This document summarizes a study on the effect of perturbations in the flame front on the approach flow velocity of a premixed flame. Numerical simulations were performed to obtain data on flame characteristics like temperature, velocity, and species concentration. This data was then analyzed using MATLAB. Relationships between the flame front shape and local approach velocity were explored based on theoretical equations derived from Navier-Stokes equations. Good agreement was found between velocities calculated from the theoretical equation and those obtained from simulation data, validating the relationship between flame front shape and local flow velocity. The work provides insight into the physical phenomena governing flame instability.

1. Siddharth Ratapani Navin | Premixed Flame Combustion | September 2, 2016
Study of the effect of an
unforced perturbation in the
flame front on the approach
flow of a premixed flame.
Under the guidance of
Dr. Donghyuk Shin

2. PAGE 1
Introduction.
Recently the fundamental research has been driven mostly by growing
environmental concerns related to decreasing emissions and by
achieving better control of combustion devices (e.g. better flame
stability in gas turbines). The environmental laws on emissions have
become more straightened.
These restrictions put upon already very advanced combustion
technologies like the lean premixed combustion prompt a need to
exploit these combustion techniques to their maximum potential.
The present work focuses on the combustion of methane which is the
main component of natural gas commonly used to power stationary
gas turbines. Gas turbines have a significant share in the total energy
production. For example in the year 2002 the worldwide production of
the electrical energy by natural gas-powered gas turbines reached
worldwide about 17% of the total produced electricity. This share is
expected to grow in the future due to many advantages of natural gas
combustion such as the possibility of achieving very low NOx
emissions.
One of the major research targets is the determination of the axial fuel
approach speed uy and description of dependence of ụ on the shape of
the flame front ξ. The study of local changes in the approach fuel speed
at the flame front due to perturbations in the front can lead to a better
understanding of the physical phenomena that govern the flame
instability.
The main goal of the research is to obtain a mathematical relation
between the axial approach speed and the shape of the flame front for
different types of unforced perturbations on the flame front. The
relation is brought about by studying flame characteristics at the front
through data obtained by numerical simulations.

3. PAGE 2
Acknowledgements
For this project I would like to thank Dr. Donghyuk Shin for entrusting
me with this project, offering me valuable advice and explaining key
concepts in such a crisp manner. Working on a project of such caliber
in a prestigious college under the supervision of Dr. Shin is a priceless
experience. I would also like to thank Dr. Andy Aspden, for providing
me with the numerical simulation programs. Dr. Aspden was also very
instrumental in helping me learn the key aspects of Linux in a very
short time. I would also like to thank Dr. Tom Bruce and my HOD
Dr. V Krishna, without whom I would not have gotten this wonderful
opportunity in the first place. Mrs. Pauline Clark from the IES was very
helpful and provided me with all the resources which I required at the
beginning of the project. My classmate and friend Shreyans sakahare
was instrumental in providing a second opinion to my programs and
outputs. Last but not the least I would like to thank my parents who
supported this endeavor, without who’s guidance I would not have
gotten this far.

4. PAGE 3
Abstract
The project was split into two stages as listed below:-
1. NUMERICAL SIMULATION PRE-PROCESSING.
Numerical simulations were executed on a Linux platform to produce
the spatial distribution of parameters such as temperature, axial
velocity, density, concentration of CH4 etc. The values of these
parameters in the Cartesian system were saved in files. All critical data
required for the study was extracted this way. Parameters such as the
wavelength and amplitude of the disturbance were the input to the
simulation to get different out puts. The data was recorded for
different time steps in the simulation.
2. DATA POST-PROCESSING
The data saved into the files was analyzed using a MATLAB code. All
different parameters were saved into 2D and 3D arrays respectively.
These arrays were used to plot all the significant graphs required for
analysis. The noise from the output graphs was filtered out leading to
clean results. The first set of results were obtained by plotting graphs
which were expected to follow theoretical trends. A theoretically
derived relationship between the approach flow velocity and Flame
front position was used to validate the dataset. The final relationship
was a result of the amalgamation of the theoretical relationships and
graphical fits obtained from theoretical simulations. The new formula
was used against different modes and amplitudes, the difference
between the formula output and real-time simulation was in close
tolerances, which was validation enough that the formula was valid for
all types of perturbations.

5. PAGE 4
Procedures used in Analysis
1. COLLECTION OF DATA FOR VARIOUS VARIATIONS IN THE
SIMULATION PARAMETERS
To plot data and obtain relations, it is imperative to have a lot of test
conditions. As the perturbations are characteristic of this research, the
properties of the perturbation in the flame front were varied i.e. their
wavelengths and amplitudes were changed.
Visualization of the 1st
mode perturbation flame front
of amplitude 6.6xE-4 m t=0.0007 s, between the unburnt gases (blue)
and burnt products (yellow).

