S Harmand
email: souad.harmand@univ-valenciennes.fr

J Pellé

S Poncet

Review of fluid flow and convective heat transfer within rotating disk cavities with impinging jet

Fluid flow and convective heat transfer in rotor-stator configurations, which are of great importance in different engineering applications, are treated in details in this review.

The review focuses on convective heat transfer in predominantly outward air flow in the rotor-stator geometries with and without impinging jets and incorporates two main parts, namely, experimental/theoretical methodologies and geometries/results. Experimental methodologies include naphthalene sublimation techniques, steady-state (thin layer) and transient (thermochromic liquid crystals) thermal measurements, thermocouples and infrared cameras, hot-wire anemometry, laser Doppler and particle image velocimetry, laser plane and smoke generator. Theoretical approaches incorporate modern CFD computational tools (DNS, LES, RANS etc). Geometries and results part being mentioned starting from simple to complex elucidates cases of a free rotating disk, a single disk in the crossflow, single jets impinging onto stationary and rotating disk, rotor-stator systems without and with impinging single jets, as well as multiple jets. Conclusions to the review outline perspectives of the further extension of the investigations of different kinds of the rotor-stator systems and their applications in engineering practice.

Nomenclature

Introduction

Fluid flow and convective heat transfer in rotor-stator configuration are of great importance in engineering practice in the turbomachinery industry, namely, in the gas turbine cooling. Another important example from the modern practice has relevance to wind generators. Discoidal rotor-stator systems do not incorporate gears in order to avoid altering the system's mechanical efficiency, and are thus able to improve power production at low rotational speeds. Reliable methods for estimation convective heat transfer intensity in the air gap can help manufacturers to better design the discoidal wind generators to prevent overheating. Different strategies are employed by designer depending on the particular engineering application.

A review of the most important results for the rotor-stator systems available so far is presented here. The review focuses on convective heat transfer in predominantly outward air flow in rotor-stator cavities with and without impinging jets. The structure of the review comprises two main parts, namely, methodology and geometries/results.

Methodologies elucidate experimental techniques most widely used for measurements in rotating-disk systems of (a) heat/mass transfer, such as naphthalene sublimation techniques, steady-state (thin layer) and transient (thermochromic liquid crystals) thermal measurements, thermocouples and infra-red cameras, and (b) flow structure using hot-wire anemometry, laser Doppler and particle image velocimetry, as well as by means of a laser plane and smoke generator. Numerical methodologies for flow and heat transfer modelling in the rotor-stator systems encompass methods of the direct numerical simulation (DNS), Reynolds-averaged Navier-Stokes equations (RANS) with the turbulence models both in the general-purpose form as incorporated in the commercial computational fluid dynamics (CFD) software, and often specially modified to be used in the rotating-disk systems. They include different modifications of standard k-ε, k-ω and Reynolds-stress models (RSM); large-eddy simulations (LES) approach is also quite widely used.

Geometries and results part is built up on the principle of ascent from simple to complex geometries. This helps understand complex fluid flow and heat transfer phenomena in the rotor-stator systems as a superposition of a number of separate effects like rotation, impingement/crossflow with different angles of attack, flow regime, wall temperature profile etc.

It starts with a free rotating disk. Both fundamental findings and recent advancements in the theoretical and experimental investigations of fluid mechanics and heat transfer over a free rotating disk in laminar, transitional and turbulent flow regimes are presented here together with the recommendations on the most reliable relations for prediction of the friction and transport coefficients including effect of the radial profile of the disk temperature.

The next more complicated geometry is the single disk in the cross-flow. Here experimental/computational results are discussed for a stationary/rotating disk of a finite and zero thickness in parallel cross-flow including the case where the disk was mounted on a side surface of cylinder with a significantly smaller radius.

Single jets impinging onto stationary and rotating disks are dealt with in the next part in both experimental and theoretical aspects involving self-similar exact solution for the stagnation point.

The following section describes in details peculiarities of fluid flow and heat transfer in the rotor/stator system without impinging jets. Presented here are maps of the flow regimes depending on the height of the gap between the rotor and stator and rotational speed. The section is completed with equations for the moment coefficient and Nusselt number.

As the next step, rotor-stator systems with the impinging jets are treated. Flow regimes including Batchelor and Stewartson flow structures are elucidated here, and predictive relations for the boundaries of the flow regimes, friction coefficients and Nusselt numbers complement this material.

In the end, rotor-stator systems with the multiple impinging jets finalize the description of the geometrical configurations in the review.

Conclusions to the particular parts of the review outline perspectives of the further extension of the investigations of different kind of the rotor-stator systems and their and their applications in engineering practice.

Methodology

Experimental methodologies for flow and heat transfer

In this part are considered methodologies which are used in studies about rotating disks with or without jet impingement, impinging jet on stationary surfaces, rotor-stator configurations with or without jet.

Convective Heat Transfer

Mass transfer. The general naphthalene sublimation technique for mass transfer measurements is described in detail in [START_REF] Souza Mendes | The naphthalene sublimation technique[END_REF] and [START_REF] Goldstein | A review of mass transfer measurements using naphthalene sublimation[END_REF]. The surface where heat and mass transfer is to be determined is covered with a thin solid layer of naphthalene. Different methods are available for coatings, but the casting procedure is the most convenient and popular method [START_REF] Goldstein | A review of mass transfer measurements using naphthalene sublimation[END_REF]. The coating thickness is usually less than 0.2mm. The air flow near the surface involves a mass and heat transfer from the wall to the air. Some works [KRE68, SPA82, TIE63, JAN86, WAN11] use that technique to determine a surface-averaged mass transfer on an entire disk by only measuring the weight before and after experiments and determining the global sublimation rate. To do that, a 0.1 mg resolution balance is often used with a 200g capacity [START_REF] Goldstein | A review of mass transfer measurements using naphthalene sublimation[END_REF]. In order to ensure accurate measurements, the run time must be such that at least 10mg of naphthalene sublimate. Other works [CHE98, CHO03, HE05, CHO01, MAB72, PRA95, SHI87] experimentally study the local convective heat transfer in rotating disk configurations using this technique. Then the mass transfer coefficient is determined with:

(1)

where and are the naphthalene vapor and air density, and is the sublimation rate.

The mass transfer is essentially isotherm and the naphthalene vapor density is assumed to be constant and equal to the one corresponding to the cooled surface temperature. In order to determine precisely, the sublimation rate has to be determined by evaluating the sublimation thickness due to the mass transfer:

(2)

where is the naphthalene solid density; is the duration of the sublimation; is the naphthalene thickness, which is sublimated during the experiment. The most accurate method for the thickness measurement is the use of a linear variation differential transformer (LVDT). An LVDT having a 0.5mm linear range and 25nm resolution is welladapted. Different tips are used for the LVDT gauge. The tip is selected so that it can measure a slope change of the surface with little error. A tip pressure that is too high produces a scratch mark on the naphthalene surface, whereas a tip pressure that is too low can introduce more measurement error [START_REF] Goldstein | A review of mass transfer measurements using naphthalene sublimation[END_REF].

The measured depends on the position over the disk. The local mass flux is then evaluated with Eq. ( 2) and is deduced with Eq. (1). The local Sherwood number is then determined by:

(

with L a reference length, the air gap thickness in the case of a rotor stator system [START_REF] Prata | Local heat transfer in axially feeding radial flow between parallel disks[END_REF] or jet diameter in [START_REF] Chen | Heat (mass) transfer between an impinging jet and a rotating disk[END_REF], and the diffusion coefficient of the naphthalene vapor in the air, which is usually obtained with an equation of the following form:

(4

)
where T is the air temperature inside the gap or near the disk. Depending on the author, a ranges from to and b from 1.75 to 1.983. The Reynolds analogy is then used to link the mass transfer to the heat transfer using the local Nusselt, Sherwood, Schmidt and Prandtl numbers:

(5)

The exponent m is independent of the Pr and Sc numbers, with the values of m being determined from the empirical results for different flow regimes. Typically, the Schmidt and Prandtl numbers in different works varied over the range Sc=[2.28-2.5] and Pr=[0.7-0.74],

respectively.

An in-depth investigation into the proper values of the exponent m based on the experimental data and theoretical models of different authors was performed in [START_REF] Shevchuk | A new evaluation method for Nusselt numbers in naphthalene sublimation experiments in rotating-disk systems[END_REF][START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF]. For laminar flow, the value of m=0.53 first suggested by [START_REF] Janotková | A naphthalene sublimation method for predicting heat transfer from a rotating surface[END_REF] was confirmed by the authors of [START_REF] Shevchuk | A new evaluation method for Nusselt numbers in naphthalene sublimation experiments in rotating-disk systems[END_REF][START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF]; for transitional flow, m=0.6 in the narrow range of

Re Ω =[1.9×10 5 -2.75×10 5 ] [ [START_REF] Shevchuk | A new evaluation method for Nusselt numbers in naphthalene sublimation experiments in rotating-disk systems[END_REF][START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF]; for turbulent flow, Eq. ( 5) was modified as [CHE98, [START_REF] He | Convection heat and mass transfer from a disk[END_REF][START_REF] Wang | Convective heat transfer of the different texture on the circumferential surface of coupling movement (rotating speed coupling with air velocity) disk[END_REF] in analogy with a non-rotating flat surface entails significant errors in recalculating the experimental Sherwood numbers to the respective Nusselt numbers using the analogy theory (Eq. ( 5)).

Sensibility of the exponent m to the flow regime and thus indirectly to the geometrical configuration that affects the flow regime makes the naphthalene sublimation technique rather inconvenient, when the task of determining heat transfer coefficients is predominant.

Also, though it was rather widely used, this technique is submitted to limitations in the case of low velocities due to the need of increasing the time of experiments which induces inaccuracies. And in the case of high velocities, the problem is that naphthalene can be worn or stripped of by mechanical erosion rather than diffusion and convection.

Electrochemical techniques employing rotating-disk electrodes were used in [DOS76, [START_REF] Mohr | Mass transfer to a rotating disk in transitional flow[END_REF][START_REF] Şara | Electrochemical mass transfer between an impinging jet and a rotating disk in a confined system[END_REF] for deduction of the mass transfer coefficient and then the Sherwood number with the ultimate objective to experimentally find out diffusion coefficients of particular electrolytes. As discussed in detail in [START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF][START_REF] Shevchuk | An Integral Method for Turbulent Heat and Mass Transfer over a Rotating Disk for the Prandtl and Schmidt Numbers Much Larger Than Unity[END_REF], for the high Schmidt numbers specific for electrochemistry applications, diffusion boundary layer is extremely thin and develops within the viscous sub-layer of the velocity boundary layer. This destroys full analogy between heat and mass transfer observed in the case of the naphthalene sublimation technique, entails e.g. higher values of the exponent at the Reynolds number for turbulent flow (0.9 instead of 0.8 for heat transfer applications) and makes it hardly possible to use electrochemical techniques to model heat transfer based on the analogy (Eq. ( 5)).

Thermal measurements. At present, most of authors use direct thermal measurements to determine the convective heat transfer. With the purpose to determine the local or mean heat transfer coefficient between radii and over a disk, it is usually defined as:

(6)

With:

(8)

The local convective heat flux q can be obtained in several ways, depending on the author.

Most of them [LYT94, CAR96, SAN97, LEE02, GAO06, LIU08, KAT08, AST08] performed steady-state experiments and used a thin layer technique by depositing a thin heat foil on the cooled surface, which delivers a known and uniform Joule heat flux during experiments.

Thin layer technique enables to assume that a 1-D resolution (i.e. heat conduction model in the direction orthogonal to the measurement surface) is sufficient, neglecting lateral conduction inside disk. To our knowledge, the use of this technique associated with a multidimensional resolution was only made by [START_REF] Liu | Application of pressure and temperature sensitive paints for study of heat transfer to a circular impinging air jet[END_REF]. The opposite face of the heat foil is usually insulated in order to decrease heat losses by the backside of the disk, except for [START_REF] Lytle | Air jet impingement heat transfer at low nozzle-plate spacings[END_REF] where convective losses occurred and were estimated and taken into account with available correlations. The Joule heat flux was also corrected via taking into consideration an estimated radiative heat flux on the cooled surface in [LYT94, CAR96, LEE02, GAO06] or for both faces in [START_REF] Katti | Experimental study and theoretical analysis of local heat transfer distribution between smooth flat surface and impinging air jet from a circular straight pipe nozzle[END_REF]. No corrections were used in [SAN97, GUE05], however related errors involved due to this were estimated to be under 5%. Owen [START_REF] Owen | Heat transfer from an air-cooled rotating disk[END_REF] and Pellé & Harmand Radiative heat flux was then estimated and subtracted to compute the pure convective heat flux in [START_REF] Pellé | Heat transfer measurements in an opened rotor-stator system air-gap[END_REF], while [START_REF] Owen | Heat transfer from an air-cooled rotating disk[END_REF] assumed the radiative heat flux to be negligible.

Transient techniques were also used in [YAN97, SANI98, AXC01, KAK09] using an analytical solution of the one-dimensional conduction with known initial and boundary conditions in order to restore the local heat transfer coefficient. In the works of [YAN97, SANI98], the disk was heated in an oven before experiments until reaching the stationary thermal state, while in [START_REF] Axcell | Convection to rotating disks with rough surfaces in the presence of an axial flow[END_REF] or [START_REF] Kakade | Accurate heat transfer measurements using thermochromic liquid crystal. Part 2: Application to a rotating disc[END_REF] the surrounding fluid is heated. Via recording of the wall temperature cooling/heating history in a short time period (usually up to 1 min) the heat flux was restored using the aforementioned one-dimensional heat conduction solution.

The measurement of the cooled/heated surface temperature can be achieved in 3 main ways.

