Sérgio F A Batista
email: sergio.batista@ifsttar.fr

Ludovic Leclercq
email: ludovic.leclercq@entpe.fr

Monica Menendez
email: monica.menendez@nyu.edu

On the calibration of time-varying trip length distributions for the aggregated traffic models

One of the key questions for Macroscopic Fundamental Diagram traffic models lies in the calibration of the trip lengths, i.e. the travel distances of vehicles in the regions. Few studies in the literature have attempted to propose methodological frameworks to calibrate the trip lengths. Batista et al. (2019) propose a framework to explicitly calculate trip length distributions. However, they do not consider the influence of the traffic conditions on the trip lengths. In this paper, we propose to extend their methodological framework to determine time-varying trip length distributions according to the changes in the traffic conditions.

Introduction

Aggregated traffic models at the city network level were early introduced by Godfrey (1969) and later reconsidered by [START_REF] Daganzo | Urban gridlock: Macroscopic modeling and mitigation approaches[END_REF] and [START_REF] Geroliminis | Existence of urban-scale macroscopic fundamental diagrams: Some experimental findings[END_REF]. These traffic models, although designed for urban areas require the partition of the city network (Figure 1 (a)) into regions (Figure 1 (b)) (see e.g., [START_REF] Saeedmanesh | Clustering of heterogeneous networks with directional flows based on "snake" similarities[END_REF][START_REF] Haddad | Optimal perimeter control synthesis for two urban regions with aggregate boundary queue dynamics[END_REF][START_REF] Lopez | Revealing the day-to-day regularity of urban congestion patterns with 3d speed maps[END_REF][START_REF] Casadei | Aggregation and travel time calculation over large scale traffic networks: An empiric study on the grenoble city[END_REF][START_REF] Ambühl | Approximative network partitioning for mfds from stationary sensor data[END_REF], where the traffic conditions are approximately homogeneous. Figure 1 (c) shows the regional network that corresponds to the city network partitioning. Let X be the set of regions that define the regional network. In each region, the traffic states are regulated by a Macroscopic Fundamental Diagram (MFD). An MFD is a relationship between the average circulating flow of vehicles and the average density. The traffic dynamics for a single region r ∈ X are governed by a conservation equation, where the vehicles' accumulation n r (t) depends on the balance between the inflow (Q in,r (t)) and the outflow (Q out,r (t)):

dn r dt = Q in,r (t) -Q out,r (t), t > 0 (1)
There are two kinds of MFD models that can be distinguished in the literature, depending on the assumptions made on Q out,r (t): the accumulation-based model [START_REF] Daganzo | Urban gridlock: Macroscopic modeling and mitigation approaches[END_REF][START_REF] Geroliminis | Existence of urban-scale macroscopic fundamental diagrams: Some experimental findings[END_REF]; and the trip-based model [START_REF] Arnott | A bathtub model of downtown traffic congestion[END_REF][START_REF] Lamotte | The morning commute in urban areas: Insights from theory and simulation[END_REF][START_REF] Mariotte | Macroscopic urban dynamics: Analytical and numerical comparisons of existing models[END_REF][START_REF] Mariotte | Mfd-based simulation: Spillbacks in multi-reservoir networks[END_REF].

One of the main components of MFD models is the setting of the vehicles' trip lengths, i.e. the travel distances in the regions. Most applications of the MFD models have been designed for testing control algorithms and strategies, where the authors consider a constant average travel distance for all vehicles in the same region (see e.g., [START_REF] Daganzo | Urban gridlock: Macroscopic modeling and mitigation approaches[END_REF][START_REF] Keyvan-Ekbatani | Exploiting the fundamental diagram of urban networks for feedback-based gating[END_REF][START_REF] Ekbatani | Urban congestion gating control based on reduced operational network fundamental diagrams[END_REF][START_REF] Haddad | Optimal perimeter control synthesis for two urban regions with aggregate boundary queue dynamics[END_REF][START_REF] Zhong | Robust perimeter control for two urban regions with macroscopic fundamental diagrams: A control-lyapunov function approach[END_REF][START_REF] Yang | Multi-scale perimeter control approach in a connected-vehicle environment[END_REF]. [START_REF] Aboudolas | Perimeter and boundary flow control in multireservoir heterogeneous networks[END_REF] and [START_REF] Kouvelas | Enhancing model-based feedback perimeter control with data-driven online adaptive optimization[END_REF] tested perimeter control strategies in real city networks, but the authors also consider a constant average travel distance for all vehicles traveling in the same region. Up to now, little attention has been paid in the literature to the challenging task of the trip lengths calibration for MFD models applications.

Figure 2 depicts the scale-up of trips in the city network into paths in the regional network as well as the associated challenges. One can observe that the green and blue trips cross a different sequence of regions, following the definition of the city network partitioning. This ordered sequence of crossed regions by a trip is called regional path. The green and blue trips have different travel distances inside each crossed region. This leads to different travel distances associated to the same regional path. They are then characterized by distributions of trip lengths inside each crossed region. An example is depicted in Figure 2 for the green regional path inside the gray region. In fact, the trip length distribution associated to the green regional path in the gray region contains the information of the plausible travel 2 distances of the green trips in the city network. The question is how to properly calibrate the trip lengths distributions based on the information of trips in the city network. [START_REF] Yildirimoglu | Approximating dynamic equilibrium conditions with macroscopic fundamental diagrams[END_REF] proposed a methodological framework to implicitly estimate time-varying average trip lengths. Recently, Batista et al. (2018) and [START_REF] Batista | Estimation of regional trip length distributions for the calibration of the aggregated network traffic models[END_REF] went one step further and proposed a methodology to explicitly calculate distributions of trip lengths for the calibration of dynamic MFD models. The methodology is based on trip patterns in the city network and different levels of information from the regional network. Since the true trip patterns in the city network are unknown and change over time, the authors constructed a virtual set of trips. For this, they randomly sampled a large number of origindestination (od) pairs in the city network and calculated the shortest-paths in terms of distance for each pair. These virtual trips were then filtered according to different levels of information from the regional network. The latter ranged from no information about the previous and next regions to be traveled by the trips, to their associated regional path. The first level of information calculates trip length distributions considering the travel distances of all trips that cross one region. It assigns a common average trip length for all vehicles traveling in the same region, independent of their regional path. The most detailed level only considers trips that define the same regional path. This allows to determine a different trip length distribution for all regional paths that cross the same region. The authors showed that the first level of information is not able to capture the trip length variability of all regional paths crossing the same region. Moreover, they also showed that the trip lengths calibration clearly influences the modeled traffic dynamics in the regions. The authors concluded that filtering the trips by their associated regional path to explicitly calculate the trip length distributions should be considered. [START_REF] Yildirimoglu | Approximating dynamic equilibrium conditions with macroscopic fundamental diagrams[END_REF] and [START_REF] Leclercq | Macroscopic traffic dynamics with heterogeneous route patterns[END_REF] showed that the vehicles' trip lengths are influenced by the traffic states. In this paper, we revisit the methodological framework proposed by Batista et al. (2018) and [START_REF] Batista | Estimation of regional trip length distributions for the calibration of the aggregated network traffic models[END_REF] to explicitly determine trip length distributions for the calibration of MFD models. We propose to extend this framework for time-varying trip length distributions. In Sect. 2, we review in more detail the methodological framework proposed by Batista et al. (2018) and [START_REF] Batista | Estimation of regional trip length distributions for the calibration of the aggregated network traffic models[END_REF]. We also discuss the extension of this methodological framework to determine time-varying trip length distributions. In Sect. 3, we test the proposed extesion on the 6 th district Lyon network, that is divided into four regions. We discuss some preliminary results for two regional paths. In Sect. 4, we outline the conclusions of this paper.

