I Diniz

E Dumur

O Buisson

A Auffèves

Ultrafast QND measurements based on diamond-shape artificial atom

Keywords:

We propose a Quantum Non Demolition (QND) read-out scheme for a superconducting artificial atom coupled to a resonator in a circuit QED architecture, for which we estimate a very high measurement fidelity without Purcell effect limitations. The device consists of two transmons coupled by a large inductance, giving rise to a diamond-shape artificial atom with a logical qubit and an ancilla qubit interacting through a cross-Kerr like term. The ancilla is strongly coupled to a transmission line resonator. Depending on the qubit state, the ancilla is resonantly or dispersively coupled to the resonator, leading to a large contrast in the transmitted microwave signal amplitude. This original method can be implemented with state of the art Josephson parametric amplifier, leading to QND measurements in a few tens of nanoseconds with fidelity as large as 99.9%.

Superconducting circuits have demonstrated in the last decade their high ability to perform coherent quantum experiments [1,2]. Relaxation T 1 and coherence T 2 times are continuously increasing [3]. In addition, these quantum systems benefit from very strong coupling with the electromagnetic field, and potential scalability. Finally the circuit parameters that define the quantum dynamics are tunable and adjustable on demand, which makes them very promising candidates to process quantum information on chip. In this framework, the ability to perform ultrafast single shot read-out of a quantum bit is highly desirable. Up to now, high one shot fidelity was obtained by switching quantum measurements using escape process [4], the intrinsic drawback of this method being its destructiveness. Quantum Non Demolition (QND) measurements are performed by coupling the qubit dispersively to a resonator [5]. The qubit acts as a state-dependent refractive index that shifts the cavity frequency, and the measurement is performed by probing the resonator with an external microwave. QND character is preserved as long as one remains in the dispersive regime, keeping the photon population n of the resonator below a critical value [6], and also limiting the incident power. Low temperature amplifiers have thus to be used to reach high fidelity. On the other hand, using non-linear resonator and bifurcation [7,8] or Jaynes-Cummings non linearity [9] allows to reach one shot high fidelity read-out, but at the price of lower QND fidelity. Thanks to recent advances in parametric amplification using Josephson junction circuits [10][11][12], single shot read-out has been demonstrated, allowing to observe quantum jumps in superconducting artificial atoms [13], high fidelity read-out [14][15][16] and everpersisting Rabi oscillations [17]. However this measurement scheme still requires several hundred nanoseconds measurement time to reach high fidelity. Consequently, further improvements are necessary in order to reach very high fidelity measurements in a few ten's of nanoseconds. New quantum measurement protocols inspired by ion traps and quantum optics were recently proposed with this purpose [18,19].

Here we propose an original method to realize ultrafast QND measurements of a qubit with large resonator linewidth and measurement bandwidth, while preserving high fidelity. Our system is a resonator coupled to a four level diamond-shape atom, which can be seen as two qubits coupled by crossed Kerr interaction. In this picture, the first qubit is the one we read out, while the second qubit plays the role of an ancilla whose frequency depends on the first qubit state. The resonator is not coupled to the qubit but only to the ancilla. This huge difference with respect to previous experiments induces important consequences on the qubit and on the resonator properties. First, Purcell effect between the qubit and the cavity is absent, and the readout performance is independent from the detuning between the qubit and the resonator. Second, the present proposal allows to eliminate the harsh constraint on the amplification and resonator bandwidth, allowing to reach fast, one-shot, high fidelity QND read-out of our qubit even with the present day amplifier technology.

We exemplify hereafter this method with a diamondshape artificial atom consisting of two transmons coupled by a large inductance. The system under study here is pictured in Fig. 1, in which the resonator is schematized by a coplanar wave guide resonator, but our result can be applied to 3D cavities or lumped element resonators. The Josephson energy E J = Φ 0 I c /2π and the charging energy E c = e 2 /2C, with I c and C the critical current and the capacitance per junction, are fixed to verify the typical ratio E J /E c ≈ 50 of the transmons limiting the decoherence effects [20]. Such devices can be described by an anharmonic oscillator with two degrees of freedom. Its quantum description is the same than the one performed in Ref. [21] for a dc SQUID in the particular case of zerocurrent bias or in Ref. [22,23] for two transmons coupled by a SQUID. It gives rise to two orthogonal modes, the symmetric and antisymmetric mode [21]. When the an- harmonicity is strong enough, we can consider just the first two levels of the two modes. The system is then reduced to two coupled two-level systems. The two first quantum states of the symmetric mode, |g and |e , provide the logical qubit σ qb z . The second two level system corresponds to the two first quantum states of the antisymmetric mode, and it will be used as an ancilla σ a z for the quantum measurement. In the absence of coupling between the two modes, the transition frequencies of the qubit and ancilla are given by of ω qb and ω a , respectively. Hereafter we restrict our study to the working point given by zero-flux bias which constitutes an optimal point for the artificial atom. The coupling between the two systems is reduced to a longitudinal interaction term σ qb z σ a z with a strength given by g zz = E c /( 1 + 2E L /E J ) [24] where

E L = (Φ 0 /2π) 2 /L with L the SQUID inductance.
This term can be viewed as an analogue of a cross-Kerr term between the two quantum systems, leading to a conditional energy transition of the ancilla which depends on the quantum state of the qubit |g and |e . The respective frequencies of the transitions |g → |m and |e → |p are (ω a + g zz ) and (ω ag zz ). The artificial atom inside a coplanar resonator is described by the following Hamiltonian, written in the rotating wave approximation:

H f ree = ω qb σ qb z /2 + (ω a -g zz σ qb z )σ a z /2 + ω r (a † a + 1/2) -i g a (aσ a + -a † σ a -) (1)
The first three terms describe the artificial atom and the fourth describes the resonator of frequency ω r . In the following we choose the frequency condition between the resonator and the ancilla: ω r = ω a + g zz . The last term couples the resonator and the ancilla when the artificial atom is localized at the center of the resonator. Indeed, at this particular place, the quantum fluctuation of the flux is maximal and the voltage fluctuations are reduced to zero. Because of zero-flux bias working point, the qubit is not affected by flux fluctuations, leading to zerocoupling between the resonator and the qubit. This way, σ qb z commutes with the Hamiltonian of the system, ensuring the non-destructive character of the measurement whatever the number of photons in the resonator.

To describe the transmission properties of the cavity as a function of the qubit state, we write a closed set of differential equations from Eq. ( 1) describing the time evolution of the system operators in the Heisenberg picture. These are deduced from input-output equations established in the case of a transmitting cavity as in [25]. We define the external fields b in (injected microwave field), b r (reflected field), and b t (transmitted field) that lead to the usual input-output equations: b r = b in + i √ κa and b t = i √ κa where κ is the resonator coupling to external transmission line modes. As we consider an overcoupled cavity, we neglect the internal losses of the resonator and thus κ entirely defines the resonator linewidth. The qubit energy relaxation and dephasing times are assumed to be very long compared to the resonator relaxation time (κT 1 ≫ 1 and κT 2 ≫ 1). The Heisenberg equations are written in the frame rotating at the frequency ω of the probe, yielding

σa z = -2g a (σ a + a + σ a -a † ) , σa -= -i(ω r -ω + δ j )σ a -+ g a σ a z a , σ qb -= -i(ω qb -g zz σ a z )σ qb -, ȧ = -i(ω r -ω)a -κa + g a σ a -+ i √ κb in , (2)
where δ j = -g zz (1 + σ qb z ) is the qubit state dependent shift, and the index j defines the qubit state (j = g or e). As expected from a QND measurement, the evolution preserves σ qb z . We are interested in the transmission properties of this system in the steady state regime established after a time much larger than 1/κ. We adopt the semiclassical approach where the quantum correlations between atomic and field operators are neglected [25]. From now on we identify the operators with their average complex values, which could be measured in a homodyne experiment. The ratio t(ω) = b t / b in can be written, in the steady state regime

t j (ω) = t 0 (ω) 1 - 1 1 + p ps 1 - 2i(ω r + δ j -ω) Γt 0 (ω) -1
(3) where Γ = 2g 2 a /κ, and t 0 (ω) = -[1 + i(ω rω)/κ] -1 is the transmission of the empty resonator. We have introduced the drive power in units of photons per second p = b † in b in , and the saturation power p s of the atomcavity system reads

ps Γ = (ωr+δj -ω) 2 Γ 2 + (ωr-ω) Γ (ωr+δj -ω) κ -1/2 2 .
- The essence of the protocol is pictured in Fig. 2, in the linear regime when p ≪ p s . In this regime the transmission is given by t j (ω) =

1 t0(ω) + iΓ 2(ωr+δj -ω) -1
. If the qubit is in state |g , δ g = 0 so that the ancilla qubit is resonant with the cavity mode and the transmission consists on two peaks located at ±g a with respect to the frequency of the resonator. If the qubit is in state |e , |δ e | = 2g zz , inducing a dispersive coupling between the resonator and the ancilla provided that g zz > g a . The transmission essentially consists in a single peak slightly shifted by L = g zz ( 1 + g 2 a /g 2 zz -1) with respect to ω r (see Fig. 2). Thus a change in the state of the qubit can now translate into a switch from dispersive to resonant coupling between the resonator and the ancilla. This is evidenced by a visible displacement in the transmission peaks by a quantity g a , which can be as high as 150M Hz, about two orders of magnitude higher than the usual dispersive ac-Stark shift [13,14]. This strong effect allows an increase in the linewidth of the resonator while keeping a high fidelity read-out. Working with a low Q cavity has important advantages. First, it drastically increases the total bandwidth of the circuit, and consequently the read-out speed. Moreover, for the same probe power, the average intracavity photon number is lower, preserving the lifetime and coherence time of the qubit [6]. The read-out is performed by the injection of a short microwave pulse of power p at the frequency (ω r + δ L )/2π. Thus, the transmitted power depends on the state of the qubit, giving rise to two conditional output signals

p t|j = b † t b t = |t j | 2 p.
When p largely overcomes p s , one recovers the transmission pattern t 0 (ω) of the empty cavity, a signature of saturation [25] which limits the information on the qubit state.