Few studies used the Thermochromic Liquid Crystals (TLC) [YAN97, SANI98, AXC01 restriction is that the shape of the initial temperature non-uniformity over the disk surface must hold during the time of measurements. According to [START_REF] Shevchuk | Unsteady conjugate laminar heat transfer of a rotating non-uniformly heated disk: application to the transient experimental technique[END_REF][START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF], this is fully fulfilled for a disk made of Plexiglas  in view of the applicability of the semi-infinite-slab approach for a wide range of both radially increasing and decreasing temperature distributions over the disk surface (including the isothermal disk). For a disk made of highly conductive material, like e.g. Aluminum, the transient experimental technique is inapplicable for all kinds of the initial surface temperature non-uniformity, except for the case of an isothermal disk [START_REF] Shevchuk | Unsteady conjugate laminar heat transfer of a rotating non-uniformly heated disk: application to the transient experimental technique[END_REF][START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF].

Thermocouples were used in [START_REF] Owen | Heat transfer from an air-cooled rotating disk[END_REF][START_REF] San | Impingement cooling of a confined circular air jet[END_REF], where the disk was made of a relatively high thermal conductivity material, in order to assume that the measured temperature inside the disk was the same as the surface temperature. The thermocouples must be also placed with a thermally-well conductive paste in order to avoid contact resistance. With the use of thermocouples on rotating systems, it is very difficult to get a sufficient spatial resolution. A rotating mercury-ring collector must be placed to transmit the signal from the rotating disk to the stationary acquisition system, with the number of channels being limited. Techniques involving the thermocouples and partially the TLC are intrusive, because they can affect the thermal properties of the cooled/heated body. The adiabatic wall temperature is the temperature that would be the surface temperature without any heat transfer, which would be also the fluid temperature near the wall. It takes into account possible overheating due to viscous friction at the wall which depends on the fluid dynamics [START_REF] Kakade | Accurate heat transfer measurements using thermochromic liquid crystal. Part 2: Application to a rotating disc[END_REF]. Usually, it is written as:

with being the temperature of the fluid entrained by the surface boundary layer and being a characteristic velocity representative for the considered flow, and f being an increasing function with V. For a laminar flow on a rotating disk, the adiabatic temperature can be easily calculated using the theory developed in the book by Owen [START_REF] Owen | Flow and Heat Transfer in Rotating-Disc Systems[END_REF], so that f(V²)=Pr 1/3 (ωr)². Cardone et al. [START_REF] Cardone | Heat Transfer Measurements on a Rotating Disk[END_REF] have measured the adiabatic wall temperature for high rotating speed up to 21000 rpm. For those velocities, their results are in agreement with the theory by Owen [START_REF] Owen | Flow and Heat Transfer in Rotating-Disc Systems[END_REF] and exhibit a temperature rise up to 10K only due to viscous dissipation. A method for determination of T ad was also proposed by Newton [START_REF] Newton | Transient heat transfer measurements using thermochromic liquid crystal. Part 1: An improved technique[END_REF]. In most papers, tangential velocities (ωr) are not so high than in [START_REF] Cardone | Heat Transfer Measurements on a Rotating Disk[END_REF], so that the authors consider that the error introduced by considering T ad =T r is negligible in comparison with the difficulty of its measurement. So, T r is often chosen as the characteristic (reference) temperature of the fluid in which the experiment is made: ambient (or jet) air temperature in the case where the measurement surface is heated [LYT94, YAN97, SANI98, LEE02, PEL07, LIU08, PEL09, PEL11] or fluid temperature when the fluid is heated in place of the disk [START_REF] Axcell | Convection to rotating disks with rough surfaces in the presence of an axial flow[END_REF]. According to [START_REF] Cardone | Heat Transfer Measurements on a Rotating Disk[END_REF], neglecting the influence of the viscous dissipation can lead to some errors in the determination of the Nusselt numbers, namely at outer radii.

The choice of the reference temperature is a very difficult point, and in order to compare the different studies of heat transfer the reference temperature measurement must be presented and justified.

Flows

Hot wire anemometry. Hot wire anemometry is a well-known method used to measure velocity profiles and Reynolds stresses in a complex flow. However, it is an intrusive method used by several authors in rotating [OWE74, ITO98, DJA98, DJA99, DEB10] or jet impingement [START_REF] Itoh | An experimental study of the radial wall jet on a rotating disk[END_REF][START_REF] Guerra | The near wall behavior of an impinging jet[END_REF] configurations.

In [START_REF] Djaoui | Heat transfer in a rotor-stator system with a radial inflow[END_REF][START_REF] Djaoui | Mise en évidence expérimentale de la sensibilité de l'écoulement de type rotor-stator aux effets de bord[END_REF], radial and azimuthal velocity components u and v as well as the turbulent correlations u'², v'², u'v' were measured. The technique involved using a special probe made of three wires, 2.5 µm in diameter, all situated on the same plane and having an angular spacing equal to 120°. In [START_REF] Debuchy | An Analytical Modeling of the Central Core Flow in a Rotor-Stator System With Several Preswirl Conditions[END_REF], the authors used the same technique to measure u, v and v'², in an opened rotor-stator cavity with an inward radial flow.

In [START_REF] Itoh | An experimental study of the radial wall jet on a rotating disk[END_REF], two hot wires were used: one had a wire mounted normal to its axis of rotation and capable of measuring the components u, v and u'², v'², u'v'; the other had a wire inclined at an angle of 50° to its rotation axis and was used for measurements of w'², u'w' and v'w'.

The determination of the mean flow field in the case of an impinging jet was made with the use of 10000 samples in [START_REF] Guerra | The near wall behavior of an impinging jet[END_REF], and each velocity profile was established with 100 measurement points. Authors also indicated that the measurement becomes independent of the number of samples for more than 5000 samples. However, it usually provides less spatial and temporal resolution than LDA/LDV techniques, except when using high speed laser and cameras. [START_REF] O'donovan | PIV Measurements and convective heat transfer of an impinging jet[END_REF] used PIV to have an idea of the main flow field before using LDA/LDV to closely look at the jet impingement area. In these experiments, the filed size was 1280×1024 pixels and minimum time between the frames was 200ns.

Laser

Boutarfa [START_REF] Boutarfa | Local convective heat transfer for laminar and turbulent flow in a rotorstator system[END_REF] measured the mean flow field in different planes between a rotor facing a stator using a water spray added to the air flow. The dimensions of the field covered by the objective were 114×110 mm 2 and the thickness of the laser plane was about 5mm. The cross-correlation between two images was carried out on a 850×150 pixels zone. The time interval ∆t between images, regulated as a function of the air velocity, ranged between 30 and 400 ns.

On a co-rotating disk configuration, Wu [START_REF] Wu | A PIV study of co-rotating disks flow in a fixed cylindrical enclosure[END_REF] used PIV with polycrystalline particles (diameter of about 30µm) with a 1.5mm thickness laser plane and a maximal acquisition frequency of 30Hz.

In LDV/LDA or PIV, the authors interested in flows in confined configurations (such as rotorstator etc.) [START_REF] Boutarfa | Local convective heat transfer for laminar and turbulent flow in a rotorstator system[END_REF][START_REF] Wu | A PIV study of co-rotating disks flow in a fixed cylindrical enclosure[END_REF] needed to use Plexiglas windows for optical access, whose thickness is usually under 10mm.

Basic

Numerical methodologies for flow and heat transfer modelling

Despite their relative geometric simplicity, enclosed rotating disk flows contain a complex physics, which makes their modelling a very challenging task for numerical methods. The flow problem presents indeed several complexities, such as the high rotation rates, confinement effects, the coexistence of laminar, transitional and highly turbulent flow regions, three-dimensional precessing vortical structures, very thin 3D turbulent boundary layers along the disks, strong curvature of the streamlines, recirculation zones etc. When an axisymmetric jet is superimposed on this flow and impinges onto the rotating disk in a confined rotor-stator system, the flow pattern becomes even more complex with the interaction between the jet and the secondary rotor-stator flow. Numerically, there are also some additional technical constraints due to firstly the singularity in the Navier-Stokes equations when dealing with cylindrical coordinates in cylindrical (and not annular) geometries and secondly the choice of the boundary conditions at the outlet to ensure the mass conservation. Thus, most of the studies up to now have been dedicated to either enclosed rotor-stator flows without jet or impinging jet flows on a flat plate without rotation and more rarely to both problems at the same time.

Direct numerical simulations (DNS)

To the best of our Re Ω =1.1×10 5 ). These DNS were based on the same pseudo-spectral method with Chebychev polynomials in the non homogeneous directions and Fourier series in the tangential direction, which provides very highly accurate results but in relatively simple enclosed annular geometries. Moreover, they require several millions of mesh points and excessive calculation times preventing from performing a parametric study in such a configuration. We can cite also Lygren and Andersson [START_REF] Lygren | Turbulent flow between a rotating and a stationary disk[END_REF], who performed a DNS of the turbulent flow (Re Ω =4×10 5 ) in an angular section of an unshrouded rotor-stator cavity assuming a periodicity of the flow in the tangential and radial directions.

Axisymmetric Reynolds Averaged Navier-Stokes (RANS) computations

In the turbulent regime, most of the numerical studies have assumed the flow pattern as being axisymmetric and steady. During the 1980s, flow modelling was based on axisymmetric RANS computations based on the isotropic linear eddy-viscosity hypothesis.

Chew [START_REF] Chew | Prediction of flow in rotating cavity with radial outflow using a mixing-length turbulence model[END_REF] Prandtl number ranging between 0.91 and 1. For all five models, they noticed a poor prediction of the velocity field in the jet region and attributed it to an overestimation of the free-jet spreading before its impingement on the rotor. The two RSM models (an ellipticblending RSM and the RSM using scalable wall functions) did not improve significantly the results of the k-ω SST, φ-f and ζ-f models, unless if we consider the radial distribution of the radial fluctuating velocity close to the disk. They concluded by the need for additional modelling development to sensitize RANS models to the effect of rotation, which has been done only by Elena and Schiestel [START_REF] Elena | Turbulence modeling of rotating confined flows[END_REF] for a Reynolds Stress Model.

Three-dimensional computations in the turbulent regime

Though the geometry in this kind of problem is strictly axisymmetric, some numerical simulations [START_REF] Tucker | Temporal behaviour of flow in rotating cavities[END_REF][START_REF] Craft | Some swirling flow challenges for turbulent CFD[END_REF] and experimental flow visualizations [START_REF] Czarny | Precessing vortex structures in turbulent flow within rotor-stator disc cavities[END_REF] have revealed the existence of precessing large scale vortical structures even in the turbulent regime. Owen [START_REF] Owen | Flow and heat transfer in rotating disc systems[END_REF] has drawn attention to difficulties in predicting the flow in rotating disk arrangements using bidimensional steady calculations and speculated that the cause might be related to the formation of these vortices. Thus, Craft et al. [START_REF] Craft | Some swirling flow challenges for turbulent CFD[END_REF] developed a 3D unsteady version of a low-Reynolds number k-ε model and applied it to a turbulent rotorstator flow in an enclosed cavity. Their results confirmed the "S-shaped vortex" regime but this strategy required a very fine mesh across the sublayers and also prohibitive time integration preventing from further calculations.

In the 2000's, Large Eddy Simulations (LES) appeared to be the appropriate level of modelling in such complex flows including 3D effects. The LES results of Séverac et al. [START_REF] Séverac | Large eddy simulation and measurements of turbulent enclosed rotor-stator flows[END_REF] were extended in the non isothermal case by Poncet and Serre [START_REF] Poncet | High-order LES of turbulent heat transfer in a rotor-stator cavity[END_REF] using the Boussinesq approximation for Rayleigh numbers up to 10 8 . These authors showed in particular that the turbulent Prandtl number is not constant within the boundary layers and that the classical value of 0.9 used by all the RANS models is valid only at a given location within the boundary layers. Tuliska-Snitzko et al. [START_REF] Tuliszka-Sznitko | LES of the non-isothermal transitional flow in rotating cavity[END_REF] developed an algorithm for LES, with subgrid modelling based on the spectral Chebyshev-Fourier approximation. They used a dynamic Smagorinsky eddy viscosity model in which the Smagorinsky coefficient is averaged along the particle trajectories. It has the advantage that it is applicable also to inhomegeneous flows and that it allows performing computations for higher Reynolds number flows, which are strongly inhomogeneous and anisotropic due to the combined effects of rotation and confinement. It was applied to 3D turbulent and transitional non-isothermal flows within a rotor-stator cavity using the Boussinesq model.

Heat transfer modeling

Since the early 1980s, a particular effort has been done to develop innovative RANS models capable of predicting accurately the mean and turbulent flow fields under isothermal conditions. On the contrary, heat transfers have been addressed only with relatively simple approaches. The first reason may be due to the lack of reliable experimental databases, the acquisition of heat transfer information being rather complex and expensive in such systems.

Thus, it has slow down the development of specific turbulence models for heat transfer predictions. The second reason is that very simple approaches, such as the one used by

Geometries and results

Free disk

Rotating disk in quiescent air is a basic configuration presented on

1.0 -1 -2 -3 -4 θ F, G, -H, ζ -5 -6 -7 -8 -9 -10 -11
1 -F, 2 -G, 3 --H, 4 - ) /( ) ( ∞ ∞ - - = θ T T T T w (subscript "∞"
relates to the flow outside of the boundary layer) for an isothermal disk (T w =const), Pr=0.71. Experiments: The nature of the flow over a rotating disk is strongly linked with the local Reynolds number Re r =Ωr²/ν. The flow becomes unstable when the rotational velocity and/or radius over the disk are high enough. For the entire disk, this means there is always a laminar flow region around the rotation axis; in other words, the flow cannot be turbulent over the entire disk.

5 -F [ITO94], 6 -F [ELK97], 7 -F [LIN96], 8 -G [ITO94], 9 -G
As pointed out in the review made in [START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF], the range of the local Reynolds number where the flow undergoes transition from laminar to turbulent regime is not wellestablished and varies a lot depending on the experimental apparatus and measurement techniques used (see also in [START_REF] Northrop | Heat transfer measurements in rotating-disc systems part 1: The free disc[END_REF]). Different authors used different criteria for determining the critical value of the Reynolds number. According to the review [START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF],

the lowest value of the Reynolds number for the beginning of transition is 1.8×10 5 [START_REF] Tien | Heat and Mass Transfer from rotating cones[END_REF] while the highest one for the end of transition is 3.

