## Abstract

This paper investigates numerically and experimentally the flow structure and convective heat transfers in an unconfined air gap of a discoid technology rotor–stator system. The cavity between the interdisk is defined by dimensionless spacing varying between G = 0.02 (Haidar et al., 2020, “Numerical and Experimental Study of Flow and Convective Heat Transfer on a Rotor of a Discoidal Machine With Eccentric Impinging Jet,” J. Therm. Sci. Eng. Appl., 12(2), 021012) and G = 0.16. For experimental data, an infrared thermography is applied to obtain a measurement of the rotor surface temperatures and a steady-state energy equation is solved to evaluate the local convective coefficients. A numerical study is performed with a computational code ansys-fluent and based to apply two different turbulence models named the Reynolds stress model (RSM) and k–ε renormalization group (RNG). The results of the numerical simulation are compared with experimental results on heat transfer for the rotational Reynolds number ranging from $2.38×105$ to $5.44×105$, the jet Reynolds numbers varying from $16.6×103$ to $49.6×103$, and for dimensionless spacing G between 0.04 and 0.16. Three heat transfer zones on the rotating disk surface are identified. A good accord between a numerical result and experimental data was obtained. Finally, a correlation relating the Nusselt number to the rotational Reynolds number, jet Reynolds number, and dimensionless spacing varying from 0.02 to 0.16 is proposed.

## Introduction

Heat exchange by convection and fluid flow through rotor–stator systems are very important in a large variety of industrial processes, for example, in the cooling of gas turbines. Another important example of current practice concerns wind turbine generators. The study of optimization of the wind turbines and alternators that they made it possible to emphasize the technology named “discoidal,” which allows in particular to improve the mechanical efficiency of this installation, due to the absence of a speed multiplier. This process places the rotary disk opposite the stationary disk and provides powerful performance at slow rotational speeds. Nevertheless, its main inconvenience consists of the air flow generated from the alternator turning which is often insufficient for efficient and comfortable system ventilation.

The air flow about a rotor was first described by Von Karman [1] in 1921. It showed that near the disk the fluid velocity has two components: One is radial, evidencing rotational inertia effect, while the other is tangential, evidencing an air viscosity effect. The heat exchange a rotating disk in resting air was also investigated by various authors [2,3].

The many studies on impinging jets have demonstrated their interest in improving convective heat transfer on rotating machinery. A number of authors examined the impingement jets influence, as Huang [4], who has investigated a variety of jet generators, as well as Fenot [5], who has investigated multiple jets simultaneously. Angioletti [6] revealed the importance of heat transfer close to the impingement zone onto the disk. The effects of jet diameter D, jet Reynolds number Rej, and jet exit to rotor spacing H/D are demonstrated by Chen [7]. The authors [8,9] then distinguished three zones onto disk surface: a zone close to the impingement site in which the effect of the jet is prevalent upon the heat transfer, a zone external to the disk in which the rotation is prominent, and finally a combined zone depending on the jet position.

In the case where the rotor placed opposite a fixed disk with the jet, the available data concern first the flow structure within the air gap. The authors [10] distinguished several possible flow regimes, based on both the dimensionless distance G = H/R and the injected air flowrate through the stator center. The air injection across the air gap center imposes a centrifugal flow over the entire width of a small radius inside the air gap. At larger radii, sufficient rotor velocity creates suction similar to that encountered in a case without a jet [11], which induces a centripetal flow close to the stator. It is possible to observe a Batchelor style flow [12] having a rotational core of fluid with zero radial velocities. A flow of the Stewartson style [13] can also take place without rotating fluid core. A separation between the laminar and turbulent flows is summarized by Daily et Nece [14] based on rotational Reynolds number as well as interval between two disks G.

Sara [15] experimentally examined both heat as well as mass transfer over a rotor stator device subjected to an axial jet. This writer demonstrated the convective exchange over the rotor increases with increasing $Reω$ and $Rej$, and this thermal exchange is dominated by the influence of the jet.

