1. DYNAMIC MODE DECOMPOSITION OF PARTIALLY
PREMIXED LAMINAR FLAMES IN MIXING LAYERS
A THESIS
Submitted by
SABARISH BHARADWAZ VADAREVU
AE08B023
for the award of the degree
of
DUAL DEGREE
DEPARTMENT OF AEROSPACE ENGINEERING
INDIAN INSTITUTE OF TECHNOLOGY, MADRAS.
MAY 2013
2. THESIS CERTIFICATE
This is to certify that the thesis titled DYNAMIC MODE DECOMPOSITION OF
PARTIALLY PREMIXED LAMINAR FLAMES IN MIXING LAYERS, submitted
by Sabarish Bharadwaz Vadarevu, AE08B023 to the Indian Institute of Technology
Madras, Chennai for the award of the degree of Dual Degree, is a bona fide record of the
research work done by him under the supervision of Prof. S R Chakravarthy. The
contents of this thesis, in full or in parts, have not been submitted to any other Institute or
University for the award of any degree or diploma.
Prof. S R Chakravarthy
Research Guide
Professor
Dept. of Aerospace Engineering
IIT-Madras, 600 036
Place: Chennai
Date: 7th
May 2013
4. ACKNOWLEDGEMENTS
First and foremost, I express my sincere gratitude to my Guide, Prof. S R Chakravarthy,
for his guidance and support from the initial to the final stages of my research work,
which enabled me to carry out this work successfully. I also gratefully acknowledge
the generosity and infinite patience shown by him.
With great pleasure, I express my gratitude to my beloved parents, Late Mr. V Venkat
Bharadwaz and Mrs. V Uma Devi for their moral support and patience.
This work would have been incomplete without their blessings.
I thank Mr. David Bhatt, research scholar, Dept. of Aerospace Engineering, IIT Madras,
for his constant help in providing me with required data, analysing my results and
guiding my work. I thank Mr. Ramgopal Sampath, research scholar, Dept. of Aerospace
Engineering, IIT Madras for allowing me to use his algorithm and Matlab® code for
Dynamic Mode Decomposition. I also thank Mr. Trinath Gaduparthi for the support he
had provided in the initial stages of the project.
I thank National Centre for Combustion Research and Development (NCCRD) for
providing me with computational resources on their computing cluster.
Last but not the least, I thank all my friends, colleagues and all others who directly or
indirectly helped me in carrying out this research work in a successful manner.
Place: Chennai Sabarish Bharadwaz
Date: 7th May 2013
5. 2
ABSTRACT
A numerical study has been conducted to investigate the stability
characteristics of the diffusion flame in a reacting laminar mixing layer. In an earlier
work by David and Chakravarthy [1], the non-linear equations for thermal energy and
species concentrations have been solved for a range of Lewis numbers (Le: 1.6, 1.7,
1.8), Damköhler numbers (Da: 10 to 1000) and amounts of pre-mixedness (ζ: non pre-
mixed to fully pre-mixed). When perturbed, the diffusion flame is found to approach a
steady state for a certain range of parameters (Le, Da, ζ), and exhibit limit cycle
oscillations in the rest of the parameter space. To extend the understanding of
thermal-diffusive instabilities (T-D instability) for diffusion flames, the instantaneous
flow fields have been subjected to Dynamic Mode Decomposition (DMD) and coherent
structures, along with their oscillatory frequency and growth rate, for the flame have
been identified.
Dominant modes, as defined in the linear stability analysis framework, along
with their corresponding oscillation frequencies and growth rates have been obtained
for the non-linear evolution through DMD. The results indicate that the non-linear
evolution, close to the final steady state or limit cycle oscillations, may be
approximated by linearly evolving modes (obtained through DMD). In the current
work, only a small sub-domain in the Le-Da- ζ parameter space has been explored to
demonstrate the applicability of a linear analysis technique such as DMD. The current
work may be extended to explore the full parameter space, interpret the linear DMD
modes to understand T-D instability, and to more complex flows such as shear flows,
flows with density stratification, backward facing step, etc. The results, without
further extension, may be further used to study interaction of the modes, with regards
to their oscillatory frequency, to other modes due to hydrodynamics, acoustics and
buoyancy.
