Understanding Kelvin–Helmholtz instability in paraffinbased hybrid rocket fuels
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Abstract
Liquefying fuels show higher regression rates than the classical polymeric ones. They are able to form, along their burning surface, a low viscosity and surface tension liquid layer, which can become unstable (Kelvin–Helmholtz instability) due to the high velocity gas flow in the fuel port. This causes entrainment of liquid droplets from the fuel surface into the oxidizer gas flow. To better understand the droplets entrainment mechanism, optical investigations on the combustion behaviour of paraffinbased hybrid rocket fuels in combination with gaseous oxygen have been conducted in the framework of this research. Combustion tests were performed in a 2D singleslab burner at atmospheric conditions. High speed videos were recorded and analysed with two decomposition techniques. Proper orthogonal decomposition (POD) and independent component analysis (ICA) were applied to the scalar field of the flame luminosity. The most excited frequencies and wavelengths of the wavelike structures characterizing the liquid melt layer were computed. The fuel slab viscosity and the oxidizer mass flow were varied to study their influence on the liquid layer instability process. The combustion is dominated by periodic, wavelike structures for all the analysed fuels. Frequencies and wavelengths characterizing the liquid melt layer depend on the fuel viscosity and oxidizer mass flow. Moreover, for very low mass flows, no wavelength peaks are detected for the higher viscosity fuels. This is important to better understand and predict the onset and development of the entrainment process, which is connected to the amplification of the longitudinal waves.
List of symbols
 \(A_{\text {cross}}\)
Area crossed by the oxidizer mass flow (\(\text {m}^2\))
 B
Constant for the perturbation expression (–)
 \(G_{\text {Ox}}\)
Oxidizer mass flux (\(\text {kg/m}^{2}\text {s}\))
 \(K_{0,1,2}\)
Dispersion relation constants (–)
 U
Gas velocity (\(\text {m/s}\))
 a, n
Regression rate parameters (–)
 g
Gravity acceleration (\(\text {m/s}^2\))
 h
Height (\(\text {m}\))
 k
Wave number (\(1/\text {m}\))
 \(\dot{m}_{\text {Ox}}\)
Oxidizer mass flow (kg/s)
 \(r_\text {f}\)
Regression rate (m/s)
 t
Time (s)
 x
Longitudinal coordinate (m)
 \(\gamma\)
Surface tension (N/m)
 \(\mu\)
Kinematic viscosity (Pa s)
 \(\rho\)
Density (\(\text {kg/m}^3\))
 \(\omega\)
Wave frequency (1/s)
Subscripts
 G
Gas
 I
Imaginary part
 L
Liquid
 R
Real part
1 Introduction
1.1 Hybrid rocket propulsion
Hybrid rocket engines (HRE) are generally made up of a solid fuel and a liquid or gaseous oxidizer. The combustion develops within the turbulent boundary layer over the fuel surface and the flame is located where the vaporized oxidizer and fuel exist in a combustible mixture (Marxman and Gilbert 1963). The heat is then radiated and convected from the flame to the fuel surface, in a selfsustained but diffusionlimited process. In a classical hybrid, the polymeric fuel pyrolyses and its vapours are transported to the diffusion flame, where it reacts with the atomized oxidizer transported from the free stream via turbulent diffusion (Chiaverini 2007). The diffusionlimited combustion process that characterizes conventional polymeric hybrid fuels, such as hydroxylterminated polybutadiene (HTPB) or highdensity polyethylene (HDPE), leads to a low regression rate and, consequently, to low thrust levels with respect to solid or liquid rocket engines. To overcome this problem, the available burning area has typically been increased through the use of multiport fuel grains. Unfortunately, this leads to complex grain geometries, low volumetric loading, increase of the residual mass of unburned fuel (which leads to a decrease in the delivered specific impulse) and uneven burning. Because of this, the use of hybrid propulsion systems has been hindered in the past despite their advantages with respect to solid and liquid engines. In fact, due to the fact that the propellants are stored in two different states of matter, hybrids are safer than solid motors. This also contributes to reduce the total costs of the engine. Moreover, they are characterized by controllable thrust, including shut off and restart capability. With respect to liquid engines, they are mechanically simpler and, consequently, cheaper. Finally, they have increased performance compared to solids and generally a specific impulse nearly comparable to liquid systems (Chandler 2012).
