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1
Lagrangian investigations on velocity
gradients in compressible turbulence:
Examination of viscous process and flow-field
topology
Nishant Parashar1 †, Sawan Suman Sinha
1
1
and Balaji Srinivasan2
Department of Applied Mechanics, Indian Institute of Technology Delhi, New Delhi 110016,
India
2
Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai
600036, India
(Received xx; revised xx; accepted xx)
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 (Lüthi 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; Ohkitani 1993; Pumir 1994;
Girimaji & Speziale 1995; O’Neill & Soria 2005; Chevillard & Meneveau 2006; da Silva
& Pereira 2008; Chevillard & Meneveau 2011) and compressible turbulence (Soria et al.
† Email address for correspondence: nishantparashar14@gmail.com
2
N. Parashar, S. S. Sinha and B. Srinivasan
1994; Pirozzoli & Grasso 2004; Suman & Girimaji 2009, 2010, 2012; Wang & Lu 2012;
Vaghefi & Madnia 2015; 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 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 Girimaji
& Pope (1990b). Yeung & Pope (1989) focused on Lagrangian statistics of velocity,
acceleration, and dissipation. Girimaji & Pope (1990b) 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 for 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 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 smallscale 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
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. 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 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.
In the second part of this work, we examine the evolution of flow-field topology in
compressible turbulence following the Lagrangian trajectories (LT) of the invariants of
the velocity gradient tensors. Topology can be visualized as the local streamline pattern,
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), but can also be readily determined by
knowing the three invariants of the VGT (Cantwell & Coles 1983; Chong et al. 1990).
Flow field topology can also be used for visualization of a flow field. Further, topology has
been shown 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 Martı́n 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 (Martı́n et al. 1998; Ooi et al.
1999; Elsinga & Marusic 2010; Meneveau 2011; Atkinson et al. 2012; Zhou et al. 2015)
and compressible flows (Chu & Lu 2013; Bechlars & Sandberg 2017). Even though
CMTs do 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 (Martı́n et al. 1998). An investigation of the exact 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. Based on such a motivation, we employ the LTs to investigate lifetime of
topologies and their interconversion processes in compressible turbulence.
To address the identified objectives of this paper, we employ direct numerical simulations of decaying isotropic compressible turbulence over a wide range of turbulent Mach
number (0.075, 1.5) and a moderate range of Reynolds number (70, 350). The Lagrangian
dynamics are obtained using an almost time continuous set of Eulerian flow fields along
with spline-aided Lagrangian particle tracker (more details in §4).
4
N. Parashar, S. S. Sinha and B. Srinivasan
This paper is organized into six sections. In §2 we present the governing equations. In §3
we provide details of our direct numerical simulations as well as the Lagrangian particle
tracker. In §4 we examine the influence of compressibility on the viscous processes of the
velocity gradient dynamics and evaluate the LLD model. In §5 we study the dynamics
of topologies, compare CMT and LT and quantify lifetimes of various flow-topologies
existing in compressible turbulence. Section 6 concludes the paper with a summary.
2. Governing Equations
The evolution of a compressible flow field is governed by the following equations:
∂ρ
∂ρ
+ Vk
=
∂t
∂xk
∂Vi
∂Vi
+ Vk
=
∂t
∂xk
∂T
∂T
+ Vk
=
∂t
∂xk
−
−
−
+
∂Vk
;
∂xk
1 ∂p
1 ∂σik
+
;
ρ ∂xi
ρ ∂xk
∂Vi
n − 1 ∂qk
T (n − 1)
−
∂xi
ρR ∂xk
n−1 ∂
(Vi σji ) ;
ρR ∂xj
ρ
p = ρRT ;
(2.1)
(2.2)
(2.3)
(2.4)
where Vi and xi represents the velocity and position vector respectively. The thermodynamic properties are represented by ρ (density), p (pressure) and T (temperature), while
R and n denotes the gas constant and specific heat ratio respectively. Equation (2.1) is
the continuity equation, (2.2) represents the momentum conservation equations, (2.3) is
the equation of energy conservation and (2.4) is the equation of state for a perfect gas.
Heat flux (qk ) and the specific heat ratio (σij ) are found using the following constitutive
relationships:
∂Vi
∂Vj
∂Vk
σij = µ
+
+ δij λ
;
(2.5)
∂xj
∂xi
∂xk
∂T
;
(2.6)
qk = −K
∂xk
where, K is the thermal conductivity, µ is the first coefficient of viscosity and λ denote
the second coefficient of viscosity (λ = − 2µ
3 ). The velocity gradient tensor is defined as:
Aij ≡
∂Vi
.
∂xj
By taking the gradient of momentum equation (2.2), the evolution equation of Aij can
be obtained:
DAij
∂
1 ∂p
= −Aik Akj −
Dt
∂xj ρ ∂xi
|
{z
}
Pij
+
∂
∂xj
|
1 ∂
∂Vi
∂Vk
2 ∂Vp
µ
+
−
δik
;
ρ ∂xk
∂xk
∂xi
3 ∂xp
{z
}
(2.7)
Υij
where, Pij is the pressure Hessian tensor and Υij is the viscous process governing the
evolution of the velocity gradient tensor. The rate of change following a fluid particle
Lagrangian statistics in compressible turbulence
is caculated using the substantial derivative, which is represented by the operator
∂
∂
∂t + Vk ∂xk ).
