Poststall prediction of multiple-lifting-surface configurations using a decambering approach

TL;DR: In this article, a novel scheme is presented for an iterative decambering approach to predict the post-stall characteristics of wings using known section data as inputs, which differs from earlier ones in the details of how the residual is computed.

Abstract: A novel scheme is presented for an iterative decambering approach to predict the post-stall characteristics of wings using known section data as inputs. The new scheme differs from earlier ones in the details of how the residual is computed. With this scheme, multiple solutions at high angles of attack are brought to light right during the computation of the residual for the Newton iteration. As with earlier schemes, multiple solutions are obtained for wings at high angles of attack and the resulting converged solution depends on the initial conditions used for the iteration. In general, the new scheme is found to be more robust at achieving convergence. Results are presented for a rectangular wing with two different airfoil lift curves and for a wing-tail configuration.

Summary

Introduction

• The ability of linear aerodynamic methods such as lifting-line theory (LLT), Weissinger’s method and vortex-lattice methods to successfully predict the lift and induced drag behavior of medium to high aspect ratio wings at small angles of attack is well established.
• There 1 are also several additional isues that currently prevent the routine use of CFD for high-alpha predictions such as the time required for generating high-quality grids for each configuration.
• Thus the search for approximate approaches for stall and post-stall prediction of wings using known section data continues to be of interest.
• In the second approach, the deviation of the airfoil nonlinear lift curve from the potential-flow linear lift curve is used to apply a correction to the local α at each section of the wing.
• A literature study of the development of flow prediction methods over the years and brief descriptions of the two approaches follows.

1.1 Literature Study

• With the remarkable success of Prandtl’s lifting-line theory (LLT) in being able to predict the flow past medium to high aspect ratio unswept wings in incompressible flow, LLT became a standard tool for computing wing aerodynamics.
• At any given spanwise location, the change in Γ is shed as trailing vorticity, which in turn causes induced velocities along the lifting line.
• For this purpose, the classical Prandtl LLT assumes a linear lift-curve slope for the airfoil sections that form the wing.
• This lift-curve slope is typically close to 2 2π per radian.
• The studies in this regard can be widely divided into two categories as described below:.

1.1.1 The Iterative Γ distribution Approach

• Tani1 is believed to have developed the first successful technique in 1934 for handling nonlinear section lift-curve slopes in the LLT formulation.
• The boundary condition of zero normal flow is applied at the control point, which is the three-quarter-chord location for each horseshoe vortex.
• In the method of Piszkin and Levinsky, at each step of the iteration, the downwash computed using the Γ distribution from the previous time step is used to compute the change in the Γ distribution using the airfoil lift curve.
• Their article provides guidelines for the design of such wings and presents results for CL-α curves that extended to very high post-stall angles of attack close to 50 deg.
• In both Refs. 8 and 9, researchers reported that no asymmetric lift distributions for symmetric flight conditions are observed even when the iterations were started with asymmetric initial lift distribution.

1.1.2 The α Correction Approach

• An entirely different approach to the use of nonlinear section data was developed by Tseng and Lan.10 More recently, an approach similar to that reported in Ref. 10 was used by van Dam, Vander Kam and Paris11 for rapid estimation of CLmax and other high-lift characteristics for airplane configurations.
• If there is significant variation in the spanwise wing geometry, then airfoil data for additional sections along the span are used.
• Hence, the initial Cl distribution along span is calculated.
• If the difference between the viscous and linear potential section Cl at a section is greater than a given tolerance then the section angle of attack is corrected by a factor of the difference in the viscous and potential Cl to the lift curve slope calculated earlier.

1.2 Current Approach

• In the current research effort, a decambering approach (Scheme 1) was developed12 for predicting post-stall aerodynamic characteristics of wings using known section data.
• This approach is somewhat similar to that developed in Ref. 10, but differs in its capability to use both the Cl and the Cm data for the section and in the use of a two-variable function for the decambering.
• It was found that Scheme 1 worked for only some airfoil lift curves.
• Chapter 3 describes the incorporation of the decambering approach into the poststall analysis of a three-dimensional wing.
• Chapter 4 presents results to validate the use of the decambering approach for post-stall prediction purposes.

