Cross-Wavelet Analysis on Radiation Emitted by the plume of a small-size Solid Rocket Motor

This work follow a previous activity that developed a non-intrusive measurements technique for particles velocity ejected from a solid rocket motor based on wavelet analysis. The aim was to detect the particles passage and calculate the axial velocity through an auto-conditioned cross-correlation algorithm. In this study, a wavelet based processing techniques are applied and an innovative methodology based on the cross-wavelet analysis is presented. The objective is to detect the impulsive events associated to the particles crossing the field of view of the acquisition system. Starting from two synthetic signals characterized by a Gaussian peak, each one located at a different time instants, cross-wavelet analysis identify an increase of energy for all the correlation time producing a structure in the time scale comparable to the delay between the peaks. On the experimental database a continuous wavelet transform and a local intermittency measurement are performed to localize high energy events with randomic occurrence in the time series that are associated to the passage of allumina particles. Cross-wavelet analysis present several correlation structures localized in time between the events highlighted by the LIM maps. Results are very promising and could be a very interesting bases for deeper analysis based on wavelet, LIM and cross-wavelet.


Introduction
One of the most widely used technologies in the space sector are the Solid Rocket Motors (from now on SRM).The best-known application is for thrust generation in the early stages of aerospace launchers.This technology is characterized by an extreme simplicity of construction, it can be represented as an energized fuel ejected at high speeds through a nozzle with a main drawback of low control during propulsion.The fuel is solid composite propellant in which 10-20% of the mass is aluminum particles and 70% is ammonium perchlorate (oxidizer) dispersed in a liquid binder.Crystalline alluminium particles are characterized by low cost, high specific impulse and more combustion stability that make it very suitable for the application.The combustion is characterized by the reaction of the burned metallic fuels with the oxidizer and the products are metal oxides like Aluminium Oxide Al 2 O 3 (allumina).Thus energized Aluminium Oxide particles are ejected at very high speed from the nozzle of the SRM and thrust is generated.The low control of the combustion is the main drawback of this technology and it represents an open question for the scientific community.It is well known that the combustion instabilities could have a strong influence on performance and safety of many tecnology like burners or turbine.When the burn rate oscillation couples with the acoustic oscillation in the combustion chamber usually instabilities are generated and amplify causing also structural damages.While most of these technologies allow to modify fuel concentration or other factor of influence on the combustion, for SRM the ignition represent the last controllable action on the combustion of solid propellant.Thus, to prevent the instability of the phenomena is fundamental to know it thoroughly in order to avoid inefficiency of malfunctions.Many information are contained in the plume of a solid rocket motor, some can come form the distribution and composition of the particles ejected from the nozzle [1].Nowadays, the space sector is strongly industrialized and these information can represent a source of deep optimization of the production processes of the manufacturer companies.Currently many space agency are investing in research and development of new tecnologies or optimization of the processes in order to reduce costs generally very expensive.New frontier of the space travel is the reuse of launchers in order to drastically reduce costs and make it accessible to the tourism, so the maximum efficiency of the SRM is fundamental.To approach high level of efficiency the physical phenomena occurring during the propulsion of a rocket have to be deeply known through innovative tecnique and instruments.At the actual state-of-the-art, in order to investigate the combustion process a series of intrusive and laboratory-sized diagnostic techniques are employed but them can not access to the information inside the plume due to the extreme chemical and physical operative condition.These considerations justifies this work since a non-intrusive method designed from previous work [2] allow to extract information from the very inaccessible plume.Regardimg the plume of SRM, the aluminium particles represent the main source of thermal radiation and emit most of it in a wavelength ranges from 0.5 to 5 µm, with peaks at 1 and 2 µm [3,4].Many non-intrusive techniques to measure the size of the particles [5,6,4] has been studied and [2,7] shown that a photodiodes can measure the velocity the aluminium particles.The experimental setup used for the acquisition of the database analyzed, described in section 2, performed single-point measurements of the velocity and particle size in the plume using an array of photodiodes along the longitudinal axis of the SRM.While the previous work processed a wavelet based conditional tecnique to process the experimental data and identify the aluminium oxide particle passage, in the present study a cross-wavelet analysis is performed to investigate its reliability to measure the particles velocity.The wavelet analysis and cross-wavelet technique is presented in section 3, and the relative results are discussed in section 4. The conclusion are given in section 5.

