Francesco Scarano

Romain Gojon

Erwin R Gowree
email: erwin.gowree@isae-supaero.fr

On the development of a turbulent boundary layer over staggered three dimensional cavities

Studies by Gowree et al. have suggested that staggered three dimensional circular cavities can potentially reduce the skin friction drag of a turbulent boundary layer developing over a flat plate. This is true up to a certain Reynolds number, after which the drag rapidly increases. The detailed mechanism behind the drag increase and the possible skin friction drag reduction was not discussed, due to the limitations in the measurement technique. In the current experimental campaign the results found by Gowree et al. have been confirmed, in addition, to more detailed insight of the turbulent boundary layer over the first rows of cavities. Through numerical simulations we aim to give a heuristic explanation of the two distinct behaviours. Here we report some quantitative evidence of the mechanism that may govern the drag increase, which is the main focus of the current study. The modification of the flow field in the presence of the cavities provides a qualitative explanation of the drag reduction benefit which is subjected for further investigation.

INTRODUCTION

Being the main contributor to the overall drag on an aircraft (> 50%), skin friction drag has undergone serious scrutiny and is still a priority in the agenda for aerodynamic design and optimisation, especially in an era where there is an urge to reduce emissions. Let alone the difficulties in implementing technologies for skin friction drag reduction, the assessment of the skin friction drag penalty from a turbulent boundary layer itself is a complex task. Especially once the wall boundary conditions deviates from the hydrodynamically smooth condition and the well-established semi-empirical relations derived for turbulent boundary layers are no longer applicable. Perforated acoustic liners found in air-breathing engines is such an example. These surfaces, usually optimised for acoustic performances, can act to the detriment of the aerodynamic performance due to the significant rise in skin friction drag which results in pressure losses from the engine. The acoustic liners used in aeronautical applications are made of a perforated facing sheet which is placed on top a honeycomb cell sandwiched by a backing sheet. The primary flow is a turbulent boundary layer that develops over the facing sheet, where the perforations act to modify the flow in the near wall resulting in significant drag rise. There are still questions regarding the effect of the aero-acoustic interaction on the sharp drag rise which needs to be addressed.

With the aim of characterising the aerodynamic performances of the acoustic liners and give an insight of the flow field characteristics, it is fundamental to refer to the simpler case of a turbulent boundary layer grazing on a perforated surface with zero pressure gradient.

During an unpublished experimental campaign, E. Gowree observed that for lower Mach number, M < 0.5, the drag rise was less significant and in some cases it could even be lower than that on equivalent smooth and unperforated surfaces. This encouraged further low speed experiment by Gowree et al. [START_REF] Gowree | Skin friction drag reduction over staggered three dimensional cavities[END_REF] to investigate how the perforated surface or the presence of the 3D circular cavities could lead to a reduction in local skin friction drag. Over the past few decades, several techniques for skin friction drag reduction from turbulent boundary layer have been proposed. Longitudinal grooves known as riblets, inspired by the dermal denticles of sharks, have demonstrated the potential of skin friction drag reduction from turbulent boundary layer, by rearranging the turbulent kinetic energy production cycle through modification of the wall boundary conditions. This technique was initially proposed by Walsh and co-workers and further detail of the flow physics was presented during the following studies [START_REF] Walsh | Drag characteristics of v-groove and transverse curvature riblets[END_REF][START_REF] Choi | Near-wall structure of a turbulent boundary layer with riblets[END_REF][START_REF] Bechert | The viscous flow on surfaces with longitudinal ribs[END_REF][START_REF] Bechert | Experiments on dragreducing surfaces and their optimization with an adjustable geometry[END_REF][START_REF] Bechert | Experiments with three-dimensional riblets as an idealized model of shark skin[END_REF][START_REF] Dean | Shark-skin surfaces for fluid-drag reduction in turbulent flow: a review[END_REF], to quote a few. In a more recent study van Nesselrooij [START_REF] Van Nesselrooij | Drag reduction by means of dimpled surfaces in turbulent boundary layers[END_REF] demonstrated that dimples could also reduce skin friction drag for certain Reynolds number regime and the physical mechanism was once again tied to the near wall production cycle modification. On the other hand, studies conduced on acoustic liners [START_REF] Wilkinson | Influence of wall permeability on turbulent boundary-layer properties[END_REF][START_REF] Jasinski | Mechanism for increased viscous drag over porous sheet acoustic liners[END_REF] reported a skin friction drag increase. This behaviour is usually associated with an increase in the turbulent activity or rather an increase of the frequency of the bursts and ejections that contribute to an increase in the turbulent kinetic energy. The physical mechanism which governs this drag increase has not been sufficiently explained yet.

Hot wire measurements, coupled with an accurate determination of the wall to probe distance [START_REF] Gowree | A simple digital-optical system to improve accuracy of hot-wire measurements[END_REF], showed, in correspondence with the first rows of cavities, a distortion in the average and root-mean-square velocity fields associated with a considerable oscillation of the boundary layer integral quantities. Even if the experimental results provide a real trend in drag in a more representative scenario, the measurement technique was limited in its ability to provide a detailed representation of the flow. Thus high fidelity Large Eddy simulations were conducted on a simplified geometry to complement and confirm the experimental finding. In this case we focused on the mechanism leading to the drag increase, but we were able, at the same time, to describe qualitatively a possible explanation of the drag benefit.

METHODS

Experimental setup

The experimental campaign has been conduced in the Sabre wind tunnel of ISAE-SUPAERO. The test models consisted in one smooth and one perforated surface, the geometrical characteristics of the latter will be reported above in the numerical setup. A Dantec P5517 hot wire probe for boundary layer attached to a two axes traversing system coupled with an optical probe to wall alignment technique [START_REF] Gowree | A simple digital-optical system to improve accuracy of hot-wire measurements[END_REF], proved to be a reliable method to obtain velocity profiles at a close distance one another and to obtain measures close to the wall.

Along the center-line of the plate, and in correspondence to the first two rows of cavities, a series of velocity profiles with a fine spacing of 2.5 mm was made, Figure 1. This led to a 2D grid covering a region of the first two cavities where the distortion of the turbulent boundary layer is expected to be more pronounced.

Numerical setup

A high-order finite-volume code is used to solve the compressible Navier-Stokes equations. It solves the spatially filtered compressible Navier-Stokes equations for conservative variables using a finite volume formulation and a control-volume based discretization on unstructured hexahedral meshes. For time stepping, an explicit third-order Runge-Kutta scheme is used. For the spatial derivative, the solver uses a fourth-order non-dissipative centred numerical scheme [START_REF] Bermejo-Moreno | Confinement effects in shock wave/turbulent boundary layer interactions through wall-modelled large-eddy simulations[END_REF]. The subgrid-scale model of Vreman [START_REF] Vreman | An eddy-viscosity subgrid-scale model for turbulent shear flow: Algebraic theory and applications[END_REF] is used to dissipate the small scale turbulent structures that are not resolved by the mesh. We are thus performing explicit LES. This solver has been used previously for several studies of turbulent flow over an airfoil [START_REF] Grebert | Investigation of wall-pressure fluctuations characteristics on a naca0012 airfoil with blunt trailing edge[END_REF][START_REF] Boukharfane | Characterization of the pressure fluctuations within a controlled-diffusion airfoil boundary layer at large reynolds numbers[END_REF] or shock boundary layer interaction [START_REF] Grébert | Simulations of shock wave/turbulent boundary layer interaction with upstream micro vortex generators[END_REF][START_REF] Grébert | Simulations of shock wave/turbulent boundary layer interaction with upstream micro vortex generators[END_REF].