Concept for Flow Past an Airfoil

• The overall objective of the current research is to arrive at a scheme for incorporating the nonlinear section lift curves in wing analysis methods such as Lifting Line Theory (LLT), discrete-vortex Weissinger’s method and vortex lattice methods (VLM).
• For this purpose the wing span is assumed to consist of several sections and for each of these sections it is assumed that the two-dimensional data (Cl-α and Cm-α) is available from either experimental or computational results.
• Nonlinear lift curve slopes in wing analysis are incorporated by finding the effective angle of attack, αeff , and the corresponding effective reduction in camber at each section of the wing.
• The procedure for calculating the twovariable decambering function and the use of this decambering function to account for the differences in the potential and viscous flows for an airfoil is explained in detail.
• Results are then presented to show the effectiveness of the decambering approach for an airfoil.

2.1 The Decambering Concept

• This section illustrates the concept of decambering by using a simple example of a two-dimensional flow past a NACA 0012 airfoil.
• For the illustration of the decambering for two-dimensional flow over an airfoil, the following procedure is used: 1. Determine the viscous Cl and Cm for the α under consideration from exper- imental or computational data for the airfoil.
• Compute the difference between the viscous and the potential-flow results: ∆Cl = (Cl)visc − (Cl)potential and ∆Cm = (Cm)visc − (Cm)potential.
• In these equations, θ2 is the angular location in radians of 13 the start point for the second decambering function shown in Fig. 2.2(b) and can be expressed in terms of its x-location, denoted by x2, as shown in Eqn. 2.3.
• The progressive increase in the decambering required to model the boundary-layer separation at higher angles of attack is observed from these 15 figures.

Post-stall prediction of a finite

• Using the overall methodology described in chapter 2, two schemes have been formulated for determining the post-stall solution of a finite wing.
• The primary difference between the two schemes is in the details of how the residuals, ∆Cl and ∆Cm, are computed at each section of the wing.
• The first scheme, introduced in Ref. 12, was found to work well for certain airfoil lift curves, but failed to converge for several other airfoil lift-curves.
• This lack of robustness provided the impetus for developing the second scheme.
• The following sections explain the decambering approach for a wing, the overall iteration procedure used to implement the decambering approach, and the two schemes in detail.

3.1.1 Vortex Lattice Method(VLM)

• In a typical VLM, the lifting surface is divided into several spanwise and chordwise lattices.
• The primary advantage of using ring vortex elements is that they can be easily implemented in a computer program.
• Also, the zero-normal-flow boundary condition is satisfied on the actual lifting surface which may have camber and different planform shapes.
• In the current work unsteady analysis is not done.
• Therefore, the wake behind the wing is not discretized and in order to satisfy the Kutta condition at the trailing edge the wake is replaced by a series of horse-shoe vortices.

3.1.2 Predicting the decambering along wing span

• As explained earlier, the lifting surface is divided into several spanwise and chordwise lattices.
• Each spanwise section j (composed of a row of chord-wise lattices) has two variables, δ1j and δ2j, for defining the local decambered geometry at that section.
• Unlike in the two-dimensional case, where the δ1 and δ2 are selected to match the differences between the potential-flow and the viscous-flow results, in the three-dimensional case, changing a δ on one section is likely to have a significant effect on the neighboring sections and on the sections of the downstream 21 lifting surfaces.
• For each step of the iteration, F and J are determined, and δx is computed using Eqn. 3.1.

3.2 The iteration procedure

• The iteration scheme can be summarized using the flow chart in Fig. 3.2, the illustration in Fig. 3.3 and the following procedure: 1. Assume starting values of the decambering defined by δ1 and δ2 for each section of the wing; for example, section j has starting values denoted by (δ1s)j and (δ2s)j; 2. Compute the wing aerodynamic characteristics using VLM3D.
• For this com- putation, the unit normal vector of each lattice is rotated to account for the decambered shape of each section of the wing; the VLM3D analysis provides the Cl and Cm of each section as output.
• Therefore, in scheme 2, the target Cl, (Clt,2)j, of section j for example, is the point of intersection between the trajectory line for section j and the airfoil lift curve, as illustrated in Fig. 3.3.
• Update the values of δ1s and δ2s by adding the correction vector δx multiplied by a user-specified damping factor D (also referred to under-relaxation factor).