The coefficients a and c depend on the flow regime, Prandtl number and the radial temperature distribution on the disk, while exponents b and d depend, in general, on the flow regime (in other words, the range of the Reynolds numbers).

Laminar flow

When the local Reynolds number Re r is lower than 1.8×10 5 -3.6×10 5 [SHE09a], the flow is laminar. In this case, the kinematic boundary layer has a constant thickness estimated from the self-similar solution as [START_REF] Owen | Flow and Heat Transfer in Rotating-Disc Systems[END_REF]. As a result, the local convective heat transfer coefficient is independent of the radial position and increases with the rotational speed as the boundary layer becomes thinner. The local Nusselt number given by Eq. ( 10) varies also with r, with the exponent b being equal to 0.5.

The mean Nusselt number defined by Eq. ( 11) is constant, with a=c and d=0.5. In other words, for laminar flow, the mean Nusselt number is by definition equal to the local Nusselt number calculated at the radius where the laminar flow ends (in case where a turbulent flow also develops) or at the external radius (in case of a laminar flow over the entire disk), denoted :

(13)

In the case of an isothermal surface, the experimentally/theoretically determined coefficient a varies for air depending on the author: ]. Differences can be attributed to inaccuracies of measurement and also to possible differences in surface roughness [START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF].

[MIL52], [HAR59, POP75], a≈0.32 [ELK97], [WAG48], [OEL78], [KRE68], a=0.36 [COB56], [GOL35], [ DOR63
The exact self-similar solution of the boundary layer equations yields for [START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF].

In the case of a radial distribution of temperature which can be interpolated by a law expressed as , Dorfman [START_REF] Dorfman | Hydrodynamic Resistance and Heat Loss from Rotating Solids[END_REF] and Owen & Rogers [OWE89] obtained a semi-empirical solution, which for air at Pr=0.72, writes:

As shown by [START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF] (see also in Table 1), this equation agrees quite well with the selfsimilar solution for n≥2, however deviates from it for n<2, with these deviations increasing for the decreasing n, and being already noticeable for an isothermal surface (n=0) and becoming very significant for negative values of n.

An alternative approximate solution was obtained by [START_REF] Shevchuk | Effect of the Wall Temperature on Laminar Heat Transfer in a Rotating Disk: An Approximate Analytical Solution[END_REF][START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF] for a wide range of the Prandtl number; for Pr=0.72, this solutions looks as:

) 2 /( 002 . 2 3486 . 0 4435 . 0 n a + + = (15)
The values of the coefficient a for Eq. (10) computed by Eq. ( 14) (column 3, Table 1) agree very well with the exact self-similar solution (column 2, Table 1) over the entire range of the values of n= [-1.5; 4].

n
Exact solution [START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF] Equation ( 14) [START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF] Equation ( 13 [OEL78] and [START_REF] Hartnett | Heat Transfet ffrom a non-isothermal disk rotating in still air[END_REF] obtained values of the coefficient a also presented in Table 1. In general, all values for n = 0 are obviously concordant with the experimental data for an isothermal surface mentioned above.

Fundamental peculiarities of the effects of the disk permeability in laminar flow being of perspective interest for the rotor/stator flows with a perforated stator were studied in [START_REF] Turkyilmazoglu | Wall stretching in magnetohydrodynamics rotating flows in inertial and rotating frames[END_REF][START_REF] Turkyilmazoglu | Thermal radiation effects on the time-dependent MHD permeable flow having variable viscosity[END_REF].

Transition from laminar to turbulent flow

The transition of the flow from laminar to turbulent flow regime is very dependent on the experimental conditions as demonstrated in [START_REF] Northrop | Heat transfer measurements in rotating-disc systems part 1: The free disc[END_REF][START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF]. Several empirical equations for the local heat/mass transfer for transitional flow are documented in [START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF]. Table 2 represents empirical equations for heat transfer in air.

Author Correlation Lowest Re Highest Re [POP75] [ELK97] [CAR96]

Table 2. Local Nusselt numbers over a rotating disk for transitional flows.

Turbulent flow

For a local Reynolds number higher than [START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF], the flow over the disk becomes turbulent. Looking at the entire rotating disk, turbulent flow region is located in the outer peripheral area (Fig. 4). The boundary layer thickness in turbulent flow can be estimated from the equation derived using the integral method by von Kármán [START_REF] Kármán | Uber laminar und turbulente Reibung[END_REF]. More detailed information on the relations for different parameters of the boundary layer over a rotating disk obtained using an integral method is documented in the monograph [START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF].

In turbulent flow, the local heat transfer coefficient, unlike that in the laminar flow, For an isothermal disk the values of these coefficients in accordance with Eqs. ( 15) and ( 16) For the isothermal disk (n=0), this solution yields a=0.0187, whilst for q w =const it results in a=0.0169; unlike the Dorfman's solution, Eq. ( 15 As mentioned above, both exact and approximate analytical solutions of the thermal boundary equations in form of the Eqs. (10), (11) were possible for the boundary condition . A new type of the wall boundary condition providing a possibility of an analytical solution of the thermal boundary layer equation for a free rotating disk was revealed and discussed in details in [START_REF] Shevchuk | A new type of the boundary condition allowing analytical solution of the thermal boundary layer equation[END_REF][START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF]. This solution represents a twoparameter family of curves that allows modeling wall temperature profiles with maxima and minima and values of d noticeably different from 0.8. As mentioned in [START_REF] Shevchuk | A new type of the boundary condition allowing analytical solution of the thermal boundary layer equation[END_REF][START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF], this solution provides much better accuracy of the agreement with the experiments [START_REF] Northrop | Heat transfer measurements in rotating-disc systems part 1: The free disc[END_REF] than the Eqs. ( 10), ( 15) and ( 16), which significantly strengthens the analytical toolkit for modeling heat transfer of a rotating disk.

are

Disk in cross flow

Rotating disks in quiescent air were elucidated very detailed in the literature. However, the configuration with a cross flow parallel to the plane of rotation was paid far less attention.

Stationary disk

In the case of a stationary disk in air crossflow, aus der Wiesche [START_REF] Der Wiesche | Heat transfer from a rotating disk in a parallel air crossflow[END_REF] between its results obtained by LES and those for the flat plate, characterized by the symmetric Nusselt number (Fig. 6a). Locally, a high convective heat transfer occurs at the beginning of the disk, in the developing boundary layer, and decreases along the disk diameter. The flow is laminar up to a critical value of , for which the transition into turbulence begins, whereas the turbulent regime is fully reached for . The LES data are correlated by Eq. ( 18) for the laminar case, and by Eq. ( 19) for the turbulent regime.

(

) (20) 19
Dennis et al. [START_REF] Dennis | The heat transfer from a rotating disc in an air crossflow[END_REF], based on their original experimental data for turbulent flow, developed an empirical correlation which yields significantly higher mean Nusselt numbers than those suggested by Eq. ( 19):

(21

)
The theoretical solution presented also by Dennis et al. [START_REF] Dennis | The heat transfer from a rotating disc in an air crossflow[END_REF] suggested the constant in Eq. ( 20) to be 0.027 rather than 0.036.

Measurements by He et al. [START_REF] He | Convection heat and mass transfer from a disk[END_REF] were described by the empirical equation:

(

which is concordant with the aforementioned theoretical solution of Dennis et al. [START_REF] Dennis | The heat transfer from a rotating disc in an air crossflow[END_REF] and still suggests significantly higher Nusselt number than Eq. ( 19).

As pointed out in the review [START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF] analysing the given above equation, the reason for the Eq. ( 19) to suggest much lower values of the Nusselt numbers than those measured by [START_REF] Dennis | The heat transfer from a rotating disc in an air crossflow[END_REF][START_REF] He | Convection heat and mass transfer from a disk[END_REF] is that the disk simulated by aus der Wiesche [WIE07] had a zero thickness, which thus prevented formation of the flow separation and reattachment region close to the leading edge of the disk and subsequent significant heat transfer enhancement.

In the recent experimental works [START_REF] Der Wiesche | Heat Transfer from Rotating Discs with Finite Thickness Subjected to Outer Air Streams[END_REF][START_REF] Trinkl | Experimental study of the convective heat transfer from a rotating disc subjected to forced air streams[END_REF], it was also shown that the LES results by [START_REF] Der Wiesche | Heat transfer from a rotating disk in a parallel air crossflow[END_REF] underestimate the real heat transfer for such configuration. Authors also point out the influence of the thickness of the disk when positioned in a cross flow. Indeed, the disk thickness affects the size of the separation zone that can strongly influence the local heat transfer in the part of the disk close to its leading edge.

Rotating disk

In the case of a rotating disk in air cross-flow, few studies are known that deal with the combined effect of rotation and air cross-flow on convective heat transfer. The most It was shown that the air cross flow has the dominant influence for the mean heat transfer while . In fact, the mean Nusselt number remains constant at the value obtained without rotation. In the second zone, corresponding to , there is a significant increase in the mean Nusselt number due to some instabilities, created between rotational and plane flows, which tend to increase the convective heat transfer. Based on the LES study, the author proposes different correlations in order to predict convective transfer on the disk.

By comparison with the mean Nusselt number without rotation, two correlations are proposed:

for :

(23

)
and for :

(

They can also be calculated with:

(25)

in the case where and , or by: (

in the case where and .

Being useful to understand the nature of mutual interaction of the rotation and cross-flow effects, Eqs. ( 22)-( 25) obtained for a zero thickness disk can not be applied for computation of heat transfer over a disk of a finite thickness. To the best of our knowledge, there is only one previous study about the mean convective exchanges around a rotating disk mounted on a cylinder in airflow [START_REF] Watel | Etude des échanges convectifs sur un cylindre aileté en rotation soumis à un courant d'air parallèle aux ailetés[END_REF]. In the case of a disk of 21-mm height and 1-mm wide mounted on a 58-mm diameter cylinder, Watel

[WAT97] developed a general correlation of the mean Nusselt number depending on Re U and Re Ω :

(27)
From this equation, different domains of influence (rotation, air cross-flow, combined rotation/air cross-flow) on convective heat transfer were determined.

Jet on stationary and rotating disks 2.3.1. Fluid flow

Stationary impinged surface. Figure 8 shows the flow which develops in a jet on a stationary plate where 3 main areas can be identified.

In the first area just after the pipe exit, a potential core and a free jet region can develops depending on the ratio e/D. The free jet region is a mixing region between the fluid in the jet and the fluid outside the jet so that vortices develop as shown by [START_REF] Angioletti | Simultaneous visualization of flow field and evaluation of local heat transfer by transitional impinging jets[END_REF] for e/D=4.5. The vortices are believed to not develop when e/D<1. For large enough aspect ratios e/D, the centreline velocity and turbulence intensity keeps quite constant and equal to the exit velocity up to 2.5 diameters from the nozzle [START_REF] O'donovan | PIV Measurements and convective heat transfer of an impinging jet[END_REF]. and , [START_REF] Minagawa | Development of turbulent impinging jet on a rotating disk[END_REF] showed that the flow is unaffected by the rotation if r/D is lower than a critical dimensionless radius, which increases as the rotational speed decreases. This phenomenon can be seen in Figure 9. The authors used a parameter , which described the ratio of the centrifugal forces and strength of the impingement. [START_REF] Minagawa | Development of turbulent impinging jet on a rotating disk[END_REF] identified three areas over the rotating disk: at α<4.7, the flow is unaffected by the rotation, for 4.7<α<9.2 the flow is affected by rotation and by the impinging jet, and finally for α>9.2, the flow is only affected by the rotation.

Heat transfer

Stationary impinged surface. Viskanta [VIS93], in his review paper, summarized the studies concerning heat transfer due to single jet on stationary surfaces performed before 1993 and did not focus on rotating systems. Several authors showed that convective heat transfer near the impingement region is strongly influenced by the aspect ratio e/D and jet Reynolds number Re j . In the region r/D<5, one or two peaks in the local Nusselt number distribution were observed. When e/D>4, the impingement between the jet and the plate is situated at the end of the potential core of the jet, leading to a maximal heat transfer at the stagnation point which monotonically decreases with the radius. For lower aspect ratios, two peaks can eventually be observed in the radial profile of local Nusselt numbers, which are not aligned with the center of the jet. The first peak is located at r/D=0.5 while the position of the secondary peak varies (as shown on Figure 10) according to the relation [START_REF] Lytle | Air jet impingement heat transfer at low nozzle-plate spacings[END_REF] (28) over the range: and .

The work from Lytle and Webb [LYT94] also clearly showed that this secondary peak is due to the transition from laminar to turbulent boundary layers which occurs earlier when the aspect ratio decreases. [START_REF] O'donovan | PIV Measurements and convective heat transfer of an impinging jet[END_REF] explains this secondary peak by the fact the boundary layer thickness reaches its minimum and that it is mixed with the free jet shear layer which increase turbulence levels at this point. Results are depicted in Figure 11. [START_REF] Chen | Heat (mass) transfer between an impinging jet and a rotating disk[END_REF] concluded that the best jet position is the center of the rotating disk, which is the position that maximizes heat transfer.

[AST08] studied jet impingement on a rotating disk for high aspect ratios (3<e/D<42.5) and rotational Reynolds numbers under the transitional one. For these conditions of laminar flow, authors correlate the stagnation Nusselt number with the relations:

(30) when is under 1, or

when the jet effect is assumed to be dominating ( ).

Under this last condition, authors also provided a correlation for the end of the jet dominated regime that pinpoints the radius over the disk surface where results start varying with the variation of the rotational speed:

Unfortunately, the aspect ratio (e /D) effect is not included in this correlation.

Though empirical correlations for the local Nusselt numbers were not developed in [START_REF] Saniei | Local Heat Transfer Characteristics of a Rotating Disk under Jet Impingement Cooling[END_REF][START_REF] Şara | Electrochemical mass transfer between an impinging jet and a rotating disk in a confined system[END_REF], surface distributions were depicted in these works.

