# Turbulence Modeling of Torsional Couette Flows

Authors: Sofia Haddadi; Sébastien Poncet
Journal: International Journal of Rotating Machinery
 (2008)

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## Abstract

The present study considers the numerical modeling of the turbulent flow inside a rotor-stator
cavity subjected or not to a superimposed throughflow. Extensive numerical predictions based
on one-point statistical modeling using a low Reynolds number second-order full stress transport
closure (RSM model) are performed mainly in the case of turbulent flows with merged boundary
layers known as turbulent torsional Couette flows and belonging to regime III of Daily and Nece (1960). The RSM model has already shown its capability of predicting accurately the mean and turbulent
fields in various rotating disk configurations (Poncet, 2005; Poncet et al., 2005, 2007, 2008). For the first time, a detailed mapping of the
hydrodynamic flow over a wide range of rotational Reynolds numbers (180000≤Re≤10000000), aspect ratios of the cavity (0.02≤G≤0.05), and flow rate coefficients (−10000≤Cw≤10000) is here provided in the turbulent torsional Couette flow regime.


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## Body

## 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 Re=Ωb2/ν based on the outer radius b of the rotating disk and to the aspect ratio G=h/b of the cavity, with h 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, atRe=107,
Daily and Nece [1] estimate that the transition between the regimes IV
and III occurs for G=0.0196.
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 G=0.0138 and two Reynolds numbers Re=2.95−6.9×105.
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 G=0.01 and Re=1.6×106.
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 Figure2 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 Ω.Figure 2
Schematic
representation of the rotor-stator configuration and notations.The mean flow
is mainly governed by four flow control parameters: the aspect ratioG,
and the curvature parameter Rm of the cavity, the rotational Reynolds number Re based on the outer radius of the rotating
disk, and the flow rate coefficient Cw defined as follows:
(i)
0.02≤G=h/b≤0.05,(ii)
Rm=(b+a)/(b−a)=1.33,(iii)
1.8×105≤Re=Ωb2/ν≤107,(iv)
−104≤Cw=Q/(νb)≤104,
where ν is the fluid kinematic viscosity, a, b the radii of the hub and of the rotating disk,
respectively, h the interdisk spacing, and Q the superimposed throughflow. Note that Q is a volume flow rate. Cw=0 corresponds  to a closed cavity. Cw>0 (resp., <0) denotes the case where a centripetal (resp.,
centrifugal) throughflow is superimposed. The dimensionless values of the
radial gaps jh/b between the hub and the stator and js/b 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 k-ϵ 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 Rij can be written as(1)dRijdt=Pij+Dij+Φij−ϵij+Tij,
where Pij, Dij, Φij, ϵij,
and Tij,
respectively, denote the production, diffusion, pressure-strain correlation,
dissipation, and extra terms.The diffusion
termDij is split into two parts: a turbulent diffusion DijT,
which is interpreted as the diffusion due to both velocity and pressure
fluctuations [20] and
a viscous diffusion Dijν,
which cannot be neglected in the low  Reynolds number region:(2)DijT=(0.22kεRklRij,l),k,Dijν=−νRij,kk.In a classical
way, the pressure-strain correlation termΦij can be decomposed as follows:(3)Φij=Φij(1)+Φij(2)+Φij(w),
where Φij(1) is interpreted as a slow nonlinear return to
isotropy and is modeled as a quadratic development in the stress anisotropy
tensor aij,
with coefficients sensitized to the invariants of anisotropy. This term is
damped near the wall(4)Φij(1)=−(c˜1aij+c1′(aikakj−13A2δij))ε,
where aij denotes the stress anisotropy tensor and c˜1 and c1′ are two functions deduced from Craft's
high-Reynolds number proposals [21] adapted for confined flows
(5)aij=Rijk−23δij,c˜1=(3.1AA2+1)(1−e−Ret2/40),c1′=3.72AA2(1−e−Ret2/40),
where A=1−9/8(A2−A3) is the Lumley's flatness parameter with A2 and A3 as the second and
third invariants of the anisotropy tensor. Ret=k2/(νε) is the turbulence Reynolds number.The linear rapid partΦij(2) includes cubic terms. It can be written as
(6)Φij(2)=−0.6(Pij−13Pkkδij)+0.3εaijPkkε−0.2[RkjRlik(Vk,l+Vl,k)−Rlkk(Rik(Vj,l+ϵjmlΩm)+Rjk(Vi,l+ϵimlΩm))]−min(0.6,A)(A2(Pij−Dij)+3amianj(Pmn−Dmn)),
with Pij=−RijVj,k−RjkVi,k and Dij=−RikVk,j−RjkVk,i.Since the slow
part of the pressure-strain correlation is already damped near the wall, a wall
correctionΦij(w) 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 k3/2ε−1 is replaced by k/ε(Rijninj)1/2 which is the length scale of the fluctuations
normal to the wall
(7)Φij(w)=0.2[(Φkm(2)+Φkm(R))nknmδij−32(Φik(2)+Φik(R))nknj−32(Φkj(2)+Φkj(R))nink]kRpqnpnqεy,
where y is evaluated by the minimal distance of the
current point from the four walls.The viscous
dissipation tensorεij has been modeled in order to conform with the
wall limits obtained from Taylor series expansions of the fluctuating
velocities [23](8)εij=fAεij*+(1−fA)(fsεRijk+23(1−fs)εδij),
with fA, fs and εij* defined as(9)fA=e−20A2e−Ret2/20,fs=e−Ret2/40,εij*=(Rij+Riknjnk+Rjknink+Rklnknlninj)(k/ε)(1+(3/2)(Rpq/k)npnq).The extra termTij=Φij(R)+Bij+DijR+Jij accounts for implicit effects of the rotation
on the turbulence field. Φij(R),
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. Bij is a spectral jamming term acting only in the
case of strong rotation. The inhomogeneous diffusion term DijR slows down the bidimensionalization of the
flow close to the walls. Finally, Jij 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 ε(10)dεdt=−cε1εkRijVi,j−cε2fεε˜εk+(cεkεRijε,j+νε,i),i+cε3νkεRjkVi,jlVi,kl+(cε4νε˜kk,i),i, where ε˜ is the isotropic part of the dissipation rate ε˜=ε−2νk,i1/2k,i1/2. cε1=1, cε2=1.92, cε=0.15, cε3=2,
and cε4=0.92 are four empirical constants and fε is defined by fε=1/(1+0.63AA2).
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(11)dkdt=−RijVi,j−ε+Tjj2+0.22(kεRijk,j+νk,i),i.
It is verified
after convergence that k is exactly 0.5Rjj.


### 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 140×80 mesh in the (r,z) frame is sufficient in most cases to get
grid-independent solutions. Nevertheless, a more refined mesh 200×80 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 200×80 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 Δ1r=1.677×10−4h and Δ1z=1.532×10−4h in the radial and axial directions,
respectively.Table 1
Influence of the mesh grid on the mean and turbulent fields forG=0.02, Re=106, and Cw=0. Comparisons with the 200×80 mesh which is chosen as the reference mesh grid.
Difference with the200×80 mesh250×80300×80250×100Mean field0.5%0.6%0.5%Turbulent field1.9%2.1%2%It is also verified that the
grid is sufficiently refined close the disks to
describe accurately the viscous sublayers. For example, the wall coordinatez+=Δ1zu*/ν (u* the friction velocity at the wall and Δ1z the size of the first mesh in the axial
direction) remains below 0.1 along both disks and for the whole radial
extent for G=0.04, Re=1.8×105,
and Cw=0.
It is quite below the classical value z+=1,
for which the viscous sublayer is described by 5 mesh points.About 20 000 iterations (about7 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 velocityVθ,
which is set to Ωr on rotating walls and zero on stationary
walls. At the inlet, Vθ is supposed to vary linearly from zero on the
stationary wall up to Ωr 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 Vz 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:z*=z/h, r*=r/(b+js), Vi*=Vi¯/(Ωr),
and Rij*=vi′vj′¯/(Ωr)2 with i,j=(r,θ,z).


### 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 k-ε model of Launder and Sharma [27] for Re=106, G=0.036,
and several values of the flowrate coefficient Cw.The axial
profiles of the mean radialVr* and tangential Vθ* velocity components are displayed at r*=0.56 in Figures 3 and 4 for centripetal and
centrifugal throughflows, respectively.Axial profiles of two mean velocity components
atr*=0.56 for G=0.036 and Re=106 with or without a centripetal superimposed
throughflow: (a) Cw=0,
(b) Cw=1976,
(c) Cw=5929,
(d) Cw=9881.
Comparisons between the present RSM model (−), the k-ϵ model (−⋅−), and LDA measurements (∘).
(a)(b)(c)(d)Axial profiles of two mean velocity components
atr*=0.56 for G=0.036 and Re=106 with or without a centrifugal superimposed
throughflow: (a) Cw=0,
(b) Cw=−1976,
(c) Cw=−5929,
(d) Cw=−9881.
Comparisons between the present RSM model (−), the k-ϵ model (−⋅−), and LDA measurements (∘).
(a)(b)(c)(d)For centripetal
throughflows (see Figure3), the k-ϵ model underestimated the mean tangential
velocity in the core of the flow around z*=0.5.
The RSM model improves also the predictions of the mean field in the Ekman
boundary layer attached to the rotating disk (z*≃0) and provides very satisfactory results
compared to the experimental data. The main discrepancy is obtained for very
large values of Cw 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 Cw,
to have a numerical code as universal as possible. In the experiments, Poncet
[11] has observed that
it slightly varies between 0.45 for Cw=0 and 0.55 for very large values of Cw.In the case of
centrifugal superimposed throughflows (see Figure4), the RSM model still
provides better results than the k-ϵ 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 componentsRrr*, Rrθ*,
and Rθθ* are presented in Figure 5 for three
characteristic values of Cw at r*=0.56.
The k-ϵ 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 Rrθ* component along the stator side for
centripetal throughflows, which is directly linked to the underestimation of
the mean radial velocity in this boundary layer.Axial profiles of three Reynolds stress tensor
components atr*=0.56 for G=0.036 and Re=106:
(a) Cw=9881,
(b) Cw=0,
(c) Cw=−9881.
Comparisons between the present RSM model (−), the k-ϵ model (−⋅−), and LDA measurements (∘).
(a)(b)(c)To conclude,
the RSM model improves very significantly the predictions of a classicalk-ε 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.


