High-order direct numerical simulations of a turbulent round impinging jet onto a rotating heated disk in a highly confined cavity

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High-order direct numerical simulations of a turbulent round impinging jet onto a rotating heated disk in a highly confined cavity R. Oguic a, S. Poncet a,b,∗, S. Viazzo a a

Aix-Marseille Université, CNRS, Ecole Centrale Marseille, Laboratoire M2P2 UMR 7340, 38 rue F. Joliot-Curie, Technopôle de Château-Gombert, 13451 Marseille, France b Université de Sherbrooke, Faculté de génie, Département de génie mécanique, 2500 Boulevard de l’Université, Sherbrooke (QC) J1K 2R1, Canada

a r t i c l e

i n f o

Article history: Received 12 January 2016 Revised 25 April 2016 Accepted 17 May 2016 Available online xxx Keywords: Impinging jet Rotor-stator Direct numerical simulation Turbulent flow Heat transfer

a b s t r a c t The present work reports Direct Numerical Simulations (DNS) of an impinging round jet onto a rotating heated disk in a confined rotor-stator cavity. The geometrical characteristics of the system correspond to the experimental set-up developed by [J. Pellé and S. Harmand. Heat transfer study in a rotor-stator system air-gap with an axial inflow. Applied Thermal Engineering, 29:1532–1543, 2009.]. The aspect ratio of the cavity G = h/Rd between the interdisk spacing h and the rotor radius Rd is fixed to 0.02 corresponding to a narrow-gap cavity. The axial Reynolds number Rej based on the jet characteristics is also fixed to Re j = 5300, while the rotational Reynolds number Re may vary to preserve the swirl parameter N∝Re /Rej (0 ≤ N ≤ 2.47) between the present simulations and the experimental data of [J. Pellé and S. Harmand. Heat transfer study in a rotor-stator system air-gap with an axial inflow. Applied Thermal Engineering, 29:1532–1543, 2009.] and [T. D. Nguyen, J. Pellé, S. Harmand, and S. Poncet. PIV measurements of an air jet impinging on an open rotor-stator system. Experiments in Fluids, 53:401–412, 2012.] for comparisons. The results are discussed in terms of radial distributions of the mean velocity components and corresponding Reynolds stress tensor components. The swirl parameter does not modify the size of the recirculation bubble developed along the stator close to the pipe exit. For N ≥ 1.237, centrifugal effects at the rotor periphery are balanced by a centripetal flow along the stator. Some spiral patterns develop then in the stator boundary layer corresponding to the SRIII instability of [L. Schouveiler, P. Le Gal, and M. P. Chauve. Instabilities of the flow between a rotating and a stationary disk. Journal of Fluid Mechanics, 443:329–350, 2001.] in an enclosed cavity. The numerical results are found to agree particularly well with the experimental data in terms of the distribution of the local Nusselt number along the rotor. Finally, a correlation for its averaged value is proposed according to the swirl parameter. © 2016 Elsevier Inc. All rights reserved.

1. Introduction The objective of the present work is to investigate by highorder Direct Numerical Simulations the turbulent flow and associated heat transfer produced by a round centered jet impinging onto a rotating heated disk inside a confined rotor-stator cavity. More generally, impinging jets are used in a wide variety of industrial applications, from the drying of textile, papers or metal plates, to the cooling of electrical components. When the impingement plate is a rotating disk, the applications are more related to harddisk drives, gas turbines or any rotating machinery. The present work has been more specifically motivated by the effective cool∗

Corresponding author. Tel.: +1 819 821 80 0 0-62150. E-mail addresses: [email protected] (R. Oguic), [email protected], [email protected] (S. Poncet), [email protected] (S. Viazzo).

ing of an alternator placed in a wind generator. It consists of a discoidal rotor-stator system, which does not use gears allowing the generators to operate at low rotational speeds while reducing energy losses. The main technological lock consists in solving the ineffective cooling due to high electrical losses dissipated for a relative low rotational speed. An improvement on the cooling of discoidal rotor-stator alternators could be obtained by using air jet impingement. The literature is abundant about the fluid flow and heat transfer in rotor-stator cavities. For an exhaustive state-of-art, the reader can refer to the review of Launder et al. (2010) or to the monographs of Owen and Rogers (1989), Shevchuk (2009) or Childs (2010). Some authors investigated the influence of a superimposed axial outward throughflow coming from the top disk at a given radial location r = 0, either experimentally (Firouzian et al., 1986; Sparrow and Goldstein, 1976) or numerically (Iacovides and Theofanopoulos, 1991; Poncet and Schiestel, 2007). The physical

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Please cite this article as: R. Oguic et al., High-order direct numerical simulations of a turbulent round impinging jet onto a rotating heated disk in a highly confined cavity, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.013

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Nomenclature D G h H N Nr , Nθ , Nz Nu Nur Pr r, θ , z Rd Rj Rej Re T Uj ur , uθ , uz

t 

κ ν ω 

∗ ing r.m.s. rotor ∞ DNS PIV POD RSM

diameter of the inlet pipe (m). aspect ratio of the cavity (− ). interdisk spacing (m). length of the inlet pipe (m). swirl parameter (− ). number of mesh points in the radial, tangential and axial directions (− ). mean Nusselt number (− ). local Nusselt number (− ). Prandtl number (− ). cylindrical coordinates (m, rad, m). radius of the rotating disk (m). radius of the inlet pipe (m). axial Reynolds number (−). rotational Reynolds number (−). temperature (K). inlet mean axial velocity (m/s). mean radial, tangential and axial velocity components (m/s). time step (s). ratio between the interdisk spacing and the inlet pipe diameter (− ). thermal diffusivity of the fluid (m2 /s). kinematic viscosity of the fluid (m2 /s). vorticity magnitude (1/s). rotation rate of the disk (rad/s). denotes a normalized quantity. ingress. root-mean-square value of the fluctuating velocity (m/s). denotes a quantity evaluated at the rotor surface (z = 0). denotes a quantity evaluated at the pipe inlet. Direct Numerical Simulation. Particle Image Velocimetry. Proper Orthogonal Decomposition. Reynolds Stress Model.

