Discharge Coefficients for Flow through Holes Normal to a Rotating Shaft

Discharge Coefficients for Flow through Holes Normal to a Rotating

Shaft

A. Alexiou†, N. J. Hills†, C. A. Long†, A. B. Turner†, L.-S. Wong†, J. A. Millward*

†Thermo-Fluid Mechanics Research Centre, School of Engineering,

University of Sussex, Brighton, BN1 9QT, Sussex, U.K.

*Rolls-Royce Plc, P.O. Box 31, Derby, DE24 8BJ, U.K.

A. Alexiou et al. / Int. J. Heat and Fluid Flow, 2000

1

Abstract

A possible design for a more compact gas turbine engine uses contra-rotating high

pressure (H.P.) and intermediate pressure (I.P.) turbine discs. Cooling air for the I.P.

turbine stages is taken from the compressor and transferred to the turbine stage via

holes in the drive shaft. The aim of this work was to calculate the discharge

coefficient characteristics of the holes in this rotating shaft, and, in particular, to

ascertain whether the sense of rotation affected these significantly.

This paper reports on experimental measurements of the discharge coefficients and

CFD modelling of this flow. The experimental results show the effects on the

discharge coefficient of rotational speed, flow rate, and co and contra rotation of the

shaft relative to the discs. The measured values of the discharge coefficient are

compared with established experimental data for non-rotating holes in the presence

of a crossflow. For stationary shaft and discs, co-rotation of the shaft and discs, and

differential rotation with the disc speed less than the shaft (in the same rotational

direction), the discharge coefficients are in reasonable agreement with this data. For

differential rotation (including contra-rotation) with the disc speed greater than the

shaft, there is a significant decrease in discharge coefficient. The CFD modelling is

in poor absolute agreement with the values of discharge coefficient (presumably due

to deficiencies in grid and turbulence model) but does reproduce these trends,

helping to explain the experimental measurements.


2

Nomenclature

a, b inner and outer radius of the cavity

Ah, Aan Hole area, annulus area between disc bore and shaft outer diameter

CD Discharge coefficient

Cp Specific heat capacity at constant pressure

d Hole diameter

L Hole depth

m Mass flow rate

M Mach Number

Ns, Nd Shaft speed, Disc speed

p Pressure

r, , z Radial, angular and axial coordinates

rf Fillet radius of hole

ro, ri Inner and outer radius of the shaft (ro – ri = L)

R Gas constant

Ro Rossby number ( = V1,z / V1, )

s Axial distance between discs

T Temperature

V Velocity

Ratio of specific heats

p Measured pressure difference across shaft hole

v Velocity head ratio ( = (pT,rel – p2) / (pT,rel – p1) )

Density

d, s Angular speed of the discs and shaft, respectively


3

Subscripts

1 Upstream of shaft holes

2 Downstream of shaft holes

rel Relative

s Static

T Total

, z Tangential and axial components


4

1. Introduction

The motivation for this work arises from design features in the latest gas turbine aero

engines. A more compact design of engine can be produced using contra-rotating

high pressure (H.P.) and intermediate pressure (I.P.) turbine discs. This reduces the

aerodynamic deflection and hence losses in the nozzle guide vanes at entry to the

I.P. turbine. Cooling air for the I.P. turbine stages passes through the bore of the H.P.

compressor stack and is then bled off through a series of holes in the rotating drive

shaft. This paper reports on experimental measurements of the discharge

coefficients of these holes in the drive shaft and accompanying CFD modelling of the

flow. Measurements were made on an experimental rig having geometric features

relevant to modern gas turbine engines and capable of operating at engine

representative conditions, including the ability to provide co-rotation and contra-

rotation of the drive shaft relative to the discs.

A considerable amount of previous work by a number of workers has been carried

out (both theoretically and experimentally) to calculate discharge coefficients for a

variety of different flow conditions and geometries. Only a brief discussion of the work

directly relevant to this investigation will be given here, and for a more

comprehensive review the reader is referred to Hay and Lampard (1998).

