1. Introduction

Enclosed rotating disks simulate conditions found in a large variety of rotating machinery, including hard disk drives, centrifugal pumps, air cycle machines, and gas turbines. The first sets of extensive and detailed measurements of the hydrodynamic flow in a rotor-stator cavity have been performed by Daily and Nece [1], who proposed also a comprehensive theoretical analysis of rotor-stator flows. They pointed out the existence of four basic flow regimes, whose approximate domains are presented in Figure 1, according to a combination of two parameters: the rotational Reynolds number based on the outer radius of the rotating disk and to the aspect ratio of the cavity, with the interdisk spacing. The four regimes are denoted as follows.

Figure 1: Diagram of the four-flow regimes according to Daily and Nece [1] with the cases considered here (cross-symbols).

(i)Regime I: laminar flow with merged boundary layers.(ii)Regime II: laminar flow with unmerged boundary layers.(iii)Regime III: turbulent flow with merged boundary layers.(iv)Regime IV: turbulent flow with unmerged boundary layers. The physics of turbulent flows with unmerged boundary layers (IV), known as turbulent Batchelor flows, has already been widely addressed both experimentally and numerically [2–4]. The reader is also referred to the experimental work of Schouveiler et al. [5] for details about the transition between regimes II and IV and between regimes I and III. The present work is mainly focused on the regime III of Daily and Nece [1], corresponding to turbulent torsional Couette flows with merged boundary layers (see Figure 1).

For example, at , Daily and Nece [1] estimate that the transition between the regimes IV and III occurs for . It means that, for a typical rotating disk radius of 250 mm [3], the interdisk spacing should be smaller than 4.9 mm to ensure merged boundary layers. Because of the complexity of making accurate flow measurements in such close clearance, there are relatively few experimental works devoted to this flow regime. A brief review of the works focused on regime III until 1989 is proposed in the detailed monograph of Owen and Rogers [6]. The main experimental investigation to our knowledge is the one of Daily et al. [7], who performed mean velocity, torque, and pressure measurements, for and two Reynolds numbers . For this set of parameters, they clearly obtained merged boundary layers. They investigated also the influence of a radial outward throughflow on the flow structure. Phadke and Owen [8] studied, using flow visualizations and pressure measurements, the effect of seven shroud geometries on the ingress of external fluid into the rotor-stator cavity for a large range of flow control parameters including the torsional Couette flow regime.

The difficulty to acquire accurate measurements in such close clearances has reduced the development of advanced numerical codes adapted to these geometries. Cooper and Reshotko [9] proposed a scalar effective viscosity technique to calculate the flow in a rotor-stator cavity of infinite radius for both close and large clearances. It provided qualitatively good results compared to the experimental data of Daily and Nece [1] obtained without throughflow. Iacovides and Theofanopoulos [10] used an algebraic modeling of the Reynolds stress tensor in the fully developed turbulence area and a mixing length hypothesis near the wall. It provided satisfactory results compared to simpler models in the case of a rotor-stator flow with and without throughflow but some discrepancies remained and the authors concluded that no single form of their models was satisfactory for all rotating disk configurations considered. Poncet et al. [2, 3, 11] compared pressure and velocity measurements with numerical predictions based on an improved version of the Reynolds stress modeling of Elena and Schiestel [12] for centripetal and centrifugal throughflows mostly in the case of large aspect ratio cavities. All the comparisons were in good agreement for both the mean and turbulent fields. Andersson and Lygren [13] performed some large-eddy simulations of enclosed rotor-stator flows (the cavity was assumed to be periodic in both the tangential and radial directions) for both the wide and narrow gap cases. They obtained a turbulent torsional Couette flow for and . At that time, three-dimensional accurate simulations (DNS or LES) are still limited to relatively low Reynolds numbers and most of all to idealized enclosed cavities without throughflow. Moreover, they require several millions of mesh points and excessive calculation times preventing from performing a parametric study in such a complex configuration. Finally, as there is no experimental or numerical evidence at that time that three-dimensional unsteady structures exist at very high rotation rates in the case of turbulent torsional Couette flows (contrary to turbulent Batchelor flows for which Czarny et al. [14] have observed precessing 3D vortexes), the RSM model used by Poncet et al. [2, 3] seems to be a valuable tool to describe such complex flows and to perform a parametric study of merged turbulent flows with strong axial throughflows.

