Diffusion in inhomogeneous flows Unique (1)

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Computers & Fluids 88 (2013) 440–451

Contents lists available at ScienceDirect

Computers & Fluids

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p fl u i d

Diffusion in inhomogeneous flows: Unique equilibrium state
in an internal flow
Tapan K. Sengupta ⇑, Himanshu Singh, Swagata Bhaumik, Rajarshi R. Chowdhury
Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India

a r t i c l e

i n f o

Article history:
Received 2 November 2012
Received in revised form 10 July 2013
Accepted 2 October 2013
Available online 12 October 2013
Keywords:
Navier–Stokes equation
Inhomogeneous flow
Enstrophy transport equation

Direct numerical simulation
Diffusion
Rectangular lid driven cavity

a b s t r a c t
The role of diffusion in creating rotationality (enstrophy) is studied here and a transport equation for
enstrophy is derived to explain this connection. As an illustration, flow instabilities and pattern formation
are investigated here for an inhomogeneous internal flow with definitive boundary conditions. Results
obtained by direct numerical simulation (DNS) of flow inside a two-dimensional rectangular lid driven
cavity (RLDC) show that diffusion is responsible in forming patterns at a post-critical Reynolds numbers.
The transport equation for enstrophy derived from the Navier–Stokes equation in Eulerian framework
helps to explain the enstrophy spectrum in flows, specially in 2D flows, where vortex stretching is absent
as the dominant energy cascade mechanism to small scales. For the 2D flow in RLDC, diffusion and
convection provide a unique equilibrium state in an intermediate post-critical range of Reynolds number
around 6000. This is independent of the geometric aspect ratio (height to width of the cavity) of the
cavity greater than or equal to two. Such equilibrium can be observed in numerical simulations, only
when special care is exercised for diffusion discretization at high wavenumbers. Another motivation in
this work is to show that diffusion and dissipation are not identical for inhomogeneous flows, as opposed
to equating these in studies of homogeneous turbulent flows. Organized enstrophy is shown as a
consequence of over-riding action of diffusion in creating rotationality in this flow.

Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction
Investigation on the true role of diffusion has remained a
problem, ever since the time when its role was considered as
stabilizing fluid flow by damping disturbances, attributed to
Kelvin, Helmholtz and Rayleigh [1]. Equating viscous diffusion with
dissipation was the sole reason for early instability studies to
ignore diffusion, as discussed in [1,2]. However, such studies were
unable to explain instability of flow over a flat plate, while the
same flow was successfully investigated by solving Orr-Sommerfield equation (OSE) [3–5], which includes viscous diffusion in
the formulation. It was thought that retaining diffusion is
equivalent to producing an appropriate phase shift for a positive
feedback, which leads to flow instability.
Doering and Gibbon [6] studied the enstrophy transport for
two-dimensional periodic flows and obtained the evolution of
integrated enstrophy over the full domain as




d 1
kxk22 ¼ mkrxk22
dt 2

⇑ Corresponding author. Tel.: +91 512 2597945.
E-mail address: tksen@iitk.ac.in (T.K. Sengupta).
0045-7930/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.compfluid.2013.10.005

ð1Þ

where x is the vorticity and m is the kinematic viscosity. Here, the
enstrophy is defined over the full periodic domain by kxk22 . Thus,
one notes the effects of diffusion as strictly dissipative for periodic
flows viewed globally. In performing DNS of flows, one discretizes
all the terms and obtains the numerical solution without any
ambiguity. However, the point of view of equating diffusion with
dissipation is often used, as given above in Eq. (1), while
interpreting DNS results of homogeneous turbulent flows [7].
However if diffusion is viewed instantaneously at any point in a

flow, then the effects of diffusion is not strictly dissipative, as will
be explained here. When one looks at the time-averaged kinetic
energy of turbulent flows globally, effects of diffusion is again seen
to be as dissipative [8,9]. As shown in Eq. (4.34) of [9], time-average
of the diffusion term of Navier–Stokes equation manifests itself as a
combination of (i) a strictly dissipation term and (ii) another viscous
transfer term. However, the viscous transfer term integrates to zero
over the whole flow by the divergence theorem. This term is sometimes
also referred to as diffusive, because it is zero for homogeneous
turbulence. The authors furthermore add that the viscous transfer
term is negligible at high Reynolds numbers, except within the thin
viscous layers very near any solid surfaces while on the other hand,
the dissipative term is of crucial importance to turbulence energetics
everywhere. Similar observations are made in Section 3.3 of [8], with
respect to time-averaged turbulent kinetic energy. In the present

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T.K. Sengupta et al. / Computers & Fluids 88 (2013) 440–451

investigation, we look at the instantaneous local behavior of diffusion term and demonstrate in a flow the existence of a unique equilibrium state where the diffusive nature dominates over the
dissipative nature of the viscous term in Navier–Stokes equation.
We also show by an appropriate analysis requiring the consideration of the time-accurate total mechanical energy (as opposed
to time-averaged property of only the kinetic energy) of the flow
for which the action of viscous terms is not directly apparent. If
one constructs an equation for the total mechanical energy, as suggested in [10] and developed in [11], then the role of diffusion becomes clearer, as described in the following. One writes the
Navier–Stokes equation in rotational form for this analysis as

!
V ~
V
@~
V ~
p ~
~
~
þ
þ mr  x
 V  x ¼ r
q

2
@t

ð2Þ

where different variables represent their usual meanings and the
viscous effects is via the last term on the right hand side, written
as the curl of the vorticity vector, multiplied by the kinematic
viscosity.
Describing the total mechanical energy (E) by

E¼

p

q

þ

~

V ~
V
2

and taking a divergence of the above Navier–Stokes equation yields
the distribution of E by the following equation

~Þ
r2 E ¼ r  ð~
V x

ð3Þ

Note that the viscous term drops out identically due to a vector
identity and the right hand side originate strictly from convection
term. However, the right hand side of the above equation can be expressed using the vector identity in further simplification of this
equation

~Þ ¼ x
~ x

~ ~
r  ð~
V x
V  ð r  xÞ
Denoting the instantaneous point property of enstrophy by
~ x
~ , Eq. (3) can be written as
X1 ¼ x