Rohde, Richards, and Metger (1969) obtained discharge coefficients for orifice plates

with the approach flow perpendicular to the orifice axis (with a Mach number of up to

0.5) and for three different inlet edge conditions (rf/d=0, 0.192, 0.487). They found a

significant increase in discharge coefficient from rounding of the inlet edge of the


5

orifice. This was considered to be due to the reduction of the separation at the

upstream edge.

The effect of rotation on discharge coefficient has primarily been studied for holes in

a rotating disc with an approach flow normal to the disc. Meyfarth and Shine (1965)

investigated the effect of rotation of an orifice on the rate of flow through it using a

rotating disc inside an annular casing. They found that as the orifice speed increases,

for a given pressure drop, the flow rate remained essentially constant until the

tangential velocity of the disc reached about one half of the axial velocity of the fluid.

Further increase in disc speed caused the flow rate to decrease. Wittig, Kim, Jacoby,

and Weissert (1996) presented experimental and numerical results of the flow

through orifices in rotating discs. It was shown that the rotational effects strongly

influence the discharge coefficient. They measured local flow velocities in front of

and behind the orifices using a 2-D Laser Doppler Velocimetry technique. Their

results revealed a very complex flow field dominated by high velocity gradients in

close vicinity to the orifices. They found that rotation reduced the discharge

coefficient due to the growing inclination angle of the inflow, leading to enlarged

separation at the orifice inlet corner.

Maeng, Lee, Jacoby, Kim and Wittig (1998) investigated the discharge coefficients of

long orifices (L / d = 10) in a disc rotating at speeds of up to 10000 rev/min. They

found that rounding the inlet gave higher (up to 20%) discharge coefficients

compared to a sharp edge orifice. They redefined the discharge coefficient to

account for the rotational momentum transfer from the disc to the orifice flow and

found that the discharge coefficient decreases with increasing rotational speed. They

also presented the additional losses due to the relative motion of the orifice in respect


6

to the upstream and downstream cavity flows taking into account the rotation and

compressibility effects.

McGreehan and Schotsch (1988) used empirical data from various investigators to

derive correlations that correct the basic discharge coefficient of the flow through a

sharp-edged orifice for Reynolds number (CD:Re = 0.5855 + 372 / Red), inlet radius

(CD:r = 1 – f1 (1 - CD:Re) and f1 = 0.008 + 0.992 exp(-5.5(rf / d) - 3.5(rf / d)2)), length

(CD:r,L = 1 – f2 (1 - CD:r) and f2 = [1 + 1.3 exp(-1.606(L / d))2] (0.435 + 0.021(L / d)))

and relative tangential velocity of approach (CD:r,L,v = CD:r,L (C1 + C2 C3) where C1 =

exp(-Rv1.2), C2 = 0.5 Rv

0.6 (CD:r,L / 0.6)-0.5, C3 = exp(-0.5 Rv0.9), Rv = (v / Uid) (CD:r,L / 0.6)-

3, v is the inlet relative tangential velocity and Uid is the orifice ideal throughflow

velocity) effects. However, it must be noted that their correlations overestimate the

effects of rf/d and L/d on the experimental discharge coefficients of Rohde et al. with

which the current experimental results are compared. This is believed to be due to

the fact that the correlations for rf/d and L/d effects were not derived from data where

the approaching flow is normal to the orifice axis. Parker and Kercher (1991)

improved these correlations to reduce the amount of iteration needed.