3.3 Multiple intersections in scheme 2

• While there is no ambiguity in determining the values of the target Cl for lines L1 and L3, there are clearly three possible choices for the target Cl for line L2.
• This illustration clearly demonstrates that it is possible to obtain multiple solutions for post-stall conditions; a fact, that was apparently first suggested by von Kármán (see Ref. 4) and has since been discussed by several researchers.
• Using the trajectory line L2 in Fig. 3.4 for example, the intersection point 1 is chosen if the logical switch lpoststall 26 for the section is unstalled and the intersection point 3 is chosen if the logical switch for the section is stalled.
• This logic removed any occurrence of unstalled regions with multiple-intersections sandwiched between two stalled regions.

Results

• The iterative decambering approach discussed in Chapter 3 has been implemented for the analysis of multiple-lifting-surface configurations in VLM3D, a custom VLM code.
• The computation of the residual has been implemented using two schemes and the effectiveness of the two schemes are compared.
• The examples in this chapter are presented in five sections as follows: 1. Section 4.1: In this section, the examples have been used to compare the predicted results from the current method with experimental results from Naik and Ostowari.
• For the examples in this section, the two-dimensional experimental data has been used as input for the method and the predicted finite-wing lift curves have been compared with the experimental data.
• Tapered wings of different taper ratios are used to study where the wing first stalls and how the stall progresses along the span with increasing angle of attack.

4.1 Experimental Validation

• Experimental two-dimensional data17 for a NACA 4415 airfoil at two Reynolds numbers of 0.5 million and 0.75 million, shown in Fig. 4.1, is used as input to generate post-stall results for rectangular wings of aspect ratios 12, 9, and 6 using VLM3D.
• In the same figure, the airfoil Cl-α curve and the wing CL-α curve from experiment 17 are also shown for comparison.
• The results from scheme 2, shown in Fig. 4.5, do not exhibit these “sawtooth oscillations” for a majority of the conditions.
• Therefore the wing CL decreases between 44 deg and 51 deg.
• Scheme 2 uses an inclined trajectory line in order to determine a solution wherein, whenever more than two solutions are possible the intermediate solutions are neglected and either the maximum or minimum solution is chosen depending upon “lpoststall”.

4.1.2 Effect of initial conditions on the iterations for

• Figure 4.9 shows the wing CL-α predicted using scheme 2 for two different starting conditions for the Newton Iteration: (a) δ1 = −40 deg for all sections at each α and (b) δ1 = 0 deg for all sections at each α.
• Figure 4.10 shows the spanwise Cl distributions for the two cases for α of 18 40 deg.
• The results clearly illustrate that multiple solutions are possible for poststall conditions.
• Furthermore, the results do not provide any clear guidelines as to which is the correct solution.
• It can however be said that the different schemes and starting assumptions predict the wing CL for a given post-stall angle of attack within a small scatter band.

4.1.3 Rectangular Wing (AR=9) with the NACA 4415

• It is evident again from Fig. 4.11 that in comparing the results of the two schemes with the experimental data, scheme 2 gives a better comparison with experiment.
• Experimental results for the spanwise Cl distributions were not available for comparison.
• Results from both schemes have numerically converged to 42 within a tolerance of 0.001 in ∆Cl and ∆Cm.

4.1.4 Rectangular Wing (AR=6) with the NACA 4415

• Figure 4.14 shows the wing CL-α curve from VLM3D using scheme 2.
• In the same figure the airfoil Cl-α curve and the wing CL-α curve from experiment17 are also shown for comparison.
• For this case, scheme 1 failed to converge for post-stall angles of attack.
• The trends are similar to those seen in sec.

4.1.5 Rectangular Wing (AR=12) with the NACA 4415

• In Fig. 4.16, the wing CL of only the converged angles of attack are plotted for scheme 2.
• As observed before, it can be seen from Fig. 4.16 that scheme 2 gives a better comparison with experiment.
• Experimental results for the spanwise Cl distributions were not available for comparison.

4.1.6 Rectangular Wing (AR=9) with the NACA 4415

• In the same figure the airfoil Cl-α curve and the wing CL-α curve from experiment17 are also shown for comparison.
• As observed in earlier examples, it can be seen from Fig. 4.19 that scheme 2 gives a better comparison with experiment.
• Experimental results for the spanwise Cl distributions were not available for comparison.
• The trends are similar to those observed for earlier cases.

