Batchelor vs Stewartson Flow Structures in a Rotor Statotr Cavity

Batchelor versus Stewartson flow structures in a rotor-stator

cavity with throughflow

Sebastien Poncet,∗ Marie-Pierre Chauve,† and Roland Schiestel‡

I.R.P.H.E., UMR 6594 CNRS-Universite d’Aix-Marseille I & II

Technopole Chateau-Gombert, 49 rue F. Joliot-Curie,

13384 Marseille cedex 13 - FRANCE - Fax. 33 (4) 96 13 97 09

(Dated: 20th April 2005)

The present work considers the turbulent flow inside a high speed rotor-stator

cavity with or without superimposed throughflow. New extensive measurements

made at IRPHE by a two component laser Doppler anemometer technique and by

pressure transducers are compared to numerical predictions based on one-point sta-

tistical modeling using a low Reynolds number second-order full stress transport

closure (RSM model). The advanced second-order model provides good predictions

for the mean flow as well as for the turbulent field and so is the adequate level of

closure to describe such complex flows. A better insight into the dynamics of such

flows is also gained from this study. Indeed the transition between a Batchelor type

of flow with two boundary layers separated by a central rotating core and a Stewart-

son type of flow with only one boundary layer on the rotating disk is characterized

in the (r∗, Ro) plane, where r∗ is the dimensionless radial location and Ro a mod-

ified Rossby number. The 5/7 power-law of Poncet et al. [1] describing the mean

centripetal flow in a rotor-stator system is extended to different aspect ratios and to

the case of centrifugal Batchelor type of flows.

∗Electronic address: [email protected],Tel.33(4)96.13.97.75†Electronic address: [email protected],Tel.33(4)96.13.97.76‡Electronic address: [email protected],Tel.33(4)96.13.97.65

2

I. INTRODUCTION

The interest in rotating disk flows is multiple and has justified many works since the

pioneering work of Ekman [2] in 1905. It has a major interest in turbomachinery, but ex-

amples where rotation and throughflow are associated can also be found in oceanography,

geophysics or astrophysics. From a fundamental point of view, the rotor-stator problem is

also one of the simplest configuration where rotation influences turbulence field and where

exact solutions of the Navier-Stokes equations can be found for laminar flows. Von Karman

[3] studied the laminar flow over an infinite rotating disk in a quiescent fluid. He simplified

the equations of motion governing the flow on the free disk to nonlinear differential equa-

tions using the assumption of axisymmetry. He showed that the flow is confined in a thin

boundary layer on the disk. He studied also the turbulent case by using momentum integral

methods with power law velocity profiles. The Von Karman analysis was subsequently fol-

lowed by Bodewadt [4] who investigated numerically the flow over a infinite stationary plane

with an outer flow in solid body rotation. Batchelor [5] solved the system of differential

equations relative to the stationary axisymmetric flow between two disks of infinite radius.

He specified the formation of a non-viscous core in solid body rotation, confined between

the two boundary layers which develop on the disks. From 1953 to 1983, this division of

the flow into three distinct zones was the subject of an intense controversy: Stewartson [6]

found indeed, in 1953, that the tangential velocity of the fluid can be zero everywhere apart

from the rotor boundary layer. The problem of the existence or not of a core in solid body

rotation justified many works until Kreiss and Parter [7] proved the existence of a multiple

class of solutions discovered numerically in fact by Mellor et al. [8]. Zandbergen and Dijk-

stra [9] showed also that the similarity equations do not generally have unique solutions and

that the two solutions advocated by Batchelor and Stewartson can thus both be found from

the similarity solutions. In the case of a laminar flow between two finite disks, Brady and

Durlofsky [10] found that flows in an enclosed cavity are of Batchelor type, while open-end

flows are of Stewartson type.

Daily and Nece [11] performed experimental and theoretical studies in a closed rotor-

stator cavity. They pointed out the existence of four flow regimes according to the Reynolds

number Re = ΩR22/ν and to the aspect ratio G = h/R2 (Ω is the angular velocity of the

rotating disk, R2 the outer rotating disk radius, ν the kinematic viscosity and h the inter-

3

disk space): two laminar and two turbulent regimes, each of which corresponding either

to merged or separated boundary layers. This classification was confirmed numerically by

Lance and Rogers [12] and Owen and Rogers [13]. Daily et al. [14] measured the average

velocity profiles and put forward the importance of the outflow on the development of the

Ekman layer and on the values of the entrainment coefficient of the fluid. Firouzian et al.

[15] performed velocity and pressure measurements for a wide range of flow rates and rota-

tional speeds. They tested also the effect of inlet geometry on the flow between two disks

enclosed by a peripheral shroud. Owen and Rogers [13] showed experimentally the dom-

inating influence of a centripetal flow in the determination of the entrainment coefficient

K of the fluid (K is the ratio between the mean tangential velocity in the central rotating

core and that of the disk at the same radius). The experiment of Itoh et al. [16] provided

a great contribution to the understanding of the turbulent flow in a shrouded rotor-stator

system. The authors measured the mean flow and all the Reynolds stress components and

brought out the existence of a relaminarized region even for high rotation rates. Cheah et

al. [17] performed detailed measurements of mean velocity and of the turbulent flow field

inside a rotor-stator system. Debuchy et al. [18] studied the radial inflow in a rotor-stator

system. They found analytical solutions and provided experimental data in good agreement

with the features of their asymptotic model. More recently, Poncet et al. [1] measured the

entrainment coefficient of the fluid K for different aspect ratios in the case of a centripetal

throughflow and showed that K depends on a local flow rate coefficient Cqr according to a

5/7 power law which has been also determined analytically.

