A Weppe

F Moreau

D Saury

Experimental investigation of a turbulent natural convection flow in a cubic cavity with an inner obstacle partially heated

This work presents the dynamical and thermal behaviors of a turbulent natural convection flow occurring in a cubic cavity with an inner and partially heated obstacle. The top, bottom, front and back walls of the cavity are adiabatic and the other side walls are at imposed uniform temperature. Velocity and temperature measurements are carried out using Particle Image Velocimetry and micro-thermocouples. Boundary layer flows are observed in both vertical channels. A recirculating zone, in the upper part of the heated channel, breaks the linearity of the thermal stratification found in the lower part of the channel and it also disturbs heat transfers. A boundary layer instability of Tollmien-Schlichting type and the subsequent formation of an oscillating buoyant jet are analyzed. The presence of internal gravity waves in the thermally stratified heated channel are highlighted.

reference temperature, T 0 = 1 2 (T h + T c ), K U, V, W dimensionless velocities (scaled by α √ Ra H /H) x, y, z physical cartesian coordinates, m X, Y, Z dimensionless coordinates, (X, Y, Z) = (x, y, z)/L Greek symbols α thermal diffusivity, m 2 .s -1 β thermal expansion coefficient, K -1 ∆T temperature difference between the isothermal walls, ∆T = T h -T c , K ε emissivity θ dimensionless temperature, θ = (T -T 0 )/∆T λ thermal conductivity, W.m -1 .K -1 µ dynamic viscosity, kg.m -1 .s -1 ν cinematic viscosity, m 2 .s -1 ρ density, kg.m -3 σ X standard deviation, (units of X) Φ X power spectral density, (units of X) 2 .Hz -1 Ω dimensionless pulsation

Introduction

Natural convection in closed and confined spaces takes place in a wide variety of industrial configurations : ranging from a car's underhood to the cooling system of nuclear power plants. Within this range of specific designs, the high Rayleigh numbers often encountered lead to turbulent regimes. Numerical simulations, usually carried out in industrial context with RANS models to achieve adapted computation time, crudely lack of accuracy in the determination of the flow and the heat transfers. The case of enclosed cavities with obstacle is often encountered in actual applications. However, turbulent phenomena, occurring in such a situation, remain challenging. In order to tackle these issues, a simplified configuration is defined which allows to analyze natural convection flows being representative of flow regimes encountered in industrial contexts and especially in the underhood of cars. In this study, a cubic and partially heated obstacle is placed into a cubic enclosure. As it will be later depicted in the results section of the paper, the flow inside the heated channel has some similarities with both flows along vertical heated plates and differentially heated cavity flows. Natural convection along a vertical heated plate has been first investigated theoretically and experimentally by Schmidt et al. [1] in laminar regime. Ostrach [2] specified the theory and compared it to experiments for various Prandtl numbers. A boundary layer type flow is found for Grashof numbers large enough and correlations for heat transfers are determined. When the Grashof number is increased up to a critical value along the heated vertical plate, the flow can become unsteady. Dring and Gebhart [START_REF] Dring | A theoretical investigation of disturbance amplification in external laminar natural convection[END_REF] highlighted a filtering frequency mechanism discussed later in this paper. The unsteady flow will eventually lead to turbulence (Jaluria et al. [START_REF] Jaluria | On transition mechanisms in vertical natural convection flow[END_REF]).

Turbulent flows inside differentially heated cavities (DHC) have also been studied both numerically and experimentally. For instance, Mergui et al. [START_REF] Mergui | Analyse des vitesse et température de l'air en convection naturelle dans une cavité carrée différentiellement chauffée à ra = 1.69 * 10 9[END_REF] or Salat et al. [START_REF] Salat | Experimental and numerical investigation of turbulent natural convection in a large air-filled cavity[END_REF] investigated a cubic differentially heated cavity of Rayleigh number up to Ra = 5.34 × 10 9 . A separated boundary layers flow is observed. More recently, Belleoud et al. [START_REF] Belleoud | Coupled velocity and temperature measurements in an air-filled differentially heated cavity at ra = 1.2 * 10 11[END_REF][START_REF] Belleoud | Etude de la convection naturelle turbulente en cavité verticale différentiellement chauffée : Analyse des structures et des transferts turbulents[END_REF] investigated a differentially heated cavity with aspect ratio A ≈ 4 (cavity's height over cavity's width) for Rayleigh number based on the cavity height Ra = 1.2 × 10 11 . Turbulent quantities have been analyzed and compared with numerical results (Trias et al. [START_REF] Trias | Direct numerical simulations of two-and three-dimensional turbulent natural convection flows in a differentially heated cavity of aspect ratio 4[END_REF]) with good agreements. The influence of an obstacle inside an enclosure was mostly studied numerically. House et al. [START_REF] House | Effect of a centered conducting body on natural convection heat transfer in an enclosure[END_REF] studied the effect of an obstacle conductivity and length in a DHC. They found that, for obstacle sizes large enough but not too large, heat transfers are decreased (respectively enhanced) in the enclosure when the ratio of the obstacle conductivity over the fluid conductivity is large (respectively small). Bhave et al. [START_REF] Bhave | Natural convection heat transfer enhancement using adiabatic block : Optimal block size and prandtl number effect[END_REF] studied the impact of a centered adiabatic obstacle inside a DHC and looked for an obstacle size where heat transfers are optimized. For P r = 0.71 heat transfer can be increased by 4% when the ratio of the obstacle area over the cavity area is near the size of the existing stagnant core of the cavity without obstacle. Other cavity shapes as, for instance, E-enclosure [START_REF] Raizah | Isph simulations of natural convection flow in e-enclosure filled with a nanofluid including homogeneous/heterogeneous porous media and solid particles[END_REF] or H-shaped cavities [START_REF] Aly | Mixing between solid and fluid particles during natural convection flow of a nanofluid-filled h-shaped cavity with three center gates using isph method[END_REF] have been investigated using ISPH numerical methods. Liu and Phan-Thien [START_REF] Liu | A complete conjugate conduction convection and radiation problem for a heated block in a vertical differentially heated square enclosure[END_REF] investigated more specifically the radiation effect of a heated obstacle in a DHC. The wall emissivity ϵ can play an important role on heat transfers ; the part of thermal radiation over the total heat transfer can reach 30% when ϵ = 0.9 and the obstacle is sufficiently heated. Ha and Jung [START_REF] Ha | A numerical study on three-dimensional conjugate heat transfer of natural convection and conduction in a differentially heated cubic enclosure with a heat-generating cubic conducting body[END_REF] pinpointed 3D effects when studying the conjugate heat transfer of a heated and conducting obstacle inside a DHC. Variation of the local Nusselt Number along the transverse direction are highlighted and induced by the conducting obstacle. Those numerical studies are still limited to laminar flow. The lack of experimental studies analyzing turbulent heat and fluid flows within an enclosure including an obstacle is a clear motivation for the present paper.