6. 3
LIST OF SYMBOLS USED
Da Damköhler number
Le Lewis Number
ζ Premixedness parameter
ω Reaction rate
φ Equivalence ratio
T Temperature
Ta Adiabatic flame temperature
T0 Inlet temperature
u Velocity
(For the above quantities, symbols with tilde represent dimensional quantities,
symbols without tilde represent non-dimensional quantities)
YF Fuel mass fraction
YX Oxidizer mass fraction
α Prandtl number
ρ Flow density
B Dimensional pre-exponential factor in the Arrhenius form of
the reaction rate’s dependence on temperature,
EA Dimensional overall activation energy of the reaction,
R Universal gas constant,
β Zeldovich number
γ Heat release parameter.
7. 4
CHAPTER 1
INTRODUCTION
An understanding of flame dynamics, especially near extinction conditions, is
very crucial in developing more powerful and efficient combustors for applications
such as in power generation, aerospace vehicles and chemical and metallurgical
processing. Depending on the flow geometry and the fluid and flow parameters such
as Lewis number, Damköhler number, Reynolds number, Rayleigh number, amount of
pre-mixedness, etc.., the flame may exhibit various kinds of unsteady behaviour such
as wrinkling or oscillations, due to thermal-diffusive (T-D) instabilities, or lock on to
other modes due to hydrodynamics, acoustics and buoyancy. Undesired unsteadiness
of flames may result in local hot spots, reduced performance, or even extinction of the
flame altogether. Despite the associated problems, unsteadiness of flames becomes
unavoidable when combustors are operated over a wide range of conditions.
Predicting the unsteady evolution of a flame hence has fundamental importance in
combustion studies.
Previous research on flame dynamics of premixed and non-premixed flames
has been outlined by Matalon [2]. Very little work has been done on the instabilities of
diffusion flames under finite rate chemistry models [3]. Papas et. al. [3] have
conducted a comprehensive study on reacting mixing and shear layers to investigate
thermo-diffusive instabilities of diffusion flames, including the role of hydrodynamics
on the instability, but only using semi-analytical profiles for the flames and employing
a linear analysis for the evolution of perturbed flames. David and Chakravarthy [1]
have investigated the T-D instability of diffusion flames (in isolation, without
hydrodynamic, buoyant or acoustic effects) by studying a reacting mixing layer with
zero density stratification and zero velocity gradient for Lewis numbers more than
unity. Their work included obtaining numerical solutions for the non-linear transport
equations for fuel and oxidizer and the energy equation, over a range of Damköhler
numbers and amounts of premixedness (identified using a parameter called the
premixedness parameter, ζ) for different Lewis numbers (1.5, 1.6 and 1.7). Their work
showed the existence of a thumb-shaped region in the Da-ζ parameter space within
which the flame exhibits limit cycle oscillations when perturbed.
8. 5
The current work is based on the numerical data obtained by David and Chakravarthy
[1] and aims to achieve the following:
1. To determine the extent to which linear stability analysis may be applied to
combustion problems involving thermo-diffusive (T-D) instability (for Le> 1)
2. Extend the understanding of thermo-diffusive instabilities by extracting and
studying coherent structures that dictate the evolution and the eventual decay,
or limit cycle oscillations, of perturbed flames.
Dynamic mode decomposition (DMD) has been used to extract these coherent
structures, and their corresponding oscillatory frequencies and growth rates from the
numerical data. DMD has been very briefly described in a later section, the reader is
referred to the paper by Schmid [4] for a complete mathematical treatment of the
technique.
9. 6
Chapter 2
FLAME DESCRIPTION
The flame investigated in the current work is the same as that studied by David
and Chakravarthy [1], namely, the diffusion flame in a reacting mixing layer. The
mixing layer has zero density and velocity gradients, hence allowing the study of the
thermo-diffusive instability in isolation. The flame geometry and boundary conditions
are shown in figure 1 (courtesy: David and Chakravarthy [1]). The inlet to the
computational domain consists of a parallel flow with zero velocity gradient, but with
gradients in fuel and oxidizer concentrations based on the amount of premixedness.
The inlet fuel and oxidizer concentrations are obtained based on a cold mixing layer
flow and has been described by David and Chakravarthy [1].
The temperatures are non-dimensionalized using the inlet temperature ( )
and the adiabatic flame temperature ( ) as T = ( - )/( – ), tilde indicates
dimensional quantities. The inlet species concentrations are normalized based on
mass fractions of fuel and oxidizer at +/- ∞. The computational domain is assumed
(and shown by David and Chakravarthy [1]) to be long enough to apply zero gradient
Neumann boundary conditions for temperature and species concentrations at the
outflow and lateral boundaries.
