Arthur Degeneve
email: arthur.degeneve@centralesupelec.fr

R Vicquelin

C Mirat

B Labegorre

P Jourdaine

J Caudal

T Schuller

Scaling relations for the length of coaxial oxy-flames with and without swirl

Keywords: Coaxial jet, Oxy-combustion, Swirl, Flame length, Turbulence

An extensive experimental study is carried out to analyze scaling laws for the length of methane oxy-flames stabilized on a coaxial injector. The central methane fuel stream is diluted with N 2 , CO 2 and He. The annular air stream is enriched with oxygen and can be impregnated with swirl. Former studies have shown that the stoichiometric mixing length of relatively short flames is controlled by the mixing process taking place in the vicinity of the injector outlet. This property has been used to derive scaling laws at large values of the stoichiometric mixture fraction. It is shown here that the same relation can be extended to methane oxy-flames characterized by small values of the stoichiometric mixture fraction. Flame lengths are determined with OH* chemiluminescence measurements over more than 1000 combinations of momentum ratio, annular swirl level and composition of the inner and outer streams of the coaxial injector. It is found that the lengths of all the flames investigated without swirl collapse on a single line, whose coefficients correspond to within 15% of flame lengths obtained for fuel and oxidizer streams at much larger stoichiometric mixture fractions. This relation is then extended to the case of swirling flames by including the contribution of the tangential velocity in the flow entrainment rate and is found to well reproduce the mixing degree of the two co-axial streams as long as the flow does not exhibit a vortex breakdown bubble. At higher swirl levels, when the flow features a central recirculation region, the flame length is found to also directly depend on the oxygen enrichment in the oxidizer stream.

Scaling relations for the length of coaxial oxy-flames with and without swirl Paper length: 8 pages, Equivalent word count: 6200 words Supplemental material: One page of supplemental material is added.

Introduction

Many industrial combustors are powered by co-axial injectors in which the oxidizer and fuel streams are injected separately and expand in a large combustion chamber. It is well known that the mixing degree between the two streams controls the flame length and therefore it has been the topic of numerous investigations [1][2][3][4][5][6][7][8][9][10]. The development of technologies such as oxy-combustion needs however to broaden the investigated range of operations and to deepen knowledge of the mixing mechanisms of coaxial jets, in which one of the jets may eventually feature a swirling motion.

It is at this point worth reviewing the current state of knowledge on parameters altering the flame length from co-axial injectors. Regarding diffusion flames, the distance from the injector needed to obtain a stoichiometric fuel and oxidizer mixture well defines the flame length [11]. The pioneering works from Hawthorne et al. [12] and Peters and Gottgens [11] have established mixing rules for free single jets exhausting in quiescent atmosphere. For non-buoyant turbulent coaxial jets, mixing is controlled far from the injector by the self-similar behavior of the resulting turbulent jet [3,13]. This socalled far-field approach allows to well reproduce the topology of methane diffusion flames with high impulsion of the fuel inner jet [4][5][6]. In this case the stoichiometric mixture fraction z st remains small, leading to long flames far from the inner jet potential core [4,5].

In many applications, like in H 2 -O 2 rocket engines or oxy-combustion, the stoichiometric mixture fraction z st takes relatively large values and the difference between the potential core and stoichiometric lengths reduces. In this regime, another mechanism prevails and mixing is controlled by engulfment of large scale instabilities developing in the shear layer between the central and annular jet streams in the vicinity of the jet potential core [8,14]. The developments made with this nearfield approach has led to the following scaling law for the stoichiometric flame length L s [8,9]:

L S d 1 = c 1 (J 1/2 X st ) -1 + c 0 ( 1
)
where c 0 and c 1 are positive constants, d 1 is the central injector diameter, X st is the molar stoichiometric mixture fraction and J = ρ 2 u 2 2 /ρ 1 u 2 1 corresponds to the momentum ratio between the two streams. This relation was found to successfully reproduce the stoichiometric mixing length L s of co-axial streams with an air jet in the center surrounded by a co-flow of H 2 or CH 4 in non reacting conditions [10].

These results established for cold flows have been extended to consider effects of heat release. The dominant effect of combustion lies in the drop of flow entrainment due to the drop of density in the burnt gases [15]. Tacina and Dahm [15,16] found that mixing of the inner fuel and the outer oxidizer streams in reacting conditions is equivalent to the non reacting mixing process of the inner fuel with an equivalent outer oxidizer characterized by an effective higher temperature T eq 2 . Assuming that the density difference between the two streams due to molar weight is small compared to the effect of temperature, the density of the equivalent jet is defined as ρ eq 2 = (T 2 /T eq 2 )ρ 2 [15]. This method is used in [9] to collapse data for O 2 /H 2 flames with hydrogen in the outer stream (z st = 0.89, X st = 0.33). Results in non reacting and reacting conditions collapse when the momentum ratio J is replaced by its equivalent J eq defined as:

J eq = ρ eq 2 ρ 1 u 2 2 u 2 1 ( 2
)
Scaling laws for the length of methane oxy-flames have been reported so far with contrasted results. Effects of dilution in both the inner and outer streams have not been systematically investigated [17][18][19][20]. In many studies, only heuristic corrections are presented when switching from air to oxygen operation [20] and investigations are often limited to regimes with velocities in the inner fuel stream much higher than in the annular stream (J 1) [17][18][19][20], resulting in long flames where the near-field mechanism does not control mixing. Few studies with higher momentum in the annular stream have been conducted with oxy-flames [19] and N 2 -diluted hydrogen flames [13], but no scaling laws are reported. A unified approach consistent with both air-and oxy-condition operations is still missing.

This lack of data has motivated this study conducted for a wide set of operating conditions in which the gas

Q 1 (fuel) Q 2,z L 2H l (a) (b) d 1 d 2 Q 2, X O 2 ,2 X CH 4 ,1
Air composition in the central and annular streams and the momentum ratio can separately be varied. The first objective of this article is to examine the length of coaxial oxy-flames with a large set of dilution rates in the oxidizer and fuel streams. Attempts are then made to collapse the data with theoretical elements. The second objective is to analyze effects of the swirl level in the annular jet and derive a model which accounts for the resulting turbulence enhancement.

Experimental setup

The injector sketched in Fig. 1 comprises a central tube of inner diameter d 1 = 6 mm and thickness e = 1 mm, wherein methane is eventually mixed with a diluent (N 2 , CO 2 or He), and an annular injection channel of outer diameter d 2 = 20 mm, with an adjustable O 2 /N 2 gas composition. A swirl motion can be imposed to the annular stream thanks to tangential slits. The axial and tangential volume flowrates injected in the annular channel are designated respectively by Q 2,z and Q 2,θ . Assuming a solid-body rotation, the geometrical swirl number S 2 in the annular channel is given by:

S 2 = π 4 Hd 2 NlL 1 -(d 2 /d 1 ) 4 1 + Q 2,z /Q 2,θ (3)
where H is the distance separating the tangential injection channels from the burner axis, l the width and L the height of the N tangential injection channels. This device was designed to produce geometrical swirl numbers ranging from S 2 = 0 to 1.73 with N = 2 slits. The central fuel stream is not swirled in this study. The coaxial jet exhausts in quiescent air at ambient pressure, above a back plane at room temperature T = 300 K, so as to avoid effects of confinement and heat transfer to the walls. The coaxial injector outlet is elevated 5 mm above the back plane to ease visualization and there is no recess between the central and outer injector outlets. This altitude defines the axial origin z = 0.

The flame length is defined as the furthest point on the axis with a combustion reaction [12,17]. In many cases, the length of diffusion flames is determined by recording its global emission in the visible band [12,19,20] or the emission of selected intermediate radicals [9,17,18]. In these studies, the flame length is commonly correlated to the stoichiometric mixing length L S . Given the large range of operating conditions targeted in this work, it is first worth exploring if the OH* emission signal is a good tracer for the length of oxy-flames and if it is well correlated to the stoichiometric mixing length L S .

Numerical simulations of steady one-dimensional counterflow methane oxy-flames were carried out with the Grimech 3.0 mechanism [21] in [22] and completed with OH* chemistry in [23]. These authors verified that the OH* peak emission from diffusion oxy-flames lies very close to stoichiometry at z = z st for a large range of oxygen enrichments and strain rates. Numerical simulations have also been carried out in this study to verify this property for selected operating points with the same Grimech 3.0 mechanism completed with the chemistry of OH* [23]. They are provided as supplementary material and confirm that both peaks of temperature and OH* lie close to the flame front at z = z st , when compared to the flame thickness. Self-similar counterflow equations were solved with an in-house code with identical transport and thermodynamics properties as in the CHEMKIN package and similar numerical algorithms.