phenomena get more complex because of the interactions between the main tangential flow due to the bottom disk rotation and the secondary flow produced by the annular and decentered jet. Such unsteady and three-dimensional flows are very challenging for numerical methods because of the coexistence of laminar, transitional and turbulent flow regions, very thin boundary layers, curvature and confinement effects, heat transfer, high geometrical aspect ratio ...For impinging jet flows onto a stationary plate, the flow dynamics is also rather complex due to highly turbulent shear layers, wall-jet interactions, the presence of secondary vortices ...as depicted in the review by Zuckerman and Lior (2006). When a centered round jet impinges a rotating disk, the resulting flow and heat transfer are somewhat different but gather all these complexities. The hydrodynamic and thermal fields are governed by many parameters, whose the axial Reynolds number Re j = U j D/ν based on the incoming jet velocity Uj and the nozzle diameter D, the rotational Reynolds number Re = R2d /ν based on the disk rotation rate  and the disk radius Rd , the aspect ratios of the rotor-stator cavity G = h/Rd (h the interdisk spacing) and the inlet nozzle, the flow and thermal boundary conditions, the type of fluid ...among other things. The injection shape greatly affects also the physical phenomena inside the cavity as shown by some authors (Cafiero et al., 2014; Meslem et al., 2013; Whelan and Robinson, 2009). Please

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In the case of a round jet impinging perpendicularly onto a rotating heated disk in an opened computational domain, the measurements of Minagawa and Obi (2004) for the hydrodynamic fields and Popiel and Boguslawski (1986) for the convective heat transfer provided very useful experimental database. Among other issues, (Popiel and Boguslawski, 1986) highlighted the existence of three distinct regions: 1- the area of the disk where the jet influence on the heat transfer is the greatest near the impingement point; 2- an intermediate zone between regions 1 and 3; 3- the area where rotation has the greatest effect far from the impingement point at high radii. These data served in the framework of the 13th ERCOFTAC/IAHR Workshop on Refined Turbulence Modeling for numerical benchmarking. The performances of axisymmetric and steady-state turbulence models have been evaluated. Manceau et al. (2009) compared five turbulence models to evaluate the model performances and to investigate the effect of rotation on turbulence and heat transfer. 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. They concluded by the need for more advanced modeling development to sensitize turbulence models to the effect of rotation. It has been later extended by Manceau et al. (2014) for Re j = 14500, h/D = 5 and four rotation rates. They provided a very detailed analysis of the flow mechanisms in the free-jet and selfsimilar wall jet regions and proposed a scenario for the appearance of rotational effects. The four turbulence models considered predict well the wall jet, while failing to predict the mean and turbulent flow fields in the free-jet area. In the following, one will focus only on the impinging jet imposed through a long cylindrical tube in an unshrouded rotorstator cavity. Sara et al. (2008) have experimentally investigated the mass transfer between an impinging jet and a rotating disk in a confined system by naphthalene sublimation for 2 ≤ h/D ≤ 8, 17, 0 0 0 ≤ Re j ≤ 53, 0 0 0 and intermediate values of the rotational Reynolds number 34, 0 0 0 ≤ Re ≤ 120, 0 0 0. For those ranges of parameters, the jet has a dominating influence and the authors concluded that the mass/heat transfers increase by increasing both Re and Rej . Pellé and Harmand (2009) measured the local convective heat transfer coefficient along the rotor by infrared thermography for Rej up to 4.16 × 104 , Re = [2 × 104 − 5.16 × 105 ], h/D = 0.25 and G = [0.01 − 0.16]. Whatever the flow parameters, the jet always enhances the heat transfer compared to the no-jet configuration. They found that the size of region 1 near the stagnation point strongly depends on G and Rej , while at outer radii, the local Nusselt number on the rotor depends on both G and Re . They proposed correlations for the average Nusselt number along the rotor for different ranges of aspect ratio. Nguyen et al. (2012) provided the first quantitative experimental database for the mean and turbulent flow fields in the case of an impinging round jet in an open rotor-stator system (G = 0.02,  = h/D = 0.25) for 0.33 × 105 ≤ Re ≤ 5.32 × 105 and 17.2 × 103 ≤ Rej ≤ 43 × 103 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. 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. For a detailed review about the fluid flow and convective heat transfer in rotating disk cavities with an impinging jet, the reader can refer to the recent work of Harmand et al. (2013). From a numerical point of view, the centered jet introduces a singularity on the axis due to the presence of the 1/rn terms (n = direct

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1, 2) in the Navier–Stokes and heat equations for solvers based on cylindrical coordinates. Dedicated numerical methods have been then to be developed to overcome this singularity but also to take into account the very low aspect ratio G of the cavity. Due to the excessive computational resources required to simulate such complex problems with advanced numerical solvers, most of the calculations were limited to date to steady-state axisymmetric calculations using two-equation turbulence models not able to capture the coherent structures responsible for high heat transfer. The present paper is an attempt to fill this gap. The geometrical parameters correspond to the experimental set-up developed by Pellé and Harmand (2009) for heat transfer measurements and by Nguyen et al. (2012) for PIV measurements. Poncet et al. (2013) used these huge experimental data to perform a numerical benchmark of some advanced turbulence models. Though a particularly good agreement has been obtained for the radial distribution of the local Nusselt number along the rotor, some discrepancies remained between the experiments and the numerics. It mainly concerned the main flow in the impinging jet flow region, and especially both the jet width at low radii and the size of the recirculation zone along the stator. The axial extent of the centripetal flow along the confinement disk is underestimated by the models. The RSM does not also capture the weak secondary peak in the Nusselt number distribution. The authors concluded about the necessity to investigate the flow field and the turbulence level inside the rotor boundary layer to highlight or not the presence of three-dimensional turbulent structures not captured by the steady and axisymmetric turbulence models and not accessible by classical PIV measurements. The main objectives of the present work are then three-fold: (1) highlight (or not) the existence of three-dimensional coherent vortices within the rotor boundary layer, which may be responsible of higher heat transfer; (2) give a detailed insight about the flow structure, especially in the impinging jet flow region, with a particular emphasis on the recirculation zone along the stationary confinement disk; (3) investigate the heat transfer process in light of the flow dynamics and propose a correlation for the average Nusselt number along the rotor according to the rotation rate. The paper is then organized as follows: the numerical modeling is briefly presented in Section 2. The influence of the swirl parameter N on the hydrodynamic and thermal fields is discussed in details in Sections 3 and 4, respectively, before some concluding remarks in Section 5. 2. Numerical modeling High-order Direct Numerical Simulations using a multidomain and parallelized solver are performed to investigate the fluid flow and heat transfer in a very elongated rotor-stator cavity with a centered round impinging jet. 2.1. Geometrical modeling and flow parameters The rotor-stator cavity corresponds to the experimental setup considered by Pellé and Harmand (2009) and Nguyen et al. (2012) sketched in Fig. 1. Air flows in the inlet cylindrical pipe of radius R j = 13 mm and length H = 52 mm before impinging a smooth rotating disk of radius Rd = 310 mm. The rotor rotates at a constant angular velocity . The flow is confined by a smooth stationary disk (the stator). The interdisk spacing h is fixed to h = 6.5 mm. The geometry may be fully characterized by the following three normalized parameters:

