Davide Frey

Rachid Guerraoui

Anne-Marie Kermarrec

Maxime Monod

Vivien Quéma

Stretching Gossip with Live Streaming

Gossip-based information dissemination protocols are considered easy to deploy, scalable and resilient to network dynamics. They are also considered highly flexible, namely tunable at will to increase their robustness and adapt to churn. So far however, they have mainly been evaluated through simulation, very often assuming ideal settings.

Instead, in this paper, we report on an extensive study of gossip protocols, deployed on a 230 Planetlab node testbed, in the context of a challenging video streaming application in environments with constrained bandwidths. More precisely, we assess the impact of varying the well known knobs of gossip, fanout and refresh rate, in various uploadbandwidth distributions and churn. Our results show that in such challenging contexts, the performance of gossip protocols may be hampered by high fanout values. We also show that the more proactive a gossip protocol, the better it copes with churn. For instance, when 20% of the nodes simultaneously crash, 70% of the remaining nodes do not suffer any loss in stream quality, while the others only experience a performance decrease for an average of 5 seconds around the churn event.

Introduction

Gossip-based protocols [START_REF] Birman | Bimodal Multicast[END_REF][START_REF] Demers | Epidemic Algorithms for Replicated Database Maintenance[END_REF][START_REF] Deshpande | CREW: A Gossip-based Flash-Dissemination System[END_REF][START_REF] Li | FlightPath: Obedience vs. Choice in Cooperative Services[END_REF] are well known for their robustness and ability to cope with network dynamics. A lot of efforts have thus been devoted to their theoretical analysis and evaluation [START_REF] Karp | Randomized Rumor Spreading[END_REF][START_REF] Sanghavi | Gossiping with Multiple Messages[END_REF]. It was for instance shown that the number of communication partners (called fanout) in a system of size n, should be greater than a threshold value ln(n) for the gossip dissemination to reach all nodes with high probability [START_REF] Kermarrec | Probabilistic Reliable Dissemination in Large-Scale Systems[END_REF].

Gossip protocols have also been shown to be effective for challenging applications like live streaming [START_REF] Bonald | Epidemic Live Streaming: Optimal Performance Trade-Offs[END_REF]. Yet, most of the work evaluating gossip has considered ideal settings; e.g., unconstrained bandwidth, no (or uniform) message loss, global knowledge about the state of all nodes. Evaluations conducted through simulation [START_REF] Bonald | Epidemic Live Streaming: Optimal Performance Trade-Offs[END_REF][START_REF] Deshpande | CREW: A Gossip-based Flash-Dissemination System[END_REF][START_REF] Massoulié | Randomized Decentralized Broadcasting Algorithms[END_REF] also assume that key parameters of gossip, such as fanout and gossip rate, can be arbitrarily tuned to improve robustness and to adequately adapt to network dynamics. Moreover, exceptions of gossip experiments in real settings assume infinite bandwidth [START_REF] Li | FlightPath: Obedience vs. Choice in Cooperative Services[END_REF], or consider applications with low bandwidth needs, such as membership maintenance [START_REF] Drost | ARRG: Real-World Gossiping[END_REF].

In the presence of heterogeneous bandwidth and greedy applications, we have no evidence that gossip will fulfill its promises. The impact of the key parameters of gossip protocols in such contexts remains unexplored. For instance, the impact of varying the fanout, an obvious knob to tune the robustness of a gossip protocol, has never been determined.

The rate at which the gossip partners are changed reflects the proactiveness of a gossip protocol. The impact of varying this rate has never been studied either. At one extreme, one might consider a gossip protocol where nodes change their neighbors at every communication step (this is the scheme typically considered in theoretical studies). At the other extreme, nodes would never change their communication partners unless they notice malfunctions; this is typically the approach underlying mesh-based systems where the unstructured overlay used to create random or deterministic dissemination trees, once constructed, is kept as is until the mesh connectivity is reduced (e.g., a node has less than a given number of neighbors), or a node feels it is incorrectly fed [START_REF] Li | Inside the New Coolstreaming: Principles, Measurements and Performance Implications[END_REF]. A wide range of schemes can be considered between these two extremes.

In this paper, we report on extensive experimentation on gossip protocols in a realistic environment and in the context of greedy applications with respect to bandwidth, namely video streaming applications. We evaluate the impact of the key parameters of gossip protocols with various churn scenarios and bandwidth capabilities. Finally, we do not consider any repair mechanism or underlying overlay structure, thus preserving the inherent simplicity of gossip.

In short, our results show that gossip can be very effective in greedy and capability-constrained applications, but only within a small window of parameter ranges. First, the resulting stream quality is very sensitive to the fanout value. The power of gossip is unleashed when the fanout is slightly larger than ln(n) but degrades drastically with higher fanout values as a result of higher contention. For example, with a bandwidth cap of 700 kbps for a stream rate of 600 kbps, gossip reaches its optimal performance with a fanout of 7 (230 nodes). Second, gossip is most effective when the set of communication partners is continuously changing, particularly in the presence of churn. When 20% of the nodes simultaneously crash, 70% of the remaining nodes do not suffer any loss in quality, while the remaining ones only experience a performance decrease for an average of 5 around the churn event. These numbers drop to 30% when communication partners are refreshed only every 2 gossip rounds.

Finally, our results show that allowing nodes to change their incoming communication partners explicitly by means of periodic requests to be fed by a new set of nodes does not provide any improvement over standard gossip techniques. This further confirms the strength of gossip protocols in their simplest form.

Gossip-based content dissemination

Consider a set of n nodes, and an event e to be disseminated in the system: e typically consists in a payload and an id. Gossip-based content dissemination protocols generally follow a three-phase push-request-push pattern as depicted in Algorithm 1. The protocol follows an infectand-die model [START_REF] Kermarrec | Probabilistic Reliable Dissemination in Large-Scale Systems[END_REF]. Each node periodically contacts a fixed number, f (fanout), of nodes chosen according to the se- lectNodes function and proposes them a set of event ids (of the events it has received) with a [PROPOSE] message (line 5 for the broadcaster and 6 for other nodes). Upon receipt of a [PROPOSE], nodes pull the content needed with a [REQUEST] message to the proposing peer. The peer being pulled then sends back the actual content (the event) in a [SERVE] message that contains the requested events. Such a protocol leverages the reliability of gossip to disseminate the event ids while avoiding redundancy in content dissemination, which would increase the consumed bandwidth.

Tailoring gossip-based dissemination

While the protocol is simple, its parameters may be tuned to improve efficiency, reliability, or resilience to churn. We explore the parameter space as follows.

Fanout. The fanout is defined as the number of communication partners each node contacts in each gossip operation. It has been theoretically shown that a fanout greater than ln(n) in an infect-and-die model [START_REF] Kermarrec | Probabilistic Reliable Dissemination in Large-Scale Systems[END_REF] ensures a highly reliable dissemination. Theory also assumes that increasing the fanout results in an even more robust (as the probability to receive an event id increases) and faster dissemination (as the degree of the resulting dissemination tree increases). In practice, however, too high a fanout can negatively impact performance as heavily requested nodes may exceed their capabilities in bandwidth constrained environments.