The OH* images are recorded with a 16 bit intensified CCD camera (ICCD, Princeton, PI-MAX4, 1024 × 1024 pixels) equipped with an UV objective (Nikkor 105 mm) and a 10 nm bandpass filter centered at 310 nm (Asahi XBPA310-Bandpass). Line of sight integrated averaged fields of OH* emission are determined from 30 snapshots each with an exposure time of 200 ms. Statistical convergence of the mean OH* intensity is systematically verified.

The Otsu thresholding method [24] is chosen here to infer the flame front location from the OH* images as in [25,26]. Examples without swirl are shown in Fig. 2 and effects of swirl are illustrated in Fig. 3. The stoichiometric mixing length L S is determined as the highest point of the Otsu contour, which is found to lie along the burner centerline, except for the highly swirled flames featuring a central recirculation zone. A sensitivity analysis has been carried out to examine changes of L S when varying the image brightness and contrast. Different exposure times and camera apertures were tested. It appears that the largest variations of L S are limited to within 5%. The method is also validated in section 4 by comparing the results with the stoichiometric mixing length determined from measurements made with acetone laser induced fluorescence [9,10].

Operating conditions

Changes of the flame length are investigated by modifying the momentum ratio J between the two streams and the stoichiometric (molar) mixture fraction X st . This is made by varying the bulk flow velocity u 1 from the central stream and the gas composition of the central and annular streams (see Tab. 1). All the following experiments are conducted at high Reynolds numbers Re 2 to get a fully developed turbulent flow in the annular channel. In this case, the flame length L S does not depend on Re 2 [9]. The fuel in the central jet is also always the deficient reactant to avoid combustion between the excess of fuel in the burnt gases with ambient air. Therefore, only flames at globally lean conditions are considered here. Aside from their effect on the flame length, O 2 enrichment and fuel dilution are known to greatly affect flame stabilization. For instance, Wanatabe et al. [27] showed that replacing N 2 by CO 2 in twin premixed oxy-flames can increase or decrease the extinction strain rate depending on the equivalence ratio. Chen and Axelbaum [28] showed that increasing the stoichiometric mixture fraction z st drastically increases the extinction limits of counterflow diffusion flames. For these reasons, only flames well attached to the nozzle are considered in this work. This has been achieved with a high degree of O 2 enrichment in the outer stream: X O 2 ,2 ≥ 0.37 (except for D6 dataset), and without using any bluff-body, confinement or pilot flame, as in [19,20]. Two decades of the momentum ratio J are covered in the following experiments. High levels of fuel dilution in the inner stream and various levels of O 2 enrichment in the outer injection channel are examined.

Typical results for non-swirling flames are presented in Fig. 2. They all feature a diffusion flame structure attached to the edge of the central injector and their length changes by a factor four. Swirled flames are shorter in Fig. 3 and their opening angle widens as the swirl number S 2 increases.

The dimensionless streamwise coordinate ξ F from [1] based on the mixing length is of the order of 0.5 indicating that buoyancy effects can be neglected for all configurations explored in this work.

Oxy-flames without annular swirl

In the following experiments the swirl level is set to zero S 2 = 0. The momentum ratio J and the methane molar fraction X CH 4 ,1 in the central stream are varied. The flame length L s deduced from the images is made dimensionless with the central injector diameter d 1 and its evolution is examined as a function of the near-field parameter Φ m = (J eq 1/2 X st ) -1 , where J eq denotes the equivalent momentum ratio from Eq. ( 2).

Figure 4a shows results from dataset D1 in Tab. 1 in which the oxygen enrichment is set to X O 2 ,2 = 0.47 in the annular stream. Data are plotted for fixed methane molar fractions X CH 4 ,1 varying from 0.40 to 0.93 in the inner jet, the fuel being diluted with nitrogen. Effects of the oxygen enrichment in the annular stream with respect to air are examined in Fig. 4b for X O 2 ,2 =0.37 to 0.67. The spread of the color legends emphasizes the broad ranges covered by X CH4,1 and X O 2 ,2 in Fig. 4.

Flame lengths L S from datasets D1 and D2 all collapse on the same curve in Fig. 4 when results are plotted as a function of Φ m . Modifying the composition X st or the momentum ratio J has a distinct influence on L S . Increasing the fraction of nitrogen N 2 in the inner stream or of oxygen O 2 in the annular jet increases the stoichiometric mixture fraction X st , which moves the stoichiometric mixing length L S closer to the potential core of the central jet. Secondly, increasing the annular momentum ratio J increases the level of turbulent diffusion, which reduces the length of the potential core of the central jet and enhances mixing of both streams. These figures confirm that the two mechanisms successfully account for the measured flame lengths through the use of Φ m . The range of stoichiometric mixture fractions 0.09 ≤ z st ≤ 0.43 covered in this study is wider than previous studies [9,10] conducted with H 2 /O 2 and 0.80 ≤ z st ≤ 0.95. The linear relationship of the flame length with Φ m is still valid for the lowest z st value corresponding to X CH 4 ,1 = 1 and X O 2 ,2 = 0.37. Equation ( 1) derived with arguments regarding the development of shear layer instabilities near the inner jet potential core, a situation where z st is large [8], appears to remain valid for a large range of z st values. This observation deserves further investigations to understand its roots.

Figures 4a and4b also indicate that the slope of L S /d 1 deviates from the linear behavior when Φ m ≥ 20. The saturation observed at high Φ m values does not appear to be a bias of diagnostic. The OH* emission is not deteriorated for flames 6 and 7 in Fig. 2 and the Otsu contour still well delineates the flame reaction front. In the asymptotical regime reached for vanishingly small impulsion ratios J → 0, turbulent mixing is solely con- trolled by the inner jet velocity u 1 and the flame length does not depend anymore on the velocity ratio u 2 /u 1 . In this limit case, one recovers the scaling law from a free single jet exhausting in quiescent air [11,12]. The transition for the relevant shear velocity from u 2 to u 1 was already mentioned in [8], but could not be observed in [9,10], where flames smaller than L S /d 1 = 55 are investigated. The deviation from the linear behavior for L S /d 1 ≥ 75 in Fig. 4 is attributed to the transition in a regime where mixing is controlled by the inner jet turbulence.

In the linear regime observed for Φ m < 20, a regression of the data D m in Figs. 4a-b yields :

D m : L S d 1 = b 1 (J eq 1/2 X st ) -1 + b 0 ( 4
)
where b 1 = 3.1 and b 0 = 10.7. These results obtained for N 2 -diluted CH 4 /O 2 -enriched flames are found to be in good agreement with measurements from Schumaker and Driscoll [9] who report b 1 = 2.7 in their H 2 /O 2 configuration, in which the fuel is the annular jet. Besides, they also found that the constant b 0 used to improve the linear fit mainly depends on the injector geometry and therefore cannot be compared. The line D m from Eq. ( 4) serves as a reference in the remaining part of this article.

Oxy-flames with annular swirl

Effects of the swirl level in the annular stream are now examined. Results are first presented for the dataset D3 in Tab. 1 obtained for a moderate swirl level S 2 = 0.4 and a slight oxygen enrichment X O 2 ,2 = 0.37 of the oxidizer stream. They are grouped in Fig. 5. Nitrogen N 2 used to dilute the central jet is here alternatively replaced by CO 2 and He exhibiting large differences in molar weights. The measured flame lengths L s collapse in these cases again on a single curve in Fig. 5 when data are plotted as a function of Φ m , which therefore remains the proper parameter to switch from one fuel stream diluent to another. The stoichiometric mixing lengths lie however below the line D m found for the non-swirling flames (S 2 = 0). This means that addition of the swirl S 2 = 0.4 has uniformly reduced the stoichiometric lengths L s of all flames of dataset D3. Swirling the annular stream is indeed known to enhance the mixing rate and reduce the stoichiometric mixing length [2,29].