H =2 D

G=

h = 0.02 Rd

=

h = 0.25 D

(1)

where D = 2R j = 0.026 m is the inlet nozzle diameter. The length of the inlet pipe, being not defined in Nguyen et al. (2012); Pellé Please

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Table 1 Oguic et al. submitted to Int. J. Heat Fluid Flow. Rej

Re

corresponding N

Pr

Present DNS

5300

17.2 × 103

0 0.153 1.237 2.47 0

0.7

Experiments (Nguyen et al., 2012; Pellé and Harmand, 2009)

0 1.01 × 104 8.2 × 104 16.36 × 104 0

0.33 × 105 2.66 × 105 5.32 × 105

0.153 1.237 2.47

0.7

and Harmand (2009), has been chosen long enough to ensure a turbulent velocity profile at the inlet of the rotor-stator cavity. The hydrodynamic flow and the heat transfer in the forced convection regime are mainly governed by the axial Rej and rotational Re Reynolds numbers and the Prandtl number Pr defined as:

Re j =

UjD

ν

Re =

R2d ν

Pr =

ν κ

(2)

where ν and κ are the kinematic viscosity and thermal diffusivity of air. The experimental studies of Pellé and Harmand (2009) and Nguyen et al. (2012) have been performed for very high values of the axial Reynolds number (Rej ≥ 1.72 × 104 ). To reach such values by DNS, very fine meshes would be required together with long integration time. For this reason, the axial Reynolds number is fixed to Re j = 5300. To enable more direct comparisons, the rotational Reynolds number has been fixed to preserve the swirl parameter N defined as:

N=

D Re × Re j Rd

(3)

Poncet et al. (2014) have shown that this parameter governs the mean hydrodynamic flow in Taylor–Couette systems with an axial throughflow. All values of the flow parameters (Rej , Re , N, Pr) are summarized in Table 1. In the absence of an impinging jet, considering only G and Re , the base rotor-stator flows would have remained in the regime I, laminar with merged boundary layers, as defined in the non-isothermal case by Pellé and Harmand .2 (2007). Note also that G = 0.02 < Glim , with Glim = 1.05Re−0 =  0.095 (Owen et al., 1974) obtained for the most constraining case Re = 16.36 × 104 . It means that the stationary disk will greatly affect the flow produced by the rotor. 2.2. Numerical method The Navier–Stokes equations written in primitive variables are discretized in cylindrical coordinates using a second-order semi-implicit Adams-Bashforth/Backward-Differentiation temporal scheme (Oguic et al., 2015). The pressure-velocity coupling is overcame through a projection scheme based on a preliminary pressure. The spatial discretization is achieved by fourth-order compact schemes in the radial and axial directions and Fourier series in the tangential direction. The axis singularity is treated by the method proposed by Sandberg (2011) and extended to the fourthorder. Appropriate symmetry conditions are imposed for each variable and wave number. The projection method reduces the Navier–Stokes equations into a set of Helmholtz and Poisson equations solved by a diagonalization technique. A multidomain approach based on the continuity influence matrix technique has been developed to deal with more complex geometries (Abide and Viazzo, 2005). A direct

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Fig. 1. Oguic et al. submitted to Int. J. Heat Fluid Flow.

Table 2 Oguic et al. submitted to Int. J. Heat Fluid Flow.

Table 3 Oguic et al. submitted to Int. J. Heat Fluid Flow.

Subdomain

1

2

3

i = 4 → 12

Number of OpenMP cores

1

2

3

4

(Nr , Nθ , NZ ) rmin rmax zmin zmax r(θ )max + rmin + rmax + zmin + zmax

(70, 256, 185) 5.2 × 10−3 2.3 × 10−2 2 × 10−3 5.1 × 10−2 2.5 × 10−2 0.71 0.92 − −

(70, 256, 70) 5.2 × 10−3 2.3 × 10−2 1.5 × 10−3 1.4 × 10−2 2.5 × 10−2 − − 0.21 0.8

(110, 256, 70) 4.9 × 10−3 2.2 × 10−2 1.5 × 10−3 1.4 × 10−2 6.1 × 10−2 − − 0.45 0.85

(150, 256, 70) 13 × 10−3 1.3 × 10−2 1.5 × 10−3 1.4 × 10−2 (5π [i − 2] )/256 − − 0.51 0.69

CPU time per iteration (s) CPU time per node and per iteration (s)

43.2 1.4 × 10−6

22.3 0.7 × 10−6

14.6 0.5 × 10−6

11.5 0.4 × 10−6

hybrid MPI/OpenMP parallelization has been also implemented with a good scalability up to 64 processors. The solver has been validated against analytical solutions and experimental or numerical data available in the literature for the vortex breakdown in a cylindrical cavity and turbulent pipe flows. The method both preserves the scheme order and does not introduce any discontinuity or spurious oscillations neither along the axis nor at the interface between subdomains. The complete solver is fully explained and validated in Oguic et al. (2015). 2.3. Numerical settings The computational domain is divided into 12 fluid subdomains, 1 for the inlet pipe and 11 for the rotor-stator cavity, as shown in Fig. 1. The azimuthal direction is discretized by Nθ = 256 mesh points. The numbers of mesh points in the radial Nr and axial Nz directions vary depending on the subdomain. The mesh grid is refined in the near-wall region to guarantee at least 8 mesh points in the viscous sublayer, such that the maximum values of the wall + + coordinates in the two inhomogeneous directions rmax and zmax are still lower than unity as shown in Table 2. The same time step t = 5 × 10−4 s (or

tU j Rj

= 2.5 × 10−4 ) has

been used in all simulations. 12 MPI tasks, one by subdomain, are associated with n = 1 → 4 OpenMP cores by MPI task, depending on the availability of the computational resources. Table 3 presents the performances of the present solver. The statistical averaging Please