Exact solution for the stagnation point

As said above, the best jet position is the center of the rotating disk, because this positioning maximizes the surface heat transfer. Shevchuk et al. [START_REF] Shevchuk | Impingement heat transfer over a rotating disk: integral method[END_REF][START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF] following commonly accepted theoretical model assumed the boundary layer in the stagnation region over a rotating disk subjected to uniform outer flow orthogonal to the disk to be fully laminar, with the radial velocity at the outer boundary of the boundary layer increasing linearly as

. The constant a according to the known exact solution equals to , where R is the outer radius of the disk. Exact self-similar solution of this problem is tabulated in As shown in [START_REF] Shevchuk | Impingement heat transfer over a rotating disk: integral method[END_REF][START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF], at κ=a / ω >1.5 heat transfer is dominated only by peculiarities of the impinging jet. Detailed comparisons with the cases of the uniform orthogonal flow onto a rotating disk and co-axial impingement of round jets of the diameter much smaller than the disk's diameter performed in [START_REF] Shevchuk | Impingement heat transfer over a rotating disk: integral method[END_REF][START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF] showed good efficiency of the above solution in simulations of such geometrical configurations.

Rotor-Stator without jet

In this section, the review is restricted to the smooth disk case in small aspect ratio cavities. High aspect ratio may indeed favour the development of the vortex breakdown phenomenon, which is not discussed here.

We consider the flow between a rotating disk (the rotor) facing a stationary one (the stator) as it can be seen in Fig. 12. Both disks have the same outer radius R and are separated by an axial gap denoted e in the following. The cavity may be enclosed or not by a fixed external cylinder (the shroud). The influence of a superimposed throughflow in an opened system will be discussed in the next section. An inner rotating cylinder (the hub) may be attached (annular cavity) or not (cylindrical cavity) to the rotor. Thus, both the hub and the rotor rotate at the same angular rotation rate Ω. The main effect of finite radial extension of the disk and most of all of the inner and outer cylinders is that the boundary conditions are not compatible with the self-similarity solutions of Ekman [START_REF] Ekman | Om Jordrotationens inverkan pa vindstrømmar I hafvet[END_REF], Von Kármán [START_REF] Kármán | Uber laminar und turbulente Reibung[END_REF] or Bödewadt [START_REF] Bödewadt | Die Drehströmung über festem Grunde[END_REF] though there may be qualitative resemblance far from the end walls. The reader can refer to the detailed monograph of Owen and Rogers [START_REF] Owen | Flow and Heat Transfer in Rotating-Disc Systems[END_REF] for an overview of the experimental, theoretical and numerical works until 1989 including heat transfer effects and to the recent review paper of Launder et al. [START_REF] Launder | Laminar, Transitional, and Turbulent Flows in Rotor-Stator Cavities[END_REF] for a detailed review of isothermal rotor-stator flows in enclosed cavities.

Base flows

The first major experimental and theoretical study of enclosed rotor-stator flows was carried out by Daily & Nece [START_REF] Daily | Chamber dimension effects on induced flow and frictional resistance of enclosed rotating disks[END_REF], who identified the four flow regimes shown in Fig. 13.

The base flow is indeed governed by two global parameters: the aspect ratio of the cavity G=e/R and the rotational Reynolds number Re Ω =ΩR 2 /ν. The flow is then either laminar In regimes II and IV, the secondary flow structure in the meridian plane is known as the Batchelor flow structure since the similarity analysis of Batchelor [START_REF] Batchelor | Note on a class of solutions of the Navier-Stokes equations representing steady rotationally-symmetric flow[END_REF], who solved the system of differential equations relating to the stationary axisymmetric flow between two infinite disks. At a given radius r, the velocity profiles are divided into three distinct regions:

two boundary layers develop on each disk separated by a non viscous core region (Figure 14a to c). It can be seen as the connection of the Bödewadt and Ekman flow problems over single infinite disks. Thus, by analogy with these prototype flows, the boundary layers along the rotor and the stator are known as the Ekman and Bödewadt layers respectively. The geostrophic core is characterized by quasi zero axial and radial velocities and a constant tangential one. In its normalized form, the tangential velocity in the core of the flow is known in the literature as either the core-swirl ratio β or the entrainment coefficient K (=V θ /(Ωr), defined as the ratio between the tangential velocity of the fluid in the core and that of the disk as the same radius). As a consequence of the Taylor-Proudman theorem, there is also no axial gradient in that region. Thus, one can readily link the core-swirl ratio β to the radial pressure gradient at the rotor, which is crucial for designing thrust bearings. In both regimes II and IV, the fluid is pumped centrifugally outwards along the rotor and is deflected in the axial direction after impingement on the external cylinder (Fig. 15). After a second impingement on the stator, it flows radially inwards along the stator, by conservation of mass, before turning along the hub and being centrifuged again by the rotating disk. This secondary flow picture is modified in the absence of the hub [START_REF] Peres | A 3D pseudo-spectral method for cylindrical coordinates. Application to the simulations of flows in rotating cavities[END_REF]. The main difference between these two flow regimes mainly consist in the values of the coefficient β as we will see in the following. to the inner and outer cylinders (Fig. 16b). This picture is not significantly modified in regime III [START_REF] Haddadi | Turbulence Modeling of Torsional Couette Flows[END_REF]. The empirical correlation for regime I fully matches with the theoretical one of Owen and Rogers [START_REF] Owen | Flow and Heat Transfer in Rotating-Disc Systems[END_REF]. Note that for regime II and in contrast with regime I, the moment coefficient increases then when G increases. Soo [START_REF] Soo | Laminar flow over an enclosed rotating disk[END_REF] provides a theoretical correlation for the moment coefficient in the turbulent regime with merged boundary layers (regime III) with a dependence on G with a power -1/4 and a constant equal to 0.0308.

From the moment coefficient correlations, we can deduce for example, the aspect ratio for which the transition between regimes III and IV is observed:

3/16 0.2112 Re G - Ω = (42)
Owen [START_REF] Owen | Fluid flow and heat transfer in rotating disk systems[END_REF] proposed two empirical laws given the minimum and maximum values for which the boundary layers are merged and separated, respectively, in the whole cavity:

G for G G for G for G π - Ω Ω Ω   > <     > × < < > × >

Instability patterns (regimes I and II)

During the transition process to turbulence in an enclosed rotor-stator cavity, a large variety of patterns can be observed according to the combination (Re Ω , G) as mapped by depending on the value of the aspect ratio and rotation rate. Schouveiler et al. [START_REF] Schouveiler | Stability of a traveling roll system in a rotating disk flow[END_REF] showed that the wave-number selection process of this secondary instability can be regarded as resulting from an Eckhaus instability that selects the number of spiral arms. The influence of a central hub on the characteristic of these two bifurcations is weak as shown by [START_REF] Poncet | Revisiting the two first instabilities of the flow in an annular rotor-stator cavity[END_REF].

The rotor layer is less well documented mainly because measurements are technically more difficult on that side. Moreover, instabilities occur at higher rotation rates, which make computations more computationally expensive due to resolution requirements. For annular cavities, Serre et al. [START_REF] Serre | Annular and spiral patterns between a rotating and a stationary disk[END_REF] showed that the instabilities developed in the Bödewadt layer are convected along the hub and induce disturbances in the rotor layer, triggering axisymmetric and then three-dimensional convective modes characterized by a negative angle ε with the tangential direction (-7.5 ≤ ε ≤ -20°).

The

Turbulent flow regimes (III and IV)

The flow in regimes II or IV exhibits a Batchelor-like structure with a rotating inviscid core separated by the Ekman layer on the rotor and the Bödewadt layer on the stator. The fluid flows outward along the rotor with a mean tangential velocity ranging from Ωr to βΩr. As there is neither radial nor axial flow in the core region, by conservation of mass, a radial inflow is created along the stator with a tangential velocity ranging from 0 on the disk and βΩr. One main difference between regimes II and IV stands in the value of the core-swirl ratio β. In regime IV, this coefficient is constant with the local radius r, as far as the two boundary layers are turbulent in the whole cavity. In regime II, it strongly varies with r.

Stépanoff [START_REF] Stépanoff | Pompes centrifuges et pompes hélices[END_REF] was the first to determine the core-swirl ratio β of the turbulent flow in a rotor-stator cavity. He suggested a value of β equal to 0. Moreover, the levels of the normal stresses are quite comparable, which is not the case for Batchelor flows, where the axial normal component of the Reynolds stress tensor is generally negligible compared to the other normal components [START_REF] Poncet | Turbulent Rotating Disk Flow with Inward Throughflow[END_REF].

Heat transfer

Nikitenko [START_REF] Nikitenko | Experimental Investigation of Heat Exchange of a Disk and a Screen[END_REF] conducted experiments with an air-filled enclosed rotor-stator system where both disks were isothermal for a wide range of the aspect ratio 0.018≤G≤0.085 and of the Reynolds number Re Ω ≤10 6 . He found no effect of the aspect ratio and correlated his results for the rotor side according to Shchukin and Olimpiev [START_REF] Shchukin | Heat transfer of disc rotating in a housing with transitional and turbulent boundary layers[END_REF] measured the mean Nusselt number in a closed rotor-stator cavity such as G=0.0646. For a radial temperature distribution on the rotor varying as r 0.25 , their results for the turbulent regime were correlated by , which is 11% higher than the free disk value. Many later works will show that both the local and averaged Nusselt numbers depend on the aspect ratio of the system.

In regime I corresponding to a laminar flow with merged boundary layers, Owen and Rogers [START_REF] Owen | Flow and Heat Transfer in Rotating-Disc Systems[END_REF] proposed a correlation for the local Nusselt number on the rotating disk, depending on the aspect ratio of the cavity and not on the rotational speed:

, 1 1

r I I r N u Nu G R G = ⇒ = (45)
The local Nusselt number proportionally decreases with the increase in G for regime I. When the Reynolds number increases, the flow becomes turbulent (regime III) and heat transfer depend on the rotational velocity and G such as:

7 / 4 1/ 4 3/ 4 1/ 4 3/ 4 , 0.0308 0.01176 Re Re r III III r Nu G Nu G R π - - Ω Ω   = ⇒ =     (46)
Dorfman [START_REF] Dorfman | Hydrodynamic Resistance and Heat Loss from Rotating Solids[END_REF] proposed correlations for the local Nusselt numbers on the rotor for regimes II and IV with unmerged boundary layers:

(47) (48)
In his experiment, the aspect ratio G was fixed, that is why it does not appear in these correlations. Owen and Rogers [START_REF] Owen | Flow and Heat Transfer in Rotating-Disc Systems[END_REF] observed that the global heat transfer over the rotating disk is minimum when the flow regime is in transition between the low spacing regimes and the high spacing regimes. When G increases, the Couette-type flow disappears and a rotating core of fluid appears which strongly decreases the shear stresses in the airgap. When G increases again, the rotating core of fluid disappears, which is the transition between Batchelor and Stewartson type flow, increasing the shear stresses so the convective heat transfer. Daily and Nece [START_REF] Daily | Chamber dimension effects on induced flow and frictional resistance of enclosed rotating disks[END_REF] proposed correlations for the mean Nusselt numbers on the rotor: From a general point of view, their results showed that none of the previous correlations obtained either by Owen and Rogers [START_REF] Owen | Flow and Heat Transfer in Rotating-Disc Systems[END_REF] for regimes I and III or by Dorfman [START_REF] Dorfman | Hydrodynamic Resistance and Heat Loss from Rotating Solids[END_REF] for regimes II and IV, provide fully satisfactory results for the whole disk surface, whatever the set of parameters (Re Ω , G). For G=0.01, the local Nusselt numbers remain very low and do not vary so much with the rotation rate. Thus, they are well approximated by the correlation relative to regime I [OWE89], which is not the case for the other values of G. For the highest interdisk spacings, the local Nusselt numbers are found to be very similar to those obtained in the case of a single disk for all considered values of the rotational Reynolds number. Pellé and Harmand [START_REF] Pellé | Heat transfer measurements in an opened rotor-stator system air-gap[END_REF] proposed also a method to calculate the averaged Nusselt numbers over the whole disk taking into account the possible transition between two regimes for a given set of parameters. They provided then correlations for the mean Nusselt numbers for each regime together with a schematic representation of the four heat transfer regimes, as Daily and Nece [START_REF] Daily | Chamber dimension effects on induced flow and frictional resistance of enclosed rotating disks[END_REF] did it before for a closed cavity under isothermal conditions (see Fig. 18).

There is much less results concerning the heat transfer along the stationary disk. We can cite the work of Yuan et al. [START_REF] Yuan | Turbulent heat transfer on the stationary disk in a rotor-stator system[END_REF], who performed measurements using the liquid crystal technique to determine the turbulent heat transfer on the stationary disk in a rotor-stator cavity opened at the periphery for 1.42×10 5 ≤Re Ω ≤3.33×10 5 . They provided also axisymmetric computations using a RNG k-ε model available within Fluent, assuming no thermal effect on the hydrodynamic flow. They focused their attention to the flow and thermal fields at the periphery of the stator. Like open-end rotor-stator flows, the mean velocity maps exhibit a Stewartson-like structure, confirming the results of Brady and Durlofsky [START_REF] Brady | On rotating disk flow[END_REF]. It has an important role on the heat transfer process. For each value of

Re Ω , their results highlighted the existence of an optimum interdisk spacing at which the averaged Nusselt number on the stator was maximum. The optimum spacing is found to decrease when the Reynolds number increases. Howey et al. [START_REF] Howey | Radially resolved measurement of stator heat transfer in a rotor-stator disc system[END_REF] developed a new experimental method for measuring stator heat transfer in a rotor-stator system partially blocked at the periphery for G=0.0106, 0.0212 and 0.0297 and 3.7×10 4 ≤ Re Ω ≤ 5.6×10 5 .

Transition at the stator was observed to occur at Re Ω > 3×10 5 for all aspect ratios. An Increase of the Nusselt numbers at the periphery was observed for all sets of parameters due to the ingress of ambient fluid along the stator because of the rotor pumping effect.