~Þ
r2 E ¼ X1  ~
V  ðr  x

ð4Þ

This equation shows the relevance of enstrophy and the diffusion
operator to be central in distributing total mechanical energy. In
[7], a similar equation has been written for the static pressure
(see Eq. (1.2) of the reference) which in present notations is given
by

r2

 
p

q

¼ ðX1  =mÞ=2

ð5Þ

where  = 2m sij sij and sij is the symmetric part of the strain tensor.
This equation is wrongly stated to be valid only for homogeneous
turbulence. Eq. (4) is written for any general flow derived from
Navier–Stokes equation without making any assumption or simplification. One notes that the term on the right hand side of Eq. (4)
2~
can be written as ~
V  rm V , in drawing an analogy with the term /m,
on the right hand side of Eq. (5), even though the right hand side

of Eq. (4) purely originates from convection term. This source of
confusion prompted the authors in [7,12,13], to equate the roles
of enstrophy and dissipation. One of the motivations here is to
highlight the connection between diffusion and enstrophy for flows.
The development and use of total mechanical energy equation to
study any flow instability is described in detail in [1,11].
In trying to understand the role of diffusion in creating rotationality, an evolution equation is also developed here for enstrophy, as
a point property and its higher powers for any flow. This exercise
explains the roles of diffusion, dissipation and creation of rotationality progressively to smaller scales. To demonstrate that this is

441

valid for any flow, we focus on a 2D flow, which does not have
the presence of vortex stretching to create smaller scales.
Reported DNS in [7], used Fourier spectral discretization in
space and second order Runge–Kutta time integration to solve Navier–Stokes equation. This space–time dependent discretization is
very restrictive in parameter space, due to its numerical instability
and also due to its high dispersion error, as shown by spectral analysis in the appendix using the 1D convection equation. It is obvious
that any method which cannot solve this simple convection equation, is practically of little use in solving more complex Navier–
Stokes equation. The dynamical equilibrium in flows is a balance
between convection and diffusion processes, both of which have
to be captured correctly in equal measure. One of the salient features of the presented results here is to show the existence of a
universal equilibrium between convection and diffusion in Navier–Stokes equation. This can be captured only by carefully designed numerical methods explained in the next section and
appendix.
There have been significant progresses made in developing high
accuracy compact schemes, which are dispersion relation preserving (DRP) and has been used for inhomogeneous flows. A similar
method has been used in [14,15] to simulate an inhomogeneous
zero pressure gradient boundary layer from the receptivity to a
fully developed 2D turbulent stage, displaying k3 spectrum for
the energy. One of the motivations here is to show that for 2D
flows, rotationality is created at different scales via the enstrophy
cascade. This establishes a link between diffusion and enstrophy
for a wall-bounded inhomogeneous flow.
Here, the flow inside a RLDC driven by uniform translation of
the top lid (U1) is used as an example to reveal the role of diffusion
in Navier–Stokes equation, where pronounced rotationality is created by simple translation of the top lid. It is well known [16] that
turbulence is characterized by many attributes, out of which the
primary ones being rotationality and broad-band energy spectrum
created by various instability mechanisms.
Flow in a square LDC has been studied and a unique topology
(triangular core vortex and gyrating satellite vortices) is described
in [17,18]. This was obtained with the help of highly accurate discretization of convection and diffusion processes in the flow. Flow
in RLDC is more complicated due to the presence of multiple cells
having distinct vortical structures. The upper cell of RLDC resembles the flow in a square LDC, which in turn, drives the cell below
and so on. The rotational flow structures seen in various cells of
RLDC are caused by the translational motion of the lid, with each
cell showing presence of vortices of both signs.
The manuscript is formatted in the following manner. In the
next section, governing equations and the numerical methods to
solve 2D flow inside the RLDC are described. This is followed by
a section describing the flow inside RLDC, with respect to the instability sequence, topology and Hopf bifurcation of the flow. To explain this instability sequence and induced rotationality,
transport equation for enstrophy has been derived in Section 4.
In Section 5, we emphasize the requirements on diffusion discretization in DNS. This is followed by summary and conclusion of the
results. In the appendix, the spectral analysis of numerical schemes
used for convection equation has been carried out.

2. Governing equations and numerical formulation
We have used the streamfunction–vorticity (w, x)-formulation
of Navier–Stokes equation to obtain numerical results reported for
the RLDC shown in Fig. 1. This formulation allows satisfaction of
solenoidality for velocity and vorticity identically. The non-dimensional form of vorticity transport equation (VTE) for 2D flows is
given by

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T.K. Sengupta et al. / Computers & Fluids 88 (2013) 440–451

(x, i)
0

0.2

similar expressions are used for the second derivative, while for
the first derivative, we have signs of the coefficients are changed
on the right hand side. For j = 2 and j = N  1, the derivatives are
evaluated by using

U∞
0.4

0.6

0.8

1

0





2b2 1
8b2 1
u1 

þ
3
2
3
3



8b2 1
2b
u2 þ ð4b2 þ 1Þ u3 
þ
u4 þ 2 u5
3
3
6

u02 ¼

(y, j)
0.5

D1

1
h





2bN1 1
8bN1 1

þ
uN 
uN1
3
3
3
2



8bN1 1
2b
uN3 þ N1 uN4 ð11Þ
þ
þ ð4bN1 þ 1Þ uN2 
3
6
3

u0N1 ¼ 

Jet
1

Spiral
Vortices

ð10Þ

H

1
h

with b2 =  0.025 and bN1 = 0.09 [18]. The second derivatives at
j = 2 and (N  1) are obtained using identical stencils used for j = 1
and N, resepctively. One simultaneously obtains the first and second
derivatives by numerical solution of the stencils along with the
closure schemes given above. Advantages of this scheme are discussed in [15,18]. Effectiveness parameters are defined based on
the spectral representation of the first and second derivatives and
their properties are studied in the appendix.

D2
1.5

W
D3
2
Fig. 1. Definition sketch of RLDC with AR = H/W = 2. Representative solution in the
form of vorticity contours is for Re = 6000 at t = 1187. Time histories of vorticity are
recorded at D1, D2 and D3.