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starts after 1800 s, while the whole integration time for all simulations is set to T = 2202 s (or T U j /R j = 1101). At the pipe inlet, a turbulent profile calculated using the numerical procedure proposed by Kempf et al. (2005) is imposed. Velocity is first decomposed into mean and fluctuating components. The objective is to construct a spatially correlated signal for the fluctuating components satisfying a given profile for the Reynolds stress tensor. A randomly distributed noise is generated then normalized to account for the size of the calculation cells. The spatial scale for the signal is of the order of half the size of the calculation cell. A diffusion equation is then solved to generate coherent struc√ tures of size L = 2π cnz, where c = Dt/z2 . n is the number of temporal iterations, D is the diffusion coefficient, z is the size of the grid cell in the axial direction and t is the time step. All details are given in Kempf et al. (2005) and Oguic et al. (2015). No-slip boundary conditions are imposed at the walls. All velocity components are set to zero except from the tangential component fixed to r on the rotating disk, where r is the local radius. The choice of appropriate boundary conditions at the outlet is still an open and crucial question. Numerical tests have shown here that Neumann boundary conditions for velocity are more appropriate than convective conditions. For the thermal field, the same boundary conditions as in Poncet et al. (2013) have been considered. Air enters the cavity at a given temperature T∞ = 20◦ C. The walls of the inlet pipe are assumed to be adiabatic. The rotor and stator temperatures are fixed to Trotor = 80◦ and T∞ , respectively. Though a large temperature gradient is imposed between the two disks, it has been verified that the Richardson number remains much weaker than unity (O(10−4 )), corresponding thus to the forced convection regime. It enables to consider temperature as a passive scalar, such that only the Prandtl number Pr is relevant here. At the cavity outlet, if some direct

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Fig. 2. Oguic et al. submitted to Int. J. Heat Fluid Flow.

Fig. 3. Oguic et al. submitted to Int. J. Heat Fluid Flow.

external fluid enters the cavity, temperature is fixed to T∞ . If not, a Neumann-type boundary condition is imposed for temperature. 3. Hydrodynamic fields The influence of the swirl parameter N is discussed in terms of the mean and turbulent flow fields and compared with previous published data (Nguyen et al., 2012; Poncet et al., 2013) for the same geometry. Finally, three-dimensional coherent structures have been identified both in the jet- and rotation-dominated areas. 3.1. Mean flow It is first important to recall that for the basic configuration of a jet impinging onto a stationary plate or a rotating disk in an unbounded domain, the flow may be divided into three zones: the free jet consisting in a potential core surrounded by an intense shear layer, the stagnation zone, where the flow turns radially outward and the wall jet flow with decreasing velocities. The potential core length ranges usually between 4 and 6 times the injection diameter D = 2R j (Zuckerman and Lior, 2006). In the present case, the interdisk spacing being only one-fourth of the injection diameter ( = 0.25), the free jet cannot develop and the hydrodynamic fields are then somehow different to the unconfined case. Fig. 2 displays the streamline patterns obtained for the four values of the swirl parameter. The air jet squirts from the center of the stator and impinges the rotor. After the impingement, the fluid is deflected and flows radially outwards along the rotor. This radial outflow, which is enhanced due to the combination of the jet flow and centrifugal effects, is confined by a large recirculation zone appearing along the stationary disk. Poncet et al. (2013) showed that the size of this recirculation bubble depends mainly on the aspect ratio G of the cavity, which is confirmed by the present simulations. Whatever the flow parameters, this jet-dominated area observed close to the rotation axis extends to a dimensionless radius r/Rd ࣃ 0.12. It is coherent with the predictions of the RSM (r/Rd ࣃ 0.14) and the fact that the k − ω SST model strongly overestimated the radial extent of the recirculation bubble (Poncet et al., 2013). The sizes of this recirculation bubble obtained by the different approaches are compared on Fig. 3. For all approaches, the aspect ratio remains the same: G = 0.02. Due to some experimental constraints, Nguyen et al. (2012) could not perform measurements for r/Rd ≤ 0.04. The size of this recirculation is much more important in the experiments both in the radial and axial directions. Downstream the recirculation area, at larger radii, the flow is purely centrifugal with streamlines parallel to the disks when roPlease

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Fig. 4. Oguic et al. submitted to Int. J. Heat Fluid Flow.

tation effects remain weak (N ≤ 0.153). Soo (1958) obtained approximate solutions for regime I and displayed the different flow patterns according to only one parameter: (r ) =

1 Re j N 2 G3

R

( rj )3 . For

= 1/90, the flow is purely centrifugal, whereas some ingress may be observed along the stator for (ring ) = 1/180. For N =

0.153, the disk should have a radius of 2.37 m to observe some ingress along the stator. But for N = 1.237 and 2.47, this relation predicts a critical radius for the appearance of a centripetal flow along the stator equal to ring /Rd = 0.59 and 0.37, respectively. It is found to be in very good agreement with the values obtained here by DNS: ring /Rd = 0.6 and 0.44 for N = 1.237 and 2.47, respectively. Though the relation given by Soo (1958) is simple and has been obtained in a rotor-stator cavity with radial outflow, it works quite well here mainly because the base rotor-stator flows are also in the regime I as defined by Pellé and Harmand (2007) and the impinging jet has no direct influence on the mean flow at larger radii. Fig. 4 presents the mean velocity fields obtained for four values of the swirl parameter. The mean axial velocity is one order weaker than the two other components within the rotor-stator cavity and the parameter N has no noticeable influence on it. As the axial Reynolds number is constant here and rotation has no effect in the impinging jet region, the field of uz is not modified there when N increases. From the ur -map, it can be seen that the jet is deviated in the radial direction after its impingement onto the rotating disk. The recirculation bubble along the stator side confines the incoming fluid along the rotor and similar accelerations are observed whatever the value of N. For N ≥ 1.237, negative radial velocities are obtained along the stationary disk confirming the reentry of the external fluid as already evoked regarding the streamline patterns. The maps of the mean tangential component highlight clearly the development of the rotor boundary layer when moving towards the periphery of the cavity due to centrifugal effects. The variation of the swirl parameter has no noticeable effect on the mean tangential component when N = 0, with negligible values for r/Rd ≤ 0.3 outside the rotor boundary layer. As shown by Poncet et al. (2013), the flow may switch from a Stewartson to a Batchelor structure depending on the rotational and axial Reynolds numbers and the local radius. Such terms are used to name the structure of the mean tangential velocity profile in the axial direction. Stewartson profiles refer to a mean tangential velocity close to zero everywhere apart in the rotor boundary layer, which is the case for 0.3 ≤ r/Rd ≤ 0.9 and N = 0.153 (Fig. 5). Similar profiles are direct