5
D
Dt (≡
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). GKM was originally
developed by Xu (2001) and has been shown to be quite robust in terms of numerical
stability. Further, GKM has the ability to capture shocks without producing numerical
oscillations. Several researchers 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 2π with a uniform grid and periodic boundary
conditions imposed on opposite sides of the domain. The initial velocity field is generated
at random with zero mean and having the following energy spectrum E(κ):
E(κ) = A0 κ4 exp −2κ2 /κ20 ,
(3.1)
where κ is wavenumber. Values for spectrum constants A0 and κ0 are provided in Table 1
for various simulations employed in this work. The relevant Reynolds number for isotropic
turbulence is the one based on Taylor micro-scale (Reλ ):
r
20
k,
(3.2)
Reλ =
3ν
where k, , and ν represent turbulent kinetic energy, its dissipation-rate, and kinematic
viscosity. For compressible isotropic turbulence, the relevant Mach number is the turbulent Mach number (Mt ):
r
2k
,
(3.3)
Mt =
nRT
where T represents mean temperature.
Following the work of Kumar et al. (2013), we have used a 4th order accurate weightedessentially-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 seven different simulations
(Simulations A-G). Descriptions of these simulations are presented in Table 1.
6
N. Parashar, S. S. Sinha and B. Srinivasan
1
0
A
B
C
D
E
F
G
k/k0
0.6
-2
A
B
C
D
E
F
G
SV
0.8
-0.5
0.4
-4
0.2
0
0
2
4
6
8
0
2
t/τ
4
6
8
t/τ
(a)
(b)
Figure 1. Evolution of (a) normalized turbulent kinetic energy k/k0 and (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 Simulations A-H. In figure 1(b), we present the evolution of skewness of the velocity derivative
(SV ) which is defined as :
SV + SV22 + SV33
SV = 11
;
(3.4)
3
where,
0 3
SV11 =
∂V1
∂x1
0
∂V1
∂x1
2 3/2
, etc.
(3.5)
Note that the time in these figures have been normalized using τ , which represents eddy
turnover time (Yeung & Pope 1989; Elghobashi & Truesdell 1992; Samtaney et al. 2001;
Martı́n et al. 2006):
λ0
(3.6)
τ= 0;
u0
where u00 and λ0 are the root mean square (rms) velocity and integral-length-scale of the
initial flow field (at time, t = 0).
To obtain Lagrangian statistics, a Lagrangian particle tracker (LPT) is used to extract
the full time-history of tagged fluid particles. Our LPT obtains the trajectory (X+ (y, t))
of a fluid particle by solving the following equation of motion:
∂X+ (t, y)
= V X+ (t, y) , t ,
∂t
(3.7)
where the superscript “+” represents Lagrangian flow variable, and y indicates the
label/identifier assigned to the fluid particle at a reference time (tref ). The initial value of
X+ at a reference time is chosen at random. Using this initial condition, we then integrate
(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. 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
Lagrangian statistics in compressible turbulence
Simulation
Reλ
Mt
Grid size
A0
A.
B.
C.
D.
E.
F.
G.
70
350
150
100
70
70
70
0.075
0.6
1.0
1.0
1.0
1.25
1.5
1283
10243
5123
5123
2563
2563
2563
0.000023
0.0015
0.0042
0.0042
0.0042
0.0065
0.0094
7
κ0
4
4
4
4
4
4
4
Table 1. Initial parameters of DNS simulations.
this purpose. Like our DNS solver, 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 (§4.1). Following a similar methodology, in §4.2 we examine
the performance of the LLD model of Jeong & Girimaji (2003).
4.1. DNS-based examination of viscous process
The viscous process Υ in the evolution equation of the velocity gradient tensor (2.7)
is:
∂Aij
ν ∂ρ
∂Akk
Υij = ν
−
+ν
∂xk ∂xk
∂xi ∂xj ρ ∂xj
} | {z
| {z
} |
ΥIij
ΥIIij
∂Aik
1 ∂Akk
.
+
∂xk
3 ∂xi
{z
}
(4.1)
ΥIIIij
In (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 EHEE 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 ):
p
ΥIij ΥIij
p
p
;
fI = p
ΥIij ΥIij + ΥIIij ΥIIij + ΥIIIij ΥIIIij
p
ΥIIij ΥIIij
p
p
,
fII = p
ΥIij ΥIij + ΥIIij ΥIIij + ΥIIIij ΥIIIij
p
ΥIIIij ΥIIIij
p
p
fIII = p
.
(4.2)
ΥIij ΥIij + ΥIIij ΥIIij + ΥIIIij ΥIIIij
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 (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
8
N. Parashar, S. S. Sinha and B. Srinivasan
on these findings, we conclude that for the range of Mach number and Reynolds number
considered in this study, the exact process Υ is almost solely represented by ΥI , itself.
Thus, in the rest of this study we focus only on ΥI and assume it to be synonymous with
Υ.
Our interest is to examine how the viscous process ΥI 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 ):
p
ΥIij (t)ΥIij (t)
,
(4.3)
r(t, tref ) = p
ΥIij (tref )ΥIij (tref )
where, ΥIij (t) and ΥIij (tref ) are values of the quantity ΥIij associated with an identified
fluid particle at an arbitrary time t and 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 to 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.
In this work fluid compressibility is parametrized based on the initial turbulent Mach
number (3.3) and the locally normalized dilatation rate (aii ). The normalized dilatation
rate (aii ) is the trace of the locally normalized velocity gradient tensor, which is defined
as:
p
(4.4)
aij = Aij / Amn Amn .