2. Experimental Apparatus and Instrumentation

The experimental tests discussed in this paper were carried out using the ‘disc and

drive cone’ rig shown in Figure 1. A full and detailed description of this rig, the

instrumentation and ancillary equipment is available in Alexiou (2000), and the reader

is referred to that thesis for further information. Designed to be geometrically similar

(approximately 70% full size) to contemporary gas turbine aero-engine H.P.

compressor design, the experimental rig aims to simulate the internal air system


7

flows inside a high pressure compressor. The external surface of the rotor is heated

by a separate supply of hot air. As in an actual engine cooling air flows axially in the

annular space between the disc bores and the drive shaft.

The titanium 318 rotor is made up from three discs and a cone section, bolted

together at the periphery, and having an outer diameter of 0.4913m. In this way the

completed rotor assembly forms two cylindrical cavities and one conical cavity. Each

cylindrical cavity has an inner radius, a = 0.0701m (1st disc) and a = 0.06655m (2nd

disc), an outer radius of b = 0.22m and is separated by an axial distance,

s = 0.0429m. The conical cavity has an outer radius of b = 0.22m and a maximum

axial gap of s = 0.1235m. As in a high pressure compressor, a steel ‘drive’ shaft of

0.1195m outer diameter, runs through the centre of the rig. This shaft has twelve

15mm diameter holes allowing some of the cooling air to pass through them. These

holes have an inlet radius of 1mm (giving rf/d=0.067) and an exit radius of 1mm.

There is an uncertainty of 10-15% in the fillet radii due to machining and inspection

difficulties. The holes are divided into two sets of six, separated by 43mm and

staggered by 30 (see Figure 2). The thickness of the shaft is 6.75mm giving a length

to diameter ratio for the orifice of L/d=0.45. The rotor and the shaft are independently

driven by two separate A.C. motors - the rotor up to 10000 rev/min, the shaft to 7000

rev/min (but in either direction).

The relevant instrumentation for the calculation of the discharge coefficient is also

shown in Figures 1 and 2. The total air temperature, TT, is measured by two rotating

K-type thermocouples (40 V/C) inside the shaft and diametrically opposite each

other. Their axial position was between the two sets of holes. The upstream static


8

pressure, p1, and the pressure drop across the holes, p, are measured by sub-

miniature, rotating Kulite pressure transducers with a sensitivity of 29 mV/bar and 58

mV/bar respectively. All the rotating instrumentation is led out to a 48-way, Wendon

slip ring unit. The instrumentation signals are read by a Solatron/Orion data logger

scanning at 40 channels/s with a resolution of 1V (corresponding to 0.025C for

the thermocouples, 3.510-5 bar for the absolute pressure transducers and 1.710-

5 bar for the differential ones).

The rotor assembly is housed within a steel casing and supported by a high precision

grease lubricated ball bearing on the upstream side and an oil lubricated cylindrical

roller bearing on the downstream side. A similar configuration is used to support the

drive shaft. Since the rig is pressurised, a number of labyrinth seals are used in

critical locations to reduce the leakage flow. These seals were designed to industry

design codes – the fins are on the rotating member, the static radial clearance is

0.1mm and the stationary member is coated with an abradable material (Apticote

800/38). The leakage through the seals was calculated using a validated method,

based on experimental and theoretical results from several sources (Curzons,

(1991)).

Air is supplied to the rig by a single stage Howden screw compressor (driven by a

275 kW electric motor) which can deliver up to 1.1 kg/s of air at 4 bar (absolute) and

200C. Before entry to the rig, the air is cooled to 30C to 55C. To eliminate

condensation, the actual value of this temperature was varied depending on the

measured relative humidity and the pressure of the cooling air. The air flow rate is

measured by a single orifice plate at inlet and by two orifice plates at exit to the rig.


9

These orifice plates are designed to BS 1042, and this particular configuration allows

separate determination of the inlet mass flow, that passing through the holes in the

shaft and, by inference (and also calculation from Curzons), the leakage through the

labyrinth seals.