Experimental set-up

Cavity description

The setup consists in a partially side-heated cubic obstacle (referred later as "obstacle") of length H obs = 0.80 m centered inside a cubic cavity (referred ) and are then considered as adiabatic. They are also covered with low-emissivity aluminum sheets (ϵ = 0.08) to minimize the radiation effects. The front and rear walls of the cavity are 6 mm-thick glass panels which enable flow visualization. Extruded polystyrene blocks are maintained in front of the glass panels to keep them thermally insulated. A window is preserved for flow visualization and measurements. The inner partially heated obstacle is held by four polymer rods which re-duce conduction heat transfer (λ = 0.25 W.m -1 .K -1 ). They are placed at each obstacle ends to limit flow disturbances in the mid-depth plane where measurements are carried out. The obstacle is made up of two 4 cm-thick aluminium plates (λ = 174 W.m -1 .K -1 ). One of the plate is heated by a heating wire at T h = T c + ∆T . Those two plates are the obstacle side walls which are in front of the cavity side walls maintained at T c . Between those two plates are a Rockwool panels succession along with Ertalon and POM-C plates. The temperature is thus decreasing from the heated plate up to the other obstacle side wall. That other wall preserves the whole mechanical strength. Front, rear, top and bottom walls are covered with 4 mm POM-C plates keeping good surface condition. Those plates are also covered with low-emissivity aluminum sheets (ϵ = 0.08). The temperature difference between the heated plate of the inner obstacle and the cavity side wall is maintained at ∆T = T c -T f = 32 K±0.3 K. This is the temperature difference between those two vertical isothermal plates that creates air motion inside the cavity. A Rayleigh number, based on the height H obs of the inner obstacle used in this study is Ra H obs = gβ∆T H 3 obs αν = 1.37(±0.1) × 10 9 , with α the thermal diffusivity, ν the kinematic viscosity and β the compressibility coefficient. x,y and z axis are made dimensionless with respect to L.

Note that the ambient temperature in the experimental hall, isothermal plate temperatures of the cavity (imposed by the two thermostated baths) as well as the temperature of the 6 surfaces of the inner obstacle are continuously monitored to ensure suitable and well-known boundary conditions. 1 Pt100 probe and 69 type-K thermocouples are calibrated and used for this purpose. Data are recorded via a data acquisition unit agilent 34972A at a frequency of 1 60 Hz.

Velocities measurements

Velocities were measured using PIV (Particle Image Velocimetry) measurement technique. A smoke generator produces particles used as the flow tracers. Those tracers are paraffine oil particles with a diameter d ≈ 5 µm leading to a Stokes number Stk = ρpd 2 v ref 18µl

= 6.2 × 10 -4 where ρ p is the density, µ is the dynamic viscosity and

V ref = α H obs
Ra H obs is a typical scale velocity for natural convection [START_REF] Gebhart | Buoyancy-Induced Flows And Transport[END_REF]. The particles can be considered as good air tracers for natural convection flow since Vg Vmax ≈ 1.5 × 10 -3 where

V g = d 2 g(ρp-ρ air ) 18µ
is the settling velocity and V max is the maximum velocity encountered in the flow. The laser sheet is generated by a pulsed Nd:YAG laser (2 × 50 mJ@100Hz). The particles are visualized with a CMOS camera, 2560 × 2160 pixels 2 (Fig. 2). The pixel pitch size is 7 µm and the field size is 120 × 105 mm 2 . 5000 pairs of images are acquired at 25 Hz and the time between two frames is ranging between 1000 µs and 8000 µs depending on the zone analyzed and the associated velocity of the flow. Image processing is carried out with LaVision PIV algorithm to obtain velocity vectors. The algorithm proceeds with interrogation area size ranging from 64 × 64 to 32 × 32 pixels 2 with 50% of overlapping. For each zone, statistics over the vertical and horizontal velocities are analyzed. The components (U, V, W ) of the velocity are made dimensionless with respect to V ref previously defined. See table 1 for the velocity uncertainty.

Temperature measurements

Temperature measurements were carried out in the vertical mid-depth plane at X = 0.50 with a type K micro-thermocouple (diameter = 12.7 µm). The cold junction is placed inside an aluminium block of large inertia. This block is placed in the experimental hall at T = T amb and its temperature is regularly measured with a PT100 probe (±0.1 K). This type-K thermocouple is mounted on two pins of 100 µm joined on a rod, out of the vertical middepth plane (X = 0.50) in order to minimize the intrusive effects (Fig. 3). For the same reasons, this rod is placed downstream of the flow and 45 • oriented regarding the flow direction. A displacement device allows the rod to move along the Y and Z axes with a 100 µm increment (±6.5 µm). For each measurement, the acquisition frequency is set at 6 Hz and the acquisition time is 10 minutes to ensure statistical convergence. Moreover, the microthermocouple cut-off frequency is 20 Hz which is one order of magnitude greater than the maximum flow frequency. The temperature measurements are presented in terms of θ = T -T 0 ∆T where T 0 = T h +Tc 2 is the reference temperature.

Uncertainties for both the temperature and velocity are reported in the following table :

u(θ) u(W ) ±0.038 ±0.004
N u = H obs L × ∂θ ∂Y wall (1)
In order to compute the Nusselt number, the micro-thermocouple is progressively moved away from the wall by 100 µm increments until a maximum distance from the wall of 800 µm in order to remain into the linear zone of the temperature profiles. For each position, a temperature acquisition is made at a frequency of 40 Hz and an acquisition time of 10 minutes. The Nusselt number is then determined with the slope of the curve at the origin obtained by ANOVA on at least 5 points. A Nusselt number is validated only when the correlation coefficient on the slope exceeds R 2 > 0.995. The confidence interval at 95% is then computed using the Student's law (n being equal to the number of considered points and n ≥ 5) :

N u -t 0.025,n-2 σ2 D xx ≤ N u ≤ N u + t 0.025,n-2 σ2 D xx with D xx = 1 H obs 2 i (Y i -Ȳ ) 2 ; σ2 = 1 n -2 i (θ i -θi ) 2
(2) where t 0.025,n-2 is the quantile of order 97.5% for n-2 degrees of freedom of the Student's law, . and . refer respectively to the estimator and the mean.