Beside the theoretical aspect, turbulent rotating disk flows can be considered as useful

benchmarks for numerical simulations because of the numerous complexities embodied by

this flow. Chew [19] was the first to study numerically the flow inside a rotor-stator cavity

with centrifugal throughflow using a low Reynolds number k− ε model. Chew and Vaughan

[20] studied this type of flow with and without a superimposed throughflow with a model

based on a mixing length hypothesis inside the whole cavity. Their results were quite com-

parable to the experimental data of Daily and Nece [11] and Daily et al. [14] apart from a

relaminarization area close to the rotating axis. The model of Iacovides and Theofanopoulos

[21] used an algebraic modeling of the Reynolds stress tensor in the fully developed turbu-

lence area and a mixing length hypothesis near the wall. It provided good results in the case

of a rotor-stator flow with and without throughflow but some discrepancies remain on the

4

Ekman layer thickness and the rotation rate in the central area. Iacovides and Toumpanakis

[22] tested four turbulence models and showed especially that the Reynolds stress model was

an appropriate level of closure to describe rotor-stator flows in a closed cavity. Schiestel et

al. [23] have used both a low-Reynolds number k− ε model near the walls and an Algebraic

Stress Model (ASM) in the core of the flow. Second order informations were also found

to be necessary in turbulence closure to get a sufficient degree of universality in predicting

highly rotating flows. Debuchy [24] realized a comparative study between experimental re-

sults relative to the flow inside a rotor-stator with a superimposed centripetal throughflow

and the computations obtained with a numerical model developed from an asymptotic ap-

proach. But the limitations inherent to the turbulence models and to the representations of

the boundary conditions, did not allow to carry out reliable predictions. Later Elena and

Schiestel [25] proposed also some numerical calculations of rotating flows based on a zonal

approach. They have also used a new modeling of the Reynolds stress tensors derived from

the Launder and Tselepidakis [26] one. It provides a better prediction than the simpler

model of Hanjalic and Launder [27]. But there also, the authors emphasized the too high

laminarization of the flow in comparison with the expected results. Iacovides et al. [28]

tested two low Reynolds number models: a classical k − ε model and a modified Reynolds

stress model (RSM model) which takes into account the effect of the rotation. Poncet et al.

[29] compared pressure and velocities measurements with numerical predictions based on an

improved version of the Reynolds stress modeling of Elena and Schiestel [30] for centripetal

and centrifugal throughflows [29]. All the comparisons were in excellent agreement for the

mean and turbulent fields. The RSM model is a valuable tool to describe the turbulent flow

with or without a superimposed throughflow compared to the DNS which are limited at

present time to small rotation rates. Since the pioneering contribution of Fromm [31], all

the DNS are restricted to simulate turbulent rotor-stator flows with Reynolds number not

higher than 3− 4× 105 [32–34].

The present paper is devoted to the study of the turbulent flow in a rotor-stator system

of large aspect ratio (regime IV [11]) when a throughflow is superimposed on the rotating

fluid. The basic flow belongs to the Batchelor type family: the two boundary layers are

separated by a central rotating core. By superimposing a centripetal flow, the core rotates

faster than in the closed cavity but the flow remains of Batchelor type. On the contrary,

by superimposing a centrifugal flow, the flow gets a Stewartson type of flow with only one

5

boundary layer on the rotating disk and a quasi zero tangential velocity outside. The pur-

pose of this work is to compare advanced turbulence modeling to the new data obtained

from extensive velocity and pressure measurements when a throughflow is superimposed and

to characterize the transition between Batchelor and Stewartson flow structures.

II. EXPERIMENTAL SET-UP

A. Apparatus

The cavity sketched in figure 1 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.

The mean flow is mainly governed by three control parameters: the aspect ratio of the

cavity G, the rotational Reynolds number Re based on the outer radius of the rotating disk

and the flow rate coefficient Cw defined as follows:

G =h

R2

Re =ΩR2

2

νCw =

Q

νR2

where ν is the kinematic viscosity of water, R1, R2 the inner and outer radii of the rotating

disk, R3 the outer radius of the cavity, R4 its central opening and Q the superimposed

throughflow. Cw = 0 corresponds to a closed cavity. Cw > 0 (resp. < 0) denotes the case

where a centripetal (resp. centrifugal) throughflow is superimposed. The interdisk space h

can vary between 3 and 12 mm (0.012 ≤ G ≤ 0.048). In the present work, the interdisk

space h and the radial gap e = R3 −R2 are respectively fixed to 9 (G = 0.036) and 3 mm.

A pump allows to impose a variable throughflow Q. The measurement of the flow rate is

performed by an electromagnetic flow-meter, located at the exit of the cavity. The rotation of

the disk is produced by a 5.5 kW electric servo-motor. A variable speed numerical controller

directs the angular velocity Ω. The accuracy on the measurement of the angular velocity

and on the throughflow is better than 1%.

In the case of centripetal throughflows, the fluid is entrained in prerotation. As the disk

rotates, water is sucked through the central opening of the cavity situated above the hub.

The incoming fluid is entrained in rotation while passing through a breakthrough crown

mounted underneath the rotor and linked to it. It enables to increase the tangential velocity

6

of the fluid, and consequently limits the influence of the non rotating cylindrical wall. This

prerotation is achieved through 48 holes having a diameter of 10 mm calibrated in order

to entrain enough fluid when the disk rotates. According to the superimposed centripetal

throughflow, the mean prerotation rate ranges between 0.43 and 0.54 [35].

In order to avoid cavitation effects, the cavity is maintained at rest at a pressure of 2 bar.

Pressurization is ensured by a tank-buffer and is controlled by two pressure gauges. The

temperature is maintained constant (23C) using a heat exchanger which allows to remove

the heat produced by friction in order to keep constant the density ρ and the kinematic

viscosity ν of water.

B. Instrumentation and measurements

The measurements are performed using a two component laser Doppler anemometer

(LDA) and also using pressure transducers. The LDA technique is used to measure from

above the stator the mean radial Vr and tangential Vθ velocities as well as the associated

Reynolds stress tensor components R∗rr = v′2r /(Ωr)2, R∗

rθ = v′rv′θ/(Ωr)2, R∗

θθ = v′2θ /(Ωr)2 in a

vertical plane (r, z). This method is based on the accurate measurement of the Doppler shift

of laser light scattered by small particles (Optimage PIV Seeding Powder, 30 µm) carried

along with the fluid. Its main qualities are its non intrusive nature and its robustness. About

5000 validated data are necessary to obtain the statistical convergence of the measurements.

It can be noticed that, for small values of the inter-disk space h, the size of the probe volume

(0.81 mm in the axial direction) is not small compared to the boundary layer thicknesses

and to the parameter h. Pressure is measured using 6 piezoresistive transducers. These

transducers are highly accurate (0.05% in the range 10 to 40C) and combine both pressure

sensors and temperature electronic compensations. They are fixed to the stator at the

following radial positions 0.093, 0.11, 0.14, 0.17, 0.2 and 0.23 m located on two rows because

of geometrical constraints. Previous pressure measurements by embedded pressure gauges

[36] showed that the pressure on the rotor and the one on the stator at the same radius are

in fact identical within 2.5% accuracy. This is in fact a direct consequence of the Taylor-

Proudman theorem which forbids axial gradients in rapidly rotating flows.