## 2.1. Geometrical Configuration

The cavity sketched in Figure2 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 Ω.Figure 2
Schematic
representation of the rotor-stator configuration and notations.The mean flow
is mainly governed by four flow control parameters: the aspect ratioG,
and the curvature parameter Rm of the cavity, the rotational Reynolds number Re based on the outer radius of the rotating
disk, and the flow rate coefficient Cw defined as follows:
(i)
0.02≤G=h/b≤0.05,(ii)
Rm=(b+a)/(b−a)=1.33,(iii)
1.8×105≤Re=Ωb2/ν≤107,(iv)
−104≤Cw=Q/(νb)≤104,
where ν is the fluid kinematic viscosity, a, b the radii of the hub and of the rotating disk,
respectively, h the interdisk spacing, and Q the superimposed throughflow. Note that Q is a volume flow rate. Cw=0 corresponds  to a closed cavity. Cw>0 (resp., <0) denotes the case where a centripetal (resp.,
centrifugal) throughflow is superimposed. The dimensionless values of the
radial gaps jh/b between the hub and the stator and js/b 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 k-ϵ 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 Rij can be written as(1)dRijdt=Pij+Dij+Φij−ϵij+Tij,
where Pij, Dij, Φij, ϵij,
and Tij,
respectively, denote the production, diffusion, pressure-strain correlation,
dissipation, and extra terms.The diffusion
termDij is split into two parts: a turbulent diffusion DijT,
which is interpreted as the diffusion due to both velocity and pressure
fluctuations [20] and
a viscous diffusion Dijν,
which cannot be neglected in the low  Reynolds number region:(2)DijT=(0.22kεRklRij,l),k,Dijν=−νRij,kk.In a classical
way, the pressure-strain correlation termΦij can be decomposed as follows:(3)Φij=Φij(1)+Φij(2)+Φij(w),
where Φij(1) is interpreted as a slow nonlinear return to
isotropy and is modeled as a quadratic development in the stress anisotropy
tensor aij,
with coefficients sensitized to the invariants of anisotropy. This term is
damped near the wall(4)Φij(1)=−(c˜1aij+c1′(aikakj−13A2δij))ε,
where aij denotes the stress anisotropy tensor and c˜1 and c1′ are two functions deduced from Craft's
high-Reynolds number proposals [21] adapted for confined flows
(5)aij=Rijk−23δij,c˜1=(3.1AA2+1)(1−e−Ret2/40),c1′=3.72AA2(1−e−Ret2/40),
where A=1−9/8(A2−A3) is the Lumley's flatness parameter with A2 and A3 as the second and
third invariants of the anisotropy tensor. Ret=k2/(νε) is the turbulence Reynolds number.The linear rapid partΦij(2) includes cubic terms. It can be written as
(6)Φij(2)=−0.6(Pij−13Pkkδij)+0.3εaijPkkε−0.2[RkjRlik(Vk,l+Vl,k)−Rlkk(Rik(Vj,l+ϵjmlΩm)+Rjk(Vi,l+ϵimlΩm))]−min(0.6,A)(A2(Pij−Dij)+3amianj(Pmn−Dmn)),
with Pij=−RijVj,k−RjkVi,k and Dij=−RikVk,j−RjkVk,i.Since the slow
part of the pressure-strain correlation is already damped near the wall, a wall
correctionΦij(w) 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 k3/2ε−1 is replaced by k/ε(Rijninj)1/2 which is the length scale of the fluctuations
normal to the wall
(7)Φij(w)=0.2[(Φkm(2)+Φkm(R))nknmδij−32(Φik(2)+Φik(R))nknj−32(Φkj(2)+Φkj(R))nink]kRpqnpnqεy,
where y is evaluated by the minimal distance of the
current point from the four walls.The viscous
dissipation tensorεij has been modeled in order to conform with the
wall limits obtained from Taylor series expansions of the fluctuating
velocities [23](8)εij=fAεij*+(1−fA)(fsεRijk+23(1−fs)εδij),
with fA, fs and εij* defined as(9)fA=e−20A2e−Ret2/20,fs=e−Ret2/40,εij*=(Rij+Riknjnk+Rjknink+Rklnknlninj)(k/ε)(1+(3/2)(Rpq/k)npnq).The extra termTij=Φij(R)+Bij+DijR+Jij accounts for implicit effects of the rotation
on the turbulence field. Φij(R),
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. Bij is a spectral jamming term acting only in the
case of strong rotation. The inhomogeneous diffusion term DijR slows down the bidimensionalization of the
flow close to the walls. Finally, Jij 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 ε(10)dεdt=−cε1εkRijVi,j−cε2fεε˜εk+(cεkεRijε,j+νε,i),i+cε3νkεRjkVi,jlVi,kl+(cε4νε˜kk,i),i, where ε˜ is the isotropic part of the dissipation rate ε˜=ε−2νk,i1/2k,i1/2. cε1=1, cε2=1.92, cε=0.15, cε3=2,
and cε4=0.92 are four empirical constants and fε is defined by fε=1/(1+0.63AA2).
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(11)dkdt=−RijVi,j−ε+Tjj2+0.22(kεRijk,j+νk,i),i.
It is verified
after convergence that k is exactly 0.5Rjj.


## 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 140×80 mesh in the (r,z) frame is sufficient in most cases to get
grid-independent solutions. Nevertheless, a more refined mesh 200×80 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 200×80 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 Δ1r=1.677×10−4h and Δ1z=1.532×10−4h in the radial and axial directions,
respectively.Table 1
Influence of the mesh grid on the mean and turbulent fields forG=0.02, Re=106, and Cw=0. Comparisons with the 200×80 mesh which is chosen as the reference mesh grid.
Difference with the200×80 mesh250×80300×80250×100Mean field0.5%0.6%0.5%Turbulent field1.9%2.1%2%It is also verified that the
grid is sufficiently refined close the disks to
describe accurately the viscous sublayers. For example, the wall coordinatez+=Δ1zu*/ν (u* the friction velocity at the wall and Δ1z the size of the first mesh in the axial
direction) remains below 0.1 along both disks and for the whole radial
extent for G=0.04, Re=1.8×105,
and Cw=0.
It is quite below the classical value z+=1,
for which the viscous sublayer is described by 5 mesh points.About 20 000 iterations (about7 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 velocityVθ,
which is set to Ωr on rotating walls and zero on stationary
walls. At the inlet, Vθ is supposed to vary linearly from zero on the
stationary wall up to Ωr 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 Vz 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:z*=z/h, r*=r/(b+js), Vi*=Vi¯/(Ωr),
and Rij*=vi′vj′¯/(Ωr)2 with i,j=(r,θ,z).


## 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 k-ε model of Launder and Sharma [27] for Re=106, G=0.036,
and several values of the flowrate coefficient Cw.The axial
profiles of the mean radialVr* and tangential Vθ* velocity components are displayed at r*=0.56 in Figures 3 and 4 for centripetal and
centrifugal throughflows, respectively.Axial profiles of two mean velocity components
atr*=0.56 for G=0.036 and Re=106 with or without a centripetal superimposed
throughflow: (a) Cw=0,
(b) Cw=1976,
(c) Cw=5929,
(d) Cw=9881.
Comparisons between the present RSM model (−), the k-ϵ model (−⋅−), and LDA measurements (∘).
(a)(b)(c)(d)Axial profiles of two mean velocity components
atr*=0.56 for G=0.036 and Re=106 with or without a centrifugal superimposed
throughflow: (a) Cw=0,
(b) Cw=−1976,
(c) Cw=−5929,
(d) Cw=−9881.
Comparisons between the present RSM model (−), the k-ϵ model (−⋅−), and LDA measurements (∘).
(a)(b)(c)(d)For centripetal
throughflows (see Figure3), the k-ϵ model underestimated the mean tangential
velocity in the core of the flow around z*=0.5.
The RSM model improves also the predictions of the mean field in the Ekman
boundary layer attached to the rotating disk (z*≃0) and provides very satisfactory results
compared to the experimental data. The main discrepancy is obtained for very
large values of Cw 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 Cw,
to have a numerical code as universal as possible. In the experiments, Poncet
[11] has observed that
it slightly varies between 0.45 for Cw=0 and 0.55 for very large values of Cw.In the case of
centrifugal superimposed throughflows (see Figure4), the RSM model still
provides better results than the k-ϵ 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 componentsRrr*, Rrθ*,
and Rθθ* are presented in Figure 5 for three
characteristic values of Cw at r*=0.56.
The k-ϵ 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 Rrθ* component along the stator side for
centripetal throughflows, which is directly linked to the underestimation of
the mean radial velocity in this boundary layer.Axial profiles of three Reynolds stress tensor
components atr*=0.56 for G=0.036 and Re=106:
(a) Cw=9881,
(b) Cw=0,
(c) Cw=−9881.
Comparisons between the present RSM model (−), the k-ϵ model (−⋅−), and LDA measurements (∘).
(a)(b)(c)To conclude,
the RSM model improves very significantly the predictions of a classicalk-ε 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 numberRe and the aspect ratio G of the cavity is
investigated in the case of an enclosed rotor-stator cavity (Cw=0).


