Kuppuraj Rajamanickam
email: kuppuraj5005@gmail.com

Franck Lefebvre

Carole Gobin

Gilles Godard

Corine Lacour

Bertrand Lecordier

Armelle Cessou

David Honoré

David Honore
email: david.honore@coria.fr

Effect of H 2 addition on the local extinction, flame structure, and flow field hydrodynamics in non-premixed bluff body stabilized flames

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Introduction

The progressive rise in demand for clean combustion leads to increased interest in developing power generation devices that can operate with low/zero-carbon fuels. Among the other options, hydrogen (H2) is emerging as a better candidate in the global transition to decarbonization, as it can be readily produced from renewables such as wind and solar energy. Therefore, using hydrogen as blended fuel with natural gas becomes an appealing strategy in the devices like gas turbines, high-temperature furnaces, domestic burners, etc. However, the difference in the thermophysical properties of H2 in comparison to conventional fuels brings additional operational challenges. For instance, although the high flame speed and wide flammability range offer improved flame stabilization, it intensifies the risk of flashback [START_REF] Ranjan | Insights into flashback-to-flameholding transition of hydrogen-rich stratified swirl flames[END_REF]. Besides, the increased extinction strain rate (𝜅𝜅 𝑒𝑒𝑒𝑒𝑒𝑒 ) due to H2 enrichment [START_REF] Shanbhogue | Flame macrostructures, combustion instability and extinction strain scaling in swirl-stabilized premixed CH4/H2 combustion[END_REF] is expected to alter the flame shape significantly. Hence, it is apparent that the widespread implementation of H2 across various practical systems needs a thorough understanding of the different physical mechanisms. This forms the motivation for this fundamental research, where the effect of H2 enrichment to the primary fuel CH4 in a canonical non premixed bluff body burner have been investigated.

Many propulsive devices and stationary combustion facilities employ bluff bodies in the fuel injection system to improve flame stabilization. In addition, these burners are often utilized to understand the fundamental turbulence-chemistry interactions. The flow field in such configuration can be globally categorized into three zones, namely, 1. Recirculation Zone (RZ), 2. Neck Zone, 3. Jet Zone. The recirculation zone is formed immediately downstream of the bluff body exit thanks to the wake effect. The hot products trapped inside the RZ provide a continuous ignition source to the incoming fresh reactants in the shear layer. Thus, the pilot flame induced in the RZ helps flame stabilization. Downstream of the RZ, the annular air flow converges towards the central fuel jet, thereby increasing the radial velocity gradients in the neck zone [START_REF] Chen | Flame lift-off and stabilization mechanisms of nonpremixed jet flames on a bluff-body burner[END_REF]. Previous studies [START_REF] Masri | Turbulent diffusion flames of hydrocarbon fuels stabilized on a bluff body[END_REF] [START_REF] Takahashi | Vortex-flame interactions and extinction in turbulent jet diffusion flames[END_REF] have shown localized flame extinctions in the neck zone due to the strong interaction of flame with high magnitudes of strain rate associated with the central fuel jet. Such localized extinctions are identified as a precursor event in the flame blow-off. Furthermore, the local variation in the gas expansion/temperature ratio (𝑇𝑇 𝑏𝑏 /𝑇𝑇 𝑢𝑢 ) caused by local extinctions is known to influence the hydrodynamics of the flow field. In general, the bluff body flows in the nonreacting case exhibit the asymmetric vortex shedding induced by Bénard -Von Karman instability (BVK). However, the BVK instability is often suppressed in the reacting flows and transitions to axisymmetric Kelvin Helmholtz instability (KH) based on the local density/temperature ratio. For instance, the experimental investigation by [START_REF] Emerson | Density ratio effects on reacting bluff-body flow field characteristics[END_REF] elucidated the noteworthy feature of the density ratio in the absolute (BVK) and convective (KH) instability transitions in bluff body stabilized flames. Interestingly, [START_REF] Nair | Near-blowoff dynamics of a bluff-body stabilized flame[END_REF] observed the emergence of BVK instability even in the reacting flows where the flame exhibits a high probability of local extinctions. Similarly, precessing vortex core (PVC) that is usually suppressed in the reacting swirl flows is found to exist in flames with a high degree of local extinctions [START_REF] An | The role of strain rate, local extinction, and hydrodynamic instability on transition between attached and lifted swirl flames[END_REF]. This delineates the influence of the local extinctions in governing the hydrodynamics of the flow field. Furthermore, the local extinctions in themselves are significantly altered by H2 enrichment as it intensifies the extinction strain rate (𝜅𝜅 𝑒𝑒𝑒𝑒𝑒𝑒 ). For instance, the recent experimental investigation by [START_REF] Datta | Impact of hydrogen addition on the thermoacoustic instability and precessing vortex core dynamics in a CH4/H2/air technically premixed combustor[END_REF] showcased the absence of local extinction and subsequent suppression of PVC in hydrogenated flames.

The objectives of this study are twofold. First, the impact of H2 addition over the local extinction and subsequent reignition of the flame fronts in the neck zone is analyzed. The second objective is to demonstrate how these changes observed in local extinctions with H2 addition determine the evolution of large-scale coherent structures in the flow field. For this purpose, simultaneous high-speed Particle Image Velocimetry (PIV) and OH Planar Laser Induced Fluorescence (PLIF) are implemented in large part of the turbulent flame. The obtained time-resolved data is meticulously analyzed to perceive several quantitative insights concerning H2 addition in a canonical non-premixed bluff body burner. The rest of the paper is organized as follows. Section 2 describes the burner configuration and operating conditions, and the details concerning the time-resolved simultaneous PIV and OH PLIF systems are outlined in section 3. The global effect of the H2 enrichment on the flow field and flame structure/reaction zone is presented using the mean quantities in section 4. Section 5 quantitatively delineates the impact of H2 addition in the evolution of the instantaneous reaction zone heights. Following this, an effect of H2 enrichment on the local extinctions and subsequent reignition/reconnection of the broken flame branches are shown in section 6. Finally, in section 7, Proper Orthogonal Decomposition (POD) and one dimensional (1D) Continuous Wavelet Transform (CWT) are implemented to quantify an interlink between the observed variations in the local extinction, reignition, reaction zone structure/height, and the flow field hydrodynamics concerning the H2 enrichment.

Experimental setup and test conditions

The schematic of the non-premixed canonical bluff body burner employed in this study is depicted in Figure 1. It comprises of annular air passage and a central fuel injection port embedded with the cylindrical bluff body. The working fluids are air, methane, and hydrogen, respectively. All the fluid flow rates are controlled using thermal mass flow meters (Bronkhorst). The fuel flow rates of CH4 and H2 are varied (Table 1) in reference to the critical experimental parameter, namely, the volumetric concentration level of H2 (𝜶𝜶 𝑯𝑯𝑯𝑯 ). Throughout all the test conditions, the thermal output of the burner (Table 1) and annular airflow rate have been maintained constant; the latter allowed us to keep the same global features of the nonreactive flow close to the burner exit.

Table 1. Experimental conditions

𝑈𝑈 𝑎𝑎𝑎𝑎𝑎𝑎 , 𝑈𝑈 𝑓𝑓 -bulk airflow and fuel jet velocity (m/s) at the burner exit

Simultaneous time-resolved PIV and OH PLIF diagnostics a. Time-resolved particle imaging velocimetry

The configuration of the time-resolved PIV system is shown in Figure 2. A double cavity, solid state Nd: YAG laser (Photonics, λ~532 nm; 30 mJ/pulse) is used as an illumination source. The laser beam is formed into a thin light sheet using spherical and cylindrical lenses (L1, L2) and passed along the burner's centreline. Oxides of zirconium particles are used as a tracing medium, and it should be noted that the annulus air and fuel have been seeded separately. The Mie scattered light from the particles is captured using a Phantom V2512 camera fitted with a Nikon lens (f/2.8, 105 mm). Besides, a 532 nm bandpass filter is coupled with the lens to reject flame emissions. The field of view for PIV measurements is selected as 90 x 110 mm 2 . The PIV image resolution is 789 x 721 pixels, while the pixel resolution is 8.05 pixels/mm. 10,000 PIV image pairs are acquired for all the test conditions, corresponding to an acquisition duration of 2 seconds. The recorded images are post-processed in Lavision's Davis 8.4 software to reconstruct the vector field. Across all the test cases, multipass decreasing interrogation window size with an adaptive PIV scheme (also with image deformation) is employed. Accordingly, the initial and final interrogation window sizes have been chosen as 24 x 24 pixels and 16 x 16 pixels (with 50 % overlap), respectively. This results in the final vector spacing of 0.99 mm. Figure 2 illustrates the optics arrangement associated with the OH PLIF measurement system. It comprises a single cavity, solid state Nd: YAG pump laser (EdgeWave; λ~532 nm), and a tunable dye laser (Sirah Credo). The dye laser is tuned to produce a laser beam of wavelength ~ 566 nm with the help of Rhodamine 6G dye dissolved with ethanol, and then, it is frequency-doubled to get a desired UV laser beam of wavelength ~ 283 nm: 0.3 mJ/pulse. The generated 283 nm beam is used to excite the OH radicals in the Q1 (6) line of 𝐴𝐴 2 ∑ + -𝑋𝑋 2 ∏(1,0) transition band. The UV laser beam is converted to a collimated sheet approximately ~ 70 mm in height using three fused silica lenses (L3, L4, L5). Then, it is precisely overlapped with the PIV laser sheet to ensure the acquisition of OH fluorescence and flow fields from the same longitudinal plane. Since the pulse energy of the UV beam is low at a high repetitive rate, the illumination area for PLIF imaging in the vertical plane is limited to ~ 70 mm x 70 mm. Furthermore, the PLIF laser pulse is temporally positioned between two PIV laser pulses to enable simultaneous measurement of OH and flow field.

