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Proper orthogonal decomposition and its applications.

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ASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING
Asia-Pac. J. Chem. Eng. 2011; 6: 120–128
Published online 13 July 2010 in Wiley Online Library
(wileyonlinelibrary.com) DOI:10.1002/apj.481
Special Theme Research Article
Proper orthogonal decomposition and its applications
Sanjeev Sanghi1 * and Nadeem Hasan2
1
2
Department of Applied Mechanics, Indian Institute of Technology Delhi, New Delhi, India
Department of Mechanical Engineering, Aligarh Muslim University, Aligarh, India
Received 18 August 2009; Revised 21 May 2010; Accepted 31 May 2010
ABSTRACT: The proper orthogonal decomposition (POD) has become a very useful tool in the analysis and lowdimensional modelling of flows. It provides an objective way of identifying the ‘coherent’ structures in a turbulent
flow. The application of POD to the case of a thermally driven two-dimensional flow of air in a horizontal rotating
cylinder is presented. The data for the POD analysis are obtained by numerical integrations of the governing equations
of mass, momentum and energy. The decomposition based on POD modes or eigenfunctions is shown to converge
to within 5% deviation of the computational data for a maximum of 15 modes for the different cases. The presence
of degenerate eigenvalues is an indicator of travelling waves in the flow, and this is confirmed by symmetry in both
space and time for the corresponding eigenfunctions. Wave speeds are also determined for these travelling waves.
Furthermore, low-dimensional models are constructed employing a Galerkin procedure. The low-dimensional models
yield accurate qualitative as well as quantitative behaviour of the system. Not more than 20 modes are required in
the low-dimensional models to accurately model the system dynamics. The ability of low-dimensional models to
accurately predict the system behaviour for the set of parameters different from the one they were constructed from is
also examined.  2010 Curtin University of Technology and John Wiley & Sons, Ltd.
KEYWORDS: natural convection; rotational buoyancy; proper orthogonal decomposition
INTRODUCTION
The proper orthogonal or Karhunen–Loève decomposition is an established technique for extracting the most
‘energetic’ or dominant spatial structures in the average sense from an ensemble of functions representing
an infinite-dimensional process. It provides a spatial
basis of a vector space onto which the average square
projection of the ensemble is maximum. The ensemble may comprise data over a series of experiments or
data obtained at different time instants during a single
experiment. Once the spatial structure of the ensemble
is captured in the spatial basis functions or modes, the
members of the ensemble can be represented as linear
superpositions of these modes. The advantage of using
the proper orthogonal or K-L basis is that finite truncations of the modal expansion of the ensemble members
yield the least error in the mean square sense compared with the other possible orthogonal bases having
the same dimension. In the context of fluid mechanics,
this means that the structure of a complex spatiotemporal field such as the velocity field, representing a
*Correspondence to: Sanjeev Sanghi, Department of Applied
Mechanics, Indian Institute of Technology Delhi, New Delhi, India.
E-mail: sanghi@am.iitd.ac.in
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Curtin University is a trademark of Curtin University of Technology
flow, could be captured in an average sense by a relatively few number of proper orthogonal decomposition
(POD) modes. Thus, a data ensemble comprising the
information of the velocity field at different instants
of time could be represented by linear superposition
of few POD modes weighted with time varying coefficients. If the partial differential equations, such as the
Navier–Stokes equations, governing the fluid flow are
projected onto these spatial POD modes, a system of
ordinary differential equations for the temporal coefficients is obtained. Hence, the time evolution of the
system could be studied in a state space involving few
degrees of freedom making it easier to investigate the
underlying physics.
The earliest attempts to apply the above ideas in
studying fluid motions came in the context of identifying and studying the role and dynamics of large-scale
average spatial structures of the fluctuating velocity field
of a turbulent flow. These large-scale structures are
the dominant ones in terms of the average energy of
the fluctuating flow and are commonly referred to as
coherent structures. However, due to lack of a precise
definition of a coherent structure, there is a considerable
amount of subjectivity in its identification in a turbulent
flow. The POD provides an objective way of identifying
such structures.
Asia-Pacific Journal of Chemical Engineering
The ideas involving extraction of dominant spatial
structures using POD and the related low-dimensional
modelling have been mostly applied to turbulent flows.