The simulations have been performed by changing the inlet condition in order to study the effect of the Reynolds number on the topology of the flow keeping fixed the geometry excluding the direction normal to the wall that is scaled with the boundary layer thickness. The wall resolved mean velocity profiles obtained experimentally have been used as inlet condition. The free-stream velocities, the boundary layer thickness at the inlet of the model δ 0 , and the ranges of Reynolds number based on the momentum thickness along the model are summarized in the Table 1. Since the measurements were performed with a hot wire probe, only one component of the Reynolds stress is available (fluctuations of the stream-wise velocity). Therefore it has been decided then to assign the average turbulent velocity profile and study its effect on the topology of the flow. The lack of synthetic turbulence allows a reductions in the computational costs; this numerical approach has already been used for similar investigations [START_REF] Bauerheim | Les of the aero-acoustic coupling in acoustic liners containing multiple cavities[END_REF].

CASE U ∞ [m/s] δ 0 [m] Re θ I 10
We tried to reproduce, numerically, the experimental model that consisted of a face sheet with a grid of staggered circular cavities. During the experimental campaign we focused our attention on the first two rows of cavities where we saw a stronger effect on the turbulent boundary layer.

As a matter of fact, the numerical geometry consisted of three rows of cavity with a stagger angle of 45 deg and a spacing of L c =22 mm as shown in Figure 2; the first row is located one diameter downstream of the inlet. A periodic boundary condition on the transverse direction The geometry and grid parameters used for the present investigation are presented in Table 2 where X is the longitudinal coordinate, Y is the wall normal coordinate, and Z is the transverse coordinate. The domain was divided in two parts, the first one goes from the inlet up to 2L c while the second goes up to the outlet. The dimensions are reported in Table 2. An unstructured grid for the first part and a structured grid for the second one were generated, the reason for this choice being the simplification of the meshing procedure when dealing with the cavities. On the second part of the mesh an increasing cell size factor of 1.05 is assigned in order to have the last cells to behave like a sponge region.

The grid consists of about 6 millions hexahedral cells. Those grids are designed from surface grids at the plate extruded in the wall normal direction. At the wall, a cell size of ∆y = 1.38 × 10 -5 m is imposed in order to have y + ≈ 1 at the wall. In the region 0 to δ , the mesh size increases by a factor 1.10, then by a factor 1.25 from δ to 2.5δ , and finally a sponge zone with cells increasing in

L x /Lc L y /δ L z /Lc 6.5 4 2
Table 2: Geometry dimensions, with δ the boundary layer thickness imposed at the inlet of the domain and L c distance between the cavities.

size by a factor 1.35 is used up to 4δ .

RESULTS AND DISCUSSIONS

Experimental results

Single profiles taken at the leading edge and trailing edge of both the perforated and the smooth surface allowed the determination of the skin friction by means of momentum deficit. The measurements taken on the smooth surface validated the technique showing a good match with the Colels-Fernholz relation for smooth walls, Figure 3.

For the perforated wall a non-negligible decrease in skin friction drag, 12% and 7%, was observed for the case of 10 m/s and 15 m/s respectively when compared with the smooth case and a sudden rise of 13% was observed at 20 m/s. From the velocity profiles, it is possible to evaluate the percentage variation between the momentum thickness of the perforated and smooth wall with the Reynolds number based on the cavity diameter.

The results reported in Figure 4 confirmed the findings of Gowree et al. [START_REF] Gowree | Skin friction drag reduction over staggered three dimensional cavities[END_REF] by consolidating the fact that the momentum thickness, which also represents the skin friction drag, is strongly correlated with the Reynolds number based on cavity diameter. This consolidates the initial arguments laid by Gowree et al. regarding the two distinct regime based on Re d , one being of drag reduction and the other of drag increment when compared with a smooth surface. From a further interpolation of the 2D grid measurements, the contour of the mean velocity and root mean square was obtained as reported in Figure 6 for the 15 m/s case. A distortion of the boundary layer is already evident just from the mean flow contour in Figure 6. This effect was accentuated in the U RMS fields where there spatial development of the flow appears to be oscillatory. This distortion is even more evident if we consider the evolution of the integral boundary layer quantities along the two rows of cavities. In particular, for the three free-stream test velocities, the evolution of the momentum thickness is reported in Figure 5 together with five terms Fourier series that approximate the trends.

The oscillations appear to be of smaller amplitude for the 20 m/s case but they start right downstream of the first cavity. For the 15 m/s and 10 m/s cases the oscillations start gradually with a relative peak that is slightly downstream respect the 20 m/s case. Both the 20 and 15 m/s cases have a sudden increase in the momentum thickness that could suggest a breakdown of the mean flow that does not seem to be present in the case of the 10 m/s case. These experimental results were very insightful in understanding the major modification that the mean flow undergoes in the vicinity of the leading cavities. The considerable distortion with increasing free-stream velocity or Re d is in line with the two regimes of Re d identified earlier, where the critical regime can be define as that experiencing an increase in momentum thickness, thus skin friction drag. Usually, a critical Reynolds number will be linked to a change of flow behaviour through a transition or breakdown process. The jump in the momentum thickness demonstrates this behaviour, however the restrictive nature of the experimental results cannot provide a larger picture of the breakdown process, hence complementary LES simulations was undertaken.

Numerical results

For the numerical results we mainly focused on the comparison between the 10 and 20 m/s cases. A first qualitative view of the structures produced by the cavities is presented in Figure 7. From the Q-criterion, the formation of semi-circular structures in correspondence to the first row of cavities is visible, but the the structures appear to increase in intensity for the 20 m/s case. When the flow encounters the circular cavity, a 3D effect is intrinsically produced. The flow enters inside the cavities and a self-sustaining recirculation region is created. The first consequence is a jetting at the downstream part of the cavities with a positive velocity normal to the wall out of the cavity. In Figure 8 the normal to wall velocity is reported for the 20 m/s case, the values and the topology for the 10 m/s differs only slightly. We can notice that the jetting is more pronounced for the cavity at the first row and it is attenuated in the second row.

The second phenomenon is the spillage of flow over the sides; this latter generates an alternating span-wise velocity pattern visible in Figure 9 which shows a slice at Y δ = 0.01 of the contour of the non dimensional span-wise mean velocity. This phenomenon is present for the three inflow conditions tested. The magnitude of this lateral velocity, evaluated at a fixed value of Y /δ , is of the order of 1% of the free-stream velocity. We identified a decreasing trend of the non-dimensional side velocity with the free-stream velocity and we estimated a decrease of 13% passing from 10 to 20 m/s.