The normalized dilatation-rate of a fluid particle (henceforth, referred to as just “dilatation”) represents the normalized rate of change in density of a local fluid particle:
1 dρ
= −aii
ρ dt0
(4.5)
p
where dt0 = dt Aij Aij represents time normalized with the local magnitude of the
velocity gradient tensor itself. A positive value of aii implies an expanding fluid element,
and a negative value of aii implies a contracting fluid element. A fluid particle with
instantaneous 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 on hr(t, tref )i, in figure 2 we present Lagrangian mean hr(t, tref )i from simulations E-G. These simulations have identical initial
Reynolds number (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 has 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.
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 stage of evolution can be
related mainly to the decay in kinetic energy. We observe that as initial Mt increases,
the peak value of hr(t, tref )i also increases.
In several previous studies like Suman & Girimaji (2010) and Parashar et al. (2017a),
it has been demonstrated that even when the unconditioned Eulerian or Lagrangian
statistics obtained using DNS data of compressible homogeneous turbulence do not show
any discernible influence of a global compressibility parameter like Mt , the same statistics
Lagrangian statistics in compressible turbulence
9
40
G
F
hr(t, tref )i
30
E
20
10
0
0
2
4
6
t/τ
Figure 2. Mach number dependence on evolution of exact viscous term (hr(t, tref )i).
5
15
- ,
+
r(t, tref )-aii
4
3
= −1.00
= −0.75
= −0.50
= −0.25
= 0.00
- ,
+
r(t, tref )-aii
aii
aii
aii
aii
aii
aii
aii
aii
aii
aii
10
2
= +1.00
= +0.75
= +0.50
= +0.25
= 0.00
5
1
0
0
0
2
4
(t − tref )/τ
(a)
6
0
2
4
6
(t − tref )/τ
(b)
Figure 3. Dependence of hr(t, tref )|aii i on aii . Results are from simulation G (tref = 0.5τ ).
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
ΥI ),
variables directly. Thus to gain further insight into the viscous diffusion process (Υ
we subject hr(t, tref )i to conditioned averaging on discrete values of aii .
In figure 3 we present conditional Lagrangian mean of hr(t, tref )i from Simulation G,
with aii at tref as the conditioning parameter. The reference time is chosen to be 0.5τ .
In figure 3(a), separate curves are presented for aii = −1.0, −0.75, −0.50, −0.25 and 0,
whereas in figure 3(b) 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 )|aii i seem to retain the two-stage evolution patterns as
shown by the unconditioned statistics hr(t, tref )i, the extent to which the magnification
of hr(t, tref )i happens (figure 3) seems to be strongly affected by the value of aii . In figure
3(a) we observe that as the dilatation level changes from being zero (volume preserving
fluid particles) to being more negative (contracting fluid particles), the peak value of
hr(t, tref )|aii i reduces. On the other hand, in figure 3(b) we observe the opposite trend:
for fluid particles with high positive dilatation (fast expanding fluid particles) the peak
value of hr(t, tref )|aii i tends to increase.
10
N. Parashar, S. S. Sinha and B. Srinivasan
In our attempt to understand and explain the behavior observed in figure 3(a) and 3(b),
in figure 4(a) and 4(b) we present the mean value of the amplification of the magnitude
of the velocity gradient tensor A itself, following the same set of fluid particles as used
in figures 3. We measure this amplification as:
Aij (t)Aij (t)
hrA (t, tref )i =
.
(4.6)
Amn (tref )Amn (tref )
We observe that over almost the entire range of dilatation considered in this work,
the trends shown by hrA (t, tref )|aii i are similar to hr(t, tref )|aii i. Like hr(t, tref )|aii i,
a more positive dilatation tends to move the peak of hrA (t, tref )|aii i higher, and a more
negative dilatation tends to lower the peak of hrA (t, tref )|aii i. This similarity in the trends
shown by hr(t, tref )|aii i in figure 3 and that shown by hrA (t, tref )|aii i in figure 4 is not
completely unexpected, and it indeed substantiates a gradient diffusion like hypothesis
which assumes ∆A ∝ A (Martı́n et al. 1998; Jeong & Girimaji 2003). However, in the
light of figure 4(a) and 4(b) our primary curiosity that why the peaks of hr(t, tref )i (in
figure 3(a) and 3(b)) rise for expanding fluid particles and reduce
p for contracting fluid
Aij Aij aii at three
particles can now possibly be explained. In figure 5 we show
timepinstants, tref , tref + τ /4 and tref + τ /2 (We observe that the one time statistics
of
Aij Aij aii does not show significant variations with time). The dependence of
p
Aij Aij aii on aii is monotonic and almost linear for aii > 0. A faster expanding
p
fluid particle (large positive aii ) is associated with smaller
Aij Aij aii 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 p
high negative
p
aii ) have a very large
Aij Aij aii . As dilatation becomes less negative
Aij Aij aii
drops
first
(till
a
≈
−0.25)
and
then
again
tends
to
be
larger.
Thus
the
dependence
ii
p
of
Aij Aij aii on aii is non-monotonic as well as (apparently) non-linear (figure 5).
In figure 6(a) we present the averaged rate of change in dilatation (ψ) of tagged fluid
particles over one Kolmogorov time:
(aii (τκ + tref ) − aii (tref ))
ψ=
;
(4.7)
τκ
where, τκ is the Kolmogorov time at tref . In figure 6(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 par ticles are associated
with negative ψ and contracting particles are associated with positive ψ. In other words,
both contracting and expanding fluid particles tend to reduce the magnitudes of their
dilatation. Thus particles with higher initial positive dilatation
tend to acquire lower
p
Aij Aij aii of particles with
positive dilatation levels. Further, since the associated
higher positive dilatation is smaller than those with lower positive dilatation (figure 4),
it is plausible to expect that the peak of hr(t, tref )i will be more for higher positive aii
than that for lower positive aii . Indeed this behavior is observed in figure 3(b).