Methodological framework

One of the key questions for the application of MFD-based models is the calibration of the trip lengths. We start this section by briefly introducing the methodological framework (Sect. 2.1) proposed by Batista et al. (2018) and [START_REF] Batista | Estimation of regional trip length distributions for the calibration of the aggregated network traffic models[END_REF] to calculate static trip length distributions considering the most detailed level of information. In Sect. 2.2, we discuss the extension of this methodological framework Batista et al. (2018) and [START_REF] Batista | Estimation of regional trip length distributions for the calibration of the aggregated network traffic models[END_REF] proposed a methodological framework to explicitly calculate trip length distributions. The methodology utilizes a representative set of trips in the city network as well as its partitioning. Let G(N, A) be a graph that represents the city network, where N is the set of all nodes and A is the set of all links. To partition the city network, one can use one of the different techniques described in the literature (see e.g., [START_REF] Saeedmanesh | Clustering of heterogeneous networks with directional flows based on "snake" similarities[END_REF][START_REF] Haddad | Optimal perimeter control synthesis for two urban regions with aggregate boundary queue dynamics[END_REF][START_REF] Lopez | Revealing the day-to-day regularity of urban congestion patterns with 3d speed maps[END_REF][START_REF] Casadei | Aggregation and travel time calculation over large scale traffic networks: An empiric study on the grenoble city[END_REF][START_REF] Ambühl | Approximative network partitioning for mfds from stationary sensor data[END_REF]. Since the full set of true trips in the city network is not known and very difficult to estimate, the authors propose to construct a set of virtual trips by randomly sampling several origin-destination nodes in the city network and then calculate the shortest-path in distance for each pair. Let Γ be this set of virtual trips. The trip length distributions are determined based on four levels of information from the regional network.

Static trip length distributions

In this paper, we focus on the most detailed level that filters the virtual trips by their associated regional path. Generically, a regional path p is defined as:

p = (p 1 , . . . , p m , . . . , p R ), ∀m = 1, . . . , R ∧ m ∈ X (2)
where R is the number of regions that define p. p 1 and p R are the Origin (O) and Destination (D) regions, respectively. In this paper, we distinguish between internal and regional paths (Batista & Leclercq, 2018). Figure 3 depicts these differences.

A regional path crosses an ordered sequence of different regions. An internal path is defined by virtual trips that travel only within the same region.

The set of trip lengths L p r of regional path p in region r is (Batista et al., 2018[START_REF] Batista | Estimation of regional trip length distributions for the calibration of the aggregated network traffic models[END_REF]:

L p r = {δ p rk l rk }, ∀k ∈ Γ (3)
where l rk is the length of virtual trip k that occurs in region r; and δ p rk is a binary variable that equals 1 if virtual trip k defines regional path p, or 0 otherwise.

The average trip length L p r of regional path p in region r is:

L p r = k δ p rk l rk k δ p rk , ∀k ∈ Γ (4)
This level of information has two limitations [START_REF] Batista | Estimation of regional trip length distributions for the calibration of the aggregated network traffic models[END_REF]. First, the average trip length L p r (see Eq. 4) is strongly influenced when the set size L p r is small, i.e. there is a low number of virtual trips associated to the regional path p.

Let N p be the number of virtual trips associated to regional path p. Second, the trip length distributions for the Origin and Destination regions of regional path p depend on the spatial distribution of the sampled od pairs inside these regions.

Time-varying trip length distributions

The set of trip lengths L p r (see Eq. 3) is calculated from virtual trips that represent shortest-paths in terms of distance, independent of the traffic conditions in the regions. However, [START_REF] Yildirimoglu | Approximating dynamic equilibrium conditions with macroscopic fundamental diagrams[END_REF] showed that the trip lengths depend on the traffic conditions and consequently change over time. In this paper, we propose to extend the methodological framework described in the previous section to calculate time-varying trip length distributions for MFD traffic models.

Generically, the set of trip lengths L p r depends on the traffic conditions:

L p r = f (v 1 (n 1 ), . . . , v m (n m )), ∀m ∈ X (5)
where v m (n m ) is the speed-MFD that regulates the traffic conditions inside a generic region m ∈ X. For a generic regional path p, L p r is a multi-dimensional function where the number of variables depend on the number of regions that define X, with vehicles' accumulation n m .

The speed-MFD of a region r ∈ X ranges between 0 and the free-flow speed

v f f r (n r ), i.e. v r (n r ) ∈ [0, v f f r (n r )]
. The first step of the proposed extension consists in building the multi-dimensional numerical grid, by discretizing the speed v r (n r ) of each region r ∈ X:

ω r (n r ) = {v 1 r (n r ), . . . , v Nr r }, ∀r ∈ X (6)
where N r is the total number of speed samples for region r; and ω r (n r ) is the set of discretized speeds for region r.

For each regional OD pair, we sample a set of origin and destination nodes in the city network that are in the Origin and Destination regions, respectively. Note that this set is fixed for each OD pair. Then, for each point of the multi-dimensional numerical grid, we calculate the time-dependent shortest path connecting each od pair in the city network using the classical Dijkstra algorithm. The travel time of link a is:

t a = l a δ ar v h r (n r ) , ∀a ∈ A ∧ ∀h = 1, . . . , N r (7)
where l a is the length of link a; and A is the set of links that define the city network.

