similar ﬁndings from incompressible simulations have been reported earlier as well ?.

Further, since we expect magnitudes of all components of Rto be small, (4.29) can be

further approximated as:

ν∇2A≈ − ν

(δX)2(αRα+βRβ+γRγ)A≈ − ν

(δX)2(αRα)A(4.31)

Now, adopting the interpretation oﬀered by ?for the quantity δX ≈λ, (??) is expressed

as:

ν∇2A≈ − ν

λ2(αRα)A(4.32)

At this point we compare (4.32) and the form of the model used by ?(4.23). Taking into

account that the DNS behaviour shows |α|>> |β|,|γ|, the approximation made in (4.31)

can also be applied to (4.23). This will reduce (4.23) to:

ν∇2A≈ − ν

λ2α1

3A(4.33)

Thus, the essential diﬀerence between the modiﬁed model (4.33) and the original LLD

model (4.32) is that while the original model has already committed to Rαbeing 1/3,

the latter has not imposed any such restriction so far.

Given that αis known to grow exponentially, and the fact that (4.33) leads to gross

overestimation of growth of hrm(t, tref )i(Figure 7), we conjecture that the quantity Rα

must follow an exponential decay towards zero so as to restrain the unrealistic growth of

Lagrangian statistics in compressible turbulence 17

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).

hrm(t, tref )iin comparison to the expected behavior represented by hr(t, tref )i:

Rα≈B|α|−m(4.34)

The exponent m, however, is expected to be a time varying quantity dependent on the

instantaneous as well as the history of the ﬂuid particle being followed. The role of mis

to provide an additional exponential modulation, so that the exponential growth shown

by the original LLD model (4.23) displayed in Figure 7 can be reined in. At this point

we present 4.34 merely as a proposal. More detailed analysis and probably DNS data

over a wide range of Mach number and Reynolds number will be required to arrive at a

concrete functional form of m. Currently, however, such an eﬀort is outside the scope of

the present work.

5. Study II: Lagrangian investigation of dynamics of velocity

gradient invariants

The topology associated with a ﬂuid element is the local streamline pattern in its

vicinity observed with respect to a reference frame which is purely translating with the

center of mass of the ﬂuid element. Topology depends on the nature of eigenvalues of

the local state of the velocity gradient tensor. However, it can also be inferred with a

knowledge of the three invariants (P,Q,R) of the velocity gradient tensor :

P=−Aii, Q =1

2P2−Aij Aji, and

R=1

3−P3+ 3P Q −Aij AjkAki.(5.1)

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

tensor (aij ) are deﬁned in terms of the normalized velocity gradient tensor (aij ):

p=−aii, q =1

2p2−aij aji, and

18 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 9. 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).)

r=1

3−p3+ 3pq −aij ajkaki.(5.2)

Chen et al. (1989) categorizes topological patterns (Table 2) that can be observed

in an incompressible ﬁeld into unstable-node-saddle-saddle (UNSS), stable-node-saddle-

saddle (SNSS), stable-focus-stretching (SFS), and unstable-focus-compressing (UFC).

In compressible ﬂows, additional four more major topologies can exist: stable-focus-

stretching (SFS) and stable-node-stable-node-stable-node(SNSNSN), which are associ-

ated with contracting ﬂuid particles; and unstable-focus-stretching (UFS) and unstable-

node-unstable-node-unstable-node (UNUNUN), which are associated with expanding

ﬂuid particles. Figure 9 shows diﬀerent regions in the p−q−rspace associated with

diﬀerent topologies. Figure 10 present schematics of these topological patterns. The

reader is referred to Chong et al. (1990) 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 element, 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 expressed

Bechlars & Sandberg (2017):

dt =P2−2Q−Sii,

dt =QP −2P

3Sii −3R−Aij S∗

ji,

Lagrangian statistics in compressible turbulence 19

(a) (b)

(e) (f)

(g) (h)

Figure 10. 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 =−Q

3Sii +P R −P Aij S∗

ji)−AikAkj S∗

ji (5.3)

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

tion 2.7). The symbol S∗

ij is the traceless part of the Sij tensor, which is deﬁned as:

Sij =−Pij +Υij , and

S∗

ij =Sij −Skk

3δij .(5.4)

Here Pij is the pressure hessian tensor and Υij represents the viscous process in the

evolution equation of the velocity-gradient tensor (2.7).