Their results highlighted so the importance of peripheral fluid inflow at the stator in determining the stator heat transfer at outer radii.

Other effects like the prerotation level or the presence of a large central opening in the stator on the heat transfer have also been considered. Dibelius and Heinen [START_REF] Dibelius | Heat transfer from a rotating disc[END_REF] investigated experimentally the heat transfer along the rotor in a quasi-isothermal-rotor / cooled-stator cavity for Re Ω =2×10 6 and three aspect ratios G=0.0125, 0.0625 and 0.1375.

They obtained two main results: the heat transfer increases as G decreases in contradiction to the application of the Reynolds analogy and most of all, the heat transfer are reduced in the presence of prerotation at the cavity inlet. Beretta and Malfa [START_REF] Beretta | Flow and heat transfer in cavities between rotor and stator disks[END_REF] studied the effect of a central opening on the stator side on the heat transfer in a rotor-stator system. They performed computations using a low Reynolds number k-ε model within Fluent for adiabatic-rotor / isothermal-stator conditions and compared them to an in-house semiempirical model developed to evaluate temperature and heat fluxes and based on mass and angular momentum balances and the Reynolds analogy. A reasonable agreement was found for both geometries: closed and opened housings. They showed in particular that the radial distribution of temperature difference between rotor and stator disks is parabolic and that, under conditions of validity of the Reynolds analogy, it is independent of the presence or not of a radial outflow. Boutarfa and Harmand [START_REF] Boutarfa | Local convective heat exchanges and flow structure in a rotor-stator system[END_REF] investigated also the influence of a central opening in the stator on the flow structure and the local convective heat transfer in a heated-rotor / isothermal-stator enclosure. They used PIV and infrared thermography respectively for 5.87×10 4 ≤ Re Ω ≤ 1.76×10 5 and 0.01 ≤ G ≤ 0.17 They proposed correlations for the local and averaged Nusselt numbers along the rotor as a function of the rotational Reynolds number and the aspect ratio.

Rotor Stator with jet

In this section, a confined round jet impinges onto a rotating disk (Figure 19). The interaction of rotation and impingement creates a complex and powerful flow capable of improving heat transfer processes considerably. The flow is confined by a stationary disk located at an axial distance e of the rotating disk. The incoming fluid is discharged from the center of the stator through an orifice of diameter D at a volumetric flow rate Q. As the fluid is the same as the fluid within the cavity, it is called submerged jet. The rotor-stator cavity may be partially enclosed at the periphery by a stationary cylinder (shrouded cavity) or opened to atmosphere (unshrouded cavity).

Figure 19: Schematic diagram of a cylindrical rotor-stator cavity with an impinging round jet.

Hydrodynamic field

Round impinging jet without rotation

The configuration without rotation of the bottom disk is often used as a reference test case to validate experimental or numerical methods. For an exhaustive review about impinging jet flow in so-called stator-stator cavities, the reader may refer to the recent A small tertiary inertia-driven circular roll is identified in the confined impinging jet for high jet Reynolds numbers. Beyond a critical value of Re j , the high jet inertia causes the vortex flow to become time dependent. A reduction in the interdisk spacing results in a significant decrease in the Rayleigh number and can substantially reduce the size and strength of the buoyancy-driven roll. The time-dependent flow behaviour has been considered in more details by Hsieh et al. [START_REF] Hsieh | Unstable vortex flow and new inertia-driven vortex rolls resulting from an air jet impinging onto a confined heated horizontal disk[END_REF] for Re j ≤1623, Ra ≤ 63420 and e=10, 20 and 30 mm. At sufficiently high Re j , the inertia-driven tertiary and quaternary rolls can be induced aside from the primary and secondary rolls. At an even higher Re j the vortex flow becomes unstable due to the inertia driven flow instability. For e = 20 mm, the flow is also subjected to a buoyancy-driven instability for this range of parameters (Figure 19). They finally proposed flow regime maps delimiting the temporal state of the flow together with correlating equations for the boundaries separating various flow regimes.

Figure 21: Time-dependent impinging jet flow for Re j =829, e=20 mm, t=1.45 s in the isothermal case after Hsieh et al. [START_REF] Hsieh | Unstable vortex flow and new inertia-driven vortex rolls resulting from an air jet impinging onto a confined heated horizontal disk[END_REF].

Transition between the Batchelor and the Stewartson flow structures

When an impinging jet is superimposed on the base rotor-stator flow, the same four flow regimes already discussed in the previous section have been observed with similar but different boundaries. In particular, a superposed radial outflow can cause a premature transition to turbulence [START_REF] Kreith | Mass and heat transfer from an enclosed rotating disk with and without source flow[END_REF][START_REF] Owen | Flow and Heat Transfer in Rotating-Disc Systems[END_REF]. The second difficulty, which arises when dealing with opened cavities, is the possibility for the external fluid from the surroundings to be pulled into the system (ingress), when the superposed flowrate is relatively weak.

Soo [START_REF] Soo | Laminar flow over an enclosed rotating disk[END_REF] obtained approximate solutions for regime I and displayed the different flow patterns according to only one parameter Ф(r/R):

(

with Ф<0 for inward superposed flow and . For Ф =1/90, the flow is purely centrifugal, whereas some ingress is observed along the stator for Ф =1/180. Köhler and Müller [KOH71] compared velocity measurements with calculations based on a finitedifference method for G=0.01 and Re Ω <1200 with a reasonable agreement between the two approaches.

In regime III, Dorfman [START_REF] Dorfman | Effect of radial flow between the rotating disc and housing on their resistance and heat transfer[END_REF] obtained approximate solutions using an integral method for the case of turbulent flow without an inviscid core. He provided interesting correlations for the moment coefficient for the two limiting cases without throughflow (Q=0) and for Q→∞. Haddadi and Poncet [START_REF] Haddadi | Turbulence Modeling of Torsional Couette Flows[END_REF] proposed RSM calculations for 0.02 ≤ G ≤ 0.05, 1.8×10 5 ≤ Re Ω ≤ 10 7 and C w ≤ 10 4 . In most cases, the flow belongs to the regime III: turbulent with merged boundary layers. Turbulence is mainly confined along the rotor and vanishes toward the stator. The maximum of the turbulence intensities is obtained at the inlet close to the rotation axis and turbulence levels slightly decrease when moving toward the periphery of the cavity. It is mainly composed of three components, whatever the values of the flowrate coefficient Cw.

In regime II with laminar and unmerged boundary layers, an inviscid rotating core may appear inside the cavity between the two boundary layers. The flow structure still depends on the Reynolds number Re Ω but also on a new parameter, the flow rate coefficient C w , which are combined to form the flow parameter κ. For a laminar flow, this parameter is defined as:

0.5 Re lam w C κ - Ω = (55)
Soo [START_REF] Soo | Laminar flow over an enclosed rotating disk[END_REF] has shown that the tangential velocity of the fluid rotating core is a function of the radial location. For radii larger than ( ) 0.5 / lam κ π the rotating core does not appear and the flow near the rotor is similar to the one obtained for a single rotating disk. Soo [START_REF] Soo | Laminar flow over an enclosed rotating disk[END_REF] has furthermore said that for ( ) ( )

0.5 / 1 lam κ π < ,
1 13/ 5 0 1 12.74 / turb r R κ β β -   = +       (57)
where β 0 is the core-swirl ratio in the case without throughflow. Poncet et al. [START_REF] Poncet | Batchelor versus Stewartson flow structures in a rotor-stator cavity with throughflow[END_REF] established a new analytical law for the core-swirl ratio β for a turbulent rotor-stator in an annular enclosure submitted either to a centripetal or centrifugal injection:

( ) 5/ 7 2 5.9 0.63 1

r Cq β = × × + - (58)
where Very recently, Nguyen et al. [START_REF] Nguyen | PIV measurements of an air jet impinging on an open rotor-stator system[END_REF] provided the first quantitative experimental database for the mean and turbulent flow fields in the case of an impinging jet in an open rotor-stator system (G = 0.02, e/D = 0.25) for 0.33×10 5 ≤ Re Ω ≤ 5.32×10 5 and 17.2×10 3 ≤ Re j ≤ 43×10 3 by the means of PIV measurements. A recirculation flow region, which was centered at the impingement point and possessed high turbulence intensities, was observed. Local peaks in root-mean-square fluctuating velocity distributions appeared in the recirculation region and near the periphery, respectively with a weak influence of the rotational speed (Fig. 22). By POD velocity decompositions, their results revealed the existence of coherent large-scale structures with an oval shape and a size comparable to the interdisk spacing. Moreover, they showed that they appear in all configurations with and without rotation of the bottom disk.

1/ 5 3 Re /(2 ) r r Cq Q r π = Ω is a local flowrate coefficient.

Confinement effects

If the rotor-stator cavity is unshrouded, the pumped flow from the rotating disk exits the cavity to the external surroundings and inflow (ingress) from the external environment may occur to supply the entrainment demands of the rotating flow. At the outer radius of the cavity there can therefore be both inflow on the stator and outflow on the rotor. The case of an unshrouded rotor-stator cavity with radial outflow has been studied for laminar merged boundary layers by Soo [START_REF] Soo | Laminar flow over an enclosed rotating disk[END_REF], for laminar but unmerged boundary layers by Vaughan [START_REF] Vaughan | A numerical investigation into the effect of an external flow field on the sealing of a rotor-stator cavity[END_REF], for turbulent merged boundary layers by Dorfman [START_REF] Dorfman | Effect of radial flow between the rotating disc and housing on their resistance and heat transfer[END_REF], and for turbulent separate boundary layers by Daily et al. [START_REF] Daily | Enclosed rotating discs with superposed throughflow: mean steady and periodic unsteady characteristics of induced flow[END_REF].

The addition of a shroud at the periphery of a rotor-stator cavity normally acts to restrict the fluid exchange between the cavity and the external environment and so ingress.

Bayley and Owen [START_REF] Bayley | The fluid dynamics of a shrouded disk system with a radial outflow of coolant[END_REF] and then Haynes and Owen [START_REF] Haynes | Heat transfer from a shrouded disk system with a radial outflow of coolant[END_REF] investigated the impact of a shroud attached at the stationary disk. For Re Ω < 4.2×10 6 , the shroud has the effect to reduce the moment on the rotor when no superposed flow is supplied; however, it has the opposite effect when a superposed outflow is supplied. This behaviour was attributed to the relative strengths of both increased angular momentum conservation due to the better sealing of the cavity and increased viscous shear as the flow exits the cavity due to the narrow gap. At very low values of the flowrates, the conservation of angular momentum in the cavity is dominant. As the flow is increased the shear in the rim region at the periphery becomes more significant. The moment on the rotor drops for large shroud to rotor gap. The overall gap between the rotor and stator disks has little impact on disk torque. As with the unshrouded case, the moment coefficient drops with increasing rotational Reynolds number [START_REF] Bayley | Flow between a rotating and a stationary disc[END_REF]. The presence of the shroud increases the pressure in the cavity above the ambient conditions. The pressure inside the cavity increases as the shroud clearance ratio and rotational Reynolds number decrease. For small gap ratios, an experimental study was undertaken by Bayley and Owen [START_REF] Bayley | The fluid dynamics of a shrouded disk system with a radial outflow of coolant[END_REF] to determine the minimum superposed flow requirements to fully seal a cavity by measuring the pressure difference across the shroud.

They assumed that the inflow was suppressed when the pressure difference between a location under the shroud and the external environment was zero. The superposed flow necessary to prevent ingestion Cw min was found to be strongly dependent on the shroud clearance G c , independent of gap ratio and well predicted by the empirical relationship: for G c =e c /R=0.0033 and 0.0067 (e c the axial clearance between the shroud and the rotor), G=0.06, 0.12 and 0.18 and 4×10 5 < Re Ω < 4.2×10 6 . The results of Bayley and Owen [START_REF] Bayley | The fluid dynamics of a shrouded disk system with a radial outflow of coolant[END_REF] for the moment coefficient were confirmed numerically by Gosman et al. [START_REF] Gosman | Prediction of recirculating, swirling, turbulent flow in rotating disc systems[END_REF] using a kε model with reasonable agreement.

Bayley and Owen [START_REF] Bayley | Flow between a rotating and a stationary disc[END_REF] studied the effects of a radial outflow in a rotor-stator cavity by comparing experimental measurements of pressure, velocity and moment coefficients with values obtained from integral solutions of the boundary layer equations. Tests were performed at Re Ω <3.8×10 6 for a wide range of aspect ratios G from 0.008 to 0.03. As already discussed, the main effect of the outward flow created by the jet is the reduction of the core-swirl ratio, which can tend to zero at larger aspect ratios, and then a Stewartson-like structure occurs. In the same way, increasing the rotational Reynolds number increases the tangential velocity within the cavity. The main new result concerned the influence of the interdisk spacing e on the flow structure. As e is increased, the tangential velocity of the inviscid core region reduced and ultimately the core region disappeared as the effects of the stator reduced and the flow began to behave as if adjacent to a single free rotating disk. In a latter work, Owen et al. [START_REF] Owen | Heat transfer from an air-cooled rotating disk[END_REF] proposed the limiting aspect ratio G beyond which the moment coefficient on the rotor is unaffected by the presence of the stator, for an unshrouded system:

0.2 1.05 Re G - Ω =
with Re Ω up to 4.1 millions, aspect ratios G in the range [0.005-0.604] and nozzle diameter to interdisk spacing ratio between 51 and 2.25. Bayley and Owen [START_REF] Bayley | Flow between a rotating and a stationary disc[END_REF] showed that narrower gaps resulted in higher moment coefficients. The pressure distribution predicted by momentum integral techniques agreed well with experimental data in the larger interdisk spacings but, as e was reduced, the effect of the outflow was overpredicted. Conversely, the prediction of moment coefficient improved for small clearances.