We filter the time-dependent virtual trips following the regional path they define and calculate the average trip lengths L p r (see Eq. 4). Let Ω OD represent the set of all regional paths gathered that connect the regional OD pair. This step allows to determine the L p r values in the multi-dimensional grid. There are two challenges to implement this methodology. First, we need to sample a significant number of od pairs in the city network for each regional OD pair. Second, the calculation of the time-dependent virtual trips for each point in the multi-dimensional numerical grid requires a large computational burden. To bypass these challenges, we make use of the Latin hypercube technique [START_REF] Tang | Orthogonal array-based latin hypercubes[END_REF] for sampling the od pairs. This allows to gather a representative subset of all possible connections between the origin and destination nodes. In one hand, using this approach significantly reduces the computational cost required. On the other hand, it also solves the question of the sample size of od pairs we should consider, i.e. the number of od pairs that we should sample to calculate the virtual trips set [START_REF] Batista | Estimation of regional trip length distributions for the calibration of the aggregated network traffic models[END_REF].

Suppose that Lp r is the average trip length that we aim to estimate given a set v * r , ∀r ∈ X of average speeds in the regional network. That is Lp

r = f (v * 1 (n 1 ), . . . , v * m (n m )), ∀m ∈ X.
To estimate Lp r , we first find the location of v * r , ∀r ∈ X in the numerical grid ω r (n r ) (see Eq. 6). We then gather the 2N r closest points as well as the corresponding calculated average trip lengths L p r . We fit a multi-dimensional linear regression model defined as:

L p r = α 0 + i∈X α i v i + j∈X i =j α ij v i v j , ∀r ∈ p ∧ ∀p ∈ Ω OD (8)
where v i are the predictors and represent the speed samples of region i previously gathered from ω r (n r ); α 0 , α i and α ij are the regression coefficients to be determined; and Ω OD is the set of all regional paths connecting the regional OD pair.

The average trip length Lp r is then estimatd as:

Lp r = α 0 + i∈X α i v * i + j∈X i =j α ij v * i v * j , ∀r ∈ p ∧ ∀p ∈ Ω OD (9)

Results and discussion

We now discuss the implementation of the methodological framework introduced in the previous section. The test network is the 6 th district of Lyon (France) depicted in Fig. 4 (a). It is composed by 757 links and 431 nodes. This city network is partitioned into four regions, for which we fitted the production-MFD and speed-MFD functions depicted in Fig. 4 (b) and Fig. 4 (c), respectively. We assume a bi-parabolic shape for the MFD (one parabola for the increasing and one for the decreasing part of the MFD, with a first derivative equal to zero for the critical accumulation that maximizes production). The free-flow speeds for regions 1 to 4 Fig. 5 -Left: Evolution of the number of time-dependent virtual trips N p associated with regional path p = {1} as function of the speeds v 2 and v 3 . Right: same but for the average trip length L 1 (in m). Fig. 6 -Evolution of the number of time-dependent virtual trips N p associated with regional path p = {124} as well as of the average trip lengths (in m) for regions 1 (L 1 ), 2 (L 2 ) and 4 (L 4 ) as function of the speeds v 2 and v 3 .

are v f f 1 = 5.2 (m/s), v f f 2 = 6.4 (m/s), v f f 3 = 6.2 (m/s) and v f f 4 = 5.8 (m/s).
We first analyze how the average trip lengths L p r are influenced by the traffic states (i.e. their relation with the regional mean speeds v r (n r ), ∀r ∈ X) for an internal path p = {1} and a regional path p = {124}. To construct the multidimensional numerical grid, we discretize the speed-MFDs (see Fig. and v 3 . We show the results for four values of v 1 = 5.2, 2.8, 1.6, 0.8 (m/s). We observe that under free-flow conditions of regions 1, 2 and 3, the number of timedependent virtual trips that define regional path p = {1} is N p ∼ 1800. A similar trend is observed when the three regions are congested (i.e. when v 1 , v 2 and v 3 are low). The average trip lengths L 1 are also not influenced by the traffic states because similar time-dependent virtual trips set are obtained for the previous two scenarios. As region 1 becomes more congested, while the adjacent regions 2 and 3 are still in free-flow conditions, the number of time-dependent virtual trips N p associated with the internal path p = {1} decrease. For lower values of v 1 , the travel times of the city network links of region 1 increase. When the speeds in region 2 and 3 are in free-flow conditions, the city network links travel times for these two regions will be inferior than the ones of region 1. This means that the time-dependent virtual trips will detour to the links in regions 2 and 3 and then define other regional paths, such as for example p = {1231} and p = {1321}.

The set of time-dependent virtual change and the average trip length L p 1 = {1} is then dependent on the traffic conditions (i.e. on the observed mean speeds in the regions).

Fig. 6 also depicts the evolution of N p as well as of the average trip lengths for regions 1 (L 1 ), 2 (L 2 ) and 4 (L 4 ) as function of v 2 and v 3 , for regional path p = {124}. One can observe similar trends as in the case of regional path p = {1}.

For low v 2 values, the link travel times of region 2 increase. When region 3 is in free-flow conditions, the time-dependent virtual trips detour from region 2 to 3 and therefore define a different regional path p = {134}. This reduces the N p associated to regional path p = {124} and influences the average trip lengths. When both regions 2 and 3 are in free-flow conditions, the time-dependent virtual trips are more probable to cross the border of region 3 when their origin nodes are closer to this region. However, when v 3 is low, the travel times of the links in this region increase and the time-dependent virtual trips detour to region 2. This increases the average trip length in region 1 as observed in Fig. 6. This is also true for the destination region 4.

We also estimate the average trip lengths for regional paths p = {1} and p = {124} (see Eq. 8 to Eq. 9), considering different set of v 1 , v 2 and v 3 . We have also determined the average trip lengths for these sets of speeds, based on the calculation of the time-varying virtual trips. The results are listed in Table 1 for regional path p{1} and in Table 2 for regional path p{124}. We observe that the estimated trip lengths show a good agreement with the calculated ones based on the time-varying virtual trips. The exceptions happen for regional path p{124} and low values of v 2 and v 3 , i.e. when the average trip lengths are more sensible to N p .

One solution is to consider a smaller δv to construct the numerical grid for smaller v r , ∀r ∈ X.

Outline

In this paper, we propose an extension of the methodological framework proposed by [START_REF] Batista | Estimation of regional trip length distributions for the calibration of the aggregated network traffic models[END_REF] to determine time-varying trip lengths for the calibration of aggregated traffic models. We show how the traffic conditions in the regions influence the average trip lengths. We also show that the proposed methodology yields good estimation results for the average trip lenghts. We plan to pursue this study for other regional paths and analyze the trip lengths estimation for larger city network partitioned into a larger number of regions.

