Tip-Clearance and Secondary

Flows in a Transonic
Compressor Rotor
G. A. Gerolymos The purpose of this paper is to investigate tip-clearance and secondary flows numerically
in a transonic compressor rotor. The- computational method used is based on the
numerical integration of the Favre-Reynolds-averaged three-dimensional compressible
I. Vallet Navier-Stokes equations, using the Launder-Sharma near-wall k-e turbulence closure. In
order to describe the flowfield through the tip and its interaction with the main flow
accurately, a fine O-grid is used to discretize the tip-clearance gap. A patched O-grid is
LEMFI, URA CNRS 1504,
used to discretize locally the mixing-layer region created between the jetlike flow through
Batiment 511,
Universite Pierre-et-Marie-Curie, the gap and the main flow. An H-O-H grid is used for the computation of the main flow.
91405 Orsay, Paris, France In order to substantiate the validity of the results, comparisons with experimental
measurements are presented for the NASA_37 rotor near peak efficiency using three grids
(of I06, 2 X 106, and 3 X I06 points, with 21, 31, and 41 radial stations within the gap,
respectively). The Launder-Sharma k-e model underestimates the hub corner stall present
in this configuration. The computational results are then used to analyze the interblade-
passage secondary flows, the flow within the tip-clearance gap, and the mixing down-
stream of the rotor. The computational results indicate the presence of an. important
leakage-interaction region where the leakage-vortex after crossing the passage shock-
wave mixes with the pressure-side secondary flows. A second trailing-edge tip vortex is
also clearly visible.

Introduction fine O-grids to predict the tip-clearance flow of a turbine rotor. The
Leakage flow through the tip-clearance gap of axial-flow com- advantage of this approach is that the fine flow details within the
pressors, and its interaction with the passage secondary flow, have gap are captured correctly. The purpose of this paper is to inves-
great influence both upon performance (Denton, 1993; Storer and tigate the tip-clearance flow in a transonic compressor rotor, using
Cumpsty, 1994) and upon stability (Adamczyk et al, 1993). Many fine grids within the gap. The flow is modeled using the Favre-
computational methods try to avoid detailed computation of the Reynolds-averaged Navier-Stokes equations with near-wall low-
tip-clearance gap, either using simplified models (Storer and turbulence-Reynolds-number k-e turbulence closure (Gerolymos
Cumpsty, 1994; Puterbaugh and Brendel, 1997) for computing the and Vallet, 1996; Gerolymos et al., 1998). This turbulence closure,
leakage massflow (Denton, 1993; Adamczyk et al., 1993; Arnone, although far from perfect, is state of the art, and, when correctly
1994) or modifying the blade-tip geometry to a sharp edge, and implemented, gives adequate quantitative results. The well-known
thus joining together the grid points on either side of the blade test case of the NASA_37 rotor was chosen for this exercise (Davis
(e.g., Hah, 1986; Dawes, 1987; Jennions and Turner, 1993; Turner et al., 1993; Strazisar, 1994). Recently the team responsible for the
and Jennions, 1993; Copenhaver et al, 1993; Hah et al., 1997). experimental investigation of NASA_37 rotor raised some doubts
These methods aim to model the influence of the leakage flow on about the validity of the experimental data near the hub (Shabbir et
the mainstream, assuming that leakage massflow is the only im- al., 1997), but it is believed that experimental data are quite
portant parameter. accurate near the tip. Computations are run on three progressively
finer grids in order to assess the grid influence on solution. After
A few modern studies are interested in the numerical computa- validating the method against experimental measurements, the
tion of the flow through the gap, as part of the solution (Rai, 1989a, tip-clearance flow is studied.
b; Liu and Bozzola, 1993; Kunz et al., 1993; Basson and Lak-
shminarayana, 1995; Copenhaver et al., 1996, 1997; Hah and
Loellbach, 1999; Ameri and Steinthorsson, 1995, 1996; Ameri et
al., 1998). With the exception of Ameri et al. (1995, 1996, 1998), Flow Model and Computational Method
these studies Use very few points within the tip-clearance gap The flow is modeled using the Favre-Reynolds-averaged
(Table 1). As a consequence the fine details of the flow over the tip Navier-Stokes equations with the Launder-Sharma near-wall low-
are not captured. turbulence-Reynolds-number k-e closure. The model is well
It is often argued that very coarse resolution of the tip-clearance known and the particular implementation is described in detail by
gap is sufficient to give the overall effects of leakage and in Gerolymos and Vallet (1996). The particular form of the mean-
particular its influence on flow structure. The satisfactory predic- flow energy equation is derived by Gerolymos and Vallet (1997).
tion of tip-clearance effects on flow structure and component The method has been extended to axial turbomachinery applica-
performance are advanced as evidence that this is true (Denton, tions by Gerolymos et al. (1998). The k-e closure used was
1993; Adamczyk et al, 1993; Suder and Celestina, 1996; Puter- chosen, not only because it is as good (or as bad) as various other
baugh and Brendel, 1997). Ameri et al. (1995, 1996, 1998) used two-equation closures, but also because it has the decisive advan-
tage that the near-wall low-turbulence-Reynolds-number terms
depend only on the turbulence-Reynolds-number ReT. The result-
Contributed by the International Gas Turbine Institute and presented at the 43rd ing equations are therefore field equations completely independent
International Gas Turbine and Aeroengine Congress and Exhibition, Stockholm, of the distance from the wall n, and the normal to the wall e„,
Sweden, June 2-5, 1998. Manuscript received by the International Gas Turbine
Institute February 1998. Paper No. 98-GT-366. Associate Technical Editor: R. E.
which greatly simplifies the numerical implementation in complex
Kielb. geometries (Gerolymos, 1990). It is noted that the tensor-invariant