Pellé and Harmand [16] evaluated the coefficient of local convective exchange in nonconfined rotor–stator model subjected to an axisymmetric jet for a Reynolds number related to the jet Rej up to $4.20×104$, a rotational Reynolds number $2 × 104≤Reω≤5.16 × 105$ and a distance dimensioned $0.01≤G≤0.16$. The researchers discovered that the zone size close to the impact point was determined by both G and $Rej$, although at the external radius, the Nu onto the rotor varies with two parameters G and $Reω$, and provided correlations giving the $Nu¯$ over the rotor for various parameters studied. Poncet [17] investigated the influence of the rotational and airflow coupling over the exchange by convection in a rotor–stator system. The models of turbulence evaluated were $k−ω SST$ and the Reynolds stress model (RSM). The writer observed the model $k−ω SST$ tendency to overpredict the characteristics of the flow near the impact. However, the RSM model produces a result which is in excellent accordance with experimental data [18]. In impinging jet configurations, one or two peaks on the radial profile of the local Nusselt number is observable next to the stagnation region. Whatever of G ratio, Pellé and Harmand [16] found a single peak while turbulence models used by Poncet et al. [17] revealed two peaks.

The most frequently encountered configurations are those with a small gap between the disks, resulting in a low cooling. Adding an impacting jet of air of speed Uj and a diameter D represents an important remedy for that problem. Previously we studied [8] the impact of the eccentric jet over the convective heat exchanges for a small dimensionless spacing between two disks equal to G = 0.02. Nevertheless, the effects of the jet, the rotation velocity and the spacing has not been determined. This paper proposes to analyze the global structure of the flow within a cavity enters two disks, in order to compare it with heat transfer onto the rotating disk. We then suggest correlations giving Nusselt number in function of three study parameters that are the rotational Reynolds number $Reω$, the jet Reynolds number $Rej$ and the dimensionless spacing between both disks G.

## Experimental Study

### Experimental Set-Up.

Figure 1 showed the experimental setup used. The rotor diameter is 620 mm and it constructed of 43 mm thick strongly conductive aluminum $(λ=200 W/mK)$, on which 2.5 mm thick zircon insulating thermal conductivity has been deposited by plasma projection. The use of aluminum served to homogenize the temperature at the interface zircon/aluminum, measured with thermocouples. Its backside is heated by four infrared radiators having a global power of 12 kW, and is cooled on its side facing the jet. A rotational Reynolds number varies between $2.38×105$ and $5.44×105$.

Fig. 1
Fig. 1
Close modal

The stator used is constructed of aluminum having an external diameter of 620 mm. The dimensionless spacing enters two disks ranges from G = H/R= 0.02 to G = 0.16. The stator is equipped with a fluorspar window to permit temperature determination on the rotor via infrared thermography. A fluorite material is selected because of its significant coefficient of transmission. The stator is perforated with a 26 mm diameter eccentric opening to permit a large cylindrical tube to which a centrifugal fan is attached and through which an eccentric jet is imposed. The jet Reynolds numbers vary from $Rej=16.6×103$ to $49.6×103$. The surface temperatures of the rotor are obtained from the thermal profile as determined by the infrared cameras. In order to improve the rotor's thermal radiation with respect to parasitic fluxes, this rotor is colored in gray paint. The emissivity of this surface is calibrated as $εr=0.937±0.01$. The estimated uncertainty in the rotor surface temperature as well as in the interface temperature between the aluminum and isolating material is assessed at T = ±0.9 K as well as ±0.3 K as, respectively. The temperature of the reference air detected with a T-type thermocouple located 1 m from the surface of the disk. The estimated absolute error rate is 0.3 K.

### Local Nusselt Number Determination.

The coefficient of local convective exchange as well as local Nusselt number can be evaluated by calculating parietal flux on the surface of the rotor. Solving the thermal equation within the insulating material gives us access to this. A mesh of the ceramic is then performed and solved using the method of finite differences. Limiting conditions are the temperatures measured using infrared thermography at the surface of the zircon exposed to the airflow T and the temperature of the aluminum/zircon interface obtained by thermocouples.