For contracting particles the dynamics seem to
pbe more complicated because of nonAij Aij |aii (figure 5). At low negative
monotonic and highly non-linear distribution of
dilatation (say aii ≈ −0.25), the dominant tendency p
of particles is to move towards
zero dilatation (figure 6). The association of higher
Aij Aij aii at zero dilatation
compared to that at low negative dilatation (figure 5) still allows hrA (t, tref )|aii i to show
a substantial magnification at early times (figure 4(a)).
For faster contracting particles (say those with initial aii ≈ −0.75), figure 6 shows
that (like other contracting particles) they begin their journey
p towards zero dilatation.
However, figure 5 suggests that as their dilatation reduces,
Aij Aij aii severely drops.
Lagrangian statistics in compressible turbulence
2
aii
aii
aii
aii
aii
1
= −1.00
= −0.75
= −0.50
= −0.25
= 0.00
aii
aii
aii
aii
aii
1.5
- ,
+
rA (t, tref )-aii
- ,
+
rA (t, tref )-aii
1.5
11
0.5
1
= +1.00
= +0.75
= +0.50
= +0.25
= 0.00
0.5
0
0
0
2
4
6
0
2
(t − tref )/τ
4
6
(t − tref )/τ
(a)
(b)
Figure 4. Dependence of hrA (t, tref )|aii i on aii . Results are from simulation G (tref = 0.5τ ).
7000
t = tref
t = tref + τ /4
t = tref + τ /2
+p
- ,
Aij Aij -aii
5000
3000
1000
-1
-0.5
0
0.5
1
aii
Figure 5.
p
Aij Aij |aii vs aii at different time instants for Simulation G (tref = 0.5τ ).
Thispdecreasing tendency, combined with the fact that initially they had a higher value
Aij Aij aii , results into a dramatic drop in hrA (t, tref )|aii i as observed in figure
of
4(a).
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. It is known that as
initial Mt is increased, more particles tend to have non-zero dilatation, and also larger
magnitudes of normalized dilatation are generated (Suman & Girimaji 2009; Lee et al.
2009). In figure 7(a) we show pdf of dilatation at the peak dissipation time in Simulation
E−G. We observe that (i) the population of fast expanding particles as well as 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 7(b) we present hrA (t, tref )|aii i at peak dissipation from simulations E−G. We
observe that as initial Mt increases, the disparity of hrA (t, tref )|aii i across aii increases.
This explains the increase in peak of hr(t, tref )i observed in figure 2.
4.2. Evaluation of the LLD model
Having examined the behavior of the exact process ΥI we examine the performance of
the LLD model of Jeong & Girimaji (2003), which intends to capture the essential physics
12
N. Parashar, S. S. Sinha and B. Srinivasan
1
1
0.8
0.5
ψ
0.6
0
0.4
0.2
-0.5
-1
-0.5
0
0.5
0
-1
1
-0.5
aii
(a)
0
0.5
1
aii
(b)
Figure 6. (a) ψ vs aii at tref = 0.5τ . (b) Fraction of particles moving towards +ve aii
(symbol +) and -ve aii (symbol o).
10 1
10
7000
E
F
G
E
F
G
0
Mt
- ,
Aij Aij -aii
5000
PDF
10 -1
+p
10 -2
3000
10 -3
10 -4
1000
-1.5
-1
-0.5
0
0.5
1
1.5
aii
(a)
Figure 7. (a) PDF of aii for Simulations E−G. (b)
-1
-0.5
0
0.5
1
aii
(b)
p
Aij Aij |aii vs aii for Simulations E−G.
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 the modeled expression and other time-dependent aspects remain
questionable−especially its anticipated exponential growth at late times (see comments
by 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: A and G.
Both these simulations have the same initial Reynolds number (70) but different Mach
numbers. Simulation A with initial Mt = 0.075 can be treated as almost incompressible,
while Simulation G has considerable compressibility with initial Mt = 1.5.
The LLD modeling approach of Jeong & Girimaji (2003) uses Lagrangian-Eulerian
change in variables to cast ΥIij as:
∂
∂Xm ∂Aij
∂ 2 Aij
=ν
,
(4.8)
ν
∂xk ∂xk
∂xk ∂xk ∂Xm
Lagrangian statistics in compressible turbulence
13
where, Xi and xi are Eulerian and Lagrangian spatial co-ordinates. Further expansion
of rhs of (4.8) leads to:
ν
∂Aij ∂ 2 Xm
∂ 2 Aij
∂Xn ∂Xm ∂ 2 Aij
+ν
.
=ν
∂xk ∂xk
∂xk ∂xk ∂Xm ∂Xn
∂Xm ∂xk ∂xk
{z
} |
{z
}
|
A
(4.9)
B
Jeong & Girimaji (2003) neglects Term B (first modeling assumption) on the rhs of (4.9)
to arrive at the following:
ν
∂ 2 Aij
∂Xn ∂Xm ∂ 2 Aij
≈ν
.
∂xk ∂xk
∂xk ∂xk ∂Xm ∂Xn
(4.10)
∂xi
and the right CauchyUsing the definition of the deformation gradient tensor Dij = ∂X
i
Green tensor Cij = Dkm Dkn , the rhs of (4.10) can be expressed in terms of the tensor
C:
ν
∂ 2 Aij
∂ 2 Aij
−1
≈ νCmn
∂xk ∂xk
∂Xm ∂Xn
(4.11)
Further, Jeong & Girimaji (2003) make the second modeling assumption wherein C−1
is approximated to be an isotropic tensor:
ν
C −1
∂ 2 Aij
∂ 2 Aij
≈ ν kk δmn
;
∂xk ∂xk
3
∂Xm ∂Xn
(4.12)
C −1 ∂ 2 Aij
∂ 2 Aij
≈ ν kk
.