Fig. 1. Schematic of the non-dimensional physical and computational domains, inlet flow and
boundary conditions (courtesy: David and Chakravarthy [1])
10. 7
Chapter 3
GOVERNING EQUATIONS
(1)
(2)
(3)
Here, T is the non-dimensional temperature, YF and Yx are the non-dimensional fuel
and oxidizer mass-fractions respectively, φ is the equivalence ratio and is the non-
dimensional reaction rate given by
(4)
where is the Damköhler number.
α is the Prandtl number,
ρ is the (constant) density of the flow,
is the (constant) velocity of the flow,
B is the dimensional pre-exponential factor in the Arrhenius form of the reaction
rate’s dependence on temperature,
EA is the dimensional overall activation energy of the reaction,
R is the universal gas constant,
The Zel’dovich number and
is the heat release parameter.
The reaction rate is assumed to be of first order with respect to each of the
reactants. In the above set of governing equations, the Lewis numbers of the fuel and
oxidizer are always taken as equal, i.e., LeF = LeX = Le.
11. 8
Chapter 4
DYNAMIC MODE DECOMPOSITION OF NON-LINEAR EVOLUTION
DMD as used for the current work is entirely based on the paper by Schmid [4],
and no further references shall be made to that paper. The technique is only briefly
described in the context of the current work, but the mathematics involved is not
covered. A complete mathematical treatment along with relevant illustrations for
DMD may be found in the aforementioned paper. The methodology that has been
adapted is described in this section. The results obtained are presented in a later
section. All of the computation: pre-processing, DMD and post-processing, has been
done using MATLAB® R2012a.
DMD is a technique that can extract coherent structures in a flow field from a
series of “snapshots”. For a temporal stability analysis, each snapshot may be an
instantaneous flow field in a particular spatial domain. For a spatial stability analysis,
the snapshots may correspond to the values of the flow field variables at successive
spatial locations. The snapshots may be a collection of measurements obtained from
experiments, a field of variables obtained from a computation, the intensity values
corresponding to an image or any other such data sets. Each snapshot is converted
into a vector and a sequence of these vectors represents the evolution of the flow field
in time or space. The current work involves a temporal stability analysis and hence
the same is discussed in the rest of the paper.
DMD is a relatively new technique and the literature available on the technique
is very limited (only a few papers are available on DMD, published by the author of the
paper already mentioned). No general guidelines are available to employ such a
technique and hence the author of the current work had to explore various options
before a finally deciding on an approach to the current problem. 7 different versions
of code have been developed, each successively correcting on the previous version,
before the algorithm is finalized. Though the work leading up to the final code has
provided valuable insights into the combustion problem and DMD, only the final
algorithm followed is discussed here.
4.0. DMD vs Linear Stability Analysis
DMD is a technique strongly related to linear stability analysis. Traditional
linear stability analyses involve assuming small perturbations about a base state
(which is already known). The evolution of the perturbations is usually posed as an
eigenvalue problem using the governing equations. The eigenvectors give the “modes”
of the system and the eigenvalues give the oscillatory frequencies and growth rates.
12. 9
Only the governing equations and a base state (usually the steady state solution to an
unsteady problem) are required.
DMD is an equation-free technique. It may be used to extract coherent
structures from a sequence of solutions without resorting to the governing equations.
DMD works on the perturbations about a base state by approximating a linear
evolution matrix for the perturbations and then obtaining the eigenvalues and
eigenvectors for the matrix.
4.1. Pre-processing 1: Total reaction rate evolution
Since DMD is a linear approximation to a non-linear solution, the results may
be expected to be accurate only when the evolution is approximately linear. The non-
linear term in the governing equations (1 through 3) is the reaction rate term. The
sum of the reaction rates at each point in the computational domain is the total heat
release rate (with some other constant factors). The total heat release rate is taken as
an indicator of the flow fields reaching a steady or an oscillatory state. The total heat
release rate is plotted as a function of time and a range of times where the total heat
release rate seems to be behaving linearly is taken for further analysis.
4.2. Pre-processing 2: Evolution of POD modes
Proper Orthogonal Decomposition (POD, also called Principal Component
Analysis) of the instantaneous flow fields is done and the most dominant modes
(based on the largest singular values) are obtained. The projection of the
instantaneous flow field on the POD modes may be obtained through simple
mathematical matrix operations and the same has been done. It has been found that
for the sequence of flow fields under study, the magnitude of the projection for the
second and third most dominant modes are 4 to 6 orders smaller than the most
dominant mode, hence demonstrating that the flow fields may be considered as small
perturbations imposed on a base flow which may be approximated by the first POD
mode.
