Study of the effect of an unforced perturbation in the flame front of a premixed flame (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

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.

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.

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.

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).

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

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

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

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

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

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

The values of 𝜕𝜉

𝜕𝑥 ,

𝜕2𝜉

𝜕𝑥2 were calculated at all points by differentiating

the fitted curve of ξ vs. x. The values of 𝜕𝑢𝑦

𝜕𝑥 and

𝜕2𝑢𝑦

𝜕𝑥2 were calculated

at each point in a similar fashion.

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=𝑢𝑦,0𝑢 + 𝑢𝑦,1

𝑢 … . (1) ; where 𝑢𝑦,1𝑢

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 𝑢𝑦,1𝑢

is proportional to -𝜕2𝜉

𝜕𝑥2 . This property was

exhibited by the data through the graphs given below.

Figure-6

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 𝑒𝑖𝑘𝑥 𝑒−𝑖𝜔𝑡 …..(2) [1] *

k=2𝜋/𝜆; .....(3)

1

𝑘𝐴𝜉

𝐴1𝑢

𝑢𝑦,0𝑢 =

1

2(

𝜎𝑝−1

𝜎𝑝 )(

𝑆𝑡2−𝜎𝑝

𝑖𝑆𝑡−1 ) …..(4) [1]

1

𝑘𝐴𝜉 𝑢𝑦,0

𝑢 =−[𝑖𝑆𝑡 +1

2(

𝜎𝑝−1

𝜎𝑝 )(

𝑆𝑡2−𝜎𝑝

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 𝑎𝑒𝑏𝑥, 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 𝐴1𝑢 and 𝐴2

𝑢 by simple substitution.

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.

𝑢𝑦,1𝑢

𝑢𝑦,0𝑢 =

1

𝑆𝑡(

𝐴1𝑢

𝑢𝑦,0𝑢 𝑒𝑘𝑦 +

𝐴2𝑢

𝑢𝑦,0𝑢 𝑒𝑖𝑆𝑡𝑘𝑦 )𝑒𝑖𝑘𝑥𝑒−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 𝑢𝑦,1𝑢 . 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

Figure-7

Figure 8

Sl. No. Mode Aξ(m)

x2.2E-4 (m)

𝐴1𝑢

(m/s)

𝐴2𝑢

(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.

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 𝜕2𝜉

𝜕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 𝑒𝑖𝑘𝑥 in equation with ∑ 𝑒𝑖𝑛𝑘𝑥8𝑛=1 . This sum can possibly fit

into all the imperfections in the graph.

References

1. Unsteady Combustor Physics- Tim C.Lieuwen

2. MATLAB Graphics and Curve-fitting- Mathworks

Study of the effect of an unforced perturbation in the flame front of a premixed flame (1)

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