The OH fluorescence intensity from the flame is captured using a Photron SA5 camera coupled with a high-speed image intensifier (LaVision GmbH; HS-IRO). In addition to the UV lens (f / 2.8, 100 mm; Cerco), the spectral filters UG11 and WG 305 are used to reject the Mie scattering from PIV tracers. The combination of spectral filtering and short gating (150 ns) achieved through an image intensifier facilitates the collection of OH fluorescence without the interference of flame emission. For all the test conditions, the PIV and OH PLIF images are recorded simultaneously at 5 kHz with an acquisition duration of 2 seconds. The PLIF camera image resolution is 775 x 706 pixels (spatial resolution -10 pixels/mm) for the above-selected acquisition rate.

Global features

For better understanding, the key terminologies that describe the topology of the mean flow field are explained before presenting the comparative results on the effect of H2 enrichment. The time-averaged flow field pertains to 𝛼𝛼 𝐻𝐻2 = 0.1 is presented in Figures 3a andb as a sample to illustrate the key aerodynamic features.

Direct photographs of flames

Three distinct flow zones are identified based on the height above the burner (Figure 3b). The first zone (Zone A) comprises a strong air Recirculation Zone (RZ) induced immediately downstream of the bluff body exit. Here, the length of RZ (𝑳𝑳 𝑹𝑹𝑹𝑹 ) is dictated by the average location of the stagnation point at the end of RZ. Furthermore, three shear layers are witnessed in Zone A (see Figure 3b), 1. Outer shear layer (OSL) formed between the outer boundary of annular air jet and ambient air; 2. Inner shear layer 1 (ISL1) separates the inner and outer boundaries of the annular air jet and RZ, respectively. 3. Inner shear layer 2 (ISL2) is induced between the inner boundary of RZ and the central fuel jet. Next to RZ, the annular airflow converges towards the central fuel jet in the small region of 40 𝑚𝑚𝑚𝑚 ≤ 𝑦𝑦 ≤ 60 𝑚𝑚𝑚𝑚. This is referred as neck zone (Zone B). Finally, zone C forms a jet-like flow structure (zone C) downstream of Zone B. Thereafter, the phrases Zone A, B, and C represents RZ, Neck zone, and Jet flow regions, respectively, will be used interchangeably in the text. The obtained flow structure is classically referred to as "central jet-dominated flow" in the non-premixed bluff body flame literature [START_REF] Roquemore | A study of a bluff-body combustor using laser sheet lighting[END_REF]; [START_REF] Martins | Simulation of the nonreacting flow in a bluff-body burner; effect of the diameter ratio[END_REF]. This corresponds to typical aerodynamic conditions ensuring turbulent flame stabilization thanks to the RZ obtained in our study for all hydrogen proportions from pure methane to pure hydrogen (Figures 3c andd).

The following is focused on the effect of H2 enrichment on the evolution of the mean flow field and flame structure. Due to the axisymmetric nature of mean quantities, only the positive side of the radial coordinate is shown in Figure 3c andFigure 4. Interestingly, despite a significant rise in the central fuel jet velocity (𝑈𝑈 𝑓𝑓 ) from 12 m/s to 35 m/s between 𝛼𝛼 𝐻𝐻2 = 0 to 𝛼𝛼 𝐻𝐻2 = 1, the global topology of mean flow fields is (relatively) similar across all the cases (Figure 3c). This is because the low-density property of hydrogen and constant annular airflow rate keeps the momentum ratio (MR) approximately the same (Table 1); thus, the flow topology remains relatively unaltered even with a high-velocity jump associated with a central fuel jet. However, a slight increase in the length of the recirculation zone (𝑳𝑳 𝑹𝑹𝑹𝑹 ) is witnessed with H2 enrichment and the underpinning mechanism will be explained in the later part of this section.

Despite similar global flow topologies (Figure 3c) and the flame appearance (as observed in the direct photographs of flames in Figure 3d) across all the values of 𝛼𝛼 𝐻𝐻2 , there exists strong variations in the mean reaction zone height (𝒉𝒉 � 𝑹𝑹 ) obtained from OH PLIF (Figure 4). For instance, the mean reaction zones of 𝛼𝛼 𝐻𝐻2 = 0, 0.1 are found to be shorter and located within RZ. On the other hand, larger reaction zones of 𝒉𝒉 � 𝑹𝑹 ~70 𝑚𝑚𝑚𝑚 are observed for the cases 𝛼𝛼 𝐻𝐻2 = 0.8, 1. A gradual increase in the axial extent of the reaction zone (𝒉𝒉 � 𝑹𝑹 ) is noticed with hydrogen enrichment. The mechanisms behind spatial variations in mean reaction zone height (𝒉𝒉 � 𝑹𝑹 ) will be explained more quantitatively in the upcoming sections. Interestingly, for all the cases, within RZ (i.e., Zone A), the flame is located in the shear layer formed between the outer and inner boundaries of RZ and the annular air jet, respectively (i.e., ISL1). This can be explained as follows, 1. First, part of the fuel from the central jet mixes with the air inside the RZ; 2. The formed mixture is then convected radially outwards by strong air-driven RZ; 3. Finally, the reaction is established in the ISL1, probably due to the local establishment of stoichiometry (or 𝜙𝜙 = 1) in that region. This is consistent with the observation made by [START_REF] Dally | Instantaneous and mean compositional structure of bluff-body stabilized nonpremixed flames[END_REF] in a similar bluff body burner configuration. Furthermore, in line with the above-made discussions concerning different flow regimes, the flame switches its path from ISL1 to ISL2 and then resembles a jet-like flame configuration in zones B and C.

Figure 4. Distribution of mean OH PLIF reaction zone (superimposed with time-averaged vector fields) across the different test cases.

Besides the variation in reaction zone height (𝒉𝒉 � 𝑹𝑹 ), the gradual increase in OH fluorescence intensity is witnessed with H2 enrichment, particularly in the region close to the burner exit (shown with purple dotted circles (Figure 4). The increase in OH concentration is known to be a key parameter in promoting reaction rate and, thus, improved flame stabilization [START_REF] Najm | Premixed flame response to unsteady strain rate and curvature[END_REF]. The detailed experimental and LES results carried out by [START_REF] Hussain | Investigating the effect of local addition of hydrogen to acoustically excited ethylene and methane flames[END_REF]) [START_REF] Guo | Effect of hydrogen enrichment on swirl/bluff-body lean premixed flame stabilization[END_REF] in hydrogenated flames revealed the increase of H radicals in the flame front and subsequent formation of intense OH radicals through chain branching reaction (𝐻𝐻 + 𝑂𝑂 2 → 𝑂𝑂 + 𝑂𝑂𝐻𝐻). Now, the possible reason behind the increase in 𝑳𝑳 𝑹𝑹𝑹𝑹 with H2 enrichment can be interpreted from the rise in OH intensity. The work by [START_REF] Namazian | Nonpremixed bluff-body burner flow and flame imaging study[END_REF] [START_REF] Lovett | A review of mechanisms controlling bluff-body stabilized flames with closely-coupled fuel injection[END_REF] [START_REF] Massey | A scaling law for the recirculation zone length behind a bluff body in reacting flows[END_REF] highlights the increase in the size of RZ as a consequence of heat added to the flow (even at constant fuel to an annular jet momentum ratio). The intense heat release (as inferred from OH intensity; Figure 4), and an increase in the height of the reaction zone (𝒉𝒉 � 𝑹𝑹 ), may induced thermal expansion. Thus, bigger RZ is observed for hydrogenated flames.

Quantitative understanding of the underpinning mechanisms behind the variations in the reaction zone height (𝒉𝒉 � 𝑹𝑹 ) across various 𝛼𝛼 𝐻𝐻2 values and how it influences the flow-flame interaction dynamics are the main targets of this paper.

Quantitative description of reaction zone height intermittency

Beyond the mean quantities, the detailed analysis of instantaneous flow-flame interactions is indispensable in understanding the evolution of flame structures in turbulent flows. The previous section highlighted the variation in the mean reaction zone height (𝒉𝒉 � 𝑹𝑹 ) with H2 addition, the temporal evolution of the reaction zone heights across different experimental test cases is explored in this section. An image-processing scheme is implemented in MATLAB to extract the instantaneous reaction zone height following the process implemented by [START_REF] Juddoo | High-speed OH-PLIF imaging of extinction and re-ignition in nonpremixed flames with various levels of oxygenation[END_REF]. For better clarity, the working details of the image processing scheme are illustrated through two different time sequences of OH contours, one without local extinction (Figure 5a) and another with local extinction (Figure 5b). First, the raw OH PLIF image is binarized (step 1) based on the user threshold. The resultant edges from the binary images are further smoothened using a Gaussian filter to avoid inaccuracies in mapping the boundaries. The image segmentation (step 2) algorithm helps to detect the objects in the resultant binarized images. For instance, the OH contour is unbroken in the image without local extinction (Figure 5a); hence only one object is detected. On the other hand, the broken flame sheet that occurred due to local extinction resulted in the detection of two objects. Finally (step 3), the physical dimensions of the detected objects are extracted from the bounding box size. The height of the first bounding box from the burner exit yields the instantaneous reaction zone height (ℎ 𝑅𝑅,𝐿𝐿 ′ ),

where R and L denote the right and left sides of the radial coordinates. Essentially, time sequences in which the obtained (ℎ 𝑅𝑅 ′ ), (ℎ 𝐿𝐿 ′ ) values equal to ~ 70 mm (i.e., the maximum height of the OH PLIF field of view) denote an absence of local extinction at that instant.

The time traces of ℎ ′ 𝑅𝑅 acquired over the short time (0.2s) window is depicted in Figure 6a. Qualitatively, it is perceived that the time delay between successive peaks is very long for 0 ≤ 𝛼𝛼 𝐻𝐻2 ≤ 0.1. However, it is significantly reduced for 0.2 ≤ 𝛼𝛼 𝐻𝐻2 ≤ 0.3. It is further interesting to note that local extinction events are no longer witnessed beyond 𝛼𝛼 𝐻𝐻2 > 0.3, hence, it yields constant values of ℎ′ 𝑅𝑅 (i.e. ℎ′ 𝑅𝑅 = 70 𝑚𝑚𝑚𝑚) for 𝛼𝛼 𝐻𝐻2 = 0.4 in Figure 6a. It should be noted that similar results are obtained for 𝛼𝛼 𝐻𝐻2 = 0.5, 0.8, 1 as local extinctions are not observed in these cases as well, however for brevity; those are not shown in the plot.