Results

Dynamical behavior of the flow around the cavity

Velocity measurements were carried out in the vertical mid-depth plane (at X = 0.5). The setup is divided into three zones in order to facilitate its description (see Fig. 1.a) :

-Zone A : the vertical differentially heated channel (X = 0.5, Y ∈ [0; 0.1] and Z ∈ [0; 1]) -Zone B : the top horizontal channel (X = 0.5, Y ∈ [0.1; 0.9] and Z ∈ [0.9; 1]) -Zone C : the vertical unheated channel. (X = 0.5, Y ∈ [0.9; 1] and Z ∈ [0; 1]) The dynamical behavior of the flow will be analyzed in the following sections via dimensionless velocity fields and horizontal or vertical dimensionless velocity profiles around the cavity.

Velocity measurements in the heated channel (Zone A)

A developing boundary layer flow along the hot plate at temperature T h can be observed in zone A (Fig. 4). In Fig. 5, the profiles of W , the vertical component of the mean velocity, show that the maximum velocity, located inside the boundary layer, as well as the thickness of this layer increase with Z. When the flow reaches Z = 0.9, it divides into two parts. It either flows into the top horizontal channel (zone B), or towards the cold plate at T c . There, the flow can either go downward along the cold plate where a boundary layer is developing, or join the top recirculation area (Fig. 4). It is worth mentioning that the recirculation area at the top of the zone A will be of primary importance regarding the thermal analysis that will be discussed later in section 3.2.1. from the lower horizontal channel and following the developing hot boundary layer. This flow comes from the mean flow depicted in figure 1.a. At Z = 0.1, the profiles of the horizontal (V ) and vertical (W ) mean velocity components show the origin of the boundary layer flow observed along the hot plate. An incoming horizontal flow (V<0) coming from the lower channel feeds the vertical developing hot boundary layer and induces a thicker boundary layer in that area. Note that it is then a rather different case from the boundary layer theory assuming a zero velocity near the leading edge of the heated vertical plate. In the end, it comes out that there is a weak interaction between the hot and cold boundary layers into the majority of zone A. In fact, the profiles of the horizontal component, V , of the mean velocity are close to 0 all along the cavity for 0.1 < Z < 0. 3.1.2. Velocity and temperature measurements in the horizontal channel (zone B) As described in zone A, a part of the flow leaving the hot side of the obstacle goes to the top horizontal channel (Fig. 6). The horizontal mean Regarding the temperatures, it can be observed that the profiles are quite flat for most Z and all along the top horizontal channel. Upstream, at Y = 0.15 and at the upper face of the obstacle (Z ≈ 0.90), a nonadiabatic behavior for the temperature profile can be observed. This is caused by the proximity of the vertical heated plate (located at Z = 0.90 and 0.10 ≤ Y ≤ 0.14). Downstream, at Y = 0.85, a similar observation can be made, the flow going to the vertical unheated channel (zone C) gets close to the unheated aluminium plate (located at 0.86 ≤ Y ≤ 0.90 and Z = 0.90). As it is hotter than the plate, a heat exchange between the fluid and the plate occurs at this location. In other locations (Y = 0.25, Y = 0.50 or Y = 0.75), wall temperature profiles (Z = 0.90 or Z = 1) exhibit a behavior which is close to the typical one observed in the vicinity of adiabatic surfaces.

Velocity measurements in the unheated channel (Zone C)

The mean velocity field (Fig. 8) shows the path of the flow coming from the top horizontal channel (Zone B).

At the top of the zone C (Z ∈ [0.90; 1]), the flow coming from zone B, either goes towards the vertical plate maintained at T c (at Y = 1), or flows back to the obstacle side (at Y = 0.90). In fact, at this elevation, both surfaces are colder than the flow (as it will be detailed later, in section 3.2.2). It results in a descending wall flow along both plates. At Z = 0.893, the profile of the horizontal component, V , of the mean velocity (Fig. 9) exhibits positive values, which confirms that the whole flow entering in zone C is coming from the top horizontal channel.

It also results in RMS fluctuations at this elevation which are significant. The vertical component, W , of the mean velocity in the upper part of the channel (Z > 0.50) confirms the development of two descending vertical boundary layers although the maximum velocity in the boundary layer at the cavity cold plate, at T c , is three times greater than the one at the obstacle's along the obstacle. The boundary layer is in fact fuelled by both a cold reversed flow from the bottom horizontal channel and the flow coming from a recirculating zone observed in the lower part of the channel. This recirculating zone is formed by the temperature difference between the cold plate and the obstacle for Z < 0.65. Firstly, these profiles depict that thermal boundary layers are developing on both the cold and hot plates constituting this vertical differentially heated channel. Secondly, the temperature increases in the core zone as Z increases. The central zone is stratified in temperature. Focusing on this variation, the mean temperature profile along the vertical central line, X = 0.50 and Y = 0.05, were plotted in figure 11.

The evolution is linear for Z ∈ [0.10; 0.50]. The stratification parameter

S(Z) = ∂θ ∂Z Y =0.05
, which is the vertical temperature gradient in the linear zone ([0.10; 0.50]), is equal to S = 0.29. This value is surprisingly lower than common values obtained for differentially heated cavity (see Belleoud [START_REF] Belleoud | Etude de la convection naturelle turbulente en cavité verticale différentiellement chauffée : Analyse des structures et des transferts turbulents[END_REF] for In order to study the heat transfer in zone A, Nusselt numbers (Eq. 1) as well as associated confidence interval (Eq. 2) were measured along the hot and cold plates (respectively Figs. 12 and13).

Along the hot plate (Y = 0.1, Z ∈ [0.10; 0.90]), the Nusselt number is strictly decreasing when Z increases (see Fig. 12). Indeed, the hot boundary layer thickens and the fluid heats up. It results in a diminution of the heat transfer with the plate. At Z < 0.20, the Nusselt spatial variations are steeper due to the twofold influence of the cold flows coming from both the cold plate and the lower horizontal channel (see Fig. 4). Thus, the temperature difference is highly increased and so are the heat transfers. The fluctuations of the Nusselt number at Z ≥ 0.80, are probably due to the disturbances coming from the cold flow driven by the top recirculation area. The Nusselt numbers on the cold plate at T c (Y = 0, Z ∈ [0; 1]) (Fig. 13) are decreasing when Z decreases. Indeed, the temperature difference between the plate and the fluid reduces : the fluid cools down as it flows downward and thus the heat transfers decrease. This decrease is steeper for Z ∈ [0.55; 0.70] ; This trend is explained by the influence of the top recirculation area (for Z ≥ 0.6, see Fig. 4).