∂xk ∂xk
3 ∂Xm ∂Xm
(4.13)
which leads to:
ν
∂2A
A
ij
Finally, the third approximation is made: ∂Xm ∂X
≈ − τLij , leading to the final form of
m
the so called Lenear Lagrangian Diffusion (LLD) model:
ν
−1
∂ 2 Aij
1 Ckk
Aij .
≈−
∂xk ∂xk
τL 3
(4.14)
Jeong & Girimaji (2003) consider the quantity τL to be a constant and interpret this
as a molecular viscous relaxation time scale. This model has been employed by Jeong &
Girimaji (2003) to close the restricted Euler equation (REE). Later, Suman & Girimaji
(2012) 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, Chevillard & Meneveau (2006) 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−1 tensor to be isotropic, the authors
∂ 2 Aij
assumed the fourth-order tensor ∂Xm ∂X
(Lagrangian Hessian of the tensor A) to be
n
isotropic, and thereafter expressed it in terms of the tensor Aij and a length scale δX
associated with the flow field at tref :
∂ 2 Aij
−1 Aij δmn
≈ −Cmn
.
∂Xm ∂Xm
(δX)2 3
(4.15)
14
N. Parashar, S. S. Sinha and B. Srinivasan
Subsequently, (4.15) when combined with (4.11) then leads to the following form:
ν
C −1 Aij
∂ 2 Aij
;
≈ −ν kk
∂xk ∂xk
3 (δX)2
(4.16)
Chevillard & Meneveau (2006) interpret δX as the characteristic length scale that a fluid
particle traverses over a Kolmogorov timescale, and thus δX ≈ λ, where λ is the Taylormicroscale of the turbulent flow field at the reference time tref . Thus, the final version
of the LLD model takes the form:
ν
∂ 2 Aij
ν C −1
≈ − 2 kk Aij .
∂xk ∂xk
λ 3
(4.17)
As mentioned in the Introduction, with access to the DNS data of decaying turbulence
and a validated Lagrangian particle tracker, we now intend to scrutinize the performance
of this model (4.17). For this study, we select the same set of particles that we used in
−1 ν
§4.1 to examine the exact ΥI . Following these particles we calculate Ckk
3λ2 Aij . Like the
exact process (4.3), we define the amplification ratio of the modeled viscous process as:
−1 p
Ckk
p
−1
Aij Aij
Aij Aij t
Ckk
3
t
rm (t, tref ) = −1 p
= p
(4.18)
Ckk
Aij Aij t
A
A
ij
ij
ref
3
tref
At each time instant the ratio rm (t, tref ) is calculated, and subsequently the mean value
hrm (t, tref )i is computed by taking averages across the identified set of particles. A direct
−1
comparison of hrm (t, tref )i is then performed against hr(t, tref )i. Note that to find Ckk
at
any arbitrary time we use the following exact evolution equation of deformation gradient
tensor Dij (Jeong & Girimaji 2003):
dDij
= Dik Akj ,
dt
(4.19)
where, both A and D are calculated using the DNS data fields at different time instants
and the Lagrangian particle tracker. Thus, our evaluation procedure uses the exact
instantaneous states of the A and the C tensors as provided by DNS.
In figure 8(a), we present evolution of both hrm (t, tref )i and hr(t, tref )i using results
of simulation A (nearly incompressible with initial Mt being 0.075 and Reλ =70). In
figure 8(b), results are shown using DNS data from Simulation G (highly compressible
with initial Mt being 1.5 and Reλ =70). 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 time. At the early stages of evolution, this monotonic growth is at least
qualitatively similar to the exact process. However, at later stages (after the dissipation
peak) the continued monotonic growth of hrm (t, tref )i is 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 actual viscous process
in both incompressible and compressible flow fields.
In an attempt to diagnose the problems of the LLD model (4.17), we revisit the LLD
modeling procedure of Jeong & Girimaji (2003) and Chevillard & Meneveau (2006). To
begin with, we avoid the assumptions made by Chevillard & Meneveau (2006) regarding
∂ 2 Aij
. Instead, we approximate the
the isotropic structure of the fourth-order tensor ∂Xm ∂X
n
Lagrangian statistics in compressible turbulence
100
15
100
hrm (t, tref )i
hrm (t, tref )i
80
80
hr(t, tref )i
hr(t, tref )i
60
60
40
40
20
20
0
0
0
2
4
6
0
t/τ
2
4
6
t/τ
(a)
(b)
Figure 8. Comparison of hr(t, tref )i and hrm (t, tref )i: (a) Simulation A (nearly
incompressible), (b) Simulation G (compressible).
Lagrangian Hessian of Aij as:
∂ 2 Aij
Aij
Rmn ;
≈
∂Xm ∂Xn
(δX)2
(4.20)
where, δX is the same length scale as described by Chevillard & Meneveau (2006). In
(4.20) 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 Chevillard & Meneveau
(2006). Now combining (4.20) with (4.11), we arrive at:
ν
∂ 2 Aij
∂ 2 Aij
−1
−1 Aij
Rmn
≈ νCmn
≈ −νCmn
∂xk ∂xk
∂Xm ∂Xn
(δX)2
(4.21)
Equation (4.21) 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
(C : R) A
(4.22)
ν∇2 A ≈ −
(δX)2
If we now express C−1 and R tensors in the eigen-system of the instantaneous C−1
tensor, then (4.22) can be expressed as:
ν
ν∇2 A ≈ −
(αRα + βRβ + γRγ ) A
(4.23)
(δX)2
where, α, β and γ are the three eignevalues of the instantaneous C−1 sorted as α >
β > γ, and these symbols when sub-scripted, imply the component of a tensor along the
corresponding eigenvector.