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Fig. 5. Oguic et al. submitted to Int. J. Heat Fluid Flow.

obtained for higher values of N. Nevertheless, one can notice that, for N ≥ 1.237 and r/Rd = 0.3, two boundary layers are observed, one on each disk, separated by a laminar flow region where the mean tangential velocity varies with 1/r. As the flow is dominated by the axial jet for N = 0.153, the mean radial velocity profile resembles the typical laminar parabolic profile usually encountered between two parallel plates. The maximum value is obtained at mid-gap for z/h = 0.5. When moving outwards, the intensity of the radial flow gets weaker as rotation effects increase. For higher values of the swirl parameter, the strong radial flow due to the jet is associated to the radial flow due to centrifugal effects to produce an asymmetric profile with higher radial velocities along the rotor side. For N = 2.47, the radial flow gets negative from r/Rd = 0.6 along the stator to balance the strong outflow on the stator side and conserve mass. The self-similarity of the ur − profiles obtained by Manceau et al. (2014) for Re j = 14, 500 and h/D = 5 in an unbounded domain is broken here for N ≥ 1.237 when rotation effects gradually increase. It may be attributed to the strong confinement imposed here, which leads to merged boundary layers between the rotor and stator sides. Fig. 6 compares the present DNS obtained for N = 0.153 with previous published data (Nguyen et al., 2012; Poncet et al., 2013) Please

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obtained for N = 0.866. Though different values of the flow parameters (Rej and N) have been considered, the objective was to confirm or not the centripetal flow measured by Nguyen et al. (2012) at low radii whatever the axial plane z/h (defined in Fig. 3). On the rotor side (z/h = 0.23), the DNS predicts a radial distribution of the mean radial velocity close to the one predicted by the k − ω SST model (Poncet et al., 2013) with a peak location in agreement with the two turbulence models. The mean radial velocity gets 1.75 times larger than the jet velocity Uj due to the addition of the centrifugal effects. This maximum value decreases to 0.8Uj when moving towards the stator. At mid-gap (z/h = 0.53), the same remarks may be done with a mean radial velocity positive for all radii. A secondary peak in the ur − distribution is besides observed around r/Rd ࣃ 0.1 due to the presence of coherent vortices (as shown in Section 3.3). Along the stator (z/h = 0.83), negative radial velocities are obtained by DNS for 0.05 ≤ r/RD ≤ 0.1 with a minimum in good agreement with the RANS models. It confirms the overestimation of the radial extent of the inflow along the stator by the velocity measurements of Nguyen et al. (2012). It confirms the previous observations about the overestimation of the size of the recirculation area, usually fixed by the value of the aspect ratio G = 0.02. For N = 0.153, the flow is purely centrifugal as direct

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Fig. 6. Oguic et al. submitted to Int. J. Heat Fluid Flow.

previously shown such that the radial velocity remains positive at large radii even on the stator side. The remaining discrepancies between the experiments and the numerics in the jet-dominated area may be attributed to the difficulties to perform accurate PIV measurements in such a narrowgap cavity. Moreover, the thicknesses of the laser planes are not negligible compared to the interdisk spacing h = 6.5 mm, providing averaged values over a too large axial volume. Laser reflexions and seeding problems usually encountered in such systems may explain partly the relatively weak negative values of the mean radial velocity component measured by Nguyen et al. (2012). Measurements are made even more difficult due to the presence of the inlet nozzle. On the contrary, the DNS results compare quite well with the predictions of the turbulence models (Poncet et al., 2013) in terms of the jet width, peak values of the radial velocity and position of the recirculation area. Nevertheless, these axisymmetric steady-state models are not able to capture the secondary peak at mid-plane due to the passing of coherent structures. 3.2. Turbulent field Fig. 7 presents the maps of the root-mean-square values of the fluctuating velocities normalized by the inlet axial velocity Uj for the four values of the swirl parameter N considered here. Turbulence is mainly produced by the impinging jet onto the rotor. The three normal components have a similar behavior. Turbulence intensities are maximum in the recirculation area for 0.08 ≤ r/Rd ≤ 0.12 with peak values between 45% and 55%. It is to be compared to the basic configuration of an impinging jet onto a stationary or Please

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Fig. 7. Oguic et al. submitted to Int. J. Heat Fluid Flow.

rotating surface in an unbounded domain, where turbulence intensities are usually lower. For example, (Fellouah et al., 2009) measured turbulent intensities ranging from less than 8% in the free direct

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Fig. 8. Oguic et al. submitted to Int. J. Heat Fluid Flow.

jet region to around 20% in the intense shear layers for a turbulent round free jet onto a stationary plate. At larger radii, turbulence intensities significantly decrease, though the local Reynolds number Rer = (r/Rd )2 Re gradually increases. Similar values for the three normal components of the Reynolds stress tensor have been obtained whatever the value of the swirl parameter. These two observations lead to the same conclusion: rotation has only a very limited effect on the turbulence production in the present case. The radial distributions of the two normal radial and tangential components of the Reynolds stress tensor are displayed in Fig. 8 at z/h = 0.23. This axial position, defined in Fig. 3, is located on the rotor side. These two components exhibit the same behavior. After the impingement, the flow is deviated radially and a wall jet is then created along the rotor. The fluid is locally accelerated up to r/Rd ࣃ 0.1 and then is decelerated. This fluid acceleration is associated with a peak in the radial distributions of the Reynolds stress tensor components around r/Rd ࣃ 0.1. The peak level remains the same for all values of the swirl parameter N considered here, which means that rotation has no influence on the turbulent field in the impinging flow region. This critical radius corresponds to the transition to turbulence in the rotor boundary layer. From r/Rd ࣃ 0.4 up to the outlet, the r.m.s. values predicted by DNS remain rather constant. This second critical radius marks the beginning of the rotation-dominated area and slightly varies between the different approaches. As example, (Nguyen et al., 2012) observed this frontier around r/Rd ࣃ 0.34. In this flow region, increasing the swirl parameter leads to higher r.m.s. values for the tangential fluctuating velocity. The peak location for the Reynolds stresses at r/Rd ࣃ 0.1 agrees very well with the experimental data of Nguyen et al. (2012) (r/Rd ࣃ 0.12) and the RSM used by Poncet et al. (2013) (r/Rd ࣃ 0.1). On the contrary, the turbulence intensities are much higher in the DNS compared to the other approaches. It can be easily explained by the presence of 3D turbulent eddies (see in Fig. 10), which are not captured by the axisymmetric steady-state models. The second reason is certainly the normalization of the r.m.s. values by the jet velocity Uj . The jet velocity is much higher for these last authors, whereas they may have obtained r.m.s. values similar to those given by the present DNS. Nguyen et al. (2012) showed besides that turbulence intensities decrease with increasing values of the axial Reynolds number. Fig. 9 shows the radial distributions of the radial and tangential normal components of the Reynolds stress tensor for the four values of the swirl parameter N at the three different axial locations defined in Fig. 3. The peak values for ur, r.m.s. are all observed in the impingement region for low radii. They are besides higher closer to the rotating disk at z/h = 0.23 and slightly lower when Please