Changes of the annular swirl level S 2 are studied in Fig. 6 for the dataset D4 in Tab. 1 obtained for a fixed O 2 enrichment X O 2 ,2 = 0.37 in the outer stream. Each dataset color corresponds to a different swirl number S 2 . For a fixed swirl level S 2 , the flame lengths L S collapse again on a single curve as J and X st vary, even at high swirl numbers S 2 > 0.6. For a fixed value of the mixing parameter Φ m , the flame length reduces monotonically as the swirl number S 2 increases. This behavior may be accounted for by considering the azimuthal velocity u 2,θ to estimate the turbulent velocity scale within the annular jet:

u ∼ |u 2 | u 2 1 + (kS 2 ) 2 1/2 ( 5
)
where k is defined as u θ,2 /u 2 = kS 2 . The definition of 0 10 20 30 40 the geometrical swirl number in Eq. ( 3) with a solidbody rotation yields k = 2/ 1 + (d 1 /d 2 ) 2 . This theoretical value is retained in Eq. ( 5) to account for turbulence enhancement due to swirl in the outer jet. This allows to define a new mixing variable Φ S 2 :

Φ m = (J eq 1/2 X st ) -1 0
Φ S 2 = Φ m 1 + (kS 2 ) 2 -1/2 (6)
The data in Fig. 6b again roughly collapse on D m established for non-swirling flames provided that Φ m is replaced by Φ S 2 in Eq. ( 4) :

L S d 1 = b 1 Φ S 2 + b 0 (7)
where the values for b 0 and b 1 are left unchanged. These results indicate that the near-field approach [8,14] can be extended to the case of swirling flames, in which the main effect of swirl is the enhancement of turbulent mixing between the central and annular jets. A closer look at the data shows that the scaling law worsen for high swirl levels. Increasing the outer stream swirl number S 2 from small to moderate values leads to an increase of turbulent diffusion between the two jets, but a sudden change of the flow topology appears at higher values due to vortex breakdown with the formation of a central recirculation zone (CRZ). As reported in [2], the flame length drastically drops down to L S /d 1 = 5 at high swirl levels. This flow transition is examined by considering experiments made for the dataset D5 in Tab. 1. In these experiments the mixing parameter Φ m (corresponding to Φ S 2 =0 ) remains constant and only the annular swirl level S 2 is varied. The flame lengths are reported as a function of S 2 in Fig. 7 at different fixed values for Φ m . Predictions from Eq. ( 7) are superimposed to the experimental data. Increasing the annular swirl S 2 leads to a decrease of the flame length L S in a nonlinear fashion, which is well reproduced by Eq. ( 7) for swirl numbers lower than S 2 ≤ 0.5.

The extended model Eq. ( 7) also well reproduces the stoichiometric mixing length of the highly swirled flames, but only at high Φ m values. When both the annular impulsion ratio J and the swirl number S 2 are high enough, vortex breakdown drastically alters the flame topology with the formation of a CRZ leading to a much shorter flame length. This is exemplified by flame a featuring a CRZ compared to flames d and e remaining in the jet-like regime in Fig. (3). This regime transition is in agreement with results from Chen and Driscoll [30], who mapped the apparition of the CRZ in a (1/J, S 2 ) diagram. They found that for S 2 > 0.7, the transition from a jet-like to a strongly recirculating regime occurs for J > J c 1, and the critical threshold J c depends on S 2 for S 2 < 0.7. This condition is reproduced in Fig. (7), where data for flames expected in a jet-like regime are represented by symbols with a thickened black contour. Flames in a regime with a CRZ according to [30] are drawn with a thin contour. Equation ( 7) reproduces the measured flame lengths only in the jet-like regime. This corroborates that Eq. ( 7) well models mixing enhancement in the jet-like regime due to the additional turbulence generated by the rotation of the outer swirling jet, but does not hold in the CRZ regime, in which mixing is mainly controlled by recirculation of the gases in the central region of the flow.

Physics based scaling relations for the flame length in highly recirculating flows have been proposed for methane/air flames by means of velocity measurements in the recirculating vortex [2,30]. It is found here that fuel dilution and oxygen enrichment further increase the complexity of the mixing dynamics for these flames featuring a central vortex bubble, for which Eq. ( 7) fails to reproduce the stoichiometric mixing length.

Complementary measurements corresponding to the dataset D6 in Tab. 1 are analyzed for a swirl number S 2 = 0.8. The O 2 enrichment in the annular stream 7). Symbols with thickened contour designate flames in jet-like regime according to [30]. is varied from X O 2 ,2 = 0.33 to 0.57. The central fuel stream is diluted with N 2 with X CH 4 ,1 = 0.30 to 0.90. Flame lengths are plotted in Fig. 8 with respect to Φ S 2 , which again appears as a suitable parameter associated to changes of the central fuel jet composition X CH 4 ,1 and momentum ratio J, but the results are found to also depend on the oxygen enrichment X O 2 ,2 in the annular stream. Flames corresponding to β, γ and δ in Fig. 8 are displayed in Fig. 3. They share the same value Φ S 2 but feature different stoichiometric mixing lengths. Flame β is small and compact in Fig. 3, whereas flame δ is much longer. As this feature is left unexplained by the Eq. ( 7) developed in this work, modifications of the mixing degree are attributed to the presence of the central recirculating bubble, as mentioned in [2,30]. Besides, the diagram established for the CRZ of methane-air flames in [30] does not account for the following observation with oxy-flames: flame γ has a higher momentum ratio J than flame β, but flame β is smaller than flame γ in Fig. 3.

A more detailed description of the interaction between the flow in the CRZ and the oxidizer annular stream is needed to explain the topology of these flames, but the O 2 concentration in the annular stream seems to be an independent parameter that needs to be considered for highly-recirculating flames.

Conclusion

Modifications of the flame length L s of methane oxygen enriched flames stabilized on coaxial injectors have been investigated over a large set of parameters using OH* chemiluminescence measurements. Effects of dilution of the central fuel stream have first been investigated by varying the fuel molar fraction 0.33 ≤ X CH 4 ,1 ≤ 1 and the diluent nature N 2 , CO 2 and He. Secondly, experiments have been reported for momentum ratio J = ρ 2 u 2 2 /ρ 1 u 2 1 covering a broad range 0.2 ≤ J ≤ 16. Third, effects of oxygen enrichment in the annular oxidizer stream have been investigated from 0.37 ≤ X O 2 ,2 ≤ 0.67. Finally, the swirl number of the annular stream has also been varied up to S 2 ≤ 1.5. This parametric analysis enabled to explore the lengths of oxy-flames through more than 1000 combinations.

In the absence of swirl, the relation proposed in [9] has been used to successfully reduce the data for the flame lengths L S on a single line as a function of a single dimensionless mixing parameter corresponding to the inverse of the product of the stoichiometric mixture fraction X st and the square root of the equivalent momentum ratio J eq . The slope of this line has been found to match the one reported for O 2 /H 2 flames with 0.80 ≤ z st ≤ 0.95 in [9,10] within 15%. The current experiments validate this relation for oxy-methane flames featuring a smaller stoichiometric mixture fraction 0.09 ≤ z st ≤ 0.43.

With swirl, the same type of relation with a correction taking into account enhancement of turbulent diffusion due to the azimuthal flow component has been found to well reproduce the length of swirling flames before the apparition of a central vortex bubble. At high swirl levels characterized by a central recirculating flow, it has been found that the flame length also separately depends on the oxygen enrichment in the oxidizer stream.

The current work provides a large database for the lengths of coaxial oxy-flames. Given the broad range of operating conditions, the extended scaling law Eq. ( 7) may be used to ease the design of co-axial injectors from oxy-burners, with eventually a slight swirl in outer stream. Furthermore, the data gathered in this work also yields a large validation database for simulation of stoichiometric mixing lengths in oxy-flames.

A.

  Degeneve a,b,1 , R. Vicquelin a , C. Mirat a , B. Labegorre b , P. Jourdaine b , J. Caudal b , T. Schuller a,c a Laboratoire EM2C, CNRS, CentraleSupélec, Université Paris-Saclay, 3 rue Joliot Curie 91192 Gif Sur Yvette cedex, France b Air Liquide, Centre de recherche Paris Saclay, Chemin de la Porte des Loges, B.P. 126, 78354 Les Loges en Josas, France c Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, INPT, UPS, Toulouse, France Colloquium: Stationnary combustion systems and control of green house gas emissions Word Count: Method 2, counting from a twocolumn formatted version of the paper.

Figure 1 :

 1 Figure 1: Sketch of the coaxial injector. (a) Axial cut. (b) Transverse cut through the swirler.

Figure 2 :

 2 Figure 2: OH* intensity distribution of selected flames from dataset D2 in Tab. 1 and increasing Φ m values. Numbers refer to Fig. 4b. White contours indicate the Otsu threshold used to determine L S .

Figure 3 :

 3 Figure 3: OH* intensity distribution of selected flames from datasets D5 and D6 in Tab. 1 with high annular swirl levels S 2 . Latin letters refer to data shown in Fig. 7. Greek letters make references to data shown in Fig. 8. White contours indicate the Otsu threshold used to determine L S .

Figure 4 :

 4 Figure 4: Flame lengths from datasets D1 (top graph) and D2 (bottom graph) in Tab. 1 obtained without swirl S 2 = 0. D m : Eq. (4).

Figure 5 :

 5 Figure 5: Flame lengths from dataset D3 in Tab. 1 obtained for X O 2 ,2 = 0.37 and S 2 = 0.4. D m : Eq. (4).

Figure 6 :

 6 Figure 6: Flame lengths from dataset D4 in Tab. 1. Top : Original scaling Φ m from[8,14]. Bottom : New model Φ S 2 accounting for the swirl level S 2 in the outer stream. D m : Eq. (4).

Figure 7 :

 7 Figure 7: Flame lengths from the dataset D5 in Tab. 1 as a function of the swirl number S 2 for different values of mixing parameter Φ m . Solid lines correspond to predictions from Eq. (7). Symbols with thickened contour designate flames in jet-like regime according to[30].