@x ~
1
þ V  rx ¼ r2 x
Re
@t

ð6Þ

where x is the non-zero component of vorticity in the z-direction
and ~
V is the velocity and Re is the Reynolds number defined with
respect to the width (W) of the cavity. This is solved along with
the streamfunction equation (SFE) given by

@2w @2w
þ
¼ x
@x2 @y2

ð7Þ

Impulsive start is assumed for the initial condition on w and x,
while no-slip condition at the walls is used as the boundary condition. The SFE (Eq. (7)) is solved using unpreconditioned Bi-CGSTAB
method [19], while the time integration of VTE (Eq. (6)) is carried
out using fourth order four-stage Runge–Kutta (RK4) method. In
all computations, we have used uniform grid with 257 points in
the x-direction and 257 ⁄ AR points in the y-direction, where AR is
the aspect ratio (height by width, H/W) of the rectangular cavity.
A time step of Dt = 1  103 is used for the reported simulations.
The first and second derivatives of x in the VTE are evaluated
using combined compact difference (CCD) scheme reported in
[18]. This scheme is used for its good diffusion discretization property and high spatial resolution as compared to other schemes [18].
General stencils of CCD scheme for an internal node are given by

7 0
h 00
15
ðu þ u0j1 Þ þ u0j 
ðu  u00j1 Þ ¼
ðujþ1  uj1 Þ
16 jþ1
16 jþ1
16h

ð8Þ

9 0
1
3
ðu  u0j1 Þ  ðu00jþ1 þ u00j1 Þ þ u00j ¼ 2 ðujþ1  2uj þ uj1 Þ
8h jþ1
8
h

ð9Þ

where, u is a function defined on a domain of N equidistant points
with grid spacing h and primes indicate derivatives with respect to
independent variables. Both the equations are used for j = 3 to
(N  2), where N is the number of grid points. With Dirichlet boundary conditions at j = 1 and j = N, we have 2N unknown derivatives.
Out of these unknowns, the derivatives at j = 1 are given as:
2
u01 ¼ ð1:5u1 þ 2u2  0:5u3 Þ=h and u001 ¼ ðu1  2u2 þ u3 Þ=h . At j = N,

3. Flow in rectangular lid driven cavity
Here Navier–Stokes equation has been solved in (w, x)-formulation using CCD scheme for spatial discretization and RK4 time
integration scheme. In Fig. 1, representative vorticity contours
are shown for Re = U1W/m = 6000 at t = 1187 to explain the geometry and selected sampling points’ location with respect to the coordinate system chosen. This definition sketch is for the RLDC with
AR = H/W = 2, with U1 in the positive x-direction. We have used W
and U1 as the length and velocity scales and time is non-dimensionalized by W/U1. Contours drawn by solid lines indicate positive vorticity and the dashed lines indicate negative vorticity in
all figures shown. The flow field is analyzed, with three points chosen at D1, D2 and D3, where time histories of x are recorded. Point
D1 is in the top cell at (x = 0.50, y = 0.75), while points D2 and D3 are
in the second cell at (x = 0.25, y = 1.25) and (x = 0.004, y = 1.996),
respectively. Indicated jet in Fig. 1 is responsible for transferring
momentum and energy from the top to bottom cells.
Figs. 2–4 show time histories of vorticity (left frames) and corresponding Fourier transforms (right frames) for D1, D2 and D3. Results shown are for Re = 5757, 6000, 6250, 6300 and 6700. The time
series at these points for Re = 5757 show an initial transient followed by temporal instability and subsequent non-linear saturation. Extensive simulations and Hopf bifurcation analysis [20,21]
for this geometry reveal a critical Reynolds number of 5752. Hopf
bifurcation indicated in the time series is described in the following sub-section. The time variation is clearly multi-periodic, as
noted in the FFTs. For Re = 5757, one notices absence of a single
dominant mode for D1. However for D2 and D3, one notices a dominant mode at f = 0.1253, while there are other secondary modes of
non-negligible amplitudes. The dynamics is distinctly different for
Re = 6000, as noted from the corresponding frames of Figs. 2–4,
which show essentially single-period dynamics. Specifically for
the representative points at D2 and D3, time variation is essentially
monochromatic with f = 0.5938, while D1 has an additional superharmonic at f = 1.184. The time series for Re = 6000 shows significantly reduced amplitude for both D2 and D3, implying very low
levels of disturbance in the second cell, while at D1 disturbance
is of similar magnitude, as for Re = 5757. With increase of Re up
to 6250 and 6300, one notices the strengthening of the spiral chain
of vortices in the second cell. While this is indicated in the time
series, the FFT indicates the presence and strengthening of other

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T.K. Sengupta et al. / Computers & Fluids 88 (2013) 440–451

Re = 5757

Re = 5757

-4

10

-6

10

1000

0.5

t

0.5

1.5

500

f

Re = 6000

10
10

1000

0.5
0.5

t

1

1.5

500

Re = 6250

-4

0.5
0.5

1

500

1.5

f

t

-4

0.5

-6

t

1

500

1.5

-4

10

1
0.5

10

-4

10

-6

0.5

10

-2

10

-4

10

-6

0.5

10

-2

10

-4

10

-6

0.5

1.5

500

1.5

1

1.5

1

1.5

1

1.5

-4

10

-6

0.5

1

1.5

f

Fig. 3. Vorticity time history (left frames) and the corresponding Fourier transform
(right frames) shown for point D2 (x = 0.25, y = 1.25).