### 3.1. Structure of the Mean Flow

Figure6
presents the axial profiles of the mean radial and tangential velocity
components for Re=106 and G=0.02 at different radial locations in the range of 0.31≤r*≤0.83.
The axial profiles of the axial mean velocity component are not shown here
because Vz*≃0 in the whole cavity except very close to the
inner and outer cylinders. According to the diagram (Re, G) 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, Vr* remains positive. By mass conservation, Vr* is then negative along the stator. Nevertheless,
from the axial profiles, two different behaviors are obtained depending on the
radial location. For r*=0.31,
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 Vθ* 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 r*≥0.44,
the region of zero Vr* has disappeared and Vr* varies linearly between the disks. By
increasing the values of r*,
the magnitudes of Vr* 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 r*=0.31 and r*=0.44 can be explained by considering the local
Reynolds number Rer=Ωr2/ν,
which is the pertinent parameter to characterize the transition between two
flow regimes. Thus, Rer=96 100 for r*=0.31,
whereas Rer=193 600 for r*=0.44.
The value Rer=96 100 is below the critical Reynolds number for the
transition between the regimes II and III [1].Axial profiles of the mean radial and
tangential velocity components forRe=106 and G=0.02 at different radial locations r*.
(a)(b)In enclosed
rotor-stator flows, the boundary layer thicknesses are an increasing function
of the radial locationr* [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 Rer-values (see Cooper and Reshotko [9]).Figure7
presents the axial profiles of Vr* and Vθ* for a given aspect ratio and various Reynolds
numbers in the range of 1.8×105≤Re≤107 at midcavity (r*=0.57). The axial velocity Vz* 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 Re=1.8×105 (Rer=58 482), there is no region of zero radial velocity
and the extrema of Vr* are quite large, which is typical of regime I.
For Re=5×105 (Rer=162 450), the flow belongs to the regime III with a
linear profile of Vr* far from the disks and a significant decrease
of the magnitudes of the Vr* extrema. By increasing further the Reynolds
number up to Re=107,
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 Re up to 1.02×107.Axial profiles of two mean velocity components
atr*=0.57 for G=0.02 and different Reynolds numbers.
(a)(b)The streamline
patterns corresponding to the cases previously considered are displayed in
Figures8 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 (z*≃0.5) as in Figure 8(a) are typical of a
Batchelor-like flow. For Re=1.8×105,
it is the case for the largest aspect ratios G=0.05 (see Figure 8(a)) and 0.04 (see Figure 8(b)). For the lower value G=0.02 (see Figure 8(c)), the streamline
patterns get curved and are characteristic of torsional Couette flows. When one
increases the Reynolds number up to 5×106,
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 G=0.04 (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.Influence of the aspect ratio on the streamline patternsΨ*=Ψ/(Ωh2) (20 contours) for various sets of flow parameters
(Re, G): (a) (1.8×105, 0.05), −0.03≤Ψ*≤5.96;
(b) (1.8×105, 0.04), −0.05≤Ψ*≤10.8;
(c) (1.8×105, 0.02), −0.31≤Ψ*≤76.89; (d) (5×106, 0.04), −0.04≤Ψ*≤8.32;
(e) (5×106, 0.02), −0.32≤Ψ*≤36.92.
(a)(b)(c)(d)(e)Influence of the Reynolds number on the
streamline patternsΨ* (20 contours) for G=0.02 and six values of Re:
(a) 1.8×105, −0.31≤Ψ*≤76.89;
(b) 5×105, −0.66≤Ψ*≤60.59;
(c) 106, −0.63≤Ψ*≤48.19;
(d) 3×106, −0.41≤Ψ*≤39.16;
(e) 5×106, −0.32≤Ψ*≤36.92;
(f) 107, −0.23≤Ψ*≤34.38.
(a)(b)(c)(d)(e)(f)For a given
aspect ratioG=0.02,
Figure 9 highlights the effect of the rotation rate reflected in the values of
the Reynolds number Re on the streamline patterns. For Re=1.8×105 (see Figure 9(a)), the flow exhibits a
torsional Couette flow behavior. By increasing the values of Re (see Figures 9(b)–9(f)), the streamlines get
aligned with the z 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 r*.
A strong curvature of the streamlines close to the walls indicate strong
extrema of the radial velocity Vr* 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. Figure10 shows the axial profiles of the six components of the Reynolds stress tensor
at given Reynolds number Re=106 and aspect ratio G=0.02 for various radii in the range of 0.31≤r*≤0.83 corresponding to the mean velocity profiles
presented in Figure 6. Close to the rotation axis at r*=0.31,
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 Rer,
which is proportional to the local radius
to power 2.
The maximum of the normal stresses, located close to the rotor, is reached at r*=0.57 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 r*.
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 10−3), which is not the case for Batchelor flows,
where Rzz* 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.Axial profiles of the six Reynolds stress
tensor components forRe=106 and G=0.02 at different radial locations r*.
(a)(b)(c)(d)(e)(f)The influence
of the Reynolds number on the six Reynolds stress tensor components is assessed
in Figure11 at r*=0.57 and for G=0.02.
As expected, when Re is increased, higher turbulence intensities
are obtained. For Re=1.8×105,
turbulence is confined along the rotor, whereas the flow remains laminar
everywhere else. For Re=5×105,
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, r*=0.57), the flow along the rotating disk is found
to be more turbulent than along the stator. For Re≥5×106,
there is no significant effect of the Reynolds number on the turbulence
intensities.Axial profiles of the six Reynolds stress
tensor components atr*=0.57 for G=0.02 and different Reynolds numbers.
(a)(b)(c)(d)(e)(f)The isovalues
of the turbulence Reynolds numberRet=k2/(νε) enable us to
visualize the turbulent regions of the flow. The flow is considered as being
laminar for Ret≤1/cμ≃11 (cμ=νtϵ/k2=0.09,
where νt is the turbulence viscosity). Figure 12
presents some maps of Ret in a (r,z) plane for various flow conditions (Re,G). Whatever the flow parameters, the maximum
of Ret,
denoted by Retmax,
is localized at the periphery of the cavity, where the highest values of the
local Reynolds number are obtained. For a moderate Reynolds number Re=1.8×105 and G≥0.02 (see Figures 12(a)–12(c)), the maximum of Ret is localized at the junction between the
shroud and the stator along the Stewartson layer attached to the external
cylinder. Retmax decreases with decreasing the values of the
aspect ratio.
If the Reynolds number is increased up to Re=5×106 (see Figures 12(d)-12(e)), Retmax is 27 (resp., 39) times larger for G=0.04 (resp., G=0.02). The flow is then turbulent in the whole
cavity with higher turbulence levels at largest radii.Influence of the aspect ratio on the isovalues
of the turbulence Reynolds numberRet=k2/(νε) (normalized by its maximum value Retmax) for various sets of flow parameters (Re,G): (a) (1.8×105, 0.05), Retmax=84.74;
(b) (1.8×105, 0.04), Retmax=78.58;
(c) (1.8×105, 0.02), Retmax=57.61;
(d) (5×106, 0.04), Retmax=2141.7;
(e) (5×106, 0.02), Retmax=2233.
(a)(b)(c)(d)(e)Figure13
presents the influence of the rotational Reynolds number, in the range of 1.8×105≤Re≤107,
on the maps of the turbulence Reynolds number Ret for a given aspect ratio G=0.02 corresponding to torsional Couette flows with
merged boundary layers. As expected, Retmax increases with increasing values of Re:
from Retmax=57.61 for Re=1.8×105 to Retmax=4128.3 for Re=107. Retmax remains localized close to the shroud but the
radial extent of the turbulent regions increases with Re too.Isovalues ofRet/Retmax for G=0.02 and six values of Re:
(a) 1.8×105, Retmax=57.61;
(b) 5×105, Retmax=145.13;
(c) 106, Retmax=464.32;
(d) 3×106, Retmax=1391.4;
(e) 5×106, Retmax=2233;
(f) 107, Retmax=4128.3.
(a)(b)(c)(d)(e)(f)Figure14 shows
the anisotropy invariant map for the Reynolds stress tensor at r*=0.7 for various sets of parameters (Re,G). The second A2 and third A3 invariants of the anisotropy tensor aij of the second moments of the fluctuations are
defined as A2=aijaji and A3=aijajkaki [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 A3=±A23/2/6 and the straight upper one corresponding to
the two-component limit A3=A2−8/9.
The influence of the aspect ratio on the anisotropy invariant map is large for Re=1.8×105 (see Figure 14(a)). For G=0.05,
the flow exhibits a Batchelor-like structure (see Figure 8(a)) and for 0.22≤z*≤0.45,
the invariant map follows the axisymmetric curve (A3<0), which is not the case for the other value of G corresponding to a torsional Couette flow
structure (see Figure 8(c)). For G=0.05 and G=0.02,
turbulence tends to the one-component limit (large values of A2 and A3) close to the disks. The influence of the Reynolds number on the
invariant anisotropy map of torsional Couette flows (G=0.02) remains weak (see Figure 14(b)). The map for Re=1.8×105 has already been discussed. In this case, the
flow is laminar at this radial location (see Figure 13(a)). If Re is increased to values larger than 5×105,
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 0.3≤r*≤0.9.
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 A 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.Anisotropy invariant maps atr*=0.7.
(a) Influence of the aspect ratio for Re=1.8×105 and G=0.05 (red), G=0.02 (blue); (b) influence of the Reynolds number
for G=0.02 and Re=5×105 (red), Re=3×106 (blue), and Re=107 (green).
(a)(b)