The relationship between the non-normalized invariants (P,Q,R) and normalized in-

variants (p,q,r) is:

p=P

pAij Aij

, q =Q

Aij Aij

, and r =R

(Aij Aij )3/2(5.5)

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

Using equations (5.3 and the relationship (5.5), the evolution equation of normalized

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

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 ,

dt =d

dtR

(Aij Aij )3/2=1

(Aij Aij )3/2

dt −3R

(Aij Aij )5/2Aij

dAij

dt .(5.6)

While following an identiﬁed 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 as well. We

refer to such a trajectory of the ﬂuid particle in p-q-rspace as the Lagrangian trajectory

(LT).

5.1. Lifetime of topology

One of the central questions to address while studying the dynamics of velocity

gradients and ﬂow ﬁeld topology is how long a topology lasts and how compressibility

inﬂuences that. In this section we address this question. We quantify the lifetime of a

topology as the time it takes for a ﬂuid particle in to change its topology relative to the

topology it had at a reference time tref . We express this time non-dimensionalized by

the the Kolmogorov time scale τκof the homogeneous ﬂow ﬁeld at tref . We refer to this

normalized lifetime of a given topology (T) as LT:

LT=t∗−tref

τκ

; (5.7)

where t∗denotes the time instant when the original topology Tassociated with a tagged

ﬂuid particle changes to another topology, and τκdenotes the Kolmogorov time scale of

the ﬂow ﬁeld at tref . Correspondingly, the mean value of LTis calculated by following a

large number of tagged particles which have the same topology Tat tref :

hLTi=ht∗−tref i

τκ

.(5.8)

In all our calculations and analysis of lifetime of topologies we employ the data ﬁelds

of Simulation A-H, and in each case tref = 4τ. The sample sizes used for calculating

the mean lifetime of topologies ranges between 50,000 −300,000. To identify the rolw of

compressibility on lifetimes, we also examine hLTiconditioned upon discrete values of

aii of the reference time: hLT|aiii.

In Figure 11 we present mean lifetimes of various topologies conditioned upon diﬀerent

initial dilatation levels: hLT|aiii. Each sub-ﬁgure corresponds to a speciﬁc topology. The

six major topologies that exist in compressible turbulence are considered (UNSS, SNSS,

SFC, SFS, UFC and UFS). Further, to identify the role of initial turbulent Mach number

Mt, we have calculated hLT|aiiiusing Simulations xx, yy and zz. These simulations have

identical initial Reynolds number (70), but diﬀerent initial Mach number (1.00. 1.25 and

1.5).

We observe that the both compressibility parameters dilatation and initial Mach

numbers inﬂuence the lifetimes of some topologies selectively. Mean lifetimes of UNSS

(Figure 11a), UFC (Figure 11c) and SFS (Figure 11d) topologies seem to be signiﬁcantly

more sensitive to both the Mach number and dilatation comnapred to those of other

topologies. As the level of dilatation increases from high negative values to zero dilatation,

Lagrangian statistics in compressible turbulence 21

-1 -0.5 0 0.5 1

aii

0.5

1.5

hLT|aii i

(a)

-1 -0.5 0 0.5 1

aii

0.5

1.5

hLT|aii i

(b)

-1 -0.5 0 0.5 1

aii

0.5

1.5

hLT|aii i

(c)

-1 -0.5 0 0.5 1

aii

0.5

1.5

hLT|aii i

(d)