Proactiveness. We define proactiveness as the rate at which a node modifies its set of communication partners. We explore two ways of modifying this set. First, the node may locally refresh its set of communication partners and change the output of selectNodes every X calls. In short, when X = 1 the gossip partners of the node change at every call to selectNodes (i.e., every gossip period), whereas X = ∞ means that the communication partners of a node never change.

Second, every Y gossip periods, the node may contact f random partners asking to be inserted in their views. When Y = 1, a node A sends a feed-me message to f random partners every gossip period asking them to feed it. Each of the random f partners replaces a random node from its current set of f partners with A.

Evaluation

We evaluate the impact of the fanout and proactiveness of a gossip protocol by deploying a streaming application based on Algorithm 1 over a set of 230 PlanetLab nodes. Our implementation is based on UDP and incorporates retransmission to recover lost stream packets (lines 14, 15, 25 in Algorithm 1).

Bandwidth constraints. PlanetLab nodes benefit from high bandwidth capabilities and therefore are not representative of peers with limited capabilities. We thus artificially constrain the upload bandwidths of nodes with three different caps: 700 kbps, 1000 kbps and 2000 kbps. To limit message loss resulting from bandwidth bursts, our bandwidth limiter also implements a bandwidth throttling mechanism.

Streaming Configuration. A source node generates a stream of 600 kbps and proposes it to 7 nodes in all experiments. To provide further tolerance to message loss (combined with retransmission), the source groups packets in windows of 110 packets, including 9 FEC coded packets. The gossip period is set to 200 ms.

Evaluation metrics. We assess the performance of the streaming protocol along two metrics: (i) the stream lag, defined as the difference between the time at which the stream is published by the source and the time at which it is actually delivered to the player on the nodes; and (ii) the stream quality, which represents the percentage of the stream that is viewable. A window is jittered if it does not contain enough packets (i.e., strictly less than 101) to fully reconstruct the window. A stream with a maximum of 1% jitter means that at least 99% of all the windows are complete. 1 Stream lag and quality are correlated notions: the longer the acceptable lag, the better the quality. We consider stream qualities corresponding to several lag values as well 1 An incomplete window does not mean that the window is unusable. Using systematic coding, a node receiving 100 out of the 101 original packets, experiences a 99% delivery in that window.

as to an infinite lag, which represents the performance obtainable by a user that downloads the stream for playing at a later stage, e.g., offline viewing. Each experiment was run multiple times. We plotted the most representative one.

Impact of varying the fanout

We start our analysis by measuring how varying the fanout impacts each of the two metrics, with a proactiveness degree of 1 (X = 1). Results are depicted in Figures 123. Figure 1 shows the percentage of nodes that can view the stream with less than 1% jitter for various stream lags in a setting where all nodes have their upload capabilities capped at 700 kbps. Optimal fanout range. The plot clearly highlights an optimal range of fanout values (from 7 to 15 in Figure 1) that gives the best performance independently of the lag value considered. Lower fanout values are insufficient to achieve effective dissemination, while larger values generate higher network traffic and congestion, thus decreasing the obtainable stream quality. The plot also shows that while the lines corresponding to finite lag values have a bell shape without any flat region, the one corresponding to offline viewing does not drop dramatically until a fanout value of 40. The bandwidth throttling mechanism is in fact able to recover from the congestion generated by large fanout values once the source has stopped generating new packets. For fanouts above 40, on the other hand, such recovery does not occur.

Critical lag value.

A different view on the same set of data is provided by Figure 2. For each value t, the plot shows the percentage of nodes that can view at least 99% of the stream with a lag shorter than t. A fanout in the optimal range (e.g., 7) causes almost all nodes to receive a high quality stream after a critical lag value (t = 5 s for a fanout of 7). Moderately larger fanout values cause this critical value to increase (t = 22 s for a fanout of 20), while for fanouts above 35, no critical value is present. Rather congestion causes significant performance degradation. With a fanout of 40, only 20% of the nodes can view the stream with a lag shorter than 60 s, and a lag of 90 s is necessary to reach 75% of the nodes. Behavior with less tight distributions. The presence of an optimal fanout value is clearly a result of operation under limited bandwidth. Figure 3 complements the picture by showing how fanout affects performance under less critical conditions: 1000 kbps and 2000 kbps of capped bandwidth. As available bandwidth increases, the range of good fanout values clearly becomes larger and larger and tends to move to the right. With an available bandwidth of 1000 kbps, which is more than 1.67 times the stream rate, it is still possible to identify a clear region outside of which performance degrades significantly. With 2000 kbps of available bandwidth, both the 10 s-lag and the offline performance figures appear to remain high even with very large fanout values. This behavior may be better understood by examining how bandwidth is actually used by the PlanetLab nodes involved in the dissemination. According to Algorithm 1, all nodes contribute to the gossip dissemination by sending their proposal messages to the same number of nodes. However, the contribution of a node in terms of serve messages depends on the probability that its proposal messages are accepted by other nodes. In general, nodes with low-latency and reliable connections (i.e., good nodes) have higher probabilities to see their proposals accepted. This is confirmed by Figure 4. The plot also shows that the heterogeneity in bandwidth utilization increases with the amount of available bandwidth. For example the lines for the 700 kbps bandwidth cap show an almost homogeneous distribution apart from a small set of bad nodes. This is because the higher latencies exhibited by good nodes when their bandwidth utilization is close to the limit causes other nodes to work more, thus equalizing bandwidth consumption. On the other hand, if we observe the lines corresponding to a fanout of 50 in the 1000 kbps and 2000 kbps scenarios and to a fanout of 100 in the 2000 kbps scenario, we see that good nodes have enough spare capacity to operate without saturating their bandwidths. As a result, the contribution of nodes remains highly heterogeneous.

Bandwidth usage with different fanouts values.

Proactiveness

Next, we present our analysis of gossip proactiveness by showing how refreshing the set of communication partners affects application performance. Figure 5 presents the results obtained by varying the view refresh rate, X, in a scenario with a 700 kbps bandwidth cap. The plot shows three lines corresponding to stream lags of 10 s and 20 s as well as to offline viewing. In all cases, results confirm that the best performance is obtained by varying the set of gossip partners at every communication round. If on the other hand, the set of communication partners remains constant for long periods of time, a small set of nodes end up having the responsibility of feeding large numbers of nodes for as long as they keep being selected early in the dissemination process. This means that their upload rates remain constantly higher than their allowed bandwidth limits, ultimately resulting in high levels of congestion, huge latencies, and message loss. In accordance with these observations, Figure 5 shows that the slope at which the curves decrease with X is most negative for a lag of 10s. This is because longer values of lag allow the bandwidth throttling mechanism more time to recover from the bursts generated by a constant set of communication partners. Nonetheless, a completely static dissemination mesh invariably yields bad performance even for offline viewing as the load becomes then concentrated on a very small set of nodes for the entire experiment.

Requesting nodes to update their views. A second way to modify the proactive behavior of the considered streaming protocol is for nodes to periodically request a new set of nodes to feed them, i.e., every Y dissemination rounds. In Figure 6 we show the results obtained with different values of Y . Results show that this technique remains inferior to the simpler approach of choosing a view refresh rate of X = 1, as discussed above. The additional messages used by this approach may in fact be lost or delayed while the node is congested, resulting in a larger Y than planned.