Low-dimensional models based on POD modes have
been used to study the dynamics of wall turbulence in
a turbulent boundary layer. These works were based on
experimental data.[1,2] Moin and Moser[3] used direct
numerical simulation (DNS) data for studying a turbulent channel flow. They employed Fourier modes
in the streamwise and spanwise directions and POD
modes in the wall normal direction. Sirovich et al .[4]
also employed the DNS data for channel flow and found
the intermittency associated with the burst-sweep process as observed in experiments. Rempfer and Fasel[5]
employed the data generated from DNS calculations
of transition of the laminar boundary layers. They
found that the empirical eigenvalues and eigenfunctions almost occur in pairs, implying that there is an
approximate translational invariance of relatively slowly
growing structures.
In an attempt to understand the fundamental mechanisms responsible for self-sustenance of turbulence,
most of the studies involving POD and low-dimensional
modelling have been performed in the domain of turbulent flows. There have been very few studies involving
the application of POD to unsteady laminar flows where
the unsteadiness is either due to inherent instabilities or
due to some unsteady forcing. More recently, however,
applications of POD to problems such as the driven
cavity flows, flow past bluff bodies and flows with heat
transfer have appeared in the literature.
Cazemier et al .[6] applied POD and low-dimensional
modelling to investigate the flow in a square lid-driven
cavity. They obtained the first 80 POD modes from a
dataset comprising of 700 snapshots taken from DNS
calculations at Re = 22,000. Jørgensen[7] applied the
ideas of POD and low-dimensional modelling to study
the axisymmetric lid-driven flow in a cylindrical cavity
with a rotating rod. The problem was investigated by
fixing the aspect ratio of the cylindrical cavity and
the ratio of rod radius to the cylinder radius. The
bifurcations exhibited by the full DNS calculations were
captured faithfully by the POD-Galerkin models.
Ma and Karniadakis[8] investigated the stability and
dynamics of three-dimensional periodic states in flow
past a circular cylinder using low-dimensional modelling. They have shown that the limit cycle is reproduced very accurately with only 20 three-dimensional
modes. Galletti et al . have considered a POD-Galerkin
model for two-dimensional vortex shedding past a confined square cylinder.[9] They investigated the validity
of such a model for Reynolds numbers and blockage
ratios that are different from those from which the lowdimensional model was derived. Their work demonstrates that reliable results can be obtained over a short
interval of time.
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
PROPER ORTHOGONAL DECOMPOSITION
There are some studies involving the application of
POD and POD-Galerkin modelling of flows with heat
transfer. Turbulent Rayleigh-Bénard convection problems were considered by Sirovich and co-workers.[10 – 12]
The change in the structure of the flowfield with
increase in Ra was studied via empirical eigenfunctions
obtained by application of the Karhunen–Loève procedure to the numerical data. Sahan et al .[13] identified
the organized spatiotemporal structures for transitional
flow with heat transfer in a periodically grooved channel. In this study, the buoyancy effects were neglected
and the eigenfunctions for velocity and thermal fields
were calculated separately. Low-dimensional models
based on four modes yielded accurate results. Podvin and Le Quéré[14] investigated the two-dimensional
buoyancy-driven flow of air in a differentially heated
tall cavity. They employed a coupled type of decomposition, where composite vector eigenfunctions having components representing the velocity and thermal
fields were obtained. In their low-dimensional models,
the effect of neglected modes is modelled through a
Heisenberg type of model. For a slightly supercritical
Rayleigh number, they have shown that a two-mode
model captures the dynamics reasonably accurately. At
some distance from the critical point, the flow becomes
chaotic and a ten-dimensional model successfully captures the dynamics. Thus, the POD analysis and the
associated low-dimensional modelling is an effective
reduction tool not only for turbulent flows but also for
transitional laminar flows.
Hasan and Sanghi[15] considered a POD analysis
and low-dimensional modelling of thermally driven
two-dimensional flow of air in a horizontal rotating
cylinder, subject to Boussinesq approximation. The
problem is unsteady due to the harmonic nature of the
gravitational buoyancy force with respect to the rotating
observer. The data for the POD analysis are obtained
by numerical integrations of the governing equations of
mass, momentum and energy. The method of snapshots
is applied to the DNS snapshots of the fluctuating flow
field. The translational symmetry in both space and
time of the pair of modes having degenerate (equal)
eigenvalues confirms the presence of travelling waves
in the flow for several cases of Ra . The shifts in space
and time of the structure of the degenerate modes are
utilized in the estimation of wave speeds in a given
direction. The governing equations for the fluctuations
are derived and low-dimensional models are constructed
employing a Galerkin procedure. For each of the
five cases of Ra , the low-dimensional models yield
accurate qualitative as well as quantitative behaviour
of the system. Sufficient modes are included in the
low-dimensional models so that the modelling of the
unresolved scales of motion is not needed to stabilize
their solution. Not more than 20 modes are required
in the low-dimensional models to accurately model
the system dynamics. The ability of low-dimensional
Asia-Pac. J. Chem. Eng. 2011; 6: 120–128
DOI: 10.1002/apj
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S. SANGHI AND N. HASAN
Asia-Pacific Journal of Chemical Engineering
models to accurately predict the system behaviour for
the set of parameters different from the one they were
constructed from is also examined for the system under
consideration.