Mass transfer data

Kreith et al. [START_REF] Kreith | Mass and heat transfer from an enclosed rotating disk with and without source flow[END_REF] performed mass transfer measurements for 0.12<G<0.6, C w ≤1.51×10 5 , 5×10 3 ≤ Re Ω ≤ 10 5 . They proposed a correlation for the averaged Sherwood number:

Heat transfer

In the literature, a huge amount of heat transfer data is available in the case of an impinging jet onto circular disks. These data are often restricted to either cooling of a stationary disk by jet impingement or by pure rotation. The effect of the combination of rotation and jet impingement has been considered in only few papers as we will see in the where Ω c and R c are the rotation rate and the radius of the confinement disk. It was found that a higher flowrate increased local heat transfer coefficient reducing the interface temperature difference over the entire disk surface. The rotational rate also increased local heat transfer coefficient under most conditions.

Rotating flows with multi-jets

Interactions between successive jets

Depending mainly on geometrical conditions, multiple jets perform very differently from a single jet striking a target surface due mainly to two types of interactions, which do not occur in single jet systems:

• the possible jet-to-jet interaction between pairs of successive jets prior to their impingement onto the plate. This type of interference is of importance for arrays with closely spaced jets and large distances between the jets and the impingement surface.

• the interaction between the impinging jets and the flow formed by the spent fluid of the neighbouring jets. These disturbances predominantly occur for arrays with small spacing between two jets, small separation distances between the jets and the plate, and large jet velocities. The mass from one jet moves in the cross- Figure 25 presents the distributions of the Nusselt number for a row of three jets obtained by [START_REF] Goldstein | Visualization of heat transfer from arrays of impinging jets[END_REF], who showed the influence of two neighbouring jets on the heat transfer coefficient for low aspect ratio e/D=2 with the appearance of secondary peaks. They disappear at larger gaps. For given separation distances, fountains as reported in Figure 26 may occur and create recirculation cells, which strongly modify the incoming jet flows. Because of all these interactions, it appears difficult to use single jet heat transfer results as base results for the design of multiple jet systems. In single jet configurations, heat transfer coefficients can be linked to the flow parameters (Reynolds, Prandtl numbers, flowrates…) using simple power-law functions. For multiple jet systems, the heat transfer rates require the consideration of a number of additional characteristic numbers. We can cite, as example, the experimental investigation of Fenot et al. [START_REF] Fenot | Local heat transfer due to several configurations of circular air jets impinging on a flat plate with and without semi-confinement[END_REF], who used an infrared thermography technique to measure the convective heat transfer on a flat plate with either a single round jet or a row of impinging jets. They focused both on the number of jets in a row and on the influence of confinement on the heat transfer coefficient and effectiveness. Their results showed the independence of heat transfer coefficient and effectiveness from the jet injection temperature. Local Nusselt number distributions remained approximately the same as for a single jet, whatever the distance between two successive jets. A slight influence of neighbouring jets on the Nusselt number is observed midway between two jets. The distance between two successive jets affects greatly the effectiveness, which decreases more rapidly for small distances. Confinement has also only an influence on effectiveness, which decreases more rapidly without confinement. Numerous experimental investigations have been performed up to now on heat transfer of jets impinging on a flat plate without rotation. As it is not the main object of the present review, the reader may refer to the review papers of Jambunathan et al. [START_REF] Jambunathan | A review of heat transfer data for single circular impingement[END_REF], Viskanta [START_REF] Viskanta | Heat transfer to impinging isothermal gas and flame jets[END_REF] and Weigand and Spring [START_REF] Weigand | Multiple jet impingement -a review[END_REF] for an extensive literature surveys on jet impingement including multi-jet flows and the influence of the different parameters up to 2009. Han and Goldstein [START_REF] Han | Jet Impigement Heat Transfer in Gas Turbine Systems[END_REF] presented also a review of the jet impingement heat transfer in turbine systems. They have discussed the heat transfer characteristics of single jet and multiple jets, the effect of Reynolds number, Prandtl number and Nusselt number, effect of jet impingement on a curved surface and a rotating disk, and effect of angle of attack of the impinging jet and other control parameters.

Effect of rotation on the multiple jet flow problem

Although many studies have been completed for multi-jets impinging a flat surface, only few have considered the effect of rotation on impinging jet flows. As example, Epstein et al. [START_REF] Epstein | Rotational Effects on Impingement Cooling[END_REF] studied the effect of rotation on impingement cooling in the leading edge of a blade.

They reported that rotation decreases the impingement heat transfer. The case where the jet direction is perpendicular to rotation direction showed lower heat transfer coefficients compared to that with a staggered angle. Glezer et al. [START_REF] Glezer | Heat Transfer in a Rotating Radial Channel with Swirling Internal Flow[END_REF] studied the effect of rotation on swirling impingement cooling in the leading edge of a blade. They found that screwshaped swirl cooling can significantly improve the heat transfer coefficient over a smooth channel and the improvement is not significantly dependent on the temperature ratio and rotational forces. Akella and Han [START_REF] Akella | Impingement Cooling in Rotating Two-Pass Rectangular Channels[END_REF] studied the effect of rotation on impingement cooling for a two-pass impingement channel configuration with smooth walls. In their case, spent jets from the trailing channel are used as cooling jets for the leading channel. They reported that whatever the direction of rotation, the heat transfer coefficient on the first pass and second pass impinging wall decreases up to 20% in the presence of rotation. Hong et al. [START_REF] Hong | Effect of jet direction on heat/mass transfer of rotating impingement jet[END_REF] investigated the local heat and mass transfer characteristics using a naphthalene sublimation technique for various conditions of impinging jets in a rotating duct, of which an inline array of round impinging jets. The rotation changes the local heat and mass transfer characteristics due to the jet deflection and spreading phenomenon. For a ratio between the nozzle to plate spacing and the nozzle diameter equal to e/D=6, the jet is strongly deflected at the leading orientation, resulting in the significant decrease in heat and mass transfer. At the axial orientation, the momentum of jet core decreases slightly due to jet spreading into the radial direction and consequently, the value of stagnation peak is a little lower than that of the stationary case. The reduction of heat and mass transfers due to rotation disappears at e/D=2.

Multiple jets in a rotating disk system

Lee et al. [START_REF] Lee | Heat Transfer Characteristics for a Confined Rotating MCM Disk With Round Jet Array Impingement[END_REF] performed experiments in the case of a confined rotating multi-chip module disk with round jet array impingement. They investigated the effects of many control parameters, such as the steady-state Grashof number, the ratio of jet to disk distance to nozzle diameter, the jet Reynolds number and the rotational Reynolds number on heat transfer and provided a new correlation for the stagnation Nusselt number in terms of the jet Reynolds number and normalized jet to disk distance. As compared to the single round jet case, the average heat transfer enhancement at the stagnation point of jet array impingement is 607%. For the rotating disk cases, the highest chip heat transfer occurs at the disk rim, and decreases sharply along the distance from the surface edge toward the center. To the best of our knowledge, there is at that time no relevant work dedicated to the fluid flow and heat transfer analysis of arrays of round jets impinging on a rotating disk. We can cite the early papers of Kuznetzov [START_REF] Kuznetzov | Heat Transfer From a Shrouded Rotating Disk at Air Jet Impingement[END_REF][START_REF] Kuznetzov | Determination of Local Heat Transfer Coefficients of a Disk Rotating in a Shroud at Air Jet Impingement[END_REF] and Devyatov [START_REF] Devyatov | Investigations of Heat Transfer for Two Versions of Turbine Disk Cooling, Aviatsionnaya Tekhnika[END_REF], who investigated the effect of multiple jet impingements on convective heat transfer for a non isothermal rotating disk in a shrouded insulated system. The single round jets were distributed along one circle either very close to the hub or at the disk rim. They considered a wide range of velocity ratios ΩR / W = 0.002 -1 for Kuznetzov [KUZ64, KUZ67] and from 0.3 to 1.65 for Devyatov [START_REF] Devyatov | Investigations of Heat Transfer for Two Versions of Turbine Disk Cooling, Aviatsionnaya Tekhnika[END_REF]. This review clearly points out the lack of experimental and numerical database in the case of multiple impinging jets in rotor-stator systems. Rotor-stator disk cavities offer a simple geometry to investigate jet to jet interaction before considering more complex geometries such as blades. It suggests an interesting direction for future research.

Conlusions

Fluid flow and convective heat transfer in rotor-stator systems are of great importance in the turbomachinery and power engineering, e.g. wind generators. A review of the most important results for the rotor-stator systems presented here elucidated the most important finding relating to investigations and predictions of convective heat transfer in predominantly outward air flow in rotor-stator cavities with and without impinging jets.

For the methodological side, the most reliable modern experimental techniques were outlined (thermocouples, infra-red cameras, laser Doppler and particle image velocimetry), and numerical were discussed, direct numerical simulation, widely used Reynolds-averaged Navier-Stokes equations techniques, large-eddy simulation approaches, self-similar solutions and integral methods.

The description of the geometries and results was structured based on the principle of mentioning first simple (free disk, disk in crossflow, single impinging jets over a stationary and a rotating disk) and finally complex rotor-stator geometries, which helps understand complex phenomena in the rotor-stator systems as a superposition of separate effects like rotation, impingement/crossflow with different angles of attack, flow regime, wall temperature profile etc.

Rotor-stator systems were outlined in full variety first without impinging jets, than with single and finally with multiple impinging jets. Maps of the flow regimes depending on the height of the gap between the rotor and stator, throughflow velocity and rotational speed help shed light onto the complex fluid flow and heat transfer phenomena in the gap.

Numerous literature references to experimental investigations and numerical simulations provides a reader much space for deepening knowledge outlined in the particular parts of the review. The material is illustrated whenever possible with equations for the moment coefficient and Nusselt number.

Particular parts of the review, especially in the rotor-stator configuration sections, outline perspectives of the further extension of both the investigations and applications of the rotor-stator systems in engineering practice.

  ]. The value m=0.4 used for all flow regimes in e.g.

[ PEL07 ,

 PEL07 [START_REF] Pellé | Heat transfer study in a rotor-stator system air-gap with an axial inflow[END_REF][START_REF] Pellé | Convective Heat Transfer in a Rotor Stator System Air Gap With Multiple Suctions of Fluid Through the Stator[END_REF] worked with the thick layer technique. In those experiments, the disk was heated on the backside by infrared emitters in [PEL07,[START_REF] Pellé | Heat transfer study in a rotor-stator system air-gap with an axial inflow[END_REF][START_REF] Pellé | Convective Heat Transfer in a Rotor Stator System Air Gap With Multiple Suctions of Fluid Through the Stator[END_REF] or by thermal resistances[START_REF] Owen | Heat transfer from an air-cooled rotating disk[END_REF] and cooled by forced convection on the front face where measurements were performed. In the method used by Pellé & Harmand[START_REF] Pellé | Heat transfer measurements in an opened rotor-stator system air-gap[END_REF], the cooling surface of a rotating aluminum disk was covered with an insulating material. Local convective heat transfer variations lead to radial temperature variations over the cooled surface. These variations in amplitude depend on the nature of the flow, the thermal conductivity of the material, and the thickness of the insulating layer. The conductive heat flux was calculated by solving the heat conduction equation by a finite difference method using measured temperatures as boundary conditions, after reaching a stationary state.

  Doppler Velocimetry. In the Laser Doppler Velocimetry (LDV) or Laser Doppler Anemometry (LDA), the difference between the frequencies of the initially sent laser signal and received signal after passing through the flow containing particles is estimated, which enables deducing velocity values and directions? Used particles are glycerine (diameter of about 5µm)[START_REF] O'donovan | PIV Measurements and convective heat transfer of an impinging jet[END_REF], olive oil (diameter of about 1µm)[START_REF] Koseoglu | The effect of flow field and turbulence on heat transfer characteristics of confined circular and elliptic impinging jets[END_REF], salt-water solution (diameter of about 1µm)[START_REF] Lytle | Air jet impingement heat transfer at low nozzle-plate spacings[END_REF] or corn statch (diameter of about 5 µm)[START_REF] Fitzgerald | A study of the flow field of a confined and submerged impinging jet[END_REF].Poncet et al.[START_REF] Poncet | Turbulent Rotating Disk Flow with Inward Throughflow[END_REF][START_REF] Poncet | Batchelor versus Stewartson flow structures in a rotor-stator cavity with throughflow[END_REF] used this experimental technique in a rotor-stator configuration with throughflow. Particle type was "Optimage PIV seeding powder" with the particle size of about 30µm. As pointed out, the main defect of this method is to provide an integrated value over the probe volume, whose size in the axial direction can be large compared to the air gap interval or at least to the boundary layer thicknesses. The rotorstator cavity in which Geis[START_REF] Geis | Flow Structures Inside a Rotor-Stator Cavity[END_REF] made flow measurements is double-shrouded with a perturbed rotor surface. [LYT94, FIT97, KOS07] used this technique to study effects of a jet impingement flow on a flat stationary surface. In[START_REF] O'donovan | PIV Measurements and convective heat transfer of an impinging jet[END_REF], for an impinging air jet configuration, areas of great interest were studied with LDV/LDA, with most of the entire flow pattern being investigated with the help of Particle Image Velocimetry.Particle Image Velocimetry. Particle Image Velocimetry (PIV) is based on the displacement measurement of particles during two laser flashes, whose time between pulses is known.

  Visualizations. Basic flow visualizations by means of a laser plane and smoke generator were used in [SOO02, HSI06] for configurations with two disks rotating independently and jet impingement onto a confined disk. Mean flow structure was identified and linked to heat transfer in [SOO02]. The transient experiment performed in [HSI06] enabled to highlight the time-dependent state of the flow and boundaries separating various flow regimes between the disks.