4.1.7 Rectangular Wing (AR=6) with the NACA 4415

• Figure 4.22 shows the wing CL-α curve from VLM3D using scheme 2.
• In the same figure the airfoil Cl-α curve and the wing CL-α curve from experiment 17 are also shown for comparison.
• For this case scheme 1 failed to converge.
• Experimental results for the spanwise Cl distributions were not available for comparison.

4.1.8 Changes to the Lift Curve with Change to Aspect

• The change to the wing CL with aspect ratio is studied.
• The results from earlier subsections for the three rectangular wings of aspect ratios of 12, 9, and 6 as shown in Fig. 4.2 have been used.
• Figure 4.24 shows the CL-α curves of the three wings from VLM3D using scheme 2.
• In the same figure the airfoil Cl-α curve from experiment 17 is also shown for comparison.
• 17It is seen that the current approach is successful in capturing all of the important trends.

4.1.9 Summary

• The spanwise section Cl distributions predicted by scheme 1 exhibit substantial undesirable sawtooth behavior.
• The spanwise section Cl distributions predicted by Scheme 2, on the other hand, are devoid of such oscillations for most of the cases studied.
• Scheme 1 also has far more convergence problems than Scheme 2.
• Scheme 2, is therefore found to be better suited for post-stall prediction purposes and results presented henceforth will only be from Scheme 2. 56.

4.2 Study of stall characteristics

• The change in stall characteristics with change in taper ratio is studied.
• The airfoil used has the hypothetical lift curve as shown in Fig. 4.28.
• This behavior occurs because at a given α, the rectangular wing has the largest section Cl among all the wings in this study.
• The outboard portion of the wing of taper ratio of 0.3 is closest to stall and with increase in taper ratio the section Cl on the outboard portion of the other wings move farther away from CLmax as seen from Fig. 4.31.
• The four wings have different stall characteristics.

4.2.1 Stall Characteristics of a Part-Tapered Wing

• As seen in the examples so far, the rectangular wing has a tendency to stall first at the root which provides stall warning and greater aileron control.
• A tapered wing, on the other hand, provides a decrease in induced drag and good structural properties.
• The rectangular inboard portion provides good stall characteristics and is cost effective.
• The airfoil used has the hypothetical lift curve as shown in Fig. 4.28.
• At this condition the entire wing remains unstalled as the local section Cl values are less than the Clmax of 1.5.

4.3 Wing-Tail Configuration

• This example illustrates the effectiveness of the method for a multiple liftingsurface configuration.
• Also shown in the same figure, is the airfoil Cl-α curve for comparison.
• It must be mentioned that the tail CL,t in this plot is nondimensionalized with reference to the tail area.
• For this analysis, the CG was located to provide 66 a static margin of 10% of the wing mean aerodynamic chord.
• To examine the cause of this nose-down pitching moment at stall onset, it is instructive to study the individual contributions of the wing and the tail to the configuration lift curve as shown in Fig. 4.39.

4.4 Wing-Canard Configuration

• This example illustrates the effectiveness of the method for another multiple lifting-surface configuration, a high aspect ratio (AR = 10) constant-chord wing geometry with a canard (AR = 10).
• Which is used only for illustration of the capability of the current method.
• Also plotted on the same figure are the wing-canard lift curve and the airfoil lift curve for comparison.
• As the α is increased beyond 20 deg most of the canard and the wing has stalled except for two small regions on the outboard portions as shown by the section Cl distribution of the wing and canard at 23 deg in Fig. 4.44.

4.5 Spanwise Asymmetry in the Initial

• The spanwise section Cl distributions obtained for these cases was symmetric.
• As stated before, while in the past, some researchers8,9 have observed no asymmetric lift distributions for symmetric flight conditions even when the iterations were started with asymmetric initial conditions, other researchers3,4, 6 have observed asymmetric lift distributions for perfectly symmetric starting lift distributions.
• The right-side planform of the rectangular wing used in this example is shown in Fig. 4.26.
• The final converged distribution of δ1 along wing span does not show any asymmetry as shown in Fig. 4.51.
• The reason for the marked asymmetry in the spanwise section Cl distributions after the first and after 100 iterations and symmetry in the final converged solutions can be traced to the distribution of the local-section effective angle of attack along span.