Performance in the presence of churn

Finally, we evaluate the impact of proactiveness in the presence of churn with Y = ∞. Results are depicted in Figures 7 and 8. The experiments consist in randomly picking a percentage of nodes and make them fail simultaneously. We compare the baseline results (e.g., when no nodes fail) with those with increasing percentages, from 10% to 80%, of failing nodes. Figure 7 shows the percentage of remaining nodes experiencing less than 1% jitter after the catastrophic failure, for values of X of 1, 2, 20 and ∞.

Figure 7 clearly shows that a completely dynamic mesh offers the best performance in terms of ability to withstand churn. With 35% churn, a proactiveness of X = 1 is able to deliver an unaffected stream to 60% of the remaining nodes, while the percentage drops to 32% for X = 2 and a stream lag of 20 seconds. The results obtained with large values of X and in particular when X = ∞ show very high degrees of variability from experiment to experiment. The resulting static or semi-static random graph may become completely unable to disseminate the stream if failing nodes are close to the source or it may appear extremely resilient to churn if failing nodes are located at the edge of the network. On average, performance is therefore in favor of a completely dynamic graph (X = 1).

It should be noted that Figure 7 only shows how many nodes manage to remain completely unaware of the churn event. To characterize the extent of the performance decrease experienced by all surviving nodes, Figure 8 shows the average percentage of decoded windows over the total number of windows streamed by the source. With X = 1, the protocol is almost unaffected by churn, and nodes correctly receive over 90% of the windows for all churn percentages lower than 80%. In addition, almost all the missing windows turn out to be concentrated in a time frame of 5 s to 10 s around the churn event when 20% and 80% of nodes fail (not shown in the plot).

Concluding remarks

The evaluations of gossip protocols for dissemination usually back up the associated theoretical claims: gossip can be tuned to achieve reliable dissemination in the presence of churn. In this paper, we challenged these results, obtained mostly through simulations in close-to-ideal settings, and evaluated a gossip-based live streaming application in a real deployment over 230 PlanetLab nodes.

Our results show that message loss and limited bandwidth significantly restrict the range of parameter values in which gossip can successfully operate. First, the fanout cannot be increased arbitrarily to improve reliability and latency, but must remain small enough to prevent bandwidth saturation. Second, the set of communication partners should be changed frequently, at every communication round, in order to minimize congestion and provide an effective response to churn.

Figure 1 :

 1 Figure 1: Percentage of nodes viewing the stream with less than 1% of jitter (upload capped at 700 kbps).

Figure 2 :

 2 Figure 2: Cumulative distribution of stream lag with various fanouts (upload capped at 700 kbps).

   shows the distribution of bandwidth utilization over all the nodes involved in the experiments sorted from the one contributing the most to the one contributing the least. The plot immediately highlights an interesting property: even though all the considered scenarios have a homoge-

Figure 3 :

 3 Figure 3: Percentage of nodes viewing the stream with less than 1% of jitter with upload caps of 1000 kbps and 2000 kbps, and different fanout values.

Figure 4 :

 4 Figure 4: Distribution of bandwidth usage among nodes with different fanout values and upload caps.

Figure 5 :

 5 Figure 5: Percentage of nodes viewing the stream with at most 1% jitter as a function of the refresh rate X.

Figure 6 :

 6 Figure 6: Percentage of nodes viewing the stream with at most 1% jitter as a function of the request rate Y .

Figure 7 :

 7 Figure 7: Percentage of surviving nodes experiencing less than 1% jitter for different values of X.

Figure 8 :

 8 Figure 8: Average percentage of complete windows for surviving nodes.

(GossipTimer(gossipPeriod)) Phase 1 -Push event ids procedure

  Algorithm 1 Standard gossip protocol

	Initialization:
	1: f := ln(n) + c
	2: eventsToPropose := eventsDelivered := requestedEvents := ∅
	3: startpublish(e) is
	4: deliverEvent(e)
	5: gossip({e.id})
	upon (GossipTimer mod gossipPeriod) = 0 do
	6: gossip(eventsToPropose)
	7: eventsToPropose := ∅	{Infect and die}
	Phase 2 -Request events
	upon receive [PROPOSE, eventsProposed] do
	8: wantedEvents := ∅
	9: for all e.id ∈ eventsProposed do
	10:	if (e.id / ∈ requestedEvents) then
	11:	wantedEvents := wantedEvents ∪ e.id
	12: requestedEvents := requestedEvents ∪ wantedEvents
	13: reply [REQUEST, wantedEvents]
	14: if (e requested less than K times) then
	15:	start(RetTimer(retPeriod, eventsProposed))

Phase 3 -Push payload upon receive [REQUEST, wantedEvents] do 16: askedEvents := ∅ 17: for all e.id ∈ wantedEvents do 18: askedEvents := askedEvents ∪ getEvent(e.id) 19: reply [SERVE, askedEvents] upon receive [SERVE, events] do 20: for all e ∈ events do 21: if (e / ∈ eventsDelivered) then 22: eventsToPropose := eventsToPropose ∪ e.id 23: deliverEvent(e) 24: cancel(RetTimer(retPeriod, events)) Retransmission upon (RetTimer(retPeriod, eventsProposed) mod retPeriod) = 0 do 25: receive [PROPOSE, eventsProposed] Miscellaneous function selectNodes(f ) returns set of nodes is 26: return f uniformly random chosen nodes in the set of all nodes function getEvent(event id) returns event is 27: return the event corresponding to the id procedure deliverEvent(e) is 28: deliveredEvents := deliveredEvents ∪ e 29: deliver(e) procedure gossip(event ids) is 30: communicationPartners := selectNodes(f ) 31: for all p ∈ communicationPartners do 32: send(p) [PROPOSE, event ids]

* Maxime Monod has been partially funded by the Swiss National Science Foundation with grant 200021-113825.


Xuan Bach Nguyen
email: xuan-bach.nguyen@ensma.fr

Didier Saury
email: didier.saury@ensma.fr

Denis Lemonnier
email: denis.lemonnier@ensma.fr

Coupled natural convection and radiation in a cubic cavity filled with an air -H 2 O mixture in the presence of a heated obstacle

Keywords: Heat transfer, Convection, Radiation, Numerical method

Natural convection in a cubical cavity with a hot obstacle located on its floor is investigated. The inner fluid is a semi-transparent mixture of dry air and water vapor, which creates a coupled convective and radiative transport within the fluid. The conservation equations are solved by a finite volume method and the radiative transfer equation by using the discrete ordinates method. The radiative properties of the mixture are accounted for by a spectral line weighted sum of gray gases model associated to the rank correlated approach. It was observed that the volume radiation has a strong influence on the thermal and dynamic fields. The nearly vertical stratification of the temperature field around the plume is broken. Radiation also accelerates the boundary layers near the lateral surfaces and the ceiling and the floor of the enclosure. The total heat transfer is decreased due to both the reduction in convective process near the vertical walls and the attenuation by radiation.