Figures 6b andc compare the histograms of ℎ′ 𝑅𝑅 , ℎ′ 𝐿𝐿 acquired over the entire experimental time sequence (i.e., 2 s). Pure methane flame (𝛼𝛼 𝐻𝐻2 = 0) exhibits higher probability (𝜌𝜌 𝑛𝑛 ) around ~ 35 -40 mm across ℎ′ 𝑅𝑅 and ℎ′ 𝐿𝐿 . On contrary, the higher probability is shifted to ~ 60 -70 mm for 𝛼𝛼 𝐻𝐻2 = 0.2, 0.3. Now, it is worthwhile to quantify the time lag between successive peaks observed in ℎ′ 𝑅𝑅 , ℎ′ 𝐿𝐿 values. To do this, autocorrelation function has been implemented over the ℎ′ 𝑅𝑅 , ℎ′ 𝐿𝐿 values. The autocorrelation (𝑅𝑅 𝑚𝑚𝑚𝑚 ) and corresponding time lag (𝜏𝜏) are calculated using equations 1 and 2. The latter is presented in Figure 7.

𝑅𝑅 𝑚𝑚𝑚𝑚 = � ��(ℎ ′ ) 𝑎𝑎 -ℎ � � -�(ℎ ′ ) 𝑎𝑎+∆𝑒𝑒 -ℎ � �� �(ℎ ′ ) 𝑎𝑎 -ℎ � � 2 𝑁𝑁-1 𝑎𝑎=0
(1)

𝜏𝜏 = 𝑙𝑙𝑙𝑙𝑚𝑚 𝑇𝑇→∞ 1 𝑇𝑇 � 𝑅𝑅 𝑚𝑚𝑚𝑚 𝑑𝑑𝑑𝑑 𝑇𝑇 0 (2)
The decay time (𝜏𝜏) is found to be reduced with increasing 𝛼𝛼 𝐻𝐻2 . This implies that H2 enrichment favors the reignition; thus, the broken flame sheet regains its height to the peak value as before extinction (i.e., 70mm) within a shorter duration. On the other hand, the lesser probability of reignition events might be a reason behind the longer time duration between two successive peaks in 𝛼𝛼 𝐻𝐻2 = 0,0.1 flames. This first-order estimate motivates further investigation of the mechanisms underlying local flame extinction and subsequent re-ignition. It is well known that the large-scale coherent structures present in the shear layers are the major driving mechanisms behind the local extinction in coaxial non-premixed flames [START_REF] Takahashi | Vortex-flame interactions and extinction in turbulent jet diffusion flames[END_REF] [START_REF] Hult | Experimental analysis of local flame extinction in a turbulent jet diffusion flame by high repetition 2-D laser techniques and multiscalar measurements[END_REF]. The conditional averaging approach has been employed to identify the correlation between local flow structures and local flame extinction events. First, as the local extinction events are majorly detected in the neck zone, thus, two axial windows are chosen: h1 = 30 -40 mm; h2 = 40-50 mm. The obtained instantaneous values of ℎ′ 𝑅𝑅 , ℎ′ 𝐿𝐿 allowed to distinguish the local extinctions that occurred in the above-chosen windows and identify the time instant (𝑑𝑑 𝑒𝑒𝑒𝑒𝑒𝑒 ) of local extinction. Next, to understand the evolution of the flow structures at the instant of local extinction, the flow fields of two-time steps before (𝑑𝑑 𝑒𝑒𝑒𝑒𝑒𝑒-2 , 𝑑𝑑 𝑒𝑒𝑒𝑒𝑒𝑒-1 ) and after (𝑑𝑑 𝑒𝑒𝑒𝑒𝑒𝑒+1 , 𝑑𝑑 𝑒𝑒𝑒𝑒𝑒𝑒+2 ) local extinctions are extracted. Finally, only these extracted flow fields are conditionally averaged to obtain the flow structures that evolved during the local extinction in the h1 and h2 windows. The total number of localized extinction events captured across h1 and h2 windows for the considered test cases are listed in Table 2 -c compare the conditionally averaged flow fields acquired at h1 and h2 windows and the time-averaged flow field of 𝛼𝛼 𝐻𝐻2 = 0.1 case. As marked with the dotted rectangle, a strong vortical structure is identified at the OSL of the conditionally averaged flow field, whereas the same is not witnessed with the time-averaged flow field (Figure 8c).

Moreover, the axial displacement in the vortex core center (VCC) of the obtained coherent structure across the two windows, i.e., h1 and h2, further confirms the dominance of coherent vortical structures induced at the OSL during the local extinction process. This phenomenon has been observed across all the cases considered here, i.e., 𝛼𝛼 𝐻𝐻2 = 0, 0.1, 0.2, and 0.3, as local extinctions are witnessed in these cases. For brevity, the identified VCC of the coherent vortical structures for the other cases is shown in the plot (Figure 8d) instead of velocity contours. The role of coherent vortical structure induced at OSL during local extinction is further explained with the aid of overlaid images of instantaneous OH contours and velocity fields. The overlaid black contours in Figure 9 represent the flame topology, and the color contours denote the 2D out-of-plane vorticity (𝜔𝜔 𝑧𝑧 ), 2D hydrodynamic strain rate (𝜅𝜅 ℎ𝑦𝑦𝑦𝑦 -norm of the strain rate obtained from the PIV velocity field) acquired at the same time. The flame boundaries shown in Figure 9 are extracted from the smoothened image (step 1 of Figure 5) using a Canny edge detection algorithm [START_REF] Canny | A computational approach to edge detection[END_REF] [START_REF] Zheng | Edge detection methods in digital image processing[END_REF]. Unlike swirl flows, 3D effects are minimal in the bluff body flows; nevertheless, the computed 2D strain from the present experiments might underestimate the entire 3D strain rate. Since the velocity of the central fuel jet is much higher than the annular air jet, the maxima of strain and vorticity magnitudes are located at the central fuel jet region (ISL 2).

It should be noted that, as shown in Figure 8, the mechanism of local extinctions is the same in all the four cases of 𝛼𝛼 𝐻𝐻2 = 0, 0.1 ,0.2 ,0.3; thus, only sequences pertain to the 𝛼𝛼 𝐻𝐻2 = 0.3 is shown in Figure 9a andb as an example. The observed cascade of events during local extinction is as follows.

Initially, at t = 0 ms (Figure 9a-i), ISL 1 (and the flame) is located away from the high strain zone of ISL2. Next, the emergence of a coherent vortex at OSL (t =1 ms; Figure 9a-i) causes inward radial displacement of ISL 1 and flame front towards ISL 2. Subsequently, the flame becomes thinner as it undergoes excessive stretch due to the larger radial velocity gradients associated with the annular airflow in that region (convective influx). This is consistent with the observations of [START_REF] Yamashita | A numerical study on flame stability at the transition point of jet diffusion flames[END_REF] [START_REF] Takahashi | Near-field turbulent structures and the local extinction of jet diffusion flames[END_REF] in the concentric non-premixed burner. Finally, the flame is locally extinguished at t = 1.8 ms due to an interaction with the high strain of 5000 s -1 in the ISL2 (Figure 9a-ii). PIV particle images overlaid with OH flame fronts (Figure 9a -iii) illustrate the above-explained sequential events.

Interestingly, during a few time instants, the sudden ejection of vortices from the central fuel jet causing the local extinctions are also observed. For example, another set of sequential events illustrated in Figure 9b clearly demarcates such a scenario. As marked with the dashed rectangle (Figure 9b-i), the flame undergoes local extinction by the vortex emerging from the central fuel jet. It should be noted that, over here, local extinction occurred in the region away from the higher strain zone (Figure 9b-ii; t=1.2 ms). [START_REF] Takahashi | Vortex-flame interactions and extinction in turbulent jet diffusion flames[END_REF] have shown that the ejection of such fuel jet vortices causes a sudden increase in the diffusive influx of the fuel over the reaction zone and results in the quenching of flame fronts in a localized region. It should be noted that, since the analysis is carried out in the neck zone, where the scenario presented in Figure 9a is majorly observed. This is consistent with the obtained vortical structures at the OSL in the conditionally averaged flow fields (Figures 8a-d).

On the contrary, as illustrated in Figure 9c (also see the comparison between 𝛼𝛼 𝐻𝐻2 = 0.1 and 𝛼𝛼 𝐻𝐻2 = 0.5 in Multimedia view), local extinctions are no longer witnessed for 𝛼𝛼 𝐻𝐻2 ≥ 0.4 despite the inward radial movement of the flame front towards the ISL 2 region followed by strong interaction with the high strain rate zone (i.e., ISL2). The increased H2 content intensifies the extinction strain resistance of flame (𝜅𝜅 𝑒𝑒𝑒𝑒𝑒𝑒 ) [START_REF] Karlis | H2 enrichment of CH4 blends in lean premixed gas turbine combustion: An experimental study on effects on flame shape and thermoacoustic oscillation dynamics[END_REF] [START_REF] Shanbhogue | Flame macrostructures, combustion instability and extinction strain scaling in swirl-stabilized premixed CH4/H2 combustion[END_REF] over the fluid dynamic strain (𝜅𝜅 ℎ𝑦𝑦𝑦𝑦 ) and thereby prevents the occurrence of local extinction events. Moreover, the increase in flammability limits with H2 enrichment further prevents the local extinction even if the flame front encounters the sudden increase in diffusive flux of the fuel due to the impingement of the central fuel jet vortex as shown in Figure 9b.