1.2 Optical investigations on hybrid rocket combustion
The discovery of liquefying hybrid rocket fuels has renewed the interest in hybrid propulsion and in optical investigations of the hybrid rocket combustion process. The theory of the Kelvin–Helmholtz instability (KHI) and of the liquid layer break up process, which leads to the fuel droplet entrainment, are wellexplained in the literature (Karabeyoglu et al. 2002; Karabeyoglu and Cantwell 2002; Funada and Joseph 2001; Amano et al. 2016). The increase in regression rate is proven at different thrust levels by Karabeyoglu et al. (2004) and some detailed optical investigations have been performed in the recent years to capture the entrainment process. Some of them are described herein.
In 2011, Nakagawa et al. investigated the dependence of the regression rate on the fuel viscosity. They performed optical tests at atmospheric pressure with different paraffinbased fuels and gaseous oxygen. Their images showed that droplets are generated during the combustion and entrained in the flow (Nakagawa and Hikone 2011). De Luca et al. also used an optical technique to investigate the hybrid combustion process. They looked inside a pressurized chamber over a mirror setup and thereby measured the instantaneous regression rate (DeLuca et al. 2011).
Many optical investigations on the combustion behaviour of both polymeric and paraffinbased hybrid rocket fuels have been done at the Stanford combustion visualization facility. In 2012, Chandler et al. investigated the combustion of paraffinbased fuels with gaseous oxygen at both atmospheric and elevated pressures. Their results showed roll waves and droplets in the atmospheric tests and filamentlike structures along the sides of the fuel grains in the tests run at elevated pressures (Chandler et al. 2012a). Moreover, they compared the combustion behaviour of paraffinbased fuels to that of classical hybrid fuels. They reported that for paraffinbased fuels entraining droplets were visible, for highdensity polyethylene (HDPE), only little droplet entrainment was seen and for hydroxylterminated polybutadiene (HTPB), no droplet entrainment was measured (Chandler et al. 2012b). In 2014–2016, many optical tests were conducted from Jens et al. with the same facility. They performed Schlieren and OH* images of the combustion of different classical polymeric and paraffinbased fuels in combination with gaseous oxygen at both atmospheric and elevated pressures (Jens et al. 2016). They reported unsteady blowing events of paraffin droplets in the tests at higher pressure, slightly above the critical pressure of their paraffin samples. Schlieren results of their tests reported a thickening boundary layer with increasing pressure (Jens et al. 2014a). Tests at atmospheric pressure were conducted also with HTPB, HDPE and PMMA (polymethyl methacrylate). No drastic change in boundary layer thickness growth was observed between paraffin wax and classical polymeric fuels (Jens et al. 2014b). On the other hand, the flame zone of the paraffin wax was found to be much thicker than that of the classical fuels (Jens et al. 2015).
In 2014, Wada et al. visualized the combustion of different polymeric fuels and paraffin. The investigated pressure range was from 1 up to 20 bar. In contrast to the other mentioned optical experiments, this setup looks at the combustion of opposing slabs of fuel mounted vertically. From their observations, they concluded that both the number and size of the entrained droplets are independent of the chamber pressure (Wada et al. 2014).
Many optical investigations have also been done at the German Aerospace Center (DLR), Institute of Space Propulsion, since 2013. Kobald et al. performed visual and Schlieren images of the combustion of paraffin wax and gaseous oxygen at atmospheric pressure. They reported visualization of droplets entrainment during startup and shutdown transients (Kobald and Schlechtriem 2013; Kobald et al. 2013). Since 2015, an automated video evaluation routine has been developed in DLR, to capture the dominant flow dynamic and combustion behaviour of paraffinbased hybrid rocket fuels during a typical test. The ignition, steadystate and extinction phases were clearly recognized with this technique. Moreover, the HTPB combustion flame showed no wavelike behaviour, in contrast to paraffinbased fuels (Kobald et al. 2015; Petrarolo and Kobald 2016).