Our DNS results from both nearly incompressible and compressible cases show that
C−1 tensor evolves to a state with its largest eigenvalue being overwhelmingly dominant
over the other two. figure 9 show the instantaneously normalized eigenvalues of C−1 from
Simulation A and G. The plotted quantities Tα , Tβ and Tγ are defined as:
α2
;
+ β2 + γ2
β2
Tβ = 2
;
α + β2 + γ2
Tα =
α2
16
N. Parashar, S. S. Sinha and B. Srinivasan
1
1
Tγ
0.8
Tγ
0.8
Tβ
Tβ
Tα
0.6
Tα
0.6
0.4
0.4
0.2
0.2
0
0
0
0.5
1
1.5
2
0
(t − tref )/τ
0.5
1
1.5
2
(t − tref )/τ
(a)
(b)
Figure 9. Evolution of Tα , Tβ and Tγ in (a) Simulation A (nearly incompressible, (b)
Simulation G (compressible).
Tγ =
α2
γ2
;
+ β2 + γ2
(4.24)
Similar findings from incompressible simulations have been reported earlier as well
Girimaji & Pope (1990a). Since the order of magnitude of the Lagrangian Hessian in
(4.23) is essentially represented by Aij /δX 2 , we expect the magnitude of the components
of R to either close to zero or of order unity. Thus it is plausible to approximate (4.23)
as:
ν
ν
ν∇2 A ≈ −
(αRα + βRβ + γRγ ) A ≈ −
(αRα ) A
(4.25)
2
(δX)
(δX)2
Now, adopting the interpretation offered by Chevillard & Meneveau (2006) for the
quantity δX ≈ λ, (4.25) can be expressed as:
ν
ν∇2 A ≈ − 2 (αRα ) A
(4.26)
λ
At this point we compare (4.26) and the form of the model used by Chevillard &
Meneveau (2006) (4.17). Taking into account that the DNS behaviour shows |α| >>
|β|, |γ|, the approximation made in (4.25) can also be applied to (4.17). This will reduce
(4.17) to:
ν
1
2
ν∇ A ≈ − 2 α
A
(4.27)
λ
3
Thus, the essential difference between the modified model (4.26) and the original LLD
model (4.17) is that while the original model has already committed to Rα being 1/3,
the modified model has not imposed any such restriction so far.
Given that α is known to grow exponentially, and the fact that (4.27) leads to gross
overestimation of growth of hrm (t, tref )i (figure 8), we conjecture that the quantity Rα
must follow an exponential decay towards zero (4.28), so as to restrain the unrealistic
growth of hrm (t, tref )i in comparison to the expected behavior represented by hr(t, tref )i:
Rα ≈ |α|−m
(4.28)
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 m
is to provide an exponential modulation, so that the exponential growth shown by the
original LLD model (4.17) displayed in figure 8 can be reined in. At this point we present
(4.28) merely as a modeling proposal. More detailed analysis and probably DNS data
Lagrangian statistics in compressible turbulence
17
Acronyms
p=0
p<0
p>0
Eigenvalues of aij
SFS
r<0
r < 0 & S2 > 0
r<0
complex
UFC
r>0
r>0
r > 0 & S2 < 0
complex
UNSS
r>0&q<0
r>0
r>0&q<0
real
SNSS
r<0&q<0
r<0&q<0
r<0
real
UFS
—
r < 0 & S2 < 0
—
complex
UN/UN/UN
—
r<0&q>0
—
real
SFC
—
—
r > 0 & S2 > 0
complex
SN/SN/SN
—
—
q>0&r>0
real
Table 2. Zones of various topologies on p − q − r space, where acronyms are:
stable-focus-stretching (SFS), unstable-focus-compressing (UFC), unstable-node/saddle/saddle
(UNSS), stable-node/saddle/saddle (SNSS), unstable-focus-stretching (UFS), unstable-node/unstable-node/unstable-node (UN/UN/UN), stable-focus-compressing (SFC), stable-node/stable-node/stable-node (SN/SN/SN).
over a wide range of Mach number and Reynolds number will be required to arrive at a
concrete functional form of m. Such an effort, however, 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 as 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 :
1
P 2 − Aij Aji , and
2
1
3
R=
−P + 3P Q − Aij Ajk Aki .
3
P = −Aii , Q =
(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 ):
1 2
p − aij aji , and
2
1
3
r=
−p + 3pq − aij ajk aki .
3
p = −aii , q =
(5.2)
Chen et al. (1989) categorize topological patterns (Table 2) of an incompressible field
into unstable-node-saddle-saddle (UNSS), stable-node-saddle-saddle (SNSS), stablefocus-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 associated with contracting
fluid particles; and unstable-focus-stretching (UFS) and unstable-node/unstable-
18
N. Parashar, S. S. Sinha and B. Srinivasan
SFS
UFC
q-axis
q-axis
SFS
S1a
S2
q 0
S1a
SFC
S2
q 0
S1b
S1b
0
0 r
(a)
r
UFC
(b)
q-axis
UFC
UFS
S1a
S2
q 0
S1b
SFS
0 r
(c)
Figure 10. Regions of different flow topologies in different p-planes: (a) p = 0, (b) p > 0 and
(c) p < 0.)
node/unstable-node (UNUNUN), which are associated with expanding fluid particles.