Re = 5757

A (f)

ω

0.002
0
-0.002

500

10

-4

10

-6

0.5

1000

t

ω

0.002

Re = 6000

500

10

-4

10

-6

0.5

1000

t
0.002

Re = 6250

500

10

-4

10

-6

0.5

1000

t
0.002

Re = 6300

500

10

-4

10

-6

0.5

1000

t
0.002

Re = 6700

500

1000

t

1.5

1

1.5

1

1.5

1

1.5

f

0
-0.002

1

f

0
-0.002

1.5

f

0
-0.002

1

f

0
-0.002

ω

distinct modes, as the Reynolds number is increased. However, as
the Reynolds number is increased further, the flow suffers a qualitative change, as evident from the vorticity time series and FFTs
for the flow at Re = 6700. This flow shows multi-periodicity in both
the cells, with higher levels of fluctuations, due to presence of multiple harmonics with side bands. Thus the flow in the bottom cell,
shows a quieter state for Re = 6000 with a single frequency, as
compared to the flow for Re = 5757. For Re = 6700, the flow is characterized by chaotic multi-periodic time variations with prominent
side-bands as shown in Figs. 2–4 – a characteristic feature discussed for low dimensional dynamical systems depicting chaos
and soft turbulence in [22].
The quieter state for Re = 6000 is studied further in Fig. 5,
where the Fourier amplitude of the most dominant mode
[Ae(f)] is plotted as a function of Re for the points at D1 (top
left), D2 (top right) and D3 (bottom). For D3, one notices a
range of Re around 6000, for which the flow is quieter. In
contrast, the amplitude of vorticity for D1 shows significantly
higher variations till Re = 6300, as compared to the other
points. The amplitude of the most dominant mode for D1
shows a relative decay with increase in Reynolds number beyond Re = 6300. This does not mean that the disturbance
amplitude in the top cell as a function of Re has reduced. This
is a manifestation of energy being distributed equally among
multiple modes in the top cell, as is readily evident from
Fig. 2 showing the presence of various modes for this Reynolds
number. This once again reveals soft turbulence, which leads to
enhanced mixing in the top cell and higher transport of energy
to the bottom cells. Note that apart from the second cell from
the top, there is an incomplete cell just below it attached to
the bottom wall in Fig. 1.

10

t

f

Fig. 2. Vorticity time history (left frames) and the corresponding Fourier transform
(right frames) shown for point D1 (x = 0.50, y = 0.75).

1000

-2

A (f)

1

10

A (f)

0.5

ω

t

10

ω

1000

1

f

A (f)

ω

A (f)

10

-6

500

1000

Re = 6700

-2

-2

-2

A (f)

Re = 6700

10

t

f

ω

-1.5

1

10

0.5

0.5

f

A (f)

ω

A (f)

10

10

1000

-6

t

-2

-2
500

1000

Re = 6300

Re = 6300

ω

-1.5

1

10
10

1000

10

f

A (f)

ω

A (f)

10

-6

500

1000

Re = 6250

-2

-2

-4

t

f

ω

-1.5

1

A (f)

-4
-6

500

10

f

Re = 6000

10

ω

A (f)

-2

1000

-2

t

-2

ω

-1.5

1

10

A (f)

500

1

A (f)

-2

10

ω

A (f)

-2

ω

-1.5

10

-4

10

-6

0

0.5

f

Fig. 4. Vorticity time history (left frames) and the corresponding Fourier transform
(right frames) shown for point D3 (x = 0.004, y = 1.996).

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T.K. Sengupta et al. / Computers & Fluids 88 (2013) 440–451
1

10

0

10 -1

(i) D1

(ii) D2

10 -2

Ae (f)

Ae (f)

10

10

-1

10

-2

10 -3

10 -4
6000

6500

6000

Re

6500

Re
10 -2

(iii) D3

Ae (f)

10 -3

10 -4

10 -5
6000

6500

Re
Fig. 5. Variation of equilibrium amplitude of the most dominant frequency (Ae(f)) plotted as a function of Re for (i) D1, (ii) D2 and (iii) D3.

3.1. Multiple Hopf bifurcations
Detailed Hopf bifurcation analysis of flow inside square LDC
was reported in [20,21]. It was shown that the time series of vorticity depicted initial transient after which the linear growth was
followed by non-linear saturation. The vorticity perturbation in linear growth region is expressed in terms of various instability
modes using Galerkin expansion. The time-dependent amplitudes
of these instability modes during linear growth are given by,
Aj ðtÞ ¼ ðconst:Þesj t , as described in [20]. Saturation of disturbance
beyond the linear region can be expressed by Landau-Stuart-Eckhaus (LSE) equation in [1,20,21] and references contained therein

dAj
¼ sj Aj þ Nj ðAk Þ
dt

ð12Þ

where Nj(Ak) accounts for all non-linear interactions among
different modes (including self-interaction). In contrast, in the Stuart-Landau equation [1,20], only one dominant mode is considered
with sj = rr + ix1, representing its linear complex temporal growth
exponent. For the non-linear interaction, Landau and Stuart considered only the self-interaction term given by, N j ¼  2l AjAj2 , where
l = lr + ili is the Landau coefficient. Writing A = jAjeih, one can rewrite
Eq. (12) for the amplitude and phase of the single dominant mode
(A) as

djAj2
¼ 2rr  lr jAj4
dt

ð13Þ

dh
li
¼ x1  jAj2
2
dt

ð14Þ

An equilibrium
amplitude can be obtained from Eq. (13) as,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jAe j ¼ 2rr =lr . This analysis by Landau and Stuart forms the basis
of classical Hopf bifurcation. For steady state equilibrium flows,
there is no vorticity variation with time and Hopf bifurcation takes
the flow from quiescent initial state to a periodic state, after the onset of starting the lid impulsively. Such bifurcation diagrams showing variation of (jAej) vs Re have been presented in [1,20] for square
LDC. These diagrams also depict multiple bifurcations, due to the
presence of several modes, in contrast to the single dominant mode