Temperature measurements (Zone C)

Horizontal profiles of temperature in the unheated channel have been plotted in figure 14. A thermal boundary layer is developing along the cold plate at T c (Y = 1, Z ∈ [0; 1]). Its thickness tends to increase. On the other side (Y = 0.90, Z ∈ [0.1; 0.9]), two opposite wall flows are observed (Fig. 8) and are strongly connected with the temperature difference between the fluid and the wall (Fig 14). At Y = 0.90 and for Z = 0.89 and Z = 0.70, a boundary layer-like temperature profile can be observed with the fluid being hotter than the obstacle. On the contrary, for Z = 0.50, Z = 0.30 and Z = 0.10, a boundary layer-like temperature profile develops with the fluid being colder than the obstacle. In fact, a part of the flow coming from the upper horizontal channel cools down along the wall of the obstacle. When it reaches the temperature of the obstacle (for Z ∈ [0.50, 0.70]) it heads towards the colder plate (at T c and Y = 1). This phenomenon is reversed for the lower Z on the obstacle. The cold flow coming from the lower recirculating zone and the lower horizontal channel (Fig. 8) moves up along the obstacle wall (Y = 0.90) before reaching its temperature for Z ∈ [0.50, 0.70]. It then heads again towards the cold plate. Note that the zone (Y = 0.90 ,Z ∈ [0.50, 0.70]) corresponds to the zone where both opposite boundary layer-like flows impinge (Fig. 8).

As for zone A, the temperature of the core of the channel decreases with Z and a thermal stratification is observed. Figure 15 shows the vertical temperature profile in the middle of the unheated channel (Y=0.95, Z ∈ [0; 1]). A linear variation is observed until Z = 0.5. For Z > 0.5, the vertical temperature gradient is increased. This increase confirms the flow behavior mentioned before and the observed crossing flow through the core of the channel (Fig. 8). In fact, the flow above the division at Z ≈ 0.65 is hotter because it mainly comes from the top horizontal channel (i.e. zone B).

Instabilities analysis (zone A) 3.3.1. Low frequency

Occurrence of internal waves in differentially heated cavities has been widely studied [START_REF] Quéré | From onset of unsteadiness to chaos in a differentially heated square cavity[END_REF][START_REF] Chorin | Heat transfer enhancement of a natural convection flow in an enclosure submitted to a small extent thermal disturbance : Influence of location and frequency[END_REF]. They are found more easily into the stratified cores of the cavities. Thorpe [START_REF] Thorpe | On standing internal gravity waves of finite amplitude[END_REF] showed that those waves are linked with the Brunt-Väisälä frequency which is defined as :

f bv = 1 2π gβ∆T S L (3)
For aspect ratios large enough, the frequency of the first modes is approximately equal to f bv . Figure 16 shows the power spectral density for the vertical velocity fluctuation W ′ (W ′ = W i -W where W i is the instantaneous vertical velocity and W is the vertical mean velocity), at different elevations, Z, in the lower half of the heated channel (zone A). The power spectral density is defined as Φ X = 1 fsnacq | X| 2 where X is the Fourier transform of the quantity X, f s is the sampling frequency and n acq is the number of acquisitions. For each Z, a low frequency stands out and fits with the order of magnitude of f bv . Indeed, regarding this study configuration, equation 3 yields f bv ≈ 0.10 Hz.

Oscillating buoyant Jet

In zone A, the boundary layer flow leaves the hot plate at Z = 0.90 and Y = 0.10 inducing a buoyant jet. This buoyant jet oscillates horizontally either towards zone A or zone B (Fig. 17 It can be observed from spectral analysis that the value of the main oscillating frequency of this buoyant jet is centered around 2 Hz. This frequency comes from a boundary layer instability that will be studied in the next section (3.3.3). The other smaller frequencies can be attributed to internal gravity waves, studied in the previous section, along with the growing complexity of the flow at this elevation.

Figure 19 shows the mean temperature evolution at different elevations, Z, in and around this oscillating jet. The maximum of the mean temperature is to be found into the jet for Z = 0.901 (just above the obstacle top corner) and for Y ≈ 0.098. This maximum decreases with Z increasing. In fact, when the jet leaves the hot plate, the flow is not heated anymore and the jet progressively widens. The jet flattens until Z = 0.983 where it drives the flow Gebhart and Jaluria [START_REF] Gebhart | Buoyancy-Induced Flows And Transport[END_REF] extensively studied the evolution of a propagating downstream periodic wave along heated vertical plates with both isothermal or uniform heat flux surface conditions. Solving the Orr-Sommerfield equations, Dring and Gebhart [START_REF] Dring | A theoretical investigation of disturbance amplification in external laminar natural convection[END_REF] showed that this periodic wave is amplified downstream and filtered into a single frequency. It will be later confirmed experimentally by Mahajan and Gebhart [START_REF] Mahajan | An experimental determination of transition limits in a vertical natural convection flow adjacent to a surface[END_REF] and Qureshi and Gebhart [START_REF] Qureshi | Transition and transport in a buoyancy driven flow in water adjacent to a vertical uniform flux surface[END_REF]. This frequency value does not depend on the position along the vertical plate and Gebhart and Jaluria [START_REF] Gebhart | Buoyancy-Induced Flows And Transport[END_REF] proposed this characteristic pulsation :

Ω = 2πf ν ν 2 gβ∆T * 2/3 (4)
where ∆T * is the temperature difference between the plate temperature and the temperature far away from it. With numerous numerical and experimental results (Hieber and Gebhart [START_REF] Hieber | Stability of vertical natural convection boundary layers : some numerical solutions[END_REF], Polymeropoulos and Gebhart [START_REF] Polymeropoulos | Incipient instability in free convection laminar boundary layers[END_REF],

Godaux and Gebhart [START_REF] Godaux | An experimental study of the transition of natural convection flow adjacent to a vertical surface[END_REF]), Gebhart and Mahajan [START_REF] Gebhart | Characteristic disturbance frequency in vertical natural convection flow[END_REF] were able to draw asymptotic Prandtl-dependant solutions, in particular for P r ∈ [0.1, 100] :