The present paper is devoted to the modeling of turbulent flows in a rotor-stator system of small aspect ratio (regime III [1]) when an axial throughflow is superimposed on the rotating fluid. The basic flow belongs to the torsional Couette type family: the two boundary layers are merged. Besides their industrial applications in turbomachinery, these flows offer a relatively simple configuration to study the influence of both rotation and shear on turbulence. The aim of this work is so to provide a better understanding of the flow dynamics in such complex flows and to quantify the influence of the flow control parameters: the Reynolds number, the flowrate coefficient, and especially the aspect ratio of the cavity. The paper is divided as follows: Section 2 is devoted to the statistical modeling including the description of the geometrical configuration, the presentation of the numerical modeling, and the validation of the RSM model against experimental data [1]. In Section 3, the effect of both the Reynolds number and the aspect ratio of the cavity on the mean and turbulent fields in the case of a closed system (without throughflow) is discussed. The influence of an inward and an outward axial throughflow on the flow structures and on turbulence is presented in Section 4 for a given aspect ratio before concluding in Section 5.

2. Statistical Modeling

2.1. Geometrical Configuration

The cavity sketched in Figure 2 is composed of a smooth stationary disk (the stator) and a smooth rotating disk (the rotor). A fixed shroud encloses the cavity. The rotor and the central hub attached to it rotate at the uniform angular velocity .

Figure 2: Schematic representation of the rotor-stator configuration and notations.

The mean flow is mainly governed by four flow control parameters: the aspect ratio , and the curvature parameter of the cavity, the rotational Reynolds number based on the outer radius of the rotating disk, and the flow rate coefficient defined as follows: (i)(ii)(iii),(iv), where is the fluid kinematic viscosity, , the radii of the hub and of the rotating disk, respectively, the interdisk spacing, and the superimposed throughflow. Note that is a volume flow rate. corresponds to a closed cavity. (resp., ) denotes the case where a centripetal (resp., centrifugal) throughflow is superimposed. The dimensionless values of the radial gaps between the hub and the stator and between the rotor and the shroud are fixed to 0.076 and 0.012, respectively.

2.2. The Differential Reynolds Stress Model (RSM)

The flow studied here presents several complexities (high rotation rate, imposed throughflow, wall effects, and transition zones), which are severe conditions for turbulence modeling methods [15, 16]. Our approach is based on one-point statistical modeling using a low Reynolds number second-order full stress transport closure derived from the Launder and Tselepidakis [17] model and sensitized to rotation effects [12, 18]. Previous works [3, 12, 19] have shown that this level of closure was adequate in such flow configurations, while the usual model, which is blind to any rotation effect presents serious deficiencies. This approach allows for a detailed description of near-wall turbulence and is free from any eddy viscosity hypothesis. The general equation for the Reynolds stress tensor can be written as where , , , , and , respectively, denote the production, diffusion, pressure-strain correlation, dissipation, and extra terms.

The diffusion term is split into two parts: a turbulent diffusion , which is interpreted as the diffusion due to both velocity and pressure fluctuations [20] and a viscous diffusion , which cannot be neglected in the low Reynolds number region:

In a classical way, the pressure-strain correlation term can be decomposed as follows: where is interpreted as a slow nonlinear return to isotropy and is modeled as a quadratic development in the stress anisotropy tensor , with coefficients sensitized to the invariants of anisotropy. This term is damped near the wall where denotes the stress anisotropy tensor and and are two functions deduced from Craft's high-Reynolds number proposals [21] adapted for confined flows where is the Lumley's flatness parameter with and as the second and third invariants of the anisotropy tensor. is the turbulence Reynolds number.

The linear rapid part includes cubic terms. It can be written as with and .

Since the slow part of the pressure-strain correlation is already damped near the wall, a wall correction is only applied to the rapid part. The form retained here is the one proposed by Gibson and Launder [22] with a strongly reduced numerical coefficient. Moreover the classical length scale is replaced by which is the length scale of the fluctuations normal to the wall where is evaluated by the minimal distance of the current point from the four walls.