PROPER ORTHOGONAL DECOMPOSITION:
THEORY
Proper orthogonal decomposition or Karhunen–Loève
expansion is a statistical technique that has been known
for quite some time but has gained popularity in the subject of fluid mechanics over the last decade. The technique provides an optimal spatial basis for capturing the
spatial structure of the most energetic fluctuations in the
flowfield in an average sense. Let Um = U(x, τm ) (m
varying from 1 to M ) be M realizations or snapshots of
the fluctuating flowfield obtained through experimentation or numerical simulations. In this case, the snapshot
Um comprises both the fluctuating velocity components
and the fluctuating temperature field with the three values (U1 , U2 , U3 ) ≡ (u , v , θ ) assigned at each point.
In order to decompose such an ensemble of snapshots,
the K-L basis functions φ are taken to be vector functions with three components at each point (φ1 , φ2 , φ3 )
associated with the two velocity components and the
thermal field. The vector space in which the decomposition is sought has an inner product defined as
According to Lumley and Poje,[16] the introduction of a
scaling factor γ is necessary to balance the velocity and
temperature fluctuation energies so that the K-L basis
captures the structure of the most energetic fluctuations
of both the temperature and the velocity fields in a
composite manner. They have shown that the proper
value of γ that maximizes the average of the square of
the projection of the data onto the basis function φ is
given as
γ =
< (u u + v v ) > dA
< θ θ > dA
(1)
The ‘< >’ denotes the ensemble average and in this
scenario is taken to be the time average as a result of
time stationarity of the flowfield. The K-L basis function
φ is governed by the Fredholm Integral of the first kind
given as
Equation (2) is an eigenvalue problem with λ as the
eigenvalue and φ ≡ (φ1 , φ2 , φ3 ) as the eigenfunction or mode. It is known that the eigenvalue problem expressed via (2) has countably infinite, nonnegative eigenvalues and corresponding eigenfunctions.
The eigenfunctions {φ}∞
1 or modes are orthogonal and
form a complete basis of an infinite-dimensional linear space spanned by them. The K-L basis is utilized
for carrying out the decomposition of the fluctuating
flowfield as
U(x, τ ) =
Rij (x , x )φj (x )dA = λφi (x ),
(i , j ) ∈ {1, 3}
(2)
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
∞
a (p) (τ )φ (p) (x),
(3)
1
where the superscripts in parentheses represent the POD
quantum numbers. Once the eigenfunctions are known,
the temporal coefficients in (3) can be readily obtained
as
(4)
a (p) (τ ) = (U, φ (p) )
The total energy captured by the expansion in the
average sense is given as
E = <(U, U) > =
< a (p) a (r) > (φ (p) , φ (r) )
=
D
Rij (x , x ) = <Ui (x , τ )Uj (x , τ ) >
p
(f1 g1 + f2 g2 + γ f3 g3 )dA
(f, g) =
The kernel of the integral in (2) is the two-point
correlation tensor defined as
r
λ(p)
(5)
(1)
If the eigenvalues {λ}∞
>
1 are ordered such that λ
(2)
(3)
λ > λ . . ., the sequence on the right of (3) converges more rapidly than with any other basis and only a
few modes are needed to obtain a good enough reconstruction of the original data ensemble Ui . The most
commonly employed criteria for truncating the infinite
sequence in (3) is the retention of the number of modes
which capture more than 90% of the average energy
in the ensemble Ui with the condition that none of the
neglected modes has more than 1% of energy of the
most energetic mode.
The discrete solution of the eigenvalue problem in (2)
involves handling a fairly large algebraic eigenvalue
problem in terms of the values of the eigenfunctions
at the discrete mesh points. To overcome this difficulty,
Sirovich[17] proposed the method of snapshots. In this
method, the eigenfunction φ is taken as a linear
combination of the snapshots given as
φ=
M
αm U m
(6)
1
Utilizing (6) and the definition of the two-point correlation tensor, it can be readily shown that the eigenvalue
Asia-Pac. J. Chem. Eng. 2011; 6: 120–128
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
PROPER ORTHOGONAL DECOMPOSITION
problem in (2) can be transformed to an equivalent
M × M algebraic eigenvalue problem,
Aα = λα
(7)
1 (Um , Un ).
where Amn = M
Since relatively few snapshots are needed to capture the temporal structure of non-turbulent flows, the
method of snapshots is preferred in most cases. This
method yields at most M modes.