Flame contours are superimposed over vorticity (i).2D strain rate (ii), and PIV raw particle image (iii);

b. Re ignition This section's primary idea is to assess how H2 addition influences the reconnection of holes/broken branches formed in the otherwise continuous reaction zone. In the present analysis, we have adopted a similar approach employed by [START_REF] Juddoo | High-speed OH-PLIF imaging of extinction and re-ignition in nonpremixed flames with various levels of oxygenation[END_REF]. Following this, the successive images containing only the two OH branches formed after the local extinction are considered. This is because, during some instants, reignition is initiated by additional OH islands appearing between upper and lower branches. These additional OH islands might be originated from out-of-plane convection. Based on the criteria stated above, nearly 30% of datasets are eliminated. As postulated by [START_REF] Steinberg | Experimental study of flame-hole reignition mechanisms in a turbulent non-premixed jet flame using sustained multi-kHz PIV and crossed-plane OH PLIF[END_REF] [START_REF] Juddoo | High-speed OH-PLIF imaging of extinction and re-ignition in nonpremixed flames with various levels of oxygenation[END_REF], the influence of such eliminations in understanding the re-ignition mechanisms is minimal. Nevertheless, the additional measurements in cross-plane will be more helpful in a complete understanding of all the events [START_REF] Cho | High-repetition-rate threedimensional OH imaging using scanned planar laser-induced fluorescence system for multiphase combustion[END_REF].

Two major events, closure and breakup, are observed upon local extinction, as depicted in Figure 10. Here, closure refers to the reconnection of the two branches in the subsequent time sequences following the local extinction (Figure 10a), and vice versa refer to the breakup situation (Figure 10b). The bounding boxes (described in Figure 5) are used to mark the boundaries of the two OH branches. The parameter '𝑦𝑦 ℎ ' that defines the distance between upper and bottom bounding boxes (aka -hole) is used to classify the closure and breakup events. For each event, five temporally successive images following the local extinction (i.e., 1 ms from 𝑑𝑑 𝑒𝑒𝑒𝑒𝑒𝑒 ) are considered for the determination of the time evolution of 𝑦𝑦 ℎ . The histograms generated (Figures 10c andd) from the obtained values of 𝑦𝑦 ℎ highlights the higher number of closure events for 𝛼𝛼 𝐻𝐻2 = 0.2, 0.3 and lesser number for 𝛼𝛼 𝐻𝐻2 = 0, 0.1 respectively. This observation clearly delineates the favorable feature of H2 addition over the reconnection of two OH branches within a shorter duration (i.e., < 1 ms). A closer inspection of the instantaneous data set reveals that, during closure events, either upward or downward movement of the upper/lower OH branch initiates the reignition. Two reignition mechanisms, edge flame propagation and turbulence mixing by adjacent vortices, are widely reported in the non-premixed turbulent jet flames. The time sequences in Figure 11a show the reconnection mainly occurred due to edge flame propagation, as there is no evidence of the deformation of broken branches by surrounding vortices. Moreover, a rapid displacement of the edges by ~ 2.5 mm in the short interval of 0.2 ms, which yields a closure rate of ~ 12.5 m/s, further confirms the dominance of edge flame propagation. Another set of sequential images in Figure 11b illustrates the reconnection by the turbulent mixing. Following the local extinction at t = 0 ms, the lower flame branch propagates around the strong vortical structure in the OSL (annotated with black circled arrows) at t = 0.2 ms (Figure 11b). Interestingly, the final reconnection shown in t = 0.6 ms (Figure 11b) is still initiated by the edge flame propagation similar to the one observed in Figure 11a. It should be noted that the contribution of edge flame propagation and turbulent convection in the re-connection of flame branches can be more quantitively described based on the criteria proposed by [START_REF] Steinberg | Experimental study of flame-hole reignition mechanisms in a turbulent non-premixed jet flame using sustained multi-kHz PIV and crossed-plane OH PLIF[END_REF]. However, such quantitative analysis is not within the scope of this current manuscript. Here, our primary idea is to identify how H2 enrichment influences the re-connection of the broken branches following the local extinction and thereby relate this to variation in the flame structure described in section 4 (Figure 4).

The measured mean rate of reconnection/edge flame speed (𝑈𝑈 � 𝑒𝑒𝑦𝑦 = ∆𝑦𝑦 ℎ /𝑑𝑑𝑑𝑑), is approximately 3 m/s, 3.5 m/s, 7 m/s, 12 m/s for 𝛼𝛼 𝐻𝐻2 = 0, 0.1, 0.2, 0.3, respectively. It should be noted that these values are much higher than the corresponding laminar flame speed of the above mixtures. For instance, the laminar flame speed (𝑆𝑆 𝐿𝐿 ) of the pure methane (𝛼𝛼 𝐻𝐻2 = 0) at T = 300 K is ~ 0.45 m/s. Following the local extinction, the edge flame propagates in the region surrounded by hot products close to the adiabatic flame temperature [START_REF] Hult | Experimental analysis of local flame extinction in a turbulent jet diffusion flame by high repetition 2-D laser techniques and multiscalar measurements[END_REF]. Henceforth edge flame speed is much higher than the laminar flame speed where the propagation occurs in the low-temperature ambient(T=300K). Furthermore, [START_REF] Briones | Effects of H2 enrichment on the propagation characteristics of CH4-air triple flames[END_REF] [START_REF] Frouzakis | From diffusion to premixed flames in an H2/air opposed-jet burner: the role of edge flames[END_REF] have shown that H2 enrichment to the methane leads to a progressive rise in the edge flame speed due to enhanced reactivity and preferential diffusion. Thus, as shown above, an increase in the edge flame speed (𝑈𝑈 � 𝑒𝑒𝑦𝑦 ) with H2 enrichment in particular with 𝛼𝛼 𝐻𝐻2 = 0.2, 0.3 leads to quicker reconnection of the broken branches that formed due to the local extinction. On the other hand, the lesser edge flame speed associated with 𝛼𝛼 𝐻𝐻2 = 0, 0.1 results in failure of reattachment of the broken OH branches in a shorter duration; this essentially causes a higher number of breakup events. An example of such breakup event can be seen in Figure 11c. As illustrated, during such breakup events, the lower part remains in the RZ, while the upper part is progressively convected downstream by the coherent shedding vortices in the OSL. Previous studies reported this periodic convection of the upper flame branch as split flashing [START_REF] Chen | Flame lift-off and stabilization mechanisms of nonpremixed jet flames on a bluff-body burner[END_REF][START_REF] Esquiva-Dano | Influence of a bluff-body's shape on the stabilization regime of non-premixed flames[END_REF] or moving fireballs/tubules [START_REF] Roquemore | Dynamic behavior of a bluff-body diffusion flame[END_REF]. The lower flame branch within the RZ is commonly referred to as a residual flame [START_REF] Zukoski | Flame stabilization on bluff bodies at low and intermediate Reynolds numbers[END_REF]. This residual flame within the RZ acted as a pilot in the reignition of the oncoming fresh reactants, thereby establishing the stable flame.

It is worthwhile to note that intermittency in the flame lift-off and reattachment (see Multimedia view) inside RZ has also been observed for 𝛼𝛼 𝐻𝐻2 = 0, 0.1. As shown in previous experimental investigations [START_REF] Pan | Chemistry and turbulence effects in bluff-body stabilized flames[END_REF] [START_REF] Chen | Flame lift-off and stabilization mechanisms of nonpremixed jet flames on a bluff-body burner[END_REF] in a similar burner configuration, this might be due to the coupled effect of an increase in the entrainment rate of fresh reactants into the RZ due to the delay encountered with the reignition of locally extinguished flame fronts.

Multimedia view

7. Flow-flame interaction dynamics a. Flow dynamics Sections 3 and 5 highlighted the differences observed in the mean (ℎ � 𝑅𝑅 ) and instantaneous reaction zone heights (ℎ 𝑅𝑅,𝐿𝐿 ′ ) as a consequence of local extinction and reignition events. In particular, as shown in Figure 4, hydrogen enrichment significantly alters the flame structure in the region downstream to the recirculation zone (RZ), which in turn changes the spatial distribution of the temperature ratio (𝑇𝑇 𝑏𝑏 /𝑇𝑇 𝑢𝑢 ) between burnt and unburnt gases. Previous studies [START_REF] Emerson | Density ratio effects on reacting bluff-body flow field characteristics[END_REF]) [START_REF] Oberleithner | Why nonuniform density suppresses the precessing vortex core[END_REF]) [START_REF] Yin | Time-frequency localisation of intermittent dynamics in a bistable turbulent swirl flame[END_REF] showcased the influential feature of a (𝑇𝑇 𝑏𝑏 /𝑇𝑇 𝑢𝑢 ) in the evolution of dominant hydrodynamic modes in the flow field. Hence, the interplay between reaction zone height and flow field dynamics is explored in this section. The influence of spatial variation in the (𝑇𝑇 𝑏𝑏 /𝑇𝑇 𝑢𝑢 ) over the dominant instability modes can be better understand from the following vorticity transport equation,

𝐷𝐷𝜔𝜔 � �⃗ 𝐷𝐷𝑑𝑑 = �𝜔𝜔 � �⃗. ∇ � �⃗ �. 𝑉𝑉 �⃗ -𝜔𝜔 � �⃗�∇ � �⃗ . 𝑉𝑉 �⃗ � - 1 𝜌𝜌 2 ∇𝜌𝜌 × ∇𝑃𝑃 + ∇ � ∇. 𝜏𝜏 𝜌𝜌 � (3)
where 𝜔𝜔 � �⃗-vorticity vector, 𝑉𝑉 �⃗ -velocity vector, 𝜌𝜌 -fluid density, 𝑃𝑃 -fluid pressure, and 𝜏𝜏 -viscous stress tensor.

In the reactive flows, the kinematic viscosity of the gas rises sharply due to the temperature difference through the term ∇ � ∇.𝜏𝜏 𝜌𝜌 �. This subsequently dampens the vorticity in combusting flows relative to the non-reactive flows. Second, the misalignment between the pressure and density gradients leads to baroclinic vorticity production (Soteriou and Ghoniem, 1994a) through the term 1 𝜌𝜌 2 ∇𝜌𝜌 × ∇𝑃𝑃. Third, the gas expansion further weakens the vorticity magnitude through the negative term 𝜔𝜔 � �⃗�∇ � �⃗ . 𝑉𝑉 �⃗ �. This is caused by variation in the gas dilation ratio (𝜌𝜌 𝑏𝑏 /𝜌𝜌 𝑢𝑢 ) across the flame.

The three mechanisms mentioned above significantly alter the vorticity magnitude in the reactive flow fields. It should be noted that it is difficult to identify each mechanism's contribution as all these interact complexly (Soteriou and Ghoniem, 1994a) [START_REF] Füri | The effect of flame position on the Kelvin-Helmholtz instability in non-premixed jet flames[END_REF]) [START_REF] Shanbhogue | Lean blowoff of bluff body stabilized flames: Scaling and dynamics[END_REF].