  Iacovides and Chew[START_REF] Iacovides | The computation of convective heat transfer in rotating cavities[END_REF], Poncet and Schiestel[START_REF] Poncet | Numerical modeling of heat transfer and fluid flow in rotor-stator cavities with throughflow[END_REF] and Manceau et al.[START_REF] Manceau | Turbulent jet impinging onto a rotating disc: a collaborative evaluation of RANS models[END_REF] based on a gradient hypothesis with a given turbulent Prandtl number, have proved sufficient to provide satisfactory results for the Nusselt number distributions. Poncet and Schiestel[START_REF] Poncet | Numerical modeling of heat transfer and fluid flow in rotor-stator cavities with throughflow[END_REF] compared indeed very favourably the predictions of the RSM of Elena and Schiestel[START_REF] Elena | Turbulence modeling of rotating confined flows[END_REF] with the experimental data of Sparrow and Goldstein[START_REF] Sparrow | Effect of Rotation and Coolant Throughflow on the Heat Transfer and Temperature Field in an Enclosure[END_REF] in the case of an impinging jet in a confined rotor-stator cavity considering a large temperature difference (75 K) between the incoming fluid and the walls. Some works using commercial CFD codes provided also reasonable results. Vadvadgi and Yavuzkurt[START_REF] Vadvadgi | Calculation of Disk Temperatures in Gas Turbine Rotor-Stator Cavities Using Conjugate Heat Transfer[END_REF] used ANSYS/FLUENT to numerically simulate the flow and the heat transfer in a rotor-stator system of aspect ratio G=0.1 and a Reynolds number equal to one million. The simulation was performed using the RSM model with a conjugate heat transfer approach. Among the few advanced heat transfer models which have been developed, we can cite the one of Abe et al.[START_REF] Abe | A two-equation heat transfer model reflecting second-moment closures for wall and free turbulent flows[END_REF], who proposed an improved version of the model developed by Nagano et al.[START_REF] Nagano | An improved two-equation heat transfer model for wall turbulent shear flows[END_REF]. It consists of a two-equation heat transfer model, which incorporates essential features of second-order modelling. They introduced the Kolmogorov velocity scale, instead of the friction velocity, to take into account the low Reynolds number effects in the near-wall region and also complex heat transfer fields with flow separation and reattachment. This model has not been yet implemented for rotor-stator flows. We can also mention Craft[START_REF] Craft | Second-moment modelling of turbulent scalar transport[END_REF], who has developed advanced transport closures for the turbulent heat fluxes.

Figure 1 .Figure 1 .

 11 Figure 1. Schematic diagram of a free rotating disk

Fig. 3 .

 3 Fig. 3. Velocity and temperature profiles in laminar flow over a free rotating disk [SHE09a]. Computations [SHE09a]: 1 -F, 2 -G, 3 --H, 4 -) /( ) (

Figure 2 .

 2 Figure 2. Flow structure over a rotating disk in still air.

  6×10 5 [ELK97]. The emergence of spiral vortices in the near-wall flow is characteristic of the transition to the turbulence and their number can be related to the local Reynolds number by the equation [MAL81]: (10) Convective heat transfer on a rotating disk has been studied by many authors, for instance, [GOL35, WAG48, MIL52, HAR59, RIC63, DOR63, KRE68, DEV75, POP75, OEL78, CAR96, OWE89, ELK97, SHE09a]. They have highlighted the fact the heat transfer on the surface of the disk strongly depends on the radial temperature profile. Authors usually correlate the local and mean Nusselt numbers by power laws such as: (11)

  increases with both the radius and rotational speed. The exponents b and d in Eqs. (10) and (11) equal to 0.8 or are close to this value. Like in the laminar flow case, the radial temperature profile over the disk surface affects noticeably the coefficients a and c in Eqs. (10) and (11).

Figure 4 .

 4 Figure 4. Local Nusselt numbers over a rotating disk in quiescent air for isothermal surface.

For and a radial

  temperature distribution approximated by and, Dorfman [DOR63] and Owen & Rogers [OWE89] obtained also a semi-empirical solution for the coefficients in Eqs. (10) and (11) for turbulent flow:

  ), both these values are in excellent agreement with the aforementioned experimental/theoretical data of [MCC70, POP75, ELK97, CAR96, TAD83].

  This configuration is presented on Figure 5. The configurations with an inclined rotating disk (with an angle of attack to the incident flow lying with the range of 0 to 90 deg) were first studied only very recently [WIE10, TRI11].

Figure 5 .

 5 Figure 5. Schematic diagram of a rotating disk in cross flow

  detailed theoretical work was performed by aus der Wiesche [WIE07] who numerically studied the flow field and the corresponding local Nusselt number distributions on the disk for and . In this study, the flow is parallel to the surface of the rotating disk that has zero thickness. By comparison between the results obtained with and without disk rotation, aus der Wiesche [WIE07] analysed the rotation effect on convective heat transfer. For high values of the cross-flow Reynolds, convective heat transfer is governed only by the cross-flow since no additional heat transfer augmentation due to rotation was observed. For low rotational and cross-flow Reynolds numbers, disk rotation affects slightly the flow field and local temperature distribution, whilst the mean heat transfer over the disk surface remains constant. Indeed, rotation increases heat convection on the co-moving side and diminishes it on the counter-moving side in such a way that the mean heat transfer is almost unaffected. While increasing , the increase of the flow turbulence leads to higher variations of the local Nusselt number, even if the typical flat plate distribution can still be identified. For the sufficiently high values of , the local Nusselt number distribution becomes uniform and mainly governed by rotational effects (Fig. 6b).

Figure 6 .

 6 Figure 6. Local Nusselt numbers over a rotating disk in cross flow [WIE07] a) Re Ω =0 b) Re Ω =50000

Figure 7 .

 7 Figure 7. 3D boundary layer separation and horseshoe vortex system in the region of interaction between mainstream boundary layer and cylinder [GOL84]

Figure 8 .

 8 Figure 8. Flow at a jet impingement

Figure 9 .

 9 Figure 9. Radial distribution (at z/D=0.032) of the near-wall velocity; (a) mean velocity components; (b) turbulent intensity; Re j =14500 and e/D=5 [MIN04]

Figure 10 .

 10 Figure10. Radial variation in Nusselt number with nozzle-plate spacing for two jet Reynolds numbers on a stationary disk[START_REF] Lytle | Air jet impingement heat transfer at low nozzle-plate spacings[END_REF]

Figure 11 :

 11 Figure 11: Local heat/mass transfer of rotating disk with impinging jet at e/D=5 [CHE98]

Figure 12 :

 12 Figure 12: Rotor-stator configuration without impinging jet.e

Figure 13 :

 13 Figure 13: The four flow regimes in an enclosed rotor-stator cavity after Daily & Nece ([DAI60]. Merged boundary layers: I (laminar) and III (turbulent). Unmerged boundary layers: II (laminar) and IV (turbulent).

Figure 14 :

 14 Figure 14: Sketches of the characteristic velocity profiles in a rotor-stator system: (a) to (c) Batchelor flow; (d) to (f) Stewartson flow, after Childs [CHI11].

Figure 15 :

 15 Figure 15: Base flow of Batchelor type with both Ekman and Bödewadt layers separated by a solid body rotating core at Re Ω = 15.000 and G=0.127. Velocity field in a meridianal plane obtained by Serre et al. [SER02] using direct numerical simulation.

Figure 16 :

 16 Figure 16: (a) Axial variation of the normalized radial velocity component u r /(Ωr) at three radial locations r/R 0.3, 0.5 and 0.85 for Re Ω =10 3 ; (b) Radial variation of the normalized azimuthal velocity component u θ /(Ωr) at mid-height of the cavity for three rotational Reynolds numbers. Regime I, after Radriamampianina et al. [RAND97].

  [START_REF] Daily | Chamber dimension effects on induced flow and frictional resistance of enclosed rotating disks[END_REF] and thenKreith [KRE68] proposed also different values of the critical Reynolds number above which the flow is turbulent in the entire cavity:

Schouveiler

  et al.[START_REF] Schouveiler | Instabilities of the flow between a rotating and a stationary disk[END_REF] in Fig.17. Three different regimes have been observed by these authors: regimes I and II already introduced and a regime with mixed boundary layers. This third regime is characterized by merged boundary layers close to the rotation axis and unmerged ones at the periphery, and represents a very narrow band in the diagram in terms of aspect ratio: 0.0179 ≤ G ≤ 0.0714.

Figure 17 :

 17 Figure 17: Transition diagram of the flow between a rotating and a stationary disk enclosed by a stationary sidewall. Curves A and B separate the mixed basic flow (BFM) from the basic flows with separated boundary layers (BFS) and with joined boundary layers (BFJ) respectively. Curves 1 and 2 are the thresholds for the circular rolls (CR) and the spiral rolls (SRI) respectively. Curve 1'is the spiral roll (SRII) threshold. Curves 1", 2" and 3" are the thresholds for the spiral rolls (SRIII), the solitary waves (SW) and the spots (SP) respectively. Curve 4" is the threshold for the simultaneous disappearing of the spiral rolls (SRIII) and of the solitary waves (SW). These thresholds have been determined by increasing the Reynolds number Re Ω , hysteresis cycles have been demonstrated for the thresholds 2" and 4", after Schouveiler et al.[START_REF] Schouveiler | Instabilities of the flow between a rotating and a stationary disk[END_REF].

  evolution to higher bifurcations in the rotor-stator cavity has received much less attention. To our knowledge, Cros et al.'s[START_REF] Cros | Transition to turbulence of the Batchelor flow in a rotor/stator device[END_REF] experiment is the only one that provides some insight into the evolution and further transitions of the stator boundary layer for Reynolds numbers up to around 74000. The transition to turbulence seems to be governed by nonlinear interactions of the circular and spiral modes leading to a kind of wave turbulence. Recent numerical studies provide some insight into transition in the rotor boundary layer. At high local Reynolds numbers (Re r > 400), the rotor layer becomes absolutely unstable, and Type I instability with spiral arms of positive inclination is observed.Large eddy simulations from Séverac et al.[START_REF] Séverac | Large eddy simulation and measurements of turbulent enclosed rotor-stator flows[END_REF] confirm the presence of spiral arms in the rotor layer at Re r ≈ 400 with all the characteristics of the absolute Type I instability and, just upstream, a turbulent region. The different scenarios for the route to turbulence in such systems are discussed in Launder et al.[START_REF] Launder | Laminar, Transitional, and Turbulent Flows in Rotor-Stator Cavities[END_REF].For the transition to turbulence in regime I with merged boundary layers (G ≤ 0.0179), the single instability pattern reported before turbulence by Schouveiler et al.[START_REF] Schouveiler | Instabilities of the flow between a rotating and a stationary disk[END_REF] and Cros and Le Gal[START_REF] Cros | Spatiotemporal intermittency in the torsional Couette flow between a rotating and a stationary disk[END_REF] using flow visualizations and image processing is related to weakly negative spirals, denoted SRIII, that occur at the periphery of the cavity at large radius. The most interesting feature observed by Cros and Le Gal[START_REF] Cros | Spatiotemporal intermittency in the torsional Couette flow between a rotating and a stationary disk[END_REF] is that transition to turbulence occurs via defect turbulence in the network of negative SRIII spirals. Increasing further the rotation rate leads to the appearance of turbulent domains within a laminar background. These instabilities, in the form of turbulent spirals (SW) are periodically located around the disk and similar to the ones observed in plane Couette flows, Taylor-Couette flows or in the Ekman-Couette problem. By increasing progressively the rotation rate, turbulent spots (SP) appear and finally invade the whole cavity. Le Gal et al.[START_REF] Gal | A statistical study of spots in torsional Couette flow[END_REF] showed by flow visualizations that these spots have a horse-shoe vertical structure and are generated in the area close to the gap between the rotating disk and the shroud. Transition mechanisms along the rotor are difficult to identify due to the speed at which turbulence ensues and due to the experimental techniques used.