## 3.1. Structure of the Mean Flow

Figure6
presents the axial profiles of the mean radial and tangential velocity
components for Re=106 and G=0.02 at different radial locations in the range of 0.31≤r*≤0.83.
The axial profiles of the axial mean velocity component are not shown here
because Vz*≃0 in the whole cavity except very close to the
inner and outer cylinders. According to the diagram (Re, G) 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, Vr* remains positive. By mass conservation, Vr* is then negative along the stator. Nevertheless,
from the axial profiles, two different behaviors are obtained depending on the
radial location. For r*=0.31,
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 Vθ* 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 r*≥0.44,
the region of zero Vr* has disappeared and Vr* varies linearly between the disks. By
increasing the values of r*,
the magnitudes of Vr* 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 r*=0.31 and r*=0.44 can be explained by considering the local
Reynolds number Rer=Ωr2/ν,
which is the pertinent parameter to characterize the transition between two
flow regimes. Thus, Rer=96 100 for r*=0.31,
whereas Rer=193 600 for r*=0.44.
The value Rer=96 100 is below the critical Reynolds number for the
transition between the regimes II and III [1].Axial profiles of the mean radial and
tangential velocity components forRe=106 and G=0.02 at different radial locations r*.
(a)(b)In enclosed
rotor-stator flows, the boundary layer thicknesses are an increasing function
of the radial locationr* [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 Rer-values (see Cooper and Reshotko [9]).Figure7
presents the axial profiles of Vr* and Vθ* for a given aspect ratio and various Reynolds
numbers in the range of 1.8×105≤Re≤107 at midcavity (r*=0.57). The axial velocity Vz* 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 Re=1.8×105 (Rer=58 482), there is no region of zero radial velocity
and the extrema of Vr* are quite large, which is typical of regime I.
For Re=5×105 (Rer=162 450), the flow belongs to the regime III with a
linear profile of Vr* far from the disks and a significant decrease
of the magnitudes of the Vr* extrema. By increasing further the Reynolds
number up to Re=107,
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 Re up to 1.02×107.Axial profiles of two mean velocity components
atr*=0.57 for G=0.02 and different Reynolds numbers.
(a)(b)The streamline
patterns corresponding to the cases previously considered are displayed in
Figures8 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 (z*≃0.5) as in Figure 8(a) are typical of a
Batchelor-like flow. For Re=1.8×105,
it is the case for the largest aspect ratios G=0.05 (see Figure 8(a)) and 0.04 (see Figure 8(b)). For the lower value G=0.02 (see Figure 8(c)), the streamline
patterns get curved and are characteristic of torsional Couette flows. When one
increases the Reynolds number up to 5×106,
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 G=0.04 (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.Influence of the aspect ratio on the streamline patternsΨ*=Ψ/(Ωh2) (20 contours) for various sets of flow parameters
(Re, G): (a) (1.8×105, 0.05), −0.03≤Ψ*≤5.96;
(b) (1.8×105, 0.04), −0.05≤Ψ*≤10.8;
(c) (1.8×105, 0.02), −0.31≤Ψ*≤76.89; (d) (5×106, 0.04), −0.04≤Ψ*≤8.32;
(e) (5×106, 0.02), −0.32≤Ψ*≤36.92.
(a)(b)(c)(d)(e)Influence of the Reynolds number on the
streamline patternsΨ* (20 contours) for G=0.02 and six values of Re:
(a) 1.8×105, −0.31≤Ψ*≤76.89;
(b) 5×105, −0.66≤Ψ*≤60.59;
(c) 106, −0.63≤Ψ*≤48.19;
(d) 3×106, −0.41≤Ψ*≤39.16;
(e) 5×106, −0.32≤Ψ*≤36.92;
(f) 107, −0.23≤Ψ*≤34.38.
(a)(b)(c)(d)(e)(f)For a given
aspect ratioG=0.02,
Figure 9 highlights the effect of the rotation rate reflected in the values of
the Reynolds number Re on the streamline patterns. For Re=1.8×105 (see Figure 9(a)), the flow exhibits a
torsional Couette flow behavior. By increasing the values of Re (see Figures 9(b)–9(f)), the streamlines get
aligned with the z 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 r*.
A strong curvature of the streamlines close to the walls indicate strong
extrema of the radial velocity Vr* 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. Figure10 shows the axial profiles of the six components of the Reynolds stress tensor
at given Reynolds number Re=106 and aspect ratio G=0.02 for various radii in the range of 0.31≤r*≤0.83 corresponding to the mean velocity profiles
presented in Figure 6. Close to the rotation axis at r*=0.31,
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 Rer,
which is proportional to the local radius
to power 2.
The maximum of the normal stresses, located close to the rotor, is reached at r*=0.57 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 r*.
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 10−3), which is not the case for Batchelor flows,
where Rzz* 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.Axial profiles of the six Reynolds stress
tensor components forRe=106 and G=0.02 at different radial locations r*.
(a)(b)(c)(d)(e)(f)The influence
of the Reynolds number on the six Reynolds stress tensor components is assessed
in Figure11 at r*=0.57 and for G=0.02.
As expected, when Re is increased, higher turbulence intensities
are obtained. For Re=1.8×105,
turbulence is confined along the rotor, whereas the flow remains laminar
everywhere else. For Re=5×105,
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, r*=0.57), the flow along the rotating disk is found
to be more turbulent than along the stator. For Re≥5×106,
there is no significant effect of the Reynolds number on the turbulence
intensities.Axial profiles of the six Reynolds stress
tensor components atr*=0.57 for G=0.02 and different Reynolds numbers.
(a)(b)(c)(d)(e)(f)The isovalues
of the turbulence Reynolds numberRet=k2/(νε) enable us to
visualize the turbulent regions of the flow. The flow is considered as being
laminar for Ret≤1/cμ≃11 (cμ=νtϵ/k2=0.09,
where νt is the turbulence viscosity). Figure 12
presents some maps of Ret in a (r,z) plane for various flow conditions (Re,G). Whatever the flow parameters, the maximum
of Ret,
denoted by Retmax,
is localized at the periphery of the cavity, where the highest values of the
local Reynolds number are obtained. For a moderate Reynolds number Re=1.8×105 and G≥0.02 (see Figures 12(a)–12(c)), the maximum of Ret is localized at the junction between the
shroud and the stator along the Stewartson layer attached to the external
cylinder. Retmax decreases with decreasing the values of the
aspect ratio.
If the Reynolds number is increased up to Re=5×106 (see Figures 12(d)-12(e)), Retmax is 27 (resp., 39) times larger for G=0.04 (resp., G=0.02). The flow is then turbulent in the whole
cavity with higher turbulence levels at largest radii.Influence of the aspect ratio on the isovalues
of the turbulence Reynolds numberRet=k2/(νε) (normalized by its maximum value Retmax) for various sets of flow parameters (Re,G): (a) (1.8×105, 0.05), Retmax=84.74;
(b) (1.8×105, 0.04), Retmax=78.58;
(c) (1.8×105, 0.02), Retmax=57.61;
(d) (5×106, 0.04), Retmax=2141.7;
(e) (5×106, 0.02), Retmax=2233.
(a)(b)(c)(d)(e)Figure13
presents the influence of the rotational Reynolds number, in the range of 1.8×105≤Re≤107,
on the maps of the turbulence Reynolds number Ret for a given aspect ratio G=0.02 corresponding to torsional Couette flows with
merged boundary layers. As expected, Retmax increases with increasing values of Re:
from Retmax=57.61 for Re=1.8×105 to Retmax=4128.3 for Re=107. Retmax remains localized close to the shroud but the
radial extent of the turbulent regions increases with Re too.Isovalues ofRet/Retmax for G=0.02 and six values of Re:
(a) 1.8×105, Retmax=57.61;
(b) 5×105, Retmax=145.13;
(c) 106, Retmax=464.32;
(d) 3×106, Retmax=1391.4;
(e) 5×106, Retmax=2233;
(f) 107, Retmax=4128.3.
(a)(b)(c)(d)(e)(f)Figure14 shows
the anisotropy invariant map for the Reynolds stress tensor at r*=0.7 for various sets of parameters (Re,G). The second A2 and third A3 invariants of the anisotropy tensor aij of the second moments of the fluctuations are
defined as A2=aijaji and A3=aijajkaki [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 A3=±A23/2/6 and the straight upper one corresponding to
the two-component limit A3=A2−8/9.
The influence of the aspect ratio on the anisotropy invariant map is large for Re=1.8×105 (see Figure 14(a)). For G=0.05,
the flow exhibits a Batchelor-like structure (see Figure 8(a)) and for 0.22≤z*≤0.45,
the invariant map follows the axisymmetric curve (A3<0), which is not the case for the other value of G corresponding to a torsional Couette flow
structure (see Figure 8(c)). For G=0.05 and G=0.02,
turbulence tends to the one-component limit (large values of A2 and A3) close to the disks. The influence of the Reynolds number on the
invariant anisotropy map of torsional Couette flows (G=0.02) remains weak (see Figure 14(b)). The map for Re=1.8×105 has already been discussed. In this case, the
flow is laminar at this radial location (see Figure 13(a)). If Re is increased to values larger than 5×105,
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 0.3≤r*≤0.9.
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 A 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.Anisotropy invariant maps atr*=0.7.
(a) Influence of the aspect ratio for Re=1.8×105 and G=0.05 (red), G=0.02 (blue); (b) influence of the Reynolds number
for G=0.02 and Re=5×105 (red), Re=3×106 (blue), and Re=107 (green).
(a)(b)


## 4. Turbulent Flow with an Axial Throughflow

In the
following, the value of the aspect ratio is fixed toG=0.02 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 coefficientCw<0,
is superimposed on the mean tangential flow due to rotation.


#### 4.1.1. Structure of the Mean Flow

Figure15
displays the velocity profiles corresponding to the case Re=106 and Cw=−2500 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 r*=0.31,
the flow exhibits a Stewartson-like structure from the Vθ*-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 |Cw|=0.219Re4/5r*13/5.
Thus, for the set of parameters considered in Figure 15, the transition should
occur for r*=0.51.
The transition between these two behaviors is continuous and appears here for r*=0.48,
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.Axial profiles of the mean radial and
tangential velocity components forRe=106, G=0.02,
and Cw=−2500 at different radial locations r*.
(a)(b)When moving
toward the periphery of the cavity, the flow switches progressively to
torsional Couette type withVθ*→0.4 for increasing values of r*,
while the flow along the stator gets centripetal. Moreover, the axial profile
of Vr* around midplane gets linear, which indicates a
strong shear around midplane. At intermediate radial locations, Vr* 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 Figure16) but
also from the streamline patterns (see Figures 17(f)–17(j)), the flow structure
switches from torsional Couette type for small values of |Cw|≤103 to Stewartson type for larger values |Cw|≥5×103 at Re=106 and r*=0.57.
The transition appears for 2500≤|Cw|≤5000,
in accordance with the empirical law of Phadke and Owen [8], which gives |Cw|=3204 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 (G, Re, Cw<0, r*) is, the mean axial velocity component
remains very weak apart along the hub and the shroud.Axial profiles of the mean radial and
tangential velocity components forRe=106, G=0.02 at r*=0.57,
and five flowrate coefficients Cw<0.
(a)(b)Figure 17
20 streamline patternsΨ* for G=0.02 and various sets of parameters (Re, Cw): (a) (1.8×105, −500), −3.67≤Ψ*≤125.06;
(b) (1.8×105, −1000), −1.56≤Ψ*≤118.96;
(c) (1.8×105, −2500), −2.32≤Ψ*≤278.67;
(d) (1.8×105, −5000), −4.65≤Ψ*≤557.79;
(e) (1.8×105, −10 000), −9.29≤Ψ*≤1116.8;
(f) (106, −500), −12.52≤Ψ*≤52.55;
(g) (106, −1000), −7.7≤Ψ*≤56.67;
(h) (106, −2500), −0.79≤Ψ*≤67.75;
(i) (106, −5000), −0.84≤Ψ*≤105.93;
(j) (106, −10 000), −1.67≤Ψ*≤201.81;
(k) (5×106, −500), −19.49≤Ψ*≤36.54;
(l) (5×106, −1000), −17.52≤Ψ*≤37.56;
(m) (5×106, −2500), −13.14≤Ψ*≤41.05;
(n) (5×106, −5000), −7.39≤Ψ*≤46.44;
(o) (5×106, −10 000), −1.06≤Ψ*≤56.02.The streamlines
are displayed in Figure17 for a given aspect ratio G=0.02 and 15 flow conditions: three values of the Reynolds
number Re=1.8×105 (see Figures 17(a)–17(e)), 106 (see Figures 17(f)–17(j)), and 5×106 (Figures 17(k)–17(o)) and five flowrate
coefficients Cw=−500,−103,−2.5×103,−5×103, and −104.(i) ForRe=1.8×105,
rotation has a dominant effect on the flow structure at the lowest value of Cw=−500 at the periphery of the cavity where the
recirculation zone is obtained (see Figure 17(a)). Closer to the rotation axis,
the flow is dominated by the outward throughflow since the streamlines are
parallel to the disks. For Cw=−103,
the recirculation zone is smaller (see Figure 17(b)) and disappears completely
for higher flowrates |Cw|≥2500 (see Figures 17(c)–17(e)). The flow is then
purely centrifugal with streamlines parallel to the disks.(ii) ForRe=106,
apart in the region very close to the rotation axis when the centrifugal
throughflow enters the cavity, the flow is mainly dominated by rotation for Cw=−500 (see Figure 17(f)). For Cw=−103,
the flow is purely centrifugal for r*≤0.4 and gets torsional
Couette type in the outer part of the cavity (see Figure 17(g)). When one
increases further the magnitude of the throughflow up to |Cw|=2500 (see Figure 17(h)), the recirculation bubble
gets smaller and disappears for |Cw|≥5000 (see Figures 17(i)-17(j)).(iii) ForRe=5×106,
the same scenario is obtained when one increases the values of |Cw|.
The flow exhibits a torsional Couette flow structure for Cw=−500 (see Figure 17(k)) and Cw=−103 (see Figure 17(l)). The size of the
recirculation bubble decreases for increasing the values of |Cw| up to 104 but it does not disappear for this high
Reynolds number. Rotation has still a dominant effect at the periphery of the
cavity, where high values of the local Reynolds number prevail.