The viscous dissipation tensor has been modeled in order to conform with the wall limits obtained from Taylor series expansions of the fluctuating velocities [23] with , and defined as

The extra term accounts for implicit effects of the rotation on the turbulence field. , which is a linear contribution in the pressure-strain correlation, is a function of the dimensionality tensor, which represents the spatial properties of the turbulent structures as the elongation of the vortices. is a spectral jamming term acting only in the case of strong rotation. The inhomogeneous diffusion term slows down the bidimensionalization of the flow close to the walls. Finally, denotes the inverse flux due to rotation, which impedes the energy cascade. All these terms are detailed in [11, 19]. They allowed some improvements of the numerical predictions of the previous RSM models [12].

Below is the proposal of Launder and Tselepidakis [17] for the dissipation rate equation where is the isotropic part of the dissipation rate . , , , , and are four empirical constants and is defined by . Note that all these numerical constants are used by the scientific community and their values have been determined once and for all by numerical optimization or experiments (decay of isotropic turbulence behind a grid, turbulent boundary layer, and so forth). The reader can refer to the work of Elena [19] for more details. No numerical optimization has been performed in the present study.

The turbulence kinetic energy equation is redundant in an RSM model but it is however still solved numerically in order to get faster convergence It is verified after convergence that is exactly .

2.3. Numerical Method

The computational procedure is based on a finite volume method using staggered grids for mean velocity components with axisymmetry hypothesis in the mean. The computer code is steady elliptic and the numerical solution proceeds iteratively. Elena [19] and then Poncet et al. [2, 3] proved that a mesh in the frame is sufficient in most cases to get grid-independent solutions. Nevertheless, a more refined mesh in the radial direction has been used in the present case to take into account the narrow gap cases. To check the grid independence of the solution, some crucial quantities for turbulent rotor-stator flows have been considered: the extrema of the mean radial velocity component along the two disks and the mean tangential velocity component at midplane for the mean field, and the maximum of the turbulence kinetic energy in the whole cavity for the turbulent field. Table 1 shows that the mesh provides the best arrangement between accuracy and calculation time compared to the three other meshes considered. For this grid, the size of the first mesh is and in the radial and axial directions, respectively.

Table 1: Influence of the mesh grid on the mean and turbulent fields for , , and . Comparisons with the mesh which is chosen as the reference mesh grid.

It is also verified that the grid is sufficiently refined close the disks to describe accurately the viscous sublayers. For example, the wall coordinate ( the friction velocity at the wall and the size of the first mesh in the axial direction) remains below along both disks and for the whole radial extent for , , and . It is quite below the classical value , for which the viscous sublayer is described by 5 mesh points.

About 20 000 iterations (about hours on the M2P2 cluster composed of 2 xeon quadcore 3 GHz) are necessary to obtain the numerical convergence of the calculation. In order to overcome stability problems, several stabilizing techniques have been introduced in the numerical procedure, such as those proposed by Huang and Leschziner [24]. Also, the stress component equations are solved using matrix block tridiagonal solution to enhance stability using non-staggered grids.

2.4. Boundary Conditions

At the wall, all the variables are set to zero except for the tangential velocity , which is set to on rotating walls and zero on stationary walls. At the inlet, is supposed to vary linearly from zero on the stationary wall up to on the rotating wall. We recall that the inlet is close to the axis of the cavity when a centrifugal throughflow is superimposed, whereas it is located at the periphery in the case of a centripetal throughflow. When a throughflow (centrifugal or centripetal) is enforced, a parabolic profile is then imposed for the axial velocity at the inlet, with a given low level of turbulence intensity. The value of the turbulence kinetic energy imposed at the inlet has only a very weak influence (less than 0.1%) on the mean and turbulent fields [11]. In the outflow section, the pressure is fixed, whereas the derivatives for all the other independent quantities are set to zero if the fluid leaves the cavity, and fixed external values are imposed if the fluid re-enters the cavity. In this case, the continuity equation is used to determine this inward or outward velocity component. The boundary condition is then of a mixed type and a special technique is used to enhance stability [11]. Note that the flow in the similarity area is practically not sensitive to the shape of profiles of tangential and axial velocity components or to the intensity level imposed at the inlet [11]. Moreover, all these choices are justified by the wish to have a model as universal as possible.

In the following, all the quantities have been normalized as follows: , , , and with .