Figure 10 shows different regions in the p−q−r space associated with different topologies.
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 (2.7), the time-evolution
of invariants (P,Q,R) can be expressed Bechlars & Sandberg (2017):
dP
= P 2 − 2Q − Sii ;
dt
dQ
2P
∗
= QP −
Sii − 3R − Aij Sji
;
dt
3
dR
Q
∗
∗
= − Sii + P R − P Aij Sji
) − Aik Akj Sji
;
(5.3)
dt
3
where, Sij is the source term in the evolution equation of velocity-gradient-tensor (2.7)
∗
and the symbol Sij
is the traceless part of the Sij tensor:
Sij = −Pij + Υij ; and
Skk
∗
Sij
= Sij −
δij .
(5.4)
3
Here Pij is the pressure hessian tensor and Υij represents the viscous process in the
evolution equation of the velocity-gradient tensor (2.7).
Lagrangian statistics in compressible turbulence
19
The relationship between the non-normalized invariants (P ,Q,R) and normalized invariants (p,q,r) is:
p= p
P
Q
R
,q =
, and r =
Aij Aij
(Aij Aij )3/2
Aij Aij
(5.5)
Using (5.3) and the relationship (5.5), the evolution equation of the normalized invariants
(p,q,r) can be derived as:
dAij
dp
d
P
1
dP
P
p
A
=
=p
−
,
3/2 ij dt
dt
dt
dt
(A
A
)
Aij Aij
Aij Aij
ij ij
d
Q
1 dQ
2Q
dq
dAij
=
=
−
Aij
,
2
dt
dt Aij Aij
Aij Aij dt
(Aij Aij )
dt
d
3R
dr
R
dQ
1
dAij
=
−
.
(5.6)
Aij
=
3/2
3/2
5/2
dt
dt (Aij Aij )
dt
dt
(Aij Aij )
(Aij Aij )
While following an identified fluid particle in physical space and tracking its invariants
information, we can track the footprints 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-r space 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 to change its topology relative to the
topology it had at a reference time tref . We express this time non-dimensionalized by
the Kolmogorov time scale (τκ ) of the homogeneous flow field at tref . We refer to the
normalized lifetime of a given topology (T ) thus obtained as LT :
LT =
t∗ − tref
;
τκ
(5.7)
where t∗ denotes the time instant when the original topology T associated with a tagged
fluid particle changes to some other topology. Correspondingly, the mean value of LT is
calculated by following a large number of tagged particles which have the same topology
T at tref :
ht∗ − tref i
.
(5.8)
hLT i =
τκ
In all our calculations and analysis of lifetime of topologies we employ the data fields of
Simulation A-E and G, 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 hLT i conditioned upon discrete values of
aii of the reference time (tref ). These conditioned lifetimes are represented as hLT |aii i.
In figure 11 we present mean lifetimes of various topologies conditioned upon different
initial dilatation levels: hLT |aii i. 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 |aii i using Simulations xx. These simulations have identical
initial Reynolds number (70), but different initial Mach number (1.00. 1.25 and 1.5).
We observe that the compressibility parameters−dilatation and initial Mach numbers
N. Parashar, S. S. Sinha and B. Srinivasan
2
2
1.5
1.5
hLT |aii i
hLT |aii i
20
1
0.5
1
0.5
-1
-0.5
0
0.5
1
-1
-0.5
aii
0.5
1
0.5
1
(b)
2
2
1.5
1.5
hLT |aii i
hLT |aii i
(a)
1
0.5
1
0.5
-1
-0.5
0
0.5
1
-1
aii
-0.5
0
aii
(c)
(d)
2
2
1.5
1.5
hLT |aii i
hLT |aii i
0
aii
1
0.5
1
0.5
0
0.5
aii
(e)
1
-1
-0.5
0
aii
(f)
Figure 11. LT |aii from different simulations (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.
influence the lifetimes of some topologies selectively. Mean lifetimes of UNSS (figure
11(a)), UFC (figure 11(c)) and SFS figure 11(d)) topologies seem to be significantly
more sensitive to both the Mach number and dilatation compared to those of the other
topologies. As the level of dilatation increases from high negative values to zero dilatation,
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
seems 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
Lagrangian statistics in compressible turbulence
21
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).
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 prospective reasons which may influence the lifetime of a topology of a fluid particle as it moves
in the p − q − r space: (i) the actual volume available to a topology in the p − q − r space,
and (ii) the velocity of the fluid particles in the p − q − r space. The velocity of a particle
in the p − q − r space 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 − r space. We
~ pqr :
denote this velocity vector by U
~ pqr = dp p̂ + dq q̂ + dr r̂;
U
dt
dt
dt
(5.9)
dq
dr
where dp
dt , dt , and dt are rates of change of invariants following a fluid particle in
accordance with (5.6). The symbols p̂, q̂ and r̂ denote the unit vectors along the three
~ pqr is indeed a
mutually perpendicular axes of p, q and r coordinates. The quantity U
measure of how fast the footprint of a fluid particle is changing in the p − q − r space.
It is observed that a smaller volume available to a topology in the p − q − r space 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
~ pqr of a fluid particle in the p − q − r
sooner. On the other hand, a high magnitude of U
space 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 the normalized invriants p,q and r. These
volumes have been reported separately on the positive and the negative side of the p
~ pqr calculated
axis. In figure 12(a-f) we present the magnitudes of the mean value of U
~ pqr over subsets of the tagged particles belonging to different
by taking the average of U
topologies and different dilatation bins. The results of simulation G has been used to
obtain these plots.