To further understand the influence of (𝑇𝑇 𝑏𝑏 /𝑇𝑇 𝑢𝑢 ) in the dominant instability modes, Proper Orthogonal Decomposition (POD) has been implemented over the instantaneous data sets of the PIV flow field and OH PLIF. For both the PIV and OH PLIF datasets, we have implemented the method of snapshots POD [START_REF] Sirovich | Turbulence and the dynamics of coherent structures. I. Coherent structures[END_REF] based on single value decomposition (SVD) [START_REF] Kerschen | The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview[END_REF]. First, the temporal data sets are arranged into the big matrix 𝑈𝑈 of 𝑚𝑚 × 𝑛𝑛, where the column 𝑛𝑛 represents the instantaneous 2D data sets. POD analysis over the big matrix 𝑈𝑈 essentially yields 𝑛𝑛 eigenmodes 𝜙𝜙 𝑗𝑗 (𝑥𝑥, 𝑦𝑦), temporal coefficients 𝑎𝑎 𝑗𝑗 (𝑑𝑑) and eigenvalues 𝜆𝜆 𝑗𝑗 . Finally, the decomposed velocity fields 𝑢𝑢 can be expressed as

𝒇𝒇 (𝒙𝒙, 𝒚𝒚, 𝒕𝒕 𝒂𝒂 ) = � 𝒂𝒂 𝒋𝒋 (𝒕𝒕 𝒂𝒂 ) 𝑵𝑵 𝒋𝒋=𝟏𝟏 𝝓𝝓 𝒋𝒋 (𝒙𝒙, 𝒚𝒚)
Where, 𝒕𝒕 𝒂𝒂 -time instant of the original data; 𝑗𝑗-mode number; 𝑁𝑁 -total number of instantaneous data sets.

Figure 12 represents the first four dominant spatial modes retrieved from the PIV flow fields. Here, the spatial modes are visualized using vorticity contours. All the four modes of 𝛼𝛼 𝐻𝐻2 = 0, 0.1 shows the dominance of large-scale vortex shedding at OSL immediately downstream of RZ. Besides, the appearance of coherent structures in the region of 30 mm < y < 50mm; -15 mm < x < -15 mm highlights the strong interaction of the annular jet with the central fuel jet in the neck zone (zone B). The dominance of vortex-shedding modes with 𝛼𝛼 𝐻𝐻2 = 0, 0.1 can be interpreted as an interlink between the local extinction and subsequent decrease in the temperature (in the region beyond RZ) due to the failure of reconnection of flame branches, as discussed in section 5.2. Many researchers [START_REF] Nair | Near-blowoff dynamics of a bluff-body stabilized flame[END_REF] [START_REF] Kiel | A detailed investigation of bluff body stabilized flames[END_REF] [START_REF] Shanbhogue | Lean blowoff of bluff body stabilized flames: Scaling and dynamics[END_REF] [START_REF] Roy | Deconvolution of reacting-flow dynamics using proper orthogonal and dynamic mode decompositions[END_REF] reported the emergence of an asymmetric sinuous mode of wake oscillations in the bluff body stabilized flames that exhibited a higher probability of local extinction events. For instance, the experiments carried out by [START_REF] Nair | Near-blowoff dynamics of a bluff-body stabilized flame[END_REF] in the near-blow-off bluff body flames reveal an increase in the localized extinction events leading to significant alteration of wake dynamics and irregular wrinkling of the flame fronts. They also found that these large-scale oscillations in the wake possess characteristic features similar to the classical Bénard -Von Karman (BVK) instability normally present in the non-reacting bluff body flows. In line with expression 3, BVK instability is often suppressed in reacting flows due to the presence of a large temperature ratio (𝑇𝑇 𝑏𝑏 /𝑇𝑇 𝑢𝑢 ) in the shear layer [START_REF] Mcmurtry | Effects of heat release on the large-scale structure in turbulent mixing layers[END_REF] (Soteriou and Ghoniem, 1994b). However, in the cases of near blow-off flames (i.e., the one with a higher degree of local extinctions/lifted flame), the decrease in the temperature ratio (𝑇𝑇 𝑏𝑏 /𝑇𝑇 𝑢𝑢 ) leads to reemergence of BVK instability in the shear layer [START_REF] Smith | LES blowout analysis of premixed flow past V-gutter flameholder[END_REF]. In line with this argument, as in the case of 𝛼𝛼 𝐻𝐻2 = 0, 0.1, the lesser probability of flame presence in the region beyond RZ, i.e., y > 30 mm (ℎ � 𝑅𝑅 ~ 35 𝑚𝑚𝑚𝑚; Figure 4) caused by a high number of breakup events (Figures 10c andd) probably resulted in a drop in temperature ratio (𝑇𝑇 𝑏𝑏 /𝑇𝑇 𝑢𝑢 ) at the OSL region downstream to RZ and, thus, the dominance of vortex shedding. domination of vortex shedding phenomenon for 𝛼𝛼 𝐻𝐻2 = 0, 0.1. On the other hand, the frequency spectrum shows broadband feature for 𝛼𝛼 𝐻𝐻2 = 0.2, 0.3 and in particular, it is intermittently oscillating between 80 to 250 Hz. Such intermittent oscillations clearly delineate the periodic switching of flow between different instability modes (e.g., vortex shedding and KH). These observations are consistent with the results of [START_REF] Emerson | Density ratio effects on reacting bluff-body flow field characteristics[END_REF] [START_REF] Erickson | The influence of reactant temperature on the dynamics of bluff body stabilized premixed flames[END_REF], who reported the occurrence of narrow-band oscillations in flames with lower temperature ratios and vice versa for higher temperature ratio flames. The above-made discussions on the role of (𝑇𝑇 𝑏𝑏 /𝑇𝑇 𝑢𝑢 ) in the suppression of vortex shedding induced at the OSL are further witnessed from the POD modes of higher hydrogen concentration flames. The increase in temperature ratio at the region downstream of RZ can be interpreted from the absence of local extinction and subsequent increase in the height of the mean reaction zone (𝒉𝒉 � 𝑹𝑹 > 𝑳𝑳 𝑹𝑹𝑹𝑹 ; 𝛼𝛼 𝐻𝐻2 = 0.5,0.8,1 Figure 4). For instance, vortex shedding becomes the least dominant mode (i.e., A4) in the case of 𝛼𝛼 𝐻𝐻2 = 0.5. In particular, for the test cases 𝛼𝛼 𝐻𝐻2 = 0.8,1, none of the first four modes (Figure 12; 𝛼𝛼 𝐻𝐻2 = 0.8,1) showed the signatures of the vortex shedding at OSL. Interestingly, for 𝛼𝛼 𝐻𝐻2 = 0.8, 1, the coherence structures are predominantly arised from the central fuel jet rather than the annular air jet. During these test conditions, the central fuel jet exhibits an intermittency feature similar to the one observed in the laminar-turbulent transition round jets [START_REF] Viggiano | Turbulence, entrainment and low-order description of a transitional variable-density jet[END_REF] and selfexcited oscillations in low-density jets [START_REF] Nichols | Self-sustained oscillations in variable-density round jets[END_REF]. The low density associated with H2 and subsequent decrease in the fuel jet Reynolds number (𝑅𝑅𝑅𝑅 𝑓𝑓𝑢𝑢𝑒𝑒𝑓𝑓 ; 𝐓𝐓𝐌𝐌𝐓𝐓𝐌𝐌𝐌𝐌 𝟏𝟏) probably induced this phenomenon.

The above said intermittent feature of the central fuel jet is evident from the instantaneous PIV raw images illustrated in the Multimedia file. These transitions mainly occurred in the region of 25 mm < y < 40 mm, and the same leads to the dominance of helical patterns in the region downstream to the RZ (the evolution of helical patterns are marked with white dotted arrows in A1 -A4 of 𝛼𝛼 𝐻𝐻2 = 0.8,1; Figure 12). Previous works [START_REF] Coenen | Absolute instability of light jets emerging from circular injector tubes[END_REF] [START_REF] Sreenivasan | Absolute instability in variable density round jets[END_REF] have also reported the dominance of such globally unstable helical modes in low-density jets.

In a nutshell, the failure of reconnection of broken flame branches following the local extinction and a subsequent decrease in the temperature ratio (as evident from the shorter reaction zone ℎ � 𝑅𝑅 in Figure 4) at OSL resulted in the dominance of vortex shedding in flames with minimal H2 enrichment (𝛼𝛼 𝐻𝐻2 = 0, 0.1). Increase in the probability of the reconnection of broken flame branches in the cases of 𝛼𝛼 𝐻𝐻2 = 0.2, 0.3 showed the transition/intermittent between vortex shedding and KH modes. On the other hand, for the high H2 enrichment (𝛼𝛼 𝐻𝐻2 ≥ 0.5), an emergence of a larger reaction zone due to the absence of the local extinction eventually led to an increase in temperature ratio in the OSL and, thus, complete suppression of the vortex shedding.

b. Flame shape oscillations

The main goal of POD implementation over OH PLIF data sets is to identify the correlation between the flow dominant modes and flame shape oscillations. The first mode (F1) of 𝛼𝛼 𝐻𝐻2 = 0, 0.1 (Figure 14) suggests flame front oscillations are predominant within the RZ. In particular, similar to the observation made by [START_REF] Yuan | Measurements in swirling spray flames at blow-off[END_REF], the intermittency in the asymmetrical flame lift-off within RZ (Multimedia view), as discussed in section 5.2, leads to the dominance of asymmetrical modal shape in the F1 of 𝛼𝛼 𝐻𝐻2 = 0, 0.1.