assumption in Landau’s and Stuart’s analysis. Such complex bifurcations and multi-periodic vorticity variations are also noted here for
the RLDC. Presence of multiple Hopf bifurcation requires the use of
Eq. (12), as discussed in details in [1].
While the flow in the top cell becomes supercritical for
Re = 5752, displayed behaviour of the dominant mode in Fig. 5(i)
for the top cell is indicative of multiple bifurcations discussed
above. However beyond Re = 6300, non-linearity and chaotic nature of the flow takes over in the top cell. The second cell receives
its energy from the top cell by the indicated impinging jet from
right to left in the intervening space between these two cells. At
the same time, this jet also shields the lower cell from the top cell
events, as evident from Fig. 5(ii) for D2. This point also displays primary Hopf bifurcation at the same Reynolds number of 5752.
Increasing Re above this critical value, one notices a very coherent
structure forming in the second cell, while the disturbance field in
the second cell dramatically decreases, as seen in Fig. 5(ii). When
Re > 5800, another instability and bifurcation is noted in the second cell, as seen in the sub-figure. To begin with, this instability
is characterized by a single-period phenomenon, as shown in
Fig. 3 for Re = 6000 at D2. However this instability is global in nature, as can be seen from Fig. 2 in the top cell. Non-linearity of disturbance field in the top cell creates the superharmonic seen for
Re = 6000 in Fig. 2. The dominant mode shown in Fig. 5(ii), causes
another bifurcation to start for Re ’ 6250, resulting in breakdown
of the coherent spiral chain of vortices in the second cell. Once
again, such an event testifies the presence of multiple Hopf bifurcations [1,20]. Fig. 5(iii) indicates the dynamics in the third incomplete cell attached to the bottom wall, which also reflects global
dynamics of the flow inside the cavity. For example, formation of
the coherent spiral chain of vortices quietens the disturbance field
globally. This is seen in Fig. 5(iii) by the constant amplitude dominant mode centered around Re = 6000. Instability of this chain of
vortices is also recorded via the growth of the dominant mode in
Fig. 5(iii) beyond Re = 6300.
In Fig. 6, flow topologies inside the RLDC are shown by vorticity
contours for AR = 2, for the indicated Reynolds numbers at t = 1187,
after the initial transients have disappeared. For Re = 5757, multicellular vortical structures are seen in the top two cells. In the top-

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T.K. Sengupta et al. / Computers & Fluids 88 (2013) 440–451

(ii) Re = 6000

x

x

0.2 0.4 0.6 0.8

1

0

0.2 0.4 0.6 0.8

(iii) Re = 6700
x
1

0

0

0.5

0.5

0.5

1

1

1

y

0

y

y

0

(i) Re = 5757

1.5

1.5

1.5

2

2

2

0

0.2 0.4 0.6 0.8

1

Fig. 6. Vorticity contours of the flow in RLDC with AR = 2 plotted for (i) Re = 5757, (ii) Re = 6000 and (iii) Re = 6700 at t = 1187.

most cell, gyrating orbital vortices with negative sign are noted
without any central core. Motion of the lid induces a recirculating
cell in the top, whose bottom edge creates a jet-like structure
transporting conserved variables from right to left. The impinging
jet on the left wall also creates the middle cell with vortices, which
trickles down from the site of the jet impinging on the left wall.
The motion of these vortices are very slow, as compared to the
gyrating vortices in the topmost cell. For Re = 6000, one notices distinctly different and almost quasi-steady structures, with coherent
vortices on a spiral up to the center of the middle cell. This is due to
a unique equilibrium between convection and diffusion, as will be
discussed in the following section. Vorticity contours for Re = 6700
in Fig. 6(iii) are characterized by the absence of the coherent spiral
vortical structure in the middle cell. The enhanced energy convected by the impinging jet, destabilizes the coherent spiral chain
of vortices in the middle cell, indicating a departure from an intermediate equilibrium state.
3.2. Universal equilibrium state for RLDC
The unique feature of flow in RLDC attaining an intermediate
equilibrium state with AR = 2 for Re = 6000, is also noted for RLDC
with higher aspect ratios shown in Fig. 7 with vorticity contours,
for AR = 2.5, 3 and 4 shown at t = 1190. Despite minor differences
in the top cell, the second cell shows identical coherent spiral vortical structures, as in Fig. 6 for AR = 2. Thus, the coherent spiral vortex chain in the second cell is a universal flow feature for RLDC
with AR P 2 for a narrow range of Re around 6000. However, increase in AR causes the incomplete bottom cell to evolve into a full
cell, while another incomplete cell attached to the bottom wall
makes its appearance, as seen in Fig. 7.

dependent variable to study transitional and turbulent flows. We
explain instabilities and pattern formations in RLDC, with the help
of enstrophy transport equation (ETE) derived from the nondimensional VTE in tensor notation given for 3D flows by

@ xi
@ xi
@ui
1 @ 2 xi
þ uj
¼ xj
þ
@t
@xj
@xj Re @xj @xj

ð15Þ

where subscripts, i, j = 1, 2 and 3, represent Cartesian axes and repeated index implies summation. Taking a dot product of Eq. (15)
with xi and using X1 = xixi to represent the local enstrophy, one
obtains its transport equation as




@ X1
@ X1
@ui
1 @ 2 X1
2 @ xi
@ xi
þ uj
 2xi xj
¼

@t
@xj
@xj Re @xj @xj Re @xj
@xj

ð16Þ

The third term on the left hand side (LHS) is due to vortex stretching
(corresponding to the first term on the right hand side (RHS) of Eq.
(15)), which is absent for 2D flows. The diffusion of xi gives rise to
RHS terms in Eq. (16). This contains a diffusion of X1 (the first term
of RHS) and the second term of RHS represents strictly a loss or dissipation term for the transport of X1.
The present study views enstrophy as a point property in the
flow domain and is different from the traditional approaches [6],
where the enstrophy over the full domain is traced. The traditional
approaches utilize the simplification brought about for problems
which are homogeneous and hence periodic. However, such
restriction to periodic flows leaves out most of the practical problems which are inhomogeneous and the present study is for those
class of problems.
Focusing on the case of 2D RLDC flow with the non-zero component of vorticity (x), the ETE can be written in vector form as

4. Enstrophy transport equation



DX1
2 1 2
¼
r X1  ðrxÞ2
Dt
Re 2

The role of enstrophy in fluid flow to create rotationality is similar to kinetic energy describing the translational motion in fluid
flow. While vorticity also describes rotationality, a measure of it
is naturally obtained via enstrophy unambiguously describing
the energy expended by the system in creating and sustaining rotationality. In all flows, physical instabilities take an equilibrium
state to another and in the process, the energy of the system is
redistributed into rotational and translation degrees of freedom.
For example, in the present flow inside RLDC, the input translational energy is partly converted into rotational form and this will
be manifested in creating enstrophy. Thus, enstrophy is a natural