Introduction

The study of natural convection in confined environments is still the subject of many research, both numerically and experimentally. In this type of problem, the different modes of heat transfer (convection, conduction, radiation) can intervene in a coupled way. However, when radiative transport is considered, a particular problem arises when the fluid behaves as a semi-transparent medium, i.e., absorbs and emits infrared radiation. It is then necessary to take into account an internal heat source within the medium, resulting from cite as: X. B. Nguyen, D. Saury, D. Lemonnier, « Coupled natural convection and radiation in a cubic cavity filled with an air-H2O mixture in the presence of a heated obstacle », Progress in Computational Fluid Dynamics, Volume 22(3) pp. 139-158, 2022. DOI: 10.1504/PCFD.2021.10042895 the difference between the radiant energy absorbed and emitted by each volume element. Many studies have investigated this phenomenon in a rectangular enclosure [START_REF] Yücel | Natural convection and radiation in a square enclosure[END_REF], [START_REF] Tan | Combined radiation and natural convection in a two-dimensional participating square medium[END_REF], Colomer et al. (2004a), [START_REF] Colomer | Coupled radiation and natural convection: Different approaches of the slw model for a non-gray gas mixture[END_REF], [START_REF] Soucasse | Numerical study of coupled molecular gas radiation and natural convection in a differentially heated cubical cavity[END_REF], [START_REF] Billaud | Numerical investigation of coupled natural convection and radiation in a differentially heated cubic cavity filled with humid air. Effects of the cavity size[END_REF], [START_REF] Hajji | Natural convection analysis through a radiatively participating media within a rectangular enclosure[END_REF]). The problem of a heat supply placed in a confined environment has also received attention, either with point source (Tetsu et al. (1973), Urakawa et al. (1983), Xin et al. (2004), Hernandez (2015), etc.) or with finite sized solid obstacles (Kuznetsov and Sheremet (2006a), Kuznetsov and Sheremet (2006b), Xu and Wang (2008), Paroncini and Corvaro (2009), Kuznetsov et al. (2013), Souayeh et al. (2015), Gibanov and Sheremet (2016), Iyi et al. (2013) Rahmati and Tahery (2018), etc.), but in most cases the heat source is set within the fluid, without any contact with the walls. Here, we address the case of a heating obstacle placed on the floor of the cavity. This constitutes a new worthy configuration of interest.

First of all, concerning the natural convection inside a cavity containing an opaque obstacle, Kuznetsov and Sheremet (2006b) have investigated the effects of the Grasshof number (10 5 -10 7 ) on the fluid motion and concluded the influence of this parameter on the thermal field in the enclosure. Kuznetsov and Sheremet (2006a) have studied the same configuration, but with a different range of the Grasshof number (10 7 -10 9 ) and pointed out that as this number increases, the flow and the heat transfer process are stabilized. Paroncini and Corvaro (2009), Kuznetsov et al. (2013) Souayeh et al. (2015), Gibanov and Sheremet (2016), Iyi et al. (2013) Rahmati and Tahery (2018) have studied different convective flow in both numerical and experimental manners with the influence of the Rayleigh number. They have reported that the increase of this parameter intensifies the fluid motion and raises the heat transfer through convective process. In addition, the study on the hot obstacle size of Paroncini and Corvaro (2009) pointed out that, when the height of the obstacle is a half of the cavity, the convective heat transfer is the worst among the analyzed cases. Besides, Bouafia and Daube (2007) have considered the unsteady behaviors in this configuration at different aspect ratio of the enclosure and concluded that the mechanism of the unsteadiness was the shear instabilities when the aspect ratio is 1 and 2. For higher aspect ratio (4), the leading mechanism was the buoyancy instabilities. Hernandez (2015) has studied a square cavity with a heat source attached to the floor. His results point out that the unsteady behavior was due to the high horizontal velocity right above the obstacle.

Next, the problem of natural convection combined with the surface radiation have been investigated by Sun et al. (2011), [START_REF] Martyushev | Surface radiation influence on the regimes of conjugate natural convection in an enclosure with local energy source[END_REF], [START_REF] Saravanan | Surface radiation effect on convection in a closed enclosure driven by a discrete heater[END_REF], [START_REF] Patil | Conjugate laminar natural convection and surface radiation in enclosures: Effects of protrusion shape and position[END_REF], [START_REF] Miroshnichenko | Turbulent natural convection combined with surface thermal radiation in a square cavity with local heater[END_REF]. They have concluded that the increase in the emissivity strengthens the fluid motion near the wall, intensifies the radiative transfer but reduces the convective transport. Sun et al. (2011) have also reported that surface radiation slows down the transition to the unsteadiness inside the cavity.

Through a detailed literature review, it is shown the lack of observations on the flow characteristics when the medium is filled by a radiatively participating gas mixture in the configuration of a cavity with a heat source inside. This work, therefore, addresses this problem by numerical simulations of the coupling natural convection and gas radiation in a cubical enclosure filled with an air -H 2 O mixture and a heated obstacle located on its floor.

Nomenclature

m 2 • mol -1 G incident radiation W • m -2 I radiative intensity W • m -2 • sr L cavity length m l obstacle size m N g number of gray gases P absolute pressure P a q inc incident radiative flux W • m -2 q net net radiative flux, q net = σT 4 w -q inc W • m -2 R ideal gas constant 8.3144621 J • K -1 • mol t time s T c absolute temperature at cold surfaces K T h absolute temperature at hot surfaces K T w absolute wall temperature K T ref reference temperature, T ref = (T h + T c )/2 K u, v, w velocity component m • s -1 U x normalized velocity component, U x = u U ref U ref reference velocity, U ref = α √ Ra L m • s -1 X, Y, Z normalized coordinates, X = x L , Y = y L , Z = z L x, y, z Cartesian coordinates m x H2O molar fraction of H 2 O in the mixture Dimensionless numbers N u c convective Nusselt number, N u c = L ∆T | ∂T ∂x | N u r radiative Nusselt number, N u r = L|q net | λ∆T N u t total Nusselt number, N u t = N u c + N u r P l Planck number, P l = λ L 4σT 3 ref P r Prandlt number, P r = ν α Ra Rayleigh number, Ra = gβ T ∆T L 3 αν Greek symbols α thermal diffusivity m 2 • s -1 β T thermal expansion coefficient K -1 emissivity η wavenumber cm -1 κ absorption coefficient m -1 λ thermal conductivity W • m -1 • K -1 ν kinematic viscosity m 2 • s -1 Ω solid angle sr Ω m discrete direction ρ density kg • m -3 σ Stefan Boltzmann constant 5.670367 × 10 -8 W • m -2 • K -4
Abbreviation ALBDF Absorption Line Blackbody Distribution Function 2 Mathematical model

Problem statement

The configuration under consideration in this research is illustrated in figure 1. It deals with a cubical cavity of size L = 0.25m having adiabatic horizontal walls, while the vertical ones are maintained at a constant and uniform temperature (T c ). A small solid cube of size l = 0.05m is located at the center of the bottom wall. It creates an opaque obstacle whose surfaces are uniformly set at a higher temperature (T h > T c ). The enclosure is filled with either dry air or a mixture of air and water vapor, depending on the case study. The medium is homogeneous in composition. The temperature difference between the obstacle and the cavity generates a natural convection flow and, when water vapor is present, H 2 O acts as an emitting-absorbing component for infrared radiation.