Ω = 0.315P r -0.065 (5)
More recently, developing perturbations along isothermal vertical plates were studied by Zhao et al. [START_REF] Zhao | Resonance of the thermal boundary layer adjacent to an isothermally heated vertical surface[END_REF][START_REF] Zhao | A PIV measurement of the natural transition of a natural convection boundary layer[END_REF][START_REF] Zhao | On the selection of perturbations for thermal boundary layer control[END_REF][START_REF] Fan | Natural convection over vertical and horizontal heated flat surfaces : A review of recent progress focusing on underpinnings and implications for heat transfer and environmental applications[END_REF]. They have also observed a growing perturbation of high frequency (that does not depend on the vertical position along the plate) which is acknowledged as the characteristic frequency. For P r = 0.7, they obtained the following correlation for the dimensional characteristic frequency (Hz) :

f c = 0.07 gβ∆T * α 2/3 ν 1/3 (6)
This correlation can be rewritten according to Gebhart's definition of the dimensionless characteristic pulsation :

Ω = 2π × 0.07P r 2/3 (7)
Table 2 shows the characteristic pulsation determined asymptotically by Gebhart (Eq. 5), the one determined by Zhao (Eq. 6) for P r = 0.7 and the one from the present study (Eq. 4). In our case, ∆T * is the temperature difference between the temperature plate and the mean temperature in the center of the heated channel between Z = 0.10 and Z = 0.90 (Fig. 11). This temperature choice is motivated by the horizontal shape of the temperature profiles in the cavity center (Fig. 10) and it appears here to be a suitable way to define ∆T * with respect to Gebhart's definition. Although configurations slightly differ, the pulsation obtained is close to the ones obtained by Gebhart and Zhao and differs only by a maximum of 11%. Furthermore, a possible range for Ω can be determined which takes into account the stratified medium and is Ω ∈ [0.31; 0.47].

From the spectral analysis of the velocity fluctuations, the amplitude A z of the oscillating frequency can be obtained. Figure 21 shows the normalized amplitude of the oscillating frequency obtained at Y = 0.09 for several elevations Z. This amplitude is normalized by the assumed neutral amplitude of the oscillating frequency along the vertical plate A 0 . The neutral amplitude chosen here, is the amplitude where the disturbance is first detected (Z 0 ≈ 0.4). The spectrum amplitude is determined with an fft algorithm There is a quite good agreement with the linear stability theory which predicts [START_REF] Dring | A theoretical investigation of disturbance amplification in external laminar natural convection[END_REF] :

A z A 0 = e -z z 0 α i dz (8)
where A 0 is the disturbance neutral amplitude, A Z the disturbance amplitude at Z farther downstream and α i the spatial amplification rate of the disturbance. Note that the theory predicts the amplitude evolution for an injected perturbation. In our case, the neutral amplitude chosen is only based on observation and cannot be accurately the neutral amplitude described by the theory [START_REF] Knowles | The stability of the laminar natural convection boundary layer[END_REF]. However, the criterion used for denoting the instability of the flow is α i < 0. For increasing Z, the criterion in the present study is then respected and leads to an unstable flow in the hot boundary layer and its vicinity.

Considering enclosed spaces, Janssen and Henkes [START_REF] Janssen | Instabilities in threedimensional differentially-heated cavities with adiabatic horizontal walls[END_REF][START_REF] Janssen | Influence of Prandtl number on instability mechanisms and transition in a differentially heated square cavity[END_REF], Le Quéré and Behnia [START_REF] Quéré | From onset of unsteadiness to chaos in a differentially heated square cavity[END_REF] and Kishor et al. [START_REF] Kishor | Flow instabilities and heat transfer in a differentially heated cavity placed at varying inclination angles : Non-intrusive measurements[END_REF] likewise observed the occurrence of travelling waves inside differentially heated cavities with multiple aspect ratios and angles. In particular, Xin and le Quéré [START_REF] Xin | Natural-Convection Flows in Air-Filled, Differentially Heated Cavities with Adiabatic Horizontal Walls[END_REF] provided critical angular frequencies for the most unstable modes in differentially heated cavities with vertical aspect ratio between 1 and 8. Their results show travelling waves frequencies which are of the same order of magnitude than the one observed in the present study. Those frequencies are determined for a critical Rayleigh number which is one order smaller than the one of the present case. It can be added that the high frequency in the present study does not stand out in the cold boundary layer while it does and stays identical in the differentially heated cavities for any aspect ratios.

Turbulence

The understanding of turbulence development is still a challenge. This is why different ways to describe it have been carried out over the years. One of them is based on the behavior of the fluctuations spectra. Over a heated vertical plate, during the transition to turbulence, the spectrum is centered around the boundary layer characteristic frequency defined earlier.

The turbulence developing downstream leads to an energy distribution from large to smaller scales. Considering the frequencies, it means that their is an upper broadening of the fluctuations spectra [START_REF] Bill | The development of turbulent transport in a vertical natural convection boundary layer[END_REF]. Figure 22 shows the cumulative power spectra of three points taken along the hot boundary layer, at Y = 0.09.

For the point located at the beginning of the transition (at Z ≈ 0.50) almost all the energy is concentrated into frequencies smaller than the filtered frequency (2 Hz). During the transition, at Z = 0.65 , there is a slight broadening around the characteristic frequency meaning that there is a start of energy distribution to the neighbouring frequencies. Farther downstream, at In particular, a substantial increase of the energy distribution for frequencies greater than the filtered frequency is observed. It means that the energy has been transferred from large eddies to smaller ones. Moreover, a large part of the energy is still concentrated around the filtered frequency (the gradient around 2 Hz is higher than for smaller frequencies). While the spectrum is broadening, it means that the process of selective frequency amplification still remains. Through the developing spectrum, an "inertial subrange" can arise. It means that the gradient of the spectral density remains constant over a certain range of wave numbers. Lumley [START_REF] Lumley | Interpretation of Time Spectra Measured in High-Intensity Shear Flows[END_REF] showed that it is possible to substitute the wave numbers by the frequencies for small enough turbulence scales and velocity gradients. Considering buoyancy-driven flows, it has been shown that the power spectral density decays with a slope of -3 for a quite large frequency range [START_REF] Gebhart | Buoyancy-Induced Flows And Transport[END_REF]. Figures 23 and24 show the power spectral density versus the frequency for two points downstream along the hot plate (Y = 0.1). At Z = 0.65, during the transition, the energy decay towards frequencies superior to the filtered frequency is non-linear. On the contrary, at Z = 0.80, a slope of -3 is observed which confirms the presence of a large inertial subrange starting near the characteristic frequency which is also the start of the upper frequencies broadening.