Φ S 2 = Φ m 1 + (kS 2 ) 2 - 1

 21221

Figure 8 :

 8 Figure 8: Flame lengths from the dataset D6 in Tab. 1 obtained for S 2 = 0.8 and different oxygen molar fractions X O 2 ,2 in the annular stream. D m : Eq. (4).

Table 1 :

 1 Ranges of operations explored grouped by dataset. N f is the number of investigated flames in each group. The central CH 4 stream and the annular O 2 stream are diluted with N 2 except for dataset D4 where fuel dilution is also made with He or CO 2 .

	Dataset	J	X CH 4 ,1	X O 2 ,2	X st	Z st	T s [K]	S 2	Re 2	N f
	D1	0.2-16	0.40-0.93	0.47	0.20-0.37	0.12 -0.31	2630-2780	0	12000	61
	D2	0.2-16	0.34-1	0.37-0.67	0.16-0.50	0.09-0.43	2570-2930	0	12000	174
	D3	0.2-16	0.33-1	0.37	0.16-0.36	0.09-0.31	2450-2660	0.4	12000	138
	D4	0.2-16	0.33-1	0.37	0.16-0.36	0.09-0.31	2450-2670	0-1	15000	294
	D5	0.5-12	0.62-1	0.37	0.16-0.23	0.09-0.17	2600-2660	0-1.5	15000	83
	D6	0.4-16	0.30-0.90	0.33 -0.57	0.12-0.37	0.09-0.31	2260-2864	0.8	15000	231

Acknowledgments

This work is supported by the Air Liquide, Cen-traleSupélec and CNRS Chair on oxy-combustion and heat transfer for energy and environment and by the OXYTEC project (ANR-12-CHIN-0001) from l'Agence Nationale de la Recherche.


Jérôme Denis
email: jerome.denis@mines-paristech.fr

David Pontille
email: david.pontille@mines-paristech.fr

The Multiple Walls of Graffiti Removal Maintenance and Urban Assemblage in Paris

In this chapter, we investigate graffiti removal as a crucial component of urban walls mundane governance. Drawing on an ethnography in Paris, we show that graffiti removal accomplishes a series of consecutive, sometimes simultaneous, ontological enactments that perform contrastive versions of urban walls. In official documents and during removal interventions, the 'same' wall can indeed be handled as the object of a specific kind of maintenance, as a quantified surface, as a controlled space of public expression, as part of an official decorum, and as an uncertain material composite. These versions offer a particularly interesting case of what Deleuze and Guattari call 'collective assemblages of enunciation', where the erasure of unofficial public inscriptions implies the practical articulation of the material properties of walls (stone, brick, paint, concrete…), graffiti (ink, paint…) and removal techniques (water, sponge, paint, sand, rags, chemical products…). Such ontological multiplicity is precious to better understand both the modes of existence of walls in the city and the politics of public order maintenance.

Pick a wall in the streets of Paris, let's say in the lively third arrondissement. Look at it. It's a nice wall, a facade of one of those old buildings that make the charm of the City of Light. Depending on which author inspires you most -Charles Baudelaire, Georges Perec, Walter Benjamin -you could use many different words to describe it, to explain what it looks like, what it is made of, and to characterize its place in the neighborhood. But let's complicate things a little and consider this wall as it looked four days ago, or at least at the wall that could be seen at the same address four days ago. Now, there are numerous inscriptions on it. Again, depending on your background and your expertise, you may or may not be able to read this lettering, though you would certainly agree to name them 'graffiti'. But let's go back to the wall. Is it the same wall? Hard to say. Maybe you shouldn't decide right away. Especially if, again, you move to a new vantage point, and find yourself two days after the apparition of the graffiti. Early in the morning, you see a man already at work, spraying some liquid on the wall and, as his intervention unfolds, you witness the progressive disappearing of the graffiti. A few minutes later, the wall is exactly as you found it at the beginning of this paragraph (that is, chronologically, two days after the man's intervention). This little thought experiment is an invitation to think, and look at, walls and graffiti in a slightly different way than usual. It shows how hard it can be to identify stable properties that satisfyingly could be used to describe urban walls. Indeed, in this story, the very materiality of the hypothetical wall changed over the days. First, one or several graffiti writers added some paint to the amalgamation of bricks, concrete, stone and paint, that the wall was made of. Then, a worker spouted water, maybe sand and chemical products, on it, progressively erasing the letters and the words the wall had borne for a few days. And even after this intervention, we can doubt that the wall simply recovered its 'original' material properties. The process certainly affected some of them and probably added new ones. All these changes occurred in just a few days -imagine if we have observed the wall for a whole year, or a decade…

As for graffiti, its traditional descriptions are shifted by our story as well. Of course, the inscriptions that once appeared on the wall take place in a social world and are the outcome of cultural practices that come with their own norms and values. They can also be studied as speech acts, or writing practices -practices that have become widespread in large cities around the world for decades now. But in our story, the graffiti is described from a very particular standpoint, foregrounding a dimension that has rarely been considered as such in social sciences: its removal.

In this chapter, we aim to investigate this removal, and the policy that sets the conditions of its accomplishment. Removal is a mundane operation, yet one that contributes crucially to the constitution of contemporary urban realities. Studying graffiti removal, we believe, is notably a way to better understand the modes of existence of walls in the city. To do so, we explore the case of Paris in this chapter, drawing on an ethnographic study during which we gathered the main legal and contractual documents of the current graffiti removal policy of the city, made observations of removal interventions and verification processes, and conducted in-depth interviews within the municipality and two of its contractors. Our exploration stands at the intersection of two growing research areas: studies focused on 'urban assemblages' on the one side [START_REF] Brenner | Assemblage urbanism and the challenges of critical urban theory[END_REF]Farías and Bender, 2010;[START_REF] Mcfarlane | Assemblage and critical urbanism[END_REF], and repair and maintenance studies on the other [START_REF] Henke | The mechanics of workplace order: Toward a sociology of repair[END_REF][START_REF] Graham | Out of order: Understanding repair and maintenance[END_REF][START_REF] Denis | Maintenance work and the performativity of urban inscriptions: The case of Paris subway signs[END_REF][START_REF] Jackson | Rethinking repair[END_REF]. Far from being a reductionist instrument of territorial control, the Paris graffiti removal policy is indeed a complex assemblage of urban maintenance that performs economical, political, technical and moral orders. The walls of Paris are ones of the crucial components of this assemblage, featuring as complex, multiple artifacts. In this chapter, we show that graffiti removal produces a series of ontological enactments [START_REF] Smith | The social construction of documentary reality[END_REF][START_REF] Woolgar | Mundane governance: Ontology and accountability[END_REF] that practically perform various versions of the walls [START_REF] Mol | Ontological politics. A word and some questions[END_REF]).

In the following sections, we first outline the main streams of research we draw on: urban assemblages, maintenance and repair, and ontological multiplicity. We then briefly describe how graffiti removal is organized in Paris. We identify five distinct ways through which graffiti removal policy enacts the walls of the city: as an object of maintenance, as a surface, as a space for public expression, as an official decorum, and as a material composite. Finally, we conclude by foregrounding how the notion of assemblage (agencement) in Deleuze and Guattari's terms (1980) helps to understand the way walls are governed and enacted in the city.

Urban Assemblages, Maintenance and Multiplicity

For several years now, at the crossroads of human geography and actor-network theory, the notion of 'assemblage' has been at the center of many debates (Farías & Bender 2010;McFarlane 2011a[START_REF] Mcfarlane | The city as assemblage: Dwelling and urban space[END_REF][START_REF] Rankin | Assemblage and the politics of thick description[END_REF][START_REF] Kamalipour | Assemblage thinking and the city: Implications for urban studies[END_REF]. Borrowed from [START_REF] Deleuze | Milles plateaux[END_REF], the term and its revival owe a lot to DeLanda, who has promoted the virtue of 'assemblage thinking' [START_REF] Delanda | A new philosophy of society: Assemblage theory and social complexity[END_REF]2016). Its use in urban studies is also largely indebted to the earlier contributions of human geography and actor-network theory, particularly to Thrift's work (Pile and Thrift, 2000;[START_REF] Amin | Cities: Reimagining the urban[END_REF]) and Latour's foray into urban fabrics [START_REF] Latour | Paris ville invisible[END_REF].

Assemblage thinking is an attitude that allows for a 'non-reductionist' approach to the city [START_REF] Delanda | A new philosophy of society: Assemblage theory and social complexity[END_REF] and has contributed to 'decentering' urban studies' traditional objects and issues (Farías and Bender 2010). It particularly draws attention to two hitherto neglected aspects of cities: their socio-material heterogeneity, and the circumstantial character of their transformations in time. Urban life, in this perspective, is anything but a stable reality that could be analyzed 'at rest'; cities are studied as composite entities constantly in the making.