3. Data Analysis

The discharge coefficient is defined as the ratio of the actual to ideal mass flow

through the holes. Hence, for twelve holes in the shaft the discharge coefficient, CD,

is given by:

h22

2D

AV

12mC

(1)

where m2 is the measured mass flow through the twelve holes and 2, V2 are the

ideal density and velocity through the hole, which are calculated assuming a one-

dimensional, isentropic compressible expansion from the total upstream relative

pressure to the downstream static pressure p2. Since the approach flow is

perpendicular to the orifice axis, there will be little dynamic pressure recovery, but

using the total upstream relative pressure (rather than upstream static pressure)

allows direct comparison with the data of Rohde et al.

The total upstream relative pressure, pT,rel, is calculated from the measured static

pressure p1, the measured total temperature TT, and the measured mass flow rate m1

by an iterative procedure. The upstream density 1 is based on the upstream static

temperature T1,s:

s,1

11

RT

p (2)


10

where R is the gas constant. The upstream static temperature is obtained from:

2

rel,1

p

Ts,1 VC2

1TT (3)

where V1,rel is the relative velocity of the cross flow to the orifice. This consists of the

tangential, V1,, and axial components, V1,z, and hence:

2

z,1

2

,1rel,1 VVV (4)

The axial velocity, V1,z, is calculated from the upstream density and mass flow rate

and using the annulus area between the shaft and 2nd disc bore. The tangential

velocity, V1,, is obtained by assuming the cross flow air to be rotating at the disc (i.e.

rotor) speed and is calculated for the mean inlet radius. This assumption is

considered reasonable since (as can be seen in Figure 1) the air is introduced

through 6 holes of 25mm diameter in the end-disc and then passes through a 3mm

thick perforated plate, attached to the end-disc, with 630 3mm diameter holes.

Hence, it is expected to be fully swirled up to rotor speed. Equations (2) – (4) are

then solved by an iterative procedure and the upstream relative total pressure is

given by (for the general case of compressible flow):

12

,11,

2

11

relrelT Mpp (5)

where the relative Mach number is:

s,1

2

rel,12

rel,1RT

VM

(6)

and is the ratio of specific heats, which here is taken as 1.4.

Finally 2 and V2 can be calculated from the following equations:


11

1

2

rel,T

T

22

p

p

RT

p (7)

and

1

rel,T

2T

p

p1

1

RT2V2 (8)

The pressure downstream of the shaft holes, p2, is found from:

p2 = p1 – [p - 0.5 1 s2 (ro

2 - ri2)] (9)

where p is the measured pressure drop through the shaft holes and 0.5 1 s2 (ro

2 -

ri2) accounts for the pressure drop across the length of the hole due to the radial

pressure gradient which sustains solid body rotation at shaft speed. This correction

ensures that zero pressure drop across the hole corresponds to zero mass flow

through the hole. The value of this correction, expressed non-dimensionally by the

quantity 0.5 1 s2 (ro

2 - ri2) / p, varied from 0.004 to 0.18.

Again for direct comparison with the data of Rohde et al., the graphs of discharge

coefficient are all plotted against velocity head ratio which is defined as:

1rel,T

2rel,T

vpp

ppΘ

(10)

A most probable error approach was used in estimating the overall error in the

calculated discharge coefficient due to the instrumentation errors (as recommended

by Kline and McKlintock (1953) for analysing the uncertainties in single sample

experiments). The errors in the measured values are given in Table 1, leading to an

estimate of uncertainty in the discharge coefficient of around 4%.


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QUANTITY ERROR

Mass Flow Rate 3%

Temperature 1C

Pressure 0.1%

Rotational Speed 1 rev/min

Linear Dimension 0.1 mm

Table 1: Errors in Measured Quantities

4. CFD modelling

The CFD modelling was carried out using a commercial CFD code (FLUENT Version

5.1). In order to provide confidence in the CFD results, a test case using the

geometry of Rohde et al. (a hole on the side of a duct with the approach flow

perpendicular to the hole axis) was modelled. The conditions were chosen to be in

the range of data given by Rohde et al. and also to be in the range considered in this

paper. A discharge coefficient within 2% of the experimental value was obtained

from the CFD calculation, giving confidence in the ability of the code to model these

types of flow.