Experimental Setup
The experimental campaign has been carried out at AVIO Colleferro test bench.In this facility, a cylindrical metallic case with 6 kg of solid propellant was equipped with an igniter and a nozzle to simulate a small-scaled solid rocket propulsion.As a combustion's reaction of aluminium and ammonium perchlorate powders, aluminium oxide particles and other products was ejected at high velocity from the nozzle.The radiant energy of the plume was sampled by two blue/UV enhanced silicon photodiodes placed at different axial distances from the nozzle (2.5 and 5 diameters) as represented in figure 1.
A lens and a tube with 20 mm length and 1 mm in diameter were used to reduce the field of view of the photodiodes in order to detect small number of particles per second.The sensors, placed outside the plume, were pointed towards the axis of simmetry of the small-size SRM and a syncronous acquisition was performed.The acquisition frequency was 100 kHz and a pressure transducer measured the pressure within the combustion chamber.

Methods
Wavelet based post processing tecnique has been one of the most used methods to analyze randomical data.Starting from 3 decades ago [8,9] one of the best application of this analysis is on turbulent flows experimental or numerical data, in the large amount of literature the authors suggest [10,11,12].Theoretical aspects of wavelet analysis are presented by [13] where also cross-wavelet analysis is introduced.The cross-wavelet analysis are deeply used in turbulent flows [14,15] and in other fields like geophysical data series [16].Briefly describing, wavelet analysis is a tool to investigate both time and frequency information in time series trough the wavelet transform.Based on the Multi Resolution Analysis (MRA) wavelet analysis evaluates multiple scales in time and frequency domain thus is it possible to represent the signal with the frequency resolution like the Fourier Transform (FT) and the time resolution of a time series.Differently from FT, wavelet analysis doesn't use sine and cosine function as orthogonal basis but has several functions available for the projection.Thus, the first step is to select one mother function in order to achieve better representation, for this case Morlet function has been selected as mother wavelet: Where t is the dimensionless time and ω 0 is the frequency center of the wavelet.The second step is generate several function by scaling and shifting the mother one.Varying the scaling and time parameter s and τ the Continuous Wavelet Transform for a time series x(t) is calculated as: where the integral represents a convolution between x(t) and the dilated and translated complex conjugate counterpart of ψ(t).
In order to compare Fourier and wavelet transform is possible to convert the time scale, identified by the scaling factor s, to a pseduo-frequency scale through the equation: In the present work the Continuous Wavelet Transform was performed using a custom made function in Matlab © enviroment.Starting from the wavelet coefficients a very useful tool to analyze the phenomena under investigation is the Local Intermittency Measurement (LIM).LIM allow to highlight high-energy events as pointed out by [9], thus similarly to the identificaton procedure introduced by [17], a passage of high-energy physical structures should induce in the LIM a burst located at the corresponding time scale location.The LIM is calculated as: Where <> denotes the time average.The LIM representation, called LIM map enhances no uniform distribution of energy in time, indeed the wavelet analysis is based on an energetic criterion [18,2].The main analysis of the present work start Considering two different signals with which is possible to define a wavelet cross-scalogram: Where w * 2 (r, t) is the complex conjugate of w 2 (r, t).What is expected is to implement a crosscorrelation analysis based on the cross-wavelet.In order to verify this feature, two synthetic signal with Gaussian peaks with a variable time delay have been generated (Fig. 2).The crosscorrelation should identify the delay between the peaks, while with cross-wavelet analysis a local measurement of correlation has been calculated.The results of cross-wavelet analysis present an increase in energy for all the correlation time, thus a structure in a timescale comparable to the delay between the peaks is produced (Fig. 3).The peaks in the synthetic signals ideally represent the peak associated to the passage of the allumina particles, so the same analysis to the real time series is expected to be successful.