4.3. Pre-processing 3: Choice of base state
DMD is to be used when only a sequence of flow fields are known. However, to
obtain perturbations, the base state must be known. In the current study, the full non-
linear solutions are available and the base state needs to be selected from the
available set of flow fields. Identifying an accurate base state strongly affects the
solution as may be intuitively expected. While the base state may usually be the
average of the instantaneous flow fields in a linear system, this is not the case in a
non-linear system.
In the current work, the formulation for a linear stability analysis involves
13. 10
defining perturbations as departures from the steady state. Hence the steady state
solution must be considered as the base state. The governing equations for the
reacting mixing layer problem ([1]) are available and they have been employed to
obtain the best approximation of a steady state. Different flow fields (“flow field” here
means the collection of the temperature and species concentration fields at a
particular time instant) are considered in calculating the steady state residue- last
time instant flow field from the non-linear solution, the mean of an oscillatory cycle of
the flame, mean of two cycles and the evolution of the dominant POD (proper
orthogonal decomposition, also called principal component analysis) mode in a
complete cycle. The field that gave the least steady state residue has been taken as the
base state.
4.4. Pre-processing 4: Extent of linearity of the evolution of perturbations
Since DMD gives a linear approximation for a non-linear flow, the extent to
which the evolution of perturbations (defined as described above) is linear is
expected to suggest the accuracy of the results. The non-linearity in the governing
equations is due to the reaction rate term. So, the perturbation in reaction rate is
linearized for small perturbations in the temperature and species concentration fields.
The linearized perturbation in reaction rate (calculated from perturbed temperature
and concentration fields) is compared to the actual reaction rate field as a measure of
the extent of linearity of the flow. The extent of linearity is defined, at each point, as
the ratio of the magnitude of linearized reaction rate to the sum of magnitudes of the
linear and non-linear parts of the reaction rate. An average of the extent of linearity at
each point is taken to be the extent of linearity for the full domain. This is then
compared for each instantaneous perturbation field. A suitable range of times where
the extent of linearity is more than a cut-off value (arbitrarily fixed, ~90% for
oscillatory flames, ~95% for steady flames) is then chosen for performing DMD.
4.5. DMD:
With a base state identified, and a range of time instants chosen (based on
extent of linearity), the perturbed fields are then subjected to DMD. DMD, in the way
the current work performs it, involves doing a POD of the perturbed fields. In the
singular value matrix obtained through POD, smaller singular values are omitted in
order to save memory and to have better conditioned matrices (the singular value
matrix has to be inverted). Except the first several singular values, the rest are too
small to affect the accuracy of the solution. In the current work, singular values upto
10-8 times the largest singular value are considered and the remaining neglected. This
is one of the very few parameters available to control the DMD technique.
14. 11
It must be noted here that the DMD modes obtained are usually complex (the
elements of the field have complex values) and the amplitudes of the modes are taken
to be more relevant physically since the growth/decay of the modes is of prime
importance to the study.
4.6. Post Processing 1: Projecting perturbations onto DMD modes
A subspace is defined with the DMD modes as its (non-orthogonal) basis. Each
instantaneous perturbation is projected onto this subspace and components
corresponding to each DMD mode is obtained.
If vi is the perturbation field (converted to a vector) at the ith time instant, and if xj is
the jth DMD mode, then vi is written as
The matrix cij is obtained using iterative matrix inversion methods and , the residual
vector, is then calculated. A residue is defined as the magnitude of the residual vector.
This residue is then normalized using the 2-norm of the perturbation field, and it is
calculated for each perturbation field. It is found that this relative residue is usually of
the order of 10-6 , hence the DMD modes capture the non-linear evolution to a
sufficient degree of accuracy.
Further, for each DMD mode, say ,the amplitude of is plotted against ‘i’,
the time. The average over ‘i’ of the magnitudes of , for each ‘j’, is taken to be
indicating the dominance of the ‘jth’ mode. The modes with the highest average of the
magnitude of the components is taken to be the most dominant mode. The evolution
of the magnitudes of over ‘i’ is plotted for the dominant modes and is found to be
exponential. This, and the small magnitude of the relative residual, conclusively show
that the DMD modes, as a complete set, capture the evolution of the flow quite
accurately.