We now introduce the model to optimize the measurement scheme. The performance of the read-out is usually quantified by two figures of merit, namely their fidelity F and speed. Speed is high when the system can be measured frequently, the delay between two measurements being inferiorly bounded by their typical correlation time τ c . τ c is related to the inertia of the circuit, since the resonator imposes τ c > κ -1 . Fidelity and correlation time depend on two independent parameters to optimize. First, the resonator linewidth should be narrow enough to give a large contrast between the two transmission patterns (κ < g a , g zz ), while large enough to allow a large transmitted signal p t|j = nj κ and therefore high speed qubit read-out for a given photon number nj inside the resonator. In the same way, the driving power p should be sufficiently low to avoid the saturation of the ancilla p << p s , but high enough to have a large p t|j .

In a typical circuit QED experiment, microwave photons are amplified before being sent through a homodyne detection scheme and digitalized within a short time interval τ , which is usually equal to τ c . For our purpose, we shall consider the field at the entrance of the amplifying chain. The chain is modeled by a perfect amplifier [11] radiating at the input of the circuit a white thermal field of effective temperature T N . This noise temperature ranges from a few hundreds of mK for the recent generation of quantum limited devices [12,15], to 4 -10K for commercial devices. The total noise power N = (k B T N / ω)B, in units of photons per second, is given by Johnson-Nyquist noise [START_REF] Johnson | [END_REF], where B is the bandwidth of the amplifier, imposing an additional lower band to the correlation time τ c > B -1 . Consequently, high speed measurements are obtained at the price of increased bandwidth and noise power N .

Estimation of the read-out fidelity is based on the photon number distributions P(n|j) conditioned to the qubit j, which we computed using the Glauber-Sudarshan Prepresentation [28]. In our case, this simply corresponds to the P-representation of a thermal field of temperature T N displaced by a coherent field of amplitude √ p t|j .

Thus we can readily calculate the generating function [START_REF] Mandel | Optical Coherence and Quantum Optics[END_REF] for the photon statistics, from which we extract the coefficients

P(n|j) = N n τ n (1 + N τ ) n+1 exp -τ p t|j 1 + N τ L n -p t|j /N 1 + N τ , (4)
where L n is the n-th order Laguerre polynomial. The histograms plotted in Fig. 3 clearly show how the amplification noise has a large effect on the statistics of the counts associated with each of the qubit states. As expected, the noise power increases with noise temperature (a vs. c and b vs. d), degrading the fidelity. By increasing the integration time, one can regain fidelity (a vs. b and c vs. d). As a matter of fact, it increases the signal, but it also allows one to operate with a lower bandwidth, reducing the noise power. We expect this protocol to yield a fidelity as high as 90% with a commercial amplifier, within a typical time of τ = 50ns. An integration as short as τ = 60ns should be enough to reach 99.9% with a state of the art amplifier with T N = 140mK.

Drive power, p(photons/ns) Resonator Linewidth, κ /2π (MHz) Fig. 4 shows optimization of the fidelity as a function of the resonator linewidth and probe power. The digitization time τ = 10ns has been chosen, compatible with a bandwidth B = 50MHz. A fidelity F = 95% can be reached with a resonator linewidth κ = 40 MHz and very small pumping power, corresponding to n = 1.8 photons. This fidelity corresponds to up to date results obtained in the dispersive measurement scheme with the same amplifier [13,15], but allows a much faster acquisition time. Indeed, in dispersive based read-out schemes the dynamics is slow because of the inertia imposed by the resonator of linewidth κ ∼ 5MHz, inducing a typical correlation time of τ c = 100ns. With our scheme, using a low temperature amplifier allows a drastic increase in the bandwidth and read-out speed. This enables a projective measurement of the qubit to be performed on a timescale much shorter than the recently measured relaxation time, T 1 = 50µs [3]. This scheme opens the path to the observation of quantum jumps in circuit QED with a very high temporal resolution, comparable to the performances achieved in recent experiments performed with Rydberg atoms [29], where the system is typically measured 10 3 times before undergoing a quantum jump.

In conclusion we propose a new read-out scheme based on a superconducting diamond-shape artificial atom which contains a logical qubit strongly coupled to an ancilla qubit by a cross-Kerr term. We predict fast highfidelity QND read-out of the transmon qubit with a commercial amplifier. Using a quantum limited amplifier, 60 ns read-out time and 99.9% fidelity are predicted. This original method overcomes the current read-out limitation of the superconducting qubits dispersively coupled to a resonator. In addition Purcell effect between the logical qubit and cavity is absent. As a side effect the intracavity population is minimal, n = 1.8 for the optimal parameters, minimizing any adverse effect on the qubit coherence properties. This opens the possibility of monitoring quantum jumps of the qubit with very high temporal resolution [29], of generating non-classical states [30] or implementating quantum error correction codes [START_REF] Nielsen | Quantum Computation and Quantum Information[END_REF] using closed feedback loops.

FIG. 1 :

 1 FIG. 1: (a) Schematic circuit for a fast QND read-out. The incident microwave signal of frequency ω is transmitted through the resonator coupled to the superconducting artificial atom. The transmitted signal is amplified and homodyne detected. The noise amplifier is illustrated by additional microwave source at the amplifier input. (b) The artificial atom, realized by two transmons coupled by a large inductance. (c)Energy spectrum of the uncoupled (left and right) and dressed (center) diamond shape artificial atom-resonator states when ωr = ωa + gzz. The blue arrows indicate the injected microwave frequency when ω = ωr.

1

  FIG. 2: Transmission coefficient at low pump power for a microwave pulse injected in a resonator containing a diamondshape artificial atom. Blue solid curve: Qubit state |g . Green dashed curve: Qubit state |e . Frequency is centered on the bare cavity. We took gzz/2π = 250 MHz , ga/2π = 150 MHz and a cavity linewidth κ/2π = 40 MHz (Q = 250) and p << ps. The black dotted line indicates the pump frequency ω.

FIG. 3 :

 3 Photon Counting distribution

FIG. 4 :

 4 FIG. 4: Single measurement fidelity for a state of the art amplifier (TN = 140mK) and an acquisition time τ = 10 ns versus drive power p in units of photons/10ns and resonator linewidth κ. Optimal value of F = 95% is reached for broad range around κ = 40MHz and p = 1photon/ns.

The authors gratefully thank N. Roch, P. Bertet, P. Milman , A. K. Feofanov and I. M. Pop for useful discussions. This work was supported by the European SOLID and the ANR-NFSC QuExSuperC projects, Nanosciences Foundation of Grenoble and CAPES.


Fabrice Devaux

Joé Mougin-Sisini

Paul-Antoine Moreau

Eric Lantz

Towards the evidence of a purely spatial Einstein-Podolsky-Rosen paradox in images: measurement scheme and first experimental results

Keywords: 42, 50, Ar Photon statistics and coherence theory -42, 50, Lc Quantum fluctuations, quantum noise and quantum jumps

We propose a scheme to evidence the Einstein-Podolsky-Rosen (EPR) paradox for photons produced by spontaneous down conversion, from measurement of purely spatial correlations of photon positions both in the near and in the far-field. Experimentally, quantum correlations have been measured in the far-field of parametric fluorescence created in a type II BBO crystal. Imaging is performed in the photon counting regime with an electron-multiplying CCD (EMCCD) camera.

Introduction

Spontaneous parametric down conversion (SPDC) of a wide monomode gaussian pump results in a strongly multimode beam: the extension of the down converted beam in the near field (image plane)is identical to that of the pump, in the limit of low gain and for a sufficiently wide crystal, while the far field (Fourier plane) extension is lim-Send offprint requests to: Fabrice Devaux Correspondence to: fabrice.devaux@univ-fcomte.fr ited by phase-matching. The etendue of the beam, i.e. the product of its transverse surface by the solid angle it subtends or the number of transverse modes in appropriate units (resolution cells in ref. [1]), has been recognized [2,3] as corresponding to the two-photon Schmidt number.

Note that the spatial extension of a mode in either the near or the far-field is proportional to the inverse of the full beam extension in the other plane. For single photon imaging, the laws of diffraction are equivalent to the Heisenberg's uncertainty relation: a photon that can be localized in one mode of the near field, for example by traversing an aperture of the size corresponding to the mode, will be detected at a random position in the entire far-field diffraction pattern. However, the laws of quantum mechanics state that a pair of signal-idler photons will be detected either in the same mode in the near field or in opposite modes in the far field, if no detection occurs in the other plane. Because the detection plane can be chosen at a time where causal interaction between photons is no more possible, these correlations are not compatible with local realism, as demonstrated first in the famous EPR paper [4], though compatible with the Heisenberg's uncertainty relation since correlations cannot be measured in both planes for the same photon pair.

For a sufficiently low SPDC rate, it can be assumed that a pair does not interact with another, which makes the system very close of that considered in the original EPR paper: the positions of photons 1 and 2 are detected in the near-field and their momenta correspond to the farfield. Howell et al. [5] have measured in both planes the probability distribution of the position of photon 2, conditioned by the detection of photon 1. The product of the conditional variances is 25 times smaller than the limit for the product of variances for a single photon given by the Heisenberg's uncertainty relation. This impressive result has been obtained by measuring temporal coincidences between cross-polarized photons in type 2 SPDC. These photons were separated by a polarizing beam-splitter: for a fixed position of a narrow slit transmitting the photon 1 to an avalanche photodiode, the level of coincidences was measured for each position of a similar slit transmitting the photon 1 to a separate similar detector. Three directions can be considered in order to add to the results of ref [5]. First, for a detection in this experiment of a photon 1, the photon 2 is not detected but at a precise position, resulting in a vast majority of single detections without coincidences. In the words of Reid et al. [6] about the Aspect et al. [7] experiments: "However, it is debatable whether this can be regarded as a rigorous EPR experiment because for the full ensemble, most counts at one detector correspond to no detection at the other". The situation is comparable in the Howell et al experiment, though with other conjugate variables (position-momentum instead of polarizations). Note that the light in several transverse modes has been simultaneously recorded in a recent experiment [8].