[T]he concept of assemblage is particularly useful for grasping the spatially processual, relational and generative nature of the city, where 'generative' refers both to the momentum of historical processes and political economies and to the eventful, disruptive, atmospheric, and random juxtapositions that characterise urban space. (McFarlane 2011a: 650-651) Moreover, far from being an univocal and consistent phenomena, urban life is made of uncountable ingredients, of extremely diverse nature. Even the most mundane objects and the simplest task become part of a more or less complex assemblage. This shared interest in small objects and ordinary practices highly contrasts with most of urban studies' traditional concerns. For instance, in this approach, even a crucial notion such as space is subject to reconsideration. Instead of being the unquestioned starting point for analysis, spatial properties are considered as heterogeneous and precarious outcomes of sociomaterial practices from which they cannot be separated [START_REF] Latour | Paris ville invisible[END_REF][START_REF] Latham | Moving cities: Rethinking the materialities of urban geographies[END_REF][START_REF] Kärrholm | The materiality of territorial production: A conceptual discussion of territoriality, materiality, and the everyday life of public space[END_REF].

This stream of research thus calls for a study of both the sociotechnical complexity of cities and their very situatedness and ordinariness, ranging from large infrastructures to mundane objects. It invites us to discover how things hold together, how they collapse, and to understand the conditions in which these more or less precarious conglomerations act, what kind of urban reality they perform, without deciding what counts as an important feature beforehand.

Among the uncountable objects and practices this interdisciplinary inquiry into urban assemblages draws attention to, some bear witness to a mostly unproblematized dimension of urban life: maintenance and repair work. Maintenance and repair activities have been of growing interest in social sciences, and have recently gathered a community of scholars from different domains: ethnomethodology, human geography, media studies, history, science and technology studies [START_REF] Henke | The mechanics of workplace order: Toward a sociology of repair[END_REF][START_REF] Graham | Out of order: Understanding repair and maintenance[END_REF][START_REF] Gregson | Practices of Object Maintenance and Repair: How consumers attend to consumer objects within the home[END_REF][START_REF] Jackson | Rethinking repair[END_REF][START_REF] Denis | Material ordering and the care of things[END_REF][START_REF] Rubio | On the discrepancy between objects and things[END_REF][START_REF] Tecnoscienza | Maintenance and Repair in Science and Technology Studies[END_REF]continent. 2017). Even though very different objects and settings have been investigated in this emerging literature, maintenance and repair studies share a common set of concerns that resonate with some of the issues raised in the discussions on urban assemblages. By exploring how fixing, mending, restoring, cleaning or up-keeping are concretely accomplished, maintenance and repair studies notably describe a world where sociomaterial order is constantly produced and reproduced, mostly through mundane practices. There is no place here for material reductionism and essentialism. Through the lens of maintenance and repair practices, the ability of an infrastructure to order behaviors, as the capacity of an object to last, or the potential for a 'mobile' to become 'immutable' cannot be reduced to intrinsic material properties. The material properties of things, in their very heterogeneity, are enacted through ceaseless maintenance and repair practices.

A few scholars have shown that maintenance and repair studies offer a great entry point to reveal the complexity of hitherto overlooked urban realities, and to pursue the exploration of urban assemblages. For instance, [START_REF] Edensor | Entangled agencies, material networks and repair in a building assemblage: The mutable stone of St Ann's Church, Manchester[END_REF][START_REF] Edensor | Vital urban materiality and its multiple absences: The building stone of central Manchester[END_REF], who has investigated the conditions in which the St Ann's Church in Manchester has been restored, explains how repair and maintenance work literally unfolds urban materiality, bringing to light both its fluidities and its stabilities. Discovering the processes of the church's restoration, we understand that the stone itself is not at all an inert and isolated object. Its repair goes through dealing with 'entangled materialities' and constantly changing entities that are connected to a wide geography of knowledge and labor. Other mundane aspects of urban life have also been studied at the intersection between assemblage thinking and maintenance and repair studies. Investigating the daily life of buildings, [START_REF] Strebel | The living building: Towards a geography of maintenance work[END_REF] has highlighted the importance of 'block checks' during which concierges use 'conversation, gestures, craft and movement (…) tools and technologies' to bring buildings to life (p. 244). The order of the building, he explains, is a complex process that maintenance workers deal with on a daily basis. We also observed the same kinds of delicate sociomaterial ordering processes within the Paris subway stations, where the wayfinding system is enacted as a standardized set of immutable and immobile signs thanks to the constant supervision and the daily interventions of maintainers. This maintenance work goes through a regular exploration of the material ecology of subway spaces and an incessant mix of assembling and disassembling operations [START_REF] Denis | Maintenance work and the performativity of urban inscriptions: The case of Paris subway signs[END_REF][START_REF] Tecnoscienza | Maintenance and Repair in Science and Technology Studies[END_REF]. The streets themselves can be understood as the fragile results of a continual maintenance work, as demonstrated by [START_REF] Shaw | Cleaning up the streets: Newcastle-upon-Tynes night-time neighbourhood services team[END_REF]. Following the cleaners who operate in the center of Newcastle at night, Shaw has emphasized that both the development of night-time leisure industry in the city and the way the neighborhood looks during the daytime are made possible by the sociomaterial competencies of these maintenance workers who, with their tools and the materials they deal with, 'form a single waste-production-machine-assemblage ' (p. 190).

The city -its neighborhoods, its infrastructures, its furniture, its activities, etc. -is thus the outcome of a continuous maintenance work that goes through more or less known practices, instruments, documents, rules, and of course workers. This is where assemblage thinking and maintenance and repair studies meet: instead of questioning urban reality as a ready-made phenomenon, or as the encounter only between planners' strategies and users' tactics, they investigate it as a series of practical enactments. What we would like to add to this landscape is attention to the multiplicity of these enactments, and thus to the ontological variations of the city components.

The recent conversation about multiplicity in social sciences has participated in an ontological moment -if not a 'turn' -that pushed a bit further certain epistemological displacements initiated by early Science and Technology Studies on differences and variability [START_REF] Woolgar | The wrong bin bag: A turn to ontology in science and technology studies?[END_REF]. Instead of considering reality as a stable given ground, where plurality can be seen as the result of various interpretations or constructions, it invites us to recognize that reality is always practically enacted, and that these enactments support a variety of ontologies [START_REF] Mol | Ontological politics. A word and some questions[END_REF]). Reality itself is then multiple. This has important consequences on how social research is carried out and what can be its outcome. The researcher here does not provide a single overarching description of a phenomenon, nor does she gather distinct perspectives on it. Rather, she explores the conditions in which different versions of the studied phenomenon are enacted. Such attention to the performativity of descriptions and practices bring to the light hitherto unnoticed or overlooked differences. In their seminal work, de Laet and [START_REF] De Laet | The Zimbabwe bush pump: Mechanics of a fluid technology[END_REF] have studied for instance a 'fluid' bush pump, an innovation that is ceaselessly reshaped and never takes the form of a closed black box. In a very different setting, Mol's inquiries into the multiple versions of atherosclerosis [START_REF] Mol | The body multiple: Ontology in medical practice[END_REF] foregrounded the technical, practical, and linguistic dimensions of their enactments. [START_REF] Woolgar | Mundane governance: Ontology and accountability[END_REF] have expanded on these works investigating the multiple ontological enactments of ordinary objects and the mundane governance it participates in. Investigating these differences not only allows for an inventory of various versions of an object but also tackles the issue of their relationships: versions may have partial connections, align with one another, one version may include or be dedicated to another, and of course, versions may ignore one another, and sometimes even be contradictory [START_REF] Law | The materials of STS[END_REF].

Because ontological enactments do not install the conditions of existence of isolated, autonomous objects, but of worlds in which the relational identities of various entities are held together, they are a particularly rich entry point to expand on the works on maintenance and urban assemblages. Describing such enactments helps us to question the performative character of urban maintenance practices, and to investigate the multiple and relational ontologies enacted in urban assemblages. It is especially a fruitful means to reconsider the mode of existence of one of the paradigmatic artifacts of territorial ordering and disciplinarization processes -the wall. If it is difficult to decide the status of the wall we took as an example in the beginning of this chapter, it is precisely because its ontology varies. Graffiti removal itself, an operation of urban maintenance that could be seen as a stabilizing and disambiguating practice, participates in these variations.

Removing Graffiti in Paris

At the end of 1999, the city of Paris launched a call for tenders for a particularly ambitious program: the removal of '90% of graffiti' within one year. Following this 'clean-walls operation', a 5-year period of maintenance was planned, during which contractors would ensure that the walls of Paris would remain free from unsolicited inscriptions. Since then, new calls for maintenance have been issued every two to five years. This framework, which has remained roughly the same safe for small adjustments, has dramatically changed the way graffiti have been taken care of in Paris. Until then, graffiti removal was an obligation that every building owner had to fulfill. The first call for tenders drew on a municipal by-law that had centralized the process and had organized it in such a way that it could be provided as a free service. Since then, a number of private companies (initially one, nowadays three) have been bestowed the task to 'clean' the walls of Paris day after day.