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moving towards the stator, along which the recirculation bubble is observed. A different behavior is obtained for the tangential component uθ , r.m.s. . As the flow is dominated by the axial jet at low radii and the mean tangential flow due to centrifugal effects is rather weak, the peak value is higher at mid-plane at z/h = 0.53 than along the rotor side. It corresponds to the edge of the recirculation zone. The maxima are obtained at r/Rd ࣃ 0.08 for z/h = 0.53, r/Rd ࣃ 0.09 for z/h = 0.84 and r/Rd ࣃ 0.1 for z/h = 0.23. The swirl parameter N has no noticeable influence on the radial distributions of the radial and tangential r.m.s. values confirming that rotation does not affect the flow in the impingement region. At higher radii, around r/Rd ࣃ 0.22, not surprisingly, uθ , r.m.s. increases from the rotor to the stator and for N ≥ 1.237 increases also with N. 3.3. Coherent vortical structures The objective is to identify the possible three-dimensional coherent structures embedded in the main flow. They may be responsible for higher heat transfer coefficients and could explain some remaining discrepancies between the experiments (Nguyen et al., 2012) and the steady-state axisymmetric turbulence models (Poncet et al., 2013). Fig. 10 presents some maps of the magnitude of the instantaneous vorticity obtained for N = 1.237 and the results of the other values of the swirl parameters lead to similar conclusions. Vorticity is maximum at the nozzle exit, where elongated coherent structures are formed before being advected by the centrifugal flow. Vorticity levels remain high up to r/Rd ࣃ 0.06 before a sudden drop at r/Rd ࣃ 0.08 followed by an increase from r/Rd ࣃ 0.1. Several spots of high vorticity levels may be observed at higher radii. Depending on the θ − plane, they are located either on the rotor or stator sides, revealing the unsteadiness and three-dimensional nature of the flow. To analyze the form of these coherent structures, Fig. 11 displays the isovalues of the Q-criterion for N = 1.237 and N = 2.47 close to the impingement point region. It confirms the previous results on the vorticity maps: turbulent structures are formed at the nozzle exit along the stator side before being advected by the jet flow and extend in the axial direction for 0.19 ≤ z/h ≤ 0.54. Being weakly affected by rotation effects, they form annular coherent vortices up to a critical radius r/Rd = 0.1. These primary vortices are characterized by a positive azimuthal vorticity and are similar to the roll-up vortices produced by the jet leaving the nozzle lips observed by Crow and Champagne (1971) for example. By POD, (Nguyen et al., 2012) obtained oval vortices, whose size is fixed by the interdisk spacing. These primary vortices generate also secondary structures with a negative vorticity located in the vicinity direct

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Fig. 9. Oguic et al. submitted to Int. J. Heat Fluid Flow.

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Fig. 10. Oguic et al. submitted to Int. J. Heat Fluid Flow.

Fig. 11. Oguic et al. submitted to Int. J. Heat Fluid Flow.

of the rotor for 0 ≤ z/h ≤ 0.19. At larger radii, they are less organized. This critical radius corresponds to the transition to turbulence of the stator boundary layer. It confirms the previous observations of Dairay et al. (2014) or Hadziabdic and Hanjalic (2008) in less constrained jet configurations. Dairay et al. (2014) noticed also smaller structures elongated in the radial direction, which are also obtained here and which roll-up around the primary vortices confirming the velocity fields previously discussed. They are also similar to the mixing layer or secondary vortices obtained close to the wall by Popiel and Trass (1991) for small nozzle to plate distances. For N = 0 and N = 0.153, the turbulent structures within the cavity remain axisymmetric, showing that rotation effects are weak for N = 0.153. For higher values, spiral patterns may appear within the cavity due to the dominating effect of rotation. Fig. 12 presents the isovalues of the Q-criterion for N = 1.237 and N = 2.47, colored by the local radius. They are located at mid-gap between the rotating and stationary disks. They are formed at r/Rd ࣃ 0.6 for N = 1.237 and increasing the rotation rate up to N = 2.47 leads Please

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to the formation of these spiral arms earlier at r/Rd ࣃ 0.5. They are closely linked to the recirculation zone already observed in Fig. 2 as they grow up at the critical radial location. These spiral vortices roll-up in the opposite rotation sense compared to the disk rotation and so are named as negative spirals. They form a weakly negative angle ε according to the tangential direction (Fig. 12c), which varies with the local radius. Thus, ε = −4.8◦ for N = 1.237 and ε = −10.2◦ for N = 2.47 at r/Rd = 0.7. These are the characteristics of type II instabilities usually encountered in rotating disk arrangements and related to the combined effects of Coriolis and viscous forces. Though having a weak intensity in the present case, they are very similar to the SRIII instability observed first by Schouveiler et al. (2001) in an enclosed rotor-stator cavity. These authors showed that, for G = 0.02, the spirals appear for rotational Reynolds numbers between Re ࣃ 6.3 × 104 and 8.5 × 104 . Here, N = 1.237 and N = 2.47 correspond to Re ࣃ 8.2 × 104 and Re = 1.636 × 105 , respectively. This last value is out of the stability domain of the SRIII instability in an enclosed direct