Second, the asymmetry between photons 1 and 2 could be relaxed [6], even if this asymmetry is present in the original EPR paper. Third, the assumption of a pure biphoton state [9] does not correspond to a general model of parametric amplification of quantum noise: beyond the limit of a very low gain, stimulated pairs remain perfectly correlated [10]. We propose in this paper a purely spatial detection scheme that has some advantages as regards the two first points, while assuming strictly spontaneous conversion without further amplification like in ref. [5]. The discussion of this third point is left for the conclusion section.

In recent papers, we have experimentally demonstrated, using an EMCCD camera in the photon-counting regime, that far-field opposite spatial fluctuations of type 1 SPDC are correlated in the quantum regime, with a variance of the photon number difference between opposite areas below the shot-noise level. This conclusion holds close to degeneracy for opposite pixels [11], as well as for broad-band SPDC for opposite angular sectors [12]. To demonstrate EPR correlations, we must also show quantum correlations of signal-idler spatial fluctuations in the near field.

Since an EMCCD cannot distinguish a pair from a single photon, a type 2 interaction in a BBO crystal is used, where the signal and idler photons can be separated because of their orthogonal polarizations. Hence, our experimental set-up is similar to that of Howell et al. until the polarization beam splitter, with a completely different detection scheme: images are formed by all detected photons for each polarization, without any temporal resolution, as in experiments of Jedrkiewicz at al. [13] and Brida et al [14] but in the photon-counting regime.

The paper is organized as follows. We propose in Section 2 a measurement procedure to evidence purely spatial EPR correlations. Section 3 is devoted to experimental results, in the far-field because presently the results in the nearfield are not still convincing. We quantify in section 4 the degree of apparent violation of the Heisenberg uncertainties that could be reached. Last, we discuss in the conclusion ways of improvement and extension to the stimulated regime.

Purely spatial EPR correlations 2.1 Spatial measurements of EPR correlations

For a detection of a photon 1 at -→ r 1 , the probability density of detection of a photon 2 at -→ r 2 can be written as:

p( -→ r 2 | -→ r 1 ) = p( -→ r 2 ) + f (∆r) (1)
where p( -→ r 2 ) is the probability density of detection of a photon of another pair (accidental coincidences) and f (∆r)

is the probability density of detection of the twin photon, with ∆r = -→ r 2 ± -→ r 1 , + holding for the far-field (correlation of momenta on opposite modes) and -for the near-field. It is assumed translational invariance (this hypothesis will be further discussed in the next subsection), circular symmetry and independence of the pairs (pure SPDC without further amplification). Hence, if N 1 is the number of photons 1 detected on a surface S 1 and N 2 the corresponding quantity for photons 2, we have:

< N 1 N 2 > = S1 dr 2 1 S2 dr 2 2 p( -→ r 1 and -→ r 2 ) (2) = S1 dr 2 1 S2 dr 2 2 {p( -→ r 1 )p( -→ r 2 ) + p( -→ r 1 )f (∆r)}
Therefore, the probability of detection in S 2 of the twin photon 2 of the photon 1 detected on S 1 is simply given by:

F (S 2 ) = S2 dr 2 2 f (∆r) = < N 1 N 2 > -< N 1 >< N 2 > < N 1 > (3)
If S 1 and S 2 have the same size, this expression can be symmetrized and becomes the normalized intercorrelation function:

F (S 2 ) = F (S 1 ) = < N 1 N 2 > -< N 1 >< N 2 > (< N 1 > + < N 2 >)/2 (4)
For independent pairs, this quantity can also be expressed as a function of the variance of the difference between N 1 and N 2 :

< N 1 >=< N 2 >=< (N 1 ) 2 > -< N 1 > 2 ⇒ (5) F (S 2 ) = 1 - < (N 1 -N 2 ) 2 > < (N 1 + N 2 ) >
Because of the translational invariance, the means in eq.

4 and 5 can be estimated by spatial averages on the different pixels in one image.

To summarize, we have linked the intercorrellation function with the conditional probabilities of detecting the twin SPDC photons. The width of the experimental intercorrelation will then be used to demonstrate an EPR violation of the Heisenberg inequalities. More details on the violation we expect to obtain are given in section 4.

Measurements of far-field spatial quantum correlations

The experimental setup is sketched in Fig. Photon counting regime was ensured by adjusting the exposure time such that the mean fluence of SPDC was between 0.1 and 0.2 photon per pixel [15].

With a pump beam diameter ∼2 mm, the coherence area in the far-field is smaller than the area of a pixel. Moreover, the use of pump pulses with 300 ps duration (much longer than the coherence time of SPDC) and an exposure time of the EMCCD of 5 ms (i.e. 5 laser shots) allow the excess noise to be limited by increasing the number of temporal modes [16]: the mean number of photons for one spatiotemporal mode is less than 10 -3 , in good agreement with the hypothesis of pure spontaneous down conversion, without any stimulated amplification. lution consists in calculating the variance of the difference of the binned pixels (Eq. 5). Fig. 3 shows a comparison between these three methods. In all methods, a correction has been applied to take into account the small proba-bility of multiple photons impinging on the same pixel : see below. The agreement is good and the total intercorrelation coefficient of 0.1 is significant: all single detected photons contribute to a deterioration of this coefficient, Induced Noise [15]. We have also verified that the variance of the difference between opposite pixels is smaller than the shot noise level, as in our preceding papers in type 1 and in ref. [13] and [14] be detected on the same physical pixel [11]. After this correction, the experimental spatial fluctuations on one signal or idler image obey a Poisson statistics: c r is calculated for each image and for different binnings. Fig. 4 shows the values of r measured on 50 images: most of the measurements for individual images are below the shot noise level and the average of r lies clearly in the quantum regime. For example, for a 5×5 binning, r = 0.916 ± 0.020, at 95% of confidence. Note that r decreases and spreads out when increasing binning. The second feature comes from the smaller number of available pixels for the statistics.

Our experimental results in the near-field are not still convincing and are therefore not reported here. The most evident supplementary difficulty is the adding of a polarizer beam-splitter that induces losses and distortions. Note that these distortions must be eliminated over all the image field, unlike in the experiment of Howell et al where only a small part of the image plane is used [5]. Moreover, it has been shown [17] that the walk-off should be taken into account and depends on the crystal thickness.

We plan to use a thinner crystal and to correct some geometrical aberrations in order to obtain convincing results in the near-field, that would allow the demonstration of a purely spatial EPR effect.

Expected widths of EPR correlations

In an ideal experiment, F tends to 1 if S 1 (and S 2 ) is greater than the coherence cell, in agreement with results of ref. [10]. We determine in this section the expected size of this coherence cell, in the far field as well as in the nearfield, and deduce the degree of violation of the Heisenberg criterion that can be attained by using conditional means.

We compare also the experimental results in the far-field with this expected size.

Far field

We first consider an experiment where an ideal narrow band interferential filter ensures a perfect frequency degeneracy: ν s = ν i = ν p /2, where ν is the frequency of respectively the signal (ν s ), the idler (ν i ) and the pump (ν p ). The biphoton function reads in the far-field [18]:

ψ(q s , q i ) = χ E p (q s + q i ) ζ(q s , q i ) ( 6
)
where χ is the coefficient of nonlinear interaction, q s and q i are the transverse spatial frequencies for the signal and the idler respectively, E p is the pump field in the far-field and ζ is the phase matching function. Eq. 6 states that the conditional probability function in the far-field is proportional to the pump amplitude in this plane, and therefore has the same width inasmuch as this width is much smaller than the width of the phase matching function. This condition is fully fulfilled in our experimental conditions: the phase matching function width is around 65 mm -1 ( see figure 1b), while the width of the pump field in the Fourier plane is around 0.5 mm -1 (see below), i.e. of the order of one pixel.

The situation is more complex if we take into account the non negligible width of the chromatic filter. Eq. 6 is still valid but the coordinates f ′ sin θ in the far-field detection plane are no more proportional to the transverse spatial frequencies. We have now sin θ s = qsc/νs 2π , sin θ i = qic/νi 2π .

This effect can be roughly quantified as follows. For a plane wave pump ensuring q s + q i = 0, the shift ∆θ of the idler photon position with respect to the position symmetrical of the signal photon is given by:

∆θ = θ s + θ i = q s c 2π (1/ν s -1/ν i ) (7)
In a type 2 crystal, the centers of the SPDC patterns of the idler and the signal are separated in a noncritical configuration by the walk-off angle [19], i.e 120 mrad at 710 nm: see fig. 1b. The transmission of the interferential filter is gaussian with a standard deviation of 4 nm. To obtain a rough estimate of the maximum value of ∆θ, we consider an equivalent rectangular filter with unity transmission, with the same total transmission after integration. The total width of this rectangular filter is 4 √ 2π = 10 nm, giving for a couple of signal-idler photons at the edges of this filter ∆θ = 120 × 10 710 = 1.70 mrad, i.e 4 pixels. Because of the absence of walk-off, the effect is much weaker in the horizontal direction (angle ϕ), explaining the asymmetry in fig. 2a.