6. PAGE 5
Visualization of the 3rd
mode perturbation flame front
of amplitude 2xE-3 m; t=0.0090s
Mode 5, amplitude 2.2 E-4 m.; t=0.0028s

7. PAGE 6
2. EXTRACTION OF FLAME FRONT PARAMETERS.
To study the variation parameters at the flame front, it is essential to
make an accurate prediction of the position of the flame front in the
Cartesian plane. To make an estimate of the flame front position
certain assumptions were made:-
1. Constant temperature of 900 °K was assumed at the flame front.
The value of 900 °K was specifically chosen as it was the mean
temperature in the flame front, from the graph given below.
2. Though the flame front has a certain thickness, the thickness
was not taken into consideration in any of the calculations.
Figure-1

8. PAGE 7
From these assumptions, the ‘y’ coordinates of the flame front for
every ‘x’ coordinate were extracted from the temperature distribution
output of the numerical simulation. Every point in the temperature
distribution with a temperature equal to 900 °K was recorded, its
respective coordinates were recorded. In a similar fashion the
approach velocity uy at the flame front was extracted from the velocity
distribution data of the same plot. The ‘y’ coordinates of the flame
front are essentially equal to ξ. The values of ξ and uy were stored into
arrays.
3. FILTERING OF DATA
From the first assumption made, the presence of noise in the
data required for analysis was apparent. Filtering of the data is
critical before any type of processing.
The data recorded for ξ had a global appearance of a sine wave
but locally consisted of steps. These steps would cause problems
in fitting the curves, and also in using numerical methods such
as FDM for finding the gradient at different points of the graph.
Figure-2

9. PAGE 8
The process used for smoothening out the graphs is called the
method of moving averages. The value at every point in the graph is
calculated as a local average over 5% of the span.
Figure-3

10. PAGE 9
The distribution of uy along the x axis was also globally sinusoidal but
exhibited a frequency distribution like characteristic locally. The
moving average method is not suitable for such data. The Savitzky-
Golay method is used as it filters frequency distribution type data and
also does not remove useful data as noise like the moving average
method.
Figure-4

11. PAGE 10
4. FITTING DATA INTO A SUM OF SINES CURVE
As the analysis required the values of the derivatives at various points,
the simplest method to go about this was by fitting the data into
curves. As all the distributions were globally sinusoidal, the sum of
sines curve was considered an ideal fit. The curves of ξ vs. x and uy vs. x
were fit into sum of sines curves. The fit was excellent and in very close
tolerances.
Figure-5

12. PAGE 11
The values of
𝜕𝜉
𝜕𝑥
,
𝜕2 𝜉
𝜕𝑥2
were calculated at all points by differentiating
the fitted curve of ξ vs. x. The values of
𝜕𝑢 𝑦
𝜕𝑥
and
𝜕2 𝑢𝑦
𝜕𝑥2
were calculated
at each point in a similar fashion.

13. PAGE 12
Results
As all data was stored into arrays and filtered, there were two ways to
proceed with the project. These methods are explained below:-
 FINDING LOCAL CHANGE IN APPROACH VELOCITY DUE TO
THE CONVEX/CONCAVE SHAPE OF THE FLAME
If we envision the flame front perturbations, we can say that if the
shape of the front is convex into the flow, the local velocity uy at this
location decreases as the inflow diverges away from the point leading
to deceleration. Whereas if the front were to be concave into the flow,
this would lead to the flow converging into the area, leading to an
accelerated flow i.e. an increase in uy.
If u=𝑢 𝑦,0
𝑢
+ 𝑢 𝑦,1
𝑢
… . (1) ; where 𝑢 𝑦,1
𝑢
is the local change in velocity due
to acceleration /deceleration.
As the concavity or convexity of a curve is determined by its second
order derivative.
We can say that 𝑢 𝑦,1
𝑢
is proportional to -
𝜕2 𝜉
𝜕𝑥2
. This property was
exhibited by the data through the graphs given below.
Figure-6