A grid with approximately the same grid density in the orifice region as for the

validation case was used to model the experimental rig. This led to a hybrid grid of

around 370,000 cells with triangular prisms used next to the solid walls to resolve the

boundary layers and tetrahedra elsewhere. A cross-section of the grid in the r-z plane

is shown as Figure 3, which also serves to illustrate the domain used. As shown, the

domain was extended downstream of the cone space as flow reversal otherwise


13

occurred at the exit to the domain. This indicates that there is greater dependence on

the conditions downstream than is ideal for the CFD modelling, but modelling the

remainder of the domain adequately would have led to much greater complexity.

The standard high Reynolds number k- turbulence model was used throughout. All

CFD models were run as incompressible flow as for the experimental conditions

considered (M1,rel<0.3) this should be a good approximation.

A CFD model was obtained for a case with upstream density 1.79 kg/m3, molecular

viscosity 1.9x10-5 kg/ms, inlet axial velocity 37.8 m/s (giving a mass flow of m1 =

0.183 kg/s), and a mass flow split of 62% through the shaft. These conditions were

taken from one of the experimental runs. The experimental value of p1 for these

conditions was 157 x 103 Pa. The pressure drop across the differential pressure

transducer locations from the CFD model was 1767 Pa compared to the

experimentally measured value of 2510 Pa. This gave a discharge coefficient from

the CFD model of 0.51, approximately 10% higher than the experimentally measured

value of 0.46.

A test of the grid independence of the solution was made by refining the grid in the

region of the shaft holes. The grid was refined by a factor of 8 (i.e. by a factor of 2 in

each direction) in the region of the shaft holes. This gave a grid of some 830,000

cells and the pressure drop across the differential pressure transducers calculated

from this model was 1828 Pa, again giving a discharge coefficient of 0.51. This

suggests that the original grid is sufficiently fine in the orifice region to model the flow

there accurately. It is thought that the discrepancy between the CFD and the


14

experimental data is due to a combination of the uncertainty in the machining of the

fillet radii, the dependence on the downstream conditions, and possibly a lack of

resolution of the interaction of the two flows (through the two separate sets of holes)

inside the shaft. However, given that the grid appears sufficiently fine in the orifice

region to model the flow accurately there, it was felt (and, as was indeed the case)

that the CFD mesh would be sufficient to capture the trends due to the effects of

rotation.

5. No Rotation Case

The first experiments were carried out with stationary discs and shaft. A plot of the

measured discharge coefficient against velocity head ratio v (Equation (10)) is

shown as Figure 4. Results from Rohde et al. are shown in Figure 4 for L/d=0.5,

M1=0.13 and rf/d=0 (a sharp corner on the hole). The value of M1=0.13 was selected

as the majority of the rig tests were carried out in the region 0.12<M1,rel<0.14. The

experimental data presented by Rohde et al. were used to derive a correction to

account for a finite corner radius. This assumes that the change in the discharge

coefficient due to the inlet edge condition is proportional to rf/d and is given by:

CD = CD,0 + 1.63(rf/d)CD,0 (11)

where CD,0 is the discharge coefficient for a sharp orifice. A curve showing values of

discharge coefficient calculated from Equation (11), with a value of appropriate to the

current experimental data, rf/d = 1/15 is also shown on the graph. After applying

Equation (11) to account for a finite fillet radius there is very good agreement

between the current experimental data and that obtained from Rhode et al. The

difference in L/d between the current experimental data and those of Rohde et al.

has an effect of less than 3% on the discharge coefficient, over the range of v


15

tested. This is based on experimental data present by Rohde et al. for different

values of L/d.