The convective heat flux density can be obtained by describing the heat balance at each point on the surface of the disk
$φcd=φcv+φray=λzir×(∂T(r,z)∂z)z=0φcv=h×(T(r,z=0)−T∞)φray=σFεrεs1−F2(1−εr)(1−εs)(T(r,z=0)4−Tstator4)$
(1)
Local Nusselt number calculated by
$Nu=hrλair=λzir (∂T(r)∂z)z=0−σ F εr εs1−F2 (1−εr)(1−εs)(T(r,z=0)4−Tstatar4)T(r,z=0)−T∞×rλair$
(2)
where F is a form factor relating to G, it is given by Ritoux [19]
$F=1+G22−G44+G2$
(3)

### Mean Nusselt Number.

The mean Nusselt number achieved by integrating the local thermal flux on a rotor surface
$Nu¯=2R×∫0RNu×(T(r,z=0)−T∞)drT(r,z=0)−T∞$
(4)

### Uncertainty Analysis.

Considering all the above-listed uncertainties, the medium relative error is assumed to be 12% for the local Nusselt number. A similar uncertainty for the global Nusselt numbers could also be computed. The absolute uncertainty of the rotational speed from the provider is 2 rpm for the interval 0–1000 rpm, implying a value of up to 7% for the rotational Reynolds number uncertainty. The approach taken to relate the average jet speed to a pressure drop across the diffuser gives maximal relative accuracy for the jet Reynolds number at 10%.

## Numerical Modeling

### Geometry.

The configuration considered is described in Fig. 2. Two disks of identical radius ($R=0.31 m$) are modeled, one rotating (Rotor) another fixed (Stator). The rotor's geometrical characteristics are those described in previous section. The stator is an aluminum with an eccentric opening. We have varied the spacing enters rotor and stator from $H=6 mm$ to $50 mm$ so as to obtain a dimensionless spacing comprised between $G=0.02$ [8] and $G=0.16$. The rotor is set in rotation up to a speed of 850 rpm, or $Reω=5.44×105.$ An injected air flowrate is imposed, with a diameter of nozzle fixed at $D=0.026 m$, to find jet Reynolds numbers indicating that $16.6×103≤Rej≤49.6×103$. Hydrodynamic flow and the heat exchange in forced convection are mostly affected by jet and rotation Reynolds numbers, defined as follows:
$Reω=Rω2/ϑair and Rej=UjD/ϑair$
Fig. 2
Fig. 2
Close modal

where $ϑair$ represents the air kinematic viscosity.

### Mesh Size.

The discretized transport equations are solved on the computational field, which is decomposed into many polyhedral elements. A polyhedral mesh was chosen as it has many advantages compared to cubic and tetrahedral meshes. The mesh of the spacing between two disks should be very fine. Indeed, for each studied case the mesh is very refinements close to the walls in order to have enough points in the boundary layer ensuring than the first layer of cells in the direction normal of wall is Y+< 5. This involves using a different mesh size when changing the injected airflow, the spacing G, and the rotational velocity of the rotor. A boundary layer of 10 meshes is modeled on each disk edge. The solution independence study from the mesh results to use a cell number ranging from $3×106$ to $9×106$ square and polyhedral meshes respectively for G = 0.04 up to G = 0.16.

#### Numerical Method.

The Navier–Stokes and the turbulent field transport equations as well as an energy equation were resolved by a direct Gauss-Seidel method and a SIMPLE algorithm evolved by Patankar [20], in this way, the problem of pressure velocity coupling can be solved by avoiding an irrational velocity and pressure field [21]. A second order spatial discretization [22] is selected for solving the convection terms and a PRESTO scheme was chosen for equation of pressure correction. This scheme offers enhanced interpolation of pressure for the force of the body or high pressure fluctuations in vortex flows [22].

### Turbulence Model.

Reynolds' formulation proposes to treat physical variables in a statistical way. These variables are divided into an average part and fluctuating part. The Reynolds decomposition is applied on Navier–Stokes equations. For the averaged quantities, the equations obtained are as follows:
$u¯j∂u¯i∂xj=−1ρ∂p¯∂xi+ν∂2u¯i∂xj∂xj−∂(ui′uj′¯)∂xj$
(5)
$uj¯∂T¯∂xj=−∂∂xj(λρcp∂T¯∂xj−T′uj′¯)$
(6)

where $Rij=ui′uj′¯$ and $T′uj′¯$ are called Reynolds tensor and turbulent heat fluxes that appear in the momentum transport (2) and energy (3) equations. The unknown's number of the problem is then greater to the number of available equations and it becomes essential to model the Reynolds stress tensor using turbulence models, in order to close the problem and relate the fluctuating field to the mean-field. For more details on turbulent flows and turbulence models, we invite the reader to consult the book by Schiestel [23].