Fig. 1 -

 1 Fig. 1 -(a) City network. (b) Partition of the city network. (c) Regional network.

Fig. 2 -

 2 Fig.2-The scale up of trips into regional paths, that are characterized by trip length distributions.

Fig. 3 -

 3 Fig. 3 -(a) Internal path. (b) Regional path.

Fig. 4 -

 4 Fig. 4 -(a) 6 th district Lyon network divided into four regions. (b) Production MFDs. (c) Speed-MFDs.

  4 (c)) of the four regions as ω 1 (n 1 ) ∈ [0.4 : δv : 5.2], ω 2 (n 2 ) ∈ [0.4 : δv : 6.4], ω 3 (n 3 ) ∈ [0.2 : δv : 6.2] and ω 4 (n 4 ) ∈ [0.2 : δv : 5.8], where δv = 0.4 (m/s).

Fig. 5

 5 Fig. 5 depicts the evolution of the number of time-dependent virtual trips (N p ) as well as of the average trip length L p 1 = {1} as function of the speeds v 2

  Estimated L1 and calculated average trip lengths L 1 (in m) for regional path p = {1} and different values of v 1 , v 2 and v 3 (in m/s). Same as in Table1, but for p = {124} and different values of v 2 and v 3 .

				Region
		v 1	v 2	v 3	1
				L 1	L1
		5.10 0.60 1.15 407 407
		2.50 5.80 6.13 369 369
		3.40 2.15 4.36 407 407
		4.33 3.77 5.20 407 407
		1.05 5.40 2.40 324 320
		0.66 0.50 5.95 406 403
		0.45 0.60 0.88 407 397
	Tab. 1 -v 2	v 3	1	Region 2	4
		L 1	L1	L 2	L2	L 4	L4
	0.60 1.15 200 270 666 652 298 679
	5.80 6.13 208 209 833 821 225 229
	2.15 4.36 214 226 801 809 232 228
	3.77 5.20 210 214 828 813 225 230
	5.40 2.40 209 231 827 754 229 229
	0.50 5.95 134 225 674 792 354 632
	Tab. 2 -

Acknowledgments

This project is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 646592 -MAGnUM project).


Augustin C Tibère-Inglesse

Sean D Mcguire

Pierre Mariotto

Christophe O Laux

Experimental study of recombining nitrogen plasmas. Part I: Vibronic population distributions and nonequilibrium molecular radiation

We present optical emission spectroscopy measurements in recombining nitrogen plasma flows at atmospheric pressure. An inductively coupled plasma torch is used to create an equilibrium plasma, which is then forced to recombine by flowing through a water-cooled tube. For certain conditions, the plasma is forced out of chemical equilibrium. The emission of 𝑁 2 (𝐵 3 𝛱 𝑔 ), 𝑁 2 (𝐶 3 𝛱 𝑢 ) and 𝑁 2 + (𝐵 2 𝛴 𝑢 + )

is studied to measure the nonequilibrium vibrational density distributions within these electronic states. These densities are found to be highly overpopulated in comparison with their corresponding equilibrium values, which is consistent with previous results in recombining flows. The measured densities are also compared with the predictions of a 2-temperature model. This 2-temperature model underpredicts the measured densities but is closer to the measured distributions than the equilibrium densities. The total measured radiation is approximately 100,000 times stronger than the corresponding equilibrium radiation. The 2-T model estimate of this radiation is much closer yet still underestimates by a factor of 10 the measured radiation. These data are intended as a new dataset to test the recombining plasma models used to simulate afterbody flows during atmospheric reentry.

Introduction

In recent years, there has been increased interest in manned Mars return missions [START_REF] Park | Review of chemical-kinetic problems of future NASA missions. II -Mars entries[END_REF]. There are numerous challenges to achieving such a mission. One such challenge is the reentry into the Earth atmosphere on the return voyage [START_REF] Brandis | Investigation of Nonequilibrium Radiation for Mars Entry[END_REF]. During reentry, the capsule enters the atmosphere at velocities above 10 km/s [START_REF] Johnston | Influence of Coupled Radiation and Ablation on the Aerothermodynamic Environment of Planetary Entry Vehicles[END_REF]. At these high velocities, the shock-heated plasma layer formed around the vehicle can result in a radiative heat flux higher than the convective flux [START_REF] Johnston | Features of Afterbody Radiative Heating for Earth Entry[END_REF] [START_REF] Johnston | Shock Layer Radiation Modeling and Uncertainty for Mars entry[END_REF]. The radiation on the forebody has been the focus of much research [START_REF] Park | Nonequilibrium hypersonic aerothermodynamics[END_REF] [START_REF] Johnston | Refinements to Afterbody Radiative Heating Simulations for Earth Entry[END_REF] [8] [START_REF] Hermann | Characterization of a Reentry Plasma Wind-Tunnel Flow with Vacuum-Ultraviolet to Near-Infrared Spectroscopy[END_REF] and is relatively well modeled. However, uncertainties on the afterbody radiative heat flux can still be as high as 200% [START_REF] Johnston | Influence of Coupled Radiation and Ablation on the Aerothermodynamic Environment of Planetary Entry Vehicles[END_REF] [START_REF] Johnston | Features of Afterbody Radiative Heating for Earth Entry[END_REF]. One of the reasons for this uncertainty is the rapid expansion into the afterbody region which forces recombination and a subsequent departure from thermochemical equilibrium conditions. A fundamental study of recombination kinetics is therefore necessary to correctly predict the degree of nonequilibrium of recombining plasmas and the resulting radiation. Recent work has focused on the kinetics and radiation of recombining plasma flows, both experimentally and numerically: see for example Refs. [START_REF] Wei | Experimental and Numerical Investigation of Air Radiation in Superorbital Expanding Flow[END_REF] [11] [12]. This paper focuses on nitrogen recombination kinetics.