#### 4.1.2. Turbulence Statistics

Figure18
exhibits the axial profiles of the six Reynolds stresses for Re=106, G=0.02,
and Cw=−2500 at five radial locations. As for turbulent
Batchelor flows (regime IV) with inward throughflow [3], turbulence intensities are
higher along the rotating disk and vanish toward the stationary disk. Contrary
to the case of a closed cavity Cw=0,
they decrease when moving toward the periphery of the cavity, whereas the local
Reynolds number is increased. Turbulence is then mainly controlled by the
imposed throughflow, whose influence on turbulence is preponderant compared to
rotation. It is confirmed by the maps (see Figure 20) of the isoturbulence
Reynolds number Ret (detailed below), which show that turbulence
is confined at the inlet of the throughflow close to the rotation axis.Axial profiles of the six Reynolds stress
tensor components forRe=106, G=0.02,
and Cw=−2500 at different radial locations r*.
(a)(b)(c)(d)(e)(f)At a given
radial locationr*=0.57,
Figure 19 shows the influence of the flowrate coefficient on the Reynolds
stresses. Turbulence intensities increase for increasing values of |Cw| but the effect of the throughflow gets weaker
for |Cw|≥5000.
The magnitudes of the normal components of the Reynolds stress tensor are quite
comparable for this range of parameters. They all vanish at the stator surface.
In the same way, the cross-components of the Reynolds stress tensor are maximum
along the rotating disk, which indicates a very strong shear stress induced by
a combined effect of the throughflow and the rotation and vanish toward the
stationary disk.Axial profiles of the six Reynolds stress
tensor components forRe=106, G=0.02 at r*=0.57 and five flowrate coefficients Cw<0.
(a)(b)(c)(d)(e)(f)Figure 20
Isovalues ofRet/Retmax for G=0.02 and various sets of parameters (Re, Cw): (a) (1.8×105, −500), Retmax=1497.7;
(b) (1.8×105, −1000), Retmax=1483.4;
(c) (1.8×105, −2500), Retmax=1293.5;
(d) (1.8×105, −5000), Retmax=2019.1;
(e) (1.8×105, −10 000), Retmax=4535.6;
(f) (106, −500), Retmax=1499.8;
(g) (106, −1000), Retmax=1499.9;
(h) (106, −2500), Retmax=1499.2;
(i) (106, −5000), Retmax=1752.4;
(j) (106, −10 000), Retmax=4410.7;
(k) (5×106, −500), Retmax=2169.4;
(l) (5×106, −1000), Retmax=2199.6;
(m) (5×106, −2500), Retmax=2282.8;
(n) (5×106, −5000), Retmax=2420.5;
(o) (5×106, −10 000), Retmax=2642.1.Compared to the
case without throughflowCw=0,
when an outward throughflow is superimposed on the mean tangential flow, the
maps of the turbulence Reynolds number Ret are clearly modified (see Figure 20). Whatever
the Reynolds number is, the maximum of Ret is an increasing function of the magnitude of
the flowrate coefficient Cw,
especially for |Cw|≥5000.
For Re=1.8×105 and 106,
the flow is mainly dominated by the imposed throughflow. Thus, Retmax is located along the hub in the inlet region.
This maximum is slightly shifted to larger radii, when |Cw| is increased. For the highest Reynolds number Re=5×106 (see Figures 20(k)–20(o)), turbulence is
produced by a combined effect of the high rotation rate and the strong incoming
jet. The highly turbulent flow regions are then located both at the inlet and
at the periphery of the cavity. The iso-Ret patterns resemble the
ones obtained in a closed cavity (see Figure 13(e)).As shown in
Figure21(b), the anisotropy invariant maps are not modified by an imposed
outward throughflow compared to the closed cavity case (see Figure 14(b)) for r*=0.7 and turbulent torsional Couette flows (Re=106, G=0.02). Turbulence is mainly composed
of three components, whatever the values of the flowrate
coefficient Cw.
It is not the case closer to the rotation axis at r*=0.3 (see Figure 21(a)), where the curves tend to
the isotropic case (A3=A2=0) for centrifugal flows, whereas we recall
that turbulence tends to the one-component limit in a closed cavity. In the
isotropic case, the flatness parameter A is equal to 1.Anisotropy invariant map at (a)r*=0.3 and (b) r*=0.7 for Re=106, G=0.02 and three flowrate coefficients: Cw=−500 (red), Cw=−2500 (blue), and Cw=−10 000 (green).
(a)(b)


### 4.2. Case of a Centripetal Throughflow

An axial outward throughflow, characterized by a positive flowrate coefficientCw>0, is now superimposed on the mean tangential flow. The inlet of the jet is then located at the periphery of the cavity and the outlet is confined between the
hub and the stator.


#### 4.2.1. Structure of the Mean Flow

Figure22 represents
the axial profiles of the dimensionless tangential and radial velocity
components for Re=106, G=0.02,
and Cw=2500.
We first analyze the influence of the radial position in the range of 0.31≤r*≤0.83.
The corresponding streamlines are displayed in Figure 24(d). In the outer half
of the cavity r*≥0.57,
where a large recirculation is observed, the flow structure is quite similar to
the one obtained in a closed cavity Cw=0:
the fluid is centrifugal due to rotation along the rotor and is centripetal
along the stator by conservation of mass. Between these two flow regions, the
axial profile of the radial velocity is almost linear and the tangential
velocity is almost constant and still inferior to the local disk velocity (Vθ*<1). For r*=0.44,
the fluid in the main part of the interdisk spacing rotates at the same angular
velocity than the rotating disk. The radial velocity along this disk is then
equal to zero. When one approaches further the rotation axis, at r*=0.31, Vθ* increases and the fluid rotates 50% faster than the rotor. Indeed, because of the
conservation of the angular momentum, a fluid particle increases its rotation
when flowing to the center. Vr* is then negative whatever the axial position
and the flow is purely centripetal. This behavior
has been observed also for turbulent Batchelor flows by Poncet et al. [3].Axial profiles of the mean radial and
tangential velocity components forRe=106, G=0.02,
and Cw=2500 at different radial locations r*.
(a)(b)When the local
radius is fixed tor*=0.57 and we vary the flowrate coefficient Cw from 2500 to 104,
the same phenomenon is obtained (see Figure 23). For low values of Cw,
the velocity profiles are quite similar to the case Cw=0.
For Cw=5000,
the fluid rotates at the same angular velocity as
the rotor and the radial velocity is then equal to
zero along the rotating disk. By increasing further Cw up to 104,
by conservation of the angular momentum, Vθ* gets larger than 1 and the flow is purely centripetal. As in the
case of a centrifugal throughflow, the mean axial velocity component remains
very weak apart along the hub and the shroud when an inward throughflow is
superimposed.Axial profiles of the mean radial and
tangential velocity components forRe=106, G=0.02 at r*=0.57 and three flowrate coefficients Cw>0.
(a)(b)20 streamline patternsΨ* for G=0.02 and six sets of parameters (Re, Cw): (a) (1.8×105, 2500), −398.68≤Ψ*≤151.17;
(b) (1.8×105, 5000), −714.6≤Ψ*≤115.9;
(c) (1.8×105, 10 000), −1274.1≤Ψ*≤82.89;
(d) (106, 2500), −127.01≤Ψ*≤37.5;
(e) (106, 5000), −117.11≤Ψ*≤23.18;
(f) (106, 10 000), −201.77≤Ψ*≤58.8.Regarding the
corresponding streamline patterns (see Figure24), it appears that the inward
throughflow has a dominant effect on the flow structure compared to the effect
of rotation. At the inlet, when the axial jet impinges the stator, one part of
the fluid is moving inwards toward the axis of rotation along the stator and
the other part is centrifugal due to the rotation of the disk. Thus, a large
recirculation is created at the periphery of the cavity (see Figure 24(d)). In
that case, the flow exhibits the same structure as in a closed cavity for large
radii and the fluid rotates faster than the disk close to the axis. For a given
Reynolds number Re=106,
when the flowrate is increased to 104 (see Figure 24(f)), the flow is dominated by
the incoming axial jet. The flow is purely centripetal and the streamlines are
then parallel to the disks as in the centrifugal case.


#### 4.2.2. Turbulence Statistics

As for
turbulent flows belonging to regime IV [1], turbulence is concentrated along the stator when an
axial inward throughflow is superimposed (see Figure 25). Nevertheless, one
characteristic feature of these flows compared to flows in large clearance
cavities is that turbulence intensities decrease when one moves toward the
periphery of the cavity. According to the maps of Ret (see Figure 27), turbulence is enhanced by the
inlet and the outlet of the axial throughflow. Besides, the
maximum of turbulence is obtained in the outlet region close to
the hub. This behavior is very similar to the one obtained in the case of a
centrifugal throughflow, previously discussed.Axial profiles of the six Reynolds stress
tensor components forRe=106, G=0.02,
and Cw=2500 at different radial locations r*.
(a)(b)(c)(d)(e)(f)The effect of
the flowrate coefficient on the turbulent field is displayed in Figure26. A
centripetal throughflow increases the turbulent intensities and there is no
threshold value for Cw contrary to the case of a centrifugal
throughflow. If the flowrate coefficient is increased by a factor 2,
turbulence levels increase by almost a factor 1.5.
For a given Reynolds number Re=106,
the aspect ratio G=0.02 and the radial location r*=0.57;
by comparing the results shown in Figures 11, 19, and 26, the maximum of
turbulence intensities is obtained along the stator when an inward throughflow
is superimposed. Note that the values of the normal Reynolds stress tensor
components are still quite comparable in the whole cavity.Axial profiles of the six Reynolds stress
tensor components forRe=106, G=0.02 at r*=0.57 and three flowrate coefficients Cw>0.
(a)(b)(c)(d)(e)(f)Isovalues ofRet/Retmax for G=0.02 and six sets (Re, Cw): (a) (1.8×105, 2500), Retmax=675.94;
(b) (1.8×105, 5000), Retmax=1557.5;
(c) (1.8×105, 10 000), Retmax=4507.4;
(d) (106, 2500), Retmax=1383.9;
(e) (106, 5000), Retmax=1615.7;
(f) (106, 10 000), Retmax=3333.The effect of
an inward throughflow on the maps of the turbulence Reynolds numberRet is highlighted in Figure 27. As in the case of
an outward throughflow, the maximum values of Ret are localized along the hub which corresponds
to the outlet region of the flow. It is a combined effect of the rotating hub
and of the outcoming jet, which creates a fully turbulent flow region at the
lowest radii. A second flow region of high turbulence levels is found at the
periphery of the cavity along the shroud and especially at the impact of the incoming
flow with the stator. Retmax is an increasing function of both Re and Cw but the influence of the flowrate coefficient
remains larger in the present case.Figure28
presents the anisotropy invariant maps when an inward throughflow is
superimposed on a turbulent torsional Couette flow (Re=106 and G=0.02) at two radii. Turbulence tends to the
one-component limit close to the disks and is fairly close to the isotropic
case at midplane. The map remains almost the same whatever the values of Cw are and for these
two radial locations.Anisotropy invariant map at (a)r*=0.3 and (b) r*=0.7 for Re=106, G=0.02 and three flowrate coefficients: Cw=2500 (red), Cw=5000 (blue), and Cw=10 000 (green).
(a)(b)


## 4.1. Case of a Centrifugal Throughflow

In this
subsection, an axial centrifugal throughflow, characterized by a negative
flowrate coefficientCw<0,
is superimposed on the mean tangential flow due to rotation.