The results of this research are focused on the connection between the Kelvin–Helmholtz instability mechanism and the entrainment in paraffinbased hybrid rocket engines. In particular, it is important to find the parameters that are mostly influencing the liquid layer instability process and to understand how this phenomenon is connected to the droplets entrainment rate. In this way, the fuel regression rate can be directly controlled just by triggering precise frequencies and wavelengths of the liquid layer.
In this paper, the influence of fuel properties and oxidizer mass flow on the Kelvin–Helmholtz instability mechanism is presented and discussed. It is demonstrated that there is a strict dependence between these parameters and the dynamics of the liquid layer. This is a first fundamental step in order to connect the instability process with the regression rate.
2 Fuel choice and characterization
2.1 Paraffinbased fuels
2.2 Paraffinbased fuels characterization
Characterization of the paraffin samples is necessary, since the entrainment process strongly depends on the fuel properties. Analytical formula for material properties like surface tension and viscosity of paraffin waxes are available in different publications, such as Marano and Holder (1997a, b, c). Unfortunately, these are often only valid for straight paraffin with distinct carbon numbers. Therefore, they cannot be applied for most of the samples used in this research, as well as for paraffinbased mixtures.
Viscosity, liquid density and surface tension measurements of the four types of paraffinbased fuels used in this research were performed in the M3 chemical laboratory at the DLR Lampoldshausen.
3 Experimental setup and methods
3.1 Test setup and data acquisition
3.2 Video analysis

video preprocessing, where the video is modified and the Snapshot Matrix is created;

decomposition of the Snapshot Matrix with the two methods: POD (proper orthogonal decomposition) and ICA (independent component analysis);

postprocessing, where the power spectral density (PSD) of spatial and temporal coefficients is performed.
In a second step, the Snapshot Matrix is decomposed with both algorithms, POD and ICA, into two matrices containing spatial and temporal coefficients. At the end, the power spectral density (PSD) of these coefficients is performed, to obtain the most excited frequencies and wavelengths during the combustion. Further details of the applied methods are given in Kobald et al. (2015) and Petrarolo and Kobald (2016).
4 Theoretical background: decomposition methods
4.1 Proper orthogonal decomposition
The proper orthogonal decomposition (POD) or principal component analysis (PCA) has been used in diverse area of research to obtain approximate, lowdimensional descriptions of turbulent fluid flows, structural vibrations and dynamical systems. It has also been extensively used in image processing, signal analysis and data compression (Chatterjee 2000).
The POD is a statistical method where an orthogonal transformation is used to convert a set of data into a set of linearly uncorrelated variables, which are called principal components. Their number is usually less than the number of the original variables. An orthogonal transformation to the basis of the eigenvectors of the sample covariance matrix is performed and the data are projected onto the subspace spanned by the eigenvectors corresponding to the largest eigenvalues (most energetic modes). So, POD gives an orthogonal basis that ranks modes according to an energy criterion (Kerschen et al. 2005). This enables us to retain only the dominant modes and to filter out the presence of the measurement noise, thus providing a good characterization of the dynamics of the problem. Finally, POD is able to explicitly separate the spatial and time information (Risvik 2007). On the other hand, the linear nature of the method can be a restriction for some data sets. Moreover, since POD removes linear correlations among variables, it is only sensitive to secondorder statics. This means that this method is able to find only uncorrelated variables (Risvik 2007).
In the present work, the POD is applied to the analysis of the luminosity field of images (scalar field) in a reactive flow. This allows for an analysis of the considered scalar field by decomposing it into mean, coherent and incoherent parts via statistical methods and to visualize the relevant morphologies. In general, the coherent part includes all fluctuations possessing a somehow structured feature over the burning process. The incoherent part includes all fluctuations for which no pattern can be identified over the burning process. It is commonly thought that the first few modes correspond to the average structure of the data, while higher order modes contain information about fluctuations (Geladi and Kowalski 1986). The nonlinear iterative partial least squares (NIPALS) algorithm is used for the principal component analysis in the POD method. The power spectral density (PSD) of the temporal and spatial coefficients is performed at the end of the algorithm to obtain the excited frequencies and wavelengths during the combustion.