GOVERNING EQUATIONS
This work applies POD and POD-Galerkin models to
study the problem of two-dimensional thermally driven
flow in a horizontal steadily rotating cylinder. Similar
application in the near wall turbulent boundary layer
was first presented by Aubry et al .[1] The problem has
been investigated numerically by Hasan and Sanghi,[18]
hereafter referred to as HS. A Cartesian frame of
reference attached to the rotating cylinder has been
employed in the study. To an observer attached to
the cylinder, the gravity vector rotates resulting in a
time-periodic gravitational buoyancy (GB) force driving
the flow. Thus, the problem involves unsteady forcing
and is therefore inherently unsteady in nature. It is
assumed that the cylinder has been rotating steadily for
a sufficient length of time so that the fluid is in a state of
solid-body rotation under isothermal conditions. For a
figure of the problem, please refer to HS. A temperature
perturbation in the form of a periodic cosine function
of the angular location of the spatial point is imposed
on the wall of the cylinder. The fluid motion relative to
the rotating cylinder is induced by the combined action
of GB and centrifugal or rotational buoyancy (RB). The
dimensionless governing equations of mass, momentum
and energy subject to Boussinesq approximation in a
Cartesian frame rotating with the cylinder are given as
Mass : ∇ · V = 0
(8)
dV
= −∇pm − Rag Prθ n̂g − Ra Prθ r
Momentum :
dτ
− 2Ta1/2 Pr(k̂ x V ) + Pr∇ 2 V (9)
Energy :
dθ
= ∇ 2θ
dτ
(10)
The vector n̂g in (9) is the unit vector indicating the
instantaneous direction of the rotating gravity vector and
is defined as
n̂g = −î sin(Ta1/2 Pr τ ) − ĵ cos(Ta1/2 Pr τ )
The dimensionless system (8)–(10) shows four dimensionless parameters on which the solution depends.
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
These dimensionless parameters are[1] gravitational
Rayleigh number (Rag ),[2] rotational Rayleigh number (Ra ),[3] Taylor number (Ta) and[4] Prandtl number
(Pr). These are defined as
g βTR 3
2 βTR 4
, Ra =
,
νκ
νκ
2 R 4
ν
Ta =
, Pr =
2
κ
ν
Rag =
(11)
The computational study of HS has correlated the
spatiotemporal dynamics and the heat transfer characteristics to the changes in Ra and Ta over a wide
range at a fixed Rag = 105 and Pr = 0.71. The computational study revealed that for Ra < 105 , the spatial
structure of the flow is quite sensitive to the changing orientation of the gravity vector in the rotating
frame. In the time domain, for Ra < 105 , the flow
exhibits a periodic behaviour with frequency locked
onto the rotation frequency of the gravity vector. It was
concluded that the temporal variations in the flow for
Ra < 105 are caused by harmonic forcing of the GB
force. For Ra > 105 , the large-scale spatial structure of
the flow is quite insensitive to the harmonic time varying GB force. Examination of the temporal structure
via fast Fourier transform revealed bifurcations from
periodic to quasi-periodic states for Ra ∈ (106 , 107 ).
The study on variation of time mean heat transfer over
the hot portion of the wall of the cylinder with Ra
revealed that significant control of flow via rotation
could be achieved. The mean heat transfer characteristic is highly non-monotonic, indicating the aiding and
mitigating effects of rotation on convection in different
ranges of Ra . It was shown that at low rotation rates
(Ra < 2 × 103 ), the flow is governed essentially by the
GB force with the centrifugal effects being negligible.
For Ra > 105 , the role of gravity progressively diminishes and the flow for Ra in the range (5 × 105 , 107 ) is
driven essentially by the centrifugal or RB forces. In this
range, the large-scale flow structure consists of a twocell convection with the interface of the cells aligned
along the horizontal diameter and an increase in Ra
brings about an increase in convection. The two cells
become increasingly distorted and multiple rolls appear
as Ra approaches 107 .
In the range given by 2 × 103 ≤ Ra ≤ 5 × 105 , the
flow is controlled by both the body forces. Under the
competing influence of the two buoyancy forces, it is
shown that the convection is in general suppressed with
very low flow velocities and associated heat transfer.