Journal of Turbomachinery Copyright © 1999 by ASME OCTOBER 1999, Vol. 121 / 751

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 form of the model satisfies Galilean invariance in the rotating 1 i i 1
frame-of-reference (Speziale, 1989).
Another advantage may be found in the fact that models that t
base their damping terms on the nondimensional distance from the -
wall n* = nuTv~] (where uT is the friction velocity and v„, the
~ R ® ©@@®
kinematic viscosity at the wall) are not well-adapted to the com- ! T^J !! -
putation of unsteady flows (Fan et al., 1993). inflow | [outflow
i

Concerning transition, it is a well-established fact that all exist- i 1U-TT— !!

ing models are inadequate. Nonetheless, the. model used was i 7l / i ii
developed to predict transition and relaminarization (Jones and LE TE
Launder, 1972; Launder and Sharma, 1974), and is one of the less
unsatisfactory models (Savill, 1994). The resulting method is
therefore always run with the turbulence model on, and depending
x ~
on flow conditions may return a laminar solution. This is not the
case with many other k-e models, including the model of Chien i 1 1 i
(1982), which does not perform satisfactorily in transitional flows
-0.2 -0.1 0 0.1 0.2
(Savill, 1994). The major deficiency of the turbulence model used
is that it underestimates boundary-layer separation effects (Geroly- Fig. 1 Meridional view of NASA_37 rotor and measurement stations
mos, 1990), and this should be kept in mind when examining the
computational results.
The mean-flow and turbulence-transport equations are written in 2). The space outside the tip-clearance gap is discretized using an
the (x, y, z) Cartesian rotating (relative) coordinate system, and H-O-H grid, consisting of three domains: the upstream UH-grid, the
are discretized in space, on a structured grid, using a third-order O-grid around the blade, which is generated biharmonically
upwind-biased MUSCL scheme with Van Leer flux-vector-splitting (Gerolymos and Tsanga, 1999), and the downstream DH-grid. The
and Van Albada limiters, and the resulting semidiscrete scheme is tip-clearance gap is discretized using an o-grid (TC-grid), also
integrated in time using a first-order implicit procedure. The mean- generated biharmonically (Gerolymos and Tsanga, 1999). The
flow and turbulence-transport equations are integrated simulta- TC-grid is stretched both near the casing and at the blade tip, in
neously. Source terms (centrifugal, Coriolis, and k-e) are treated order to describe correctly the boundary layers. To the authors'
explicitly. The local-time-step is based on a combined convective knowledge this is rarely the case in published studies. A jetlike
(Courant) and viscous (von Neumann) criterion (MacCormack, structure emanates from the tip-clearance gap on the sucjion-side
1982). For steady turbomachinery computations, CFL = 20 local- of the blade and mixes with the main flow. This creates a mixing-
time steps, give good convergence and ensure stability in all the layer, which persists far away from the blade. One possibility to
cases studied. Further details on the computational method are compute this part of the flowfield accurately is to stretch the blade
given by Vallet (1995), Tsanga (1997), and Gerolymos et al. o-grid in the same way. This would require a radially refined grid,
(1998). even at regions away from the blade where this is not necessary. It
was therefore prefered to use the oz-grid (Fig. 2) patched within
the o-grid of the blade. This grid is stretched both at the casing and
Comparison With Measurements at the blade tip, thus accurately describing the mixing of the