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Fig. 12. Oguic et al. submitted to Int. J. Heat Fluid Flow.

cavity. Nevertheless, it may be easily explained by two arguments. Firstly, in the enclosed cavity, the stability diagram of these spirals strongly depends on the aspect ratio: for example, for G ࣃ 0.021, they can subsist up to Re ࣃ 1.2 × 105 . Secondly, the stability of such rotating flows is mainly governed by the local Rossby number Ro(= uθ (z/h = 0.5 )/(r )), which is equal to Ro ࣃ 0.5 for Schouveiler et al. (2001). In the present case, the centrifugal flow due to the impinging jet tends to stabilize the main tangential flow and Ro, though being a function of the local radius, remains around 0.2 or below (see Fig. 5). 4. Heat transfer The optimal configuration for the heat transfer corresponds to

 = 6 for a jet issuing from a long circular nozzle after Baughn and Shimizu (1989). Though  is particularly low in the present

case (= 0.25), it may be useful for engineering applications to determine the heat transfer coefficient in this specific geometry of alternator. 4.1. Local Nusselt number distributions The heat transfers are discussed first in terms of the local Nusselt number calculated along the rotating disk only as:

−2R j Nur = T



∂T ∂z



(4) rotor

with T = Trotor − T∞ . To enable direct comparisons with Pellé and Harmand (2009) and Poncet et al. (2013) though different values of Rej and Re have been considered, the local Nusselt number Nur has been normalized by its maximum value Numax . Due to mechanical constraints, (Pellé and Harmand, 2009) were not able to measure this parameter at the impinging point but proposed the following correlation valid for 0.01 ≤ G ≤ 0.02 to evaluate the maximum value of the Nusselt number at the impinging point: Numax = 0.025G−0.1 Re0j .85 . As shown in Fig. 13, the present DNS confirms the three different flow regions identified earlier by Pellé and Harmand (2009) then Poncet et al. (2013) when increasing the radius. Heat transfer is maximum near the jet impingement where high turbulence intensities are obtained. The slight increase after the impingement point is due to the high values of the local radial velocPlease

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Fig. 13. Oguic et al. submitted to Int. J. Heat Fluid Flow.

ity in the rotor boundary layer, which induce high shear stresses. A first peak in the Nusselt distribution is then observed at r/Rd ࣃ 0.042, which corresponds to a distance equal to 0.5D. This value remains the same whatever the approach and the value of the swirl parameter N. The location of this absolute maximum value matches perfectly with the one at 0.5D proposed by Lytle and Webb (1994) or Lee and Lee and Lee (2010) for a turbulent free jet. A secondary peak is also obtained in the present simulations at a distance equal to 1.037D (r/Rd ࣃ 0.087) from the stagnation point in agreement with Lee and Lee (2010); Lytle and Webb (1994) for turbulent round or elliptic free jets onto a stationary plate, with low nozzle-to-plate distances and high Reynolds numbers. Hadziabdic and Hanjalic (2008) attributed such maximum value to the interaction of the shear layer vortices with the wall and to the presence of secondary vortices. In the present case, this secondary peak is more related to the transition to turbulence in the rotor boundary layer as shown regarding the Reynolds stress distributions in Fig. 8 and to the local increase of the vorticity magnitude displayed in Fig. 10. A similar explanation has been provided by Lytle and Webb (1994) and O’Donovan and Murray (2007), who attributed this local maximum to the rise of the turbulence intensities in the wall boundary layer. Surprisingly, the k − ω SST model used by Poncet et al. (2013) predicts also this direct

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Table 4 Oguic et al. submitted to Int. J. Heat Fluid Flow. N

0

0.153

1.237

2.47

NuN,DNS NuN,cal (m = 0.85) NuN,cal (m = 0.78)

5.25 5.25 5.25

5.73 5.75 5.51

6.12 8.25 6.61

7.46 10.65 7.58

peak at a larger radius 1.78D, whereas it is not captured by the RSM. This secondary peak has not been obtained by the different turbulence models of Manceau et al. (2014) for Re j = 14500 and h/D = 5 in an unbounded domain. After this secondary peak, the local Nusselt number decreases exponentially with the local radius. In this mixed region at intermediate radii, rotational and jet effects coexist. This drop is associated to a decrease of the turbulence intensities and of the mean radial velocity along the rotor when moving radially outwards. Finally, for r/Rd ≥ 0.4 (or 4.77D), rotational effects dominate and lead to higher heat transfer than in the case without rotation. The local Nusselt number distributions do not depend anymore on the radial location. Though its effect is rather weak, increasing the swirl parameter N leads to higher values of the heat transfer coefficient in this region. For a turbulent round jet onto a rotating heated disk in an unbounded domain, Manceau et al. (2014) showed that the rotation-dominated area appears for Rer ≥ /16 , which corresponds here to r/Rd ࣃ 0.49 and 0.34 for 0.25Re21 j N = 1.237 and 2.47, respectively. This flow region should not be observed here for lower values of the swirl parameter. This difference is a direct effect of the confinement disk. 4.2. Average Nusselt number distribution

 −0.037  h D

1 − 0.168

R2 Rd + 0.08 d2 D D



(5)

The value Nu0,DNS = 5.25 obtained here by DNS for N = 0 is much lower than the one provided by Eq. (5), Nu0,cal = 9.38. It may be easily explained by the values of the different parameters h/D = 0.25, Re j = 5300 and Rd /D = 12.5 in the present simulations, which are out of the validity range of Eq. (5). Poncet et al. (2013) proposed the following correlation Nu0 = 0.017Re0j .72 obtained for the same geometrical parameters but much higher values of Rej and Re . Such correlation leads to Nu0 = 8.16 in the present case. It has to be considered with a certain caution as these last authors consider few different values of Rej . To investigate the rotation effects on the average Nusselt number, the approach proposed by Poncet et al. (2013) has been considered. It consists of decoupling the influence of the constant parameters included in Nu0,DNS and investigate only the influence of the rotational Reynolds number Re through the following equation:

NuN,cal = Nu0,DNS + 2 × 10−4 Rem 

(6)

Table 4 presents the average values of the Nusselt number along the rotor obtained for four values of the swirl parameter N either Please