In the Figure 10 a comparison of the magnitude of the total U RMS /U ∞ is presented. The scale of the contour for the 20 m/s case is ten times larger than the one of 10 m/s. We can then notice immediately that inside the cavity, for the 20 m/s case, the fluctuations are larger compared to the 10 m/s case. For both free-stream velocities we can notice an effect of the first cavity to the second one, in fact the U RMS is larger inside the cavity on the second row. In addition, for the 20 m/s case we can notice that the U RMS increases even in the region downstream the second cavity close to the wall.

The profiles taken experimentally on the smooth part right after the first and the second cavities confirm that this effect is indeed more pronounced for the 20 m/s case, Figure 11. We stress again here that the U RMS in the simulation is purely due to the presence of the cavity and not to the free-stream turbulence, as no synthetic turbulence was injected at the inlet. In Figure 12 a comparison of the U RMS of the streamwise component is reported for the cases 10 and 20 m/s. The contours show a slice from the top taken at Y /δ = 0.01. One can notice a rearrangement of the flow in the cavities both passing from 10 to 20 m/s and passing from the first to the second row. For the second row at 20 m/s the rearrangement is evident with the formation of three distinguishable lobes that suggest the presence of coherent structures inside the cavities.

On the experimental turbulent spectra at 20 m/s, downstream the second cavity, we noticed a sharp peak at a Strouhal number around Sr = 0.3 where Sr = f t U ∞ . Using a numerical probe at the same position we were able to retrieve a similar peak in the numerical spectra of the average stream-wise velocity component at Sr ∼ = 0.29, as it is visible in Figure 13. This would suggest we correctly captured a modulation of the mean flow due to the first rows of cavities.

Discussion

Based on the experimental results that confirmed Gowree's findings and hypothesis, two different regimes, depending on the Reynolds number could potentially be identified: the "drag benefit regime" in which the cavities seem to produce a beneficial effect for the skin friction and a regime in which the skin friction increases, the "drag penalty" regime. We aim here to provide an explanation of the possible mechanism that governs these two behaviours while relying on the flow topology that arises from the numerical results. We identified a span-wise velocity pattern between the cavities that matches quite well with the drag benefit explanation given by van Nesselrooij for staggered dimples. They postulated, in fact, that staggered dimples are able to create an alternating span-wise velocity pattern which is the cause of of a reduction in turbulence induced friction. They stressed the importance of the staggered configuration and the strong interaction between the dimples.

According to this hypothesis, the flow, when passing between the cavities in the inner layer, is subjected to an alternating span-wise velocity. It could then be confined in span-wise direction to follow a particular path, in other words the span-wise velocity would act as if it was a physical constraint. This mechanism could potentially reduce the span-wise movement and propagation of rolls and could weaken the sweep and the burst activity, responsible for turbulent kinetic energy production in the boundary layer. The final consequence of this process would be the reduction in skin friction found experimentally. A similar mechanism is indeed the one of the riblets, but what is described here has potentially a closer resemblance with drag reduction trough span-wise wall oscillation described by Karniadakis [START_REF] Karniadakis | Mechanisms on transverse motion in turbulent wall flows[END_REF] and Yakeno [START_REF] Yakeno | Modification of quasi-streamwise vortical structure in a drag-reduced turbulent channel flow with spanwise wall oscillation[END_REF]; the wall oscillation in fact generates a span-wise velocity that hinders the rolls and reduces the hairpin vortex activity.

According to van Nesselrooij the drag coefficient decreases with the Reynolds number while in our case we notice a clear increasing trend. This can be ascribed to the different geometry considered (shallow dimples versus circular cavities); in addition, in our case another mechanism can play an important role in counteracting the drag benefit and produce a drag increase.

On the other hand, in fact, the sudden increase of skin friction at 20 m/s can be explained as an effect of the breakdown of the turbulent structures created by the cavities. The Reynolds number dependence may suggest that the cause of this may be an instability mechanism generated by the cavities.

At this purpose it is very interesting to underline the similarities between our results and the results reported by Brès and Colonius [START_REF] Brès | Threedimensional instabilities in compressible flow over open cavities[END_REF]. They investigated square cavities of infinite span-wise dimension and different length to depth ratio d t . A stability analysis assessed the presence of a 3D centrifugal instability. For cavities with 1 < d t < 2 they found an unstable mode at a Strouhal number around 0.31. Even without performing a stability analysis we were able to assess both in the experimental and numerical results an oscillation characterized by similar values of Strouhal number. In addition to that, the arrangement of the flow in lobes that we identified in the contour of stream-wise U RMS have close resemblance with the structures inside the cavity described by Brès and Colonius which are an expression of the aforementioned instability.

For all the velocities we noticed a strong influence of the first cavity on the second one: the disturbances produced by the first cavity interfere with the second cavity and this leads to a partial rearrangement of the flow and an increase in turbulence activity. We may infer that this instability, in some cases, when the modes are amplified up to a certain level, is the cause of the occurrence of the aforementioned break of the structures. This can explain the drag increase found experimentally and the increase in U RMS shown before.

CONCLUSION & FURTHER WORKS

Within reasonable confidence, coupling experimental and numerical results, we demonstrated the presence of a distortion and oscillation of the mean flow field of a turbulent boundary layer caused by the presence of circular cavities in staggered arrangement. We presented here some evidence of an instability mechanism that could govern this oscillation and, in certain conditions, be responsible for the drag increase found experimentally.

To summarize, the intrinsically three-dimensionality of the case studied may generate a span-wise flow pattern and an instability mechanism. Finally the overall effect on drag will be a balance between these two effects. In some cases the flow results to be stable or the instability does not produce any modification in the turbulent structures and so the overall effect is beneficial for the skin friction; when the instability occurs it can overcome the effect of the span-wise flow enhancing the burst and leading to a skin friction increase.

A stability analysis will be necessary in order to provide more quantitative evidences of the instability mechanism that we think governs the drag increase for certain flow regimes; this should bring to a more precise evaluation of a critical Reynolds number. In addition, the application of methods such as proper orthogonal decomposition and dynamic mode decomposition should better reveal the arrangement of the structures generated by the cavities. Furthermore, with these techniques it should be possible to detect modes that contribute to the increase in turbulent kinetic energy production and how these affect the skin friction drag.

On parallel, more quantitative investigations are planned in order to generate further insight into the drag reduction mechanism and to confirm the hypothesis described here and reported in the literature.

Figure 1 :Figure 2 :

 12 Figure 1: Experimental setup, a) perforated surface model (in red the measurements locations), b) side view of the 2D grid.

Figure 3 :

 3 Figure 3: Friction coefficient for smooth and perforated model, comparison with Coles-Fernholtz relation valid for smooth walls.

Figure 4 :

 4 Figure 4: Momentum thickness percentage variation between perforated and smooth surface, measurements taken at the trailing edge of the model, comparison with Gowree et al. [9] and linear trend with Reynolds based on the diameter of the cavity.

Figure 5 :

 5 Figure 5: Percentage evolution of the momentum thickness along the first two rows of cavities for the three velocities tested.