Results
The acquired thermal radiation of the 2 sensor is shown in fig. 4 and presents several peaks with very high energetic content compare to the mean energy.The previous work [2] demonstrate that these intermittent phenomena can be associated to the particles crossing the field of view of the photodiodes.Indeed, asimmety in PDFs confirmed the presence of intermittent events and Local Intermittent Measurement calculation, based on wavelet transform, confirm that these events can be associated to the same physical phenomena.Thus, a conditioned cross-correlation identified a mean velocity of the alluminium particles.In this work a cross-wavelet analysis is applied on the database to investigate if this algorithm could be an additional and innovative method to calculate the cross-correlation between the signals and, for the case under investigation, the velocity of aluminium particles.First of all the wavelet transform of the signals has been calculated and are reported in fig 5 .The wavelet spectrograms identify several impulsive energy events associated with the passage of particles in both signals.These energetic structures are at different time scales and present randomic occurrence.This random occurrence explain why a wavelet transform is a very useful tool to analyze the database.Indeed a Fourier analysis could not localize in time these randomic highly energetic events.
As described in section 3, LIM can be a suitable tecnique to select the energetic events associated to the passage of allumina particles, thus the LIM map of the time series is presented in figure 6.
The LIM maps identify some structures associated to high-energy events, as the thermal radiation emitted from the allumina particles that cross the field of view of the sensors.With this analysis it is possible to localize in time the coherent events betweeen the time series and where the cross-wavelet analysis is expected to show peaks of energy.
Therefore, by calculating the cross-wavelet we obtain the cross-spectrogram in the figure 7.
From the validation done with the synthetic signals, it is possible to conclude that the energy structures identify the correlation events.These structures are located at the time scale of correlation.This result represents a very interesting information in order to develop an innovative tool to analyze cross-correlation between time series.For the case under investigation these results could represent a basis for the development of an auto-conditioned cross-wavelet  based method to identify the portion of the data with aluminium particle passage information.Thus, with this separation a windowed cross-correlationn could be processed to obtain very interesting and accurate results.

Summary, conclusions and future perspectives
Following a previous work aimed to the velocity measurement of aluminium oxide particle ejected from a small-scale solid rocket motor, an innovative method based on the cross-wavelet analysis is performed to investigate its efficacy to detect these events.The validation starts from the application of cross-wavelet analysis on two synthetic signals characterized by a Gaussian peak each one located at a different time instants.These signals represent an ideal experimental acquisition of an impulsive events like the passage of an alumina particle.Many cases have been investigated with different time delays between the peaks of the two synthetic time series and the cross-spectrograms are presented.The cross-spectrograms show an increase of energy for all the correlation time producing a structure in the time scale comparable to the delay between the peaks.On the experimental database a wavelet based analysis is performed and the scalograms identify several impulsive energy events associated with the randomic passage of particles in both signals.
In order to select information only from uncoherent and high energy events a Local Intermittency Measurement is performed.LIM maps allow to localize in time the coherent events betweeen the time series and where cross-wavelet is expected to produce interesting results.Thus, performing cross-wavelet analysis on the experimental database some correlation events and structures are identified.The validation of this method allow to extract the time scales of correlation and could be a very interesting bases for deeper analysis like an autoconditioned cross-wavelet windowing to select portions of the signal where to perform cross-correlation and calculate important information as the velocity of the alluminium oxide particles.

Figure 1 .
Figure 1.Representation of the experimental setup.

Figure 2 .
Figure 2. Synthetic signals with gaussian peaks with different time delays of 1 second (a), 2 seconds (b) and 4 seconds (c).

Figure 3 .
Figure 3. Cross-Wavelet Analysis applied on synthetic signals with gaussian peaks with different time delays of 1 second (a), 2 seconds (b) and 4 seconds (c).

Figure 4 .
Figure 4. Acquired signal of the thermal radiation emitted by the plume of the SRM with several peaks generated the radiation scattered by the aluminium oxide particles.

Figure 5 .
Figure 5. Wavelet transform of the thermal radiation time series of the first sensor (a) and the second sensor (b).

Figure 6 .
Figure 6.Local Intermittency Measurement Map of the time series of the first (a) and second (b) sensor.

Figure 7 .
Figure 7. Cross-Wavelet transform of the two thermal radiation time series with higlighted structure of correlation event.