7

III. STATISTICAL MODELING

A. The differential Reynolds Stress Model (RSM)

The flow studied here presents several complexities (high rotation rate, imposed through-

flow, wall effects, transition zones), which are severe conditions for turbulence modeling

methods [37–39]. 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 [26] model and sensitized to rotation effects [30, 40]. Previous works [29, 30, 41]

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:

dRij

dt= Pij + Dij + Φij − εij + Tij (1)

where Pij, Dij, Φij, εij, and Tij respectively denote the production, diffusion, pressure-strain

correlation, dissipation and extra terms.

The diffusion term Dij is split into two parts: a turbulent diffusion DTij, which is inter-

preted as the diffusion due to both velocity and pressure fluctuations [42] and a viscous

diffusion Dνij, which can not be neglected in the low-Reynolds number region:

DTij = (0.22

k

εRklRij,l),k (2)

Dνij = −νRij,kk (3)

In a classical way, the pressure-strain correlation term Φij can be decomposed as below:

Φij = Φ(1)ij + Φ

(2)ij + Φ

(w)ij (4)

Φ(1)ij is interpreted as a slow nonlinear return to isotropy and is modeled as a quadratic

development in the stress anisotropy tensor, with coefficients sensitized to the invariants of

anisotropy. This term is damped near the wall:

Φ(1)ij = −(c1aij + c

′1(aikakj − 1

3A2δij))ε (5)

8

where aij denotes the stress anisotropy tensor and c1 and c′1 are two functions deduced from

Craft’s high-Reynolds number proposals [43] adapted for confined flows:

aij =Rij

k− 2

3δij (6)

c1 = (3.1√

AA2 + 1)(1− e−Re2t40 ) (7)

c′1 = 3.72

√AA2(1− e−

Re2t40 ) (8)

A = 1− 9/8(A2 −A3) is Lumley’s flatness parameter with A2 and A3 the second and third

invariants of the anisotropy tensor. Ret = k2/(νε) is the turbulent Reynolds number.

The linear rapid part Φ(2)ij includes cubic terms. It can be written as:

Φ(2)ij = −0.6(Pij − 1

3Pkkδij) + 0.3εaij

Pkk

ε

− 0.2[RkjRli

k(Vk,l + Vl,k)− Rlk

k(Rik(Vj,l + εjmlΩm)

+ Rjk(Vi,l + εimlΩm))]−min(0.6, A)(A2(Pij −Dij)

+ 3amianj(Pmn −Dmn)) (9)

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

correction Φ(w)ij is only applied to the rapid part. The form retained here is the one proposed

by Gibson and Launder [44] 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:

Φ(w)ij = 0.2[(Φ

(2)km + Φ

(R)km )nknmδij − 3

2(Φ

(2)ik + Φ

(R)ik )nknj

− 3

2(Φ

(2)kj + Φ

(R)kj )nink]

k√

Rpqnpnq

εy(10)

9

y is evaluated by the minimal distance of the current point from the four walls.

The viscous dissipation tensor εij has been modeled in order to conform with the wall

limits obtained from Taylor series expansions of the fluctuating velocities [45]:

εij = fAε∗ij + (1− fA)(fsεRij

k+

2

3(1− fs)εδij) (11)

with fA, fs and ε∗ij defined as followed:

fA = e−20A2

e−Re2t20 (12)

fs = e−Re2t40 (13)

ε∗ij =(Rij + Riknjnk + Rjknink + Rklnknlninj)

kε(1 + 3

2Rpq

knpnq)

(14)

The extra term Tij accounts for implicit effects of the rotation on the turbulence field, it

contains additional contributions in the pressure-strain correlation, a spectral jamming term,

inhomogeneous effects and inverse flux due to rotation, which impedes the energy cascade.

This term allowed some improvements of results [30] in the Itoh et al. [16] calculation.

Below is the proposal of Launder and Tselepidakis [26] for the dissipation rate equation

ε:

dε

dt= −cε1

ε

kRijVi,j − cε2fε

εε

k+ (cε

k

εRijε,j + νε,i),i

+ cε3νk

εRjkVi,jlVi,kl + (cε4ν

ε

kk,i),i (15)

ε is the isotropic part of the dissipation rate ε = ε−2νk1/2,i k

1/2,i . cε1 = 1, cε2 = 1.92, cε = 0.15,

cε3 = 2, cε4 = 0.92 are four empirical constants and fε is defined by: fε = 1/(1+0.63√

AA2).

The kinetic turbulent energy equation is redundant in a RSM model but it is however

still solved numerically in order to get faster convergence:

dk

dt= −RijVi,j − ε +

Tjj

2+ 0.22(

k

εRijk,j + νk,i),i (16)

It is verified after convergence that k is exactly 0.5Rjj.

10

B. 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. A 140 × 80 mesh in the

(r, z) frame proved to be sufficient in most of the cases considered in the present work to get

grid-independent solutions [41]. However a more refined modeling 200× 100 is necessary for

higher rotation rates such as Re = 4.15× 106. About 20000 iterations (several hours on the

NEC SX-5, IDRIS, Orsay, France) 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

[46]. Also, the stress component equations are solved using matrix block tridiagonal solution

to enhance stability using non staggered grids.

C. Boundary conditions

At the wall, all the variables are set to zero except for the tangential velocity Vθ, 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. In the outflow

section, the pressure is fixed, whereas the derivatives for all the other independent quantities

are set to zero if the fluids 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 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. By multiplying by a factor 3 the turbulence intensity level imposed at

the inlet, the change is about 0.08% on the maximum of the turbulent kinetic energy in the

whole cavity. Moreover, these choices are justified by the wish to have a model as universal

as possible.

11

IV. TURBULENT FLOW IN A CLOSED CAVITY

We study first the turbulent flow in a closed rotor-stator system of aspect ratio G = 0.036

with no throughflow. This basic flow belongs to the Batchelor type family: the two boundary

layers are separated by a central rotating core. Two values of the rotational Reynolds number

are here investigated: Re = 106 and Re = 4.15× 106. We define the following dimensionless

quantities: r∗ = r/R2, z∗ = z/h, V ∗r = Vr/(Ωr), V ∗

θ = Vθ/(Ωr) and V ∗z = Vz/(Ωr). Note

that z∗ = 0 corresponds to the rotor side and z∗ = 1 to the stator side.