The fluctuation levels are large at low rotation rates
(Ra ∼ 102 ) and decrease progressively as the influence
of the rotating gravity vector decreases with increase in
Ra . The fluctuation levels become extremely small to
render the flow to be almost steady at Ra ∼
= 105 . At
5
higher rotation rates (Ra > 10 ), the fluctuation levels
Asia-Pac. J. Chem. Eng. 2011; 6: 120–128
DOI: 10.1002/apj
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S. SANGHI AND N. HASAN
again start to increase as an outcome of bifurcations to
quasi-periodic states.
As summarized above, there is a wide variation
in the spatiotemporal structure of the flow as Ra
is varied. This study is motivated by the fact that
for a complex unsteady flow, such as the one under
consideration, the POD analysis can provide useful
information regarding the flow physics by representing
the flow in terms of few modes. Furthermore, it is of
interest to examine the feasibility of constructing lowdimensional models and to assess the performance of
such models for the class of problem under study. The
procedure for carrying out the decomposition and the
results of the decomposition are presented in the next
section.
POD ANALYSIS
From this point onwards, unless stated otherwise,
it is understood that snapshot refers to the instantaneous spatial distribution for the fluctuating flow
field.
The fast convergence of POD modes is observed
as a small number of modes are needed to capture
almost 99% energy of the snapshots in the average
sense. The distribution of the spectrum is shown in
Fig. 1a–e where the respective eigenvalues are plotted. The spectrum of eigenvalues in each of these cases
decays rapidly. At Ra = 102 , seven modes capture
more than 99% of the total energy. For Ra = 103 ,
eight modes are needed to capture more than 99% of the
total energy. For Ra = 104 and Ra = 105 , only two
modes are sufficient. For Ra = 106 , only six modes
are needed. From a qualitative viewpoint, it is shown
that the spatial structure of the instantaneous flow field
as obtained by DNS calculations can be faithfully captured using few modes. For the case of Ra = 102 ,
ten modes appear to capture the instantaneous flow
structure very well. Another observation from the POD
reconstructions is that while 10–15 modes capture the
relevant scales for the case of Ra = 102 and 103 , only
two modes are needed for the cases of Ra = 104 and
105 . This can be correlated to the temporal structure
of the flow field for these cases, investigated via Fast
Fourier Transform (FFT) of the time histories, in the
earlier work of the authors. As shown in the computational study of HS, the flows for Ra = 102 –105 are
periodic, and the frequency spectrum exhibits a fundamental frequency equal to the rotation frequency of
the gravity vector with high-frequency low-power harmonics. These harmonics represent less energetic small
scales in the flow. As Ra is increased from 102 to
105 , the effects of gravity are reduced and the centrifugal buoyancy starts to affect the flow. The number of harmonics progressively decreases, indicating
the reduction in the scales of the fluctuating motion.
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pacific Journal of Chemical Engineering
The POD reconstructions also reflect the phenomenon
of reduction in scales as Ra is increased from 102
to 105 , with lesser number of modes needed to capture the structure of the fluctuating flow in the average
sense.
It is observed that for Ra = 103 and 106 , a few number of eigenvalues are degenerate, i.e. occur in pairs.
The former case is discussed in detail in this work.
The eigenvalues for modes[3,4] ,[7,8] and[9,10] are some
of the more energetic degenerate pairs. The degeneracy
of the eigenspectrum has its roots in the symmetries
of the flow solution as demonstrated in the works of
Aubry and co-workers.[1,19,20] To observe such symmetries, time histories of the temporal coefficients of a
degenerate pair of modes (a (q) (t), a (q+1) (t)) and the
contributions of the individual modes of the selected
degenerate pair to the flow field at an instant are compared. It is observed that the contour patterns are shifted
in the circumferential direction. This clearly suggests
a translational shift in the circumferential direction in
the spatial structure of the degenerate pair of modes.
Some of the structures experience some distortion along
with a shift in the circumferential direction. The distortion is expected due to nonlinear convective interactions
and diffusive effects. It is also seen that the pair of
coefficients a (3) and a (4) as well as other pairs are
almost identical except for a phase shift in time. It
is observed that the spatial structures of φ3(3) and φ3(4)
appear to exhibit a translational symmetry in the circumferential direction. Furthermore, while mode 3 is
shifted in the counter clockwise direction relative to
mode 4 in space, mode 3 also leads mode 4 in time.
This clearly suggests that the degenerate pair[3,4] represents a propagating structure or a travelling wave in the
clockwise direction with the flow. Rempfer and Fasel[5]
also exploited the space-time symmetry of a propagating
flow structure to demonstrate the existence of travelling waves in a transitional boundary-layer flow over
a flat plate. Similar space-time symmetries are exhibited by the other degenerate pairs. The deviations from
the perfect translational symmetry in space and time
can be attributed to the fact that the propagating structures represent modulated travelling waves. Thus, it can
be argued that at Ra = 103 , the flow exhibits travelling waves propagating in the circumferential direction.