## Turbulence Closure

### The k–ε Renormalization Group Model.

The present model is an improvement of the standard kε using techniques derived from renormalization band theory [24]. Analytical calculations obtained by this approach yield a model with different constants than the standard model, as well as additional terms in the $ε$ transport equation
$uj¯∂k∂xj=∂∂xj[(ν+νtσj)∂k∂xj]+νt(∂ui¯∂xj+∂uj¯∂xi)∂ui¯∂xj−ϵ$
(7)
$uj¯∂ϵ∂xj=∂∂xj[(ν+νtσϵ)∂ϵ∂xj]+Cϵ1Cμk(∂uj¯∂xi+∂ui¯∂xj)∂ui¯∂xj−Cϵ2*ϵ2kAvec Cϵ2*=Cϵ2+Cμη3(1−η/η0)ϵ2(1+βη3)kAvec Cϵ2*=Cϵ2+Cμη3(1−η/η0)ϵ2(1+βη3)k$
(8)

### The Reynolds Stress Model.

The basis of our approach is statistical modeling at one point using a second-order closure with small Reynolds number drifted from Launder's model [25] and which is sensitive to rotational impacts [26].

The Reynolds' general tensor Rij components is written, in relative terms
$uj¯∂ui′uj′¯∂xj=Pij+Ωij+Φij+Dij−εij+Oij$
(9)
where $Pij$,$Φij$, $Dij$, $εij$ simultaneously denote the following terms: production, diffusion, the pressure-deformation correlation term and the dissipation term. The Ωij is a Coriolis source term from the convection and production terms in the reference frame change. The term $Oij$ which takes account the implicit influences of rotation on turbulence is divided into four parts
$Oij=Φij(R)+DijR+Bij+Jij$
(10)

$Φij(R)$ is a derivative of pressure deformation correlation. The second term $DijR$ represents an inhomogeneous diffusion term that in presence of a wall decreases the tendency to bidimensionalization. The $Bij$ denotes a homogeneous source, corrected for $Φij$. In particular Cambon [27] has taken this into account when modeling homogeneous turbulence submitted to rapid rotation. The $Jij$ represents a corrective term of $εij$, thus enhancing the level of turbulence in the flow's central part.

Forced convection was effectively the primary heat transfer mechanism for the current system This scheme was successfully applied to similar rotor stator cavity applications [28] and to Taylor–Couette-Poiseuille flows [29]. The next temperature equation is given as follows:
$∂T∂t+VjTj=αTjj−Fj,jt$
(11)
where $FiT$ is turbulent thermal flux estimated using assumption gradient with tensor diffusion factor
$Fit=−ctkεRijTj where ct=0.01$

The RSM provides a level of closure adequate for description of the medium and turbulent ranges of these complex flows, and thus seems to be the best suited to study these flows. The problem remains, however, that computations using RSM turbulence models are lengthy and difficult to converge. For this reason, we chose an approach based on the two models Kϵ renormalization group (RNG) and RSM (Reynolds Stress Model), which allowed us to find a better compromise between computation velocity and results accuracy.

### Boundary Conditions.

The Limit conditions are determined below:

• In inlet: a speed and the profile of temperature (Uj et T∝) are imposed. A turbulence intensity equivalent to 5% is also imposed.

• On the walls $Ur=Uz=0$, except the tangential speed is equivalent to $Uθ=ωr$ (disk velocity at radius r) on rotor and $Uθ=0$ on stator.
$Uθ|rotor=ωrUθ|wall=0$
• In the exit, a condition of “output” is required which implies that the same flow rate required at the entry is achieved at the output.

The air is entering the cavity at T = 20, which also is the imposed temperature at the stator surface. A condition of adhesion to the cooled surface and a temperature of Trotor = 80 at the heated surface are imposed on the rotor. When exiting the domain, if outer fluid reenters into the cavity (the fluid velocity is negative), the temperature is fixed at T.

## Results and Discussion

### Flow Structure.

Numerical study of flow structure is still a major key to a greater understanding of local convective exchange phenomena into the rotating disk for the different dimensionless spacing studied. Figure 3 represents mean speed ranges in the air gap between both disks for G = 0.08, $2.38×105≤Reω≤5.44×105$, and $Rej=33×103$. Figure 3(a) shows proximity the impingement zone, the radial velocity field (Ur/Uj) showed positive over-speeds on rotor as well as negative over-speeds on stator. These overspends correspond to the direction change of the flow and its confinement by the recirculation area. For large radii r/R > 0.7, we notice that the radial velocities are negative. Such speeds refer to an area of recirculation representing an external fluid inlet from outside. This results in a centripetal flow on stator and a centrifugal flow on rotor (Fig. 3(b)).