We note that the first term on RHS of Eq. (17) is missing from Eq.
(1), due to periodicity of the flow. Strictly negative RHS in Eq. (1)
implies the action of viscosity to dissipate energy globally. In
contrast for inhomogeneous flows, the first term on RHS of Eq.
(17) can be either positive or negative. Therefore, the diffusion
term in ETE can create or destroy rotationality, determined by
the sign of RHS in Eq. (17). The two terms on RHS taken together
determine net growth or decay of X1. Thus, the diffusion term of
the VTE should not be identified strictly as dissipative for general
flows. Diffusion and dissipation have been used interchangeably
[7], which is true for homogeneous flow in the global sense. In
ETE, X1 is strictly positive and then RHS being positive would

ð17Þ

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T.K. Sengupta et al. / Computers & Fluids 88 (2013) 440–451

x
0

0.5

(iii) AR = 4

(ii) AR = 3

(i) AR = 2.5

x
1

0
0

0.5

0.5

1

1

x
1

0

0.5

1

0

1

y

y

0

0.5

1.5

2

2

2.5

2.5

y

1.5

2

3

3

4
Fig. 7. Vorticity contours for Re = 6000 plotted for RLDC flow with (i) AR = 2.5, (ii) AR = 3 and (iii) AR = 4 at t = 1190.

indicate the diffusion to cause local instability. It would act as a
sink of X1, where RHS is negative. This provides a mechanism of
creating rotationality at different scales by diffusion and is
distinctly different from the concept of creating smaller scales
by vortex stretching, as the only dominant mechanism of generating small eddies relevant only for 3D flows. We also note that
this role of diffusion in creating new length scales is ubiquitous
for flows in both 2D and 3D.
To explain the mechanism of enstrophy creation at multiple
scales simultaneously, one can plot the right hand side of the
ETE. In Fig. 8, the regions where the RHS of Eq. (17) is positive

(i) Re = 5757

(ii) Re = 6000

x
0

0.5

are shown by dark shades for Re = 5757, 6000 and 6700 at
t = 1187 for the RLDC with AR = 2. The blank regions in the figure
indicate zones where RHS is negative, i.e., where diffusion leads
to loss of X1. Snapshot for Re = 6000 reveal the unique spiral vortical chain to form exactly at those places where RHS is positive,
indicating the correspondence of regions where enstrophy is created
by diffusion as explained by ETE. Creation of such coherent structures, simultaneously quietens the flow in the neighborhood of
the spiral, as indicated in Figs. 4 and 5 for the representative point
D3. This unambiguously establishes that at selective Reynolds
numbers, diffusion assists in creating rotational flow structures.

(iii) Re = 6700

x
1

0

0.5

x
1

0

0.5

0.5

0.5

1

y

0

y

0

y

0

0.5

1

1

1

1.5

1.5

1.5

2

2

2

Fig. 8. Regions where right hand side of enstrophy transport equation is positive shown as black regions for RLDC with AR = 2 for (i) Re = 5757, (ii) Re = 6000 and (iii)
Re = 6700 at t = 1187.

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T.K. Sengupta et al. / Computers & Fluids 88 (2013) 440–451

(i) Contours of F
800

800
7

(ii) Contours of FU

A
0.1

0.0037

600

5.5

6

600

4.5
3

400

400

0.0061

3.5

2
2. 5

1

200

0.0037

ky

ky

4

0.014

0.027

5

0.

01

9

6.5

200

1.5

0.011
0.0079
0.0061

0.019

0.5
0.0037

0.05
0.1

200

400

600

800

200

kx

400

600

B
800

kx

Fig. 9. Contour plots of F and FU for Re = 6000 of AR = 2 cavity at t = 1480 in the (kx, ky)-plane.

To further investigate effects of diffusion in Eq. (17) at multiple
scales, one can derive transport equations for higher powers of X1.
n1

Multiplying Eq. (17) with X1 and defining Xn ¼ X21
for X2 the following transport equation



DX2
1
¼ 2Re1 r2 X2  ðrX1 Þ2  X1 ðrxÞ2
2
Dt

, one obtains

ð18Þ

X2
X1
Noting further that DDt
¼ 2X1 DDt
, one can write Eq. (18) as the ETE,
i.e.. an evolution equation for X1. Multiplying Eq. (18) with X2 and
simplifying one can obtain transport equation for X3, which can be
used to write the ETE involving X1, X2 and X3. This process can be
generalized to obtain the transport equation for Xn as

DXn
1
¼ 2Re1 r2 Xn  ðrXn1 Þ2  C
2
Dt




ð19Þ

where
n2
X
C¼
2nk1
k¼0

n1
Y

!

Xj ðrXk Þ2

j¼kþ1

ð20Þ

P
Q
and X0 = x with
indicating summation over all k’s and indicating the product of all the jth elements.
Also, the substantive derivative of Xn can be written and simplified as

!
n1
Y
DXn
DXn1
DXn2
DX1
n1
¼ 2Xn1
¼ 2Xn1  2Xn2
¼2
Xj
Dt
Dt
Dt
Dt
j¼1
which can be further simplified to give

DXn
DX1
¼ 2n1 Xb
1
Dt
Dt

ð21Þ

where b = 1  2n2. Using above relations, one can rewrite Eq. (19)
as the ETE given by



DX1 Re1 Xb1 1 2
r Xn  ðrXn1 Þ2  C
¼ n2
Dt
2
2

ð22Þ

One notes that while writing the transport equation for Xn, the diffusion term from the transport equation for Xn1 contributes two
terms; one of which is strictly dissipative (dependent on Xn2)
and the other as a diffusion term for Xn1. The diffusion term
involving Xn1 can be furthermore expressed into two terms
involving a strictly dissipative term with Xn1 and another diffusive
term involving Xn2. This process can cascade indefinitely in Eq.
(22), for increasing n with the leading term as a diffusion term
and the rest are strictly dissipative. Higher order moments of enstrophy will contribute more for higher wavenumbers, implying that

the order of even moments of enstrophy will be restricted by the
energy supplied to the flow.
For 3D flows as well, the RHS of Eq. (22) is present as the forcing
term. However, in this case, the vortex stretching term is retained.
The ETE for 3D flow is same as given by Eq. (16). Following a similar approach as in deriving the transport equation for Xn for 2D
flows, the transport equation for Xn can be derived for 3D flows
to be given by