The contracts signed by the municipality and the private companies are performancebased. Besides identifying the areas of intervention (all building facades of a specific neighborhood, up to 4 metres high from street level), they set objectives (a maximum threshold of graffiti presence in each area) and intervention deadlines (10 days to remove a graffiti after its detection). The service is operated in two steps. First, a report is sent via an online form by either a civil servant, a company worker or private citizen. Second, reports are gathered every morning and distributed to the removal squads. Four main removal techniques are employed: paint, solvent, high-pressure water, and sandblaster. Each month, the contracted companies activity is verified. One of its managers and three or four representatives of the City organize a walk in the streets of Paris, during which they collectively check whether the amount of graffiti remains above the contractual threshold, or not -if not, the companies face failure penalties.

Besides private companies, a team from the municipality remains in charge of specific places not included in contracts, such as public gardens, schools and some courtyards. The private and public agencies work hand in hand. The managers of the local team run the inspections, and on exceptional occasions, such as when the interventions take place after a protest, the teams work altogether.

Seen from a distance, this removal policy might look like a monolithic 'disciplinary apparatus' [START_REF] Foucault | Discipline and punish[END_REF] that places walls under an intense scrutiny. From such a perspective, the removal policy could be examined as a powerful device of governance that results in growing social control of urban spaces, aimed at preventing any disruptions of the order in place -notably, regarding who is authorized to display public inscriptions. Yet, after reading carefully the rules and regulations, and shadowing the workers during their interventions, things appear more nuanced and complex. The 'mundane governance' [START_REF] Woolgar | Mundane governance: Ontology and accountability[END_REF] at the core of this removal policy is multiple, and its enactment does not reduce either walls or graffiti to unequivocal objects.

An Object of Maintenance

First, the Paris graffiti removal policy enacts walls as objects of a specific kind of maintenance. To understand this, we have to go back to the early 1980s when a theoretical and political model of maintenance known as the 'broken-window theory' [START_REF] Wilson | Broken windows[END_REF] emerged. This theory famously influenced graffiti removal programs in the US, notably the 'war on graffiti' that was initiated in the NYC Subway by Mayor Ed Koch in the early 1980s, and then expanded to the whole city by Rudolph Giuliani during the 1990s. Giuliani was himself a great inspiration to Jean Tiberi, the Paris Mayor who launched his own removal policy in 2000. Still today, broken-window theory is considered as the cornerstone of most graffiti removal initiatives [START_REF] Stewart | Cracks and contestation: Toward an ecology of graffiti and abatement[END_REF].

But what exactly is 'broken-window theory', and how is it related to graffiti? In their landmark paper, [START_REF] Wilson | Broken windows[END_REF] -respectively, a professor of Government at Harvard, and a former director of the evaluation team of the Police Foundation -addressed the issue of community life quality and, more generally, public order. The authors insisted on the importance of foot police patrols and their 'order-maintenance function.' We shouldn't forget, they claimed, that policemen are essential to such maintenance, and this goes beyond the sole fight against major crimes. To explain why a day-to-day order maintenance matters, they relied on a psychosocial experiment conducted in the 1960s and, more importantly to our concern, on a seminal policy paper discussing the effects of graffiti on subway trains in New York [START_REF] Glazer | On subway graffiti in New York[END_REF]. They drew two main conclusions from these sources. First, public order is a matter of visibility. Some things, such as broken windows and graffiti, work as signs of disorder, 'signs of official failure', as Glazer wrote. These symptoms should be eradicated, Wilson and Kelling held, not only because they can be the cause of a sense of fear within communities, but also, and this is the second foundation of the theory, because they are contagious. '(I)f a window in a building is broken and is left unrepaired, all the rest of the windows will soon be broken' (Wilson and Kelling 1982: 31, emphasized in the original). Understood as signs that 'no one cares', each single instance of broken urban furniture, each piece of graffiti, is seen as an encouragement that may lead to other cases. Each is a potential step toward the deliquescence of social order .

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The authors contrasted the broken-window approach with the usual criminological focus on crime solving. Their theory calls for a constant supervision of places focused on signs of disorder, and it requires numerous and regular interventions to get rid of them as soon as possible. This way of dealing with graffiti has important and concrete consequences for how the walls of the city are treated. Indeed, if graffiti are signs of disorder, it means that walls do have a normal, 'ordered' condition which graffiti disrupt. The removal interventions are a way to bring back walls to this original shape, and thus to restore order. Yet, configured as a maintenance activity, graffiti removal is an endless process that entails thousands and thousands of interventions per year. It is an ongoing task that can virtually never stop. Hence, what is regarded as the 'normal condition' is not a prior stable condition for walls, but the ephemeral outcome of a continual undertaking: clean walls are both a premise and a result in the graffiti removal assemblage.

The idea that walls have an 'original' condition, and that graffiti removal should lead to its restoration is at stake in each intervention. Letting the walls as if neither graffiti nor removal interventions happened is not an easy task though. One of the difficulties in using paint to cover a graffiti is, for instance, to find the right hue that will match exactly the color of the wall. And even if they manage to obtain the right color, workers still have to find a way to delimit the zone of their intervention, so that the new painting remains as invisible as possible. Otherwise, they would produce what they call a 'cleanness stain.' The same kind of problem occurs when using high-pressure water or a sandblaster: if graffiti is removed simply following the outlines of the letters and drawing, what workers call a 'ghost' would remain. The ink or the paint used to write the graffiti would be erased, but its shape would still be distinguishable. Bearing such traces, the wall would not be considered cleaned, and maintenance work would not be successfully accomplished. Moreover, graffiti removal interventions are also steered by a certain ideal of 'invisible work' [START_REF] Star | Layers of Silence, Arenas of Voice: The Ecology of Visible and Invisible Work[END_REF]. As one of the workers we followed told us:

For a more general discussion on the role of observable traces in the continuous upkeep of urban 1 spaces, see [START_REF] Murphy | Dwelling Together: Observable Traces and Controls in Residential Urban Spaces[END_REF].

I have this kind of philosophy: that you do not see anything. Ideally, no one would notice that we came, and no one would even remember there was graffiti. (Interview with John) What is at stake in each intervention and in the constantly reiterated 'restoration' of wall order is thus the production of a double invisibility: of course, invisibility of the removed graffiti, but also invisibility of its removal, namely the erasure of any traces of materials used (e.g. chemical, water, paint, sand…), of actions performed during the intervention, and of the workers themselves.

A Surface

The constant (re)production of clean walls is a requirement explicitly stated in the contracts the City Hall signs with contractors. The organization of the supervision of walls, and the strict deadlines each company has to meet once a graffiti is detected, are a direct translation of the broken-window theory principles. Yet, these principles are not organized in terms of rhythm of interventions only. Maintenance contracts also define the graffiti removal process in spatial terms. The presence or absence of graffiti is measured in Euclidian units and interventions are described in square meters and 'thresholds'. The question, 'how much graffiti should be removed?' is thus translated into 'how many square meters of graffiti should be erased?' Through this ontological enactment, walls are enacted as surfaces, and the city is handled as a quantified territory. Walls are comprehended as two-dimensional areas, converted into planes that can be measured, added and combined, so as to be statistically commensurable. Before the first call for tenders, an 'Observatory of graffiti' had been set up to do first measurements and split the city into different zones. Henceforth, quantification has been integrated within the maintenance process itself through the constant production of statistical data. Each intervention begins with a picture of the graffiti about to be removed (figure 1). The picture is saved in a database, as well as a form providing additional information: the nature of the wall, its location, the type of graffiti, and its size. Once the intervention is finished, another picture of the 'empty' wall is added to the file.

The data are regularly sent by contractors to the municipality, where monthly and annual reports are edited. In these reports, two indicators are particularly important: the average quantity of cleaned surface during a certain amount of time, and the 'recurrence rate' (after how long a cleaned surface has to be treated again). Not only are these indicators accounting for the activity of the contractors, but they also provide precious information to the city's staff in order to recalibrate the zones that will appear in the next call.

Measured surfaces are employed to control the contractor's activity, too. As said earlier, every month, an inspection is set up to verify if the terms of contracts are met. This inspection is conceived of as a representative sample of the whole area: a new point of departure is picked randomly each time, from which a systematic path is followed for a specified length (first street to the right, next to the left, next to the right, and so son). During this collective walk, every spotted graffiti is noted on a paper form, along with its 'size', that is the amount of surface it occupies. At the end of the tour, the sum of all the affected surfaces is calculated (figure 2). If the total exceeds the required threshold for the zone, the contractor is charged a fee. Walls are also enacted as surfaces in a further sense. The removal policy tackles the interfaces between public and private spaces, which divide and connect the streets and the buildings, to determine what falls under the municipality's responsibility and what under the tenants'. Interestingly, redefining the conditions in which theses surfaces are 'cleaned', the Paris removal policy has dramatically transformed the role of walls. Previously seen as the responsibility of building owners, graffiti removal has now been turned into a municipal matter. This was made possible thanks to a by-law that has rendered wall surfaces 'more public' than in the past.