Figure 6 :

 6 Figure 6: Distortion in the non dimensional mean velocity and U RMS field at 15 m/s, interpolation of hot wire measurements.

Figure 7 :

 7 Figure 7: Iso-surface of the Q-criterion, comparison of a) 10 m/s and b) 20 m/s case.

Figure 8 :Figure 9 :

 89 Figure 8: Jetting phenomenon, contour of U Y /U ∞ at 20 m/s, slice normal to the wall at Y /δ = 0.01.

Figure 10 :

 10 Figure 10: Magnitude of U RMS , comparison and a) 10 m/s and b) 20 m/s.

Figure 11 :Figure 12 :

 1112 Figure 11: Experimental U RMS of stream-wise velocity taken after the first and second cavity, comparison and a) 10 m/s and b) 20 m/s.

Figure 13 :Figure 14 :

 1314 Figure 13: Power Spectral Density for 20 m/s, a) LES simulation b) experiment; probes located downstream of the second cavity at Y /δ = 0.01.

ACKNOWLEDGEMENTS

The authors thank the AID (Agence Innovation Défense) for their financial support to the FriDA project. This work was performed using HPC resources from GENCI-IDRIS and GENCI-CINES on Jean Zay, Occigen (Grant A0082A07178) and CALMIP on Olympe (Grant 2020-p1425).


Alina Wildenberg

Yann Fenard
email: yann.fenard@pcfc.rwth-aachen.de

Maxime Carbonnier

Alan Kéromnès

Benoîte Lefort

Zeynep Serinyel

Guillaume Dayma

Luis Le Moyne

Philippe Dagaut

Karl Alexander Heufer

An experimental and kinetic modeling study on the oxidation of 1,3-dioxolane

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Humanity faces climate change. To address this problem a promising approach is to replace fossil fuels by biofuels obtained by second generation biomass. Within the cluster of excellence Fuel Science Center (FSC) [START_REF] Kiernan | Formaldehyde, formalin, paraformaldehyde and glutaraldehyde: what they are and what they do[END_REF] 1,3-dioxolane (structure see Fig. 1) was identified as a promising biofuel as its reactants ethylene glycol and formaldehyde can be yielded from biomass [START_REF] Li | Synthesis of 1,3-Dioxolane from Aqueous Formaldehyde Solution and Ethylene Glycol: Kinetics and Reactive Distillation[END_REF][START_REF] Wang | 2-Propylene Glycol and Ethylene Glycol Production from Lignocellulosic Biomass[END_REF][START_REF] Hader | Formaldehyde From Methanol[END_REF][START_REF] Demirbas | Biomethanol production from organic waste materials, Energy Sources, Part A Recover[END_REF]. Currently, the FSC is working on a catalytic pathway to produce dioxolane from CO 2 and ethylene glycol. Furthermore, 1,3-dioxolane has already shown soot reducing properties as a diesel additive [START_REF] Song | Comparison of the Impact of Intake Oxygen Enrichment and Fuel Oxygenation on Diesel Combustion and Emissions[END_REF]. To examine the capability of 1,3-dioxolane to be used as a biofuel its kinetics must be investigated under engine relevant conditions. So far the kinetics of the oxidation of 1,3-dioxolane by ȮH radicals was studied under atmospheric conditions by Sauer et al. [START_REF] Sauer | Atmospheric Chemistry of 1,3-Dioxolane: Kinetic, Mechanistic, and Modeling Study of OH Radical Initiated Oxidation[END_REF]. It was observed that the two possible radicals, 1,3-dioxolan-2-yl and 1,3-dioxolan-4-yl, are formed in equal amounts and then are reacting quickly with O 2 to form peroxyl radicals indicating a possible low temperature chemistry of combustion. Cutler et al. [START_REF] Cutler | Kinetics and mechanism of the vapor phase pyrolysis of 1,3-dioxolane in steam[END_REF] studied the pyrolysis of 1,3-dioxolane in steam at atmospheric pressures. Under these conditions they also observed the formation of two possible radicals but with the radical at the ethylene side being formed twice as much as the other radical at the methylene side (structure see Fig. 1). The authors concluded that this branching is due to the fact that the ethylene position has twice as many hydrogen atoms than the oxymethylene side. Cutler et al. further showed, that the main products of the pyrolysis are H 2 , CO, CO 2 , methane, ethane and ethylene [START_REF] Cutler | Kinetics and mechanism of the vapor phase pyrolysis of 1,3-dioxolane in steam[END_REF]. The thermal decomposition of 1,3-dioxolane was examined by Fernández-Sánchez et al. [START_REF] Fernández-Sánchez | de la Iglesia-Gayá, Radical-sensitized thermal decomposition of 1,3-dioxolane in the gas phase[END_REF] and Guenther et al. [START_REF] Guenther | The Thermal Decomposition of Dioxolane[END_REF] at 373-575 K and 455-525 K, respectively, and the photodecomposition was examined by B. C. Roquitte [START_REF] Roquitte | The photochemistry of 1,3-Dioxolane[END_REF]. All three examined that dioxolane mainly reacts to the pyrolysis products observed by Cutler et al. [START_REF] Cutler | Kinetics and mechanism of the vapor phase pyrolysis of 1,3-dioxolane in steam[END_REF]. The gas phase oxidation of 1,3-dioxolane under atmospheric pressure was examined by Molera et al. [START_REF] Molera | Gas-phase Oxidation of 1,3-Dioxolan[END_REF]. They also observed that 1,3-dioxolane reacts to two different radicals, whereby the radical at the energetically unfavorable ethylene side is formed more because it is statistically more likely. However, these studies do not address the combustion of 1,3-dioxolane under engine relevant conditions.

In this work a detailed kinetic model for the oxidation of 1,3-dioxolane is developed and validated against ignition delay times (IDT) and mole fraction profiles of the stable oxidation intermediates of dioxolane obtained in a jet stirred reactor (JSR). The model is used to get a deeper insight in the oxidation of dioxolane through sensitivity and reaction pathway analysis.