To take into account more realistically the gaussian character of the filter and the phase matching function, the intercorrelation function in the far-field can be written as:

< N s N i >= χ 2 T (ν s )T (ν p -ν s )dν s dϕ p × (8)
I p (ϕ p , θ p )dθ p dϕ s dθ s | ζ(θ p , ν s , ϕ s -ϕ p , θ s )| 2
Where T is the transmission coefficient of the filter and I p is the pump intensity in the far-field. We have numerically calculated this intercorrelation function and determined its width by fitting it with a two-dimensional gaussian function. As expected, its shape is asymmetrical, with a standard deviation of 0.57 pixel in the horizontal direction and of 0.91 pixel in the vertical direction, for a theoretical value of 0.13 pixel at perfect degeneracy. In our experimental conditions, one pixel corresponds to 4.04 h/mm in momentum units. The same procedure applied to the experimental intercorrelation of fig. 2 gives an horizontal standard deviation of 1.12 pixel and a vertical one of 1.74 pixel. It seems that some geometric aberrations in the optical system enlarge the correlation peak. Nevertheless, the asymmetry due to the non perfect degeneracy in temporal frequencies is visible with similar characteristics in simulation and experiment. Moreover, it can be shown that the mean probability of transmission by the filter of the twin photon is 0.56. This relatively low number is one of the effect leading to an experimental total intercorrelation coefficient of 0.1. The other important causes of reduction of this coefficient are the global quantum efficiency of the optical system, including all optics and the camera, the false positive or negative detections due to the camera [15], and probably also some residual fluorescence of the optical components.

Near field

We suppose first a perfect degeneracy of temporal frequencies. In the output plane of the crystal (we discuss at the end of this subsection how to take into account the imaging system), the biphoton function reads:

ψ(x s , x i ) = χE p (x)ζ(x s + x, x i + x)dx (9)
Let us first assume that the pump beam has a constant amplitude. In this case, the phase matching function in the direct space depends only of the difference of the coordinates [18]:

ζ(x s + x, x i + x)dx ≡ γ(x s -x i ) (10)
where γ(x) is the inverse Fourier transform of ζ(q s , -q s ).

We now justify our hypothesis: the width of ζ is 65 mm -1 , leading to a non negligible signal-idler correlation only for distances not much greater than 1/65 mm. On this scale, we can safely assume that the 2 mm wide pump beam has a constant amplitude.

We have to find the standard deviation of |γ(x)| 2 , proportional to the conditional probability P (x s |x i ). We have first:

ζ(q s , -q s ) = exp(i∆k z L) -1 i∆k z (11)
where ∆k z is the longitudinal wave vector mismatch and L the crystal length. By neglecting the index variation of the extraordinary wave versus the internal angle, ∆k z becomes a purely geometrical term that reads, for perfect collinear phase matching:

∆k z = q 2 s λ 4π 1 n s + 1 n i (12)
with n s,i the indices of the signal and the idler. The first zero of ζ(q s , -q s ) is obtained for q s0 = 2π λL(

1 ns + 1 ni ) -1/2
.

We can now determine numerically the standard deviation σ qs of the SPDC in the far-field:

σ qs = | ζ(q s , -q s )| 2 q 2 s dq s 1/2 = 0.69q s0 (13)
and the standard deviation σ xi|xs of the conditional probability in the direct space:

σ xi|xs = |γ(x)| 2 x 2 dx 1/2 (14)
We find σ xi|xs = 1.89σ -1 qs . Note that the same relation (with 1.88 instead of 1.89) has been given in [5], with no detail on the exact computation process.

Unlike in the far-field, a non perfect degeneracy does not modify these results: even if not of the same color, the twin photons are created at the same place, with an uncertainty proportional to the inverse of their spatial frequency bandwidth.

The effect of the imaging system is much more subtle. It has been analyzed in detail in [17]. A brief summary is as follows. Because the signal and idler wave vectors have directions shifted from the walk-off, the twin photons travel in the same direction in the crystal (the Poynting vectors are identical, see also [19]). However, a direction shift does exist in the imaging system and the images are laterally shifted. Moreover, to conserve a minimum σ xi|xs , the transverse plane at the middle of the crystal (at equal distance between the input and the output face) must be imaged on the camera. With these precautions, the resolution found in eq. 14 is unaffected.

Heisenberg violation using conditional probabilities

The standard deviations in intensity of gaussian beams in the direct space σ x and in the Fourier space σ q obey the Heisenberg relation σ x σ q = 1/2. In other words, if we admit that the intensity in a gaussian beam is proportional to the probability of presence of a photon, the Heisenberg uncertainty relation becomes equivalent to the standard diffraction theory. With the results of the preceding subsections, the ratio R of the Heisenberg variance product to the EPR variance product can be expressed as :

R = (σ x σ q ) 2 (σ xi|xs σ qi|qs ) 2 = 2π0.69 1.89 2 σ 2 x λL( 1 ns + 1 ni ) (15)
In our experimental conditions : σ x =1 mm, L=1 mm and Once more, we stress that R 1/2 can also be interpreted, at perfect degeneracy, as the number of resolution cells in one transverse direction [1], or as the number of Schmidt modes [3].

Conclusion

We have proposed a scheme to demonstrate experimentally a purely spatial EPR paradox. Experimental results in the far-field are compatible with this objective and we have good hope to obtain soon the lacking results in the near-field. However, these results cannot easily be extended to higher fluxes because they are established with the assumption of independent pairs, or in more technical terms of a pure biphoton state. This limitation seems curious since even at high flux the signal-idler spatial fluctuations remain perfectly correlated, as demonstrated in numerous papers (see for example [17,20]). On the other hand, the demonstration of the EPR paradox for spatial properties of beams is based on a criterion implying the amplitude and phase quadratures [21,22]. While these quadrature operators are conjugate, with a ih commutator, the field operators respectively in the image plane and in the Fourier plane are Fourier transform each from the other and an EPR criterion in the continuous variable regime seems more difficult to establish. Of course, this curious asymmetry between the spatial and the temporal variables does not preclude the demonstration of purely spatial EPR, because experimentally the recorded intensities do correspond to purely spontaneous down conversion with totally negligible stimulated conversion.

  . The pump pulse at 355 nm provided by a passively Q-switch Nd:YAG laser (mean power: 27 mW, pulse duration: 300 ps, repetition rate: 1 kHz), illuminates a 1 mm long type 2 BBO nonlinear crystal. The far-field image of the parametric fluorescence is formed in the focal plane of a lens by a back-illuminated EMCCD camera from Andor Technology (Model iXon+ DU897-ECS-BV) with a quantum efficiency greater than 90% in the visible range. The detector area is formed by 512×512 pixels, with a pixel size of 16×16 µm 2 , or 0.46×0.46 mrad 2 after division by the focal length. We used a readout rate of 10 MHz at 14 bits and the camera was cooled to -85 • C. Measurements at degeneracy were performed for a crystal orientation corresponding to collinear phase matching by using a narrowband interferential filter centered at 710 nm (∆λ=4 nm).

Fig. 1 .

 1 Fig. 1. (a): Far field experimental set-up. D : dichroic mirror, L: lens with focal length f'=35 mm, F1: broad-band red filter, F2: narrow-band interferential filter .(b): Sum of 50 far-field images of SPDC. The dotted circles denote signal and idler regions of interest (ROI) where statistics are performed.

Fig. 2 .

 2 Fig. 2. (a):Average intercorrelation between signal and idler patterns.(b): Average intercorrelation between signal and idler patterns that come from different images.

Fig. 3 .

 3 Fig. 3. Intercorrelation with respect to the pixel size (square pixels)

Fig. 4 .

 4 Fig. 4. Far-field variance of difference for individual images with respect to the mean number of photons in binned pixels, for binnings from 1×1 to 7×7.

1 .

 1 It can be easily shown that the same correction must be applied to the correlation coefficient (eq.4).

λ=710

  nm give σ xi|xs = 0.013 mm, hσ qi|qs = 0.52 h/mm and R=5940. If we admit that the interferential filter enlarges the correlation function in the far-field by a factor of 5 (see above), R ≈ 200. Of course, other factors will probably diminish the experimental value by enlarging the correlation in the near field, like distortions due to the polarizing beam-splitter, other geometrical distortions, not perfect imaging of the middle plane of the crystal, etc... Nevertheless, eq.15 gives the conditions that allow a large violation of the Heisenberg limit, a thin crystal allowing a large phase-matching range and a wide pump beam.


Pierre Quillatre

Olivier Vermorel

Thierry Poinsot

Philippe Ricoux

Large Eddy Simulation of Vented Deflagration

In order to understand gas explosion phenomena in industrial buildings, a reduced-scale vented combustion chamber is investigated numerically. In this configuration, a flame is ignited in an initially quiescent flammable mixture and propagates past solid obstacles, generating a strong pressure increase. The aim of this numerical study is twofold: The first objective is to show how large eddy simulation manages to reproduce the parameters of critical relevance for this multiscale problem, in particular the overpressure generated during the flame propagation. The second objective is to highlight that, even if large-to small-scale turbulence effects play a crucial role in the flame development and the resulting overpressure, it is also needed to correctly account for thermo-diffusive scale phenomena.

■ INTRODUCTION

Accidents due to gas explosions in industrial buildings such as offshore oil and gas producing platforms or oil storage facilities are a major safety issue. Although the primary objective is of course to prevent them, being able to understand the mechanisms which control them is a societal, economical, and physical challenge of prime importance.