Conclusions

• A decambering approach has been developed for the post-stall prediction of multiplelifting-surface configurations using known section data.
• Scheme 2, thus, brings to light the existence of multiple solutions right during the iteration process.
• The two-dimensional Cl-α curves from the experimental data is used as input to the current method and the predicted finite-wing characteristics from both the schemes are compared with the corresponding experimental results.
• The fifth example in this study was used to examine if such asymmetric behavior was seen with the current method.
• Good comparison with experimental data for pre-stall conditions and fair comparison with the experimental data for post-stall conditions indicates that the method can be used with reasonable confidence for the difficult task of rapid post-stall prediction of wing characteristics.

Figures

Figure 2.6: Effectiveness of the decambering for α of 10 deg.

Figure 4.52: Spanwise δ1 distributions after 1 iteration.

Figure 4.24: Wing CL-α predicted from VLM3D for rectangular wings of aspect ratios 12, 9 and 6 using a NACA 4415 airfoil at Reynolds number of 0.5 million.

Figure 4.25: Wing CL-α predicted from experiment for rectangular wings of aspect ratios 12, 9 and 6 using a NACA 4415 airfoil at Reynolds number of 0.5 million.

Figure 4.46: Initial δ1 distribution along wing span.

Figure 4.10: Spanwise section Cl predicted for a rectangular wing of aspect ratio 12 using a NACA 4415 airfoil at Reynolds number of 0.5 million for different starting values of δ1.

Figure 3.2: Flow chart of the iterative decambering approach for a wing(s). 28

Figure 2.1: Flow separation from an airfoil at a high angle of attack.

Figure 4.6: Wing CL variation with number of iterations for a rectangular wing of aspect ratio 12, using a NACA 4415 airfoil at Reynolds number of 0.5 million from scheme 2.

Figure 4.53: Spanwise δ1 distributions after 100 iterations.

Figure 4.11: Wing CL-α predicted using schemes 1 and 2 for a rectangular wing of aspect ratio 9, using a NACA 4415 airfoil at Reynolds number of 0.5 million.

Figure 4.36: Spanwise Cl distributions for the part-tapered wing.

Figure 2.5: The α s chosen to illustrate effectiveness of decambering.

Figure 4.42: Planform of the wing-canard configuration (RHS shown).

Figure 4.15: Spanwise Cl distribution predicted for a rectangular wing of aspect ratio 6, using a NACA 4415 airfoil at Reynolds number of 0.5 million from scheme 2.

Figure 4.47: Wing CL with an initial asymmetric distribution of δ1.

Figure 4.30: Change in the spanwise section Cl distribution with taper at α of 10 deg using scheme 2.

Figure 4.29: Wing CL-α for wings of different taper ratios, each of aspect ratio 10 using scheme 2.

Figure 4.14: Wing CL-α predicted using scheme 2 for a rectangular wing of aspect ratio 6, using a NACA 4415 airfoil at Reynolds number of 0.5 million.

Figure 4.43: Lift curves of the wing-canard configuration.

Figure 2.4: Cl and Cm using potential-flow with decambering.

Full text

ABSTRACT
RINKU MUKHERJEE. Post-Stall Prediction of Multiple-Lifting-Surface Con-
figurations Using a Decambering Approach. (Under the direction of Dr. Ashok
Gopalarathnam.)
A novel scheme is presented for an iterative decambering approach to predict
the post-stall characteristics of wings using known section data as inputs. The new
scheme differs from earlier ones in the details of how the residual in the Newton
iteration is computed. With earlier schemes, multiple solutions are obtained for
wings at high angles of attack as the final converged solution depends on the
initial conditions used for the iteration. With this scheme, multiple solutions
at high angles of attack are brought to light right during the computation of
the residuals for the Newton iteration. In general, the new scheme is found to
be more robust at achieving convergence. Experimental validation is provided
using experimental airfoil lift curves from Naik and Ostowari for three different
aspect ratios of rectangular wings. Results are presented from a study of the
stall characteristics of wings of different planform shapes and two configurations
of a wing-tail and a wing-canard configuration. Results are also presented from
a study to investigate possible asymmetric lift distributions when the iterations
were started with an initial asymmetric distribution of the decambering.