2.5. Validation of the RSM Model

The predictions of the RSM model have already been widely validated in various rotating disk configurations: turbulent Batchelor flows in a rotor-stator cavity (regime IV of [1]) with or without an imposed axial throughflow [2, 3], with heat transfer effects [25], turbulent Stewartson flows in an open cavity with throughflow [2, 3], and turbulent Von Kármán flow between counterrotating disks equipped or not with straight blades [26]. Nevertheless, to show the performances of the present model, the results of the RSM code are compared to both velocity measurements performed at IRPHE by a two-component laser Doppler anemometer and numerical predictions using the model of Launder and Sharma [27] for , , and several values of the flowrate coefficient .

The axial profiles of the mean radial and tangential velocity components are displayed at in Figures 3 and 4 for centripetal and centrifugal throughflows, respectively.

Figure 3: Axial profiles of two mean velocity components at for and with or without a centripetal superimposed throughflow: (a) , (b) , (c) , (d) . Comparisons between the present RSM model (), the model (), and LDA measurements ().

Figure 4: Axial profiles of two mean velocity components at for and with or without a centrifugal superimposed throughflow: (a) , (b) , (c) , (d) . Comparisons between the present RSM model (), the model (), and LDA measurements ().

For centripetal throughflows (see Figure 3), the model underestimated the mean tangential velocity in the core of the flow around . The RSM model improves also the predictions of the mean field in the Ekman boundary layer attached to the rotating disk () and provides very satisfactory results compared to the experimental data. The main discrepancy is obtained for very large values of especially along the stator side. It can be explained by considering the prerotation level of the fluid. In the RSM code, this prerotation level, which is the mean tangential velocity of the fluid in the radial gap between the rotor and the external fixed cylinder, is fixed to 0.5, whatever the value of , to have a numerical code as universal as possible. In the experiments, Poncet [11] has observed that it slightly varies between 0.45 for and 0.55 for very large values of .

In the case of centrifugal superimposed throughflows (see Figure 4), the RSM model still provides better results than the model, which overestimates the mean tangential velocity outside the Ekman layer. The RSM model predicts quite well the transition between the Batchelor flow structure with two unmerged boundary layers (see Figure 4(b)) and the Stewartson flow structure with only one boundary layer on the rotating disk (see Figure 4(c)).

Comparisons for the three associated Reynolds stress tensor components , , and are presented in Figure 5 for three characteristic values of at . The model, which overestimates the turbulence intensities, does not provide good profiles. The RSM model provides very satisfactory results even close to the disks, where the highest turbulence levels are obtained. A weak discrepancy is obtained for the component along the stator side for centripetal throughflows, which is directly linked to the underestimation of the mean radial velocity in this boundary layer.

Figure 5: Axial profiles of three Reynolds stress tensor components at for and : (a) , (b) , (c) . Comparisons between the present RSM model (), the model (), and LDA measurements ().

To conclude, the RSM model improves very significantly the predictions of a classical model, which is blind to any rotation effects. A very good agreement is obtained between the RSM model and LDA measurements for both the mean and turbulent fields. Some discrepancies remain along the stator in the case of very strong inward throughflows because of the different prerotation levels used in the model and measured in the experiment [11]. Considering also the previous validations in various interdisk cavities [2, 3, 25, 26], the RSM model can now be used with confidence for a parametric study of turbulent rotor-stator flows.

3. Turbulent Flow in a Closed Cavity

The influence on the mean and turbulent fields of both the Reynolds number and the aspect ratio of the cavity is investigated in the case of an enclosed rotor-stator cavity ().

3.1. Structure of the Mean Flow

Figure 6 presents the axial profiles of the mean radial and tangential velocity components for and at different radial locations in the range of . The axial profiles of the axial mean velocity component are not shown here because in the whole cavity except very close to the inner and outer cylinders. According to the diagram (, ) of Daily and Nece [1], the flow is turbulent with merged boundary layers (regime III, see Figure 1) for these parameters. Whatever the flow parameters, rotation induces a centrifugal effect along the rotor. Thus, remains positive. By mass conservation, is then negative along the stator. Nevertheless, from the axial profiles, two different behaviors are obtained depending on the radial location. For , the flow exhibits some typical characteristics encountered in laminar rotor-stator flows with unmerged boundary layers (regime II, see Figure 1). The region of zero radial velocity and constant tangential velocity enclosed by two boundary layers developed on each disk is typical of Batchelor flows with unmerged boundary layers, whereas the local maximum of at the edge of the stator boundary layer as well as the very large boundary layers obtained here is typical of laminar Batchelor flows. For larger radii , the region of zero has disappeared and varies linearly between the disks. By increasing the values of , the magnitudes of decrease and the tangential velocity far from the disks increase. This increase is characteristic of transitional torsional Couette flows. The transition between these two behaviors observed between and can be explained by considering the local Reynolds number , which is the pertinent parameter to characterize the transition between two flow regimes. Thus, for , whereas for . The value is below the critical Reynolds number for the transition between the regimes II and III [1].