Conservation equations

Several conservation equations govern the flow motion and the transfer processes in the enclosure. They express a local balance in mass, momentum, energy and composition within the fluid:

• Continuity equation

If density variations are neglected, the total mass conservation reads:

∇ • u = 0 (1)
• Momentum equation

ρ 0 ∂u ∂t + ρ 0 u • ∇u = -∇p + ρ 0 β T (T 0 -T )g + µ∇ 2 u (2)
The source term ρ 0 β T (T 0 -T )g accounts for the buoyancy force that sets the fluid into motion (here expressed under the Boussinesq approximation).

• Energy equation

ρ 0 C p ∂T ∂t + ρ 0 C p u • ∇T = λ∇ 2 T -∇ • q (3)
The divergence term -∇ • q is the internal radiative source resulting from the difference between the absorbed and emitted radiant energy in each elementary volume of fluid. It takes non-zero values whenever H 2 O is present in the mixture.

Radiative transfer equation

The spectral radiative intensity I η (s, Ω) represents the radiant flux (per unit angle solid and per unit wavenumber) that propagates at point s = (x, y, z) in the direction Ω at the wavenumber η. The local change of intensity is described by the radiative transfer equation, which, for a non-scattering medium, is expressed as:

Ω • ∇I η (s, Ω) = -κ η (s)I η (s, Ω) + κ η (s)I bη (T (s)) (4)
where κ η (s) is the local spectral absorption coefficient defined as κ η (s) = N (s) Y (s) C η (φ(s)). In this expression, C η (φ(s)) is the spectral absorption cross section, depending on the local thermodynamic state, φ(T, P, C), Y (s) the mole fraction and N (s) the molar density of the absorbing species [START_REF] Denison | An absorption-line blackbody distribution function for efficient calculation of total gas radiative transfer[END_REF]). The radiative intensity depends on three position coordinates, two direction variables (either two polar angles or two direction cosines) and the wavenumber. The total intensity can be found by integration over the whole spectrum as:

I(s, Ω) = ∞ 0 I η (s, Ω)dη (5)
The source term -∇ • q, which appears in the energy equation, is the total divergence of the radiative flux. This flux can be calculated from the total intensity by the expression:

q(s) = 4π 0 I(s, Ω)ΩdΩ = 4π 0 ∞ 0 I η (s, Ω)ΩdηdΩ (6)
and, as a result [START_REF] Modest | Radiative heat transfer[END_REF]):

-∇ • q(s) = ∞ 0 4π 0 κ η (s)I η (s, Ω)dΩdη -4π ∞ 0 κ η (s)I bη (T (s))dη (7)

Spectral model

When gas radiation is considered, the actual spectral behavior of the fluid (absorption) must be accounted for. In this work, the Spectral Line-based Weighted-sum-of-gray-gases models (SLW) [START_REF] Denison | An absorption-line blackbody distribution function for efficient calculation of total gas radiative transfer[END_REF]) has been selected. This model, according to [START_REF] V Goutiere | An assessment of real-gas modelling in 2d enclosures[END_REF] and [START_REF] V Goutière | Comparative performance of nongray gas modeling techniques[END_REF], provides the best compromise between accuracy and computational cost.

The SLW model involves a finite number (N g ) of gray gases and one clear (transparent) gas. The absorption coefficient of j th gas is calculated as:

κ j = N • Y • C j (8)
where C j is the absorption cross section. Knowing N and Y, the remaining problem is to determine C j . This parameter is here determined as C j = Cj-1 Cj where

Cj = C min (C max /C min ) j/Ng is the supplemental cross section selected in the range [C min ,C max ].
The weight of the j th gray gas a j corresponds to the difference in the absorption line black body distribution function ALBDF at the two supplemental absorption cross sections that define the j th interval [ Cj-1 , Cj ] ( [START_REF] Denison | k-distributions and weighted-sum-of-gray-gases-a hybrid model[END_REF]):

a j = F ( Cj , φ g , T b ) -F ( Cj-1 , φ g , T b ) : for j = 1,...,N g a 0 = F ( C0 , φ g , T b ) (9)
where T b is equal to the local temperature in the fluid and to the wall temperature at the boundaries.

The ALBDF is evaluated as the integral of the Planck function calculated at a source temperature T b over the wavenumber intervals such that the absorption cross section C η (φ g ) at a gas thermodynamic state φ g is below a prescribed C-value, namely:

F (C, φ g , T b ) = 1 E b (T b ) η:Cη(φg)<C E bη (T b )dη = π σT 4 b η:Cη(φg)<C I bη (T b )dη (10)
The ALBDF is determined by performing integrations over the whole spectrum at high resolution (line by line) and different pressures, temperatures and compositions. The resulting data are made available for the main participating species (H 2 O, CO 2 , CO) either as mathematical correlations or in look-up tables. The most recent contribution is reported in [START_REF] J T Pearson | Efficient representation of the absorption line blackbody distribution function for H2O, CO2, and CO at variable temperature, mole fraction, and total pressure[END_REF] and is based on the HITEMP-2000 spectral database. In this research, the look-up tables provided by [START_REF] Solovjov | Tabulated data for the absorption line black body distribution function[END_REF] have been used along with the rank-correlated approach [START_REF] V P Solovjov | The rank correlated slw model of gas radiation in non-uniform media[END_REF]) to handle the fluid temperature variations.

Finally, the radiative transfer equation for the j th gray gas is written as :

Ω • ∇I j (s) = -κ j I j (s) + a j κ j I b (s) (11)
The total intensity and radiative source term are then obtained by the following expressions:

I(s, (Ω)) = Ng j=1 I j (s, (Ω)) (12) ∇ • q(s) = Ng j=1 (κ j 4π 0 G j (s) -4πa j κ j I b (s)) (13)
where G j (s) = 4π 0 I j (s, Ω)dΩ is the incident radiation related to the j th gray gas.

Boundary conditions

Kinematic, mass and thermal fields

All the surfaces of the heated obstacle, except the one in contact with the floor of the cavity, are black and set at constant and uniform temperature:

T = T h , = 1 (14)
Regarding the cavity:

• Vertical walls are black and uniformly maintained at constant temperature:

T = T c , = 1 (15)
• Horizontal walls are adiabatic and assumed to behave as fully reflective surfaces:

∂T ∂z = 0 with z = 0, L. = 0 (16)
Zero velocities are applied to all the walls of the cavity and all the surfaces of the obstacle.

Radiative intensity field

Intensity at the bounding walls is prescribed for all directions Ω pointing inward to the cavity that is for Ω • n > 0 where n is the local unit vector. Assuming gray diffuse surfaces, the boundary condition reads:

I η (s, Ω (Ω•n>0) ) = I bη (T (s)) + (1 -) q inc η π (17)
where:

q inc η = Ω•n<0 I η (s, Ω)|Ω • n|dΩ (18)
In this study, the surfaces of the obstacle and the vertical walls of the cavity are black ( = 1) and the horizontal walls of the enclosure are purely reflective ( = 0).

Methodology

All the governing equations of our problem were solved by using Code Saturne-version 5.0.4 [START_REF] Archambeau | Code Saturne: A finite volume code for the computation of turbulent incompressible flows, industrial applications[END_REF]), an open source software for CFD calculation developed by EDF. A built-in radiative module is available, in which we have implemented our own data for directional integration and a specific module for gas radiation according to the SLW model.