14. PAGE 13
Although the data did exhibit inverse characteristic, there was no
other common trend observed in the graphs for different modes and
amplitudes. Though this approach was theoretically accurate, a
numerical relation through a common formula could not be achieved.
Hence the following method was used.
 FINDING THE LOCAL CHANGE IN APPROACH VELOCITY
THROUGH NAVIER STOKES EQUATION.
This approach used a set of equations which were derived from the
Navier Stokes equations by applying different constraints to simplify
the equation. To check the validity of these equations, the dataset
needed to match the values obtained from the equation. The set of
equations used in this method is given below:-
 ξ =A ξ 𝑒 𝑖𝑘𝑥
𝑒−𝑖𝜔𝑡
…..(2) [1] *
 k=2𝜋/𝜆; .....(3)

1
𝑘𝐴 𝜉
𝐴1
𝑢
𝑢 𝑦,0
𝑢 =
1
2
(
𝜎 𝑝−1
𝜎 𝑝
)(
𝑆𝑡2−𝜎 𝑝
𝑖𝑆𝑡−1
) …..(4) [1]

1
𝑘𝐴 𝜉 𝑢 𝑦,0
𝑢 =−[𝑖𝑆𝑡 +
1
2
(
𝜎 𝑝−1
𝜎 𝑝
)(
𝑆𝑡2−𝜎 𝑝
𝑖𝑆𝑡−1
)] …..(5) [1]
The first step in this method is to find the value of ω. The
amplitudes for a particular mode and amplitude of disturbance are
plotted against time. The variation of amplitudes with time is
exponential. The values are fitted into a graph of the type 𝑎𝑒 𝑏𝑥
, the
coefficient b is essentially equal to the value ω. For every particular
mode and amplitude of perturbation, there exists one value of ω.
The value of the wave number k is calculated by the standard
formula. The next step is to find the value of velocity amplitude
coefficients 𝐴1
𝑢
and 𝐴2
𝑢
by simple substitution.

15. PAGE 14
The last step for this method is the one which determines the
fluctuation of the velocity for the given mode and amplitude at a given
time interval by using this formula.
𝑢 𝑦,1
𝑢
𝑢 𝑦,0
𝑢 =
1
𝑆𝑡
(
𝐴1
𝑢
𝑢 𝑦,0
𝑢 𝑒 𝑘𝑦
+
𝐴2
𝑢
𝑢 𝑦,0
𝑢 𝑒 𝑖𝑆𝑡𝑘𝑦
)𝑒 𝑖𝑘𝑥
𝑒−𝑖𝑤𝑡
….. (6)
Y, in this equation is the distance of the point of interest from the
flame front parallel to the Y axis. As we want to find the velocities very
close to the flame front we take y=0.001m.
We can find the magnitude of 𝑢 𝑦,1
𝑢
. Further we can the find the
magnitude of uy by equation 1.
On plotting the newly obtained values of uy from the formula and
comparing it the values from the data set. The following graphs are
obtained.
Figure-7

16. PAGE 15
Figure-7
Figure 8

17. PAGE 16
Sl. No. Mode Aξ(m)
x2.2E-4
(m)
𝐴1
𝑢
(m/s)
𝐴2
𝑢
(m/s)
ω
(rad/sec)
1. 1 1 -0.0323-0.22i 0.0323-0.408i 58.5424
2. 2 1 -0.049-0.2i 0.049-0.56i 71.58
3. 3 3 -0.1397-0.59i 0.1397-1.58i 109.87
4. 4 3 -0.153-0.854i 0.153-1.75i 193.645
5. 5 1 -0.0213-0.0929i 0.0213-0.248i 244.76
6. 6 3 0.0055+0.017i -0.055-0.32i 253.47
The table (1) given above shows the variation of the different
parameters of equation (6) at different modes and amplitudes of
perturbance.

18. PAGE 17
Conclusion and Inference
As we can see from the graphs shown in the previous section, the
values of uy given by the formula are in close agreement with the
values obtained from the numerical simulation.
From the above formula, as mentioned in earlier hypothesis that the
approach velocity uy follows the same characteristic as the shape of
flame front perturbation given by
𝜕2 𝜉
𝜕𝑥2
. We can see that by comparing
equation (2) and equation (6), both the equations are the same but
with different amplitudes, by this we can also say that variation in uy
follows the variation of ξ i.e. the shape of the variation of uy with x will
be the same as the variation of ξ with x.
Another important observation is that equation (6) always outputs
graph which are always perfectly sinusoidal which is not the case in the
actual data. The outputs can be obtained at even closer tolerances by
replacing 𝑒 𝑖𝑘𝑥
in equation with ∑ 𝑒 𝑖𝑛𝑘𝑥8
𝑛=1 . This sum can possibly fit
into all the imperfections in the graph.

19. PAGE 18
References
1. Unsteady Combustor Physics- Tim C.Lieuwen
2. MATLAB Graphics and Curve-fitting- Mathworks