4.7. Post Processing 2: Calculating and printing reaction rate contours
The reaction rates for the dominant DMD modes are calculated in two ways: 1)
The DMD mode is multiplied by the average of the amplitude of the components and
then added to the base state. The reaction rate is calculated using this new field and
the reaction rate for the base state is subtracted from it to obtain the reaction rate
field for the DMD mode. 2) The reaction rate is calculated using the linearized reaction
rate equation (for small perturbations).
15. 12
4.8. Post Processing 2: Correlating DMD modes to POD modes and within
themselves
The DMD modes have been correlated to the second and third dominant POD
modes based on the simple vector scalar product ( . The correlation to
the POD modes is significant (~0.65) for most of the dominant DMD modes. This
further reinforces the faith in the solutions. The DMD modes have also been correlated
among themselves and some of the DMD modes are quite well correlated to the first
few most dominant DMD modes.
4.9. Post Processing 3: Comparing frequencies
The oscillatory frequency of dominant DMD modes (as given by the
eigenvalues) are compared to the frequency of the total heat release rate. Most of the
oscillating DMD modes are found to have the same frequency or they are a higher or a
lower harmonic of the total heat release rate.
4.10. Movies
Movies showing the evolution of the following fields have been made: reaction
rate contours - instantaneous, instantaneous – base, and linearized reaction rate;
extent of linearity.
4.11. Repeating for Sub-domains
The entire process mentioned above is first done for the full computational
domain as used by David and Chakravarthy [1]. However, when dealing with small
perturbations, 1) the flame region is more important and 2) the zero gradient
Neumann boundary conditions are not exact. To reduce the effect of errors due to the
above reasons, the above steps are repeated for a sub-domain of the entire
computational domain used by David and Chakravarthy [1]. The size of the domain is
based on the DMD modes and the reaction rate movies obtained.
16. 13
Chapter 5
RESULTS
In the current work, the flow fields across the bifurcation boundary (from oscillatory state
to steady state, the parameter space map is shown in figure 2) corresponding to Le=1.6, Da=100,
and from ζ = 0.65 to ζ = 0.75 have been investigated and the results for these cases are present
here. For each of the steps described in the previous section, the resulting plots for each ζ are
presented. The dominant DMD modes are presented in the end. It must be noted here that the
goal of the project has been to explore the applicability of linear stability analysis to combustion
problems.
Fig. 2. Oscillatory regime boundaries in the Da-ζ space for Le = 1.6. ‘O’ represents oscillatory state,
‘S’ represents steady state (courtesy: David and Chakravarthy [1])
In the following sub-sections, the results corresponding to the final runs for each ζ (Le
=1.6, Da=100) are presented. After carrying out the analysis for the full computational domain, as
used by David and Chakravarthy [1], which has 241x481 grid points, spaced 0.0625 non-
dimensional length units apart (non-dimensionalized by the thermal diffusive length scale, α/ ).
The region which seems to strongly interact with the flame in the DMD modes and the reaction
rate videos are then considered to narrow down the domain of analysis. The final sub-domain for
which results are presented is usually of the size 101x221, the exact number of grid points
depending on ζ.
17. 14
5.1. Total heat release rate of instantaneous flow fields
The sum of the reaction rates at each grid point (of
the instantaneous fields) is plotted against time.
Except for ζ =0.72, all cases are run for 800 time
instants, with a time interval of 0.025 seconds, i.e.
for a total time of 20 seconds. The first two cases are
within the oscillatory regime boundary and they
exhibit oscillations. The latter 3 cases are steady
cases and the total reaction rate may be seen to
reach a steady value. The initial time in the 20
seconds shows non-linearity, so some of it is
omitted.
(a) ζ = 0.6, t = 165 to 800
(b) ζ = 0.65, t = 165 to 800 (c) ζ = 0.7, t = 65 to 800
(d) ζ = 0.72, t = 315 to 1,000 (e) ζ = 0.75, t = 0 to 800
18. 15
5.2. Evolution of POD modes
The evolution of components of the POD
modes 1 to 3 for the last few cycles (of total
heat release) of the flow fields have been
plotted. They components have been
normalized by the norm of the flow field
(converted into vector). The curve for POD1 is
almost 1 and lies at the top boundary of the plot
and hence cannot be seen. POD2 and POD3
oscillate between 10-2
and 10-8
for different ζ.