A typical explosion scenario begins with an ignition of a quiescent flammable mixture inside a confined or semiconfined area. Then, the flame propagates and accelerates by interaction with obstacles (process equipment, structures, piping, ...), generating a strong increase of pressure. This overpressure is the parameter of critical interest for safety studies, since it can induce the destruction of the facilities involved. In order to understand explosions in confined and semiconfined areas, small-to medium-scale configurations of vented chambers are usually considered. Experimental 1-5 and numerical [START_REF] Gubba | LES modeling of premixed deflagrating flames in a small-scale vented explosion chamber with a series of solid obstructions[END_REF][START_REF] Ibrahim | Calculations of explosion deflagrating flames using a dynamic flame surface density model[END_REF][START_REF] Di Sarli | Using Large Eddy Simulation for understanding vented gas explosions in the presence of obstacles[END_REF] studies of explosion have extensively been performed in order to point out the mechanisms involved in flame acceleration and pressure increase generation. Effects of geometric features or fuel type on this overpressure have also been studied. [START_REF] Masri | A Comparative Study of Turbulent Premixed Flames Propagating Past Repeated Obstacles[END_REF][START_REF] Di Benedetto | Explosion behavior of CH4/O2/N2/CO2 and H2/O2/N2/ CO2 mixtures[END_REF][START_REF] Gubba | Measurements and LES calculations of turbulent premixed flame propagation past repeated obstacles[END_REF] Thanks to the growing computational power, computational fluid dynamics (CFD) appears as an interesting alternative to experiments, which remain expensive and dangerous and bring limited material for diagnostic and understanding of explosions. Explosions are a typical multiscale and multiphysics problem for which adapted numerical methods must be used. Up to now, safety related studies of explosions for industry have mainly been carried out using unsteady Reynolds averaged Navier-Stokes (URANS) methods. [START_REF] Arntzen | Modelling of turbulence and combustion for simulation of gas explosions in complex geometries[END_REF][START_REF] Zalosh | Explosion Venting Data and Modeling Literature Review[END_REF] However, the emergence of large eddy simulation (LES) on massively parallel computers has considerably improved the ability to precisely simulate fully unsteady flows in complex geometries [START_REF] Gubba | Measurements and LES calculations of turbulent premixed flame propagation past repeated obstacles[END_REF][START_REF] Di Sarli | Large eddy simulation and PIV measurements of unsteady premixed flames accelerated by obstacles[END_REF] and appears as a promising tool for explosion studies.

The ultimate objective of this research is to develop LES tools for realistic explosion cases where the sizes will be of the order of 10-100 m and Reynolds numbers will be very large. To develop these LES tools and make sure that they can capture all phases of the flame development, the strategy is to begin with a smaller size experiment (0.25 m) installed at the University of Sydney. [START_REF] Masri | A Comparative Study of Turbulent Premixed Flames Propagating Past Repeated Obstacles[END_REF] In this kind of configuration, the flow is largely turbulent during most of the explosion. The large spectrum of turbulent scales present in this flow interacts with the flame and must be properly reproduced in order to accurately simulate the flame propagation in this environment. As shown by Di Sarli et al., [START_REF] Di Sarli | Large Eddy Simulation of transien premixed flamevortex interactions in gas explosions[END_REF][START_REF] Di Sarli | Sensitivity to the Presence of the Combustion Submodel for Large Eddy Simulation of Transient Premixed Flame-Vortex Interactions[END_REF][START_REF] Di Sarli | Sub-grid scale combustion models for large eddy simulation of unsteady premixed flame propagation around obstacles[END_REF] the predictive ability of LES strongly relies on the model implemented to take into account the flame-turbulence interaction. It is well-known that the role played by the combustion submodel is dependent on both the grid resolution used and the combustion regime experienced by the propagating flame. However, the main aim of the present paper is to point out that, in LES, in addition to the importance to account for large-to small-scale turbulent combustion effects, it is also needed to correctly account for thermodiffusive scale phenomena. Reaching a correct reproduction of all of these multiscale phenomena is the prerequisite in order to accurately predict the global quantities of critical interest for safety related studies, in particular the overpressure generated in the chamber.

■ TEST CASE

The configuration studied in this work was set up at the University of Sydney. [START_REF] Masri | A Comparative Study of Turbulent Premixed Flames Propagating Past Repeated Obstacles[END_REF] It consists of a square cross section premixed combustion chamber (25 cm × 5 cm × 5 cm) with solid obstacles. The obstacles consist of three removable turbulence generating grids and one fixed central obstacle, as shown in Figure 1 (left). The bottom end of the chamber is closed, and the top end is opened out to the atmosphere. The vessel is initially filled with a premixed mixture of air and fuel at atmospheric pressure and temperature. Three different fuels are available: CNG (88% CH 4 ) at equivalence ratio Φ = 1, LPG (95% C 3 H 8 ) at Φ = 1, and H 2 at Φ = 0.7. The explosion scenario begins with an ignition of the quiescent premixed mixture at the closed end of the chamber. Ignition is followed by a laminar propagation phase and then an interaction with solid obstacles. When touching the obstacles, the initially laminar flame gets wrinkled and becomes a turbulent flame propagating at high speed. Additional details of the setup may be found in Masri et al. [START_REF] Masri | A Comparative Study of Turbulent Premixed Flames Propagating Past Repeated Obstacles[END_REF] As mentioned earlier, the turbulence generating grids can be added or removed so that several configurations can be studied. Three configurations are studied here, where the number of grids is progressively increased from one to three, located farthest from the ignition point. Illustrations of the investigated configurations can be found in Figure 1 (right). Two major aspects can therefore be studied using this experimental configuration:

• the influence of the fuel on the overpressure • the influence of the geometric features on the flame propagation and the resulting overpressure

■ NUMERICAL METHODS FOR EXPLOSION

PHENOMENA

Explosions are a multiphysics problem which requires one to account for chemical kinetics, fluid flow, turbulence, wall phenomena, and acoustics considering that all of these phenomena, which occur over a large range of scales, are strongly coupled. The numerical method must thus be able to take into account all of these scales. Explosions are consequently a typical multiscale problem (Figure 2). Chemical reactions taking place in flames occur in very thin fronts, typically less than 100 μm, and their kinetics is controlled by thermo and mass diffusive effects across these flame fronts. The smallest turbulent structures can reach less than 10 μm, while the largest scales of the flow correspond to vortices with sizes of the order of the building size (10-100 m).

In order to reproduce explosion phenomena, various numerical approaches are available to solve the compressible reactive Navier-Stokes equations. These methods mainly differ by the way turbulence is accounted for:

• Direct numerical simulation (DNS) is a bruteforce approach: all turbulent scales are resolved in space and time. This method requires meshes with a cell size smaller than the smallest turbulent scale which is the Kolmogorov scale. For explosion configurations, this would consequently lead to meshes of several hundreds of billion points which is not affordable with the current CPU power. This method is in practice dedicated to academic configurations (e.g., homogeneous isotropic turbulence), in very simple and small configurations. [START_REF] Grout | Direct numerical simulation of flame stabilization downstream of a transverse fuel jet in cross-flow[END_REF] • Currently, CFD codes used for safety studies mainly rely on URANS methods. [START_REF] Arntzen | Modelling of turbulence and combustion for simulation of gas explosions in complex geometries[END_REF] This is a statistic approach where only averaged values are resolved, consequently considerably reducing the CPU cost. However, all turbulent scales have to be modeled, which can potentially lead to errors in their prediction.

• The emergence of large eddy simulation (LES) on massively parallel computers has considerably improved the ability to precisely simulate fully unsteady flows in complex geometries. [START_REF] Enaux | Large Eddy Simulation of a Motored Single-Cylinder Piston Engine: Numerical Strategies and Validation[END_REF][START_REF] Boileau | LES of an ignition sequence in a gas turbine engine[END_REF] Indeed, the LES technique is a filterered approach which relies on a scale analysis of the turbulence: according to Kolmogorov, [START_REF] Kolmogorov | The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers[END_REF] turbulence may be considered in a fractal way where the smallest scale eddies energy can be deduced from the larger ones. The LES method relies on this statement: The energetic largest turbulent scales are explicitly computed on the mesh points, while the smallest are modeled from the resolved large scales. The overall model contribution in LES is thus reduced compared to (U)RANS methods where all turbulent scales are modeled. If the cell size is small enough, the model contribution naturally tends to zero and LES degenerates toward DNS. These properties allow a priori a better description of turbulent flows even though it requires more computational time. It is also worth noting that, by construction, LES is expected to be much less mesh dependent than (U)RANS as far as the LES cutoff length falls inside the inertial range of the energy spectrum.

LES has already shown its ability to give more reliable predictions than URANS in unsteady complex configurations such as piston engines [START_REF] Enaux | Large Eddy Simulation of a Motored Single-Cylinder Piston Engine: Numerical Strategies and Validation[END_REF] or gas turbines [START_REF] Boileau | LES of an ignition sequence in a gas turbine engine[END_REF] which present many analogies with gas explosions in terms of turbulent combustion or confinement.

The LES solver used in this study is the AVBP code. [START_REF] Gourdain | High performance parallel computing of flows in complex geometries: I[END_REF],23 AVBP solves the unsteady compressible and reactive Navier-Stokes equations on unstructured grids. The present simulations are performed with a finite volume Lax-Wendroff convective scheme. [START_REF] Lax | Difference schemes for hyperbolic equations with high order of accuracy[END_REF] Navier-Stokes characteristic boundary conditions (NSCBC) [START_REF] Granet | Comparison of nonreflecting outlet boundary conditions for compressible solvers on unstructured grids[END_REF] are used at the outlet of the plenum which is located at the open end of the chamber in order to mimic the atmosphere. The solid walls that represent the obstacles and the explosion chamber are adiabatic nonslip walls.