Next, the occurrence of alternative opposite signed intensity patches in the axial direction (highlighted with dashed rectangles; F2 -F4; Figure 14; 𝛼𝛼 𝐻𝐻2 = 0, 0.1) resembles the strong coupling of flame fronts with the coherent shedding vortices identified with the flow POD modes (A1-A4 of 𝛼𝛼 𝐻𝐻2 = 0, 0.1; Figure 12). This result corroborates well with the previous observations of [START_REF] Nair | Near-blowoff dynamics of a bluff-body stabilized flame[END_REF], i.e., wrinkling of flame fronts by an intensification of vortex shedding in the annular air jet shear layer, particularly in flames in which local extinctions are prominent. Overall, the flow and OH PLIF modes of 𝛼𝛼 𝐻𝐻2 = 0, 0.1 delineates the intensification of vortex shedding as a result of a decrease in the temperature ratio (𝑇𝑇 𝑏𝑏 /𝑇𝑇 𝑢𝑢 ) due to failure of reconnection of flame branches that formed after local extinction (Figures 10c andd; Figure 11c).

The first two POD flow modes (A1 and A2) of the 𝛼𝛼 𝐻𝐻2 = 0.2, 0.3 (Figure 12) showed the dominance of vortex shedding at the OSL and the coherent structures in the neck zone. Interestingly, none of the first four OH PLIF POD modes of 𝛼𝛼 𝐻𝐻2 = 0.2, 0.3 (Figure 14)

shows alternate opposite signed intensity fluctuations in the OSL region just downstream to RZ. Rather, it highlights the strong flapping motion of flame fronts in the ISL 2 region. Unlike 𝛼𝛼 𝐻𝐻2 = 0, 0.1, the quicker reconnection of locally extinguished flame branches during 𝛼𝛼 𝐻𝐻2 = 0.2, 0.3 (Figures 10c andd; Figures 11a andb) causes the flame to be predominantly stabilized in the ISL2 region beyond RZ. It should be noted that for 𝛼𝛼 𝐻𝐻2 = 0, 0.1 test cases, as shown in Figure 10, Figure 11c, the failure of reignition leads to convection of flame fronts by shedding vortices in the OSL (in other words coupling of the flame fronts with the flow shedding modes) and results in the intensity fluctuations at the OSL as highlighted by F2 -F4 of 𝛼𝛼 𝐻𝐻2 = 0, 0.1 in Figure 14. On the other hand, as inferred from the OH -PLIF POD modes of 𝛼𝛼 𝐻𝐻2 = 0.2, 0.3, the flame only experiences radial flapping motion due to the strong interaction of annular air jet with central fuel jet in the neck zone.

The modal structures of 𝛼𝛼 𝐻𝐻2 = 0.8, 1 also exhibit the dominance of flapping motion in the neck zone; however, the modes F2 and F3 (Figure 14; F2, F3 of 𝛼𝛼 𝐻𝐻2 = 0.8, 1) show asymmetrical flapping in the downstream region of RZ. The emergence of such asymmetrical motions in the flame fronts is caused by helical structures shown in the flow POD modes (; Figure 12; 𝛼𝛼 𝐻𝐻2 = 0.8, 1).

Figure 14 Spatial distributions of coherent flame structures in the first four POD modes

[Multimedia view fo 𝛼𝛼 𝐻𝐻2 = 0] 8. Conclusions

The impact of H2 addition in a canonical non-premixed bluff-body stabilized turbulent burner has been studied using simultaneous time-resolved (5 kHz) PIV and OH -PLIF. Such high-fidelity measurements enabled us to explore the links between H2 addition, local extinction, reignition, and hydrodynamics of the flow field. The bluff-body burner is operated with increasing levels of H2 addition (0%,10%, 20%, 30%, 50%, 80%, and 100%) in the H2 -CH4 mixture. The key findings of this paper are summarized in Figure 15. The increase in the extinction strain rate over the local flow-induced hydrodynamic strain rate (𝜅𝜅 𝑒𝑒𝑒𝑒𝑒𝑒 > 𝜅𝜅 ℎ𝑦𝑦𝑦𝑦 ) for the test cases with H2 addition > 30 % (𝛼𝛼 𝐻𝐻2 > 0.3) resulted in the absence of local extinctions in the neck zone. Interestingly, although local extinctions are observed for 0 < 𝛼𝛼 𝐻𝐻2 ≤ 0.3, the significance of H2 enrichment is witnessed during the reconnection/reignition of broken flame branches. An increase in the edge flame speed with the cases of 𝛼𝛼 𝐻𝐻2 = 0.2, 0.3 leads to a quicker reconnection of the broken branches formed during the local extinction. On the other hand, lesser reignition probability is observed in the low hydrogen concentration flames (i.e., 𝛼𝛼 𝐻𝐻2 = 0, 0.1). This phenomenon alters the flame structure/reaction zone for different H2 enrichments (in other words, temperature/density ratio in the flowfield). Proper orthogonal decomposition (POD) and 1D Continuous Wavelet Transform (CWT) have been employed to gain insights into the influence of flame structures in the hydrodynamics of the flow field. Vortex shedding is found to be a dominant mode of oscillation in 𝛼𝛼 𝐻𝐻2 = 0,0.1 test cases. Such vortex shedding modes are commonly observed in the non-reacting bluff body flows and are usually suppressed in reacting flows. The observed high probability of the failure of reconnection of the broken flame branches following the local extinction leads flames to be predominantly stabilized in the RZ region. Consequently, the flow downstream to the RZ region possesses a characteristic feature similar to the non-reacting flow and, thus, the dominance of vortex shedding. On the other hand, quicker reconnection of the broken flame branches due to the H2 enrichment in the test cases of 𝛼𝛼 𝐻𝐻2 = 0.2, 0.3 resulted in the dominance of combined vortex shedding and KH modes. Whereas for the H2 enrichment greater than 30% (𝛼𝛼 𝐻𝐻2 > 0.3), the absence of local extinctions and the further increase in the flame presence beyond the RZ resulted in the complete suppression of vortex-shedding modes. The obtained results emphasize the significant implications of H2 addition in the hydrodynamics of the flow field in a nonpremixed bluff body burner and then in the stabilization modes of such highly turbulent flame.

Figure 15 Schematic illustration of key summary on the relationship between H2 enrichment, local extinction, re-ignition/reconnection, and flow dynamics

Figure 1 .

 1 Figure 1. Configuration of the canonical non-premixed bluff-body burner (𝑫𝑫 𝒂𝒂 = 𝟓𝟓𝟓𝟓 𝒎𝒎𝒎𝒎;𝑫𝑫 𝒃𝒃 = 𝟒𝟒𝟒𝟒 𝒎𝒎𝒎𝒎;𝑫𝑫 𝒇𝒇 = 𝟒𝟒 𝒎𝒎𝒎𝒎; 𝒕𝒕 𝒃𝒃 = 𝑯𝑯𝟐𝟐 𝒎𝒎𝒎𝒎)

Figure 2 .

 2 Figure 2. Schematic of time-resolved (5 kHz) simultaneous PIV and OH PLIF optical diagnostic systems (L1 = 1000 mm; L2 =-12.4mm; L3 = 500 mm; L4 = -6 mm; L5 = 300 mm) b. Time-resolved OH-Planar Laser-Induced Fluorescence (PLIF)Figure2illustrates the optics arrangement associated with the OH PLIF measurement system. It comprises a single cavity, solid state Nd: YAG pump laser (EdgeWave; λ~532 nm), and a tunable dye laser (Sirah Credo). The dye laser is tuned to produce a laser beam of wavelength ~ 566 nm with the help of Rhodamine 6G dye dissolved with ethanol, and then, it is frequency-doubled to get a desired UV laser beam of wavelength ~ 283 nm: 0.3 mJ/pulse. The generated 283 nm beam is used to excite the OH radicals in the Q1 (6) line of 𝐴𝐴 2 ∑ + -𝑋𝑋 2 ∏(1,0) transition band. The UV laser beam is converted to a collimated sheet approximately ~ 70 mm in height using three fused silica lenses (L3, L4, L5). Then, it is precisely overlapped with the PIV laser sheet to ensure the acquisition of OH fluorescence and flow fields from the same longitudinal plane. Since the pulse energy of the UV beam is low at a high repetitive rate, the illumination area for PLIF imaging in the vertical plane is limited to ~ 70 mm x 70 mm.

Figure 3

 3 Figure 3. a. Time-averaged vector fields of 𝜶𝜶 𝑯𝑯𝑯𝑯 ~𝟐𝟐, 𝟐𝟐. 𝟓𝟓, 𝟏𝟏; b. Schematic illustration of various flow zones and key terminologies in the flow field; c. Time-averaged flow fields; d. Direct photographs of flames

Figure 5 .Figure 7 .

 57 Figure 5. Demonstration of image processing steps involved in the extraction of instantaneous flame sheet/reaction zone height �ℎ 𝑅𝑅,𝐿𝐿 ′ � from OH PLIF images

Figure 8 .

 8 Figure 8. Conditionally averaged flow field of 𝜶𝜶 𝑯𝑯𝑯𝑯 = 𝟐𝟐. 𝟏𝟏 (a, b -h1, h2 windows); c. Time-averaged flow field of 𝜶𝜶 𝑯𝑯𝑯𝑯 = 𝟐𝟐. 𝟏𝟏; d. Vortex core center (VCC) position of the identified coherent structure from the conditionally averaged flow fields.

Figure 9 .

 9 Figure 9. Typical time evolution of sequential events observed during local extinction process; a. Sequence illustrating the local extinction that occurred due to high strain in the fuel jet ( 𝜶𝜶 𝑯𝑯𝑯𝑯 = 𝟐𝟐. 𝟑𝟑);b.sudden ejection of the vortex from the central fuel jet ( 𝜶𝜶 𝑯𝑯𝑯𝑯 = 𝟐𝟐. 𝟑𝟑); c. Sequence illustrating the absence of local extinction (𝜶𝜶 𝑯𝑯𝑯𝑯 = 𝟐𝟐. 𝟓𝟓; 𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌 𝐯𝐯𝐌𝐌𝐌𝐌𝐯𝐯).Flame contours are superimposed over vorticity (i).2D strain rate (ii), and PIV raw particle image (iii);

Figure 10 .

 10 Figure 10. Illustration of closure (a) and breakup events (b); Histograms of a breakup and closure events obtained across right (c) and left (d) side windows.

Figure 11 .