Oscillatory states show higher POD2, POD3
amplitudes while steady states show less.
(a) ζ = 0.6
(b) ζ = 0.65 (c) ζ = 0.7
(d) ζ = 0.72 (e) ζ = 0.75
19. 16
5.3. Steady state residuals for potential base states
The first 3 points on x-axis correspond
respectively to flow field at last time instant,
mean of 1 cycle (of total heat release) and
mean of 2 cycles. The remaining points on x-
axis correspond to 1 cycle of POD1- the POD1
mode multiplied by its component at each time
instant (since reaction rate is non-linear, the
component must be considered and not just
the mode). The norm of the vector is shown
on the y-axis. The field with the least residual
norm is taken as base state.
(a) ζ = 0.6
(b) ζ = 0.65 (c) ζ = 0.70
(d) ζ = 0.72 (e) ζ = 0.75
20. 17
5.4. Extent of linearity
The extent of linearity for each instantaneous
perturbation field, as described in section 4.4, is
plotted against time. It can be seen that the
linearity increases significantly during the initial
part of evolution and then approximately
flattens out. The flow fields in this flattened
range of linearity are considered for DMD
calculations.
(a) ζ = 0.6
(b) ζ = 0.65 (c) ζ = 0.70
(d) ζ = 0.72 (e) ζ = 0.75
21. 18
5.5. Contour map for extent of linearity
The contours for the extent of linearity are
plotted. The contours do not remain as shown,
but evolve in time. The ones shown here are
only representative. The dark spots for the
oscillatory cases actually do not have linearity
close to 1, but have close to zero. Different
colour levels (from red to blue) are merged in a
very small area and the low resolution shows
the spots as black, all the colour levels, till black
may be seen if zoomed in. These low linearity
contours move from left to right as time
progresses, along the same curve of dark spots
that can be seen in the figures.
(a) ζ = 0.6
(b) ζ = 0.65 (c) ζ = 0.70
(d) ζ = 0.72 (e) ζ = 0.75
22. 19
5.6. Residual norm when perturbed fields projected onto DMD mode subspace
Each of the instantaneous perturbed fields is
projected onto the subspace defined by DMD
modes as a basis and the norm of the difference
between the original perturbed field and its
projection on the DMD subspace is plotted. This
residual norm is quite small, ranging between
10-6
to 10-3
, showing that the DMD mode
subspace is a decent approximation for the
evolution of the perturbations. Since the steady
state was only approximately identified (section
5.3), the residual is quite high for the steady
states towards the end, where the solution is
very close to the actual steady state
(a) ζ = 0.60
(b) ζ = 0.65 (c) ζ = 0.70
(d) ζ = 0.72 (e) ζ = 0.75
23. 20
5.7. Correlating (linearized) reaction rates of dominant DMD modes to POD2, POD3 modes
As mentioned in sub-section 4.7, reaction rate
may be calculated in two ways. Here, it is
calculated by linearizing it for small
perturbations and using the linearized
equation to calculate DMD mode reaction
rate. This reaction rate is correlated to POD2
and POD3 reaction rates and the correlation is
plotted for 8 most dominant modes. Multiple
modes correlate to POD2 at around 0.6
(a) ζ = 0.60
(b) ζ = 0.65 (c) ζ = 0.70
(d) ζ = 0.72 (e) ζ = 0.75
24. 21
5.8 . Correlating reaction rates of DMD modes to POD2, POD3 modes
In 5.7, reaction rate field for DMD modes is
calculated using a linearized equation for
reaction rate. In the current section, the DMD
modes are multiplied by their average
component and added to the base state. The
reaction rate field for the base state is
subtracted from this new field to obtain the
reaction rate field for DMD modes, and
correlated to POD2, POD3 reaction rate fields.
In this case also, multiple DMD modes correlate
to POD modes at around 0.5.
(a) ζ = 0.60
(b) ζ = 0.65 (c) ζ = 0.65
(d) ζ = 0.72 (e) ζ = 0.75
25. 22
5.9 . Correlating (linearized) reaction rates of DMD modes to DMD1, DMD3
The linearized reaction rate fields of the 8
most dominant modes are correlated to the
linearized reaction rate fields of DMD1, DMD3.