0 0.5 1

aii

0.5

1.5

hLT|aii i

(e)

-1 -0.5 0

aii

0.5

1.5

hLT|aii i

(f)

Figure 11. Variation of life of topology LTwith 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 B, C, D, E and G respectively.

lifetimes of these topologies increase. On the other side, however, as we move from zero to

positive dilatation, lifetimes seem to decrease again. An increase in initial Mach number

seem to pronounce these variations more as evident in Figures 11 a, c, and d. In contrast to

UNSS, UFC and SFS topologies the other three major topologies exiting in a compressible

ﬂow ﬁeld- SNSS, UFC and SFS seem to last for more-or-less the same time showing not

much sensitivity to either dilatation rate or Mach number.

To further understand the behavior observed in Figure 11, we investigate two prospec-

tive reasons which may inﬂuence the lifetime of a topology of a ﬂuid particle as it moves

in the p−q−rspace: (i) the actual volume available to a topology in the p−q−rspace,

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

p (−aii) UNSS SNSS SFS UFC UFS SFC SNSNSN UNUNUN

-ve 0.043 0.011 0.075 0.10 0.08 0 0 0.006

+ve 0.011 0.043 0.010 0.075 0 0.08 0.006 0

Table 3. Volume of available region for diﬀerent topologies in p-q-r space (non-dimensional).

and (ii) the velocity of the ﬂuid particles in the p−q−rspace. The velocity of a particle

in the p−q−rspace can be deﬁned as the rate at which its three invariants, p, q and r,

change with time. This rate is quantitifed as a velocity vector in the p−q−rspace. We

denote this velocity vector by ~

Upqr. This vector can be computed as:

Upqr =dp

dt ˆp+dq

dt ˆq+dr

dt ˆr; (5.9)

where dp

dt ,dq

dt , and dr

dt are rates of change of invariants following a ﬂuid particle in

accordance with Equation 5.3. The symbols ˆp, ˆqand ˆrdenote the unit vectors along

the three mutually perpendicular axes of p,qand rcoordinates. The quantity ~

Upqr is

indeed a measure of how fast the footprint of a ﬂuid particle is changing in the p−q−r

space.

We expect that a smaller volume available to a topology in the p−q−rspace will be a

contributing factor towards decreasing the lifetime of a topology, because a particle even

if moving slowly will tend to crossover to the territory of neighboring topology quicker.

On the other hand, a higher magnitude of ~

Upqr of a ﬂuid particle in the p−q−rspace will

tend to bring the particle closer to the bounding surfaces quickly and thus contributing

in reducing the lifetime of the topology associated with that ﬂuid particle.

In Table 3 we present the volumes associated with the six major topologies that exist

in compressible turbulence in the space of normalized invriants p,q and r. These volumes

have been reported separately on the positive and the negative side of the paxis. In

Figure 12 a-f we present the magnitude of the mean value of ~

Upqr calculated by taking

the average of the magnitude of ~

Upqr over subsets of the tagged particles belonging to

diﬀerent topologies and diﬀerent dilatation bins. This averaging is done using the ﬂow

ﬁeld from Simulation G at tref = 4.0τ.

5.1.1. UNSS and SNSS

Referring ﬁrst to the UNSS and SNSS topologies, we examine if the volume measures

available in Table 3 and the conditional mean values of ~

Upqr available in Figure 12 can help

us understand the variation of lifetime with dilatation reported in Figure 11. In Figure

12 we observe that mean value of the magnitude of ~

Upqr is higher at positive/negative

dilatations than what it is at at zero dilatation. Moreover, if we compare only the high

positive and high negative dilatations, mean velocity is somewhat more at negative

dilatations. On the other hand, Table 3 shows that the available volume of UNSS topology

on the negative side is less than what it is one the positive side. A higher mean velocity

associated with a smaller volume on the aii <0 side, allows a particle with initial UNSS

topology to quickly cross over to the territory of the neighboring topologies making its

lifetime low as observed in Figure 11a. At zero dilatation, the velocity drops signiﬁcantly

thus allowing the ﬂuid element to stay inside the UNSS territory for a longer duration.