A. Structure of the mean flow

Figure 2 shows the structure of the mean flow in a closed cavity for Re = 106. The flow

is of Batchelor type. It is indeed clearly divided into three distinct zones: a centripetal

boundary layer on the stator (Bodewadt layer), a central rotating core and a centrifugal

boundary layer on the rotor (Ekman or Von Karman layer). In the Bodewadt layer, the

mean radial velocity is negative and the mean tangential velocity ranges between 0 and

KΩr, with K the entrainment coefficient of the fluid. When one approaches the axis of

the cavity, the Bodewadt boundary layer thickness decreases and the minimum value of the

radial velocity increases. The central core is characterized by a quasi zero radial velocity

and a constant tangential velocity KΩr. K increases slightly from 0.38 at r∗ = 0.44 to 0.45

at r∗ = 0.8, whereas the size of the core decreases slightly with r∗. The Ekman layer is

always centrifugal: the mean radial velocity is positive and the mean tangential velocity

ranges between Ωr and KΩr. When one approaches the axis of the cavity, the Ekman

boundary layer thickness decreases and the maximum value of the radial velocity in this

layer increases. The corresponding streamlines are displayed in figure 3a. Note that, for

these three radial locations, there is an excellent agreement between the experimental data

and the model predictions (fig.2). These results are quite comparable with the predictions

of Iacovides et al. [28] using a low-Re differential second moment closure and the data of

Cheah et al. [17] for Re = 1.6×106 and G = 0.127. Itoh et al. [16] found that K is sensitive

to the radial location for Re = 106 and G = 0.08. It varies from 0.31 at r∗ = 0.4 to 0.42 at

r∗ = 0.94. These authors notice that, for r∗ ≤ 0.6, the Ekman boundary layer is not fully

turbulent and the entrainment coefficient K is then lower than 0.4.

12

By increasing the Reynolds number (fig.3b, fig.4), the structure of the mean flow does

not change significantly at the radial location r∗ = 0.56. It always belongs to the Batchelor

family with an entrainment coefficient of the rotating core K also close to 0.45. Nevertheless,

it can be noticed that the centrifugal force is weaker. Indeed, when the Reynolds number

increases, the maximum value of the radial velocity in the Ekman layer decreases. The radial

velocity profile is in very good agreement with the predictions of the algebraic stress model

of Iacovides and Theofanopoulos [21] (Re = 4.4× 106, Cw = 0, G = 0.0255, r∗ = 0.765): the

maximum value of the radial velocity in the boundary layers are well predicted. They also

bring out the existence of a rotating core with a quasi zero radial velocity. Nevertheless,

the entrainment coefficient determined by the authors K = 0.55 seems to be overestimated

compared to the results of Poncet et al. [1].

B. Pressure distributions

To complement the comparisons, we performed also pressure measurements by means

of 6 pressure transducers. We choose to take as a reference the pressure measured at the

outer radial position r∗ = 0.92 and we define the following pressure coefficient: Cp =

P ∗(r∗)− P ∗(0.92). The dimensionless pressure is given by: P ∗ = 2P/(ρΩ2R22). In figure 5,

the pressure coefficient is plotted versus the dimensionless radial position for Cw = 0 and

two Reynolds numbers. As expected, the pressure decreases towards the center of the cavity:

Cp is then always negative. At a given radius, it can be observed that Cp is almost the same

in this range of Re. The case Re = 106 seems to show a greater discrepancy (fig.5). However

the normalization used in Cp magnifies this apparent deviation. Indeed, the physical value

of the pressure difference between measurement and calculation is very small and remains

within experimental accuracy.

In a closed cavity, the radial velocity in the core is almost zero (fig.2). For all the

cases considered here, the axial velocity is almost zero too (see figure 4), that means that

the mean flow is quasi two dimensional. So the Navier-Stokes equation for the tangential

component reduces to the balance of the centrifugal force and of the radial pressure gradient:

ρV 2θ /r = ∂P/∂r. Using dimensionless quantities, the resulting equation linking the pressure

coefficient and the entrainment coefficient is:

dCp(r∗)/dr∗ = 2K2r∗ (17)

13

K being constant for both Reynolds numbers (fig.2,4), the radial distributions of the pressure

coefficient are logically rather the same. This relation is a useful check to compare velocity

and pressure measurements.

C. Turbulence statistics

In this section, comparisons between the model results (RSM) and experimental data are

given for three components of the Reynolds stress tensor. Poncet et al. [29] have already

showed that the RSM model is the adequate level of closure to describe such complex

flows, compared to a classical k − ε model, which is blind to any rotation effect. Indeed it

overestimates the turbulent intensities and fails to mimic the right profiles.

Comparisons for three components of the Reynolds stress tensor are given in figure 6

for Re = 106 at three radial locations. The RSM model provides good results compared

to the experimental data even in the boundary layers but some discrepancies occur mainly

in the low turbulence intensity regions where relaminarization is expected. In general, the

turbulent intensities are rather weak in the considered case in the whole cavity.

The turbulent intensity is mostly concentrated in the boundary layers. The Ekman layer

is besides more turbulent than the Bodewadt layer. In the rotating core, the Reynolds

stresses are weak. R∗12 is indeed almost zero, that means that there is no turbulent shear

stress in that zone. In the same way, there is no molecular viscous shear stress as the axial

gradient of the radial velocity is zero in the core (fig.2). Close to the axis (r∗ = 0.44), we can

notice a laminarization of the flow, which becomes locally turbulent, when one approaches

the periphery of the cavity.

Figure 7 gives an overview of the turbulent field for Re = 4.15 × 106 at r∗ = 0.56.

By increasing the Reynolds number (fig.6b,7), the turbulent intensities increase outside the

Ekman layer. The three Reynolds shear stress tensor components R∗rθ, R∗

rz = v′rv′z/(Ωr)2,

R∗θz = v

′θv

′z/(Ωr)2 are negligible in the core. It confirms the existence of the inviscid rotating

core. It means also that the production is almost zero in that area and that turbulence is

only due to the diffusion phenomenon. The R∗zz = v′2z /(Ωr)2 component is rather weak too.

The model predictions are there in excellent agreement with the velocity measurements.

14

V. TURBULENT FLOW WITH CENTRIPETAL THROUGHFLOW

In this section, we study the turbulent flow inside a rotor-stator cavity of aspect ratio

G = 0.036 when a centripetal throughflow is superimposed on the basic flow. Three values

of the flow rate coefficient are studied: Cw = 1976, 5929, 9881 as well as two values of the

rotational Reynolds number: Re = 106 and Re = 4.15× 106.