In order to validate the computational method, a systematic leakage flow with the passage flow. The blade o-grid is thus
comparison with experimental data for NASA_37 rotor (Fig. 1) was stretched only near the casing and the hub, independently of the
undertaken. This configuration has been computed by numerous tip-clearance gap. This has the additional advantage of being able
authors (Denton, 1996). to change the grid locally, only at the neighbourhood of the blade
Most of these computations used simplified models for the tip if computations with different tip-clearance sizes are run. All
clearance, and a mixing-length turbulence model. Suder and Celes- stretchings of the grid near the solid walls are geometric. Details of
tina (1996) studied the interaction of the leakage flow with the the grid-generation procedure are given by Gerolymos and Tsanga
main stream, both experimentally and computationally, using the (1999). The computational method structure and the boundary
Navier-Stokes solver described by Adamczyk et al. (1990), with a conditions between domains are described by Gerolymos et al.
mixing-length closure, and a simplified model for the flow within (1998).
the tip-clearance. Hah and Loellbach (1999) presented computa- Computations were run on three different grids (Table 2). By
tions for this configuration using the near-wall k-e closure of today's turbomachinery CFD standards even the coarsest grid_B is
Chien (1982). These authors descretized the tip-clearance gap a rather fine grid (Denton, 1996; Hah and Loellbach, 1999). It has
using ten radial stations and ten points tangentially across the blade 65 radial surfaces in the flowpath, and 21 radial stations in the
(Table 1). tip-clearance gap (note that the radial distributions in the flowpath
The grid used in the present work consists of five domains (Fig. o-grid and in the tip-gap TC-grid are independent). A very impor-
tant parameter of grid quality is n* (n+ of the first node away from
the wall), which to give grid-independent results at transonic flow
Table 1 Computational grids used within the tip-clearance gap
conditions (Vallet, 1995) must be < \ (note that this must be
verified after the computations). For grid_B this condition is sat-
grid-type blading-type radial* across'
Rai (1989a, 1989b) o turbine rotor 5 21 isfied on the blade surfaces, but not on the casing and the hub
Liu and Bozzola (1993) H turbine cascade* 8 7 where n* ~ f. This is also the case on the blade tip. To improve
Kuoz et al. (1993) H compressor cascade' 11 17
Basson and Lakshminarayana (1995) H turbine cascade' 15 21
this situation, grid_c has 101 radial surfaces in the flowpath, and
Copenhaver et al. (1996) H compressor rotor 10 10 31 radial stations in the tip-clearance gap. The resulting grid has
Copenhaver et al. (1997) H compressor rotor 6 7 n* ~ X on the casing and the hub, and is satisfactory on the blade
Hah and Loellbach (1999) H compressor rotor 10 10
Amerl and Steinthorsson (1995, 1996) O turbine rotor 20-40 33 tip. The final grid_D has 161 radial surfaces in the flowpath, and
Ameri et al (1998) 0 turbine rotor 20-40 115 satisfies n* < j everywhere. Within the tip gap it has 41 points
present; grid-B 0 compressor rotor 21 21
present; grid_c 0 compressor rotor 31 33
less stretched than the 31 points of grid_c. Note that the three grids
present; grid_D 0 compressor rotor 41 33 have almost identical blade-to-blade grids and differ mainly in the
' points radially from the blade-tip to the shroud radial refinement.
* points tangentially across the blade
' linear non-rotating cascade Computed and measured performance for the NASA__37 rotor
(Fig. 3) were evaluated from the computational results between