Fig. 3
Fig. 3
Close modal

So as to represent globally the flows generated within the air gap separating the two disks the whole speed field is explored.

In Figs. 46, we represented the axial profiles of the radial and tangential components of air speed inside the air gap between the rotor (Z/H = 0) and the stator (Z/H = 1), obtained numerically for dimensionless spacing between G = 0.02 [24] and 0.16, a jet Reynolds number varying from $Rej=16.6×103$ to $Rej=49.6×103$ and rotational Reynolds varying from $Reω=2.38×105$ to $Reω=5.44×105$. The radial and tangential components of the velocity are normalized to the jet speed $Uj$ and the local tangential velocity of the rotor $ωr$, respectively.

Fig. 4
Fig. 4
Close modal
Fig. 5
Fig. 5
Close modal
Fig. 6
Fig. 6
Close modal

For two dimensionless spacing $G=0.04$ and 0.08 (Figs. 4 and 5), the radial velocity component values for radii higher than $r/D>6$ are extremely important next to the rotor, where the centrifugal forces are highest. Close to stator, the radial velocity decreases strongly depending on of “Z” and becomes negative. This corresponds to air entering the air gap from the outside near the stator. The flow structure is more similar to Batchelor style [12] with a centrifugal flow around the rotor and another centripetal around the stator (a limit layer on the surface of each disk). For dimensionless radii less than $r/D<6$, the radial velocities are negative, indicating centripetal flow all along the air gap. This also implies a recirculation area near the impingement point where the flow is dominated by the eccentric airflow flux.

Near the rotor, a centrifugal flow is mainly generated by the inertial force exerted to fluid in the limit layer associated with rotating disk. This radial centrifugal flow is thus accelerated from the rotating axis to the periphery. Near the stator, a centripetal flow is created in the limit layer related to the stator where the flow is in crown shaped. This flow has formed a fluid-flow inlet from stator limit layer to rotor limit layer. As a result, we perceive an augmentation of radial velocity at the boundary layer related to the rotor and a reduction of this velocity at the boundary layer related to the stator.

The values of the tangential component obtained for different radii are very high near the rotor (Z/H = 0) where the flow is mainly entrained by the disk rotation. It then reduces away in the direction normal to the rotating disk (Z) reaching zero at the surface of the stator (Z/H = 1). Tangential component of speed at the rotor surface increases with radial position for $r/D>6$ and decreases for $r/D<6$.

When G increases again, a single boundary layer forms next to the rotor disk. . The flow tends toward the Stewartson [13] type flow, whose radial and tangential components of speed are almost null when moving away from the rotor. Thus for G = 0.16 and $Reω=5.44×105$ (Fig. 6), we notice that the radial and the tangential components are very great next the rotor then reduce quickly with 'z' to become almost zero in the midplane between both disks. The velocity tangential component is a growing function with radial position.

## Local Nusselt Numbers

### Generally.

Several authors have observed that the injected air-flow increases the local Nusselt numbers. This increase is more important as the flow of air injected through the stator is greater because the renewal of the air inside the cavity becomes more rapid. Figures 7(a), 8(a), and 9 illustrate the profiles for local Nusselt number at constant rotational Reynolds number. These profiles show there is an area near the impingement point where heat transfer is dominated by the jet-related. This corresponds to the observations of different authors, notably by Chen [7] and Popiel [9]. In addition, this area becomes larger as the G spacing between the two disks increases. Figures 7(b), 8(b), and 10 show the profiles of local Nusselt numbers for a jet Reynolds number fixed. It seems all profiles are inclined to the same boundary, highlighting that at external radii there exists a region in which the convective heat exchange is dominated by disk rotation. Indeed, the jet effects decrease as the radius increases while the rotation becomes more and more influential.

Fig. 7
Fig. 7
Close modal
Fig. 8
Fig. 8
Close modal

We present an experimentally and numerically local Nusselt numbers evolution on the rotating disk surface according to the dimensionless radius (r/D). The measurements are carried out for dimensionless spacing G between 0.04 and 0.16 and for jet Reynolds numbers between $16.5×103$ and $49.5×103$. The rotation effect on the convective transfers is examined for $2.38×105≤Reω≤5.44×105$.