Experimental

The ignition delay time measurements were performed in a high pressure shock tube (ST) and in a rapid compression machine (RCM) for stoichiometric conditions at 20 and 40 bar. The ST in DRIVE, used to measure IDTs at high temperatures, was previously described in [START_REF] Merhubi | A high pressure experimental and numerical study of methane ignition[END_REF]. It consists of a 4 m-long driver section and a 5 mlong driven section, divided by a double diaphragm, and connected to the mixture preparation tanks by a manifold. To avoid fuel condensation, the complete system is heated up to 60 °C. In the ST the IDT is defined as the time interval between the arrival of the reflected shock at the end wall and the ignition event, defined as the maximum slope of the pressure increase. To detect the position of the shock wave and its velocity, four piezoelectric pressure transducers (PCB 113B22) are installed in the driven section. The impact of the shock wave and the ignition event are detected by a pressure transducer (Kistler 603B1) at the endwall. The temperature behind the reflected shock wave (T 5 ) was calculated with the help of the chemical equilibrium software Gaseq [14]. Uncertainties in reflected shock temperatures are estimated to ±1 % (±10-15 K) [START_REF] Merhubi | A high pressure experimental and numerical study of methane ignition[END_REF]. For both pressures a mixture with 2 % fuel was prepared using a diluent/O 2 ratio of 13 with argon as bath gas. The used gases had a purity higher than 99.995 % and 1,3-dioxolane, purchased from Sigma Aldrich, had a purity of 99.8 %. Intermediate and low temperature conditions were measured in the RCM at PCFC, RWTH Aachen University, which was previously described in [START_REF] Lee | On the Chemical Kinetics of Ethanol Oxidation : Shock Tube , Rapid Compression Machine and Detailed Modeling Study[END_REF]. It consists of a reaction chamber, a hydraulic section and a pneumatically driven piston and is connected by a manifold to the mixture preparation vessel. The reaction chamber is equipped with a dynamic pressure sensor (Kistler 6125C-U20) in order to measure the pressure during the experiment. Uncertainties are estimated to ±0.15 bar and ± 5 K on the compressed conditions [START_REF] Ramalingam | An RCM experimental and modeling study on CH 4 and CH 4 / C 2 H 6 oxidation at pressures up to 160 bar[END_REF]. The RCM used in this study is equipped with an adjustable endwall in order to vary the compression ratio. In the RCM the IDT is defined as the time between the end of compression and the ignition of the fuel, defined as the maximum increase in pressure. Measurements in the RCM were done in diluted conditions with a diluent ratio of 7.52 using argon 4.6 and nitrogen 5.0 as bath bas and oxygen 5.0, all purchased from Westfalen. 1,3-Dioxolane was purchased from Sigma-Aldrich with a purity of 99.8 %. Measurements in non-diluted conditions were not possible as the ignition delay times were getting shorter than 2 ms, the lower limit measurable in the RCM.

Experiments were also performed in a fused silica jet-stirred reactor (JSR) already described in details and used in previous studies [START_REF] Dagaut | A jet-stirred reactor for kinetic studies of homogeneous gas-phase reactions at pressures up to ten atmospheres (∼1 MPa)[END_REF]. The reactor itself is surrounded by a regulated electrical oven and can be heated up to 1250 K. It is located inside a pressure-resistant jacket, allowing the pressure to rise up to 10 atm. The liquid fuel (CAS 4362-13-4, 99.8 % purity from Sigma Aldrich) was atomized by a nitrogen flow and vaporized in a heated chamber. The fuel+N 2 mixture is carried to the reactor by a heated quartz capillary. The oxidizing stream (O 2 +N 2 ) flowed separately to avoid oxidation before reaching the 4 injectors providing stirring. Gaseous flowrates are regulated by thermal mass flow controllers (Brooks 5850E) whereas the liquid fuel is injected by an HPLC pump (Shimadzu LC10 AD VP) with an online degasser (Shimadzu DGU-20 A3). A low-pressure sonic probe was used to freeze the reactions and take samples for online and off-line analyses. Samples were analyzed online by FTIR or stored at low-pressure for offline GC analysis. The carbon balance was checked after each experiment and was found to be within ±10 %. Uncertainties on the species mole fractions are due to several factors described in [START_REF] Serinyel | Kinetics of oxidation of cyclohexanone in a jet-stirred reactor: Experimental and modeling[END_REF] and are estimated to be around ±15 %. All experiments were performed at 10 atm between 700 and 1180 K after checking at that no low-temperature reactivity was observed between 500 and 700 K.

Theoretical calculations

The thermochemical data of some species, important to model the oxidation of 1,3-dioxolane, were calculated. Species calculated are the 1,3-dioxolan (RH) and the two radicals 1,3-dioxolan-2-yl and 1,3dioxolan-4-yl (Ṙ2 and Ṙ4), the β-scission products of the radicals 1,3-dioxolan-2-yl and 1,3-dioxolan-4-yl, (O=CCOĊ, C=COCȮ and O=COCĊ) and the dioxolanyl peroxyl radicals (R2Ȯ 2 and R4Ȯ 2 ). The thermodynamic data of the peroxyl radicals are particularly important as the addition to oxygen Ṙ + O 2 ⇌ RȮ 2 relies on the thermodynamic data of the reactants and products. Moreover, the reaction rate coefficients of the β-scission reactions of the radicals 1,3-dioxolan-2-yl and 1,3-dioxolan-4-yl were calculated. The geometries, frequencies and one dimensional hindered rotors were calculated with 10 degree resolution using the method B3LYP/6-31G(2df,p). The G4 combined method implemented in Gaussian 09 [19] using the atomization model was employed to calculate the standard heat of formation as recommended by Somers and Simmie [START_REF] Somers | Benchmarking Compound Methods (CBS-QB3, CBS-APNO, G3, G4, W1BD) against the Active Thermochemical Tables: Formation Enthalpies of Radicals[END_REF]. Comparing the calculated standard heat of formation for 1,3-dioxolane of 71.08 kcal/mol to the measured one of 72.10 kcal/mol [START_REF] Pihlaja | Enthalpies of Formation of Cyclic Acetals. 1,3-Dioxolane, 2-Methyl-1,3-Dioxolane, and 2,4-Dimethyl-1,3-Dioxolanes[END_REF] only a small deviation is visible justifying the usage of the mentioned method. The method B3LYP/6-31G(2df,p) was used for the statistical thermodynamic. The thermodynamic properties were calculated with the python package TAMkin, detailed by Burke et al. [START_REF] Burke | A comprehensive experimental and kinetic modeling study of butanone[END_REF]. Other thermodynamic data of species, specific to the submechanism of 1,3-dioxolane, were estimated with RMG. The standard heat of formation was calculated from the enthalpies calculated with the G4 method. The MESS software was used [START_REF] Georgievskii | Reformulation and solution of the master equation for multiple-well chemical reactions[END_REF] to calculate the reaction rate parameters of the ring opening reactions of the radicals 1,3-dioxolan-2-yl and 1,3-dioxolan-4-yl. Rotational enantiomers where included when necessary. The collisional energy transfer was modeled with nitrogen as bath gas via 〈∆𝐸 〉 200 cm • 𝑇 300 ⁄ K . , where commonly chosen parameters were used [START_REF] Döntgen | Reactions of Chemically Activated Formic Acid Formed via HCO + OH[END_REF]. The Lennard-Jones parameters for nitrogen σ = 3.681 Å and ε = 67.89 cm -1 were used as recommended by Xu et al. [START_REF] Xu | Temperature and Pressure Dependent Rate Coefficients for the Reaction of Ketene with Hydroxyl Radical[END_REF]. The Lennard-Jones parameters for the dioxolanyl radicals were taken from the RMG database [START_REF] Gao | Reaction Mechanism Generator: Automatic construction of chemical kinetic mechanisms[END_REF]. The reaction rate coefficients were calculated using the zero-curvature Eckart tunneling and then fitted with the modified Arrhenius equation k T A•T n •exp E a RT within a temperature range from 500 to 2200 K. The pressure dependence was calculated for all computed reactions. For the βscission of 1,3-dioxolane-2-yl the degeneracy was taken into account by multiplying the A-factor by 2. The rates are plotted in the supplemental material (Fig. S. [START_REF] Demirbas | Biomethanol production from organic waste materials, Energy Sources, Part A Recover[END_REF][START_REF] Song | Comparison of the Impact of Intake Oxygen Enrichment and Fuel Oxygenation on Diesel Combustion and Emissions[END_REF][START_REF] Sauer | Atmospheric Chemistry of 1,3-Dioxolane: Kinetic, Mechanistic, and Modeling Study of OH Radical Initiated Oxidation[END_REF].