In a previous paper [START_REF] Tibère-Inglesse | Validation cases for recombining nitrogen and air plasmas[END_REF], we presented experiments for recombining nitrogen and air plasmas flows using the plasma torch facility at the EM2C laboratory. The plasma torch provides an equilibrium plasma at atmospheric pressure. The plasma then flows through a water-cooled tube at relatively high velocity (~ 1 km/s). The plasma exits the tube in a state of chemical nonequilibrium. The resulting radiation at the exit of the tube was found to be up to 100,000 times higher than if the plasma were in equilibrium at the measured gas temperature determined via Raman spectroscopy measurements of the N 2 (X 1 Σ g + ) ground state. Finally, CFD codes failed to predict any of the measured temperatures under any of the experimental conditionseven those where the plasma remained in equilibrium [START_REF] Tibère-Inglesse | Validation cases for recombining nitrogen and air plasmas[END_REF]. This same discrepancy between measurements and CFD codes was previously observed by other authors [START_REF] Candler | Numerical Simulation of a Nonequilibrium Nitrogen Plasma Experiment[END_REF] [START_REF] Nagulapally | Numerical Simulation of Nonequilibrium Nitrogen and Air Plasma Experiments[END_REF].

These measurements therefore raised questions about the ability of the CFD codes to predict this temperature drop, the level of nonequilibrium, and the resulting radiation.

In this paper, we analyze the radiation of the plasma over the range 250-900 nm to obtain the absolute densities of vibrational states of N 2 (B 3 Π g ), N 2 (C 3 Π u ) and N 2 + (B 2 Σ u + ) . This was previously done only for the N2 first positive system [START_REF] Tibère-Inglesse | Validation cases for recombining nitrogen and air plasmas[END_REF] up to 800 nm. We study two experimental conditions corresponding to two total mass flow rates. The mass flow rate has a significant impact on the degree of nonequilibrium observed in the plasma. In section 1, we present the experimental setup and conditions. The measured nonequilibrium radiation coming from the plasma is presented in section 2. The emission spectra are analyzed to provide information on the absolute densities of the vibrational levels of various electronic states of N2 and N2 + and on the total radiative heat flux of the recombining plasma flows.

I) Experimental setup

The plasma torch facility is a TAFA Model 66 inductively coupled plasma (ICP) torch powered by a 120-kVA radio-frequency (4 MHz) LEPEL Model T-50-3 power supply. The power supply can generate a maximum of 12 kV DC and 7.5 A to the oscillator plates. Details of the plasma torch facility are provided in previous publications [START_REF] Laux | State-to-state modeling of a recombining nitrogen plasma experiment[END_REF] [16] [START_REF] Laux | Optical diagnostics and radiative emission of air plasmas[END_REF]. The plasma produced within the torch exits through a 1-cm diameter nozzle at a speed between 400 and 800 m/s. The plasmas studied here are comprised of 1.2 or 1.9 g/s of N2 or air, premixed with 1.5 g/s of argon to provide stable operating conditions for the torch. A water-cooled tube made of brass is mounted on top of the plasma torch exit nozzle. The length of the tube is either 10 or 15 cm. Emission and Raman measurements are performed at the entrance and exit of the modular tube. The plasma is forced to recombine by imposing rapid cooling via the cold wall (maintained at approximately 300 K). This setup is therefore intended as a simplified experimental model of a recombining flow in atmospheric reentry. Details of the experimental setup are presented in figure 1.

Figure 1: Water-cooled test-section mounted on top of the plasma torch exit nozzle

Optical emission spectroscopy was performed for several experimental conditions with the setup presented in Figure 2. The spectrometer is an Acton Instrument 2500i of 50-cm focal length with a 600 gr/mm grating blazed at 300 nm. We measured spectra from 250 to 900 nm to capture the emission bands of the first and second positive systems of N2, and of the first negative system of N2 + . The emission spectra are calibrated from pixel counts to absolute intensities using an Optronics Laboratories tungsten lamp (model OL550) between 400 and 900 nm, and a calibrated argon mini-arc for wavelengths below 400 nm. Spectra are taken each 0.5 mm along the line of sight and are Abel-inverted to get spatially resolved local plasma emission. The three different mixtures considered in this study are summarized in Table 1. Mass flow rates and conditions for the different cases studied. The chemical conditions are taken from Ref [START_REF] Tibère-Inglesse | Validation cases for recombining nitrogen and air plasmas[END_REF].

II) Nonequilibrium radiation of N2(B), N2(C) and N2 + (B)

This section discusses the experimental results for each of the three cases outlined in Table 1.

Air/Ar plasma:

The test conditions for this test are presented in Table 1. The inlet gas is air (1.9 g/s) premixed with argon (1.5 g/s). The plasma is expected to stay in LTE throughout the test-section [START_REF] Tibère-Inglesse | Validation cases for recombining nitrogen and air plasmas[END_REF]. We measured a spectrum from 250 nm to 800 nm and compared it with the LTE spectrum computed with SPECAIR [START_REF] Laux | Optical diagnostics of atmospheric pressure air plasmas[END_REF]. Results at 15 cm are presented in Figure 3. The good agreement between the experimental and computed LTE spectra supports the equilibrium assumption for this case. Note that, for this case, all temperature measurements agree within another. We present here only the spectrum measured at the center of the tube (i.e. the Abel-inverted spectrum at radial position r=0 cm), but we observed similar agreement across the tube for all Abel-inverted spectra.

Mass flow rate

Reynolds number based on the tube diameter (1 cm) In this case, 1.2 g/s of nitrogen is premixed with 1.5 g/s of argon. In previous work [START_REF] Tibère-Inglesse | Validation cases for recombining nitrogen and air plasmas[END_REF], we measured the emission spectrum between 500 and 800 nm and we observed a departure from chemical equilibrium at 10 and 15 cm leading to an overpopulation of the vibrational levels of N2(B). Here we extend these measurements from 250 to 900 nm in order to (a) validate the previous measurements, (b) access vibrational level v = 1 of N2(B) at 870 nm, and (c) measure the transitions of N2(C) → N2(B) and N2 + (B) → N2 + (X) between 280 and 450 nm. We used the CEA code [START_REF] Mcbride | Computer Program for Calculation of Complex Chemical Equilibrium Compositions and Applications II. User's Manual and Program Description[END_REF] to calculate the equilibrium composition of the plasma at the temperature measured using Raman spectroscopy. Details on the Raman spectroscopy measurements are given in our previous publication [START_REF] Mcguire | Ultraviolet Raman spectroscopy of N2 in a recombining atmospheric pressure plasma[END_REF]. We use the radiation code SPECAIR to compute equilibrium spectra with the given composition and temperature. We observe a large difference between the measured spectrum and the spectrum computed assuming equilibrium. Using an optimization algorithm coupled with Specair, we adjusted the populations of the vibrational levels of N2(C), N2 + (B) and N2(B) to match the measured emission spectra. This optimization algorithm minimizes the L2 error between the experimental and computed spectra using a truncated Newton method.