Conclusions and perspectives

We have studied a turbulent natural convection flow whose Rayleigh number is similar to those encountered in industrial applications like the underhood of cars. Understanding the complex flow behavior around an inner heated obstacle in an enclosed space is a crucial issue for industries which rely on optimized heat transfers to ensure their materials thermal resistance.

The dynamical and thermal behavior of a flow in an enclosed cavity with a partially heated obstacle is studied in the work. The heated channel is thermally stratified and a recirculating zone is observed in its upper part. The recirculating zone disturbs the linearity of the stratification and the heat transfers on both the hot and cold plates. Leaving the heated plate of the obstacle, the flow takes the form of an oscillating jet. The occurrence of a travelling wave along the heated plate and internal gravity waves in the core of the heated channel is shown. The boundary layer instability frequency is a of Tollmien-Schlichting type and has been found experimentally around 2 Hz. It is in good agreement with the linear stability theory on a heated vertical plate. Remarkably, this is also the frequency of the oscillating jet.

The gravity wave frequency is approximately equal to the Brunt-Väisälä frequency. Turbulence develops downstream with an upper broadening of the fluctuation spectrum over the travelling wave frequency. An inertial zone of -3 slope appears over the filtered frequency.

It is worth mentioning that data provided in this paper can be useful for further validations of numerical codes used when turbulent flows where dominant buoyancy forces are encountered. In addition, the boundary conditions considered in this study are stationary. A perspective for this investigation is to apply unsteady boundary conditions to the heated plate. Actually, this could help to simulate unsteady cases encountered in industrial systems as car engines, after for instance a sudden stop of the car, when the engine is no longer cooled by a forced external flow.

Figure 1

 1 Figure 1 -a) 2D scheme of the experimental set-up and of the mean flow. b) 3D scheme of the experimental set-up and of the mean flow. c) Picture of the experimental set-up

Figure 2 -Figure 3 -

 23 Figure 2 -PIV set-up

Figure 4 --Figure 5 -

 45 Figure 4 -Norm of the mean dimensionless velocity field and streamtraces ( ) for zone A

8 .

 8 The RMS velocity fluctuations are quantified in figure 5 via the vertical bars. Most of the vertical velocity fluctuations are located inside the boundary layers with greater values in the hot boundary layer. On the contrary, the fluctuations of the horizontal velocity are mostly near the outer cold boun-dary layer. It shows that the cold boundary layer fluctuates more horizontally than the hot one. At Z = 0.893, the fluctuations of the horizontal velocity are greater all along Y because of the recirculation area effects.

Figure 6 -

 6 Figure 6 -Norm of the mean dimensionless velocity field and streamtraces ( ) for zone B as well as upper parts of zones A and C.

Figure 7 -

 7 Figure 7 -Mean horizontal dimensionless velocity and mean dimensionless temperature profiles for zone B ; θ (•) ; V (×)

Figure 8 -x x xxx x x x x x x xx xx xxxxxxxxxx xxxxxx x x x x x x x xFigure 9 -

 89 Figure 8 -Norm of the mean dimensionless velocity field and streamtraces ( ) for zone C

3. 2 .Figure 10 -

 210 Figure 10 -Mean dimensionless temperature profiles for different Z in zone A. Vertical bars ( ) are of length 2σ θ .

Figure 11 -

 11 Figure 11 -Vertical profile of the mean dimensionless temperature at Y = 0.05

Figure 12 -

 12 Figure 12 -Nusselt numbers along the heated vertical plate (Zone A, Y = 0.1, Z ∈ [0.10; 0.90], X = 0.50)

Figure 13 -

 13 Figure 13 -Nusselt numbers along the cold plate (Zone A, Y = 0, Z ∈ [0; 1], X = 0.50)

Figure 14 -

 14 Figure 14 -Mean dimensionless temperature profiles for different elevations, Z, in zone C. Vertical bars ( ) are of length 2σ w,v .

Figure 15 -

 15 Figure 15 -Vertical profile of the mean dimensionless temperature at Y = 0.95 for zone C

Figure 16 -

 16 Figure 16 -Power spectral density for W ′ at different Z and Y = 0.05

  ).

Figure 17 -

 17 Figure 17 -Norm of the instantaneous velocity. Screenshots are taken at different times around the jet located near the top edge of the heated plate (zone A).

Figure 18

 18 Figure18shows the power spectral density of the horizontal fluctuations for a point located in this oscillating jet at Z = 0.94 and Y = 0.09.It can be observed from spectral analysis that the value of the main oscillating frequency of this buoyant jet is centered around 2 Hz. This frequency comes from a boundary layer instability that will be studied in the next section(3.3.3). The other smaller frequencies can be attributed to internal gravity waves, studied in the previous section, along with the growing complexity of the flow at this elevation.Figure19shows the mean temperature evolution at different elevations, Z, in and around this oscillating jet. The maximum of the mean temperature is to be found into the jet for Z = 0.901 (just above the obstacle top corner) and for Y ≈ 0.098. This maximum decreases with Z increasing. In fact, when the jet leaves the hot plate, the flow is not heated anymore and the jet progressively widens. The jet flattens until Z = 0.983 where it drives the flow

Figure 18 -Figure 19 -

 1819 Figure 18 -Power spectral density for V ′ at Z = 0.94 and Y = 0.09 (located in the oscillating buoyant jet)

Figure 20 -

 20 Figure 20 -Power spectral density for V ′ at Z = 0.65 and Y = 0.09

Table 2 -

 2 Comparison of the characteristic pulsation Ω (Eqs. 4, 5 and 6) for P r = 0.71 and f = 2 Hz applied on the normalized horizontal velocity fluctuations (v ′ /U c (z)) with U c (z) = gβ∆T * (z -0.1) (here 0.1 refers to the elevation of the heated plate leading edge). U c (z) allows to account for the fluctuations growth induced by the base flow (Dring and Gebhart[START_REF] Dring | A theoretical investigation of disturbance amplification in external laminar natural convection[END_REF]).