5.1.1. UNSS and SNSS
Referring first to the UNSS and SNSS topologies, we examine if the volume measures
~ pqr available in figure 12 can help
available in Table 3 and the conditional mean values of U
us understand the variation of lifetimes reported in figure 11. In figure 12 we observe that
~ pqr is higher at positive/negative dilatations than what
mean value of the magnitude of U
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 11(a). 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,
N. Parashar, S. S. Sinha and B. Srinivasan
2
2
1.5
1.5
-~ -Upqr -
-~ -Upqr -
22
1
0.5
-1
1
0.5
-0.5
0
0.5
1
-1
-0.5
aii
2
2
1.5
1.5
1
0.5
0.5
1
0.5
1
0.5
-0.5
0
0.5
1
-1
-0.5
0
aii
(c)
(d)
2
2
1.5
1.5
-~ -Upqr -
-~ -Upqr -
1
1
aii
1
0.5
-1
0.5
(b)
-~ -Upqr -
-~ -Upqr -
(a)
-1
0
aii
1
0.5
-0.5
0
aii
(e)
0.5
1
-1
-0.5
0
aii
(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)
even though the velocity is high, a significant increase in the volume of UNSS allows the
lifetime to decrease moderately as evident in figure 11(a). 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 negative to a very low value
at zero dilatation (figure 12(b)). 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 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
Lagrangian statistics in compressible turbulence
Simulation
A
E
G
UNSS
0.23
0.88
1.01
SNSS
0.28
1.37
1.57
SFS
0.23
0.97
1.13
UFC
0.26
1.03
1.18
Table 4. Average velocity of particles in p-q-r space (|Upqr | = | ∂p
p̂ +
∂t
simulations (Table 1)
23
UFS
0.87
0.90
∂q
q̂
∂t
+
∂r
r̂|)
∂t
SFC
1.46
1.76
for various
thus even a decrease in volume results into only a little increase in the lifetime (figure
11(b)).
Similar explanation can be provided for the variation in lifetimes observed in figure
11(c) and 11(d) for SFS and UFC topology using the volume data and mean magnitude
~ pqr of these topologies in figure 12(c) and 12(d).
of U
For the UFS and SFC topologies we do not observe any change in lifetimes in
figure 11(e). 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 11(e) 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.
5.1.2. 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
~ pqr in
trend is provided by Table 4, wherein we have included the mean magnitude of U
simulations A, E and G at tref = 4.0τ . We observe that in general, an increase in initial
turbulent Mach number increases mean velocity in the p − q − r space, consequently
reducing hLT i as evident in figure 11.
5.1.3. Role of Reynolds number
In figure 11 we have also shown results from simulation C-E. These simulations (C-E)
have identical Mt and different Reλ : 70, 100 and 150 respectively. We observe that the
lifetimes recorded for all these similations are almost similar. Thus we conclude that,
Reynolds number has negligible influence on the lifetime of topology.
Overall, our results in figure 11 clearly show that in terms of longevity, 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
population percentage of each of the six topologies at t = 4τ 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
24
N. Parashar, S. S. Sinha and B. Srinivasan
UNSS
26.07
SNSS
10.10
SFS
27.44
UFC
21.16
UFS
10.49
SFC
4.32
Table 5. Percentage topology composition for compressible simulation case G.
Composition %
Lifetime (κτ )
Percentage % of time spent
in different topologies CMTs Martı́n et al. (1998)
UNSS SNSS SFS UFC
25.2
5.4 43.5 25.9
1.80 0.53 3.32 2.08
53
21.5 20 5.5
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.
(Table 5) in other contexts (Suman & Girimaji 2010), 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 (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)
(Martı́n 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 (Martı́n
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 treated practically as an
incompressible flow.
The percentage composition in terms of the four topologies in this flow field is included
in Table 6 (at t = 4τ ). We observe that the mean lifetimes 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 Martı́n et al. (1998) using their
CMT approach. Since Martı́n et al. (1998)’s CMTs do show a cyclic change in topology
(U F C → U N SS → SN SS → SF S → U F C), 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 aspects of the dynamics of velocity gradients) we present a simple
Lagrangian statistics in compressible turbulence
25
0.25
0.4
q
0.2
RMS
q
0.2
0
r
0.15
0.1
-0.2 high concentration
(Vieillefosse tail)
0.05
-0.4
-0.15
-0.1
-0.05
0
r
(a)
0.05
0.1
0.15
0
0
1
2
3
4
5
(t − tref )/τ
(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.
comparison. In figure 13(a) we show a conditional mean trajectory (solid line) originating
at point (q = 0.3, 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, Martı́n
et al. (1998) estimate the characteristic ”cycle time” of topology interconversion as three
eddy turnover time. Next, we tag the same particles (in total 5,000) which were having
their initial invariants in the vicinity of the (q = 0.3, r = −0.05) at the reference time
of four eddy-turnover time and track their movement on the q − r plane. The locations
of these 5,000 particles after just one eddy turnover time has been shown in figure 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
a 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, Vieillefosse (1982)). Further, in figure 13(b), we present the root mean
squared value of q and r of the same sample of 5,000 particles which had their q, r in the
vicinity of (q = 0.3, r = −0.05) at the reference time of four eddy turnover time. The rms
values start increasing and within just eddy-turnover time 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 is indeed not appropriate.
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
26
N. Parashar, S. S. Sinha and B. Srinivasan
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 turbulent 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 − r space and the disparity in the speed of the fluid particles in the p − q − r space.
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 HighPerformance Computing Center (HPC) of the Indian Institute of Technology Delhi,
India.
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