DXn
1
¼ 2Re1 r2 Xn  ðrXn1 Þ2  C
Dt
2


n1
Y
@ui
þ 2n Xk xi xj
@xj
k¼1

ð23Þ

where expression for C is same as in Eq. (20). Using Eq. (21) one can
rewrite the ETE for 3D flows as

"
#
DX1 Re1 Xb1 r2 Xn
@ui
2
¼ n2
 ðrXn1 Þ  C þ 2xi xj
Dt
2
@xj
2

ð24Þ

One notes that the diffusion term gives rise to the enstrophy cascade for both 2D and 3D flows, for which the contribution at higher
wavenumbers depends upon the value of n, decided by the energy
supplied to the fluid dynamical system. However in 3D flows, vortex stretching is also present which provides an additional mechanism of energy redistribution process. This indicates that in 3D
flows, generation of different scales of vorticity is due to enstrophy
cascade via the diffusion term and energy cascade is via the vortex
stretching which is implicit in convection process. In 2D flows, it is
only the diffusion term which gives rise to enstrophy (and hence
vorticity) at different scales.
5. Requirements on diffusion discretization
Accurate diffusion discretization is very important as it helps in
controlling aliasing [18]. This is also important from the point of
view of determining correct dynamics of the enstrophy. For the
present case, the VTE given by Eq. (6) can be integrated over the
whole cavity and using Gauss’ divergence theorem, one can show
that the convection term has zero contribution, as shown below

Z Z

r  ð~
V xÞ dx dy ¼

I

^ dl ¼ 0
ð~
V xÞ  n

ð25Þ

^ is the unit normal vector at the boundary and dl is the elewhere, n
mentary tangential length along the boundary. Hence, the time rate
of vorticity integrated over the whole domain is only dependent
upon the diffusion of vorticity integrated over the whole domain.

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T.K. Sengupta et al. / Computers & Fluids 88 (2013) 440–451

The diffusion of vorticity (r2x) integrated over the whole 2D domain can be written, again using Gauss’ divergence theorem as

Z Z
If

r2 x dx dy ¼

Z Z

r  ðrxÞ dx dy ¼

I

rx  n^ dl

ð26Þ

the

vorticity is represented in spectral plane as,
RR
Uðkx ; ky Þeðikx xþiky yÞ dkx dky , then the above equation
can be simplified for the RLDC with AR = 2 to obtain

xðx; yÞ ¼
Z Z

r2 x dx dy ¼
¼

Z Z

Z Z



 ikx

 kx ky
Uðkx ; ky Þ dkx dky
e  1 e2iky  1
þ
ky kx

Fðkx ; ky ÞUðkx ; ky Þ dkx dky

ð27Þ




where Fðkx ; ky Þ ¼ eikx  1 e2iky  1 ðkx =ky þ ky =kx Þ. The factor F and
the quantity FU are plotted in frames (i) and (ii) of Fig. 9, respectively, in the (kx, ky)-plane. One can perform similar analysis for
any AR case, the present analysis for AR = 2 is simply for the purpose
of illustration, where we consider flow for Re = 6000 at t = 1480.
One notes that the factor F in frame (i) is maximum when both kx
and ky are maximum. This immediately suggests that one must
choose a method which does not filter out these high wavenumber
components, as also evident from frame (ii) of Fig. 9. These high
wavenumber combinations give rise to contributions to higher aliasing error, the region marked above the dotted line AB. This highlights that the numerical scheme must be able to resolve high
wavenumber components in (kx, ky)-plane, as the method of [18].
Here, the same CCD scheme has been used, which accurately captures the high wavenumber components. How different numerical
methods handle diffusion discretization is discussed in the appendix, by comparing the CCD scheme with second order central difference scheme (CD2) as a reference. It is noted that many
conventional schemes for diffusion discretization do not have the
ability to represent diffusion operator accurately.
6. Summary and conclusion
The role of diffusion and its relation to creating enstrophy (or
rotationality) during flow instabilities and forming patterns are
investigated here for an inhomogeneous flow. We have chosen
the problem of flow inside RLDC with definite boundary conditions
representing an inhomogeneous flow shown in Fig. 1. We have
used results obtained by DNS of flow inside the RLDC to show that
diffusion is responsible for forming a pattern of spiral vortices in
the second cell from the top, at a universal Reynolds number of
Re = 6000, as noted in Figs. 1 and 6 for aspect ratio, AR = 2. The reason for formation of the special flow structures is explained with
the help of vorticity time series in Figs. 2 to 4, and its Fourier transform. It is also noted that following the first Hopf bifurcation for
the RLDC with AR = 2 for Re = 5752, the flow exhibits a narrow
range of Reynolds number around 6000, where the second cell of
the cavity is dominated by the coherent pattern formation with
low levels of disturbance elsewhere, as noted in Fig. 5. Role of subsequent bifurcations in creating chaotic flow with multi-periodic
disturbances are also noted for Re = 6700, which is indicative of
soft turbulence in internal flows [20,22]. The bifurcation analysis
and the study of instabilities in this work are carried out based
on the solution of full Navier–Stokes equation and not on linearized theories.
The spiral vortex chain is also shown to exist for higher aspect
ratio of the cavity, shown for AR = 2.5, 3 and 4 in Fig. 7 for
Re = 6000. This implies a universal equilibrium state determined
by the supply of energy (represented by Re) and not on the geometry (represented by AR) of the cavity. To explain the creation of
rotationality by the diffusion operator, we have developed a transport equation for enstrophy (ETE) from the Navier–Stokes equation