A. Structure of the mean flow

Figure 8 shows the mean velocity profiles for Re = 106 and Cw = 5929 at three radial

locations. When an inward throughflow is superimposed, the structure of the mean flow at

the periphery preserves the properties of the flow in a closed cavity [13]. At r∗ = 0.8, the flow

is indeed of Batchelor type with two boundary layers separated by a central rotating core.

When one approaches the axis of the cavity, the Ekman layer, which was centrifugal, becomes

centripetal and the core rotates faster than the rotating disk. The flow is then centripetal

whatever the axial location. K increases close to the axis because of the conservation of the

angular momentum. The limit case is obtained for r∗ = 0.56. A stagnation line is created on

the rotor: the Ekman layer disappears and the core rotates at the same velocity as the rotor

(fig.10b). It is similar to that observed by Dijkstra and Van Heijst [47] and by Iacovides and

Theofanopoulos [21]. When a radial centripetal throughflow Cw = 3795 is superimposed of

the turbulent basic flow Re = 6.9× 105, G = 0.0685, these last authors predict a limit case

at r∗ = 0.47.

At a given radial location r∗ = 0.56 and for Re = 106, figure 9 shows the influence of

an increasing centripetal throughflow on the structure of the mean flow. For Cw = 1976,

the flow is of Batchelor type. The Ekman layer is centrifugal and the core rotates slower

than the rotor. By increasing the inward throughflow Cw = 9881, the Ekman layer becomes

centripetal and the core rotates then faster than the rotating disk. The mean radial velocity

is negative almost everywhere. The agreement between the experimental data and the model

results is very satisfactory except for the prediction of K, which is slightly underestimated

by the RSM model.

When the value of the rotation rate increases from Re = 106 (fig.9b, Cw = 5929, r∗ =

0.56) to 4.15×106 (fig.11), the entrainment coefficient K decreases from 1 to 0.64. Figure 11

15

shows the mean velocity profiles for Re = 4.15× 106 and Cw = 5929 at r∗ = 0.56. The flow

is of Batchelor type: the Ekman becomes gets again centrifugal and the core rotates slower

than the rotor. The axial velocity component is almost zero. The corresponding streamlines

are displayed in figure 10d.

B. The entrainment coefficient K

The predictions of the RSM model have been also validated on experimental data mea-

sured using the two-component LDA. Poncet et al. [1] have shown analytically and ex-

perimentally that the entrainment coefficient K of the rotating fluid can be correlated, in

the case of a centripetal throughflow, to a local flow rate coefficient: Cqr = Q2πr3Ω

(Ωr2

ν)1/5

according to a 5/7 power law:

K = 2(a× Cqr + b)5/7 − 1 (18)

with a and b experimental constants. In figure 12, several points deduced from the modeling

results are plotted against the mean experimental K-curve. The constants deduced from

the model a = 5.3 and b = 0.63 are very close to those obtained by the measurements

a = 5.9 and b = 0.63. The maximum discrepancy in K-values between experiments and

the numerical predictions is quite weak (less than 9%) and remains within the uncertainty

range of both experimental and numerical approaches. Many subtle influences coming from

uncertainties in prerotation level or boundary layer prediction may contribute to explain

this small discrepancy. This law is not sensitive to variations of the aspect ratio G, as far as

the flows remain in the regime IV defined by Daily and Nece [11] (turbulent with separated

boundary layers). These two coefficients depend strongly on the prerotation level. Debuchy

[24] studied indeed the case of a weak prerotation. These velocity measurements can be

fitted by the relation (18) with a = 2.8 and b = 0.46.

For high values of Cqr, the core rotates faster than the rotating disk (K > 1). By

increasing the throughflow or decreasing the angular velocity of the rotor or by approaching

the center of the cavity, the value of K increases: it could be due to the vortex stretching

phenomenon. In our experiments, the core can rotate two times faster than the rotating

disk.

Debuchy et al. [18] have studied the turbulent flow (Re = 1.47 × 106) in a rotor-stator

16

cavity of large aspect ratio (G = 0.08) when a weak inward throughflow (Cw = 188) is

superimposed. The entrainment coefficient is then very sensitive to the shrouding. For a

small clearance, K is close to 0.38 at the periphery and depends slightly on Cw. On the

contrary, at r∗ = 0.53, K depends strongly on Cw and reaches 0.9 for Cw = 188.

C. Pressure distributions

In figure 13, the pressure coefficient Cp is plotted versus the dimensionless radial location

for three centripetal throughflows. As expected, the pressure decreases towards the center of

the cavity: Cp is then always negative. At a given radius and for a given Reynolds number

Re, it can be observed that Cp increases for increasing values of the flow rate Cw (in absolute

value). That is in agreement with the measurements and the theoretical model of Debuchy

et al. [18].

According to relation (17), we can determine the entrainment coefficient from the value

of Cp. We first perform a polynomial fit of the curve Cp versus r∗ and then, we calculate by

finite difference the derivative of Cp to obtain K. Figure 14 compares the radial variations

of K for the data series obtained by pressure and velocity measurements in the cases of

centripetal throughflows. As it can be seen, the results are in excellent agreement. The

small differences come in fact from the calculation of the derivative of Cp and from the

hypothesis of zero radial velocity, which is less relevent for strong centripetal throughflows.

The pressure coefficient Cp enables then to know the axial thrusts on the rotor.

D. Turbulence statistics

Figure 15 shows the axial profiles of three components of the Reynolds stress tensor for

Re = 106 and Cw = 5929 at three radial locations. The model results are in excellent

agreement with the experimental data. The turbulent intensities are well predicted apart

from the R∗rθ component, which is underestimated in the Bodewadt layer. For a given

Reynolds number and a given flow rate coefficient, the flow is more turbulent close to the

axis of rotation than at the periphery of the cavity. The Bodewadt layer is besides more

turbulent than the Ekman layer. The viscous shear stress is quasi zero in this layer, whereas

it is large on the stator. The value of the normal Reynolds stress tensor components are

17

quite comparable in the whole cavity.

The effect of the flow rate coefficient on the turbulent field is displayed in figure 16. A

centripetal throughflow increases the turbulent intensities. The turbulent levels are higher

for a strong inward throughflow (Cw = 9881) than for a weak throughflow outside the Ekman

layer. Indeed, the maximum values of the Reynolds stresses in the Bodewadt layer and the

turbulent levels in the core increase with increasing values of Cw.