Post-Stall Prediction of Multiple-Lifting-Surface
Configurations Using a Decambering Approach
by
Rinku Mukherjee
A dissertation submitted to the Graduate Faculty of
North Carolina State University
in partial fulfillment of the
requirements for the Degree of
Doctor of Philosophy
Aerospace Engineering
Raleigh, NC
2004
APPROVED BY:
Dr. Ashok Gopalarathnam
Advisory Committee Chairman
Dr. Jack R. Edwards Dr. Fen Wu
Advisory Committee Member Advisory Committee Member
Dr. Zhilin Li
Advisory Committee Member

To Sudipto......
ii

BIOGRAPHY
Rinku Mukherjee had her primary education in India. She graduated with
a Bachelor’s in Civil Engineering from Jadavpur University, Kolkata, India in
1998. She got her Master’s in Ocean Engineering and Naval Architecture from
the Indian Institute of Technology Kharagpur, India in 2001. She joined North
Carolina State University in 2001 to pursue a PhD in Aerospace Engineering.
iii

ACKNOWLEDGEMENTS
I extend my deepest gratitude to my advisor Dr. Ashok Gopalarathnam for
his untiring support and encouragement without which this thesis would not be
complete. He has taken personal efforts to provide guidance and both emotional
and academic support during my time under him as a graduate student. Whether
it was studying courses, writing conference papers, preparing for presentations,
writing resumes or plain handling oneself professionally, he has always extended
his undivided attention to every minute detail and helped me to become better
than myself. His constant perseverance to do things better has been a great source
of learning. In spite of his very busy schedules he found the time to give personal
attention to every student in his continually growing research group. The greatest
lesson that I have learnt from him as a researcher is to say “I do not know”. To
acknowledge that one “does not know” and not fear the unknown can only lead
to further and rigorous research and is perhaps one of the greatest qualities to
possess as a researcher. Also, Dr. Gopalarathnam continually refers to research
as “fun” and C
l
-α plots as “beautiful” !!! That added so much colour to life and
made research lot more fun.
I am grateful to Dr. Jack R. Edwards, Dr. Fen Wu and Dr. Zhilin Li for being
on my doctoral committee.
I thank Dr. Jack Edwards for his suggestions and invaluable advice in dealing
with some of the very tricky problems in this research.
This research effort was supported under Grant NAG-1-01119 from the NASA
Langley Research Center. This support and helpful discussions with the technical
monitor, SungWan Kim, are gratefully acknowledged.
I thank my colleagues in my research group for making my graduate days less
stressful.
Thanks are overly due to my mother for her unconditional love and unrelenting
iv

Frequently asked questions

Q1. What are the contributions in "Post-stall prediction of multiple-lifting-surface configurations using a decambering approach" ?

A novel scheme is presented for an iterative decambering approach to predict the post-stall characteristics of wings using known section data as inputs. Results are presented from a study of the stall characteristics of wings of different planform shapes and two configurations of a wing-tail and a wing-canard configuration. Results are also presented from a study to investigate possible asymmetric lift distributions when the iterations were started with an initial asymmetric distribution of the decambering. 

Q2. What are the future works mentioned in the paper "Post-stall prediction of multiple-lifting-surface configurations using a decambering approach" ?

Future work will involve improving upon the modeling technique used to get a measure of the effective decambering due to separated flow in the present VLM3D. However, using scheme 2, possibilities of multiple solutions arise and the Newton ’ s method fails to converge for a few cases, and follows an undamped periodic motion. This can be done by defining the separated flow using a cubic camberline instead of two linear functions. This can be avoided by using a root bracketing method like the bisection method or the regula falsi method. 

Q3. What is the greatest lesson that The authorhave learned from Dr. Gopalarathnam?

To acknowledge that one “does not know” and not fear the unknown can only lead to further and rigorous research and is perhaps one of the greatest qualities to possess as a researcher. 

Q4. Who is the author of this essay?

Thanks are overly due to my mother for her unconditional love and unrelentingivsupport for everything that The authorhave pursued in my life. 

Q5. Where did she get her Master’s in Ocean Engineering and Naval Architecture?

She got her Master’s in Ocean Engineering and Naval Architecture from the Indian Institute of Technology Kharagpur, India in 2001. 

Q6. What is the name of the author?

She joined North Carolina State University in 2001 to pursue a PhD in Aerospace Engineering.iiiI extend my deepest gratitude to my advisor Dr. Ashok Gopalarathnam for his untiring support and encouragement without which this thesis would not be complete. 

Q7. Who is the author of this thesis?

The authorwould not be even writing this thesis without the love and care of my lifelong friend, my husband, Sudipto and The authorcould never thank him enough for always being there for me.