To quantify the uncertainties, we used the following method: first a random noise was added to the computed spectra. The amplitude of the noise was adjusted to match the experimental uncertainty, found as the addition of the measured random noise and the uncertainties coming from the Abel inversion algorithm. The uncertainties vary from experiment to experiment, but the typical value is 5 % at the centerline of the tube and below 30 % elsewhere. Then the optimization algorithm was run on these new spectra and the error from the best fit was computed. This was performed a total of 100 times for each spectrum. Above 100 times both the standard deviation and the mean of the error did not change significantly and therefore 100 times was found to be sufficient. The standard deviation of the results was taken as the uncertainty on the measurement. This procedure tests the sensitivity of our algorithm to the experimental noise and yields a good estimate of the uncertainty of the fits. In figure 4, we plot the measured spectrum, the computed spectrum using SPECAIR assuming equilibrium, and the spectrum resulting from the optimization algorithm at the exit of the 10-cm tube. We also show the fitted populations of the vibrational levels of N2(C), N2 + (B) and N2(B) used for the SPECAIR calculations.

The plasma is near equilibrium but there is a slight overpopulation of the vibrational levels of N2(C), N2 + (B) and N2(B). We performed the same measurements at the exit of the 15-cm tube. Results are shown in Figure 5. We observe larger vibrational overpopulations of N2(C), N2 + (B) and N2(B) than for the 10-cm tube. Chemical nonequilibrium therefore increases with axial position. Note the presence of NO systems between 250 and 300 nm. This is explained by the presence of small quantities of water vapor impurities in the injection gas. Results are in good agreement if we add approximately 1% mole fraction of water into the plasma. We only present here the results for the centerline of the tube. However, this fitting procedure was performed for all spectra and as a function of radius. We also performed this analysis for all spectral bands observed, which correspond to emission from various vibrational levels of N2(B), N2(C) and N2 + (B). This enabled us to obtain the absolute populations of these states as a function of both radius and axial distance (i.e. tube length). The results for N2(B) are similar to the results of Ref. [START_REF] Tibère-Inglesse | Validation cases for recombining nitrogen and air plasmas[END_REF] and therefore will not be presented here. For N2(C) and N2 + (B), we present the radial distribution of vibrational levels densities for each axial position in Appendix A.

From a modeling perspective, it is important to correctly predict the populations of the excited states as the total radiation is proportional to these populations. Hypersonic CFD codes for atmospheric reentry often use multi-temperature models such as the 2-temperature model proposed by Park [START_REF] Park | Nonequilibrium hypersonic aerothermodynamics[END_REF] [21] [22] [23]. This model uses a translational-rotational temperature Ttr and a vibrational-electronic temperature Tve. The R-T relaxation times are assumed to be fast enough to consider a single temperature for translation and rotation, and likewise for the vibrational and electronic temperature. Vibrationaltranslational (VT) relaxation is determined using the Landau-Teller equation:

𝒅𝑬 𝒗𝒆,𝒔 𝒅𝒕 = ∑ (𝒙 𝒄 𝑬 𝒗𝒆,𝒔 (𝑻 𝒕𝒓 )-𝑬 𝒗𝒆,𝒔 (𝑻 𝒗𝒆 ) 𝝉 𝑽-𝑻 𝒔-𝒄 ) 𝒄 (1)
where 𝐸 𝑣𝑒,𝑠 is the vibrational-electronic energy of species s, 𝑥 𝑐 the mole fraction of species c, and 𝜏 𝑉-𝑇 𝑠-𝑐 the relaxation time of species s colliding with species c in their ground state. The relaxation time is determined using the correlations given in the literature [24] [25]. Note that we make the simplifying assumptions that the VT time is the same for all electronic states of the considered species, and that the electronic states are in Boltzmann equilibrium with the ground state at the given Tve.

We performed a 0-D simulation with the 2-temperature model from 0 to 15 cm along the centerline of the flow. The axial profile of translational temperature is obtained from a cubic interpolation of the temperatures measured with Raman spectroscopy at 0, 10, and 15 cm (see Figure 6). The radial velocity profile is obtained by self-similarity with the temperature profile, as discussed and calculated in Ref. [START_REF] Tibère-Inglesse | Validation cases for recombining nitrogen and air plasmas[END_REF]. The centerline velocity at x= 0 cm is about 800 m/s and about 500 m/s at x=15 cm. Three species are considered, N2, N, and argon. We consider uncertainties on the measured translational temperatures (about 5% at 0 cm, and 10% at 10 and 15 cm), on the shape of the interpolated temperature profile, and on the velocity (about 20%). Given these uncertainties, the Landau-Teller model gives Tve = 5950 ± 200 K at 10 cm and 5000 ± 150 K at 15 cm. The different translational temperature profiles taken in the calculations and the resulting vibrational temperature profiles are shown in Figure 6. Figure 7 shows the measured vibrational population distributions of the N2 states at 0, 10 and 15 cm, along with the distributions corresponding to equilibrium and those obtained with the 2-temperature model. The populations of the first three vibrational levels of N2(X) were measured by Raman spectroscopy in our previous work [START_REF] Mcguire | Ultraviolet Raman spectroscopy of N2 in a recombining atmospheric pressure plasma[END_REF] and the rotational and vibrational temperatures of N2(X) were found to be equal. Therefore, the rotational and vibrational distribution of N2(X) are assumed to follow a Boltzmann distribution at the measured temperature. In all cases, the equilibrium distribution largely underestimates the measured populations. At 10 cm, the populations computed using the 2-T model underestimate the measurements by a factor of about 3. At 15 cm, better agreement is found for the first vibrational levels of N2B and N2C. However, a large discrepancy is observed for the higher vibrational levels of N2B. This is because the N2(B,v=13) vibrational level is populated by inverse predissociation of N( 4 S), and therefore its overpopulation with respect to equilibrium varies as the square of the overpopulation of nitrogen atoms, as explained in Ref [START_REF] Laux | State-to-state modeling of a recombining nitrogen plasma experiment[END_REF]. Since the atomic nitrogen population is nearly frozen along the axis of the tube, the overpopulation of N( 4 S) increases strongly along the axis of the tube. Therefore, the departure from the 2-temperature model predictions becomes stronger at 15 cm than at 10 cm. This mechanism of inverse predissociation is of course not considered in the 2-T model. The radiation emitted by the plasma is proportional to the population of the excited states. Figure 8 compares the integrated intensity from 250 to 900 nm for the measured, equilibrium, and 2-T spectra. At 15 cm, the measured total radiation is about 4 times stronger than equilibrium, and 1.5 times larger than with the 2-T model. Thus, the 2-T model improves the radiation predictions. This experiment represents a case where the plasma stays relatively close to equilibrium but with still non-negligible nonequilibrium radiation. This is therefore a useful test case for numerical codes to check their ability to predict the radiation when a small departure from equilibrium is induced.