### 4.1.1. Structure of the Mean Flow

Figure15
displays the velocity profiles corresponding to the case Re=106 and Cw=−2500 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 r*=0.31,
the flow exhibits a Stewartson-like structure from the Vθ*-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 |Cw|=0.219Re4/5r*13/5.
Thus, for the set of parameters considered in Figure 15, the transition should
occur for r*=0.51.
The transition between these two behaviors is continuous and appears here for r*=0.48,
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.Axial profiles of the mean radial and
tangential velocity components forRe=106, G=0.02,
and Cw=−2500 at different radial locations r*.
(a)(b)When moving
toward the periphery of the cavity, the flow switches progressively to
torsional Couette type withVθ*→0.4 for increasing values of r*,
while the flow along the stator gets centripetal. Moreover, the axial profile
of Vr* around midplane gets linear, which indicates a
strong shear around midplane. At intermediate radial locations, Vr* 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 Figure16) but
also from the streamline patterns (see Figures 17(f)–17(j)), the flow structure
switches from torsional Couette type for small values of |Cw|≤103 to Stewartson type for larger values |Cw|≥5×103 at Re=106 and r*=0.57.
The transition appears for 2500≤|Cw|≤5000,
in accordance with the empirical law of Phadke and Owen [8], which gives |Cw|=3204 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 (G, Re, Cw<0, r*) is, the mean axial velocity component
remains very weak apart along the hub and the shroud.Axial profiles of the mean radial and
tangential velocity components forRe=106, G=0.02 at r*=0.57,
and five flowrate coefficients Cw<0.
(a)(b)Figure 17
20 streamline patternsΨ* for G=0.02 and various sets of parameters (Re, Cw): (a) (1.8×105, −500), −3.67≤Ψ*≤125.06;
(b) (1.8×105, −1000), −1.56≤Ψ*≤118.96;
(c) (1.8×105, −2500), −2.32≤Ψ*≤278.67;
(d) (1.8×105, −5000), −4.65≤Ψ*≤557.79;
(e) (1.8×105, −10 000), −9.29≤Ψ*≤1116.8;
(f) (106, −500), −12.52≤Ψ*≤52.55;
(g) (106, −1000), −7.7≤Ψ*≤56.67;
(h) (106, −2500), −0.79≤Ψ*≤67.75;
(i) (106, −5000), −0.84≤Ψ*≤105.93;
(j) (106, −10 000), −1.67≤Ψ*≤201.81;
(k) (5×106, −500), −19.49≤Ψ*≤36.54;
(l) (5×106, −1000), −17.52≤Ψ*≤37.56;
(m) (5×106, −2500), −13.14≤Ψ*≤41.05;
(n) (5×106, −5000), −7.39≤Ψ*≤46.44;
(o) (5×106, −10 000), −1.06≤Ψ*≤56.02.The streamlines
are displayed in Figure17 for a given aspect ratio G=0.02 and 15 flow conditions: three values of the Reynolds
number Re=1.8×105 (see Figures 17(a)–17(e)), 106 (see Figures 17(f)–17(j)), and 5×106 (Figures 17(k)–17(o)) and five flowrate
coefficients Cw=−500,−103,−2.5×103,−5×103, and −104.(i) ForRe=1.8×105,
rotation has a dominant effect on the flow structure at the lowest value of Cw=−500 at the periphery of the cavity where the
recirculation zone is obtained (see Figure 17(a)). Closer to the rotation axis,
the flow is dominated by the outward throughflow since the streamlines are
parallel to the disks. For Cw=−103,
the recirculation zone is smaller (see Figure 17(b)) and disappears completely
for higher flowrates |Cw|≥2500 (see Figures 17(c)–17(e)). The flow is then
purely centrifugal with streamlines parallel to the disks.(ii) ForRe=106,
apart in the region very close to the rotation axis when the centrifugal
throughflow enters the cavity, the flow is mainly dominated by rotation for Cw=−500 (see Figure 17(f)). For Cw=−103,
the flow is purely centrifugal for r*≤0.4 and gets torsional
Couette type in the outer part of the cavity (see Figure 17(g)). When one
increases further the magnitude of the throughflow up to |Cw|=2500 (see Figure 17(h)), the recirculation bubble
gets smaller and disappears for |Cw|≥5000 (see Figures 17(i)-17(j)).(iii) ForRe=5×106,
the same scenario is obtained when one increases the values of |Cw|.
The flow exhibits a torsional Couette flow structure for Cw=−500 (see Figure 17(k)) and Cw=−103 (see Figure 17(l)). The size of the
recirculation bubble decreases for increasing the values of |Cw| up to 104 but it does not disappear for this high
Reynolds number. Rotation has still a dominant effect at the periphery of the
cavity, where high values of the local Reynolds number prevail.


### 4.1.2. Turbulence Statistics

Figure18
exhibits the axial profiles of the six Reynolds stresses for Re=106, G=0.02,
and Cw=−2500 at five radial locations. As for turbulent
Batchelor flows (regime IV) with inward throughflow [3], turbulence intensities are
higher along the rotating disk and vanish toward the stationary disk. Contrary
to the case of a closed cavity Cw=0,
they decrease when moving toward the periphery of the cavity, whereas the local
Reynolds number is increased. Turbulence is then mainly controlled by the
imposed throughflow, whose influence on turbulence is preponderant compared to
rotation. It is confirmed by the maps (see Figure 20) of the isoturbulence
Reynolds number Ret (detailed below), which show that turbulence
is confined at the inlet of the throughflow close to the rotation axis.Axial profiles of the six Reynolds stress
tensor components forRe=106, G=0.02,
and Cw=−2500 at different radial locations r*.
(a)(b)(c)(d)(e)(f)At a given
radial locationr*=0.57,
Figure 19 shows the influence of the flowrate coefficient on the Reynolds
stresses. Turbulence intensities increase for increasing values of |Cw| but the effect of the throughflow gets weaker
for |Cw|≥5000.
The magnitudes of the normal components of the Reynolds stress tensor are quite
comparable for this range of parameters. They all vanish at the stator surface.
In the same way, the cross-components of the Reynolds stress tensor are maximum
along the rotating disk, which indicates a very strong shear stress induced by
a combined effect of the throughflow and the rotation and vanish toward the
stationary disk.Axial profiles of the six Reynolds stress
tensor components forRe=106, G=0.02 at r*=0.57 and five flowrate coefficients Cw<0.
(a)(b)(c)(d)(e)(f)Figure 20
Isovalues ofRet/Retmax for G=0.02 and various sets of parameters (Re, Cw): (a) (1.8×105, −500), Retmax=1497.7;
(b) (1.8×105, −1000), Retmax=1483.4;
(c) (1.8×105, −2500), Retmax=1293.5;
(d) (1.8×105, −5000), Retmax=2019.1;
(e) (1.8×105, −10 000), Retmax=4535.6;
(f) (106, −500), Retmax=1499.8;
(g) (106, −1000), Retmax=1499.9;
(h) (106, −2500), Retmax=1499.2;
(i) (106, −5000), Retmax=1752.4;
(j) (106, −10 000), Retmax=4410.7;
(k) (5×106, −500), Retmax=2169.4;
(l) (5×106, −1000), Retmax=2199.6;
(m) (5×106, −2500), Retmax=2282.8;
(n) (5×106, −5000), Retmax=2420.5;
(o) (5×106, −10 000), Retmax=2642.1.Compared to the
case without throughflowCw=0,
when an outward throughflow is superimposed on the mean tangential flow, the
maps of the turbulence Reynolds number Ret are clearly modified (see Figure 20). Whatever
the Reynolds number is, the maximum of Ret is an increasing function of the magnitude of
the flowrate coefficient Cw,
especially for |Cw|≥5000.
For Re=1.8×105 and 106,
the flow is mainly dominated by the imposed throughflow. Thus, Retmax is located along the hub in the inlet region.
This maximum is slightly shifted to larger radii, when |Cw| is increased. For the highest Reynolds number Re=5×106 (see Figures 20(k)–20(o)), turbulence is
produced by a combined effect of the high rotation rate and the strong incoming
jet. The highly turbulent flow regions are then located both at the inlet and
at the periphery of the cavity. The iso-Ret patterns resemble the
ones obtained in a closed cavity (see Figure 13(e)).As shown in
Figure21(b), the anisotropy invariant maps are not modified by an imposed
outward throughflow compared to the closed cavity case (see Figure 14(b)) for r*=0.7 and turbulent torsional Couette flows (Re=106, G=0.02). Turbulence is mainly composed
of three components, whatever the values of the flowrate
coefficient Cw.
It is not the case closer to the rotation axis at r*=0.3 (see Figure 21(a)), where the curves tend to
the isotropic case (A3=A2=0) for centrifugal flows, whereas we recall
that turbulence tends to the one-component limit in a closed cavity. In the
isotropic case, the flatness parameter A is equal to 1.Anisotropy invariant map at (a)r*=0.3 and (b) r*=0.7 for Re=106, G=0.02 and three flowrate coefficients: Cw=−500 (red), Cw=−2500 (blue), and Cw=−10 000 (green).
(a)(b)


## 4.1.1. Structure of the Mean Flow

Figure15
displays the velocity profiles corresponding to the case Re=106 and Cw=−2500 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 r*=0.31,
the flow exhibits a Stewartson-like structure from the Vθ*-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 |Cw|=0.219Re4/5r*13/5.
Thus, for the set of parameters considered in Figure 15, the transition should
occur for r*=0.51.
The transition between these two behaviors is continuous and appears here for r*=0.48,
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.Axial profiles of the mean radial and
tangential velocity components forRe=106, G=0.02,
and Cw=−2500 at different radial locations r*.
(a)(b)When moving
toward the periphery of the cavity, the flow switches progressively to
torsional Couette type withVθ*→0.4 for increasing values of r*,
while the flow along the stator gets centripetal. Moreover, the axial profile
of Vr* around midplane gets linear, which indicates a
strong shear around midplane. At intermediate radial locations, Vr* 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 Figure16) but
also from the streamline patterns (see Figures 17(f)–17(j)), the flow structure
switches from torsional Couette type for small values of |Cw|≤103 to Stewartson type for larger values |Cw|≥5×103 at Re=106 and r*=0.57.
The transition appears for 2500≤|Cw|≤5000,
in accordance with the empirical law of Phadke and Owen [8], which gives |Cw|=3204 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 (G, Re, Cw<0, r*) is, the mean axial velocity component
remains very weak apart along the hub and the shroud.Axial profiles of the mean radial and
tangential velocity components forRe=106, G=0.02 at r*=0.57,
and five flowrate coefficients Cw<0.
(a)(b)Figure 17
20 streamline patternsΨ* for G=0.02 and various sets of parameters (Re, Cw): (a) (1.8×105, −500), −3.67≤Ψ*≤125.06;
(b) (1.8×105, −1000), −1.56≤Ψ*≤118.96;
(c) (1.8×105, −2500), −2.32≤Ψ*≤278.67;
(d) (1.8×105, −5000), −4.65≤Ψ*≤557.79;
(e) (1.8×105, −10 000), −9.29≤Ψ*≤1116.8;
(f) (106, −500), −12.52≤Ψ*≤52.55;
(g) (106, −1000), −7.7≤Ψ*≤56.67;
(h) (106, −2500), −0.79≤Ψ*≤67.75;
(i) (106, −5000), −0.84≤Ψ*≤105.93;
(j) (106, −10 000), −1.67≤Ψ*≤201.81;
(k) (5×106, −500), −19.49≤Ψ*≤36.54;
(l) (5×106, −1000), −17.52≤Ψ*≤37.56;
(m) (5×106, −2500), −13.14≤Ψ*≤41.05;
(n) (5×106, −5000), −7.39≤Ψ*≤46.44;
(o) (5×106, −10 000), −1.06≤Ψ*≤56.02.The streamlines
are displayed in Figure17 for a given aspect ratio G=0.02 and 15 flow conditions: three values of the Reynolds
number Re=1.8×105 (see Figures 17(a)–17(e)), 106 (see Figures 17(f)–17(j)), and 5×106 (Figures 17(k)–17(o)) and five flowrate
coefficients Cw=−500,−103,−2.5×103,−5×103, and −104.(i) ForRe=1.8×105,
rotation has a dominant effect on the flow structure at the lowest value of Cw=−500 at the periphery of the cavity where the
recirculation zone is obtained (see Figure 17(a)). Closer to the rotation axis,
the flow is dominated by the outward throughflow since the streamlines are
parallel to the disks. For Cw=−103,
the recirculation zone is smaller (see Figure 17(b)) and disappears completely
for higher flowrates |Cw|≥2500 (see Figures 17(c)–17(e)). The flow is then
purely centrifugal with streamlines parallel to the disks.(ii) ForRe=106,
apart in the region very close to the rotation axis when the centrifugal
throughflow enters the cavity, the flow is mainly dominated by rotation for Cw=−500 (see Figure 17(f)). For Cw=−103,
the flow is purely centrifugal for r*≤0.4 and gets torsional
Couette type in the outer part of the cavity (see Figure 17(g)). When one
increases further the magnitude of the throughflow up to |Cw|=2500 (see Figure 17(h)), the recirculation bubble
gets smaller and disappears for |Cw|≥5000 (see Figures 17(i)-17(j)).(iii) ForRe=5×106,
the same scenario is obtained when one increases the values of |Cw|.
The flow exhibits a torsional Couette flow structure for Cw=−500 (see Figure 17(k)) and Cw=−103 (see Figure 17(l)). The size of the
recirculation bubble decreases for increasing the values of |Cw| up to 104 but it does not disappear for this high
Reynolds number. Rotation has still a dominant effect at the periphery of the
cavity, where high values of the local Reynolds number prevail.