4.2 Independent component analysis
The independent component analysis (ICA) is a statistical signal processing technique whose main applications are blind source separation, blind deconvolution and feature extraction (Hyvarinen 1997). One application with combustion was demonstrated by Bizon et al. (2013b, 2013a). They applied ICA to 2D images of combustionrelated luminosity, to identify leading independent structures.
The ICA is a statistical and computational technique for revealing hidden components that underlie the observed data. The latent variables are assumed to be nonGaussian and mutually independent in space and/or time: they are called the independent components of the observed data. The transformed variables correspond to the underlying components that describe the essential structure of the data and that correspond to some physical causes involved in the process.
Some weak points have to be highlighted also for ICA: unlike for the principal components of the data, which are ordered according to their variance, no intrinsic order exists for the independent components. Moreover, ICA provides a solution only up to a multiplicative constant. In other words, the order, the signs and the scaling of the independent components cannot be determined: indeterminacy is an inherent property of this analysis (Hyvarinen and Oja 2000).
The aim of the present work is to identify independent spatial structures evolving in time. Therefore, the spatial ICA is applied to the analysis of the luminosity field of the combustion process in a hybrid engine. This allows the identification of the leading independent structures during the burning process. The FastICA algorithm (Hyvarinen and Oja 2000) is employed. The power spectral density (PSD) of the temporal and spatial coefficients is performed at the end of the algorithm to obtain the excited frequencies and wavelengths during the combustion.
5 Kelvin–Helmholtz instability
6 Results and discussion
In this paper, the influence of the oxidizer mass flow and fuel viscosity on the entrainment process is discussed. These are the two parameters that are expected to have the biggest influence on the droplets entrainment process, as shown in Fig. 1 (the oxidizer mass flow is not explicitly included in the arguments of the equation in Fig. 1, but it is influencing the entrainment mass flow through the dynamic pressure, with flow density and speed). Therefore, two main test campaigns were performed: pure paraffin 6805 and the same paraffin with 5% polymer were analysed with an oxidizer mass flow varying in a range from 10 to 120 g/s. In a previous test campaign (Petrarolo et al. 2017), the influence of the forward facing ramp of the fuel slab was investigated with a fixed oxidizer mass flow and the main frequencies and wavelengths characterizing the entrainment process were identified. Therefore, in these test campaigns, it was possible to study the influence of the fuel viscosity and oxidizer mass flow on the liquid layer break up process, independently from the fuel configuration. Fuel slabs with 20° forward facing ramp were used, due to the better flame quality (i.e., better flame holding and continuous flame front).
High speed videos of the combustion tests were recorded and analysed, as explained in Sect. 3. By analysing the PSD results of both POD and ICA, it is possible to obtain the most excited frequencies and wavelengths for each of them. POD enables the recognition of the main energetic structures of the flow field, ICA identifies the leading independent structures underlying the data. It is important to underline that, to better characterize the dynamic of the process, a combination of the two decomposition techniques was necessary. In fact, both methods yield a whole range of different frequencies and wavelengths, which are amplified during the combustion. Some of them are related to the main dynamics of the combustion process, others are just random appearing vortices or not so energetic periodic signal (such as noise). To understand which frequencies and wavelengths are actually related to the main combustion events, it was necessary to compare the results of the two methods. If a frequency peak appears only in the POD, this is most likely related to a random energetic vortex (POD recognizes the most energetic structures in the flow field). On the other hand, if a frequency peak appears only in the ICA, this is most likely a periodic but not energetic signal, so not important (ICA recognizes periodic independent structures). Those peaks which appear in both methods are periodic and energetic signals, so related to the main events during the combustion process. So, at the end, only those frequency and wavelength peaks appearing in both methods were considered. The analysis was carried out on 1 s (10,000 frames) during the steadystate. Frequency and wavelength peaks were taken for each fuel formulation and oxidizer mass flow and then compared.