The enthalpies at 0 K were obtained from the G4 combined method and used to calculate the bond dissociation energies of all C-H bonds, C-O-bond and the C-C bond of 1,3-dioxolane. The enthalpy of dioxolane was subtracted from the enthalpies of the resulting radicals after the dissociation of the bond. The bond dissociation energies of the C-H-bond at the methylene side of the 1,3-dioxolane is 1.8 kcal/mol lower compared to the similar pattern found in dimethoxymethane (DMM) [START_REF] Luo | Handbook of bond dissociation energies in organic compounds[END_REF]. The O-C-O bond angle in 1,3dioxolane is 120° same as the corresponding dioxolanyl radical. Therefore, it does not need to rearrange when forming the radical. In contrast, the O-C-O angle changes from 109° to 120° for a linear structure when forming a radical, explaining the lower BDE of dioxolane compared to DMM. Also, the C-O-bond dissociation energy of dioxolane is lower than for DMM and can be attributed to the ring strain in dioxolane. As shown in Fig. 1, the C-H-bond of the ethylene group has a very similar bond dissociation energy to tetrahydrofuran (THF) [START_REF] Fenard | A model of tetrahydrofuran low-temperature oxidation based on theoretically calculated rate constants[END_REF], which is also a five member ring. In addition, Fig. 1 shows the numbering system used in this paper for naming the species in the model. The carbon atom of the methylene site is referred to as carbon 2. The dioxolanyl radical with the radical side at this position is therefore called 1,3-dioxolan-2-yl (Ṙ2). The species Q 4OOH5 for example has a radical on carbon 4 and a hydroperoxyl group on carbon 5.

Kinetic modeling

The model development in this work is based upon AramcoMech 3.0 [START_REF] Zhou | An experimental and chemical kinetic modeling study of 1,3-butadiene combustion: Ignition delay time and laminar flame speed measurements[END_REF] as base mechanism. The reaction classes for 1,3-dioxolane are derived from the work of Curran et al. [START_REF] Curran | A Comprehensive Modeling Study of n-Heptane Oxidation[END_REF].

For the unimolecular decomposition reactions analogies to a five membered cyclic ether [START_REF] Bruycker | Experimental and modeling study of the pyrolysis and combustion of 2-methyl-tetrahydrofuran[END_REF] are assumed. The rate parameters for H-atom abstraction reaction by Ḣ, ĊH 3 and Ċ 2 H 5 radicals were chosen similar to the case of diethoxy methylene [START_REF] Kröger | Ab initio kinetics predictions for H-atom abstraction from diethoxymethane by hydrogen, methyl, and ethyl radicals and the subsequent unimolecular reactions[END_REF]. The rate parameters for H-atom abstraction reactions by ȮH radical were deduced from the work of Dagaut et al. [START_REF] Dagaut | Flash photolysis resonance fluorescence investigation of the gasphase reactions of hydroxyl radicals with cyclic ethers[END_REF] on 1,4-dioxane. The reaction rate constants for 1,3-dioxolane + HȮ 2 ⇌ 1,3-dioxolan-2-yl + H 2 O 2 are taken similar as in the case of dimethoxy methylene divided by a factor of two and for 1,3-dioxolane + HO 2 ⇌ 1,3-dioxolan-4-yl + H 2 O 2 , the rate parameters from di ethyl ether calculated by Mendes et al. [START_REF] Mendes | Rate constant calculations of H-atom abstraction reactions from ethers by HȮ 2 radicals[END_REF] were used. The H-atom abstraction by CH 3 Ȯ 2 is treated as for HȮ 2 , however reduced by a factor of 1.5 to take into account the stabilization effect [START_REF] Curran | A Comprehensive Modeling Study of n-Heptane Oxidation[END_REF]. The rate parameters for the ring opening reactions of the radicals 1,3-dioxolan-2-yl and 1,3-dioxolane-4-yl calculated in this work are added to the model and also used to describe the β-scission reactions of the Q OOH radicals, with an activation energy reduced by 2 kcal/mol to take into account the influence of the hydroperoxyl group.

In all other reaction classes, the reactions concerning carbon in position 2 of 1,3-dioxolane (Fig. 1) were treated as for the methylene carbon of dimethoxy methane. Despite Fig. 1 showing a difference of 1.8 kcal/mol between the BDE of 1,3-dioxolane and DMM this difference was not considered for the reaction rates deduced from DMM as the Evans-Polanyi plot for different similar molecule showed no clear trend (Fig.

S.1).

For the carbon in position 4 and 5 (Fig. 1) the reaction parameters were determined in analogy to the rate coefficients of H-atom separation on the primary radical position of dimethoxy methane [START_REF] Jacobs | Detailed kinetic modeling of dimethoxymethane . Part II : Experimental and theoretical study of the kinetics and reaction mechanism[END_REF]. The rates from dimethoxy methane were corrected, applying the ratio between secondary and primary position in alkane systems [START_REF] Zhang | An updated experimental and kinetic modeling study of n-heptane oxidation[END_REF], therefore correspond to diethoxy methane, which is directly comparable to dioxolane. The reaction classes concerned are the bimolecular initiations, the H-atom abstraction by Ö and CH 3 Ȯ, the additions of radicals 1,3-dioxolan-2-yl and 1,3-dioxolan-4-yl on O 2 , the internal H-atom migration RȮ 2 ⇌ Q OOH, the concerted elimination RȮ 2 ⇌ unsaturated counterpart + HȮ 2 , the HȮ 2 elimination from Q OOH, the cyclic ether formation, the ketohydroperoxide (KHP) formation ȮOQOOH ⇌ KHP + ȮH and the decomposition of the KHP to products and a hydroxyl radical.

It is worth mentioning that only the formation of the bi-cyclic species 4,5-epoxy-1,3-dioxolane is considered as 4-hydroperoxy-1,3-dioxolan-5-yl is expected to behave similar to alkanes. 2-hydroperoxy-1,3-dioxolan-4-yl and 4-hydroperoxy-1,3-dioxolan-2-yl are expected to behave like THF [START_REF] Fenard | A model of tetrahydrofuran low-temperature oxidation based on theoretically calculated rate constants[END_REF] where no cyclic ether formation was observed as this reaction cannot compete with the other branching reactions of the hydroperoxy species Q OOH. Overall, the β-scission reaction of the Q OOH radical is widely favored. The detailed kinetic model and the thermodynamic data are given in the supplementary materials.