752 / Vol. 121, OCTOBER 1999 Transactions of the ASME

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 casing

*«s
hu

UH-grid

Fig. 2 Computational grid for NASA_37 rotor (grid_D)

stations 1 and 4 (Fig. 1) following exactly the averaging proce- rotor. Although it is not certain whether this was the case in the
dures used in the experiment (Davis et al., 1993). Note however experiment, it might explain why all computational results fail to
that, in disagreement with Denton (1996), mass-averaging over all reproduce this deficit in pm (Denton, 1996). The present authors
grid points does not significantly change the efficiency 17^. All the believe that the deficiency of the computations is presumably a
computations were run until full convergence of massflow in, combination of these two effects. Note also that the finest grid
total-to-total pressure-ratio 7rT.T and loss. In all the cases residuals gives the fullest total pressure distribution at the hub. The differ-
were reduced by three orders of magnitude. Turbulence intensity at ence in computed and measured flow-angle is due to the difference
inflow was T„ = 3 percent. Computations overestimate TTT.T and in massflow (Hah and Loellbach, 1999).
underestimate -r/,s. Note that grid_D computes slightly higher mass-
Results obtained using the three grids are in quite good agree-
flow than grid_B and grid_c. Detailed comparisons are presented
ment, except at the tip. Refining the radial grid resolution and the
" for point A, corresponding to 98 percent of choke massflow mCH-
In order to correct for the differences in operating diagram be- tip-clearance grid reduces the total temperature values near the tip,
tween computation and experiment, the corresponding computed improving the prediction with respect to measurements. This is a
point is taken at the same value of linr^ (Fig. 3), corresponding significant result because it shows that the model used for the
to roughly the same work done by the rotor. The resulting mass- tip-clearance flow has an impact on work and efficiency.
flow difference is within measurement accuracy of ±0.14 kg s"1 Detailed comparisons of computed relative Mach number M„lBi,d
(Strazisar, 1994). (Fig. 5) and absolute flow-angle a,, (Fig. 6) show state-of-the-art
agreement with measurements. Note that plotted quantities were
Comparison of computed and measured pitchwise-averaged
quantities (Davis et al., 1993) at stations 1, 3, and 4 (Fig. 4) show calculated using only the x and 6 components of velocity, and
satisfactory agreement except concerning the pIM deficit near the assuming rothalpy conservation, as was done for the experimental
hub at station 4. Hah and Loellbach (1999) attribute this deficit to values (Davis et al, 1993; Strazisar, 1994). Although the wake
important corner stall, which could have been underestimated by depth is overestimated as in all published results (Denton, 1996),
the present k-e model. Shabbir et al. (1997) argue that this is due the prediction of the flow at 20 percent axial-chord \s is quite
to massflow leakage emanating from a small gap between the satisfactory, giving the correct M , level behind the shock. The
stationary and rotating parts of the hub flowpath upstream of the experimental shock-wave seems more smeared at 95 percent span
than the computed one, which might be due to both unsteady
effects in the experimental flow and to a slightly underestimated
Table 2 Summary of computational grids clearance gap in the computations, where the nominal clearance
UH" 0» DH" Trji ozi points 1 "i
STC = 0.356 mm (Strazisar, 1994) was used. Comparison between
grid-S 49x41x 65 201x45x 65 81x61x 65 201x11x21 201x21x31 1 149 421 < 1.5 the three grids used shows that these results are indeed grid
grid_c 49x41x101 201x53x101 81x61x101 201x17x31 201x21x41 1 955 587 < 1.0
grid-D 49x41x161 201x53x161 81x61x161 201x17x41 201x21x61 3 067 042 <0.5 independent, contrary to what happens above 95 percent span.
* anialxlangentialxradial
' around the bladexaway from bladeXradial
A better understanding of the radial penetration of the tip-
' without 3-grid points ove flapped by the o& grid clearance flow is given by considering relative Mach-number

Journal of Turbomachinery OCTOBER 1999, Vol. 121 / 753

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 0.95
! 1 ! !
0.925 -

0.9
+ +f + experiment
0.875 + a grid_B
i i e i Fxri^f X grid„c
0.85 ;+ :

'
A grid-P

0.825
r°r°"Tl i

'
0.8 i i i 'i m (kg E
19 19.2519.519.75 20 20.2520.520.75 2 1 -

2.3 I 2.3 i —
2.25 2.25 - •
2.2 2.2 -••• ~ax&g
' H a g £
2.15 2.15 -.
+
2.1 ""«'" + .t.. + .. . . I 2.1 - -
• % „

2.05 ^ TT.T 2.05 —•

2
1.95
- | """ J t^- 2 -
1.95 -••
...%.