At positive dilatations, even though the velocity is high, a signiﬁcant increase in the

volume of UNSS allows the lifetime to decrease moderately as evident in Figure 11a. In

the case of SNSS topology, as dilatation increases from high negative dilatation to high

positive dilatations, volume decreases. Velocity, however, deceases from a high value at

Lagrangian statistics in compressible turbulence 23

-1 -0.5 0 0.5 1

aii

0.5

1.5

Upqr

(a)

-1 -0.5 0 0.5 1

aii

0.5

1.5

Upqr

(b)

-1 -0.5 0 0.5 1

aii

0.5

1.5

Upqr

(c)

-1 -0.5 0 0.5 1

aii

0.5

1.5

Upqr

(d)

-1 -0.5 0 0.5 1

aii

0.5

1.5

Upqr

(e)

-1 -0.5 0 0.5 1

aii

0.5

1.5

Upqr

(f)

Figure 12. Variation of average velocity in p-q-r space |Upqr |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.

(Simulation G)

negative to a very low value at zero dilatation. The drop in the available volume seems to

be oﬀset by a decrease in velocity keeping the average lifetime of topology more-or-less

at the same level as it was a at high negative dilatation. When dilatation increases to

positive values, velocity increases (Figure 12)−though not as much as it was at negative

dilatations and thus even a decrease in volume results into only a little increase in the

lifetime (Figure 11b).

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

Simulation UNSS SNSS SFS UFC UFS SFC

A 0.23 0.28 0.23 0.26 - -

B 0.58 0.88 0.60 0.66 0.62 0.96

C 0.86 1.39 0.95 1.00 0.82 1.55

D 0.84 1.35 0.92 0.99 0.80 1.49

E 0.88 1.37 0.97 1.03 0.87 1.46

G 1.01 1.57 1.13 1.18 0.90 1.76

Table 4. Average velocity of particles in p-q-r space (|Upqr |=|∂p

∂t ˆp+∂q

∂t ˆq+∂r

∂t ˆr|) for various

simulations (Table 1)

5.1.2. SFS and UFC

The behavior of SFS and UFC topologies can also be explained in a similar manner.

SFS topology is associated with (i) very high mean velocity values and (ii) a small

volume at negative dilatations resulting into very low lifetime as evident in Figure 11c.

As dilatation increases, the mean velocity drops down−both factors contributing to a

steep increase in hLTi. As dilatation further increases, there is increase in volume, but

there is also an increase in velocity. The gain in velocity seem to be more dominating

as compared to increase in volume resulting into decrease in lifetime again at positive

dilatations (Figure 11c). Similar explanation can be provided for the lifetime of UFC

topology.

5.1.3. UFS and SFC

For the UFS and SFC topologies we do not observe any change in lifetime in Figure

11e. While UFS exists only at positive dilatation, SFC exists only at negative dilatations.

For both these topologies the volumes increase as the magnitude of dilatation increases.

Further, Figure 11e clearly shows that their velocities also increase as the magnitude of

normalized dilatation increases. For both these topologies, the increase in volume (which

favors high lifetime) seem to be eﬀectively counteracted by increase in mean velocity

(which favors low lifetime) resulting into an almost dilatation-independent lifetime as

evident in Figure 11.

5.1.4. Role of Mach number

As observed earlier, in general, the inﬂuence of increasing initial turbulent Mach

number is to decrease the lifetime of topologies (Figure 11). The explanation of this

trend is provided by Table 4, wherein we have included the mean magnitude of ~

Upqr

in various simulations at tref = 4.0τ. In general, an increase in initial turbulent Mach

number increases mean velocity in the p−q−rspace, consequently reducing hLTias

evident in Figure 11.