Figure 6: Axial profiles of the mean radial and tangential velocity components for and at different radial locations .

In enclosed rotor-stator flows, the boundary layer thicknesses are an increasing function of the radial location [11]. Thus, for given flow parameters, transitions between regimes II and III or between regimes II–IV and then III can occur when one moves from the rotation axis to the periphery of the cavity because of the combined increase of the boundary layer thicknesses and the turbulence levels reflected in the -values (see Cooper and Reshotko [9]).

Figure 7 presents the axial profiles of and for a given aspect ratio and various Reynolds numbers in the range of at midcavity (). The axial velocity is still very weak for these parameters. The transition between the regimes I and III is here observed by varying the rotational Reynolds number. For (), there is no region of zero radial velocity and the extrema of are quite large, which is typical of regime I. For (), the flow belongs to the regime III with a linear profile of far from the disks and a significant decrease of the magnitudes of the extrema. By increasing further the Reynolds number up to , the tangential velocity around midplane increases, which indicates higher turbulence levels, and the extrema of the radial velocity decrease. It agrees with the calculated results of Cooper and Reshotko [9] in the narrow gap case for up to .

Figure 7: Axial profiles of two mean velocity components at for and different Reynolds numbers.

The streamline patterns corresponding to the cases previously considered are displayed in Figures 8 and 9 to highlight the influence of the aspect ratio of the cavity and the Reynolds number, respectively. It is noteworthy that the aspect ratio of the cavity is not maintained, when one represents the streamline patterns to enable a better visualization of the flow structure. Streamlines parallel to the rotation axis around midcavity () as in Figure 8(a) are typical of a Batchelor-like flow. For , it is the case for the largest aspect ratios (see Figure 8(a)) and 0.04 (see Figure 8(b)). For the lower value (see Figure 8(c)), the streamline patterns get curved and are characteristic of torsional Couette flows. When one increases the Reynolds number up to , the flow structure is not so clear. Close to the rotation axis, streamlines are parallel to the axis (the boundary layers are unmerged) and get inclined at the periphery (the boundary layers are merged) both for (see Figure 8(d)) and 0.02 (see Figure 8(e)). Whatever the flow parameters, a recirculation bubble appears along the external cylinder around midplane.

Figure 8: Influence of the aspect ratio on the streamline patterns (20 contours) for various sets of flow parameters (, ): (a) (, ), ; (b) (, 0.04), ; (c) (, ), ; (d) (, ), ; (e) (, ), .

Figure 9: Influence of the Reynolds number on the streamline patterns (20 contours) for and six values of : (a) , ; (b) , ; (c) , ; (d) , ; (e) , ; (f) , .

For a given aspect ratio , Figure 9 highlights the effect of the rotation rate reflected in the values of the Reynolds number on the streamline patterns. For (see Figure 9(a)), the flow exhibits a torsional Couette flow behavior. By increasing the values of (see Figures 9(b)–9(f)), the streamlines get aligned with the direction close to the axis, which characterizes a flow with unmerged boundary layers. In the same time, the flow remains with merged boundary layers at largest radii. A small recirculation bubble due to confinement is still observed along the shroud. One noticeable effect is also the curvature of the streamlines close to the disks and for small . A strong curvature of the streamlines close to the walls indicate strong extrema of the radial velocity in the boundary layers, which is typical of laminar rotor-stator flows, whereas a weak curvature is associated with turbulent flows. This change is quite visible by comparing the flow structures close to the axis between Figures 9(c) and 9(e), for example.