As expected, by increasing the Reynolds number from Re = 106 to 4.15 × 106, the

turbulent intensities increase noticeably (fig.17). The shape of the turbulent profiles does

not change: the Bodewadt layer is still more turbulent than the Ekman layer. But the values

of the three Reynolds stresses have increased by a factor two.

VI. TURBULENT FLOW WITH CENTRIFUGAL THROUGHFLOW

In this section, we study the turbulent flow inside a rotor-stator cavity of aspect ratio

G = 0.036 when a centrifugal throughflow is superimposed on the basic flow. Three values

of the flow rate coefficient are studied: Cw = −1976,−5929,−9881 as well as two values of

the rotational Reynolds number: Re = 106 and Re = 4.15× 106.

A. Structure of the flow

Figure 18 shows the influence of the radial location on the mean flow when a centrifugal

throughflow is superimposed for Re = 106 and Cw = −5929. At the periphery r∗ = 0.8, the

flow preserves the properties of the flow without throughflow: two boundary layers separated

by a rotating core but, in this case, the core rotates slower than the rotor and than the case in

the closed cavity (Cw = 0). The Ekman layer is centrifugal (Vr > 0) whereas the Bodewadt

layer is centripetal (Vr < 0). By approaching the center of the cavity, the core disappears and

the Bodewadt layer gets centrifugal. The flow is then fully centrifugal (Vθ ' 0 and Vr > 0

everywhere) whatever the axial location z: it is a Stewartson type of flow. We can consider

it as the connection of two flows over a single disk: a Bodewadt type of flow (fixed disk)

and a Von Karman type of flow (rotating disk). The Stewartson flow structure is composed

of a single boundary layer on the rotor and a quasi zero tangential velocity outside (the

corresponding streamline patterns are shown in figure 20b). This flow structure has been

18

observed by Chew [19] using a k − ε turbulence model for Re = 3.4× 106, Cw = −5.4× 104

at r∗ = 1.

In the same manner, when a weak centrifugal throughflow is imposed (fig.19a, fig.20a),

the flow keeps the same characteristics as in a closed cavity: two boundary layers separated

by a central core, that is known as a Batchelor type of flow. The entrainment coefficient

K decreases. Note that the Batchelor type profile can appear only if the Bodewadt layer

is centripetal. By increasing the flow rate coefficient Cw (fig.19b, fig.20b), the central core

disappears and the flow gets centrifugal everywhere (Vr > 0). The profiles for the tangential

velocity are then of Stewartson type. For stronger throughflows (fig.19c), the radial velocity

profiles get closer to a channel flow like profile. Daily et al. [14] have observed two noticeable

effects of a centrifugal throughflow (Cw = −3510) on the turbulent flow (Re = 6.9× 105) in

a rotor-stator cavity of large aspect ratio (G = 0.069): the reduction in core rotation and

the Ekman boundary layer growth with decreasing radius.

The effect of the Reynolds number Re is shown in figure 21 compared to figure 18.

By increasing Re, the Bodewadt layer becomes centripetal and the central core reappears

(K ' 0.2). It slows down the transition between a Batchelor and a Stewartson flow structure

as when r∗ decreases (fig.19a, fig.21). The axial velocity component is almost zero.

The transition between the Batchelor and Stewartson flow structures is observed either

when one approaches the center of the cavity (r∗ decreases) or by decreasing the flow rate

coefficient Cw (by increasing the centrifugal throughflow) or by decreasing the Reynolds

number. This transition is mainly due to the radial velocity, which is positive whatever the

axial location for a Stewartson type of flow and, which vanishes for an axial location for a

Batchelor type of flow. Note that all the model results are in excellent agreement with the

experimental data in the case of a centrifugal throughflow.

B. Transition diagram between a Batchelor and a Stewartson type of flow

In the present work, the Stewartson type of flow has never been observed in the closed

cavity nor in the case of a centripetal throughflow. On the contrary, when a centrifugal

throughflow is superimposed, the transition between the Batchelor and the Stewartson flow

structure can be characterized by considering a Rossby number defined as Ro = Q/(2πR22eΩ)

versus the radial location r∗ (fig.22) and for a given aspect ratio G = 0.036. Ro depends

19

on r∗ according to a third degree polynomial fit, which has been determined empirically

and, which collapses all the experimental and numerical data in a single correlation law:

Ro = 0.0088− 0.0998r∗ + 0.3048r∗2 − 0.4646r∗3.

The whole experimental data (centripetal and centrifugal cases) can be merged into a

single curve giving the variations of K versus Cqr (fig.23). The law (18) has been validated

for some centripetal throughflows (Batchelor type of flow) and four aspect ratios G. When

a centrifugal throughflow is superimposed, both type of flow structures are observed. As

long as the flow is of Batchelor type, the equation (18) law is still valid (low absolute value

of Cqr). We can then calculate the radial pressure gradient and also know the axial thrust

on the rotor. For stronger centrifugal throughflows (higher negative values of Cqr), the

symmetry of the Batchelor type of flow is broken and the flow is then of Stewartson type. It

is enclosed in the Ekman boundary layer. The variation law of K changes but the transition

between these two types of flow is carried out in a continuous manner. Note that there is

no core in the Stewartson type of flow and it is difficult to define an entrainment coefficient.

So we can take K as the mean dimensionless tangential velocity outside the Ekman layer.

Finally, for strong centrifugal throughflow, K goes to an asymptotical value close to 0.

Nguyen et al. [48] have shown numerically that a cavity with a large aspect ratio furthers

a Stewartson type of flow, whereas a cavity with a small aspect ratio furthers a Batchelor

flow structure. Our experiments and the ones of Poncet et al. [49] for G = 0.012 and

G = 0.036 do not have brought out any difference in the flow structure and the equation

(18) law is still valid for these two aspect ratios. G does not seem to be a relevant parameter

in the present study and for this range of aspect ratios.

C. Pressure distributions

In the case of a centrifugal flow with Re = 106, the pressure coefficient is very close to

zero. In figure 24, the pressure coefficient is plotted versus the dimensionless radial location

for Re = 4.15 × 106. As expected, the pressure decreases towards the center of the cavity.

At a given radius and for this Reynolds number, it can be observed that Cp decreases for

increasing values of the flow rate, which is contrary to the case with a centripetal throughflow.