#### ▪  G = 0.04

For G = 0.04, a fixed rotational Reynolds number $Reω$ (Fig. 7(a)), and different values of the jet Reynolds number ranging from $Rej=16.5×103$ to $Rej=49.5×103$, the local convective exchanges are lightly influenced by the injected air flows for low radii $(r/D <4)$. In the proximity of the impingement point, and for $4≤r/D≤8$, the influence of the injected air current is very strong and causes a strong increase in heat exchange corresponding to the presence of two peaks. The two peaks simultaneously are located at $r/D=5$ and $r/D=6.5$. For fixed $Rej$ (Fig. 7(b)), the Nu is weakly dependent on $Reω$ for small radii $(r/D<4)$. In the proximity of the impingement point, heat transfers meet. This means that the rotation loses its influence to the jet. For radii $r/D>9$, the local Nusselt profiles for the different rotational velocities have diverged, which shows the decreasing jet effect on the heat exchange on the rotating disk surface. However, the greater the injected air flow $Rej$, the greater the r/D at which the local Nusselt number Nu curves differ.

#### ▪  G = 0.08

For G = 0.08 and Reω fixed (Fig. 8(a)), the general evolution of local Nusselt number remains similar. At fixed $Reω$, We have also observed variations in the local Nusselt number profile on the rotating disk surface with the variation of the spacing G. Notably, an increase in the impingement area size $(3.5≤r/D≤8)$ was noted by the increase both in the injected air flow and the dimensionless spacing G. For a jet Reynolds number $Rej$ fixed and larger radii $r/D≥8$, the local Nusselt numbers is larger at high speeds.

#### ▪  G = 0.16

For a wide spacing configuration G = 0.16, the results are similar to those achieved on an only rotating disk submitted an impinging air jet, a configuration studied previously [8]. Thus, when $Reω$ fixed (Fig. 9), the impingement zone where a single peak of the convective exchange occurs is quite large. It can be clearly distinguished for $Reω=5.44×105$. It is also noted that, at the air-gap outlet, the curves relating to the various injected flows tend to converge. This means that the jet decreases its influence to favor of the rotation. The local Nusselt number profile becomes similar to that achieved for the single rotor. For a constant injected mass flowrate (Fig. 10), we show the results independent of the rotational Reynolds number $Reω$ at low radii and close to the impingement point. For higher radii, the curves separate, as expected from the literature. This shows the decreasing influence of the jet on the local convective heat exchange at outer radii where the rotational velocity becomes increasingly influential. However, we also note the greater the injected flowrate in the air gap, the higher the separation of the curves becomes for dimensionless radii high r/D. The results of our numerical and experimental studies reveal identical trends for radial evolution the local Nusselt numbers and mean relative deviation is less than 7%.

Fig. 9
Fig. 9
Close modal
Fig. 10
Fig. 10
Close modal

### Synthesis.

The previous studies showed that the parameters G, $Rej$, and $Reω$ affect local convective heat exchanges for a rotating disk.

• For a spacing G and a jet Reynolds number $Rej$ fixed, an increase of $Reω$ leads to an increase of a local Nusselt number at the outer radii. However, for lower radii, the rotation velocity has a slight influence on a local Nusselt number.

• For spacing G and a rotational Reynolds number $Reω$ fixed, an increase of $Rej$ results in an increase of the local convective heat exchanges next to the point of impact where the effect of the jets is greatest.

• For $Reω$ and $Rej$ fixed, an augmentation of G leads to an enlargement of the impingement zone in which the effect of the jet is prevalent.

### Mean Nusselt Numbers $Nu¯$⁠.

The mean Nusselt number evolution according to the three studied parameters $G, Reω, Rej$ is represented in Fig. 11. We represented on this graph our numerical as well as experimental work results achieved from different G values. The results achieved on the unique rotating disk with a jet were also represented on this curve.

Fig. 11
Fig. 11
Close modal

### Influence of $Rej$⁠.

In general, we find that increasing in the flowrate injected into the air gap results in a significant improvement in the overall heat exchange. The mean Nusselt number curve obtained numerically are increasing functions of $Rej$ what is consistent with the experimental data. We also show that, at a fixed spacing, regardless of rotational velocity, the slopes are similar. However, for a chosen spacing, the slopes are different. We discussed the sensitivity of the results to a change in the dimensionless spacing G in the next section.