It is perhaps more convenient to consider the velocity head ratio as a measure of the

ratio of the mean velocity through the holes to the external velocity (which in general

has contributions from both axial and tangential velocity components). A small value

of v implies that the external velocity is much larger than the velocity through the

holes. The low values of CD at low values of v therefore reflects the relatively high

momentum of the external flow and the associated pressure drop required to redirect

this into the hole. Conversely, at larger values of v the discharge coefficient is

larger. For v > 3, the discharge coefficients show less of an appreciable increase,

since, by definition, the external flow now has progressively less influence on the flow

through the holes. The measured discharge coefficients tend to level off at CD 0.6,

a value that is consistent with flow through plane orifices and achieved due to the

formation of the vena contracta.

The variation of the discharge coefficient with the hole Reynolds number, defined as

Red = Vh d / where Vh = (m2/12) / ( Ah), is shown in Figure #. For all the

experiments, Red > 2104 and there is no Reynolds number effect on the discharge

coefficient. This is consistent with the findings of other workers (see for example

Lichtarowicz, Duggins and Markland (1965), Chu, Brown and Garrett (1985)).

As discussed in Section 4, a base CFD model was run for a zero rotation case. The

flow pattern for the hole region in the r-z plane is shown as Figure 5 (Ns = Nd = 0

rev/min, m1 = 0.183 kg/s, m2/m1 = 62% and p1 = 1.6 bar). The external flow is shown


16

moving from the left to the right. Flow enters the hole and separation from the

upstream edge can clearly be seen.

6. The Effect of Disc Rotation with the Shaft Held Stationary

The variation of measured discharge coefficients with velocity head ratio for rotating

discs and a stationary shaft is shown in Figure 6. The effect of disc rotation is shown

by grouping the experimental data in discrete bands of the inverse Rossby number

1/Ro, where Ro is defined as the ratio of the axial to the relative tangential velocity

(Ro = V1,z/ V1,). A zero value of 1/Ro therefore corresponds to no rotation of the

discs or co-rotation at the same rotational speed of the discs and shaft. Also plotted

on the figure are the data from the experiments carried out with a stationary shaft and

discs, and the data from Rohde et al. corrected with equation 11.

Inspection of Figure 6 shows that for a constant velocity head ratio, rotation causes a

reduction in the discharge coefficient, which decreases as 1/Ro (or rotational speed

of the disc) increases.

With rotation of the discs, the external velocity now comprises both axial and

tangential components. The tangential component is caused by rotation of the discs

and therefore affects the angle of the flow to the hole. For flow through a round orifice

located in a flat plate, there can be no dependence on flow angle since the geometry

is then symmetrical with respect to the flow direction. Hence, for the geometry

considered in this paper, the dependence on Rossby number (which is equivalent to

flow angle) must come from either the additional terms in the Navier-Stokes

equations on transforming to cylindrical polar coordinates (corresponding to the effect


17

of the geometric curvature), or from differences in the physical boundary conditions.

For the experimental conditions considered, the magnitude of the additional terms in

the Navier-Stokes equations is significantly smaller (typically 4% of p/L) and does

not account for the differences in discharge coefficient shown in Figure 6. However,

for a rotationally symmetric geometry, the physical boundary condition for the flow

variables in the -direction is periodic. This implies that any flow through the hole

must be provided by the axial component of the velocity and, in particular, if the

approach flow was purely tangential, there would be zero mass flow through the hole.

Hence, for a constant vale of V1,rel, the same amount of separation is created for any

value of Rossby number, since this is governed by the magnitude, and not the

direction of the external flow. A reduction in CD therefore occurs when Ro > 0 as the

tangential component of the external flow is not available to contribute to the mass

flow through the hole.