to explain the true role of diffusion for general flows. Utility of the
developed ETE is established by plotting the right hand side of this
equation in Fig. 8, which shows the direct correlation of the patterns formed by the rotating cells with the vorticity contours in
Fig. 6. For the flow in RLDC, unique equilibrium state is shown as
a consequence of non-negligible diffusion along with convection
in an intermediate post-critical range of Reynolds number around
6000. Organized enstrophy is shown as a consequence of over-riding action of diffusion in creating rotationality in this flow. Such
equilibrium can be observed in numerical simulations, only when
special care is exercised for diffusion discretization. In this work,
we have also shown that diffusion and dissipation are not identical
for inhomogeneous flows, as opposed to equating these for simulation results of homogeneous turbulent flows [7]. It is furthermore
explained from ETE how different scales are created, shown by the
various powers of enstrophy in Eqs. (20) and (22) for 2D flows and
in Eq. (24) for 3D flows.
Thus, the present research was conducted with the sole aim of
showing the physical and numerical aspects of diffusion process
for the accurate simulation of Navier–Stokes equation for inhomogeneous flows. It is emphasized that physically the role of diffusion
for inhomogeneous flows is not strictly dissipative, as is the case
for homogeneous turbulent flows. This is achieved by developing
ETE, with enstrophy characterizing the rotational energy of a flow.
By developing the ETE, in terms of higher even moments of X1, we
identified the index n in this equation, which is indirectly fixed
from the total energy imparted to create the flow. This approach
of viewing how smallest scale is fixed is entirely different to the logic employed for the dissipation of kinetic energy to heat, as in
explaining Kolmogorov’s scale.
In explaining the numerical significance of accurate discretization of diffusion operator, we have developed a balance equation
for 2D flows given as

Z Z

Z Z
 ikx


@x
1
e  1 e2iky  1
dx dy ¼
Re
@t


kx ky
þ
Uðkx ; ky Þ dkx dky
ky kx

ð28Þ

The relevance of diffusion operator compared to convection process
is demonstrated by this equation, where we show that convection
does not contribute at all due to the wall boundary conditions,
while the instantaneous rate of change of vorticity in the full domain (LHS of Eq. (28)) is completely determined by the diffusion
process (given by the RHS of Eq. (28)). The importance of diffusion
discretization carried over the whole domain is explained with the
help of Fig. 9, which clearly establishes the role of high wavenumber components. This figure shows the inadequacy of conventional
diffusion discretization term as compared to the CCD scheme employed here.
Appendix A. Spectral analysis of numerical methods for DNS
Here we compare two space–time discretization schemes based
on their ability to correctly propagate energy and phase of created
disturbance field maintaining neutral stability, for the purpose of
performing DNS. These two schemes have been used respectively
to simulate homogeneous [7] and inhomogeneous wall bounded
turbulent flows starting from receptivity stage [14,15]. We have
chosen 1D convection equation as the model to characterize these
schemes for the usefulness of simulating the convection terms of
Navier–Stokes equation. This model equation has exact non-dispersive solution and hence provides a tough case for calibrating
numerical schemes for space–time discretization. Let us consider
1D convection equation convecting a signal to the right with speed
c given by

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T.K. Sengupta et al. / Computers & Fluids 88 (2013) 440–451

ðA:1Þ

Eq. (A.1) admits a unique non-dispersive and non-dissipative solution, for which the group velocity Vg is equal to the phase speed (c).
Considering a grid with uniform spacing of h, the unknown quantity
u at any jth node can be represented by its Fourier transform given
by

uj ¼ uðxj ; tÞ ¼

Z

Uðk; tÞ eikxj dk

ðA:2Þ

The exact spatial derivative of u can be obtained from above as
@u
R
¼ ikU eikxj dk. When Eq. (A.1) is solved numerically by dis@x exact
crete methods, the spatial derivative (denoted by u0j ) can be represented by
num

¼

Z

ikeq U e

ikxj

dk

ðA:3Þ

In a finite-domain, the spatial derivative u0 can also be expressed as
P
[23], fu0 g ¼ 1h ½Cfug which can also be given by, u0j ¼ 1h Nl¼1 C jl ul ,
R
where ul ¼ Uðk; tÞ eikxl dk is the lth nodal value of the unknown
and N is the total number of discrete nodes. Here, [C] matrix is
dependent on the spatial discretization scheme for the first derivative and this incorporates boundary closure schemes as well. As given in [23], essential numerical properties for space–time
discretization schemes are: (i) Numerical amplification factor
defined as, G ¼

Uðk;tnþ1 Þ
;
Uðk;tn Þ

with cN denoting numerical phase speed. One notes that cN is
dependent on k, i.e.. the numerical solution is dispersive as opposed
to the non-dispersive exact solution. From these relations, one obxN
[23]. Thus, one obtains expressions for the normaltains V gN ¼ @@k
ized numerical phase speed and numerical group velocity as [23]

hc i
N

c

j

¼

ðA:8Þ



V gN
1 @bj
¼
c j hN c @k

(a) RK2-FS
3
25

8.0179

99

5
1.19498

1

1
1.00227

0.9999
0.999
1

1.00005

1

Dt
¼ yn þ ðk1 þ 2k2 þ 2k3 þ k4 Þ
6

1

3

4
37
72
.08
-0

-0.0494054

0.517571
0.9

2

2
-0.370126

min = -0.9863
max = 1.0

cN/c
3

kh

2

Nc

3

-0.160798

0.98925

-0.160798

1.0721

1

0.999

1

-0.160798

1.01027
1

For RK4 time integration, the numerical amplification factor at any
jth node is given by [23]

1

A0 ðkÞ eikxj dk

ðA:6Þ

3

1

Nc

3
0.0495147

-4
-1

0.419744

kh
1.11677

0.

1.01617

0.205056

0.9

1

97
93

1
2.07448

3

99

n

the general solution at any arbitrary time t can be obtained as

0

2

9
0.

1

3

min = -19.6509
max = 4.1336

VgN /c

3

2

2

Nc
min = 0.0209
max = 1.2071

VgN /c

ðA:5Þ

P
where Aj ¼ Nc Nl¼1 C jl eikðxl xj Þ and Nc is the Courant-Friedrich Lewy
(CFL) number defined as, Nc