When Cw = −5929, the radial pressure gradient is close to zero for r∗ lower than 0.56. The

flow is then of Stewartson type and for greater r∗, the flow is of Batchelor type. That is

20

confirmed by the velocity profiles (fig.21).

D. Turbulence statistics

Figure 25 exhibits the axial profiles of three Reynolds stresses for Re = 106 and Cw =

−5929 at three radial locations. As for the case with centripetal throughflows, the turbulent

intensities increase from the periphery to the center of the cavity. At r∗ = 0.44, R∗rr and R∗

θθ

are almost constant in the central core, whereas their profiles are nearly linear for greater r∗.

Contrary to the case with centripetal throughflows, the maximum of the turbulent intensities

is confined in the Ekman layer and they vanish in the Bodewadt layer.

The influence of a centrifugal throughflow on the turbulent field is quite negligible for

this range of flow rate coefficients (fig.26). R1/2∗rr and R

1/2∗θθ are respectively close to 0.06

and 0.07 in the Ekman layer. These two Reynolds stresses are quite comparable for these

centrifugal throughflows. R∗rθ is maximum in the rotating disk boundary layer and reaches

−0.002 whatever the value of Cw.

When an intermediate centrifugal throughflow is superimposed Cw = −5929, the axial

profiles of the three Reynolds stresses are almost the same for Re = 106 and Re = 4.15×106

(fig.25b, fig.27) in the middle of the cavity r∗ = 0.56. To conclude, the turbulent intensities

in a rotor-stator cavity with outward throughflow, depend essentially on the radial location.

For all the considered cases, the RSM predictions are in very good agreement with the

experimental data, even in the boundary layers.

Figure 28 sums up the influence of the throughflow on the turbulent field considering

the repartition of the turbulent Reynolds number Ret = k2/(νε) in the whole cavity for

G = 0.036, Re = 106 and three values of the flowrate coefficients Cw. In a closed cavity

(fig.28b), a laminar region subsists (r∗ ≤ 0.4). It appears also that the Bodewadt layer

becomes turbulent closer to the axis than the Ekman layer. That confirms the previous

results of Cheah et al. [17]. Because of geometrical constraints, we can not perform LDA

measurements in the low turbulence intensity region where relaminarization is expected.

Nevertheless, figure 6a suggests that the radial location r∗ = 0.44 is possibly the beginning

of this laminar area. The present low Reynolds number closure gives an account of this

zone. But, when a centrifugal (fig.28a) or centripetal (fig.28c) throughflow is superimposed,

strong gradients due to this throughflow prevent the subsistence of a laminar area close to

21

the axis of the cavity.

VII. CONCLUSION

New experimental investigations and extensive measurements have been performed and

compared to numerical predictions to describe the flow in a rotor-stator system with or

without throughflow according to a large range of the flow control parameters: a rotational

Reynolds number Re, a flow rate coefficient Cw and the aspect ratio of the cavity G. In

a closed cavity as well as in the case of a centripetal throughflow, the flow belongs to the

Batchelor family. It is divided into three distincts zones: two boundary layers separated by a

central rotating core. The dimensionless tangential velocity in the core K depends on a local

flow rate coefficient (function of Re and Cw) according to a useful general 5/7 power law

determined by Poncet et al. [1]. When a centrifugal throughflow is superimposed, Batchelor

and Stewartson flow structures can be found. A Stewartson type of flow is confined in the

rotating disk boundary layer. The tangential velocity is almost zero outside and the radial

velocity is positive everywhere. The results for the mean flow can be synthetized by the

hodograph on figure 29, which shows, for Re = 4.15 × 106 and three flow rate coefficients,

the Ekman spirals describing the changes in direction of the velocity vector through the

cavity at a given radial location r∗ = 0.56. The tangential velocity in the central core is

attained when the radial velocity is approximately zero, where a concentration of points

occurs. The profiles falls between the typical fully turbulent behavior and the laminar Von

Karman solution presented by [34]. Moreover it confirms that the Stewartson flow is confined

in the Ekman layer whereas the Batchelor flow structure is more symmetric. We succeeded

in characterizing the transition at a given radial location between these two flow structures

according to a Rossby number. We showed that it does not depend on the aspect ratio

G. This continuous transition occurs as soon as the radial velocity is negative for at least

one axial location. The study includes also turbulence measurements, which were seldom

possible in previous works of the literature. It appears that the turbulent intensities R∗rr and

R∗rθ in the Bodewadt layer increase from the periphery to the center of the cavity when a

throughflow is superimposed. On the contrary, in a closed cavity, these turbulent intensities

increase by increasing r∗ (fig.30). The predictions of the present RSM turbulence model

are found here in excellent agreement with the velocity and pressure measurements over a

22

wide range of parameters including strong outflow and inflow. The RSM model appears as

a valuable tool for describing the mean and turbulent fields of such complex flows.

VIII. ACKNOWLEDGEMENT

Numerical computations were carried out on the NEC SX-5 (IDRIS, Orsay, France).

Financial supports for the experimental approach from SNECMA Moteurs, Large Liquid

Propulsion (Vernon, France) are also gratefully acknowledged.

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26

• Fig.1: Schematic representation of the experimental set-up and notations: R1 = 38,

R2 = 250, R3 = 253, R4 = 55 and 3 ≤ h ≤ 12 mm.

• Fig.2: Mean velocity profiles for Re = 106 and Cw = 0 at three radial locations: (a)

r∗ = 0.44, (b) r∗ = 0.56, (c) r∗ = 0.8; (−) RSM model, () experimental data.

• Fig.3: Effect of the Reynolds number on the streamlines Ψ∗ = Ψ/(ΩR22) for Cw = 0

(RSM), 15 regularly spaced intervals: (a) Re = 106, 0 ≤ Ψ∗ ≤ 0.017, (b) Re =

4.15× 106, 0 ≤ Ψ∗ ≤ 0.014.

• Fig.4: Mean velocity profiles for r∗ = 0.56, Cw = 0 and Re = 4.15 × 106; (−) RSM

model, () experimental data.

• Fig.5: Radial pressure distributions for Cw = 0 and two Reynolds numbers: (×,−)

Re = 106 and (4,−−) Re = 4.15× 106 (the symbols represent the experimental data

and the lines the model results).