Two things should be noted, though. First, the territory quantified as measurable surfaces, and taken care of by the municipality and its contractors, is far from exhaustive. The city captured by this policy is fragmentary. Several walls -notably those owned by certain public and private companies -slip out this policy. Second, this framework is not a matter of contracts and legal texts only, rather, it has to be constantly reenacted in the streets. During the monthly verification, for instance, the definition of what counts as a cleaned surface, and how it is measured, are discussed and adjusted. Likewise, the frontier between public and private spaces is regularly re-enacted. Sometimes, the workers might have to negotiate the access to a wall with local inhabitants and tenants. Metal shutters are also an excellent example to understand the necessity of such reenactments: when they are down, they count as surfaces, and the graffiti they bear have to be removed. This means that they have to be taken care of in the early hours of the morning, before the stores open and they disappear as walls .

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A Space for Public Expression

Obviously, graffiti removal is not simply based on measuring surfaces and calculating rates of recurrence. As public inscriptions, graffiti are also considered an expressive means: any particular drawing or written word can be examined for its expressive and aesthetic features . This is another important feature of the Parisian graffiti removal assemblage: the 3 meaning, forms and colors of all graffiti are scrutinized in their own right. This close examination draws on particular, official characterizations. Such an approach towards public inscriptions entails a specific ontology of walls as spaces for public expression.

A first characterization is the difference between authorized graffiti, often called by removal workers 'artworks' or 'murals', and the rest. To make this distinction possible, local authorities regularly send to contractors an updated list of walls with graffiti that should not be removed. This list works as a supplementary document to the initial contract signed with the municipality: it designates exceptions. Such a distinction is also made through specific demands from the municipality. For instance, contractors sometimes receive pictures similar to the one in Figure 3, where the graffiti that have to be removed are clearly distinguished from the 'mural' that should remain intact.

For a broader elaboration on the diverse rhythms of walls, see [START_REF] Brighenti | Walled urbs to urban walls -and return? On the social life of walls[END_REF].

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The aesthetic examination has been at stake since the very beginning of NYC graffiti writing 3 [START_REF] Cresswell | The Crucial "Where" of Graffiti: A Geographical Analysis of Reactions to Graffiti in New York[END_REF] This characterization is at play during day-to-day interventions as well. As we were following removal workers in the streets of Paris, they regularly explained to us why this or that inscription should not be erased, mobilizing situated criteria such as 'It's clear that this inscription is intended to be here' or 'This one is signed'. Sometimes, the situation is not obvious, though. Each time workers incur into doubts, they take a picture of the wall and, once back to their headquarters, they show it to their manager, so as to discuss and make a decision. Some of these doubts may also be dispelled in situ by the tenants. For instance, a shopkeeper may ask the workers to spare what she regards as an acceptable beautification of a metal shutter or a front gray wall close to her shop. The characterization of public inscriptions is thus far from automatic and the list of authorized graffiti is supplemented by situated judgments, hesitations and collective discussions made on a daily basis. Therefore, even though each wall constitutes a unique 'where of graffiti' [START_REF] Cresswell | The Crucial "Where" of Graffiti: A Geographical Analysis of Reactions to Graffiti in New York[END_REF], depending on the conclusion these adjustments lead to, it may be enacted in different ways: from a dirty surface that has to be cleaned to a display for artworks meant to last.

A second official characterization, which refers to a very different aspect of expressivity, plays an important role in the graffiti removal policy. The municipal contract identifies a distinct form of graffiti: those that are 'offensive, pornographic or prejudicial to public order'. Such categorization has tremendous consequences. It calls for a specific treatment and a particularly demanding contractual framework. Indeed, whereas 'regular' graffiti have to be removed within ten days after their detection, the contractors are compelled to erase the offensive ones within three hours, 24/7. Each time insults appear, walls are thus treated as discursive devices, and their cleansing becomes part of a 'politics of the performative' [START_REF] Butler | Excitable speech. A politics of the performative[END_REF]).

Yet, once again things are not linear. Even though most graffiti do not raise any doubts, some may appear equivocal, while some others may remain indecipherable for a certain period of time. For instance, during our fieldwork a worker came back from his daily tour impressed by 'the amount of graffiti the artist ACAB wrote' in a certain area. His manager promptly retorted that 'ACAB' actually meant 'All Cops Are Bastards' and was thus an offensive graffiti that had to be removed immediately. She called another worker, still around this neighborhood, and asked him to intervene. Since then, a notice has been displayed in the office recalling the meaning of 'ACAB'. The injurious categorization is therefore an ongoing open process, with its hesitations and its reconsiderations. Again, the notion of assemblage is useful here, since it helps recognize the situatedness of the policy and its ontological enactments.

An Official Decorum

In some cases, notably when a politician or a public official is scheduled to appear in a specific location for a special occasion, such as the celebration of a national holiday, or the inauguration of a building, graffiti removal takes on a very particular pace. Public celebrations and the presence of authorities involve specific removal interventions, which enact the walls of the city in a manner that dramatically differs from the ones we previously described.

These interventions are exceptional in many respects. First, they are not included in the contracts that define the day-by-day task of contractors, and are charged separately. Graffiti removal is here handled on a case basis. The location or the route that has to be cleansed is sent to the concerned contractual company several days in advance, and a reinforced staff of operatives is often called forth. Second, the contractors do not work alone. The municipal team dedicated to graffiti removal participates in these operations, as well as the teams that are in charge of other aspects of street-cleaning. Lastly, and most importantly, these "highimpact operations" are not in line with the broken-window theory framework. Indeed, what is at stake here is not the eradication of a phenomenon considered as harmful to the public order through a regular supervision and fast, repeated interventions. Instead, graffiti removal takes part in 'one shot' operations, and temporarily enacts the walls of Paris as an official decorum, a pristine scene on which no sign of dirt, of any kind, must be seen.

Theses situations recall that of the renovated and repainted graffiti-free subway trains especially prepared in 1984 for the four hundred Europeans invited for a weekend stay in New York City by a tour agency based in London. This anecdote, with which Austin (2001:2-3) starts his book on the history of NYC graffiti, reminds us that the public image of a city is of crucial concern, and that its control is sometimes considered so important that it leads to the production of real theatrical performances, in Goffman's sense of the expression (1959). The walls that can be seen in the background of an official celebration are crucial components in the decor of the event. Free of graffiti, they display a stabilized and cleaned environment, meant to manage the impression of the participants: passersby and active members, officials themselves, security squads, but also the audience of the newspapers and websites that display photos or videos of the event.

Such enactment of walls as a ceremonial decor has important consequences on the urban reality that graffiti removal performs. Even though the locations of public appearances for inaugurations or memorials obviously change from one event to another, they do not take place all over the city. In Paris, only a few arrondissements are concerned. This means that, even from the standpoint of graffiti removal, not all walls of the city are similar. Inequalities emerge between those that exceptionally, but regularly, fall into officials' program and those, cleaned on the regular basis of the graffiti removal policy, that are merely dedicated to the everyday life of inhabitants, and non-VIP tourists.

Two kinds of territories are performed in this asymmetry. The first one is the one of the 'regular' city, which is managed by the regime of broken-window theory, and covers virtually the totality of the streets and the neighborhoods. Walls are here taken care of on a daily basis, cleaned so as to recover a state considered as 'normal', that is devoid of graffiti. The second territory, which is performed by exceptional interventions, is ephemeral. It pops up once in a while like a fragment detached from the day-to-day urban ecology. It takes place in the assemblage that performs the local setting of an official staging, and that is also meant to circulate in multiple visual representations -newspapers, television, websites, etc. The walls of this second territory are enacted as components of a legitimate image of Paris. This enactment goes through rendering them temporarily pristine, a condition way beyond 'normal.'

A Material Composite

Either routinely accomplished, or exceptionally operated on an event basis, graffiti removal is anything but a simple activity. As interesting as the vocabulary of erasure may be, walls are not paper sheets from which graffiti could be wiped off with a rubber. It is, actually, a challenge. Beyond the risk of generating what workers call 'cleanness stains', most removal techniques represent themselves a non-negligible threat to the integrity of walls. How these threats are taken into consideration highlights a fifth crucial ontological enactment made by the graffiti removal policy: walls are treated as material composites.