3.2. Turbulence Statistics

The influence of the different parameters on the turbulent field is now investigated. Figure 10 shows the axial profiles of the six components of the Reynolds stress tensor at given Reynolds number and aspect ratio for various radii in the range of corresponding to the mean velocity profiles presented in Figure 6. Close to the rotation axis at , turbulence is mainly concentrated in the boundary layers (as the boundary layers are unmerged at this radius), whereas the core of the flow is laminar. When one moves toward the periphery, the turbulence intensities increase because of the increase of the local Reynolds number , which is proportional to the local radius to power 2. The maximum of the normal stresses, located close to the rotor, is reached at and remains the same for larger radii. On the other hand, there is a significant increase of the normal stresses both along the stationary disk and around midplane, when one increases . Finally, at the periphery of the cavity, turbulence intensities are almost constant along the axial direction apart very close to the disks, where they vanish. Finally, contrary to turbulent Batchelor flows (regime IV [1]) where turbulence is concentrated in the boundary layers, whereas the core of the flow is laminar [3, 11], turbulence in torsional Couette flows (regime III [1]) is distributed along the axial direction. Moreover, the levels of the normal stresses are quite comparable (the order of ), which is not the case for Batchelor flows, where is generally negligible compared to the other normal components [3, 11]. The cross-components of the Reynolds stress tensor are also not so weak here as in Batchelor flows. Their levels increase when moving toward the periphery.

Figure 10: Axial profiles of the six Reynolds stress tensor components for and at different radial locations .

The influence of the Reynolds number on the six Reynolds stress tensor components is assessed in Figure 11 at and for . As expected, when is increased, higher turbulence intensities are obtained. For , turbulence is confined along the rotor, whereas the flow remains laminar everywhere else. For , the flow along the stationary disk gets turbulent. It is a noticeable difference with turbulent Batchelor flows. When the aspect ratio is sufficiently large to ensure unmerged boundary layers, the Bödewadt layer attached to the stator is more unstable than the Ekman layer attached to the rotor [3, 11] and gets turbulent for smaller values of the local Reynolds number. On the contrary, for torsional Couette flows (at midcavity, ), the flow along the rotating disk is found to be more turbulent than along the stator. For , there is no significant effect of the Reynolds number on the turbulence intensities.

Figure 11: Axial profiles of the six Reynolds stress tensor components at for and different Reynolds numbers.

The isovalues of the turbulence Reynolds number enable us to visualize the turbulent regions of the flow. The flow is considered as being laminar for (, where is the turbulence viscosity). Figure 12 presents some maps of in a () plane for various flow conditions (). Whatever the flow parameters, the maximum of , denoted by , is localized at the periphery of the cavity, where the highest values of the local Reynolds number are obtained. For a moderate Reynolds number and (see Figures 12(a)–12(c)), the maximum of is localized at the junction between the shroud and the stator along the Stewartson layer attached to the external cylinder. decreases with decreasing the values of the aspect ratio. If the Reynolds number is increased up to (see Figures 12(d)-12(e)), is 27 (resp., 39) times larger for (resp., ). The flow is then turbulent in the whole cavity with higher turbulence levels at largest radii.

Figure 12: Influence of the aspect ratio on the isovalues of the turbulence Reynolds number (normalized by its maximum value ) for various sets of flow parameters (): (a) (, 0.05), ; (b) (, 0.04), ; (c) (, 0.02), ; (d) (, 0.04), ; (e) (, 0.02), .

Figure 13 presents the influence of the rotational Reynolds number, in the range of , on the maps of the turbulence Reynolds number for a given aspect ratio corresponding to torsional Couette flows with merged boundary layers. As expected, increases with increasing values of : from for to for . remains localized close to the shroud but the radial extent of the turbulent regions increases with too.

Figure 13: Isovalues of for and six values of : (a) , ; (b) , ; (c) , ; (d) , ; (e) , ; (f) , .