6.1 Fuel viscosity
From what concerns the fuel viscosity, it is possible to notice that the most excited frequency values at the same oxidizer mass flow become lower as the viscosity increases. On the other hand, the longitudinal wavelengths become longer in the formulation with the polymer addition. So, the amplified frequency and wavelength values connected to the KHI are influenced by the liquid viscosity, as expected from the entrainment (Karabeyoglu et al. 2002; Karabeyoglu and Cantwell 2002) and Kelvin–Helmholtz (Funada and Joseph 2001) theories. In particular, by increasing the viscosity the frequency values decrease and the longitudinal wavelengths increase. This means that the higher the viscosity, the more stable the liquid layer is. This leads to a lower number of released and, consequently, entrained droplets and so to a lower regression rate.
6.2 Oxidizer mass flow
As already seen in Sect. 5, the critical speed depends on the liquid layer viscosity: the higher the viscosity, the higher its damping effect on the waves and, consequently, the higher the critical speed is. This is the reason why, for the pure paraffin samples, it is still possible to detect waves at \(\dot{m}_{\text {Ox}} \simeq 10\) g/s, while with 6805+5% polymer (which has a higher viscosity) the last detectable waves are at \(\dot{m}_{\text {Ox}} \simeq 25\) g/s. This means that the critical speed for the pure paraffin is lower than that of the same paraffin with 5% polymer (see also Fig. 14) and, consequently, the range of unstable waves is also wider for 6805 with respect to the blend. So, at flow speeds below the critical one, the disturbances of the liquid layer are damped by the liquid viscosity. On the other hand, at flow speeds above the critical one the same disturbances are able to produce waves which grow. In this way, a range of wavelength is excited and the wavelength having the highest growth rate ultimately dominates. Moreover, at oxidizer mass flows just above the critical speed, longer waves (around 1–2 cm) are mostly appearing. These waves tend to be stable since their phase speed is higher and, thus, the relative flow speed for a given mass flow is lower. On the other hand, viscous damping in the liquid dominates the very shortest waves, making them stable. Therefore, only waves with an intermediate range of wavelengths may be unstable (see also Fig. 13).
Frame images taken from the recorded combustion videos are shown in Fig. 19. They show the combustion flame of the paraffin blend at very low, intermediate and very high oxidizer mass flows. The frames are taken at the same point in time, around 2 s after the ignition. Thus, they all show a steadystate combustion flame. From these images, it can be noticed the difference in the flame stability for the different mass flows. In the test with \(\dot{m}_{\text {Ox}}=10\) g/s, the flame appears to be more flat, with some points of local extinction especially in the rear of the fuel slab, where the flame appears to be made up of elongated filaments. The dynamics of the processes in the combustion chamber is slower and the turbulence level is lower. In the case with \(\dot{m}_{\text {Ox}}=120\) g/s, many short waves appear on the flame surface and the typical shape of Kelvin–Helmholtz waves can be recognized in some of them. The flame is burning continuously along the fuel surface with no points of local extinction. The process dynamics is faster and the turbulence is higher. Finally, in the combustion tests at \(\dot{m}_{\text {Ox}}=50\) g/s waves can be seen on the flame front, but they are longer with respect to the case with higher mass flow and they seem to be more stable. Still, typical Kelvin–Helmholtz waves can be seen, like in this frame at the rear of the fuel slab. The flame front is continuous, without point of local extinction and wellattached to the fuel surface.
7 Conclusion
The combustion behaviour of different paraffinbased 2D fuel slab samples burning with GOX was investigated with an optical combustion chamber. High speed video imaging enables the collection of a huge amount of data, which needs to be analysed in detail. Therefore, two automated data evaluation techniques, based on POD and ICA, were applied to the analysis of the luminosity field of images in a reactive flow. This allows for an analysis of the considered scalar field by identifying leading components during the burning process. POD and ICA were applied separately to the same luminosity data. The results obtained prove the robustness of the two decomposition methods and the effectiveness of the video analysis process. It was shown that the combustion of paraffinbased hybrid rocket fuels is dominated by wavelike structures and that the most excited frequencies and wavelengths characterizing the liquid melt layer depend on the fuel viscosity and oxidizer mass flow. This is important to better understand the onset and development of the entrainment process, which is connected to the amplification of longitudinal unstable waves caused by the high velocity gas flow over the fuel surface.