N2/Ar high flowrate plasma:

In this case, 1.9 g/s of nitrogen is premixed with 1.5 g/s of argon. Using the spectrum measured between 250 and 900 nm, we extract with our fitting routine the populations of the vibrational levels of N2(C), N2 + (B) and N2(B). Figure 9 shows the results at the exit of the 10-cm tube. We observe a larger departure from equilibrium than in the low flowrate case. This was already observed for N2(B) in Ref. [START_REF] Laux | State-to-state modeling of a recombining nitrogen plasma experiment[END_REF] and there is a similar trend for N2(C) and N2 + (B). Results at 15 cm are presented in Figure 10. The vibrational levels of excited states of nitrogen are highly overpopulated relative to their equilibrium populations.

The vibrational population distributions of N2(C) and N2 + (B) measured at each radial and axial position are presented in detail in Appendix A. The Boltzmann plot shown in Figure 11 summarizes the population distributions of the various states measured, namely N2(X), N2(B) and N2(C). We performed the same analysis as for the low flowrate case using a 2-T model and obtained Tve = 4700 ± 300 K at 15 cm. The corresponding distribution is also plotted in Figure 11. The chemistry is partly frozen in the tube, because the excited state densities do not change significantly along the axis of the tube. Note also that the population density remains fairly continuous between the N2(B) and N2(C) states. The 2-T model is closer to the experimental values than the equilibrium distribution, but discrepancies still appear at 10 cm and for higher energy levels at 15 cm. The non-Boltzmann distribution cannot be predicted using a 2T model. Therefore, a detailed vibrational specific kinetic model must be used in order to predict the measured distributions. Figure 12 shows the volumetric emission at the centerline of the plasma flow, integrated between 250 to 900 nm, obtained from the measurements, from the 2-T model, and from equilibrium population distributions. At 10 cm, the equilibrium radiation is 250 times smaller than the measurements and about 100,000 times smaller at 15 cm. The 2-T model improves the agreement with an underprediction by a factor of 13 at 10 cm and a factor of about 6 at 15 cm. The 2-T model is therefore unable to correctly predict the radiation coming from such recombining plasma flows.

Discussion

In both N2/Ar mixtures studied, the excited states of nitrogen were found to be highly overpopulated as compared to both equilibrium and 2-T model predictions. The main reason is believed to be the inverse predissociation of N( 4 S) into N2(B,v=13) as explained in ref [START_REF] Laux | State-to-state modeling of a recombining nitrogen plasma experiment[END_REF].

Potentially, other reactions could also play a role as discussed in other papers. For example, Capitelli et al [START_REF] Capitelli | Non-Maxwell Behavior of Electron Energy Distribution Functions in Expanding Nitrogen Arcs[END_REF] showed that the population distribution of excited electronic states is sometimes affected by superelastic electron collisions. However, they simulated conditions close to those of our experiments and found that superelastic collisions have no influence under our conditions.

Plain et al [START_REF] Plain | On the coupling between Nd2(B3Πg,ν') and N2(X1Σg,ν) vibrational distributions in a glow discharge column in flowing N2[END_REF] reported that coupling effects between high-lying vibrational levels of N2(X) and N2(B) could affect the excited electronic state population distributions. This cannot be proven for the moment as our measurement of the vibrational level of N2(X) using Raman spectroscopy only yields results for low vibrational levels. A more sensitive diagnostic, e.g. CARS [START_REF] Eckbreth | Laser Diagnostics for Combustion Temperature and Species[END_REF], would be necessary to gain access to higher-lying vibrational levels. In addition, measuring precisely the rotational distribution of N2(X) could also be relevant to confirm the predictions of Panesi et al [START_REF] Panesi | Rovibrational internal energy transfer and dissociation of N2(1Σg+)-N(4Su) system in hypersonic flows[END_REF] who found the rotational and vibrational temperatures of N2(X) to differ under certain conditions.

Finally, it should be noted that the chemistry in our experiments is partly frozen and therefore the recombination kinetics are slow. This explains why the 2-T model improves the agreement between the experiments and the predictions. However, this model is expected to give worse results in a case where recombination is faster. A detailed kinetic model is currently under development to simulate the experimental results reported here.

Conclusions

New experiments were performed in our plasma torch facility to measure emission from both equilibrium and nonequilibrium N2/Ar and Air/Ar plasmas. Optical emission spectroscopy was used to measure spectra of the plasmas between 250 and 900 nm for three different test cases: a mixture of Air/Ar and two cases with N2/Ar at low and high flowrates. For the Air/Ar mixture, the emission spectra agreed with those computed assuming equilibrium. This was expected as the plasma stays in equilibrium for this case. For the N2/Ar mixtures, the plasma was found to be out of equilibrium. We were able to determine the vibrational population distributions of N2(B), N2(C) and N2 + (B) using SPECAIR and the measured spectra. These states were found to be overpopulated relative to equilibrium.

Finally, the measured radiation was integrated over the measured range of wavelength. This allows us to get the radiative heat flux which is the important property to model for reentry applications. Hypersonic CFD codes often use multiple temperatures models. Here, a 2-temperature model was employed to compare predictions with experiments. This was done using the Landau-Teller equation to compute the relaxation of the vibrational temperature to the translational one. Due to the rapid decrease in translational-rotational temperature, the vibrational temperature does not relax fast enough to equilibrate with the translational temperature. We compared the measured radiation with the 2temperature model predictions and with equilibrium predictions. The measured radiation was found to be up to 100,000 times higher than equilibrium. The agreement with the 2-T model was much better, however this model still underestimates the measured radiation by an order of magnitude and is expected to be worse in a real entry scenario. This is an important consideration for reentry applications. Note also that due to the non-Boltzmann vibrational distribution of the population of states such as N2B, only a vibrational specific kinetic model coupled to a CFD code will be able to correctly predict the radiation in recombining flows. case. Energy (cm -1 )

mm

Figure 2 :

 2 Figure 2: Experimental setup for optical emission spectroscopy (here at the tube exit)

Figure 3 :

 3 Figure 3: Measured (black) and computed (red) spectrum at the exit of the 15-cm tube for the Air/Ar plasma case. N2/Ar low flowrate plasma:

Figure 4 :

 4 Figure 4: Top: comparison between measured and computed spectra at the exit of the 10-cm tube for the N2/Ar low flowrate case at the center of the tube (r=0 cm). Equilibrium spectrum (blue) and fitted spectrum (red). Bottom: equilibrium and measured populations of emitting levels used for SPECAIR calculations.