In the numerical study of HS, it was argued on the
basis of spatial flow structure that at Ra = 103 , the
centrifugal buoyancy force is too weak to exert any
significant influence on the flow. The flow is essentially controlled by the periodic GB force. Thus, the
travelling waves found are generated due to the combined action of gravity, the fluid inertia and the viscous forces. Such waves have not been reported in the
earlier studies on rotating rectangular containers.[21,22]
This also demonstrates the power of POD analysis
in detecting organized structures from a computational
database.
Asia-Pac. J. Chem. Eng. 2011; 6: 120–128
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
PROPER ORTHOGONAL DECOMPOSITION
(b)
104
104
103
103
102
102
101
101
λ
λ
(a)
100
100
10-1
10-1
10-2
10-2
10
20
10
30
20
(c)
(d)
104
103
103
102
102
101
101
100
100
10-1
λ
λ
30
p
p
10-1
10-2
10-2
10-3
10-4
10-3
10-5
10-4
5
10
p
(e)
15
20
10-6
5
10
p
15
20
104
103
102
101
λ
100
10-1
10-2
10-3
10-4
10
20
30
p
Figure 1. pth eigenvalue, λ, as a function of p at (a) Ra = 102 , (b) Ra = 103 , (c) Ra = 104 , (d) Ra = 105 and
(e) Ra = 106 .
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pac. J. Chem. Eng. 2011; 6: 120–128
DOI: 10.1002/apj
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S. SANGHI AND N. HASAN
Asia-Pacific Journal of Chemical Engineering
LOW-DIMENSIONAL MODELS
The low-dimensional model refers to the system of
Ordinary Differential Equation (ODE) governing the
evolution of temporal coefficients in (9). The standard procedure for obtaining such system of ODEs is
Galerkin projection. In this procedure, the governing
equations for the flow field are projected onto the individual POD modes or basis functions.
The Galerkin projection requires the basis functions
to fulfil certain conditions. In the context of a Galerkin
procedure for an incompressible flow, the basis functions should be capable of representing all solenoidal
velocity fields that satisfy the boundary conditions. In
this regard, the POD eigenfunctions provide an excellent choice of basis functions. As a consequence of the
fact that the POD eigenfunctions can be represented
as linear combinations of the flowfield snapshots, any
property or boundary condition of the flow, which is
expressed via linear homogenous equations, is passed
on to these individual basis functions. The incompressibility constraint, no-slip boundary conditions and periodic boundary conditions are some of the examples of
such properties. In this study, since the POD eigenfunctions have been obtained for the fluctuating component
of the flow field, the homogeneous boundary conditions
for the fluctuating or unsteady flow (u = v = θ = 0)
are passed onto each eigenfunction making it vanish
at the domain boundary. Since each individual eigenfunction is capable of satisfying the boundary condition,
no compatibility constraints are imposed on these basis
functions.
The Galerkin projection leads to an infinite system of
ODEs for the temporal modal coefficients of the form
da (p)
= F (p) (a (1) , a (2) , . . . , τ ),
dτ
p = 1, 2, . . . , ∞
(12)
The infinite system in (12) is truncated at some level
of the quantum number p to yield the low-dimensional
model with the hope that the effect of truncation on the
system dynamics is not significant. More would be said
regarding the truncation of the infinite system in (12)
when the validation and performance assessment of the
low-dimensional model is presented.
In order to carry out the Galerkin projection for the
present problem, the governing equations for the fluctuating or unsteady flow have to be obtained from the full
set of Eqns (8)–(10). Following the Reynolds decomposition approach, the instantaneous flow variables in
Eqns (8)–(10) are considered to comprise a steady or
mean part and a fluctuating component. Symbolically,
this is expressed as
vi (x , y, t) = Umi (x , y) + ui (x , y, t)
θ (x , y, t) = m (x , y) + θ (x , y, t)
p(x , y, t) = Pm (x , y) + p (x , y, t)
(13)
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Substituting in Eqns (8)–(10) and time averaging, the
terms lead to equations governing the mean flow.