Figure 21 -

 21 Figure 21 -Disturbance amplitude evolution. (.) experimental data ; (-) exponential fitted curve (γ ≈ 6)

Figure 22 -

 22 Figure 22 -Cumulative power spectra for W ′ . (•) datas at Z = 0.80 ; (•) Z = 0.65 ; (×) Z = 0.50

Figure 23 -Figure 24 -

 2324 Figure 23 -Power spectral density for W ′ at Z = 0.65 and Y = 0.09

Table 1 -

 1 Velocity and temperature uncertainties

Acknowledgments

The authors would thank the ANR MONCACO_2025 grant (ANR-17-CE06-0005) for supporting this work as well as the "transport" CPER/FEDER Program. The authors would also like to thank H. Arlaud who built microthermocouples and the experimental set-up as well as J.-C. Fraudeau, N. Papin and C. Fuentes for her support with laser techniques.

[2] S. Ostrach, L. F. P. Laboratory, and U. S. N. A. C. for Aeronautics, An Analysis of Laminar Free-convection Flow and Heat Transfer about


Peter A Cistulli
email: peter.cistulli@sydney.edu.au

Jeff Armitstead

Jean-Louis Pepin

Holger Woehrle

Carlos M Nunez

Adam Benjafield

Atul Malhotra

Charles

Short-term CPAP adherence in obstructive sleep apnea: a big data analysis using real world data

Background

Effective treatment of obstructive sleep apnea (OSA) is an important goal due to its impact on quality of life, and the known cardiovascular and neurocognitive consequences of this common chronic disease. OSA is also associated with an increasing economic and social burden, estimated to be over $12 billion in the US in 2015 [START_REF] Watson | Health care savings: the economic value of diagnostic and therapeutic care for obstructive sleep apnea[END_REF]. Continuous positive airway pressure (CPAP) has excellent efficacy for controlling OSA, but its effectiveness is limited by variable adherence to therapy. The fact that CPAP devices objectively record usage data provides a unique opportunity to monitor adherence. A recent systematic review demonstrated that CPAP adherence did not increase across clinical trials over the last 20 years despite continuous improvement in technical innovation and support [START_REF] Rotenberg | Trends in CPAP adherence over twenty years of data collection: a flattened curve[END_REF]. Studies included in the systematic review were generally small and often performed under close monitoring conditions that are not reflective of real world usage. Therefore, such approaches may not be generalizable or sustainable. Moreover, randomized controlled clinical trials typically exclude severe patients, including those with profound sleepiness or at high risk of complications, making participants less representative of typical clinical practice. Advances in technology, intensive support, and patient/provider education are all thought to improve adherence to CPAP therapy. Recent digital health innovation using patient engagement tools has also been associated with improved CPAP use compared with usual care [START_REF] Malhotra | Patient engagement using new technology to improve adherence to positive airway pressure therapy: a retrospective analysis[END_REF]. There is a need for physicians, payers and policy makers to have access to real world data on CPAP adherence in order to opmitize models of care and patient outcomes. Hence, the aim of this study was to define CPAP adherence over the first 90 days using data from a large cloud database, as reflective of current technology and real-world clinical care.

Methods

We assessed de-identified data from a large cloud database of positive airway pressure (PAP) users to examine adherence to therapy during the period 1-Oct-2014 to 31-Oct-2017. AirView (ResMed Inc., San Diego, USA) is a HIPAA-compliant web-based solution for healthcare specialists intended to transfer and display device and therapeutic information that has been transmitted remotely from the patient's therapy device. All Air10 platform PAP devices (ResMed Inc., San Diego, USA) have an on-board mobile communications chip which connects to the secure cloud infrastructure automatically. Patient compliance and therapy data are uploaded daily, typically within one hour after each therapy session. All AirView communication and storage is encrypted to meet required international privacy and security standards. The AirView database is hosted in a secure facility in the USA. Patients on the AirView platform in the USA are managed by private and academic sleep centres, home medical equipment providers (HMEs) and primary care offices, with a wide geographical distribution. Compliance data are automatically provided to the HME to facilitate patient care, and the patient can also nominate their caring physician to access the data. The information provided includes usage data and therapy issues (eg, high mask leak, high residual AHI or central apnea index). Approximately 98% of Air10 devices setup with patients have data in AirView. The HME gives permission to ResMed to de-identify protected health information in accordance with 45 C.F.R. § 164.514(b) and use such deidentified data for analytics by ResMed. The HME is responsible consenting each patient and approximately 96% of records are thereby included. The Air10 platform became available in August 2014, and was rapidly adopted such that older devices currently represent <0.2% of devices setup in the USA. Mobile communications costs are not borne by the patient but are purchased by the HME as part of their population management process. Furthermore, a patient engagement tool, My Air (ResMed Inc, San Diego, USA), became available in October 21 2014 and the iOS app became available August 2016. Patients do not pay for access to MyAir irrespective of HME signup.

Patients were included if they were adults >18 years, started treatment on the AirSense/ AirCurve 10 (ResMed Corp) PAP platform (Air10) within the US, were registered on the cloud database by their HME, and if they used a single PAP modality for ≥1 hour during the first 90 days of therapy. The one hour threshold was chosen to minimise selection bias while ensuring that machine calculated parameters such as AHI and respiratory rate are averaged over a sensible minimum timeframe. The primary outcome was adherence, which was defined using the US Center for Medicare and Medicaid Services (CMS) definition (≥4 hours' PAP use on 70% of nights in a consecutive 30-day period in the first 90 days of therapy). The time to achieve CMS compliance was taken as the number of days taken from first day of use to the day at which the compliance threshold was met. The study was reviewed by an Institutional Review Board (IRB) and deemed exempt from IRB oversight.

All data were analysed in a secure, anonymised database physically separate from the main production server. Statistical analyses were performed using the R language (R development core team). Missing compliance data are imputed as zero. The first day of therapy is taken as the patient AirView setup date. For other data fields, including clinical metrics and respiratory events, missing data are not imputed and not included in calculations of mean values, standard deviations and proportions.

Results

During the period of analysis, 2.62 million patients met the inclusion criteria; 23.4% of patients were excluded for the following reasons: >1 HME (1.9%), invalid data entry (10.1%), more than one PAP modality used or data from SD card (9.2%), age ≤18 years (0.7%), <1 hour usage in first 90 days (1.4%). The mean (±SD) age was 55.7 ± 13.8 years. The following single PAP modalities were used: APAP (50%), CPAP (41%), bilevel (5.6%), bilevel Auto (2.4%), ASV (0.7%), ASV Auto (0.5%).