Overall, our results in Figure 11 clearly show that in terms of longevity, overall, the six

major topologies existing in compressible turbulence (Simulation G) can be arranged in

the following descending order: SFS>UNSS>UFC>SNSS>UFS>SFC. Accordingly, it is

plausible to expect that at a typical instant of this simulation, compressible decaying

turbulence should have the highest population of particles associated with the SFS

topology, lowest with the SFC topology and the populations of other four topologies

falling in the same order as the order of their lifetimes. In Table 5 we present the

Lagrangian statistics in compressible turbulence 25

UNSS SNSS SFS UFC UFS SFC

26.07 10.10 27.44 21.16 10.49 4.32

Table 5. Percentage topology composition for compressible simulation case G.

population percentage of each of the six topologies at the peak dissipation instant in

Simulation G. We observe that indeed the percentage population of the six topologies

decrease exactly in the same order as the order shown by them in terms of lifetimes.

While previous studies have also reported the percentage population of various topologies

in compressible turbulence (Table 5) in other contexts Suman & Girimaji (2010b), the

analysis presented in this work has provided a clear explanation of the same based on

careful particle tracking and calculations of invariant dynamics using actual Lagrangian

trajectories.

5.2. CMT versus LT

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 at diﬀerent time instants, but uses the

averaged value of the RHS of Equation 5.6 conditioned on a chosen set of p, q, r merely

at one single time instant. The statistics thus obtained are essentially the conditional

averages of the rate of change of the invariants with the conditional parameters being the

local values of p,q,r. The trajectories thus obtained are the 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 the past studies can only be

justiﬁed as a surrogate tool in the absence of adequate computational resources (Mart´ın

et al. 1998).

To further underline the signiﬁcance of our present work using Lagrangian trajectories,

we present Table 6, wherein we have included mean values of the lifetime of UNSS,

SNSS, SFS and UFC topologies computed using ﬂow ﬁelds of Simulation A. Note that

Simulation A has very low initial Mach number and can be practically treated as an

incompressible ﬂow.

The percentage composition in terms of the four topologies in this ﬂow ﬁeld is included

in the Table 6. We observe that the mean lifetime calculated using our Lagrangian

approach for these four topologies are in almost the same proportion as the percentage

population of the topologies. In the last row of Table 6 we have included the percentage

of time spent in various topologies as calculated by Mart´ın et al. (1998) using their CMT

approach. SinceMart´ın et al. (1998)’s CMTs do show a cyclic change in topology (UFC-

¿UNN-¿SNSS-¿SFS-¿UFC), the time spent in various topologies can be interpreted as the

CMT-based estimate of lifetime of topologies. We ﬁnd that the proportion of lifetimes

calculated using CMTs are in gross disagreement with the percentage composition of

incompressible turbulence.

To elucidate why CMTs fail signiﬁcantly in capturing topology lifetimes (and probably

other time dependents aspect of the dynamics of velocity gradients) we present a simple

comparison. In Figure 13(a) we show a conditional mean trajectory (solid line) originating

at point (q= 0.24, r =−0.05). This CMT has been generated using one-time Eulerian

ﬁeld from Simulation A at τ= 4 eddy-turnover time. The CMT shows a spiraling path

around the origin. Performing a procedure of line integrations along such trajectories, Ooi

et al. (1999) estimate characteristic cycle time of topology interconversion as three eddy

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

UNSS SNSS SFS UFC

Composition % 25.2 5.4 43.5 25.9

Lifetime (κτ) 1.80 0.53 3.32 2.08

Percentage % of time spent 53 21.5 20 5.5

in diﬀerent topologies CMTs Mart´ın et al. (1998)

Table 6. Comparison of the performances of CMTs versus the approach adopted in this work.

The composition and the computed lifetime (data in the ﬁrst two rows) are from Simulation A

of this work.