Computational Fluid Dynamics

Code Saturne uses a finite volume method to solve the governing equations of fluid motion and heat and mass transfer. For the momentum equations, Code Saturne resorts to the SIMPLEC algorithm (Semi-Implicit Methods for Pressure Linked Equations Consistent). Different discretizations in space and in time are also available.

Temporal discretization

The time scheme is implemented in Code Saturne is a θ -scheme with: θ = 1 for an implicit first order Euler scheme, θ = 1 2 for second order Crank-Nicolson scheme.

(

) 19
There are two options for setting the temporal step: constant and variable. In the latter case, the code automatically calculates the time step after each iteration that satisfies the CFL criterion. In our study, we have used the implicit first order backward Euler scheme along with the variable time step option.

Spatial discretization

Code Saturne proposes different schemes of first order (Upwind) and second order (Centered or Second-Order-Linear-Upwind (SOLU)) for spatial discretization and. In this study, we have selected the centered second order scheme.

The Discrete Ordinates Method

The discrete ordinates method (DOM) has been used to solve the radiative transfer equation corresponding to each gray gas of SLW model. This method consists in replacing the angular integrals by a summation over a set of discrete directions such as:

4π 0 f (Ω)dΩ ≈ M m=1 ω m f (Ω m ) ( 20
)
where M denotes the number of directions in the set and ω m is the weight attributed to the m th element. Consequently, the distribution of incident radiation and radiative flux are approximated by:

G P η = 4π 0 I P η (Ω)dΩ ≈ M m=1 ω m I P η,m (Ω) (21)
q P m = 4π 0

I P η (Ω)ΩdΩ ≈ M m=1 ω m I P η,m (Ω)Ω m (22)
where P here denotes the center of a control volume.

There are several ways to define the discrete direction sets. A more general overview on this problem can be found in [START_REF] Koch | Evaluation of quadrature schemes for the discrete ordinates method[END_REF]. We have here selected the level symmetric S N quadrature using the improved dataset suggested by [START_REF] Balsara | Fast and accurate discrete ordinates methods for multidimensional radiative transfer. part i, basic methods[END_REF].

Code Validation

Differentially heated cavity

In this section, Code Saturne and its improved radiative model is validated through the simulations of both pure thermal convection and double diffusive convection coupled with radiation in a differentially heated cavity (see figure 2).

Pure Thermal Convection In that case, there is no radiative effects (transparent medium, non-emitting walls). The Prandtl number is 0.71 and the computations were run over a uniform grid of 80 2 cells. In this configuration, many references are available [START_REF] Le | Accurate solutions to the square thermally driven cavity at high Rayleigh number[END_REF], [START_REF] Tric | A first incursion into the 3d structure of natural convection of air in a differentially heated cubic cavity, from accurate numerical solutions[END_REF]...).

Here, the solutions are compared against the data provided by [START_REF] De | Natural convection of air in a square cavity: a bench mark numerical solution[END_REF] and are presented below. A good agreement is found between our results and this reference (see Table 1). Regarding 3-D cases, we have performed calculations in the configurations studied by Colomer et al. (2004b) and [START_REF] Fusegi | Laminar and transitional natural convection in an enclosure with complex and realistic conditions[END_REF]. A uniform 81 3 grid was used still with P r = 0.71. Here too, a good agreement is observed with the reference data (see 2) for coupled thermal convection and radiation in a gray gas inside a 2D differentially heated square cavity. In [START_REF] Yücel | Natural convection and radiation in a square enclosure[END_REF], the authors have used a non-uniform 50 2 grid (but they did not specify of which type) and performed the calculations at different values of the overall optical thickness τ = κ • L (L is the cavity size and κ the gray absorption coefficient). The Rayleigh number was fixed at 5 • 10 6 with P r = 0.72 and the Planck number defined as P l = λ 4σLT 3 ref was set to P l = 0.02. All the bounding walls are black and the temperature ratio (θ 0 = T ref /∆T ) is set to 1.5. A nonuniform 81 2 grid is used, like in reference Laouar- [START_REF] Laouar-Meftah | Modélisation de la convection naturelle de double diffusion dans un mélange de gaz absorbant et émettant le rayonnement[END_REF]. The results are listed in Table 3. They show that the difference between our calculations and the reference does not exceed 4 %. This difference may come from different interpolation schemes used in the DOM (the Lathrop variable scheme in Laouar- [START_REF] Laouar-Meftah | Modélisation de la convection naturelle de double diffusion dans un mélange de gaz absorbant et émettant le rayonnement[END_REF] Table 3 Mean Nusselt number on the hot wall (P r = 0.71, P l = 0.02, θ0 = 1.5, i = 1)

In addition, we have performed different validation tests in a 3-D differentially heated cavity filled with a real participating gas (that is, accounting for the actual absorption spectrum of the medium). We have reproduced the works of [START_REF] Billaud | Numerical investigation of coupled natural convection and radiation in a differentially heated cubic cavity filled with humid air. Effects of the cavity size[END_REF] by considering a cubic enclosure filled with humid air (air -H 2 O mixture). Different case studies based on the radiative behavior of the bounding walls and the medium have been investigated. They are described in Table 4. Our calculations have been carried out using a non-uniform 91 3 grid like in reference [START_REF] Billaud | Numerical investigation of coupled natural convection and radiation in a differentially heated cubic cavity filled with humid air. Effects of the cavity size[END_REF]. Our results are also compared to the data provided by [START_REF] Soucasse | Natural convection in a differentially heated cubical cavity under the effects of wall and molecular gas radiation at Rayleigh numbers up to 3× 109[END_REF] and presented in Table 5. The comparisons show a fairly good agreement between our predictions and the two references.

Case

A B C D Isothermal walls = 1 = 1 = 1 = 1 Adiabatic walls = 0 = 0 = 1 = 1 Gas nature
Transparent Participating Transparent Participating

Cavity with a hot obstacle

Our purpose is now to test the ability of our simulation code to handle geometries involving obstacles. The first validation test considers a 2-D square cavity with a heat source located at the center of the bottom wall (see figure 3). The enclosure is 0. The comparisons of our results with the reference show a good agreement, the maximum difference being less than 1 % when comparing the two numerical simulations. Differences are larger with respect to measured quantities (but this is also true in the reference work). Some thermal leakage through the plexiglas plates (imperfect insulation) may explain these variations.

Figure 4 Second validation test

The second validation considers a hot obstacle located at the center of a 2D square cavity (see figure 4). Now, two vertical walls of the cavity are kept adiabatic while the horizontal one are maintained at a constant lower temperature compared to the isothermal heated obstacle. The enclosure is filled with dry air. The simulation is run at a Rayleigh number of Ra = 2 • 10 5 and a Prandlt number of Pr = 0.71. The comparisons with the results of Sun et al. ( 2011) reported in table 7 prove that our code can efficiently handle this type of configuration, both for the flow description and thermal transport (the maximum difference is less than 0.5 %).