## 4.1.2. Turbulence Statistics

Figure18
exhibits the axial profiles of the six Reynolds stresses for Re=106, G=0.02,
and Cw=−2500 at five radial locations. As for turbulent
Batchelor flows (regime IV) with inward throughflow [3], turbulence intensities are
higher along the rotating disk and vanish toward the stationary disk. Contrary
to the case of a closed cavity Cw=0,
they decrease when moving toward the periphery of the cavity, whereas the local
Reynolds number is increased. Turbulence is then mainly controlled by the
imposed throughflow, whose influence on turbulence is preponderant compared to
rotation. It is confirmed by the maps (see Figure 20) of the isoturbulence
Reynolds number Ret (detailed below), which show that turbulence
is confined at the inlet of the throughflow close to the rotation axis.Axial profiles of the six Reynolds stress
tensor components forRe=106, G=0.02,
and Cw=−2500 at different radial locations r*.
(a)(b)(c)(d)(e)(f)At a given
radial locationr*=0.57,
Figure 19 shows the influence of the flowrate coefficient on the Reynolds
stresses. Turbulence intensities increase for increasing values of |Cw| but the effect of the throughflow gets weaker
for |Cw|≥5000.
The magnitudes of the normal components of the Reynolds stress tensor are quite
comparable for this range of parameters. They all vanish at the stator surface.
In the same way, the cross-components of the Reynolds stress tensor are maximum
along the rotating disk, which indicates a very strong shear stress induced by
a combined effect of the throughflow and the rotation and vanish toward the
stationary disk.Axial profiles of the six Reynolds stress
tensor components forRe=106, G=0.02 at r*=0.57 and five flowrate coefficients Cw<0.
(a)(b)(c)(d)(e)(f)Figure 20
Isovalues ofRet/Retmax for G=0.02 and various sets of parameters (Re, Cw): (a) (1.8×105, −500), Retmax=1497.7;
(b) (1.8×105, −1000), Retmax=1483.4;
(c) (1.8×105, −2500), Retmax=1293.5;
(d) (1.8×105, −5000), Retmax=2019.1;
(e) (1.8×105, −10 000), Retmax=4535.6;
(f) (106, −500), Retmax=1499.8;
(g) (106, −1000), Retmax=1499.9;
(h) (106, −2500), Retmax=1499.2;
(i) (106, −5000), Retmax=1752.4;
(j) (106, −10 000), Retmax=4410.7;
(k) (5×106, −500), Retmax=2169.4;
(l) (5×106, −1000), Retmax=2199.6;
(m) (5×106, −2500), Retmax=2282.8;
(n) (5×106, −5000), Retmax=2420.5;
(o) (5×106, −10 000), Retmax=2642.1.Compared to the
case without throughflowCw=0,
when an outward throughflow is superimposed on the mean tangential flow, the
maps of the turbulence Reynolds number Ret are clearly modified (see Figure 20). Whatever
the Reynolds number is, the maximum of Ret is an increasing function of the magnitude of
the flowrate coefficient Cw,
especially for |Cw|≥5000.
For Re=1.8×105 and 106,
the flow is mainly dominated by the imposed throughflow. Thus, Retmax is located along the hub in the inlet region.
This maximum is slightly shifted to larger radii, when |Cw| is increased. For the highest Reynolds number Re=5×106 (see Figures 20(k)–20(o)), turbulence is
produced by a combined effect of the high rotation rate and the strong incoming
jet. The highly turbulent flow regions are then located both at the inlet and
at the periphery of the cavity. The iso-Ret patterns resemble the
ones obtained in a closed cavity (see Figure 13(e)).As shown in
Figure21(b), the anisotropy invariant maps are not modified by an imposed
outward throughflow compared to the closed cavity case (see Figure 14(b)) for r*=0.7 and turbulent torsional Couette flows (Re=106, G=0.02). Turbulence is mainly composed
of three components, whatever the values of the flowrate
coefficient Cw.
It is not the case closer to the rotation axis at r*=0.3 (see Figure 21(a)), where the curves tend to
the isotropic case (A3=A2=0) for centrifugal flows, whereas we recall
that turbulence tends to the one-component limit in a closed cavity. In the
isotropic case, the flatness parameter A is equal to 1.Anisotropy invariant map at (a)r*=0.3 and (b) r*=0.7 for Re=106, G=0.02 and three flowrate coefficients: Cw=−500 (red), Cw=−2500 (blue), and Cw=−10 000 (green).
(a)(b)


## 4.2. Case of a Centripetal Throughflow

An axial outward throughflow, characterized by a positive flowrate coefficientCw>0, is now superimposed on the mean tangential flow. The inlet of the jet is then located at the periphery of the cavity and the outlet is confined between the
hub and the stator.


### 4.2.1. Structure of the Mean Flow

Figure22 represents
the axial profiles of the dimensionless tangential and radial velocity
components for Re=106, G=0.02,
and Cw=2500.
We first analyze the influence of the radial position in the range of 0.31≤r*≤0.83.
The corresponding streamlines are displayed in Figure 24(d). In the outer half
of the cavity r*≥0.57,
where a large recirculation is observed, the flow structure is quite similar to
the one obtained in a closed cavity Cw=0:
the fluid is centrifugal due to rotation along the rotor and is centripetal
along the stator by conservation of mass. Between these two flow regions, the
axial profile of the radial velocity is almost linear and the tangential
velocity is almost constant and still inferior to the local disk velocity (Vθ*<1). For r*=0.44,
the fluid in the main part of the interdisk spacing rotates at the same angular
velocity than the rotating disk. The radial velocity along this disk is then
equal to zero. When one approaches further the rotation axis, at r*=0.31, Vθ* increases and the fluid rotates 50% faster than the rotor. Indeed, because of the
conservation of the angular momentum, a fluid particle increases its rotation
when flowing to the center. Vr* is then negative whatever the axial position
and the flow is purely centripetal. This behavior
has been observed also for turbulent Batchelor flows by Poncet et al. [3].Axial profiles of the mean radial and
tangential velocity components forRe=106, G=0.02,
and Cw=2500 at different radial locations r*.
(a)(b)When the local
radius is fixed tor*=0.57 and we vary the flowrate coefficient Cw from 2500 to 104,
the same phenomenon is obtained (see Figure 23). For low values of Cw,
the velocity profiles are quite similar to the case Cw=0.
For Cw=5000,
the fluid rotates at the same angular velocity as
the rotor and the radial velocity is then equal to
zero along the rotating disk. By increasing further Cw up to 104,
by conservation of the angular momentum, Vθ* gets larger than 1 and the flow is purely centripetal. As in the
case of a centrifugal throughflow, the mean axial velocity component remains
very weak apart along the hub and the shroud when an inward throughflow is
superimposed.Axial profiles of the mean radial and
tangential velocity components forRe=106, G=0.02 at r*=0.57 and three flowrate coefficients Cw>0.
(a)(b)20 streamline patternsΨ* for G=0.02 and six sets of parameters (Re, Cw): (a) (1.8×105, 2500), −398.68≤Ψ*≤151.17;
(b) (1.8×105, 5000), −714.6≤Ψ*≤115.9;
(c) (1.8×105, 10 000), −1274.1≤Ψ*≤82.89;
(d) (106, 2500), −127.01≤Ψ*≤37.5;
(e) (106, 5000), −117.11≤Ψ*≤23.18;
(f) (106, 10 000), −201.77≤Ψ*≤58.8.Regarding the
corresponding streamline patterns (see Figure24), it appears that the inward
throughflow has a dominant effect on the flow structure compared to the effect
of rotation. At the inlet, when the axial jet impinges the stator, one part of
the fluid is moving inwards toward the axis of rotation along the stator and
the other part is centrifugal due to the rotation of the disk. Thus, a large
recirculation is created at the periphery of the cavity (see Figure 24(d)). In
that case, the flow exhibits the same structure as in a closed cavity for large
radii and the fluid rotates faster than the disk close to the axis. For a given
Reynolds number Re=106,
when the flowrate is increased to 104 (see Figure 24(f)), the flow is dominated by
the incoming axial jet. The flow is purely centripetal and the streamlines are
then parallel to the disks as in the centrifugal case.