The observed 90-day PAP adherence rate using the CMS definition was 75%. Participants used PAP on a median 93% of nights for a mean of 5.1 ± 2.5 (SD) hours/night across all nights and 6.0 ± 2.0 h/night on nights used (see Table 1). The median proportion of days with non-zero usage was 93.3%. A histogram of CPAP usage is shown in Fig. 1. The mean residual AHI was 3.2 ± 4.5 events/hour. The mean 95th percentile pressure (CPAP and APAP) was 10.9 ± 2.8 cmH2O. The mean 95th percentile leak of 22.8 ± 19.6 L/min.

Discussion

To our knowledge this is the largest analysis of objective CPAP usage ever undertaken using real world data. The short term adherence rate of 75% in this analysis appears higher than is generally acknowledged in clinical practice and reported in the literature. The observed treatment characteristics in our study demonstrate excellent efficacy during therapy, with a mean residual AHI of 3.2/hour and acceptable leaks. The mean usage of 5.1 hours is also higher than that found in a meta-analysis of 82 clinical trials (weighted pooled mean of 4.6 h/night) [START_REF] Rotenberg | Trends in CPAP adherence over twenty years of data collection: a flattened curve[END_REF]. Among larger CPAP trials, a study of a social cognitive theory intervention involving 206 patients found that 55.2% of patients achieved >4hours use per night with standard care at three months [START_REF] Bartlett | Increasing adherence to obstructive sleep apnea treatment with a group social cognitive therapy treatment intervention: a randomized trial[END_REF]. The APPLES study (n = 443 active CPAP) found that 42% achieved the MCS adherence threshold at six months [START_REF] Kushida | Effects of continuous positive airway pressure on neurocognitive function in obstruction sleep apnea patients: the apnea positive pressure long-term efficacy study (APPLES)[END_REF]. More recently, a randomized controlled trial of telehealth strategies in 1455 patients found that 54% in the usual care group achieved MCS adherence at 90 days, rising to 73% in the telemedicine group, who received education and automated feedback [START_REF] Hwang | Effect of telemedicine education and telemonitoring on continous positive airway pressure adherence[END_REF].

Given that OSA is a chronic disorder associated with considerable morbidity and mortality, there is an imperative to optimize adherence to therapy. The observed adherence rate in this study compares favorably to that reported for pharmacotherapy in other chronic diseases. For example, high non-adherence rates have been reported for the use of inhalers in asthma (>50%) [START_REF] Mes | Pharmacists and medication adherence in asthma: a systematic review and met-analysis[END_REF] and COPD (77%) [START_REF] Sulaiman | Objective assessment of adherence to inhalers by patients with chronic obstructice pulmonary disease[END_REF], and medication use in hypertension (45%) [START_REF] Abegaz | Nonadherence to antihypertensive drugs: systematic review and meta-analysis[END_REF], diabetes (7 -64%) [START_REF] Cramer | A systematic review of adherence with medication for diabetes[END_REF], and epilepsy (38%) [START_REF] Getnet | Antiepileptic drug nonadherence and its predictors among people with epilepsy[END_REF]. Poor adherence to prescribed medications accounts for substantial worsening of disease, death, and increased health care costs.

The long-term effects of suboptimal PAP adherence bring to light the health related impact of untreated OSA. It is not sufficient simply to prescribe a PAP machine and consider the patient to be treated. There are many factors influencing PAP adherence, both medical and non-medical (eg, psychosocial), and that these need to be considered when both prescribing CPAP as well as designing cost-effective interventions to improve adherence. We believe that big data analysis such as the one reported here can offer new and important insights to enhance PAP adherence.

There is a need for health care systems to embrace digital information that can enhance patient engagement and outcomes of therapy. Although there are many challenges related to ethics, data ownership, storage, usage, and reimbursement that require clarification [START_REF] Randerath | Challenges and perspective in obstructive sleep apnoea[END_REF], sleep medicine is well placed to develop new chronic disease models of care, enabled by digital health data such as CPAP adherence. It is critical that progress is based on ethical patient-centered principles, to safe-guard against the misuse of "big data." It has been proposed that academic-industry partnerships offer a path to maximise the benefits that are achievable from appropriate use of such data, provided there are rigorous controls on who can access the data and for what purpose it is used [START_REF] Jain | Is big data the new frontier for academic-industry collaboration[END_REF].

Although our study has a large sample size there are important limitations that need to be considered. The observational nature of the study raises potential for important biases. First, we do not have data about patients who may have been prescribed treatment but did not proceed with it. Prior studies suggest that up to 24% of patients may not fulfill their PAP prescription [START_REF] Lin | Treatment compliance in patients lost to follow-up after polysomnography[END_REF] and a proportion of patients may not even turn on their PAP machine. Exclusions related to invalid data entry may have further biased the sample. In addition, we lack detailed knowledge about participants' demographic and clinical characteristics because these data are not available in the database and cannot be linked to existing data due to privacy issues. Notwithstanding these notable limitations, using a big data approach we observed excellent adherence among patients initiating PAP therapy, with 75% of patients achieving the CMS compliance threshold. This adherence rate compares favorably to those reported in the OSA literature, as well as with other chronic medical therapies. The large sample size in this study suggests that the findings are likely to reflect real world clinical care in the US. Future work is needed to examine longer term adherence to PAP therapy and outcomes. Histogram of PAP usage.

Fig. 1 .

 1 Fig. 1.

Sleep Med. Author manuscript; available in PMC 2020 January 01.

Acknowledgements and disclosures

The authors wish to thank the following for statistical analysis: Haixiang Shi, PhD and Yang Yan, M. Sc. Stat. Authors JA and AB had full access to all the data in the study and takes responsibility for the integrity of the data and the accuracy of the data analysis. ResMed Corp, USA funded the study and their employees (JA, AB, CMN) contributed to the design and conduct of the study, analysis, and interpretation of the data and preparation of the manuscript for publication. PAC has an appointment to an endowed academic Chair at the University of Sydney that was established from ResMed funding. He has received research support from ResMed, SomnoMed, Australia, and Zephyr Sleep Technologies, Canada. He is a consultant/adviser to Zephyr Sleep Technologies, and Narval, France. He has a pecuniary interest in SomnoMed related to a previous role in R&D (2004). JLP is supported by the French National Research Agency, France in the framework of the "Investissements d'avenir" program (ANR-15-1DEX-02). His department has received research support from Philips Respironics, USA, Fisher and Paykel, New Zealand, and ResMed. HW has received consulting/speaking fees from ResMed and Inspire Medical. AM relinquished all outside personal income as an Officer of the ATS in 2012. ResMed gave a philanthropic donation to UC San Diego, but AM receives no personal income from ResMed.

Table 1

Adherence data from the first 90 days of therapy.