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by DNS (NuN,DNS ) or using Eq. (6) with m = 0.85 as proposed by Poncet et al. (2013) and m = 0.78. Fig. 14 displays the influence of the swirl parameter N on the average Nusselt number along the rotor. As expected, increasing N leads to higher global heat transfer. More interesting is that the exponent m = 0.78 provides a much better fit of the DNS data. This value falls between the value m = 0.8 usually encountered in most of the rotating disk arrangements and the value m = 0.746 obtained for rotor-stator flows with unmerged boundary layers (Childs, 2010; Harmand et al., 2013; Owen and Rogers, 1989; Shevchuk, 2009). 5. Conclusion

More interesting for engineering applications is to try to propose some correlations for the average Nusselt number along the rotor side as a function of the global parameters: G, Rej , Re and Pr. The objective here is to quantify the influence of the rotational Reynolds number Re as all the other parameters have been fixed to a prescribed value. Sagot et al. (2008) proposed the following correlation for a round impinging jet onto a stationary disk (N = 0) at a prescribed temperature, which is valid for 2 ≤ D ≤ 6, 3 ≤ Rd /D ≤ 10 and 104 ≤ Rej ≤ 3 × 104 :

Nu0,cal = 0.0622Re0j .8

Fig. 14. Oguic et al. submitted to Int. J. Heat Fluid Flow.

et

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High-order

High-order Direct Numerical Simulations have been performed to investigate the effective cooling of a rotating heated disk by an impinging air jet in a highly confined rotor-stator cavity. The geometrical characteristics of this narrow-gap cavity G = 0.02 match those of the experimental set-up developed by Pellé and Harmand (2009) and Nguyen et al. (2012) with h/D = 0.25. The axial Reynolds number based on the jet characteristics is fixed to Re j = 5300 lower than those considered experimentally. Nevertheless, the values of the rotational Reynolds number have been varied to preserve the same values of the swirl parameter N = [0 − 2.47] to enable comparisons with the experiments (Nguyen et al., 2012; Pellé and Harmand, 2009). The DNS results have been also compared to the previous RANS turbulence models from Poncet et al. (2013). Three different flow regions have been identified concerning the heat transfer confirming the previous experiments of Pellé and Harmand (20 07, 20 09): a recirculation zone close to the jet impingement characterized by high heat transfer and weak rotational effects, a mixed flow area and a flow region outwards, for r/Rd ≥ 0.4, characterized by weak heat transfer and dominating rotational effects. In the jet-dominated area, a large recirculation zone has been obtained along the stator side, whose size does not depend on the swirl parameter N. Its size is larger in the experiments than in the DNS, which results in a centripetal flow also along the rotor in the experiments. This zone is characterized by high turbulent intensities with peak values around r/Rd ࣃ 0.1. Primary structures appearing as circular rolls along the stator are accompanied with threedimensional unsteady secondary vortices mostly observed along the rotor side, responsible for higher heat transfer. In the rotation-dominated area, for r/Rd ≥ 0.4, the mean tangential velocity profile may switch from a Stewartson to a Batchelor flow structure depending on the value of the swirl parameter direct

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N. For N ≥ 1.237, a centripetal flow along the stationary disk compensates the strong centrifugal flow along the rotor. Negative spiral arms appear then at mid-gap. Their characteristics are very similar to the SRIII instability of Schouveiler et al. (2001) first observed in an enclosed narrow-gap rotor-stator cavity. In this region, turbulent intensities are weaker than in the jet-dominated area but they slightly increase with N. The radial distributions of the local Nusselt number along the rotor agree particularly well with previous published data for the four values of the swirl parameter N considered here. The DNS highlights the existence of a secondary peak in the impinging jet region, which is not captured either in the experiments (Pellé and Harmand, 2009) nor by the advanced RSM (Poncet et al., 2013). Finally, it is shown that the average Nusselt number depends on the rotational Reynolds number to the power 0.78 in close agreement with the typical values around 0.8 found in similar rotating disk systems (Owen and Rogers, 1989). Further calculations by Large Eddy Simulations are now required to reach higher values of the axial and rotational Reynolds numbers to enable direct comparisons with the experiments (Nguyen et al., 2012; Pellé and Harmand, 2009). The multidomain solver is also capable of handling coupled heat transfer. Thus, some simulations are currently running with heat transfer by conduction in the rotating disk and forced convection within the air gap. Acknowledgments The authors would like to thank the LABEX MEC (ANR-11-LABX0092, project HYDREX) for its financial support. Calculations have been performed using the HPC center of Aix-Marseille Université under the project Equip@Meso (ANR-10-AQPX-29-01). S. Poncet would like to acknowledge the financial support of the NSERC Chair in Industrial Energy Efficiency, established at Université de Sherbrooke in 2014, with the support of Hydro-Québec, CanmetEnergie, Rio Tinto Alcan and the Natural Sciences and Engineering Research Council of Canada. References Abide, S., Viazzo, S., 2005. A 2d compact fourth-order projection decomposition method. J. Comput. Phys. 206, 252–276. Baughn, J., Shimizu, S., 1989. Heat transfer measurements from a surface with uniform heat flux and an impinging jet. J. Heat Transf. 111 (4), 1096–1098. Cafiero, G., Discetti, S., Astarita, T., 2014. Heat transfer enhancement of impinging jets with fractal-generated turbulence. Int. J. Heat Mass Transf. 75, 173– 183. Childs, P.R.N., 2010. Rotating flow. Butterworth-Heinemann, Oxford, UK. Crow, S.C., Champagne, F.H., 1971. Orderly structure in jet turbulence. J. Fluid Mech. 48 (3), 547–591. Dairay, T., Fortuné, V., Lamballais, E., Brizzi, L.E., 2014. LES of a turbulent jet impinging on a heated wall using high-order numerical schemes. Int. J. Heat Fluid Flow 50, 177–187. Fellouah, H., Ball, C., Pollard, A., 2009. Reynolds number effects within the development region of a turbulent round free jet. Int. J. Heat Mass Transf. 52 (17–18), 3943–3954. Firouzian, M., Owen, J.M., Pincombe, J.R., Rogers, R.H., 1986. Flow and heat transfer in a rotating cylindrical cavity with a radial inflow of fluid. Part 2: Velocity, pressure and heat transfer measurements. Int. J. Heat Fluid Flow 71 (1), 21– 27. Hadziabdic, M., Hanjalic, K., 2008. Vortical structures and heat transfer in a round impinging jet. J. Fluid Mech. 596, 221–260.

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ing jet onto a rotating heated disk in a highly confined cavity, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.013