As it has been mentioned, effects smaller than the mesh size have to be modeled:

• Sub-grid-scale turbulence is modeled by the WALE viscosity based model. [START_REF] Ducros | Wall-Adapting Local eddy-Viscosity models for simulations in complex geometries[END_REF] • When dealing with turbulent combustion configurations, another aspect of modeling is to be able to handle flame fronts which are generally much thinner than the cell size. If the flame structure is not resolved, it cannot be computed. For combustion, no theory enables one to predict flame properties from large scales, since combustion is inherently controlled by molecular processes. A method similar to LES for turbulence consequently cannot be applied. However, combustion theory [START_REF] Poinsot | Theoretical and Numerical Combustion[END_REF] tells us how flame fronts can be thickened in space by a factorand still propagate at the same speed. Here the thickened flame method [START_REF] Colin | A thickened flame model for large eddy simulations of turbulent premixed combustion[END_REF] (TFLES) allows one to make these fronts large enough to be resolved on the mesh. Flameturbulence interaction at subgrid scale is reproduced by multiplying the resolved reaction rate by an efficiency factor , which models the loss of wrinkling due to flame thickening. , is defined from the formulation of Colin et al. [START_REF] Colin | A thickened flame model for large eddy simulations of turbulent premixed combustion[END_REF] as the ratio of sub-grid-scale flame front wrinkling Ξ Δ between a nonthickened and a thickened flame:

δ δ = Ξ Ξ Δ Δ ( ) ( ) l 0 l 0 , - (1)
where δ l 0 is the laminar flame thickness of the nonthickened flame. The sub-grid-scale flame front wrinkling Ξ Δ is defined as

β δ Ξ = + - Γ Δ ′ ′ Δ Δ Δ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ c Re u S u S 1 2 ln(2) 3 [ 1] , ms t 1/2 l 0 l 0 l 0 (2) with Γ(Δ/δ l 0 , u Δ ′ /S l 0 ) = 0.75 exp[-1.2/[u Δ ′ /S l 0 ] 0.3 ](Δ/δ l 0 ) 2/3
, Δ the filter size, Re t a Reynolds number based on an estimation of the turbulent integral length scale L T (L T = 5 × 10 -3 ), β = 0.3, and c ms = 0.28. Note that this set of parameters has been kept constant for all the simulations presented in this paper.

• In the context of LES, due to their prohibitive computational cost, integration of detailed kinetics is impossible. In our work, chemistry is modeled by reduced schemes. [START_REF] Franzelli | A two-step chemical scheme for keroseneair premixed flames[END_REF] Thus, only the principal species and reactions are considered. Several one-step and two-step reduced schemes have been developed for this work. Details on these schemes and their impact on the results are given in the section "Results Sensitivity to LES Submodels".

• Ignition processes are very complex (energy transfer by laser to gas, plasma formation, shock waves, ...) and their modeling is a task that extends far beyond the scope of this study. In order to avoid this part of the modeling, calculations are consequently initialized by a small sphere of burnt gases (radius 1 cm) at the ignition point which is sufficiently strong to initialize the combustion process without any transient phase.

LES simulations were carried out on meshes of 20 million tetrahedra with a typical cell size of 0.5 mm in the chamber. This mesh density has been chosen in order to ensure that the flame, even thickened, remains thinner than the distance between the bars of the baffle plate.

■ RESULTS

The aim of this section is to check the ability of LES to simulate flame propagation past repeated obstacles and capture the critical physical quantity related to safety related studies, i.e., the overpressure generated in the chamber. Besides the understanding of explosion phenomena using LES results, a particular interest is brought to the influence of the fuel and of the number of turbulence generating grids. Finally, the focus is on the sensitivity of LES results to the submodels used to reproduce small-scale chemistry and thermo-diffusive effects.

Phenomenological Study. The primary objective of this section is to understand the phenomena occurring in explosions and to highlight the mechanisms responsible for the pressure increase. Results from LES of Config2 are investigated in order to understand the typical flame behavior during the explosion.

Figure 3 shows LES images of flame propagation compared with experiments. In the early stage of propagation, the flame is laminar and develops with a hemispherical shape and then it transitions to a "finger" shape when it approaches the walls, still being laminar. This long laminar phase controls the flame shape and strength just before it touches the obstacles. Finally, the flame front hits the obstacles, generating strong turbulence which accelerates the flame. All phases of flame propagation observed in experiments are qualitatively well reproduced by LES. With the time interval between two successive images being kept constant between experiments and LES, this highlights that the simulation correctly reproduces the flame arrival time even though the fourth LES image shows a slightly faster flame acceleration than experiments.

As mentioned earlier, the explosion induces a large pressure increase in the chamber which is the parameter of critical interest for safety related studies. Here we propose to investigate the mechanisms responsible for this increase using the LES results.

Although turbulent combustion flows are inherently controlled by small-scale physics (chemistry processes, interaction with turbulence, diffusive effects, ...), pressure increase in the chamber can be explained from a more global point of view, considering quantities integrated in the whole chamber. As explained by Di Sarli et al., [START_REF] Di Sarli | Using Large Eddy Simulation for understanding vented gas explosions in the presence of obstacles[END_REF] the pressure increase observed in the chamber is due to the competition between two opposite phenomena: the combustion rate and the venting rate. On the one hand, the burnt gases accumulated inside the chamber by combustion processes have a lower density than the fresh gases they replace and consequently take up a bigger volume, inducing a pressure increase. On the other hand, gases exhausting through the vent opening contribute to decrease the pressure. Working on a volume balance, a fine diagnostic has been set up in order to understand and highlight the mechanisms responsible for the pressure increase inside the chamber.

First, the venting rate V ̇vent is simply defined as the volume flow exhausting from the chamber:

∫ ̇= V U S d S vent n outlet (3)
where S outlet is the section area of the chamber outlet and U n is the velocity normal to this section.

Considering that the fresh gases (density ρ f ) which have been replaced by burnt gases (density ρ b ) took up a volume V f = ρ b / ρ f V b , the combustion rate V ̇comb can be written as the variation of the burnt gas volume minus the variation of the consumed fresh gases:

ρ ρ ̇= ∂ ∂ - ∂ ∂ = ∂ ∂ - V V t V t V t (1 / ) comb b f b b f (4)
The variation of the burnt gas volume can be obtained from the mass reaction rate of any combustion product, here H 2 O:

ω ρ ∂ ∂ = V t Y b H O b H O b 2 2 ( 5
)
where Y H 2 O b is the H 2 O mass fraction in the burnt gases and

ω̇H 2 O
is the production rate of species H 2 O integrated over the whole chamber.

When V ̇comb < V ̇vent , the created volume is larger than the volume exhausting the chamber and the pressure increases. When V ̇comb < V ̇vent , the trend is opposite and the pressure decreases. Figure 4 (top) compares V ̇comb and V ̇vent to the overpressure (bottom) extracted from the LES results. We can notice the perfect match between the change of sign of V ̇comb < V ̇vent and the maximum of overpressure around T = 12 ms (first vertical dotted line). Beyond the peak of overpressure, acoustic reflections inside the chamber are also reproduced by this diagnostic. Arrows referenced as g2, g3, Obs, and Out, respectively, refer to the time at which the flame reaches the turbulence generating grids number 2, number 3, the central obstacle, and the chamber outlet.

Thanks to this diagnostic, a first scenario of the explosion may be developed. In a first time, before hitting the obstacles, the flame is laminar and the combustion rate remains relatively low so that the "additional" volume created by the combustion process can correctly be exhausted through the vent opening. Then, following the contact with obstacles, the combustion rate strongly increases, whereas these same obstacles constrain the flow which cannot properly exhaust from the chamber. The delay between these two phenomena induces the pressure increase. After the first overpressure peak, the flame reaches the chamber outlet and the combustion rate stops increasing. However, fresh gases which have been trapped in recirculation zones behind obstacles and in the corners of the chamber keep on burning, consequently maintaining the combustion rate. In the same time, acoustic reflections modify the venting rate and make the overpressure oscillate toward an equilibrium state at P = P atm alternating V ̇comb > V ̇vent and V ̇comb < V ̇vent . These oscillations are reproduced in agreement with the frequency observed in experiments.

Numerical Accuracy. As mentioned previously, all simulations were performed on the same mesh compound of 20 million tetrahedra with a typical cell size of 0.5 mm inside the chamber. An important point is to be able to estimate the quality of the results and especially to check if this resolution is adequate or not.

A first indication is to measure the respective contribution of the resolved and sub-grid-scale parts of the combustion rate. Figure 5 shows the contribution of the resolved part defined as the ratio between the resolved part of the reaction rate ω̇r es and the total reaction rate ω̇t ot when the flame interacts with the central obstacle:

In areas where the flame is almost laminar (near the closed end of the chamber, before the second baffle plate), the flame wrinkling is fully resolved on the grid points and the contribution of the sub-grid-scale combustion model is of course nearly 0%. On the other hand, in areas where the flame is fully turbulent, typically after the third grid, the resolved part of the combustion rate hardly reaches 25% of the total combustion rate. For the whole chamber and for Config3 studied here, the mean ratio is 55%. This highlights the great importance of the sub-grid-scale combustion model for this kind of simulation, as it has been shown by Di Sarli et al., [START_REF] Di Sarli | Large Eddy Simulation of transien premixed flamevortex interactions in gas explosions[END_REF][START_REF] Di Sarli | Sensitivity to the Presence of the Combustion Submodel for Large Eddy Simulation of Transient Premixed Flame-Vortex Interactions[END_REF] even if the conditions are here very favorable to LES (wellrefined mesh and relatively "large" laminar flame thickness compared to other fuels or other thermodynamic conditions). Values of the efficiency function , and thickening factoraveraged over the flame front for the different simulations performed are listed in Table 1. These values are calculated at the time at which the flame wraps around the main central obstacle, i.e., when the flame is fully turbulent. It means that these values are representative of the maximum values obtained in the simulations. When going from CH 4 to C 3 H 8 and to H 2 , the global flame thickening factor increases as the laminar flame thickness decreases. The efficiency factor increases as well when going from CH 4 to C 3 H 8 and to H 2 . Indeed, when the laminar flame speed of the mixture increases, the flame propagation past the obstacles generates more turbulent structures, which will interact with the flame and makes it even quicker and more wrinkled. The subgrid contribution of the combustion rate consequently becomes higher when going from CH 4 to C 3 H 8 and to H 2 . Finally, as expected, when the number of grids is increased, the turbulence level is increased and so is the global efficiency factor.