Two main test campaigns were performed with two different fuel compositions: pure paraffin 6805 and the same paraffin with 5% polymer. Fuel slabs with 20°forward facing ramp were used, due to the better flame holding and continuous flame front. In both test campaigns, the oxidizer mass flow was varied in a range from 10 to 120 g/s. From the results, it is shown that the fuel viscosity influences the most amplified frequencies and longitudinal wavelengths connected to the liquid layer instability process. In particular, the higher the viscosity, the lower the values of the excited frequencies and the longer the longitudinal waves. This means that the higher the viscosity, the more stable the liquid layer. Moreover, a dependency of the instability process on the oxidizer mass flow was also found. In this case, the values of the excited frequencies become higher as the oxidizer mass flow increases, while the excited longitudinal wavelengths become smaller. Both fuel viscosities show the same trend. For very low mass flows, no distinct wavelength peaks are detected for the paraffin blend fuels. This is due to the fact that, at very low oxidizer mass flows, the flow speed is below the critical one that is needed to achieve instability. The two fuel formulations have different critical speed, due to their different viscosity values: the critical speed for the pure paraffin is lower than that of the same paraffin with 5% polymer and, consequently, the range of unstable waves is also wider for 6805 with respect to the blend. A comparison of the experimental results with the neutral curves coming from the KHI model was performed. Here it is clearly shown that all the experimental values for pure paraffin are in the unstable region. On the other hand, for the paraffin with 5% polymer the experimental values connected to the lower mass flows (\(\dot{m}_{\text {Ox}}\simeq\) 10–20 g/s) are below the neutral curve and, consequently, in the stable zone.
The finding of a dependence between the oxidizer mass flow and the Kelvin–Helmholtz frequencies and wavelengths brought to the demonstration that the fuel regression rate (droplets entrainment rate) is directly connected to the liquid layer instability process. In fact, from the hybrid rocket combustion theory (Marxman et al. 1963; Marxman and Gilbert 1963), it is known that the regression rate is connected to the oxidizer mass flux \(G_{\text {Ox}}\), through the law \(r_\text {f} = a {G_{\text {Ox}}}^n\). The parameters a and n depend on the propellants formulation and were determined experimentally in the framework of this research for the used fuel formulations. The oxidizer mass flux is defined as \(G_{\text {Ox}}=\frac{\dot{m}_{\text {Ox}}}{A_{\text {cross}}}\), where \(A_{\text {cross}}\) is the area crossed by the oxidizer mass flow during the combustion process, which, in the case of a fuel slab, is almost constant. At this point, the connection between the regression rate of the fuel slab and the excited frequencies and wavelengths of the liquid layer is straightforward, since they both depend on the oxidizer mass flow [(see also Petrarolo et al. (2017)].
To the author’s knowledge, this is the first case in literature that proved the connection between the Kelvin–Helmholtz instability mechanism and the entrainment in paraffinbased hybrid rocket engines, by means of optical investigations on combustion tests. Although the stability of liquid layers has been extensively studied in the past (Craik 1966; Chang and Russel 1965; Nayfeh and Saric 1971), even under strong blowing conditions (Gater and L’Ecuyer 1970; Nigmatulin et al. 1996; Ishii and Grolmes 1975), the behaviour of films under real combustion conditions has not been explored yet. The presented work confirms the results of the linear stability theory for a liquid film postulated by Karabeyoglu et al. (2002) and Karabeyoglu and Cantwell (2002), which led to a general empirical expression for the entrainment rate of liquid droplets in terms of the relevant properties of the hybrid motor (i.e., mass flux, liquid layer thickness, surface tension and liquid viscosity, see Fig. 1). In addition to that, in the framework of this research, a direct dependency of the droplets entrainment rate (i.e., regression rate) on the liquid layer instability mechanism has been found, for every analysed liquid fuel viscosity and oxidizer mass flow. This means that it is possible to directly control the fuel regression rate by triggering precise frequencies and wavelengths of the liquid layer. The next step will be the investigation of the chamber pressure influence on the liquid layer stability and, consequently, on the entrainment process.
Footnotes
Notes
Acknowledgements
This work was partially funded by the DLR project ATEK (Antriebstechnologien und Komponenten fuer Traegersysteme: Propulsion Technologies and Components for Launcher Systems). The support of the M11 team and the propellants department is greatly acknowledged.
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