-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)

(a)

012345





0.05

0.25

RMS

(b)

Figure 13. (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.

turnover times. Next, we tag the same particles (in total xx) which were having their

initial invariants in the vicinity of the (q= 0.24, r =−0.05) at the reference time of four

eddy -turnover time and track their movement on the q−rplane. The locations of these

xx particles after just one eddy turnover time have been shown on 13(a) itself. It can be

observed that the tagged population of particles spreads vastly over the q-r plane. Indeed

the spread of this population sample is identical to the characteristic global distribution

of particles of the entire ﬂow ﬁeld. This characteristic distribution has tear-drop shape

with the bulk of data concentrated on one side in the SFC region and on the other side

along the curve separating the UFC and UNSS regions (the so -called vieillefosse tail, ?).

Further, in Figure 13(b), we present the root mean squared value of qand rof the

same sample of particles which had their q, r in the vicinity of (q= 0.24, r =−0.05) at

the reference time. The rms values start increasing and within just eddy-turnover times

reach thier asymptotic states. The rms of q reaches an asymptotic state of 0.24, and

the rms of r reaches an asymptotic state of 0.05. Indeed, these values match with the

unconditioned rms values of q and r of all the particles that are present in the ﬂow ﬁeld.

Thus, even one eddy turnover time is long enough a duration for particles to completely

forget their initial association with a particular point in the q-r plane. Thus, employing

CMTs to estimate phenomena which happen over this time scale may not be reasonable.

Lagrangian statistics in compressible turbulence 27

6. Conclusions

We investigate dynamics of velocity gradients in compressible decaying turbulence

employing the Lagrangian approach of following a set of identiﬁed ﬂuid particles. Well

resolved direct numerical simulations over a wide range of turbulent Mach number and

Reynolds number along with a well validated Lagrangian particle tracker are employed

for this study. In the ﬁrst part of our work, we focus on the viscous diﬀusion process

incumbent in the exact evolution equation of the velocity gradient tensor. Speciﬁcally,

we investigate the inﬂuence of Mach number on this process. We ﬁnd that the initial

turbulent Mach number has considerable inﬂuence on the Lagrangian statistics of the

viscous process. We provide evidence and explain that this intensiﬁcation is attributable

to the development of a large disparity in the magnitude of the velocity gradients

associated with contracting and expanding ﬂuid particles combined with the overall

preferences of these contracting and expanding ﬂuid particles to change their dilatation

rate (Section 4.1). Our investigation of the exact process is followed up by evaluation of

the so called Linear Lagrangian Diﬀusion model (LLDM) of the viscous process (section

4.2). Using Lagrangian tracking we clearly demonstrate that the LLD model shows an

unphysical exponential behavior which is in gross disagreement with the exact Lagrangian

evolution of the viscous process seen in DNS, especially in the late stages of the turbulence

decay. We argue that the unrealistic behavior of the LLD model is attributable to the

assumption that the fourth order tensor representing the Lagrangian Hessian of the

velocity gradient tensor remains isotropic at all times. Finally, we propose a possible

modeling approach with which this shortcoming can eb addressed.

In the second part of the study we examine the dynamics of the invariants of the

velocity gradient tensor using the Lagrangian approach and compute lifetimes of various

topologies in compressible turbulence. In particular, we identify the role of initial turbu-

lent Mach number and the normalized dilatation rate on topology lifetimes. Explanation

of the identiﬁed trends are then provided in terms of the geometric constraint of the

p−q−rspace and the disparity in the speed of the ﬂuid particles in the p−q−rspace.

Further, using our Lagrangian data and analysis we clearly demonstrate the limitation

of the so-called conditional mean trajectory (CMT) in explaining certain aspects of the

dynamics of velocity gradient invariants.

7. Acknowledgments

The authors acknowledge the computational support provided by the High-

Performance Computing Center (HPC) of the Indian Institute of Technology Delhi,

India.

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