Our work

Results and Discussion

The enclosure under study is filled with a real air -H 2 O mixture. It involves an absorbingemitting component (H 2 O) diluted at different concentrations into a transparent gas (dry air). The temperature of the hot obstacle is set to T h = 580K and the vertical walls of the cavity are maintained at T c = 530K. The thermophysical properties of the mixture are calculated at the reference temperature according to the perfect gas law for density and the Wilke-Wassiljewa formulas for dynamic viscosity and thermal conductivity, as suggested in [START_REF] Meftah | Coupled radiation and double diffusive convection in nongray air-CO2 and air-H2O mixtures in cooperating situations[END_REF]. The actual absorption spectrum is accounted for by using the SLW model.

The characteristic parameters of the simulations are presented in Table 8.

x Table 8 The three configurations under study

In all these configurations, a steady state flow is obtained. In each case, the results are compared to the reference (transparent) values, which are generated using the same gas mixture but without including any radiation effects in volume.

We present below the cross section planes and the crosslines used to display the results. The dimensionless temperature and velocity fields are plotted in the median vertical plane of the cavity (Y = 0.5 or y = 0.125m) (see figure 5). We also consider the profiles of these quantities along different crosslines in the plane Y = 0.5: From the result displayed above, we consider that the simulations are converged with respect to the spatial meshing when using a 100 3 grid. A same observation was made in configurations involving gas radiation. Therefore, we have selected this mesh for our subsequent calculations.

All the subsequent calculations have been carried out using a uniform 100 3 grid with the S 8 discrete ordinates method and a 5-gray gas SLW model. As a whole, radiation does not significantly change the thermal profile along the centerline (Z = 0.5) (see figures 10 d,e,f) and in the lower half of the cavity (Z = 0.1) (see figures 10 g,h,i). But at a higher position (Z = 0.8) (see figures 10 a,b,c), a slight decrease of temperature in the re-circulation zone between the plume and the wall boundary layer can now be observed. In this region of low convective transport, the gas radiates toward the cold surfaces and the colder parts of the fluid (it emits more than it absorbs): this results in a negative radiative source within the fluid (see figures 12 a, 13 a and 14 a). Similar modifications in dynamic and thermal field were reported by [START_REF] Billaud | Numerical investigation of coupled natural convection and radiation in a differentially heated cubic cavity filled with humid air. Effects of the cavity size[END_REF] for a differentially heated cavity: gas radiation was found to accelerate the global circulation and to set into motion some parts of the fluid that were stagnant in the transparent case. The plots in the figure 23 clearly show that the convective Nusselt number along the centerline of any vertical wall of the enclosure is decreased close to the roof of the cavity and increased elsewhere. The reason is that, in the upper part, the fluid was cooled down (figures 12 a, 13 a and 14 a) before reaching the lateral walls. On the other hand, in the lower part, absorption dominates over the emission (figures 12 b, 13 b and 14 b), thus, the medium is heated up and the thermal gradient is increased. However, the increased part does not compensate the decreased one. Therefore, overall, the average convective Nusselt number over the vertical wall of the enclosure is reduced (see table 12). This decreasing trend in the local convective Nusselt number is also found when considering the vertical hot surfaces of the obstacle. This is illustrated in figure 24. The reason is that the thermal gradient is reduced when the radiation is considered. Indeed, the fluid is warmed up (by the absorption) along the floor of the enclosure before arriving at the vertical hot surfaces of the obstacle. Consequently, the average values of N u C displayed in table 14 are decreased when the radiation is taken into account. However, it is observed that the convective transfer along the horizontal upper surface of the obstacle increases. When radiation is present, the fluid layer near the hot surfaces becomes cooler due to the emission (it is evidenced by the negative values of the radiative source in the figures 12, 13 and 14). Consequently, the thermal gradient in this region is enhanced, which induces the increase in local values of Nusselt number with the appearance of the radiation (figure 25). This explains the higher average convective Nusselt number obtained in table 14. The values of average convective Nusselt number and average total Nusselt number, which are defined as N u = 1 A A N u(s)dA, are presented in the tables below:

Thermal and dynamic fields

Heat Transfer

x Regarding the total heat transfer, the table 12, 13 and 14 reveal that, compared to the transparent cases, the average total Nusselt number along any black surfaces is decreased. This, along the vertical walls, is due to the drop off in the convective Nusselt number. In addition, the attenuation of radiative transfer by the absorbing medium also contributes to the reduction of total thermal transport, especially, when considering this quantity along the upper surfaces of the obstacle, where the convective transfer is accelerated.

Conclusion

The coupling of natural convection with volume radiation in a cubical cavity with a heat source located on its floor has been investigated for an air -H 2 0 mixture at different concentration. The radiant sources created by the gas absorption and emission in the infrared range have been computed using the discrete ordinates method and SLW model associated with the rank-correlated approach. The other conservation equations have been solved using a finite volume method implemented in the Code Saturne software. The obtained results

Figure 1

 1 Figure 1 Configuration of the calculation

Figure 2

 2 Figure 2 Differentially heated cavity

  05m long. The heat source is 0.01 m wide and 0.025 m high. The enclosure is filled by dry air (transparent) and the Prandtl number is set to 0.71. The heat source is maintained at T h = 301.16 K while two lateral walls of the cavity are prescribed at T c = 291.16 K. The remaining surfaces of the enclosure are adiabatic. The tests are performed at different Rayleigh numbers, using a 100 2 uniform grid and the results are compared to the experiments and the simulations of Paroncini and Corvaro (2009).

Figure 3

 3 Figure 3 First validation test

Figure 5 Figure 6

 56 Figure 5 Median plane (Y = 0.5) and crosslines used for the results display

Figure 7

 7 Figure 7 2D-contours of temperature T -T ref T h -Tc in the median plane of the cavity (Y = 0.5) at xH 2 O = 0.05

  Figure 10 Temperature T -T ref T h -Tc profiles along different Z-cross lines in the median plane (Y = 0.5) at different concentrations of water vapor

Figure 12

 12 Figure 12 Distribution of radiative source term in the median plane (Y = 0.5) at xH 2 O = 0.05. Sources are normalized by 4σT 4 ref /L.

Figure 13

 13 Figure 13 2D-contour of radiative source term in the median plane (Y = 0.5) at xH 2 O = 0.10. Sources are normalized by 4σT 4 ref /L.

Figure 14

 14 Figure 14 2D-contour of radiative source term in the median plane (Y = 0.5) at xH 2 O = 0.20. Sources are normalized by 4σT 4 ref /L.