Some of the modes that do not correlate well
to DMD1 or DMD3 correlated well to POD2,
POD3, implying that a part of POD2, POD3 are
captured in DMD1, DMD3 and the other in the
other modes. Since the elements of DMD
modes are complex, there may be conjugate
pairs of DMD modes, hence DMD1, DMD3
considered instead of DMD1, DMD2.
(a) ζ = 0.60
(b) ζ = 0.65 (c) ζ = 0.70
(d) ζ = 0.72 (e) ζ = 0.75
26. 23
5.10. Comparing frequencies of dominant DMD modes to frequency of total heat release
The natural logarithm of (Schmid, [4]) imaginary
part of the eigenvalues from DMD represent the
frequency of the corresponding DMD mode. This
frequency is normalized using the frequency of
total heat release (from plots in section 5.1). The
normalized frequencies and the real parts of the
eigenvalues (positive values indicate growth) are
plotted for 12 most dominant modes, along with
the averaged amplitude of the components of
DMD modes (section 4.6). The DMD mode
frequencies are either zero, or close to
higher/lower harmonics of total heat release
frequency.
(a) ζ = 0.60
(b) ζ = 0.65 (c) ζ =0.70
(d) ζ = 0.72 (e) ζ = 0.75
37. 34
Chapter 6
CONCLUSION
The pre-processing steps: sections 4.1 through 4.5 have shown that the data
sets from the non-linear solution may be used to obtain an approximate linear
evolution for the non-linear flow. For the case of oscillatory flames (as seen from plots
in section 5.1 and from figure 2), the amplitude of oscillations, i.e. the perturbations
from the base state, are significant, ~ 10% of the total heat release. The possible
anharmonic nature of the non-linear oscillations must have been captured by the
presence of different DMD modes, with similar shape, having different eigenvalues.
DMD has produced sufficiently accurate results: 1) The frequency of
oscillations is captured by the eigenvalues of the DMD modes (section 5.10), 2) the
perturbations can be projected with sufficient accuracy on DMD mode subspace
(section 5.6), 3) dominant DMD modes correlate well with POD2 and POD3 modes
(sections 5.8, 5.9), and 4) Shapes of DMD modes represent the extent of non-linearity
contours (sections 5.5, 5.11, 5.12).
For steady flame cases, only small perturbations have been considered.
However, owing to the initial conditions that were used by David and Chakravarthy
[1], some of the steady case non-linear solutions have reached very close to the final
steady state while others were still approaching them. Since the magnitude of
perturbations is small, as compared to oscillatory cases, the choice of base state
strongly affects the accuracy of the results. As seen in section 5.3, the chosen base
state is only an approximation to the actual steady state. Despite a slightly inaccurate
base state the obtained results are encouraging. As with oscillatory modes, steady
modes also give a sufficiently accurate description of the non-linear evolution.
The current work has successfully demonstrated that linear evolution may be
approximated for non-linear combustion problems. Further work based on this may
include:
1) Doing a traditional linear stability analysis of the base states chosen in the
current work and comparing the results to the DMD modes and
eigenvalues. The (degree of) success of this would imply that traditional
linear stability analysis would (to some degree) predict the non-linear
evolution of combustion systems.
2) Use the DMD modes and corresponding growth rates and frequencies to
predict interaction with other instability modes such as hydrodynamics,
buoyant and acoustic.
38. 35
3) Extend to more complex systems such as shear layers, flows with density
stratification, flows with acoustic waves, backward facing step or other
complex geometries.
The suggested list of extensions is not an exhaustible list. Due to the
exploratory nature of the current work, and the relative infancy of the technique used,
no set of guidelines were available or be developed to carry on further work.
However, a robust code has been developed that may be of use to future researchers.
39. 36
REFERENCES
(1)David S. Bhatt and Chakravarthy S. R. , (2012) ‘Nonlinear dynamical
behaviour of intrinsic thermal-diffusive oscillations of laminar flames with
varying premixedness’, Combustion and Flame, 159 (2012) 2115–2125
(2)Matalon Moshe, (2009) ‘Flame Dynamics’, Proceedings of the Combustion
Institute 32 (2009), 57 -82
(3)Papas, Paul, Rais, Redha M., Monkewitz, Peter A. and Tomboulides,
Ananias G.(2003) 'Instabilities of diffusion flames near extinction',
Combustion Theory and Modelling, 7: 4, 603 — 633
(4)Peter J. Schmid, (2010) ‘Dynamic mode decomposition of numerical and
experimental data’, J. Fluid Mech. (2010), vol. 656, pp. 5–28