Results and discussion

Ignition delay times

Figure 2 presents the measured and simulated ignition delay times. For the simulations of the ignition delay times the kinetic solver Cantera was used [START_REF] Goodwin | Cantera: An Object-oriented Software Toolkit for Chemical Kinetics, Thermodynamics, and Transport Processes[END_REF]. In shock tube conditions constant volume simulations were performed. For the RCM experiments the heat loss was considered using effective volume profiles [START_REF] Sung | Using rapid compression machines for chemical kinetics studies[END_REF]. The ignition delay times simulated with the kinetic model lay within the uncertainty of the experimentally measured values. At high temperatures, IDTs measured in the ST show an Arrhenius behavior. At low to intermediate temperatures, IDTs show a weak negative temperature coefficient behavior, which is more pronounced at 40 bar than at 20 bar as commonly observed for alkanes for instance. In order to have a deeper insight into the reaction pathways influencing the ignition delay times the most, a brute force sensitivity analysis was performed at 20 bar and an equivalence ratio of 1.0 for 1200 K at ST conditions and at 710 K at RCM conditions (Fig. 3). The sensitivity coefficients are defined as S= log τ i + τ i -⁄ log 2.0 0.5 ⁄ ⁄ with 𝜏 being the ignition delay time computed with the reaction rate constant k i multiplied by a factor of two and 𝜏 the ignition delay time corresponding to k i divided by 2. If a reaction is denoted with a positive sensitivity coefficient, an increase in the rate of this reaction leads to an increase in the IDTs (i.e. decrease in reactivity) and a negative coefficient depicts a decrease in the ignition delay time (i.e increase in reactivity). The sensitivity analysis shows for both temperatures that the H-atom abstraction by ȮH and HȮ 2 forming 1,3-dioxolan-2-yl and the H-atom abstraction by HȮ 2 forming 1,3-dioxolan-4-yl increase the reactivity as they consume the 1,3-dioxolane the most. Additionally, HȮ 2 is reacting to produce hydrogen peroxide at 710 K that can undergo an O-O bond dissociation with increasing temperature giving two ȮH radicals that react with the fuel which ultimately triggers the ignition. At 1200 K the hydrogen peroxide immediately decomposes into two ȮH radicals promoting the reactivity by chain branching reaction. The formation of 1,3-dioxolan-4-yl by H-atom abstraction by an ȮH radical decreases reactivity, because it consumes a very reactive ȮH radical while the decomposition of 1,3-dioxolan-4-yl does not produce reactive radicals. The decomposition of 1,3-dioxolan-2yl meanwhile gives ȮCHO that reacts to yield Ḣ and CO 2 with Ḣ being a chain branching agent. The H-atom abstraction by Ḣ at both sides decreases the reactivity at high temperatures. At high temperatures the Ḣ is responsible for the chain branching reaction Ḣ + O 2 ⇌ Ȯ + ȮH increasing reactivity as it forms two active radicals. When reacting with 1,3-dioxolane at high temperature, the H radical is not available for this chain branching reaction anymore and thereby decreases the overall reactivity although it consumes the 1,3dioxolane. In ST conditions (1000 K to 1300 K) the β-scission reaction forming C 2 H 4 and ȮCHO is increasing the reactivity while the isomerization from ethyl-2yl formate (O=COCĊ) to ethyl formyl (O=ĊOCC) has the opposite effect. The decomposition of ȮCHO gives CO 2 and a Ḣ while ethyl formyl decomposes to CO or CO2, which are stable and unreactive, and Ċ 2 H 5 or C 2 H 5 Ȯ that are less reactive than a hydrogen radical, which is responsible for the chain branching reaction with O 2 . At a lower temperatures of 720 K the most sensitive reaction on the IDTs increasing reactivity is the addition of 1,3-dioxolan-2-yl to O 2 as it is the first step leading to the low temperature chain branching process [START_REF] Battin-Leclerc | Detailed chemical kinetic models for the low-temperature combustion of hydrocarbons with application to gasoline and diesel fuel surrogates[END_REF]. The reaction competing with this low temperature chain branching is the ring opening reaction of 1,3-dioxolan-2-yl to ethyl format-5-yl (O=COCĊ) which, consequently, decreases the reactivity. The β-scission of the radical 4-hydroperoxy-1,3-dioxolan-2-yl (Q 2OOH4) forming the stable intermediate 2-oxoethyl formate (O=COCC=O) is decreasing reactivity as it also stands in competition to low temperature branching. In order to investigate the pathways consuming the 1,3-dioxolane in the low temperature region a reaction pathway analysis was done at 720 K for 20 bar and an equivalence ratio of 1.0 (Fig. 4 and Fig. 5). The reaction pathway analysis shows that the 1,3-dioxolan-2-yl radical is formed faster than the 1,3-dioxolan-4-yl radical. It is attributed to the lower bond dissociation energy of the C-H bond at the methylene side compared to the C-H bond at the ethylene side. Under these conditions the ratio is roughly 3/2 in favor of 1,3dioxolan-2-yl. The 1,3-dioxolan-4-yl radical then mainly reacts via addition to O 2 forming the peroxyl radical 4peroxly-2,4-dioxolane (R4Ȯ 2 ) while the 1,3-dioxolan-2-yl radical mainly decomposes via β-scission (ring opening reaction) instead of following the low temperature chain branching pathway. The reaction enthalpy for the β-scission of 1,3-dioxoalen-4-yl is positive and can therefore not compete against the addition to O 2 . The β-scission of 1,3-dioxoalen-2-yl is, as the addition to O 2 , an exothermic reaction and takes place quickly (Fig.S.2). As the branching is determined by the unimolecular reaction the addition to O 2 is suppressed nearly entirely. This observation also stands for the Q OOH species for the second addition to O 2 . The 4-hydroperoxy-1,3-dioxolan-2-yl (Q 2OOH4) mainly reacts via β-scission forming 2-oxoethyl formate while the 2-hydroperoxy-1,3-dioxolan-4-yl (Q 4OOH2) mainly reacts by addition to O 2 . The β-scission of Q OOH species shows a negligible effect in the low temperature chemistry of linear alkanes [START_REF] Bugler | An ignition delay time and chemical kinetic modeling study of the pentane isomers[END_REF]. However, this class of reaction is sensitive on the reactivity in the case of 1,3-dioxolane the same way it was observed for tetrahydrofuran [START_REF] Fenard | A model of tetrahydrofuran low-temperature oxidation based on theoretically calculated rate constants[END_REF]. The reaction pathway analysis for 1,3-dioxolan-4-yl shows that 4-peroxly-1,3-dioxolane (R4Ȯ 2 ) only isomerizes to 4hydroperoxy-1,3-dioxolan-2-yl(Q 2OOH4) as a 6-membered transition in the H-atom migration is favored compared to a 5-membered transition due to lower ring-strain.

The reaction pathway analysis shows that the two oxygen in the ring structure impede the addition to O 2 if both oxygens are in α position to the radical position. In this case also the equilibrium of the Q OOH species of dioxolane is shifted to the β-scission reaction and not to the olefin formation as for alkanes.