 11 Figure 11. Typical time sequences illustrating the various phenomenon observed during post local extinction / re-ignition phase; a. re-connection of flame branches predominantly by edge flame propagation (𝛼𝛼 𝐻𝐻2 = 0.2); b. re-connection of flame branches by the combination of turbulence mixing and edge flame propagation (𝛼𝛼 𝐻𝐻2 = 0.2); c. periodic

Figure 13 .

 13 Figure 13. Magnitude of frequency spectrum scalograms obtained from CWT.

Table 2 . Number of extinction events in two regions of the flame for various α values Test cases No. of extinction events in h1, h2 windows (Right side) No. of extinction events in h1, h2 windows (Left side)

 2 .

	𝜶𝜶 𝑯𝑯𝑯𝑯 = 𝟐𝟐	65 ;53	70 ;49
	𝜶𝜶 𝑯𝑯𝑯𝑯 = 𝟐𝟐. 𝟏𝟏	80 ;91	76 ;88
	𝜶𝜶 𝑯𝑯𝑯𝑯 = 𝟐𝟐. 𝑯𝑯	95 ; 182	85 ; 171
	𝜶𝜶 𝑯𝑯𝑯𝑯 = 𝟐𝟐. 𝟑𝟑	79 ; 161	84 ; 135
	Figures 8a

ACKNOWLEDGEMENTS

The authors would like to acknowledge the funding from the Normandie Regional Council and the European Union through the European Regional Development Fund (ERDF) in the framework of the RAPHYD (Grant no. RIN 2018) project.

The experimental investigation carried out by [START_REF] Emerson | Density ratio effects on reacting bluff-body flow field characteristics[END_REF] in the premixed bluff body flames reveals the transition from global BVK (vortex shedding) to convective KH instability at higher temperature ratios. They have also reported intermittency in switching flow states between KH and BVK instability modes, even at fixed operating conditions. We have observed such scenarios in the test cases of 𝛼𝛼 𝐻𝐻2 = 0.2; 0.3, where only the first two modes (A1, A2) show the dominance of vortex shedding in OSL. The other two modes (i.e., A3, A4) exhibit the spatial structures similar to the KH/shear instability, i.e., over here, the alternate pattern of vortices is absent at the OSL, contrary to 𝛼𝛼 𝐻𝐻2 = 0; 0.1.

The reconstructed flow fields (see Multimedia view) based on POD modes (only with the first ten modes) suggest the dominance of vortex shedding for 𝛼𝛼 𝐻𝐻2 = 0, 0.1, and periodic switching between vortex shedding and KH for 𝛼𝛼 𝐻𝐻2 = 0.2, 0.3. To further investigate the role of vortex shedding and periodic switching phenomenon, we have implemented continuous wavelet transform (CWT). CWT is chosen instead of FFT on POD time coefficients because the former allows us to characterize the system oscillating between different dynamic states.

Here, CWT has been applied over the instantaneous vorticity values extracted from the small vector region of 7 x 7. The chosen region is located at the OSL, where the vortex shedding is prominent. In the present study, wavelet transform is computed using the 1D CWT function available in MATLAB. More details on the used CWT function can be found in [START_REF] Torrence | A practical guide to wavelet analysis[END_REF] [START_REF] Lilly | Generalized Morse wavelets as a superfamily of analytic wavelets[END_REF].

The results obtained across the left and right side of the burner's longitudinal central axis depicts a similar feature; hence, for brevity, only the CWT scalogram of the right side is shown here (Figure 13). For test cases 𝛼𝛼 𝐻𝐻2 = 0, 0.1, the frequency values are prominently concentrated at ~ 85 Hz (Figure 13). Furthermore, the obtained frequencies correspond to a Strouhal number (𝑆𝑆𝑑𝑑 = 𝑓𝑓𝑑𝑑 𝑏𝑏 /𝑈𝑈 𝑎𝑎𝑎𝑎𝑎𝑎 ) of 0.25, which is known to be characteristic frequency of vortex shedding [START_REF] Prasad | The instability of the shear layer separating from a bluff body[END_REF]. Hence, the results of CWT confirm the

AUTHOR DECLARATIONS Conflict of Interest

The authors have no conflicts to disclose.

AUTHOR CONTRIBUTIONS


Marco Ripà
email: marcokrt1984@yahoo.it

General conjecture on the optimal covering trails in a k-dimensional cubic lattice

Keywords: Conjecture, Link length, Covering Trail, Nine-dot problem, Optimization. 2020 Mathematics Subject Classification: Primary 05C57, 05C38; Secondary 68R10

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Introduction

In the present preprint we formulate a very general conjecture on the infamous nine-dot problem [START_REF] Loyd | Sam Loyd's Cyclopedia of 5000 Puzzles, Tricks and Conundrums with Answers[END_REF] extended to any given n × n × • • • × n grid in the Euclidean space R k .

Let N -{0} be the set of (strictly) positive integers so that, for any k ∈ N -{0, 1, 2}, we denote as N k the k-dimensional cubic lattice N k := {(x 1 , x 2 , . . . x n ) :

x i ∈ N ∧ n ≥ 3}.
Under the well-known nine-dot problem rules [START_REF] Chein | Working memory and insight in the nine-dot problem[END_REF], our goal is to visit all the n k nodes of the set H(n, k) := {0, 1, 2, ..., n -1} × {0, 1, 2, ..., n -1} × • • • × {0, 1, 2, ..., n -1} with a covering trail (i.e., any polygonal chain joining all the aforementioned points) having the minimum number of links.

Since this NP-complete problem (see [START_REF] Bereg | Traversing a set of points with a minimum number of turns[END_REF], p. 514) has been fully solved only for a few, very special, cases (i.e., n ≤ 3 ∧ k ∈ Z + [START_REF] Rip À | Solving the 106 years old 3 k points problem with the clockwise-algorithm[END_REF][START_REF] Rip À | Minimum-Link Covering Trails for any Hypercubic Lattice[END_REF] and n ∈ Z + ∧ k ≤ 2 [START_REF] Keszegh | Covering Paths and Trees for Planar Grids[END_REF]), we mainly base our approach on the proven bounds, linking the conjectured formula to recent results and many other evidences (e.g., combinatorial inference on graphical models, pattern recognition constraints, and case-by-case analysis [START_REF] Rip À | Solving the n 1 × n 2 × n 3 Points Problem for n 3 < 6[END_REF]).

Let us denote by h(n, k) the minimum link length of any covering trail for the set H(n, k). 6), so h u (4, 3) = 23 ⇒ h(4, 4) ∈ [85, 95] and h u (5, 3) =

Consequently, let us indicate a valid upper bound as

h u (n, k) so that h(n, k) ≤ h u (n, k) holds by definition. Since n k -1 n-1 ≤ h(n, k) ≤ h u (n, k), ∀k ≥ 3 (see [9], Theorem 2.1), let h u (n, k) := (h u (n, 3) + 1) • n k-3 -1 (by [10] Equation
1 36 ⇒ h(5, 5) ∈ [781, 924], while h(n, 2) = 2 • (n -1), h(2, k) = 3 • 2 k-2 , and h(3, k) = 3 k -1
2 have been formally proven in recent years [START_REF] Keszegh | Covering Paths and Trees for Planar Grids[END_REF][START_REF] Rip À | Solving the 106 years old 3 k points problem with the clockwise-algorithm[END_REF][START_REF] Rip À | General uncrossing covering paths inside the axis-aligned bounding box[END_REF]. Now, it is natural to suppose that the general solution follows a simple recursive pattern, as it has been shown to be true for the lower order k-dimensional grids, so we can single out the formula for the link length of any optimal covering trail for the set H(n, k) by verifying ex-post that h(4, 3) ̸ = 21, arguing a fortiori that Conjecture 2.1 by Reference [START_REF] Rip À | Minimum-Link Covering Trails for any Hypercubic Lattice[END_REF] always holds.

Thus, in order to finally decide whether to conjecture h(n, k)

= n k -1 n-1 + n -3 or h(n, k) = n k -1 n-1 + (k -1) • (n -3
), it would be very helpful to implement an efficient computer algorithm which can find (if any) a valid 22-link covering trail for H(4, 3), supporting the strong claim that h(n, k) = n k -1 n-1 + n -3. In Section 4 of the present preprint, we will describe a theoretical brute force approach that is possible to take as a starting point, going for a more sophisticated Monte Carlo method.

Full table of results

In general, given any pair of strictly positive integers (n, k), what we currently know on the open problem of covering the set H(n, k) with a minimum-link trail, is given by Equations ( 1) to [START_REF] Rip À | General uncrossing covering paths inside the axis-aligned bounding box[END_REF].

h(n, 1) = 1, ∀n. (1)
h(n, 2) = 2 • (n -1), ∀n ≥ 3. (2)
h(1, k) = 1, ∀k. (3)
h(2, k) = 3 • 2 k-2 , ∀k ≥ 2. ( 4
) h(3, k) = 3 k -1 2 , ∀k. (5)
h(n, k) ≤ 3 2 • n 2 - n -1 4 + n + 1 4 - n + 2 4 + n 4 +n-1 •n k-3 -1, ∀n ≥ 3∧∀k ≥ 3 (6)
(see Equation [START_REF] Loyd | Sam Loyd's Cyclopedia of 5000 Puzzles, Tricks and Conundrums with Answers[END_REF] in Reference [START_REF] Rip À | The n × n × n Dots Problem: An Improved 'Outside the Box' Upper Bound[END_REF]).

h(n, k) ≥ n k -1 n -1 , ∀n ≥ 3 ∧ ∀k. (7) h(n, k) ≤ (h u (n, 3) + 1) • n k-3 -1, ∀n ≥ 3 ∧ ∀k ≥ 3. (8)
Lastly, we have that h(4, 3) ≤ 23 and h(5, 3) ≤ 36 (see Section 3, Figure 1 and Figure 2,

respectively), while h(n, k) ≤ 3 2 • n 2 -n-1 4 + n+1 4 -n+2 4 + n 4 + n -1 • n k-3 -1 is
a weak upper bound which holds for any n, k ∈ N -{0, 1, 2} (see Equation ( 6)).