The mean flow equations are then subtracted from
Eqns (8)–(10) to yield the equations for the fluctuations. The governing equations for the fluctuating flowfield in Cartesian tensor notation are
∂ uj
∂ xj
=0
(14)


1/2
{ (m + θ ) sin(Ta Pr τ ) >}δi 1
1/2
∂


Pr τ )
= Rag Pr  − < θ sin(Ta

1/2
∂ τ
+{ (m + θ ) cos(Ta Pr τ ) >}δi 2
−<θ cos(Ta1/2 Pr τ )
∂p ∂Umi
− uj
− 2Ta1/2 Pr ei 3j uj − Ra Prθ xi −
∂xi
∂xj
ui
∂ < ui uj >
∂ 2 ui
∂ui
∂ui
− Umj
− uj
+
+ Pr
∂xj
∂xj
∂xj
∂xj ∂xj
(15)
θ uj
>
∂<
∂m
∂θ
∂θ
∂θ
= −uj
− Umj
− uj
+
∂τ
∂xj
∂xj
∂xj
∂xj
+
∂ 2θ ∂xj ∂xj
(16)
In equations (14)–(16), (i , j ) ∈ (1, 2).
The POD expansions for the fluctuating variables
(u1 , u2 , θ ) are substituted in (15) and (16), and then
the Galerkin projection is carried out to obtain the
system of ODEs involving the temporal coefficients. For
notational convenience, the mean and the fluctuating
thermal field are denoted as Um3 and u3 . The various
correlations, i.e. <ui uj > needed to close the system
of equations have been expressed in terms of the POD
modes in the following manner:
< a (r) a (s) > φi(r) φj(s)
<ui uj >=
=
λ(r) φi(r) φj(r) , i = 1, . . . , 3, j = 1, 2
r
(17)
The Galerkin projection yields the following system of
equations:
da (p)
= Lpr (τ )a (r) − Qprs a (r) a (s)
dt
+ Fp (τ ), p, r, s = 1, 2, . . . , ∞ (18)
Truncating the above system by retaining the first N
modes in the system in (18) yields an N -dimensional
model.[15] The term Fp (τ ) in (18) is an harmonic forcing
function associated with the rotating gravity vector.
Thus, the system in (18) has a character of a nonlinear
forced oscillator.
Asia-Pac. J. Chem. Eng. 2011; 6: 120–128
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
In order to validate and assess the performance of the
low-dimensional models, the five cases of Ra which
have been analyzed using POD are considered. Comparison between the temporal coefficients obtained through
projection of the POD modes onto the computational
data and the solutions of the POD-Galerkin models is
taken to be the criterion for valid and accurate solutions of these low-dimensional models. The truncation
of the POD-Galerkin models for accurate capturing of
the system dynamics is guided by the convergence characteristics of the POD mode based reconstructions.
Another issue related to the truncation of the PODGalerkin models is the effect of neglected modes on the
long-term behaviour of the low-dimensional model. It
is known that if sufficient number of modes are not
included then the required amount of energy of the
system is not dissipated and accumulates in the first
few modes or the large scales of the motion. Thus, in
the large time limit, the system becomes unstable. The
remedy is to include appropriate number of modes or
model the effect of neglected modes. The modelling
approach introduces additional parameters in the model
which have to be suitably calibrated. In contrast, a
model-free approach is employed in this study where
enough number of modes has been utilized not only to
stabilize the low-dimensional system but to accurately
reproduce the system dynamics.
For Ra = 102 , a five-dimensional model is found
to be sufficient to carry out integrations of the lowdimensional model for very long times (∼400 dimensionless units) without any blowing up of the solution. Thus, sufficient amount of dissipation is achieved
even for a five-mode low-dimensional model. The lowdimensional models for the different cases are constructed from a few modes greater than the modes
needed to achieve reconstructions of the flow field
within 5% of the DNS values. The temporal evolution of the modal coefficients from the low-dimensional
models is compared with that of the modal coefficients
obtained via projection of the eigenfunctions onto the
DNS snaps. The initial conditions for the system of
first-order ODEs is taken to be the values of the temporal coefficients obtained by projecting the corresponding
eigenfunctions onto the first DNS snap.
At this stage, it is worth discussing an important
aspect of the definition of the inner product defined for
the purpose of a combined decomposition of the velocity and the thermal fields. The inner product involves
the scaling factor γ . The choice of the scaling factor that
maximizes the projection of the data ensemble onto a
fixed set of modes is given as in (1). In the work of
Podvin and Le Quéré,[14] it has been mentioned that an
ad hoc value of unity for the scaling factor also works
reasonably well and that the value based on the criterion given in (1) only serves to provide slightly better
performance of the low-dimensional model. However,
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
PROPER ORTHOGONAL DECOMPOSITION
when a value of unity was tried in this study, the eigenfunctions failed to capture the dominant structures of
the fluctuating velocity and thermal fields accurately.