Mole fraction profiles

Figure 6 presents the measured and simulated mole fraction profiles of stable intermediates produced during the oxidation of 1,3-dioxolane for an equivalence ratio of ϕ = 1.0. The JSR simulations were performed with Cantera under the assumption of a perfectly stirred reactor. The most abundant species are the 1,3-dioxolane itself, carbon monoxide, formaldehyde, methane, hydrogen, ethylene, carbon dioxide and water. Ethane and acetaldehyde were detected in low amounts. All these species are formed through the high temperature reaction pathways. No specific low temperature species of 1,3dioxolane were observed during the JSR experiments. As an illustration a reaction pathway analysis was performed at 10 bar and 850 K (Fig. 7). The radicals of 1,3-dioxolane decompose only via β-scission and no low temperature branching is observed. The high dilution of the reactant in nitrogen in the JSR experiments quenches the low temperature chain branching as no cool flame was observed. Ethylene and formaldehyde are mainly produced by β-scission of the radicals 1,3-dioxolan-2-yl or 1,3-dioxolan-4-yl, respectively. The experiments show that ethylene is formed approximately twice as much as formaldehyde. This is also represented by the model. The reaction pathway analysis shows that 61 % of the fuel reacts to the 1,3-dioxolan-2-yl radical while 37 % react to the 1,3-dioxolan-4-yl radical similar to the undiluted case discussed before. As the predicted mole fraction profile of ethylene and formaldehyde lay within the uncertainty bars of the experiments the formation ratio of 1,3-dioxolan-2-yl and 1,3-dioxolan-4-yl by H-atom abstraction seems to be correct. The main reaction partner in the H-atom abstraction is the hydroxyl radical forming water. The simulated mole fraction profile of water is in good agreement with the experimental data for temperatures under 1000 K. A comparable behavior is observed for H 2 . Until 1000 K the mole fraction profile is within the error bars whereas at higher temperatures the amount of hydrogen is underpredicted by the model. Both water and hydrogen are mainly formed from either dioxolane or formaldehyde by H-atom abstraction via Ḣ or ȮH. Formaldehyde can also react via H-atom abstraction by ĊH 3 forming CH 4 and HĊO. This is the second most important reaction pathway leading to methane. As depicted in Fig. 6, the peak mole fraction of methane is predicted at lower temperatures than measured experimentally. The ĊH 3 reacting to CH 4 is mainly formed by the decomposition of ethoxy radicals (C 2 H 5 Ȯ) to CH 2 O and ĊH 3 . C 2 H 5 Ȯ comes from the decomposition of ethyl formyl (O=ĊOCC) to which the ethyl-2-yl formate (O=COCĊ) can isomerize. A possible reason for the discrepancy between the current model and the measured results could be that the decomposition of C 2 H 5 O to CH 2 O and ĊH 3 is too fast and it should mainly decompose to CH3CHO and Ḣ.

Conclusion

The ignition delay times of stoichiometric 1,3-dioxolane/O 2 /inert mixtures were measured in a rapid compression machine and shock tube at pressures of 20 and 40 bar covering a temperature range of 630 K to 1300 K. Furthermore, the mole fraction profiles of the stable intermediates produced during the oxidation of 1,3-dioxolane were measured in a jet-stirred reactor for 10 atm at ϕ = 1.0. The most abundant stable intermediates measured during the oxidation of 1,3-dioxolane in a jet-stirred reactor are ethylene, formaldehyde, CO, CO 2 , water and methane. In the low temperature range the influence of the low temperature chemistry is observed on the ignition delay times whereas it is not observed on the mole fraction profiles in the highly diluted conditions of the jet-stirred reactor. A detailed kinetic mechanism for the oxidation of 1,3-dioxolane was developed in this work and validated against ignition delay times and mole fraction profiles. The model showed that the 1,3-dioxolane mainly yields the two primary radicals 1,3dioxolan-2-yl and 1,3-dioxolan-4-yl, whereby the 1,3-dioxolan-2-yl radical is formed faster for all investigated conditions. This is caused by the difference in bond dissociation energies for the R2-H (91.1 kcal/mol) and R4-H (92.9 kcal/mol) sides. The calculated C-H-bond dissociation energies are similar to the ones from tetrahydrofuran in α position to the oxygen (93.8 kcal/mol [START_REF] Fenard | A model of tetrahydrofuran low-temperature oxidation based on theoretically calculated rate constants[END_REF]) and dimethoxy methane (92.9 kcal/mol [START_REF] Luo | Handbook of bond dissociation energies in organic compounds[END_REF]) at the methylene side. Comparing the low temperature chemistry behavior of the two primary radicals 1,3dioxolan-2-yl and 1,3-dioxolan-4-yl the addition on O 2 is not efficient if two oxygens are in α position to the carbon because of the energetically hindrance of the two oxygen. Moreover, it was observed that the Q OOH species quickly undergo a β-scission inhibiting the reaction to olefins + HȮ 2 and cyclic ethers + ȮH. This behavior differs from alkanes and explains the almost temperature independent ignition delay times in the negative temperature coefficient range.

Fig. 1 :

 1 Fig. 1: Calculated bond dissociation energies for 1,3-dioxolane compared to bond dissociation energies from tetrahydrofuran and dimethoxy methane given in kcal/mol shown in black. The numbering of carbon atoms is shown in grey, respectively.

Fig. 2 :

 2 Fig. 2: Simulated (lines) and measured (symbols) ignition delay times for 20 and 40 bar for ϕ = 1.0. The ST measurements were carried out for 2 mol-% fuel with a dilution Ar/O 2 of 13. The RCM experiments were performed for a dilution inert gas/O 2 of 7.52.

Fig. 3 :

 3 Fig. 3: Brute force sensitivity analysis at ϕ = 1.0 and 20 bar for ST conditions at 1200 K (left) and for RCM conditions at 710 K (right).

Fig. 4 :

 4 Fig. 4: Reaction pathway analysis at 720 K, 20 bar and ϕ = 1.0 performed in a constant volume ideal gas reactor for the radical 1,3dioxolan-2-yl (Ṙ2) with a diluent/O 2 ratio of 7.52 at 10 % fuel consumption.

Fig. 5 :

 5 Fig. 5: Reaction pathway analysis at 720 K, 20 bar and ϕ = 1.0 performed in a constant volume ideal gas reactor for the radical 1,3dioxolan-4-yl (Ṙ4) with a diluent/O 2 ratio of 7.52 at 10 % fuel consumption.

Fig. 6 :

 6 Fig. 6: Simulated (lines) and experimental (symbols) mole fraction profiles for the oxidation of 1,3-dioxolane/O 2 /N 2 mixtures in a JSR at 10 atm for ϕ = 1.0 and τ = 700 ms.

Fig. 7 :

 7 Fig. 7: Reaction pathway analysis for 1,3-dioxolane for 10 bar at 850 K in a JSR.

Acknowledgements

This work at PCFC was performed as part of the Cluster of Excellence "The Fuel Science Center" (EXC 2186), which is funded by the Excellence Initiative by the German federal and state governments to promote science and research at German universities. The work in DRIVE was supported by Bourgogne Council under the PARI2 program and Labex Caprysses.