A second way to characterize the quality of the LES is to quantify the amount of turbulent kinetic energy resolved. According to Pope's criterion, [START_REF] Pope | Ten questions concerning the large-eddy simulation of turbulent flows[END_REF] LES should resolve 80% of the total turbulent kinetic energy. The contribution of the resolved part of the turbulent kinetic energy, ε, can be expressed as (8) where k res and k sgs are, respectively, the resolved part and the sub-grid-scale part of the total turbulent kinetic energy k tot . The resolved turbulent kinetic energy k res is defined as

ε = = - k k k k 1 res tot sgs tot
= ′ + ′ + ′ k u u u 1 2 ( ) x y z res 2 2 2 (9)
where u x ′, u y ′, and u z ′ are the velocity fluctuations in each direction, expressed as

′ = ⟨ ⟩-⟨ ⟩ u U U i i i 2 2 (10)
with U i being the resolved instantaneous velocity in direction i and ⟨ ⟩ denoting time averaging. With the phenomenon studied here being transient, defining a time-averaging procedure to extract fluctuating values may be tricky. Here, following Di Sarli et al., [START_REF] Di Sarli | Large Eddy Simulation of transien premixed flamevortex interactions in gas explosions[END_REF] a 0.1 ms time interval has been chosen for time averaging. This enables one to minimize the variations of the mean fields due to the transient nature of the phenomenon, while the size of the data sample is kept sufficiently large to get meaningful statistics.

The sub-grid-scale turbulent energy has been estimated from the sub-grid-scale viscosity ν sgs : 31

ν = Δ k C 1 ( ) sgs m 2 sgs 2 (11)
where the constant C m is defined as

C m = [(2/3) 1/2 A]/(πK 0 2/3
). The coefficient K 0 = 1.4 corresponds to an infinite inertial spectrum for a homogeneous isotropic turbulence in the framework of Kolmogorov's hypothesis. The constant A is evaluated as A = 0.44 in the eddy-damped quasi-normal Markovian (EDQNM) theory. [START_REF] Aupoix | Modeles simples de tension de sousmaille en turbulence homogeǹe et isotrope[END_REF] Figure 6 shows the contribution of the resolved part of the turbulent kinetic energy in the flame front zone. In the most turbulent zones, after the turbulence generating grids, the criterion is fulfilled with about 100% of the total kinetic energy resolved. Close to the walls, at the closed end of the chamber, nearly 0% of the turbulent kinetic energy is resolved in the flame. However, the amount of total kinetic energy in these areas is lower than 0.1% of the total kinetic energy in turbulent areas, showing that the level of turbulence in these areas is extremely low and the flame quasi-laminar. This is corroborated by the results previously shown in Figure 5. It can be consequently concluded that Pope's criterion is reasonably fulfilled, especially in areas where turbulence is a phenomenon of critical interest.

LES Validation. The overpressure generated during the flame propagation is extracted from a probe located at the closed end of the chamber in both experiments and LES. It should be noted that, due to the experimental variability on the timing between the different experimental realizations, experimental signals have been shifted in time in order to perform the averaging procedure and to extract mean signals and statistical envelopes. Consequently, validation of LES results will not be performed on a timing criteria but only on the magnitude and the trend of overpressure signals.

Figure 7 shows LES results relative to explosions performed with C 3 H 8 for Config1-3 together with experimental mean signal and envelope of individual realizations. First, it can be noticed that LES is able to precisely reproduce the overpressure magnitude for any of the three configurations, from one to three turbulence generating grids. As mentioned earlier, postmaximum overpressure oscillations are correctly captured as well. The influence of the number of turbulence generating grids is also correctly reproduced: the flame is accelerated when adding grids, and the resulting overpressure is consequently higher. This is mainly due to a higher global level of turbulence in the chamber, as already mentioned by Masri et al. [START_REF] Masri | A Comparative Study of Turbulent Premixed Flames Propagating Past Repeated Obstacles[END_REF] Figure 8 shows the results relative to the fuel influence for explosions performed in Config3. Although compressible natural gas (CNG) and liquified petroleum gas (LPG) were used in experiments, these mixtures have been approximated by their main respective component in LES: CNG has been replaced by CH 4 (88% of CNG volume) and LPG by C 3 H 8 (95% of LPG volume). Once again, LES is also able to predict correctly the generated overpressure. C 3 H 8 explosions generate higher overpressures than CH 4 ones. As shown experimentally by Masri et al., [START_REF] Masri | A Comparative Study of Turbulent Premixed Flames Propagating Past Repeated Obstacles[END_REF] H 2 , even at a lower equivalence ratio of Φ = 0.7, generates a much higher overpressure than C 3 H 8 and CH 4 . This can be mainly related to its higher laminar flame speed S L 0 , as shown in Table 2.

These first results clearly show that LES seems an appropriate tool to deal with the multiscale physics of explosion phenomena. Its ability to predict the critical parameters related to building safety issues has been shown:

• the maximum overpressure magnitude is correctly estimated • postmaximum pressure oscillations are captured • the influence of the obstacles is reproduced • the influence of the fuel type is reproduced

■ RESULTS SENSITIVITY TO LES SUBMODELS
Although a large part of the overpressure is due to the turbulent character of the flame, it is highly influenced by the initial laminar propagation phase which controls the flame strength and shape before it hits the obstacles. This laminar phase is mainly controlled by chemical and thermo-diffusive small-scale effects. In the following, it is proposed to estimate the errors committed if rough submodels are used to reproduce these effects. Two issues in particular are investigated:

• chemistry modeling • transport modeling Chemistry Modeling Influence. As mentioned earlier, the integration of detailed kinetics into turbulent flame simulations is impossible due to their prohibitive computational cost. Indeed, typical detailed mechanisms for C 3 H 8 -air combustion such as GRI-MECH 3.0 33 involve more than 50 species and 350 reactions. Tabulated chemistry methods [START_REF] Gicquel | Laminar premixed hydrogen/air counterflow flame simulations using flame prolongation of ILDM with differential diffusion[END_REF] have demonstrated their great potential to replace detailed kinetics. [START_REF] Galpin | Large-eddy simulation of a fuel-lean premixed turbulent swirl-burner[END_REF] However, these methods may become difficult to handle when dealing with complex industrial configurations [START_REF] Franzelli | A two-step chemical scheme for keroseneair premixed flames[END_REF] and curvature and strain effects (important in the laminar phase) cannot be included easily. Another alternative solution is to use reduced chemical schemes. [START_REF] Franzelli | A two-step chemical scheme for keroseneair premixed flames[END_REF][START_REF] Westbrook | Simplified reaction mechanisms for the oxidation of hydrocarbon fuels in flames[END_REF] In these methods, the number of species and reactions is reduced to the main ones with the minimum objective to reproduce the laminar flame speed and the adiabatic flame temperature at least. Reduced mechanisms are employed here for all calculations presented in this paper. In the results presented up to now, a two-step mechanism (Table 3) has been used. This mechanism was built to reproduce the same laminar flame speed and burnt gas temperature as detailed mechanisms at ϕ = 1, P atm , and T atm (Table 5). Although the range of validity of this two-step mechanism is restricted, it is

q = Ae (-E a /RT) ∏(ρY k /W k ) n k
, where E a is the activation energy of the reaction and W k and n k are, respectively, the molecular mass and reaction exponent for species k.

adapted to this study, since the flammable mixture is initially perfectly premixed and pressure variations are too small to modify flame properties. However, in order to reduce the CPU cost, one could try to reduce even more the chemistry modeling to only one reaction: the fuel oxidation by O 2 (Table 4). However, when removing the CO-CO 2 equilibrium reaction, it is well-known that the prediction of burnt gas temperature may become rough. Even though the one-step mechanism has been built to give exactly the same laminar flame speed as the two-step mechanism, it consequently overestimates the burnt gas temperature (Table 5).

Both C 3 H 8 -air schemes have been tested on Config3 with our LES solver. The resulting overpressures are compared to the experimental data in Figure 9 (left) and the flame front positions (referred to as the position of the furthest flame front point from the ignition point) obtained with the two different kinetic mechanisms are shown in Figure 9 (right). When the two-step mechanism used up to is replaced by the onestep mechanism, the flame is accelerated and the overpressure is increased. The flame propagation speed S d obtained with the one-step mechanism is considerably higher than the one obtained with the two-step mechanism all along the flame propagation. S d is about 10% higher during the initial laminar phase and also during the turbulent propagation phase. Both propagation regimes are thus impacted, resulting in a 26% sensitivity of the overpressure to the chemical modeling. This result is directly related to the burnt gas temperature overestimation by the one-step mechanism. Since the fresh to burnt gas density ration ρ f /ρ b is overestimated, the resulting flame propagation speed is overestimated

ρ ρ = S S d f b L 0 ( 12
)
and the generated overpressure as well. This shows that thermodynamics flame properties have a strong impact on global explosion parameters and should be correctly reproduced using an appropriate kinetic mechanism. Transport Modeling Influence. Knowing how fuel particles diffuse inside the mixture is characterized by its Lewis number Le F . More generally, the Lewis number Le k of any species k controls the competition between thermal and mass diffusion effects: Le k = α T /D k , where α T and D k are respectively the thermal and mass diffusivities. A common simplification when using reduced schemes in reactive CFD codes is to set Lewis numbers Le k to one for all species, consequently neglecting the competition between thermal and mass diffusion but also neglecting preferential diffusion between the different species.

A second important consequence of this simplification is that the flame response to stretch is strongly modified. Indeed, when the flame is stretched, it can be shown that the consumption speed S c (defined as the integral of the fuel burning rate across the flame front [START_REF] Poinsot | Theoretical and Numerical Combustion[END_REF] ) is strongly impacted. In the limit of small curvature terms, the consumption speed may be written as [START_REF] Bush | Asymptotic analysis of laminar flame propagation for general Lewis numbers[END_REF][START_REF] Clavin | Dynamic behavior of premixed flame fronts in laminar and turbulent flows[END_REF]

where L a c is the Markstein length for the consumption speed and κ = (1/S) dS/dt is the flame curvature with S being the flame surface area.