A CFD model with disc rotation of 5000 rev/min was run (with the same flow

conditions and mass flow rates as in Section 4). These conditions gave a value of

1/Ro of 0.87. A pressure drop of 2760 Pa was calculated across the pressure

transducer locations, giving a discharge coefficient of 0.40 and a velocity head ratio

of 2.22. This result was compared to that from another CFD model with the same

total relative pressure and mass flow through the hole but with no rotation (and so a

value of 1/Ro of zero). This was given by an axial velocity of 50.17 m/s and a mass

flow split of 46.71% through the shaft. This gave a pressure drop of 2160 Pa and a

discharge coefficient of 0.42 and a velocity head ratio of 1.96. Hence from the CFD

results the effect of disc rotation was to reduce the discharge coefficient by


18

approximately 5%. Although this is somewhat less than the effect measured

experimentally (~9%), the trend is certainly consistent with the experimental results.

7. The Effect of Disc and Shaft Rotation

A limited number of tests were carried out with the shaft and discs co-rotate (rotating

in the same direction) at the same speed. The results of these tests are shown in

Figure 7. Also plotted on the figure are the data from the tests with no rotation

(1/Ro=0) and the data from Rohde et al. corrected with Equation (11). As can be

seen, there is virtually no noticeable change in discharge coefficient compared with

the measurements for a stationary disc and shaft.

The results for tests where both the rotor and shaft co-rotate but with a speed

differential (Ns Nd) are shown in Figure 8. For Nd > Ns, the discharge coefficients

are similar to those where the shaft was stationary and the discs rotate (Figure 6).

There is also a significant reduction in the discharge coefficient compared to the

values obtained without rotation (1/Ro = 0). However, when the shaft rotates in the

same direction but faster than the discs (Ns > Nd) there does not appear to be any

reduction in discharge coefficient and the values of CD are similar to those measured

with a stationary shaft and discs.

Finally, the measured values of discharge coefficient for contra-rotation (when the

shaft rotates in the opposite direction to that of the discs) are shown in Figure 9. This

again shows a reduction in the discharge coefficient relative to the no rotation case.


19

In order to understand these experimental results, it is necessary to consider

everything in the frame of reference rotating at the shaft speed. The additional terms

in the momentum equations can be written as:

2 x v + x ( x r)

where is the rotation vector, v is the fluid velocity vector, and r is the position

vector. The first term is the Coriolis force and the second term is the centrifugal force.

Clearly, it is only through the first of these terms in the equations that the difference

between the rotor rotating faster or slower than the shaft can be explained. Since in

the frame of reference of the shaft this will merely appear as flow approaching the

hole from the left or from the right. For flow inwards through the shaft hole the

Coriolis force will be in the direction of the shaft rotation. Hence for the discs rotate

faster than the shaft, the Coriolis force will tend to accelerate the approach flow and

hence increase separation. Conversely, when the discs rotate slower than the shaft,

the Coriolis force will tend to decelerate the flow and suppress separation. This is

confirmed in the measurements of discharge coefficient (Figure 8). When the discs

rotate slower than the shaft, the discharge coefficient does not decrease relative to

the zero rotation case.

CFD models were run with a shaft speed of 7,500 rev/min and varying rotor speeds.

The remaining experimental conditions were the same as for the base model of

Section 4. Plots of the velocity vectors relative to the shaft in the r- plane for cases

with rotor speeds of 10,000, 5000, and 2,500 are shown in Figures 10 to 12. These

disc speeds correspond to differential (disc to shaft) speeds of 2,500, -2,500, and –

5000 rev/min respectively. Since the velocity vector (in the shaft reference frame) in

the shaft hole is directed inwards the Coriolis force acting on the fluid in the hole is in


20

the direction of shaft rotation in all three cases. Hence in Figure 10, the effect of the

Coriolis force is to increase (compared with Ns = 0) the separation from the

(tangentially) upstream edge. In contrast, in Figure 11, the Coriolis force is acting to

suppress the separation and it can be seen that there is considerably less separation

than Figure 10 although both cases have the same magnitude of differential speed.