### Influence of G.

In general, at low spacing, the mean Nusselt numbers are significantly increased by adding a jet. For higher spacing, we show that the values increase slightly as the spacing increases. Thus, contrary to the no-jet case [11], the mean Nusselt number values do not reach a limit value. However, the spacing is not yet sufficient for the air-jet addition to having a negligible influence on the transfers to the rotor. For higher spacing than those tested here, the mean Nusselt numbers, at $Reω$ and $Rej$ fixed, tend toward this same limit. In fact, to find identical results to the single disk configuration, the spacing must be greater than the injected flowrate. However, we note differences depending on the rotational velocity, particularly with regard to the transition between low and high spacing. We will now see the sensitivity of the results to a variation in the rotational Reynolds number $Reω$.

### Influence of $Reω$⁠.

At a low dimensionless spacing G = 0.02, the mean Nusselt numbers change slightly with the rotational speed [10]. Viscosity influence are therefore more important than inertial effects. For dimensionless spacing $G=0.04$ and $G=0.08$, the heat exchange at the surface of the disk generally improves with increasing Rew, which is consistent at the same observations made by Pellé [16] and Poncet [17]. This increase in Rew is accompanied by a more efficient cooling of the rotor. We also show that the increase slopes are higher for the highest spacing. For a G = 0.16, we can notice that the mean Nusselt number values improve with increasing rotation speed and tend toward the results achieved for a single disk case with jet [8]. In fact, the stator's influence is not so important and the inertial forces become more important than those of viscosity.

### Flow and Heat Transfer Similarity.

The computational study of the structure of the flow within the cavity between rotor and stator provides fundamental details about the radial evolution in the local convective transfer on the rotor. For a dimensionless spacing and for the full gamut of rotational Reynolds numbers $(2.38×105≤Reω≤5.44×105)$, a zone close to the impingement point where the flow is primarily centrifugal-centripetal, and where the phenomena is dictated by the air jet. This will increase the wall stresses over the rotor and improve the convective exchange. As the flow goes away from the point of impact, the fluid gradually becomes rotated due to the viscosity effects. The radial centrifugal flow is then gradually confined near the rotor. At high radii, a centrifugal flow appears near the stator. Therefore, fresh air is supplied via the stator opening and the stator periphery. Parietal stresses are increased by the confinement of the radial centrifugal flow close to the rotor and the heat transfers are also increased. As the spacing increases, the tangential velocity component decreases and the centrifugal flow tends to disappear. The flow structure then tends toward that achieved on an only rotating disk with impinging jet.

Concerning the $Nu¯$, we had noticed that the increase of $Rej$ resulted in increasing the overall exchange for different dimensionless spacing G and different rotational velocities. It is primarily a result of increased local exchange in the impact area, and thus also increases the size. Indeed, the flow is essentially centrifugal in a major section of the air gap. The $Nu¯$ depends on $Reω$ provided the rotational speed is sufficient for a rotational-dominated regime. When $Reω$ is large enough, $Nu¯$ increases with $Reω$, which is owing to the increase in local heat exchange at the exit of the air gap where the heat transfer is controlled by rotation.

## Correlations

### Local Nusselt Numbers $Nu$⁠.

The jet's influence on local heat transfer is revealed by an important augmentation of the convective exchange in the impingement region. For this reason, correlation research was nevertheless carried out in order to give an idea of the maximum local Nusselt number encountered in the jet-dominated zone. In this zone, the effects of the disk rotation on heat transfer are negligible. The correlation should therefore not involve $Reω$. However, the amplitude of the convective exchange varies according to G and $Rej$. The evolution of this maximum with spacing is not linear. We have therefore divided the search for correlation into three groups ($0.02,$0.04, and $0.08). The search was carried out so that changes in the Nusselt maximum are expressed by power-type laws involving G and $Rej$ as follows:

• For 0.02 < G < 0.04
$Numax=4.49G0.02Rej0.43$
(12)
• And for 0.04< G < 0.08
$Numax=7.99×G0.21×Rej0.43$
(13)
• Finally, for 0.08 < G < 0.16
$Numax=4.54G−0.02Rej0.43$
(14)

The variation of Reynolds has a similar influence on local Nusselt number whatever the spacing G considered, in effect, the power does not vary. On the other hand, a variation in G spacing has different effects depending on the range where one is. The correlations give perfectly acceptable results since the average relative difference with the numerical results is less than 7%.