  5 and independent of the radial location. Then Schultz-Grunow[START_REF] Schultz-Grunow | Der Reibungswiderstand rotierender Scheiben in Gehausen[END_REF] proposed a theoretical value of β=0.512 and measured an experimental value equal to 0.357. He interpreted this discrepancy as due to the existence of a shear stress in the small radial gap between the rotating disk and the fixed cylindrical endwall. Using velocity and pressure measurements, Poncet et al.[START_REF] Poncet | Turbulent Rotating Disk Flow with Inward Throughflow[END_REF] obtained β=0.438, which coincides with the asymptotic value given by their analytical model without throughflow. This coefficient appears to be independent of the interdisk spacing (the flow remaining in regime IV with unmerged boundary layers) and of the local radius but sensitive to the core-swirl ratio at the periphery of the cavity (prerotation level). For comparisons with available values of β in shrouded or unshrouded systems, the reader can refer to the works of Randriamampianina et al.[START_REF] Randriamampianina | Numerical prediction of laminar, transitional and turbulent flows in shrouded rotor-stator systems[END_REF] and Poncet et al.[START_REF] Poncet | Lois analytiques pour les écoulements en cavité rotor-stator[END_REF].One other main difference between regimes II and IV is the thicknesses of the boundary layers. They behave globally like the thickness of the boundary layer over a single infinite rotating disk, , both boundary layers get thinner for increasing rotation rates but not in the same way. The boundary layer thickness of the rotating disk is thinner than that of the stator. Both thicknesses decrease when one approaches the axis of the cavity for low radii r, as confirmed by the experimental laws of Daily and Nece[START_REF] Daily | Chamber dimension effects on induced flow and frictional resistance of enclosed rotating disks[END_REF]: layer. The boundary layers are thus much thicker in enclosed systems in comparison with the boundary layer over a single rotating disk.The experiment of Itoh et al.[START_REF] Itoh | Experiments on turbulent flow due to an enclosed rotating disk[END_REF] provided a great contribution to the understanding of the turbulent flow in a shrouded rotor-stator system in regime IV. The authors measured the mean flow and all the Reynolds stress components for G=0.08 and brought out the existence of a relaminarized region even for high rotation rates. Cheah et al.[START_REF] Cheah | Experimental investigation of enclosed rotor-stator disk flows[END_REF] performed detailed measurements of mean velocity and turbulence intensities inside a rotor-stator system for G=0.127. These two experiments have been used as a reliable experimental database to evaluate the performances of turbulence models as in Elena and Schiestel[START_REF] Elena | Turbulence modeling of confined flow in rotating disk systems[END_REF] or in Iacovides et al.[START_REF] Iacovides | Turbulent flow computations in rotating cavities using low-Reynolds-number models[END_REF] for examples. More recently, Poncet et al.[PON05b] compared the predictions of the RSM model of Elena and Schiestel[START_REF] Elena | Turbulence modeling of rotating confined flows[END_REF] with velocity and pressure measurements in the case of a closed rotor-stator system of aspect ratio G=0.036 and rotational Reynolds numbers up to Re Ω =4.15×10 6 . From all these works, it appears that turbulence is mainly concentrated within the boundary layers with very low turbulence levels in the inviscid core. The Ekman layer is besides more turbulent than the Bödewadt layer. The Bödewadt layer gets turbulent at lower radii than the Ekman layer. Itoh et al.[START_REF] Itoh | Experiments on turbulent flow due to an enclosed rotating disk[END_REF] attributed the difference to a greater stability of the radially outward flow in the Ekman layer than the return flow in the Bödewadt layer. For Cheah et al.[START_REF] Cheah | Experimental investigation of enclosed rotor-stator disk flows[END_REF], it is due to the effects of the convective transport of turbulence. The fluid in the Bödewadt layer comes from the periphery of the cavity where the highest turbulence levels are observed due to the highest values reached by the local Reynolds number Re r =Ωr 2 /ν. Then, it flows radially inward along the stator with decreasing intensity levels. On the contrary, the fluid in the Ekman layer comes from the rotation axis, where the lowest values of Re r are obtained.For sufficiently high rotation rates, both boundary layers are turbulent in the whole cavity and the rotational Reynolds number Re Ω has no significant influence on the turbulence intensities. Reynolds stress tensor components are strongly anisotropic near the walls and the anisotropy then diminishes in the core region.Regime III is also less documented than regime IV. The main experimental investigation is the one of Daily et al.[START_REF] Daily | Enclosed rotating discs with superposed throughflow: mean steady and periodic unsteady characteristics of induced flow[END_REF], who performed mean velocity, torque, and pressure measurements, for G=0.0138 and two Reynolds numbers Re Ω =2.95-6.9×10 5 . For this set of parameters, they clearly obtained merged boundary layers. Phadke and Owen[START_REF] Phadke | Aerodynamic aspects of the sealing of gas-turbine rotor-stator systems-part 1: the behavior of simple shrouded rotating-disk systems in a quiescent environment[END_REF] studied, using flow visualizations and pressure measurements, the effect of seven shroud geometries on the ingress of external fluid into the rotor-stator cavity for a large range of flow control parameters including the torsional Couette flow regime. More recently, Andersson and Lygren[START_REF] Andersson | LES of open rotor-stator flow[END_REF] performed some LES of enclosed rotor-stator flows for both the wide and narrow gap cases. They obtained a turbulent torsional Couette flow for Re e =160, which corresponds to three sets of parameters: (G=0.02, Re Ω =4×10 5 ), (G=0.01, Re Ω =1.6×10 6 ) and (G=0.01265, Re Ω =10 6 ). The first set has been studied experimentally by Itoh[START_REF] Itoh | Experiments on the turbulent flow in the narrow clearance between a rotating and a stationary disk[END_REF]. For this value of Re e , the dimensionless mean tangential velocity exhibits an Sshaped profile, with a value consistently below 0.5 midway between the disks. In the narrow-gap cases, u θ /(Ωr) is 0.43, 0.46 and 0.47 in the three different cases and thereby tended towards the plane Couette flow value 0.5 with increasing values of Re Ω . The profiles of mean radial velocity show that the crossflow is substantially weaker than in the laminar case and reduces with increasing Re Ω . Haddadi and Poncet[START_REF] Haddadi | Turbulence Modeling of Torsional Couette Flows[END_REF] proposed extensive computations using the RSM of[START_REF] Elena | Turbulence modeling of rotating confined flows[END_REF] for a wide range of the flow control parameters (1.8×10 5 ≤ Re Ω ≤ 10 7 , 0.02≤ G ≤ 0.05) mainly covering regime III. The axial profiles of the mean tangential velocity case remain very similar to the ones obtained for turbulent Batchelor flows (regime IV). On the contrary, the profiles of the radial velocity are quite different. It is always linear in the main part of the flow as in classical plane Couette flows. Contrary to turbulent Batchelor flows, where turbulence is concentrated in the boundary layers, turbulence in torsional Couette flows is distributed along the axial direction.

  one, with Nu r the local Nusselt number.

Figure 18 :

 18 Figure 18: The four heat transfer regimes depending on G and Re as established by Pellé and Harmand [PEL07].

  . The originality of the system, previously considered by Harmand et al.[START_REF] Harmand | Local convective heat exchanges from a rotor facing a stator[END_REF] for 2×10 4 ≤ Re Ω ≤ 1.47×10 6 and G = 0.01, consisted in a large central opening in the stator. By comparisons with the previous results of Daily and Nece[START_REF] Daily | Chamber dimension effects on induced flow and frictional resistance of enclosed rotating disks[END_REF], they highlighted the influence of the central opening, which increases the heat transfer coefficient, whatever the values of Re Ω and G. For a narrow gap cavity G=0.01, the flow exhibits a torsional Couette-like structure with an outward flow in the gap, which enhances the heat transfer by comparison with large aspect ratio cavities. For G=0.02, the flow switches to a Batchelor structure, which induces a decrease of the tangential shear stresses on the rotor and consequently of the Nusselt number. When G increases further, the Nusselt number on the rotor is close to that obtained for a single disk.

  works of [CHEN02, HSI06] for examples. Confined submerged liquid jets in either axisymmetric or planar share the common feature of a small stagnation zone at the impingement surface whose size is of the order of the nozzle diameter D or slot dimension, with the subsequent formation of a wall jet region. Cheng et al. [CHEN02] established the transition diagram in a (Re j , Ra) plane and provided extensive flow visualizations of the flow patterns due to the combined effects of buoyancy and jet impingement onto a heated disk in a closed cylindrical system. They investigated in details the influence of the flowrate, the temperature difference and the pressure inside the cavity. The typical flow pattern appears in the form of two axisymmetric circular vortex rolls, one around the air jet along with another roll near the side wall of the enclosure. Both rolls have deformed cross-sections (Figure 20). The rolls are generated by the reflection of the jet from the impingement surface and by the deflection of the wall boundary layer flow along the surface by the upward buoyancy due to the heated plane. A rise in the flowrate causes the inner vortex roll to become substantially larger and the outer vortex roll becomes correspondingly smaller. Increasing the temperature difference between impingement surface and the air jet has an opposite effect.

Figure 20 :

 20 Figure 20: Flow visualizations of the impinging jet flow for Re j =520.1, P=450 Torr and ∆T=15 K after Cheng et al. [CHE02]. Hsieh and Lin [HSI05] performed flow visualizations and temperature measurements in order to investigate the influence of the jet to disk separation distance on the flow patterns.

  the flow exhibits a Batchelor-like profile and for switches to a Stewartson-like structure, with only one boundary layer on the rotating disk. In regime IV (turbulent flow with unmerged boundary layers), the structure of the turbulent flow is similar to the previous case. Some authors have defined a new flow parameter as being: characterize the flow, Owen and Rogers [OWE89], like Soo [SOO58] for configuration II, have defined a parameter, such that ( ) core region becomes negligible, and the flow is a Stewartson-type flow. Both Batchelor and Stewartson flow structures have been obtained experimentally and numerically in only very few works. We can cite, among others, Elena and Schiestel [ELE93a], who proposed calculations using both a k-ε and an algebraic stress model (ASM) for the turbulent impinging jet in a rotor-stator cavity with a central hub. In their case, the jet impinges the rotor at a given radial distance of the rotation axis to model the experiments of Daily et al. [DAI64]. They obtained satisfactory results for Re Ω =2.9x10 5 and 6.9x10 5 and C w =3530 and 7061. For Re Ω >6.9x10 5 , the flow structure gets of Stewartson type with an outward flow whatever the axial position and a quasi zero tangential velocity outside the rotor layer. For C w =3530, there is no Ekman layer on the rotor at Re Ω =10 5 and the radial velocity varies linearly in the core. When increasing the rotational Reynolds number, the Ekman layer develops and a small inward throughflow appears along the stator. When increasing further the rotation rate, the Batchelor structure is more evident and the coreswirl ratio tends to the limit 0.4 obtained by Poncet et al. [PON05a] for Re Ω =10 7 . The same observations can be made at constant rotation rate by decreasing the flowrate coefficient. The transition between the Batchelor and the Stewartson flow structures has been considered in details by Daily et al. [DAI64] and more recently by Poncet et al. [PON05b]. Daily et al. [DAI64] correlated their experimental results for the core-swirl ratio by:

Figure 22 :

 22 Figure 22: Non-dimensionalised r.m.s fluctuating velocities obtained by PIV measurements near the rotor for various rotational speeds and jet velocities after Nguyen et al. [NGUYEN12].

  is the averaged mass transfer coefficient, D υ the diffusion coefficient for naphthalene in air and Γ a fourth order polynomial law. Sara et al. [SAR08] used the electrochemical limiting diffusion current technique to measure the mass transfer from an impinging jet to a rotating disk for rotational Reynolds numbers in the range Re Ω =[3.4×10 4 --1.2×10 5 ], jet Reynolds number Re j between 1.7×10 4 and 5.3×10 4 and non-dimensional jet-todisk spacing e/D=[2-8]. They found that the jet impingement resulted in a substantial enhancement in the local and averaged mass transfer coefficients compared to the case without jet. The effect of jet impingement is dominant compared to the one of the rotation rate in this range of flow parameters. Mass transfer rates increase with increasing e/D, have a maximum value for e/D between 5 and 6, and then decrease with increasing e/D depending on both Re Ω and Re j (Figure 23).

Figure 23 :

 23 Figure 23: Average Sherwood number as function of non-dimensional jet spacing for stationary and for rotating disk after Sara et al. [SAR08].

Figure 24 :

 24 Figure 24: Four characteristic regions for the heat transfert in the case of an impinging jet onto a rotating disk in an open cavity, after Pellé & Harmand [PEL09].

  jet flow direction, and this flow can significantly alter the performance of neighboring jets. The crossflow attempts to deflect a jet away from its impinging location on the flat plate. In situations with very strong crossflow and sufficiently large jet to plane distance, the crossflow can completely deflect the jet away from the impingement surface[START_REF] Koopman | Local and Average Transfer Coefficients Due to an Impinging Row of Jets[END_REF].

Figure 25 :

 25 Figure 25 : Local Nusselt number of a row of three jets after [GOLD82].

Figure 26 :

 26 Figure 26: Typical circulation pattern in the confined jet array after [ZUC06].

Table 1 . Averaged Nusselt numbers over a rotating disk for various temperature differences.

 1

	)	Oehlbeck	Hartnett
	[DOR63]	[OEL78]	[HAR59]

  a=0.0196 and c=0.0151.

	Having experimentally investigated heat transfer of an isothermal disk with air (n=0,
	Pr=0.72), the authors of [COB56, NIK63] determined the values of the coefficient a in Eq.
	(10) equal to 0.0197 and 0.0194, respectively. Experimental values of the coefficient c found
	by [BUZ66, КUZ62, COB56, DEN70] are equal to 0.15. De Vere [DEV75] also came to Eq. (11)
	a	=	99 . 34	n 33 + . 6 48 . 2 1 +	(18)

with c=0.015. These experimental data agree with the estimates in accordance with Eqs. (

15

) and (

16

). However, experimental data

[START_REF] Mccomas | Temperature profiles and heat transfer associated with a single disc rotating in still air[END_REF]

with a=0.0179, and especially more recent

[START_REF] Popiel | Local heat-transfer coefficients on the rotating disk in still air[END_REF]

with a=0.0188 and

[START_REF] Elkins | Heat transfer in the rotating disk boundary layer[END_REF]

with a=0.0189 testify in favour of lower values of a than those predicted by Eq. (

15

). Especially salient becomes this tendency for the boundary condition on the disk surface with q w =const, where Eq. (

15

) predicts the value a=0.0186, while the recent and very accurate experimental data [ELK97, CAR96] again indicated a lower value of a=0.0163. Numerical simulations

[START_REF] Tadros | Heat and momentum transfer in the turbulent boundary layer near a rotating disk[END_REF]

for the boundary condition q w =const are generalized by Eqs. (10) with b=0.83 and a=0.0111; this curve lies below the estimate of Eq. (

15

) and close to the experiments of

[START_REF] Elkins | Heat transfer in the rotating disk boundary layer[END_REF][START_REF] Cardone | Infrared heat transfer measurements on a rotating disk[END_REF]

.

An improvement of the agreement of the theory and experiments provides an alternative approximate solution derived by

[START_REF] Shevchuk | Turbulent Heat Transfer of Rotating Disk at Constant Temperature or Density of Heat Flux to the Wall[END_REF][START_REF] Shevchuk | Convective Heat and Mass Transfer in Rotating Disk Systems[END_REF]

and valid for a wide range of the Prandtl numbers; for Pr=0.72, this solutions looks as:

  This law is still valid for a weak centrifugal injection as long as the flow remains turbulent with unmerged boundary layers (regime IV). It has been validated against velocity and pressure measurements and also against the predictions of a RSM model[START_REF] Poncet | Turbulent Rotating Disk Flow with Inward Throughflow[END_REF]. It has been extended to the laminar case by Poncet et al.[START_REF] Poncet | Lois analytiques pour les écoulements en cavité rotor-stator[END_REF]. For high values of the superimposed outflow, the flow structure switches to a Stewartson-like structure depending on the Rossby number being the radial gap between the rotor and the shroud. For G=0.036, the critical value of Ro for the Batchelor / Stewartson transition in regime IV is given by

	/(2 π Ω , j [PON05b] as: ) Ro Q jR =
	Ro	=	0.0088 0.0998 / r R -	+	( ) 0.3048 / r R	2	-	( ) 0.4646 / r R	3	(59)