• Fig.6: Profiles of the Reynolds stress tensor components for Re = 106, and Cw = 0 at

three radial locations: (a) r∗ = 0.44, (b) r∗ = 0.56, (c) r∗ = 0.8; (−) RSM model, ()experimental data.

• Fig.7: Axial profiles of the six Reynolds stress tensor components for Cw = 0 and

Re = 4.15× 106 at r∗ = 0.56; (−) RSM model, () experimental data.

• Fig.8: Mean velocity profiles for Re = 106, Cw = 5929 at three radial locations: (a)

r∗ = 0.44, (b) r∗ = 0.56, (c) r∗ = 0.8; (−) RSM model, () experimental data.

• Fig.9: Mean velocity profiles for Re = 106, r∗ = 0.56 and three flow rate coefficients:

(a) Cw = 1976, (b) Cw = 5929, (c) Cw = 9881; (−) RSM model, () experimental

data.

• Fig.10: Streamlines Ψ∗ = Ψ/(ΩR22) patterns for some centripetal throughflow values

(RSM), 20 regularly spaced intervals: (a) Re = 106, Cw = 1976, −0.009 ≤ Ψ∗ ≤ 0.016,

(b) Re = 106, Cw = 5929, −0.027 ≤ Ψ∗ ≤ 0.012, (c) Re = 106, Cw = 9881, −0.045 ≤Ψ∗ ≤ 0.007, (d) Re = 4.15× 106, Cw = 5929, −0.009 ≤ Ψ∗ ≤ 0.008.

• Fig.11: Mean velocity profiles for r∗ = 0.56, Cw = 5929 and Re = 4.15 × 106; (−)

RSM model, () experimental data.

27

• Fig.12: Mean K curve: () experimental data, (−) analytical law of Poncet et al. [1],

(4) model results, (−−) numerical correlation curve K = 2× (5.3×Cqr +0.63)5/7−1.

• Fig.13: Radial pressure distributions for Re = 106 and three centripetal throughflows:

(−) model results and experimental data for () Cw = 1976, (4) Cw = 5929, (¦)Cw = 9881.

• Fig.14: Radial profiles of K for Re = 106 and three centripetal throughflows; Com-

parison between pressure and velocity measurements.

• Fig.15: Profiles of the Reynolds stress tensor components for Re = 106, Cw = 5929 at

three radial locations: (a) r∗ = 0.44, (b) r∗ = 0.56, (c) r∗ = 0.8; (−) RSM model, ()experimental data.

• Fig.16: Effect of the flow rate on the axial profiles of the Reynolds stress tensor

components for Re = 106 at r∗ = 0.56: (a) Cw = 1976, (b) Cw = 5929, (c) Cw = 9881;

(−) RSM model, () experimental data.

• Fig.17: Axial profiles of the Reynolds stress tensor components for Cw = 5929 and

Re = 4.15× 106 at r∗ = 0.56; (−) RSM model, () experimental data.

• Fig.18: Mean velocity profiles for Re = 106 and Cw = −5929 at three radial locations:

(a) r∗ = 0.44, (b) r∗ = 0.56, (c) r∗ = 0.8; (−) RSM model, () experimental data.

• Fig.19: Mean velocity profiles for Re = 106, r∗ = 0.56 and three flow rate coefficients:

(a) Cw = −1976, (b) Cw = −5929, (c) Cw = −9881; (−) RSM model, () experimental

data.

• Fig.20: Effect of the flow rate on the streamlines Ψ∗ = Ψ/(ΩR22) for Re = 106 (RSM),

15 regularly spaced intervals: (a) Re = 106, Cw = −1976, 0 ≤ Ψ∗ ≤ 0.024, (b)

Re = 106, Cw = −5929, 0 ≤ Ψ∗ ≤ 0.032, (c) Re = 106, Cw = −9881, 0 ≤ Ψ∗ ≤ 0.046,

(d) Re = 4.15× 106, Cw = −5929, −0.001 ≤ Ψ∗ ≤ 0.019.

• Fig.21: Mean velocity profiles for Cw = −5929 and Re = 4.15× 106 at r∗ = 0.56; (−)

RSM model, () experimental data.

• Fig.22: Transition diagram (r∗,Ro) between Batchelor and Stewartson flow structure:

() experimental data, (4) RSM model.

28

• Fig.23: Experimental mean K curve compared to the analytical law (−) of Poncet

et al. [1] for four aspect ratios: (×) G = 0.012, (2) G = 0.024, () G = 0.036, (4)

G = 0.048.

• Fig.24: Radial pressure distributions for Re = 4.15×106 and three centrifugal through-

flows: (−) model results, (2) Cw = −1976, () Cw = −5929, (4) Cw = −9881.

• Fig.25: Axial profiles of three components of the Reynolds stress tensor for Re = 106

and Cw = −5929 at three radial locations: (a) r∗ = 0.44, (b) r∗ = 0.56, (c) r∗ = 0.8;

(−) RSM model, () experimental data.

• Fig.26: Effect of the flow rate on the axial profiles of three components of the Reynolds

stress tensor for Re = 106 at r∗ = 0.56: (a) Cw = −1976, (b) Cw = −5929, (c)

Cw = −9881; (−) RSM model, () experimental data.

• Fig.27: Axial profiles of three components of the Reynolds stress tensor for Cw =

−5929 and Re = 4.15× 106 at r∗ = 0.56; (−) RSM model, () experimental data.

• Fig.28: Effect of the flow rate on the iso-turbulent Reynolds number Ret = k2/(νε)

for Re = 106 (RSM), 20 regularly spaced intervals: (a) Cw = −5929, Ret ≤ 1942, (b)

Cw = 0, Ret ≤ 352, (c) Cw = 5929, Ret ≤ 7206.

• Fig.29: Hodograph, r∗ = 0.56, Re = 4.15 × 106: (−,4) Cw = −5929, (−,) Cw = 0,

(−,2) Cw = 5929.

• Fig.30: Radial profiles of the Reynolds stress tensor components in the Bodewadt layer

at z∗ = 0.946 for Re = 106: (−.,2) Cw = −5929, (−,) Cw = 0, (−−,4) Cw = 5929.

29

STATOR

ROTOR

M

TANK

breakthrough crown

Q

r

z

hInter−disk space

Ω

R1

R2

R3

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Figure 1: Poncet et al., Phys. Fluids.

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Batchelor vs Stewartson Flow Structures in a Rotor Statotr Cavity

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Transcript of Batchelor vs Stewartson Flow Structures in a Rotor Statotr Cavity