Figure 5 :

 5 Figure 5: Top: comparison between measured and computed spectra at the exit of the 15-cm tube for the N2/Ar low flowrate case at the center of the tube (r=0 cm). Equilibrium spectrum (blue) and fitted spectrum (red). Bottom: equilibrium and measured populations of emitting levels used for SPECAIR calculations.

Figure 6 :

 6 Figure 6: Translational temperature profiles (solid lines) based on the measured Raman temperatures, and vibrational temperature profiles (dashed lines) obtained from the Landau-Teller model for the N2/Ar low flowrate case. The black curves correspond to the nominal case. The red and blue curves represent the upper and lower uncertainty bounds, respectively.

Figure 7 :

 7 Figure 7: Comparison between the population distributions computed using the 2-temperature model (green) and the measurements (symbols) for the N2/Ar low flowrate case. The green shaded swaths represent the uncertainty bands on the computed Tve.

Figure 8 :

 8 Figure 8: Radiative intensity (integrated between 250 and 900 nm) of the measured, equilibrium and 2-T spectra for the N2/Ar low flowrate case.

Figure 9 :Figure 10 :

 910 Figure 9: Top: comparison between measured and computed spectra at the exit of the 10-cm tube (at r = 0 cm) for the N2/Ar high flowrate case. Equilibrium spectrum (blue) and fitted spectrum (red). Bottom: equilibrium and measured populations of emitting levels used for SPECAIR calculations.

Figure 11 :

 11 Figure 11: Comparison between the population distributions computed using the 2-temperature model (green) and the measurements (symbols) for the N2/Ar HF case. The green shaded swaths represent the uncertainty bands on the computed Tve.

Figure 12 :

 12 Figure 12: Comparison between the integrated intensity assuming equilibrium (at the measured temperature) and the measured one between 250 and 900 nm for the N2/Ar HF case.

Figure A1 :

 A1 Figure A1: Radial distribution of measured and equilibrium densities of N2B (top), N2C (bottom left) and N2 + B (bottom right) for the N2/Ar LF case at the exit of the 10-cm tube.

Figure A3 :Figure A4 :

 A3A4 Figure A3: Radial distribution of measured and equilibrium densities of N2B (top), N2C (middle) and N2 + B (bottom) for the N2/Ar HF case at the exit of the 10-cm tube.

Table

Acknowledgments

This work has been supported by Ariane Group, under a CIFRE Ph.D. grant (number 42701092/20160218/JSE), with Dr. Laurent Visconti as Technical Monitor.

Appendix A: Radial profiles of vibrational populations of N2(B), N2(C) and N2 + (B)

In this appendix we present the radial distributions of vibrational populations for each case considered. This intends to serve as a test-case database for computational simulations. In Figures A1 andA2, we present the radial distribution of vibrational populations for the N2/Ar low flowrate (LF) case at 10 and Energy (cm -1 )

mm


Radiation of Nonequilibrium Recombining Plasma Flows

Reasons for discrepancy

• Recombination of N into rovibrationally excited N 2 is less exothermic than into ground state N 2 . Therefore, the temperature may decrease more than predicted using the simple kinetic rate • Radiant conductivity may increase radial diffusion • Temperature measured with emission spectroscopy of N 2 + rotational lines may differ from gas temperature: an independent measurement of temperature is needed.

Comparison between Raman model and experimental spectrum

1

  Airbus Safran Launchers, 66 Route de Verneuil, 78130 Les Mureaux 2 Laboratoire EM2C, CNRS, CentraleSupélec, Université Paris-Saclay Grande Voie des Vignes, Châtenay-Malabry, France A.C. Tibère-Inglesse 1,2 , P. Mariotto 2 , S. McGuire 2 , , M.Y. Perrin 2 , C.O. Laux 2 ICP Plasma Torch (1 atm) • Recombining flow is produced in a modular water-cooled brass tube (from 0 to 65 cm) • Emission and Raman scattering spectroscopy are used to measure temperature 3. Previous measurements (Stanford 1997-2003, Laux et al [2at 7200 K v = 1.5 km/s Fast cooling imposed by water-cooled wall at 300 K N 2 Plasma at 4850 K Inlet : LTE Outlet: Nonequilibrium • Large discrepancy at 15 cm exit between simulations and experiments [3] • Same discrepancy with Eilmer3 (CFD code from UQ [4]) Residence time : 100 µs Chemistry : Park model Temperature profile at the inlet and the outlet from Candler et al [C.O. Johnston and A. M. Brandis. Features of afterbody radiative heating for earth entry. Journal of Spacecraft and Rockets, 52(1):105-119, 2014 [2] C. O, Laux, L. Pierrot, R. J. Gessman. State-to-state modeling of a recombining nitrogen plasma experiment. Chemical Physics [3] G. V, Candler et al. Numerical Simulation of Nonequilibrium Nitrogen and Air Plasma Experiments. 29 TH AIAA, Plasmadynamics and lasers conference [4] P. A, Jacobs et al. Eilmer's Theory Book: Basic Models for Gas Dynamics and Thermochemistry. Mechanical Engineering Report 2010/09 • LTE plasma at 7200 K Experimental data are needed to validate radiation models in recombining flows • There are large uncertainties (~200%) on the prediction of radiative fluxes on afterbody surfaces • The radiative heat flux to an afterbody surface is governed by recombining plasma dynamics • Raman measurements at test-section inlet: 𝑇 𝑟𝑜𝑡 = 𝑇 𝑣𝑖𝑏 = 6500 ± 400𝐾 close to temperature measured with emission spectroscopy • At exit of 15 cm test-section: 𝑇 𝑟𝑜𝑡 = 𝑇 𝑣𝑖𝑏 = 3100 ± 250𝐾 much lower than temperature measured with emission spectroscopy Future work: • Measurements at known gas temperature condition for validation of Raman model • Measurements for various values of 𝜏 𝑟𝑒𝑠𝑖𝑑𝑒𝑛𝑐𝑒 𝜏 𝑐ℎ𝑒𝑚𝑖𝑠𝑡𝑟𝑦 , ranging from equilibrium to nonequilibrium conditions Error between Raman model and experimental spectrum Test-section inlet Exit of 15 cm test-section Spectrum at exit of 15-cm test-section • Nonequilibrium plasma at 4850 K • Radiation is higher than LTE: x6 for N 2 (1+), x30 for N 2 (2+), x10 for N 2 + (1-)