Figure 14 shows the anisotropy invariant map for the Reynolds stress tensor at for various sets of parameters (). The second and third invariants of the anisotropy tensor of the second moments of the fluctuations are defined as and [28]. The results of the RSM model respect the realizability diagram of Lumley [28] as they remain within the region delimited by the two lines corresponding to the axisymmetric flow and the straight upper one corresponding to the two-component limit . The influence of the aspect ratio on the anisotropy invariant map is large for (see Figure 14(a)). For , the flow exhibits a Batchelor-like structure (see Figure 8(a)) and for , the invariant map follows the axisymmetric curve (), which is not the case for the other value of corresponding to a torsional Couette flow structure (see Figure 8(c)). For and , turbulence tends to the one-component limit (large values of and ) close to the disks. The influence of the Reynolds number on the invariant anisotropy map of torsional Couette flows () remains weak (see Figure 14(b)). The map for has already been discussed. In this case, the flow is laminar at this radial location (see Figure 13(a)). If is increased to values larger than , the flow gets turbulent at this radius (Figures 13(b)–13(f)) and the anisotropy invariant map is then changed. Turbulence follows mainly a three-component behavior and the one-component tendency close to the disks has disappeared. The same maps are obtained for radial locations in the range of . Close to the cylinders, the curves tend to the two-component limit as the wall normal fluctuations are damped more effectively than fluctuations parallel with the cylinders. Thus, the flatness parameter is equal to 0. Note that Poncet [11] showed that the structural anisotropy is axisymmetric whatever the flow conditions, which is due to the choice of the dimensionality tensor in the RSM model. Thus, turbulence is two dimensional with cigar-shaped vortex aligned with the rotation axis.

Figure 14: Anisotropy invariant maps at . (a) Influence of the aspect ratio for and (red), (blue); (b) influence of the Reynolds number for and (red), (blue), and (green).

4. Turbulent Flow with an Axial Throughflow

In the following, the value of the aspect ratio is fixed to and the influence of an axial flow is investigated on the mean and turbulent fields for various radial locations, Reynolds numbers, and flowrate coefficients.

4.1. Case of a Centrifugal Throughflow

In this subsection, an axial centrifugal throughflow, characterized by a negative flowrate coefficient , is superimposed on the mean tangential flow due to rotation.

4.1.1. Structure of the Mean Flow

Figure 15 displays the velocity profiles corresponding to the case and at five radial positions and Figure 17(h) brings out the corresponding streamlines. Depending on the radial location, the flow belongs to the torsional Couette or Stewartson type of flow. For , the flow exhibits a Stewartson-like structure from the -profiles with only one boundary layer on the rotor. The corresponding radial velocity which is always positive becomes significant compared to the tangential velocity: the flow is then fully centrifugal and the mean radial velocity profile gets closer to a Poiseuille-like profile at this radius. Phadke and Owen [8] found that the flowrate coefficient necessary to obtain the transition between torsional Couette and Stewartson flow structures can be scaled as . Thus, for the set of parameters considered in Figure 15, the transition should occur for . The transition between these two behaviors is continuous and appears here for , which is in excellent agreement with the empirical value given by Phadke and Owen [8]. The small discrepancy may be attributed to the different geometries considered. In their case, there is no hub attached to the rotor and the throughflow is pulling out radially at the outlet of the cavity.

Figure 15: Axial profiles of the mean radial and tangential velocity components for , , and at different radial locations .

When moving toward the periphery of the cavity, the flow switches progressively to torsional Couette type with for increasing values of , while the flow along the stator gets centripetal. Moreover, the axial profile of around midplane gets linear, which indicates a strong shear around midplane. At intermediate radial locations, is larger along the rotor, which is due to the combine effect of the centrifugal superimposed throughflow and the centrifugal effect due to the rotating disk. For this set of parameters, a large recirculation is observed at the corner between the shroud and the stator (see Figure 17(h)).

As it can be seen from the axial profiles of the mean velocity profiles (see Figure 16) but also from the streamline patterns (see Figures 17(f)–17(j)), the flow structure switches from torsional Couette type for small values of to Stewartson type for larger values at and . The transition appears for , in accordance with the empirical law of Phadke and Owen [8], which gives for the transition. When the flow exhibits a Stewartson flow structure, the flow is mainly dominated by the superimposed throughflow, whose effect is dominant compared to the one of rotation. For a torsional Couette flow structure, rotation has a preponderant effect compared to the axial throughflow. Note that whatever the set of parameters (, , , ) is, the mean axial velocity component remains very weak apart along the hub and the shroud.

Figure 16: Axial profiles of the mean radial and tangential velocity components for , at , and five flowrate coefficients .

Figure 17: 20 streamline patterns for and various sets of parameters (, ): (a) (, ), ; (b) (, ), ; (c) (, ), ; (d) (, ), ; (e) (, ), ; (f) (, ), ; (g) (, ), ; (h) (, ), ; (i) (, ), ; (j) (, ), ; (k) (