The value $Numax$ can also be located on the rotor surface. The position of this maximum value is not influenced by the flowrate injected through the stator. It only depends on the dimensioned spacing G. A correlation was searched to represent this
$(r/D)max=6.28×exp (1.23×G)$
(15)

This is an increasing function with G, which reflects the expansion of the jet-related at the exit of the tube, which is greater as the G spacing is greater.

### Mean Nusselt Numbers.

Correlation research was carried out on mean Nusselt numbers so that the predominance of the jet effects or rotational effects can be observed in a simple way. Power laws were found for three dimensionless spacing ranges G, for $Reω≥2.38$ and $16.5×103≤Rej≤49.5×103$. The search was primarily carried out for the lowest spacing G = 0.02, the law found is such that:
$Nu¯=0.52Reω0.314Rej0.27$
(16)
• For $0.04≤G≤0.08$, we obtain:
$Nu¯=0.6 G0.22Reω0.43Rej0.17$
(17)
• Finally for G = 0.16, we found
$Nu¯=0.086Reω0.56Rej0.15$
(18)

In the three ranges presented, the powers at the level of $Rej$ decrease with the increase of G while those at $Reω$ increase. This highlights the inversion of the preponderant effects for heat transfer with increasing spacing. The proposed correlations are in good agreement with our results with an average relative deviation of 10%. Some comparisons between the results obtained numerically and those obtained with the correlations are presented in Fig. 12.

Fig. 12
Fig. 12
Close modal

## Conclusion

This paper investigated the jet influence on convective heat transfer onto a rotor surface in a discoidal system.

The study of flow structure across both disks showed that three zones are distinguished. A first for a small radius where flow is centripetal along the entire cavity and another, for a higher radii where the flow structure is nearest the Batchelor style flow with a centrifugal flow near the rotor and another centripetal near the stator. When G increases, the flow structure between the disks tends toward the Stewartson type with a single limit layer near the stator.

Analysis of the numerical and experimental results of local Nusselt number identified three zone on the rotor surface. One zone near the air inlet where heat exchanges are independent of the rotation velocity, a second zone, called mixed, where heat exchanges are low influenced by the airflow rate injected and the rotational speed, and a third where the influence of the jet becomes negligible and where only the rotation velocity determines the local heat exchanges. The mean Nusselt numbers correlated with a relative difference of less than 10% by a power law.

Comparison of the Local and Mean Nusselt Number profiles over the rotating disk surface shows that the numerical results provided by both turbulence models kε RNG as well as RSM are in good accordance with experimental results. An approach based on the two models Kε RNG and RSM has allowed us to find a better compromise between the computational rapidity and the precision of results.so as to enhance this study and to better the cooling, an investigation of a rotor–stator system using a multiple jet could be carried out.

## Nomenclature

D  =

jet diameter (m)

H  =

rotor-stator spacing (m)

h  =

convective heat transfer coefficient (W m−2 K−1)

R =

outer radius of rotor (m)

Rj =

r =

radial position on the rotor (m)

$T ∞$ =

atmospheric temperature (K)

$T rotor$ =

temperature of the heated rotor surface (K)

Uj =

mean velocity of the jet (ms−1)

Ur =

radial component of velocity (ms−1)

Uθ =

tangential component of velocity (ms−1)

Uz =

axial component of velocity (ms−1)

z =

axial position

### Dimensionless numbers

Dimensionless numbers
G =

dimensionless spacing between rotor and stator

Nu =

local Nusselt number

$Nu¯$ =

mean Nusselt number

Reω =

rotational Reynolds number

Rej =

the jet Reynolds number

### Greek Symbols

Greek Symbols
$λair$ =

air conductivity (W m−1 K−1)

$λzir$ =

Zircon conductivity (W m−1 K−1)

σ =

Stefan–Boltzman constant (W m−2 K−4)

$εr$ =

rotor emissivity

$εs$ =

stator emissivity

$C$ =

air transmission coefficient

$φ$ =

heat flux density (W m−2)

### Subscripts and Superscripts

Subscripts and Superscripts
cd =

conduction

cv =

convection

j =

jet

r =

rotor

ray =

RNG =

renormalization Group

RSM =

Reynolds stress transport

SST =

shear stress transport

s =

stator

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