This formulation is dependent on the Lewis number through L a c . Several expressions of L a c can be found in the literature. Clavin and Joulin [START_REF] Clavin | Premixed flames in large scale and high intensity turbulent flow[END_REF] give

∫ δ β = - - + - L Le T T T x x x 1 2 ( 1) ln(1 ) d T T T a c F f b f 0 ( )/ b f f (14)
where T b and T f are, respectively, the burnt and fresh gas temperatures and δ is the unstretched flame thickness. The parameter β = (T b -T f )T a /T b 2 measures the activation energy, with T a being the activation temperature. Equations 13 and 14 show that the Lewis number has a direct influence on the consumption speed when the flame is stretched. The consumption speed of a fuel with Le > 1 (such as C 3 H 8 ) will therefore be significantly reduced for high curvatures, as it is typically the case in the early times after ignition. On the other hand, assuming that Le = 1 will lead to a zero value for the Markstein length and a flame speed independent of stretch to first order.

The burnt gas temperature T b is also affected by stretch. In the presence of stretch, the burnt gas temperature can be written as 40

κ - = - ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ T T T Le D S ( ) 1 1 b ad ad L 0 2 (15)
where D is a characteristic diffusivity. This shows that, for a fuel with Le > 1, when the flame is stretched, the burnt gas temperature will be significantly reduced with regard to its adiabatic value T ad for an unstretched flame. The Le = 1 assumption will lead to a burnt gas temperature independent of stretch. Using the simplification Le = 1 for a fuel whose actual Lewis number is larger than 1 consequently leads to an overestimation of the burnt gas temperature which as a main consequence overestimates the flame propagation speed, as explained by eq 12.

In the results presented up to now, a realistic transport model has been used in our calculations, i.e., a transport mechanism for which the response of the thickened flame to stretch is correctly reproduced. This can be easily done since the flame thickening is almost constant during the whole computation and may be known a priori. In order to investigate the consequences of the Le = 1 simplification, another two-step mechanism has been built (Table 6), replacing the transport model (used in the two-step mechanism of Table 3) by a simplified transport for which the Lewis numbers Le k have been set to 1 for all species. It should be noted that both mechanisms have been built to give exactly the same laminar flame speed and burnt gas temperature at ϕ = 1, P atm , and T atm .

These two mechanisms have been tested on Config3. The resulting overpressures are compared to the experimental data in Figure 10 (left), and the flame front positions obtained with the two different transport models are shown in Figure 10 (right). When the realistic transport model is replaced by the simplified transport model, the flame is accelerated and the overpressure is overestimated. The flame propagation speed obtained with the simplified transport mechanism is higher than the one obtained with the realistic transport mechanism by a factor varying around 9% during the whole flame propagation. Both laminar and turbulent propagation phases are impacted. It means that, even though the flow is largely turbulent during most parts of the explosion, molecular and thermal transports are important and have a significant impact on the results. The consequence is a 23% sensitivity of the overpressure to the transport modeling.

This result is consistent with the conclusions which have been drawn from the theory: using the simplified transport model for C 3 H 8 -air combustion accelerates the flame because the effects of curvature on flame speed and burnt gas temperature are not captured. This overestimation of the flame propagation speed considerably modifies the flame dynamics and raises significantly the peak overpressure in the chamber. For CH 4 (Le ∼ 1), further simulations have shown that, in agreement with the theory, the impact of such a modification is almost zero (+0.05%). For H 2 , the opposite behavior is observed; i.e., the flame propagation is slowed down and the overpressure is underestimated (-9%).

Although explosion processes are driven by turbulent combustion, the generated overpressure has been shown to be highly influenced by the initial laminar phase. A correct reproduction of this phase is a prerequisite to be able to precisely predict the overpressure increase inside the chamber. Even though thermodynamic and diffusive flame properties take place at very small scale, these effects must be correctly accounted for. This can be done using appropriate kinetic mechanisms and transport modeling.

■ CONCLUSION

The ability of LES to handle the multiscale problem of gas explosion in a vented chamber has been shown. Explosion characteristic properties such as the maximum overpressure inside the chamber have been precisely reproduced. Geometric and fuel modification effects on the generated overpressure have been correctly captured as well. LES enabled us to perform a diagnostic of explosions in order to improve understanding of phenomena which drive them.

Even though LES seems a promising tool for explosion studies with higher predictive capacities compared to URANS, it remains very dependent on the quality of its submodels and of the associated assumptions or simplifications. In this work, the strong influence of two LES submodels has been highlighted. This shows that, even though explosions are due to turbulent combustion processes, proper kinetic and transport modeling reproducing small-scale effects must be used in order to reproduce the correct initial laminar propagation phase which is crucial for the overpressure prediction.

Since the final objective of this work is to apply LES in realsize industrial buildings, it is now intended to extend the validity of these results to higher scale configurations. Scaled-up setups of the chamber studied in this paper are currently about to be studied experimentally and numerically to confirm these results.

■ AUTHOR INFORMATION

Corresponding Author *E-mail: quillatre@cerfacs.fr.

Notes

Figure 1 .

 1 Figure 1. Left: Explosion chamber configuration of Masri et al. 9 The vessel is orientated vertically in the experiment: the bottom end of the vessel is on the left of the figure and the top end on the right. Right: Classification of the studied configurations.

Figure 2 .

 2 Figure 2. Multiscale aspect of explosion phenomena.

Figure 3 .

 3 Figure 3. Images of flame propagation in Config2 using C 3 H 8 . Left: experimental video captured images (false colorized) extracted from Gubba et al. 6 Right: LES images (colored by reaction rate). Time intervals between two successive images are kept constant between experiments and LES.

  ω̇s gs being the sub-grid-scale reaction rate. These values can be related to the efficiency function , by

Figure 4 .

 4 Figure 4. Up: Comparison between combustion rate V ̇comb and venting rate V ̇vent as a function of time. Bottom: Overpressure inside the chamber as a function of time.

Figure 5 .

 5 Figure 5. Resolved amount of reaction rate η. Equation 6 -Config3 -C 3 H 8 .

Figure 6 .

 6 Figure 6. Contribution of the resolved part of the turbulent kinetic energy ε in the flame front -Config3 -C 3 H 8 .

Figure 7 .

 7 Figure 7. Comparison of overpressure signals between LES (plain) and experiments. 9 Mean signal (dashed) and statistical envelope (gray background) from 5 to 20 ms -C 3 H 8 -Config1, 2, and 3.

Figure 8 .

 8 Figure 8. Comparison of overpressure signals between LES (plain) and experiments. 9 Mean signal (dashed) and statistical envelope (gray background) from 0 to 15 ms -Config3 -CH 4 , C 3 H 8 , and H 2 .

a

  Coefficients are related to the Arrhenius formulation of the reaction rate:

Figure 9 .

 9 Figure 9. Influence of chemistry modeling. Left: Comparison of overpressure signals between LES and experiments. 9 One-step mechanism LES (plain with triangles), two-step mechanism LES (plain with circles), experimental mean signal (dashed) and statistical envelope (gray background). Right: Flame front position for LES calculations. One-step mechanism LES (plain with triangles), two-step mechanism LES (plain with circles) -Config3 -C 3 H 8 .

Figure 10 .

 10 Figure 10. Influence of transport modeling. Left: Comparison of overpressure signals between LES and experiments. 9 Simplified transport mechanism LES (plain with squares), realistic transport mechanism LES (plain with circles), experimental mean signal (dashed), and statistical envelope (gray background). Right: Flame front position for LES calculations. Simplified transport mechanism LES (plain with squares), realistic transport mechanism LES (plain with circles) -Config3 -C 3 H 8 .

Table 1 .

 1 Averaged Values inside the Flame Front of Thickening and Efficiency Factors

			thickening -			efficiency ,
		Config1 Config2 Config3 Config1 Config2 Config3
	CH 4			5.9			1.78
	C 3 H 8	7.2	7.2	7.2	1.38	1.67	1.84
	H 2			21.5			1.94

Table 2 .

 2 Laminar Flame Speed for Different Fuels fuel type CH 4 at Φ = 1 C 3 H 8 at Φ = 1 H 2 at Φ = 0.7 S L 0 (cm/s) at P atm and T atm 36.3 38.4 128

Table 3 .

 3 Two-Step Reduced Mechanism for C 3 H 8 -Air Combustion a

	n	reaction	A (cm 3 /mol•s) E a (cal/mol)
	1	C 3 H 8 + 3.5O 2 → 3CO + 4H 2 O	1.33 × 10 12

Table 4 .

 4 One-Step Reduced Mechanism for C 3 H 8 -Air Combustion

	n	reaction	A (cm 3 /mol•s) E a (cal/mol)
	1 C 3 H 8 + 5O 2 → 3CO 2 + 4H 2 O	3.09 × 10 12	3.347 × 10 4
	forward: n C3H8	F = 0.569 and
	n O2 F = 1.097

Table 5 .

 5 Chemistry Mechanism Properties

	properties at Φ = 1, P atm , and T atm	laminar flame speed S L 0 (m/s)	burnt gas temperature T ad (K)
	detailed mechanism (GRI-MECH 33 )	38.4	2275
	reduced two-step	38.4	2289
	mechanism
	reduced one-step	38.4	2400
	mechanism

Table 6 .

 6 Two-Step Reduced Mechanism with Simplified Transport for C 3 H 8 -Air Combustion

	n		reaction		A (cm 3 /mol•s) E a (cal/mol)
	1	C 3 H 8 + 3.5O 2 → 3CO + 4H 2 O	1.41 × 10 12	4.15 × 10 4
		forward: n C3H8	F = 0.55 and n O2	F = 0.9
	2	CO + 0.5O 2 ↔ CO

The authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

This work was performed using HPC resources from GENCI-IDRIS. This work was supported by Total and ANRT under the CIFRE-2010-597. The authors thank Professor A. Masri (University of Sydney) for providing experimental data.