This is not surprising when one considers the values of
γ obtained using (1) for the five cases of Ra . For the
five cases in this study, the value of γ was found to
vary between 104 and 107 . Such values exist primarily
due to the choice of the velocity and the temperature
scales in this study. In the study by Podvin and Le
Quéré[14] , the scalings employed were such that both
the velocity and temperature are O[1] and therefore the
value of γ was reported to be 4.68. Evidently, a value of
unity being of comparable order still yielded reasonable
results from their low-dimensional models. However, in
this case low-dimensional models provide proper results
only when the appropriate value of γ is used.
At Ra = 102 , the low-dimensional model obtained
by retaining the first 20 modes in (18) captures the
temporal evolution of the modal coefficients quite faithfully. In fact, the time evolution of the modal coefficients as predicted by the model and the corresponding
DNS values for the first five modes is virtually indistinguishable. For the 10th and 15th modes, the agreement
is still quite good in the qualitative sense. The time
period of the various temporal coefficients is found to
be 0.088 dimensionless units. The period of rotation of
the gravity vector at this Ra is 0.0885 dimensionless
units. Therefore, the phenomenon of frequency locking is captured quite well. Another feature that can be
observed in the solution of the low-dimensional model
is the fact that while the more energetic structures have
a simple periodic structure in time, the less energetic
structures (10th and 15th modes) have a greater amount
of complexity in their temporal structure. This supports the proposition made earlier that the low-power
high-frequency harmonics revealed by the FFT in the
temporal structure of the flow are reflections of the
dynamic evolution of the less energetic structures or
smaller scales of flow.
As shown in the study of HS, the phenomenon of frequency locking persists for Ra ≤ 105 . This is reflected
in the solutions of the low-dimensional models for
Ra = 103 , 104 and 105 . The time periods of the various
temporal coefficients for Ra = 103 , 104 and 105 are
found to be 0.0278, 0.00886 and 0.0028 dimensionless
units, respectively. These agree quite well with the corresponding periods of the rotating gravity vector given
as 0.02798, 0.00885 and 0.002798, respectively. As
mentioned earlier, for Ra ≤ 105 , the fluctuation levels
of the flow decrease with increase in Ra . This feature
is also readily captured by the low-dimensional models.
The decreasing complexity in the flow structure, as confirmed by the POD analysis, is also confirmed by the
fact that the size of the low-dimensional model needed
to accurately model the system dynamics progressively
decreases from 20 to 4 with increase in Ra upto 105 .
For Ra = 106 , the flow is quasi-periodic and the level
Asia-Pac. J. Chem. Eng. 2011; 6: 120–128
DOI: 10.1002/apj
127
128
S. SANGHI AND N. HASAN
of fluctuation again begins to rise. The trajectories of
the temporal coefficients in the time domain obtained
from a ten-dimensional model reproduces the dynamics
quite nicely as the trajectories follow their DNS paths.
CONCLUSIONS
This study demonstrates the effectiveness of POD
as a tool for the analysis of a complex fluid flow
phenomenon involving unsteady flow with heat transfer.
It has been demonstrated that relatively few modes are
needed to capture the structure of the unsteady flowfield
within a specified level of accuracy.
The presence of travelling waves in the flow has
been detected by employing POD decomposition to the
computational data at Ra = 103 . At Ra = 103 , the
waves propagate predominantly in the circumferential
direction in the clockwise manner and are believed to
be caused by the interaction between the GB, the fluid
inertia and the viscous forces as the centrifugal or RB
is shown to have a negligible effect on the flow at this
value of Ra . These travelling waves are actually found
to exist for values of Ra in the range 103 –1.9 × 103 .
The space-time symmetries of the degenerate pairs of
modes is utilized to estimate the average circumferential
wave speeds for these cases.
The low-dimensional models based on POD modes
have been constructed and issues involved in their
closure arising out of the specific nature of the problem
under consideration have been successfully resolved.
An important aspect of POD analysis that has not
received due importance for non-isothermal flows is
the definition of the inner product space in which the
combined decomposition of velocity and temperature
field is desired. It has been shown that for situations
where the order of magnitude of velocity and the
temperature fields are not comparable, the choice of
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pacific Journal of Chemical Engineering
the proper value of the scaling factor γ is critical to the
success of the low-dimensional model. Having included
sufficient number of modes in the low-dimensional
models so that sufficient amount of dissipation is
present to stabilize their solution for several hundred
rotation cycles of the gravity vector, not more than 20
mode models were required to accurately capture the
temporal dynamics of the system for the five cases under
consideration.
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Asia-Pac. J. Chem. Eng. 2011; 6: 120–128
DOI: 10.1002/apj
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