### 4.2.2. Turbulence Statistics

As for
turbulent flows belonging to regime IV [1], turbulence is concentrated along the stator when an
axial inward throughflow is superimposed (see Figure 25). Nevertheless, one
characteristic feature of these flows compared to flows in large clearance
cavities is that turbulence intensities decrease when one moves toward the
periphery of the cavity. According to the maps of Ret (see Figure 27), turbulence is enhanced by the
inlet and the outlet of the axial throughflow. Besides, the
maximum of turbulence is obtained in the outlet region close to
the hub. This behavior is very similar to the one obtained in the case of a
centrifugal throughflow, previously discussed.Axial profiles of the six Reynolds stress
tensor components forRe=106, G=0.02,
and Cw=2500 at different radial locations r*.
(a)(b)(c)(d)(e)(f)The effect of
the flowrate coefficient on the turbulent field is displayed in Figure26. A
centripetal throughflow increases the turbulent intensities and there is no
threshold value for Cw contrary to the case of a centrifugal
throughflow. If the flowrate coefficient is increased by a factor 2,
turbulence levels increase by almost a factor 1.5.
For a given Reynolds number Re=106,
the aspect ratio G=0.02 and the radial location r*=0.57;
by comparing the results shown in Figures 11, 19, and 26, the maximum of
turbulence intensities is obtained along the stator when an inward throughflow
is superimposed. Note that the values of the normal Reynolds stress tensor
components are still quite comparable in the whole cavity.Axial profiles of the six Reynolds stress
tensor components forRe=106, G=0.02 at r*=0.57 and three flowrate coefficients Cw>0.
(a)(b)(c)(d)(e)(f)Isovalues ofRet/Retmax for G=0.02 and six sets (Re, Cw): (a) (1.8×105, 2500), Retmax=675.94;
(b) (1.8×105, 5000), Retmax=1557.5;
(c) (1.8×105, 10 000), Retmax=4507.4;
(d) (106, 2500), Retmax=1383.9;
(e) (106, 5000), Retmax=1615.7;
(f) (106, 10 000), Retmax=3333.The effect of
an inward throughflow on the maps of the turbulence Reynolds numberRet is highlighted in Figure 27. As in the case of
an outward throughflow, the maximum values of Ret are localized along the hub which corresponds
to the outlet region of the flow. It is a combined effect of the rotating hub
and of the outcoming jet, which creates a fully turbulent flow region at the
lowest radii. A second flow region of high turbulence levels is found at the
periphery of the cavity along the shroud and especially at the impact of the incoming
flow with the stator. Retmax is an increasing function of both Re and Cw but the influence of the flowrate coefficient
remains larger in the present case.Figure28
presents the anisotropy invariant maps when an inward throughflow is
superimposed on a turbulent torsional Couette flow (Re=106 and G=0.02) at two radii. Turbulence tends to the
one-component limit close to the disks and is fairly close to the isotropic
case at midplane. The map remains almost the same whatever the values of Cw are and for these
two radial locations.Anisotropy invariant map at (a)r*=0.3 and (b) r*=0.7 for Re=106, G=0.02 and three flowrate coefficients: Cw=2500 (red), Cw=5000 (blue), and Cw=10 000 (green).
(a)(b)


## 4.2.1. Structure of the Mean Flow

Figure22 represents
the axial profiles of the dimensionless tangential and radial velocity
components for Re=106, G=0.02,
and Cw=2500.
We first analyze the influence of the radial position in the range of 0.31≤r*≤0.83.
The corresponding streamlines are displayed in Figure 24(d). In the outer half
of the cavity r*≥0.57,
where a large recirculation is observed, the flow structure is quite similar to
the one obtained in a closed cavity Cw=0:
the fluid is centrifugal due to rotation along the rotor and is centripetal
along the stator by conservation of mass. Between these two flow regions, the
axial profile of the radial velocity is almost linear and the tangential
velocity is almost constant and still inferior to the local disk velocity (Vθ*<1). For r*=0.44,
the fluid in the main part of the interdisk spacing rotates at the same angular
velocity than the rotating disk. The radial velocity along this disk is then
equal to zero. When one approaches further the rotation axis, at r*=0.31, Vθ* increases and the fluid rotates 50% faster than the rotor. Indeed, because of the
conservation of the angular momentum, a fluid particle increases its rotation
when flowing to the center. Vr* is then negative whatever the axial position
and the flow is purely centripetal. This behavior
has been observed also for turbulent Batchelor flows by Poncet et al. [3].Axial profiles of the mean radial and
tangential velocity components forRe=106, G=0.02,
and Cw=2500 at different radial locations r*.
(a)(b)When the local
radius is fixed tor*=0.57 and we vary the flowrate coefficient Cw from 2500 to 104,
the same phenomenon is obtained (see Figure 23). For low values of Cw,
the velocity profiles are quite similar to the case Cw=0.
For Cw=5000,
the fluid rotates at the same angular velocity as
the rotor and the radial velocity is then equal to
zero along the rotating disk. By increasing further Cw up to 104,
by conservation of the angular momentum, Vθ* gets larger than 1 and the flow is purely centripetal. As in the
case of a centrifugal throughflow, the mean axial velocity component remains
very weak apart along the hub and the shroud when an inward throughflow is
superimposed.Axial profiles of the mean radial and
tangential velocity components forRe=106, G=0.02 at r*=0.57 and three flowrate coefficients Cw>0.
(a)(b)20 streamline patternsΨ* for G=0.02 and six sets of parameters (Re, Cw): (a) (1.8×105, 2500), −398.68≤Ψ*≤151.17;
(b) (1.8×105, 5000), −714.6≤Ψ*≤115.9;
(c) (1.8×105, 10 000), −1274.1≤Ψ*≤82.89;
(d) (106, 2500), −127.01≤Ψ*≤37.5;
(e) (106, 5000), −117.11≤Ψ*≤23.18;
(f) (106, 10 000), −201.77≤Ψ*≤58.8.Regarding the
corresponding streamline patterns (see Figure24), it appears that the inward
throughflow has a dominant effect on the flow structure compared to the effect
of rotation. At the inlet, when the axial jet impinges the stator, one part of
the fluid is moving inwards toward the axis of rotation along the stator and
the other part is centrifugal due to the rotation of the disk. Thus, a large
recirculation is created at the periphery of the cavity (see Figure 24(d)). In
that case, the flow exhibits the same structure as in a closed cavity for large
radii and the fluid rotates faster than the disk close to the axis. For a given
Reynolds number Re=106,
when the flowrate is increased to 104 (see Figure 24(f)), the flow is dominated by
the incoming axial jet. The flow is purely centripetal and the streamlines are
then parallel to the disks as in the centrifugal case.


## 4.2.2. Turbulence Statistics

As for
turbulent flows belonging to regime IV [1], turbulence is concentrated along the stator when an
axial inward throughflow is superimposed (see Figure 25). Nevertheless, one
characteristic feature of these flows compared to flows in large clearance
cavities is that turbulence intensities decrease when one moves toward the
periphery of the cavity. According to the maps of Ret (see Figure 27), turbulence is enhanced by the
inlet and the outlet of the axial throughflow. Besides, the
maximum of turbulence is obtained in the outlet region close to
the hub. This behavior is very similar to the one obtained in the case of a
centrifugal throughflow, previously discussed.Axial profiles of the six Reynolds stress
tensor components forRe=106, G=0.02,
and Cw=2500 at different radial locations r*.
(a)(b)(c)(d)(e)(f)The effect of
the flowrate coefficient on the turbulent field is displayed in Figure26. A
centripetal throughflow increases the turbulent intensities and there is no
threshold value for Cw contrary to the case of a centrifugal
throughflow. If the flowrate coefficient is increased by a factor 2,
turbulence levels increase by almost a factor 1.5.
For a given Reynolds number Re=106,
the aspect ratio G=0.02 and the radial location r*=0.57;
by comparing the results shown in Figures 11, 19, and 26, the maximum of
turbulence intensities is obtained along the stator when an inward throughflow
is superimposed. Note that the values of the normal Reynolds stress tensor
components are still quite comparable in the whole cavity.Axial profiles of the six Reynolds stress
tensor components forRe=106, G=0.02 at r*=0.57 and three flowrate coefficients Cw>0.
(a)(b)(c)(d)(e)(f)Isovalues ofRet/Retmax for G=0.02 and six sets (Re, Cw): (a) (1.8×105, 2500), Retmax=675.94;
(b) (1.8×105, 5000), Retmax=1557.5;
(c) (1.8×105, 10 000), Retmax=4507.4;
(d) (106, 2500), Retmax=1383.9;
(e) (106, 5000), Retmax=1615.7;
(f) (106, 10 000), Retmax=3333.The effect of
an inward throughflow on the maps of the turbulence Reynolds numberRet is highlighted in Figure 27. As in the case of
an outward throughflow, the maximum values of Ret are localized along the hub which corresponds
to the outlet region of the flow. It is a combined effect of the rotating hub
and of the outcoming jet, which creates a fully turbulent flow region at the
lowest radii. A second flow region of high turbulence levels is found at the
periphery of the cavity along the shroud and especially at the impact of the incoming
flow with the stator. Retmax is an increasing function of both Re and Cw but the influence of the flowrate coefficient
remains larger in the present case.Figure28
presents the anisotropy invariant maps when an inward throughflow is
superimposed on a turbulent torsional Couette flow (Re=106 and G=0.02) at two radii. Turbulence tends to the
one-component limit close to the disks and is fairly close to the isotropic
case at midplane. The map remains almost the same whatever the values of Cw are and for these
two radial locations.Anisotropy invariant map at (a)r*=0.3 and (b) r*=0.7 for Re=106, G=0.02 and three flowrate coefficients: Cw=2500 (red), Cw=5000 (blue), and Cw=10 000 (green).
(a)(b)


## 5. Conclusion

New extensive
numerical predictions using a second-order statistical modeling have been
performed to describe the flow in a rotor-stator system of very small aspect
ratio with or without throughflow according to a large range of the flow
control parameters: the rotational Reynolds number1.8×105≤Re≤107,
the flow rate coefficient −104≤Cw≤104,
and the aspect ratio of the cavity 0.02≤G≤0.05.In a closed
cavity, depending on the parameters, the flow exhibits a torsional Couette flow
structure with merged boundary layers for small clearances and a Batchelor
structure with unmerged boundary layers for large ones. This study was mainly
focused on regime III defined by Daily and Nece [1]; turbulent torsional Couette
flows with merged boundary layers. The axial profiles of the mean tangential
velocity in that case remain very similar to the ones obtained for Batchelor
flows (regime IV). On the contrary, the profiles of the radial velocity are
quite different. It is always linear in the main part of the flow as in
classical plane Couette flows. Turbulence is almost homogeneous along the axial
direction and vanishes toward the disks. When one
moves from the axis of rotation to the periphery of the cavity, the boundary
layers thicken and turbulence intensities increase as the local Reynolds number
increases. Thus, for increasing values of the radius r*,
for a given set of parameters (Re, G), the successive transitions between regimes
II and IV and then between regimes IV and III can be obtained in agreement with
the scenario proposed by Cooper and Reshotko [9].When an axial
outward throughflow (Cw<0) is superimposed on the main tangential flow,
the flow structure switches to a Stewartson behavior for large values of |Cw| or for small radii r*,
as in regime IV [2].
The tangential velocity is almost zero outside the region attached to the
rotating disk and the radial velocity is positive everywhere. Turbulence is
confined along the rotor and vanishes toward the stator. The maximum of the
turbulence intensities is obtained at the inlet close to the rotation axis and
turbulence levels slightly decrease when moving toward the periphery of the
cavity.In the case of
an inward throughflow, the flow keeps the same characteristics than flows
belonging to the regime IV. Depending on the flow control parameters, the fluid
can rotate faster than the rotating disk because of the conservation of the
angular momentum. The flow is then purely centripetal. It appears for large
values ofCw or/and close to the rotation axis. Turbulence
is mainly confined along the stator and vanishes
toward the rotor. It is enhanced by the incoming axial jet.The predictions
of the present RSM turbulence model have been found here in good agreement with
the velocity measurements performed at IRPHE. It strongly improves the
predictions of thek-ϵ model of Launder and Sharma [27]. Nevertheless, more
experimental data are now required, especially for the turbulent field, to enable
more extensive comparisons and the improvement of the turbulence models.



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*Source: 635138-2009-01-13.xml*