One only needs to look at the first move that inaugurates every intervention to measure the importance of the material side of graffiti removal. Each time removal workers enter into action, they make the same gesture before proceeding further: they stretch their hand out and touch the surface they are about to treat (figure 4). By doing so, they told us, they assess the material properties of both the wall and of the graffiti. This first gesture provides direction for the following operations.

Thanks to this material appraisal, workers are able to properly engage in their intervention. Above all, they can choose the right technique, and dismiss those incompatible with the materials to be faced. In the presence of a porous stonewall, for instance, it is inconceivable to use high-pressure water or sandblasting. As one of the workers explained to us, this is due to the behaviors of both the stone and the graffiti:

With porous material, it's really complicated. The graffiti migrates within the stone very quickly, it is almost impossible to remove it entirely. And the stone itself is extremely fragile. If you use high pressure, you carve into the wall, really. The damages would be irreparable.

(Interview with

Steven)

There is only one solution in this case: the use of paint in order to cover the unwanted inscription. When the surface is already painted, other problems occur. The workers have to evaluate the consistency of the paint, and its capacity to resist a certain dose of chemicals. They have to find the right balance between the preservation of the original paint and the ability for the product to erase the paint or the ink of the graffiti. But the material enactment of walls is not done only by a visual and tactile evaluation. It also goes through the workers' corporeal postures and their bodily adjustments during the removal intervention. The use of high pressure water, for instance, requires the right distance. Workers need to tune their gestures with a series of meticulous trials and errors during the first seconds of the operation that allows them to gently erase the inscriptions without deteriorating the wall that bears it. Likewise, applying chemicals require a careful use of sponges and rags. It is of critical importance to deploy them in the right space and at the right time. If the product is let for too long on the wall, it would irremediably damage the wall and prevent any 'normal' state to be restored.

Several aspects are particularly interesting in this ontological enactment. It notably shows that walls are fragile. This is an important dimension of maintenance practices: they foreground the material vulnerability of things and their need to be taken care of [START_REF] Denis | Material ordering and the care of things[END_REF]. Even though walls may appear as one of the most solid material components of cities, their sturdiness should not be considered an intrinsic and universal property. Our exploration of graffiti removal can be seen as an invitation to study other kinds of walls maintenance, such as renovation, refacing, or more mundane operations. Furthermore, beyond this generic vulnerability, we have noticed that when it comes to materials, to the 'stuff the walls are made of', things may get hazy. During removal interventions, what constitutes a wall is actually not an obvious matter. What workers are dealing with is a material composite, a composite that is not the mere addition of previously identifiable things, but foregrounds a 'material ecology' (notably made of chemicals, water, sand, stones, concrete, plaster, bricks, mural paint, graffiti paint, ink, etc.) with which the workers have to engage in order to successfully proceed to removal. By doing so, they follow, without knowing it, Ingold's plea for taking the materials complexity of things into consideration:

We see the building and not the plaster of its walls, the words and not the ink with which they were written. In reality, of course, the materials are still there and continue to mingle and react as they have always done… (Ingold 2007: 9) What is particularly striking in this aspect of the removal process is that the distinction between the wall, on one side, and the graffiti, on the other, is not given beforehand; rather, it is a situated achievement, the outcome of the interventions themselves. Above all, this outcome is difficult to obtain. The whole operation could fail -and in some situations it does. A bit of what eventually proves to be the graffiti can remain in place. A bit of what eventually proves to be the wall may disappear. Yet, bringing walls back to their 'normal state' entails removing only 'the stuff graffiti is made of'. Graffiti removal requires therefore a delicate material ontological operation that simultaneously enacts what the wall is and what the graffiti is.

Conclusion

In this chapter, we showed that graffiti removal in Paris, as in other cities, is a crucial part of a politics of public order maintenance. The removal policy, a paradigmatic application of broken-window theory, is indeed organized as a daily never-ending process, through which what is considered as visible signs of disorder are constantly wiped off of walls and taken away from the sight of inhabitants and passersby. Taking into consideration what could be considered as insignificant maintenance practices profoundly changes the traditional understanding of the modes of existence of walls in the city, and the sociomaterial order they participate to perform.

First, from legal and contractual documents to the removal techniques, and workers' gestures and words, we showed that graffiti removal enacts walls. This is obviously not the only domain of activity that participates in the definition and the conditions of existence of walls, yet this enactment is extremely important: without a graffiti removal policy, a great majority of walls in Paris would neither look nor exist the way they do today.

Yet, we also showed that such enactment is nothing but unidirectional. Even though graffiti removal is conceived of as a way to maintain public order straight on the walls, its exploration foregrounds the multiplicity of ontological enactments that are performed to do so. The 'same' wall can be handled as the object of a specific kind of maintenance, as quantified surface, as controlled space of public expression, as part of an official decorum, and as an uncertain material composite. The 'same' wall is, in other words, the result of the consecutive, sometimes simultaneous, enactments of very contrastive versions that each raises specific concerns.

Investigating the enactments of walls as crucial parts of a specific urban assemblage is important in many respects. It is a means to emphasize how these enactments make language and matter hold together. Following Deleuze and Guattari, assemblages are intrinsically made of words-and-matter. Graffiti removal is particularly telling in this regard. It deals with traces and inscriptions unofficially displayed in the city, whose erasure implies the articulation of the material properties of walls (stone, brick, paint, concrete…), graffiti (ink, paint…) and removal techniques (water, sponge, paint, sand, rags, chemical products…). Graffiti removal is thus a singular case of 'collective assemblages of enunciation' [START_REF] Deleuze | Milles plateaux[END_REF] in which what is produced and reproduced is the absence, rather than the presence, of some enunciative entities. Silencing, too, is an essential part of the collective enunciation of the city.

We also showed that, as part of an urban maintenance assemblage, the ontological enactments that occur in graffiti removal are relational. Indeed, a specific enactment of walls implies, at least, a specific enactment of graffiti, of order maintenance, and of urban reality. Moreover, these relational enactments are not inert, and cannot be described as a mere set of rules and principles. The original French word used by [START_REF] Deleuze | Milles plateaux[END_REF] is precisely agencement, a term that insists not only on material and linguistic heterogeneity, but also on agency. Graffiti removal, and urban maintenance in general, are processes; obviously, they draw on documents, laws, standards that are extremely important, but they would remain lettre morte without workers, interventions, exceptional operations and verification processes. These situated actions perform a certain territorialization, but they are also subject to deterritorialization processes. Indeed, we saw that the removal interventions cannot be automatized. They imply improvisations, adjustments and situated judgments. They also have to deal with the environment of the city, for instance with weather conditions that constraint the use of certain removal techniques. The practice is also subject to changes: new kinds of walls materials regularly appear (such as porous coating or rusty metal surfaces making graffiti removal tricky), innovative graffiti removal techniques are tested (for instance, to intervene on trees without damaging them), and so on.

Finally, our exploration foregrounds the connection between graffiti removal and other assemblages, other ontological enactments. This is obvious when it comes to streets cleaning, as we have seen in the case of the enactment of an official decor. But two other assemblages, briefly encountered in our descriptions, are also directly connected to graffiti removal. The first one enacts graffiti as artworks or murals. The lists of authorized pieces that are transmitted to the municipality contractors are one of the numerous ingredients of this assemblage, in which graffiti writers themselves, specific budgets and numerous other things are also crucial. If graffiti writers are almost absent from the removal assemblage, they are at the center of the 'cultural' assemblage, in which they are enacted as artists. An important aspect is that, even though they differ dramatically in rhythm and components, both assemblages participate in the same ontological enactment of walls as sites of public expression. A second assemblage is worth mentioning, which is also essential to order maintenance: the legal 'struggle' against graffiti. Here, graffiti is enacted as a crime. Graffiti writers, again, appear to be at the center of preoccupations, yet this time as delinquents who have to be tracked, arrested, and punished. The connection with removal interventions is made through the database where the pictures of each operation are stored. Sometimes, the police ask access to this database. The walls on these occasions are not enacted as objects of maintenance, surfaces, sites of public expressions, decor, or material composite, but as evidence of crime.

Hence, the answer to the question, 'what is an urban wall?' is by no means simple, direct, and univocal. Walls are enacted in practices. They take place in, and contribute to, a variety of assemblages. Each assemblage does not perform single and distinct ontological enactments of walls, though. Some partially share the same version, while others bring different versions to existence. Graffiti removal and urban maintenance in general, though they may seem univocal ordering processes, are of this kind. They enact contrasted ontologies of walls and of urban reality.

Figure 1 .Figure 2 .

 12 Figure 1. Taking a picture before removing the graffiti (© J. Denis & D. Pontille)

  , and is still disputed nowadays (McAuliffe and Iveson 2011 ; Brighenti 2016).

Figure 3 .

 3 Figure 3. Making a clear distinction between graffiti and artwork (© J. Denis & D. Pontille)

Figure 4 .

 4 Figure 4. Touching the wall and the graffiti (© J. Denis & D. Pontille)