In Figure 12, the Coriolis force is again acting to suppress the separation, but the

relative tangential velocity is sufficiently large that significant separation still occurs.

The above points are summarised in Figure 13. For co-rotation but with the discs

rotating faster than the shaft (Nd – Ns > 0), the discharge coefficient is decreased

relative to the co-rotation case (Nd – Ns = 0). For the discs rotating slower than the

shaft (Nd – Ns < 0), there is a region where the discharge coefficient does not

decrease, where, presumably the effect of the Coriolis force is sufficient to overcome

the tendency to increased separation. For the disc speed sufficiently slow relative to

the shaft, the discharge coefficient starts to decrease, when the effect of the Coriolis

force is no longer sufficient to overcome the increased (relative) momentum of the

approach flow.

This explains why the decrease in discharge coefficient was not seen experimentally

when the shaft rotates in the same direction but faster than the discs. It is simply that,

due to the experimental arrangement, the speed differential was not great enough.

Similarly, the reason that contra-rotation of the shaft relative to the discs always

showed a reduced discharge coefficient, was that (by the very nature of contra-

rotation), the speed differential was always large and hence missed the region where

the discharge coefficient stays constant. For comparison with the CFD results,


21

measured discharge coefficients for shaft speeds of 5000 and –5000 rev/min and

varying disc speeds are also shown in Figure 13. For these tests m1 0.5 kg/s,

m2/m1 40% and p1 2.6 bar. As can be seen the experimental trends are

consistent with the CFD results and lend support to the above hypothesis.

Finally it should be noted that a number of tests were carried out under non-

isothermal conditions, where the outer surface of the rotor is heated to approximately

100C. It is known (Alexiou et al. (2000)) that heating, and indeed the level of heating

can affect the flow structure inside the cone space, and the heat transfer from the

discs and inner surface of the cone. It was, however, found that heating did not affect

the value of the discharge coefficients for flow through the holes in the drive shaft.

9. Conclusions

This paper has reported on the measurement of the discharge coefficients of the flow

through twelve 15mm holes in a rotating shaft of 0.1195m outer diameter with an

approach flow normal to the holes. The shaft passed through the centre of a titanium

rotor comprising a number of discs, forming two cylindrical and one conical cavity.

The holes were equally spaced around the circumference of the shaft at two axial

locations towards the centre of the conical cavity. This configuration simulates the

part of an internal air system in a gas turbine engine where cooling air from the H.P.

compressor is bled through holes in the I.P. shaft. As a possible design for a more

compact gas turbine engine uses contra-rotating H.P. and I.P. turbine discs, the

designer needs to know whether contra-rotation affects the discharge coefficients

significantly. Tests were carried out for the range of rotor speeds 0<Nd<10000

rev/min, a shaft rotational speed of –5000<Ns<7500 rev/min, an axial throughflow


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rate of 0.1<m1<0.8 kg/s, a mass flow ratio of 9<m2/m1<95 %, an inlet pressure of

1.3<p1<3.5 bar and an inlet temperature of 300<TT<335 K.

Experimentally, it was found that when the rotor and shaft are stationary, co-rotate at

the same speed or the shaft rotates faster that the rotor then the discharge

coefficients agree with established experimental data for non-rotating holes in the

presence of crossflow. When the rotor speed is greater than that of the shaft

(including contra-rotation) there is a significant decrease in the discharge coefficient.

From the CFD modelling, it was apparent that the effect of the Coriolis force due to

shaft rotation was to suppress the effects of separation due to the relative rotation of

the approach flow to the holes. However, for large enough relative rotation the

magnitude of the Coriolis force is insufficient; separation occurs and a decrease in

discharge coefficient was again observed.

Acknowledgements

The authors wish to express their thanks to the Engineering and Physical Sciences

Research Council and Rolls-Royce Plc for supporting this work.

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Discharge Coefficients for Flow through Holes Normal to a Rotating Shaft

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