Figure 15

 15 Figure 15 Profiles of vertical velocities w U ref at different Z-cross sections in the median plane (Y = 0.5) at different concentrations of water vapor

Figure 16

 16 Figure 16 Profiles of horizontal velocities u U ref at different X-cross sections in the median plane (Y = 0.5) at different concentrations of water vapor

Figure 17 Figure 18 Figure 19 Figure 20 Figure 21 Figure 22

 171819202122 Figure 17 Vector field in the median plane (Y = 0.5) at xH 2 O = 0.05

Figure 23

 23 Figure 23 Local convective Nusselt number along the vertical centerline of any lateral wall of the cavity

Figure 24 Figure 25

 2425 Figure 24 Local convective Nusselt number along the vertical center line of each lateral surfaces of the obstacle

F

  Tetsu, M Itsuki, and U Haruo. Buoyant plume above a horizontal line heat source. International Journal of Heat and Mass Transfer, 16(4):755-768, 1973. K Urakawa, I Morioka, and M Kiyota. Swaying motion of the buoyant plume above a horizontal line heat source. In Proc. 1st ASME-JSME Thermal Eng. Conf, pages 215-220, 1983. S Xin, M C Duluc, F Lusseyran, and P Le Quéré. Numerical simulations of natural convection around a line-source. International Journal of Numerical Methods for Heat & Fluid Flow, 14(7):830-850, 2004. R H Hernandez. Natural convection in thermal plumes emerging from a single heat source. International Journal of Thermal Sciences, 98:81-89, 2015. G V Kuznetsov and M A Sheremet. Two-dimensional problem of natural convection in a rectangular domain with local heating and heat-conducting boundaries of finite thickness. Fluid Dynamics, 41(6):881-890, 2006a. G V Kuznetsov and M A Sheremet. Conjugate heat transfer in a closed domain with a locally lumped heat-release source. Journal of Engineering Physics and Thermophysics, 79(1):57-64, 2006b. M Xu and Z Wang. The natural convection of an isolated plate in enclosures. Progress in Computational Fluid Dynamics, An International Journal, 8(7-8):549-553, 2008. M Paroncini and F Corvaro. Natural convection in a square enclosure with a hot source. International journal of thermal sciences, 48(9):1683-1695, 2009. G V Kuznetsov, V I Maksimov, and M A Sheremet. Natural convection in a closed parallelepiped with a local energy source. Journal of Applied Mechanics and Technical Physics, 54(4):588-595, 2013. B Souayeh, N B Cheikh, B B Beya, and T Lili. Prediction of unsteady natural convection within a square cavity containing an obstacle at high Rayleigh number value. International Letters of Chemistry, Physics and Astronomy, 55:19, 2015. N S Gibanov and M A Sheremet. Effect of the buoyancy force on natural convection in a cubical cavity with a heat source of triangular cross-section. In IOP Conference Series: Materials Science and Engineering, volume 124, page 012057. IOP Publishing, 2016. D Iyi, R Hasan, and R Penlington. Numerical simulation of 2d turbulent natural convection of humid air in a cavity filled with solid objects. Procedia engineering, 56:538-543, 2013. A R Rahmati and A A Tahery. Numerical study of nanofluid natural convection in a square cavity with a hot obstacle using lattice boltzmann method. Alexandria engineering journal, 57(3):1271-1286, 2018. M Bouafia and O Daube. Natural convection for large temperature gradients around a square solid body within a rectangular cavity. International Journal of Heat and Mass Transfer, 50(17-18):3599-3615, 2007. H Sun, E Chénier, and G Lauriat. Effect of surface radiation on the breakdown of steady natural convection flows in a square, air-filled cavity containing a centered inner body. Applied Thermal Engineering, 31(6-7):1252-1262, 2011.

Table 1

 1 Mean Nusselt number on the hot wall

Table 2

 2 ).

	Ra Our work Colomer et al. (2004b) Fusegi and Hyun (1994)
	10 4	2.059	2.030 (1.42%)	2.100 (1.95 %)
	10 5	4.365	4.334 (0.71 %)	4.361 (0.09%)
	10 6	8.717	8.862 (1.63%)	8.770 (0.60 %)

Table 2

 2 Mean Nusselt Number on the hot wall (Pr = 0.71)

  , the STEP scheme in the present work).

	τ	S N	Convective Nusselt Our work Ref.1 Ref.2 Our work Ref.1 Ref.2 Total Nusselt
	0.2 S 4	36.01	37.40 37.40	46.50	46.11 46.05
	1	S 8	31.88	31.25 31.25	39.38	38.93 38.81
	5	S 4	24.58	23.64 23.57	31.47	31.76 31.59

Table 4

 4 Radiative boundary conditions and radiative properties of the medium

	Case Our work Billaud et al. (2017) Soucasse et al. (2016)
	A	8.64	8.65 (0.11 %)	8.64 (0.0 %)
	B	7.24	7.42 (2.42 %)	7.55 (4.10 %)
	C	7.93	8.10 (2.09 %)	8.47 (6.37 %)
	D	8.48	8.01 (5.86 %)	8.48 (0.0 %)

Table 5

 5 Mean convective Nusselt number on the hot wall (air -H2O mixture, Ra = 10 6 , P r = 0.707)

Table 6

 6 First validation test: Mean Nusselt number on the lateral wall of the heated obstacle. The reference length is the cavity size (H).

Table 7

 7 Second validation test: Mean Nusselt number on different walls (Ra = 2 • 10 5 , Pr = 0.71)

			Sun et al. (2011) Relative difference (%)
	Side A'B' or C'D'	3.7129	3.7174	0.121
	Side A'D' (bottom)	5.6527	5.6347	0.319
	Side B'C' (top)	9.4989	9.4614	0.396
	Bottom wall	2.5342	2.5346	0.015
	Top Wall	6.4966	6.4778	0.290

Table 9

 9 Maximum horizontal and vertical velocities for different mesh sizes.

	Mesh Front wall Back wall Left wall Right wall
	80 3	1.690	1.690	1.690	1.690
	100 3	1.682	1.682	1.682	1.682
	120 3	1.681	1.681	1.681	1.681

Table 10

 10 Mean Nusselt numbers at the bounding surfaces of the enclosure.

	Mesh Front wall Back wall Left wall Right wall Top wall
	80 3	38.664	38.664	38.664	38.664	14.343
	100 3	38.508	38.508	38.508	38.508	14.166
	120 3	38.492	38.492	38.492	38.492	14.132

Table 11

 11 Mean Nusselt numbers at the bounding surfaces of the hot source.

Table 12

 12 Average convective and total Nusselt numbers along any vertical wall of the cavity

	x H2O (%) Convective Nusselt number	Total Nusselt number
		Transparent Participating Transparent Participating
	5	38.28	37.09	252.07	250.25
	10	38.34	36.75	252.19	249.31
	20	38.57	36.59	252.59	247.93

Table 13

 13 Average convective and total Nusselt numbers along any vertical wall of the obstacle

	x H2O (%) Convective Nusselt number	Total Nusselt number
		Transparent Participating Transparent Participating
	5	14.12	14.50	239.91	238.26
	10	14.17	14.90	240.03	236.92
	20	14.30	15.85	240.34	234.74

Table 14

 14 Average convective and total Nusselt numbers along the horizontal upper wall of the obstacle

(a) Transparent medium (b) Participating medium

have been compared to the case where the medium is considered as transparent. These comparisons show that:

• Radiation tends to non-uniformly accelerate the boundary layers along the cavity and the hot inner obstacle. It also sets into motion some parts of the fluid that were stagnant in the transparent case. This results in making the plume flow and its recirculation along the walls interfere and create a shear flow.

• Radiation partly modifies the thermal gradient near the bounding surfaces of the cavity: the convective Nusselt values are increased in the upper half and decreased in the lower half. However, an inverse trend is observed at the horizontal surface of the obstacle.

• Radiation modifies the nearly vertical thermal stratification outside the plume and slightly uniformizes the medium temperature.

• The presence of radiation reduces the total thermal transfer, especially the convective part on the vertical walls and the radiative part along the upper surface of the obstacle.

• All these effects were found to increase when the medium becomes more absorbing (in the considered range of H 2 O molar fractions).