Then, the proven results by Equations ( 1) to ( 8) are summarized in Table 1. n-1 for any n > 3 by Reference [START_REF] Keszegh | Covering Paths and Trees for Planar Grids[END_REF] and given the fact that h(3, k) = 3 k -1

n k 1 2 3 4 5 . . . h(n, k) 1 1 1 1 1 1 . . . h(1, k) = 1 2 1 3 6 12 24 . . . h(2, k) = 3 • 2 k-2 , ∀k ≥ 2 3 1 4 13 40 121 . . . h(3, k) = 3 k -1 2 4 1 6 [21, 23] [85, 95] [341, 383] . . . 4 k -1 3 ≤ h(4, k) ≤ h u (4, k) 5 1 8 [31, 36] [156, 184] [781, 924] . . . 5 k -1 4 ≤ h(5, k) ≤ h u (5, k) . . . . . . . . . . . . . . . . . . . . . . . . h(n, k) 1 2 • (n -1), ∀n ≥ 3 h(n, 3) h(n, 4) h(n, 5) . . . n k -1 n-1 ≤ h(n, k) ≤ h u (n, k), ∀n ≥ 3 Table 1: Proven bounds for the general {0, 1, 2, . . . , n -1} × {0, 1, 2, . . . , n -1} × • • • × {0, 1, 2, . . . , n -1} points problem in R k .
3-1 by [START_REF] Rip À | Solving the 106 years old 3 k points problem with the clockwise-algorithm[END_REF], in the present section we will assume that h(n, k) < n k -1 n-1 + n -3 should be false for any n ≥ 3 ∧ k ≥ 2 so that h(4, 3) = 22 ∨ h(4, 3) = 23.

Then, we observe how h(n, k) = n k -1 n-1 implies that each pair of consecutive line segments of any theoretical 21-link covering trail for H(4, 3) would join at least 6 of its nodes [START_REF] Rip À | Minimum-Link Covering Trails for any Hypercubic Lattice[END_REF], while, since we disregard this (very unlikely) possibility here, we need to take into account, as a worst-case scenario, that an optimal covering trail could consist of 22 links of which at least 19 have to necessarily join 2 or more unvisited points each (e.g., it would be possible to join 4 nodes with the first link, to have a triplet of consecutive pairs of links visiting 1 + 4 new nodes for three times in a row, and to finally visit the remaining 45 nodes of H(4, 3) by spending the last 15 line segments available, since we could use all the 15 residual links to join 3 nodes each time, or to have some of those pairs that visit 2 nodes with one link and 4 nodes thanks to the next one, as exemplified by the polygonal chain (0, 3, 0)-(5, 3, 0)-(-1, 0, 0)-(5, 0, 0)-(-1, 0, 3)-(5, 0, 3)-(-1, 3, 3)-(3, 3, 3)).

The constraint above trivially follows by considering that we cannot visit all the 4 3 nodes of H(4, 3) with 22 consecutive links if we spend more than three links to fit only one new point at a time, since [START_REF] Rip À | Minimum-Link Covering Trails for any Hypercubic Lattice[END_REF], proof of Theorem 2.1).

4 3 ≤ 4 + 3 • (22 -1 -2 • m) + (1 + 4) • m if and only if m ≤ 3 (see Reference
A very basic algorithm, to try to (constructively) find by brute force a 22-link covering trail for H(4, 3), is as follows.

1.

Step 1. Take one by one each pair of elements of the set H(4, 3) and draw the (unique) straight line through those two points. Do the same for every pair of nodes of H(4, 3) and denote as I 2 (4, 3) the set of all those crossing points. Now, it is easy to observe that if h(4, 3) < 23, then it would necessarily be possible to find optimal covering trails inside the axis-aligned bounding box (AABB [START_REF] Rip À | General uncrossing covering paths inside the axis-aligned bounding box[END_REF]) [START_REF] Rip À | The n × n × n Dots Problem: An Improved 'Outside the Box' Upper Bound[END_REF][START_REF] Rip À | Minimum-Link Covering Trails for any Hypercubic Lattice[END_REF]× [START_REF] Rip À | The n × n × n Dots Problem: An Improved 'Outside the Box' Upper Bound[END_REF][START_REF] Rip À | Minimum-Link Covering Trails for any Hypercubic Lattice[END_REF]× [START_REF] Rip À | The n × n × n Dots Problem: An Improved 'Outside the Box' Upper Bound[END_REF][START_REF] Rip À | Minimum-Link Covering Trails for any Hypercubic Lattice[END_REF]. Thus, the set I 2 (4, 3) has a total of 13307 elements (i.e., |I 2 (4, 3)| = 13307) and there are 1492 maximal line segments, as shown by Reference [START_REF] Alekseyev | Finding an optimal covering trail for the set {0, 1, 2, 3} × {0, 1, 2[END_REF].

2.

Step 2. Consider only the polygonal chains of link length q ∈ {1, 2, . . . , 22}, having all the vertices belonging to the set I 2 (4, 3), and such that each of their links joins at least 2 (unvisited) nodes of H(4, 3). Furthermore, from the above-mentioned class of trails, let us also disregard all the polygonal chains of length q that do not visit at least 3 • q -q 7 nodes of H(4, 3).

3.

Step 3. From the very efficient polygonal chains returned by Step 2, take only into account the trails with link length 21 (if we get only a void set of solutions, then abort the process, since the test is failed). By construction, they visit 60, 61, 62, or 63 nodes (otherwise, we would have found a 21-link covering trail for the set H(4, 3), proving that h(4, 3) = 21, and we do not expect that this could happen).

4.

Step 4. Although Step 3 does not allow us to perform an exhaustive search (and a failed test cannot prove in any way that h(4, 3) = 23), we are now ready to sort the polygonal chains with 21 links (returned by Step 3) in ascending numerical order basing on how many nodes of H(4, 3) they visit.

5.

Step 5.

(a) Case (a). If we have already found a 21-link trail joining 63 nodes, the search is over, since we will spend the last line segment to join together an endpoint of that optimal polygonal chain and the last unvisited node of H(4, 3). If not, let us move on to Step 5, Case (b).

(b) Case (b). If the best polygonal chains returned by Step 4 visit (exactly) 62 nodes, then we take them and start to check, one by one, whether the line that fits the pair of residual nodes intersects one of the two lines that include the first&second or the second-last&last vertices of the 21-link trail that we have originally selected. If there is an external match with respect to the right half-line containing the first/last link of the selected polygonal chain, then we have constructively proven that h(4, 3) < 23. Otherwise, let us move on to Step 5, Case (c).

(c) Case (c). There are 3 (or maybe even 4) unvisited nodes left. If they are not collinear points, then the test is failed; otherwise, they are collinear and we repeat the same pattern described in Step 5, Case (b), with only one difference: if there is no right external match between the above mentioned fitting line and the extensions of the first/last link belonging to the 21-link trail we have selected, then the test is failed and the process stops.

If the described test fails, then we cannot improve the current bound 21 ≤ h(4, 3) ≤ 23, and also we cannot conjecture that h(n, k) = n k -1 n-1 + (k -1) • (n -3), since the theoretical existence of a 22-link covering trail cannot be excluded by a partial search as the above.

On the contrary, if the test has succeeded, returning a 22-link covering trail for H(4, 3), then a general conjecture stating that h(n, k) = h(n, k) = n k -1 n-1 + n -3 would receive a relevant confirmation.

Lastly, to speed up the algorithm, we point out how it would be possible to run a Monte Carlo experiment looking for the longest trails with q links that visit a total of 3 • q + 1 nodes of H(4, 3), and then going forward.

In the meantime, let us state our conjecture on h(n, k) as follows.

Conjecture 1. For any n ∈ N -{0, 1, 2} and k ∈ N -{0}, we conjecture that h

(n, k) = n k -1 n-1 + n -3 ∨ h(n, k) = n k -1 n-1 + (k -1) • (n -3).
Time will tell.

Conclusion

After a dozen years spent working on this family of problems, basing our deductions on the evidence discussed in the present preprint, we are finally ready to conjecture that, if n is greater than 2, then h(n, k) = n k -1 n-1 + n -3 or h(n, k) = n k -1 n-1 + (k -1) • (n -3), since it seems to be very unlikely the evenience that, while h(n, 2) > n 2 -1 n-1 if and only if n ≥ 4 [START_REF] Keszegh | Covering Paths and Trees for Planar Grids[END_REF], at the same time, h(n, 3) is not greater than n 3 -1 n-1 for some n ≥ 4 (taking also into account that, for any k ∈ Z + , h(3, k) = 3 k -1 3-1 = 3 k -1 3-1 + (3 -3) = 3 k -1 3-1 + (k -1) • (3 -3)). Consequently, we expect h(n, k) to be n k -1 n-1 + n -3 if it will be proven h(4, 3) < 23 or, alternatively, h(5, 3) < 35, whereas we can trust in h(n, k) = n k -1 n-1 + (k -1) • (n -3) after any proof that h(4, 3) > 22 or h(5, 3) > 34.

Even though our solid belief has been clearly stated in the present preprint, the research on this topic is far from being concluded, and different answers, not previously contemplated, are conceivable at present.

3

  Proof that h(4, 3) ≤ 23 and h(5, 3) ≤ 36 A covering trail with 23 links for the set H(4, 3) is (3, 3, 1)-(1, 3, 1)-(-2, 0, 1)-(4, 0, 1)-(3, 0, 3)-(3, 3, 3)-(0, 0, 0)-(3, 0, 0)-(0, 3, 3)-(0, 0, 3)-(3, 3, 0)-(-1, 3, 0)-(2, 3, 3)-(2, -1, 3)-(-1, 2, 0)-(4, 2, 0)-(1, -1, 3)-(1, 4, 3)-(4, 1, 0)-(-1, 1, 0)-(3, 3, 2)-(3, -2, 2)-(0, 7, 2)-(0, 0, 2), Figure 1.

Figure 1 :Figure 2 :

 12 Figure 1: Constructive proof that h(4, 3) ≤ 23 [10].

Acknowledgements

The author is thankful to the MathOverflow user Max Alekseyev, for his valuable answer concerning the brute force algorithm described in Section 4, and to all the active followers of his YouTube channel, whose comments and constant support have contributed to improve the overall quality of this preprint.