1.9 1.9 -••

1.85 1.85 -•
1.8
i 1
'(kg r ) 1.8 -
19 19.5 20 20.5 21— 8 8.5 9 9.5 10 10.5 11 •

Fig. 3 Computed and measured operating map for NASA_37 rotor (point A: mirf-l = 9.85 kg s" 1 )

contours Mw at 95 percent span, 98 percent span, and 99 percent Tip-Clearance and Secondary Flows
span (Fie. 7). The tip vortex is clearly seen at 98 percent span and , , ., , . . .
„>. ' ,-, ., i j i • » »• • A quite good representation of leakage anow and its interaction
99 percent span, while the low speed clearance interaction region ., . , f , „ . , . r ,
wlth
is visible also at 95 percent span. mterblade-passage flow is obtained through entropy contours
In conclusion, the method gives satisfactory results over the (Copenhaver et al, 1996). Levels of entropy
upper 70 percent of the span. The discrepancy near the hub does
not seem to influence the flow at the tip, which is the main theme w_ _ , •* > _ „ , P< ,,.
of the present work. * SlSA Cp
288.15 s
101325 {>

3. 0.1067 m

*»s

— —_ — n —
gridJD \
grid„c

+
gridJ3
experiment
0.65 0.7 0.75 0.8 0.85 0.9 0.05
i
-0.0419 m x = 0.0457 m X = 0.1067K1
=•»* • — '
:? . 1

r? t +

i? s<
si!
(f? < •

o.a o.a
R — Rhub \ o.e 0.6
.c i
Rtip — Rhub 0.4 0.4
% 02

I
0.2

r
I 32 34 36 38 40 42 44 46 48 50
-2 0 2 4 6 8 10 40 42 444648(30525455581

R - Rhub
S^Z
Rtip ~ Rhub

TtM

0.98 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 1.15 1.2 1.25 1.3 1.35 1.2 1.25 1.3 1.3

R — Rhub
^ t( —*w

hj>
{S<
—
iia
Rtip ~ Rhub
< s s 1
• *

N
PtM

0.8 0.85 0.9 0.95

- r 1 1.05 1.1 1.15 1.2 1.9 1.95 2
ft
2.05 2.1 2.15 2.2 2.25 2.3 1.9 1.95 2
rr-l y\
2.05 2.1 2.15 2.2 2.25 2.3
Pi IS.

Fig. 4 Computed and measured radial distributions of pitchwise-averaged flow-angle «„„,,, total-pressure
p,M, and total-temperature Tm, for NASA_37 rotor (mwf.r = 9.85 kg s - 1 ; 7"„ = 3%; STC = 0.356 mm)

754 / Vol. 121, OCTOBER 1999 Transactions of the ASME

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 v? + w*
Mv,
(7-I)['K,S, -|(i4 2 + if) + nw«]

grid_D
grid_c
" * J»
4- experiment

20% x X ~ = 0.0457 m x = 0.1016 m

m *
85
fl*
86
87
m
%
':•/'
*'Z
:*.:*+* c< 95% span
89
90
TS. 1 '
91

ft*
A
I Mw
nn
0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8 1 1.2 0.4 0.6 0.8 1 13

f. A

.... \
1
80
* ;. i
•

^ <kL
82
90% span
84 IF
86 |
tt....„,
1.2 1.4
m

1.6
- 68

0.2 0.4 0.6 0.8 i 1.2 1.4 0.4 0.6

I
0.8
*•*

1 1.2 1,4
Af„

70% span

0.6 0.8 1 1.2 1.4 1.6 0.2 0.4 0.6 0.8 1 1.2 1.4 0.4 0.6 0.8 1 1.2 1.4
M„

-- |
T
-&»3*W^, y!-€&
80 80

\
1
82 82
50% span
<\
I
84 84

86
| —I -
i Zi
0.2 0.4 0.6 0,8 1 1.2 1.4
88
1
0.4 0.6
*
0.8
I **»«,-

1
f 1
«c^
/
By f 30% span

"II 1
•1- M,
0.2 0.4 0.6 0.8 1
|
1.2 0.4 0.6 0.8 I 1.2 1.4

Fig. 5 Comparison of computed and measured pitchwise distributions of M„ x o a d for NASA„37 rotor (mirf.^
9.85 kg s 1 ; Tu = 3 percent; 8TC = 0.356 mm)

where s is the entropy, slSA the entropy at standard conditions flows (Fig. 8). The projection on the axial planes of the relative
(which are the inflow conditions for the present case), cp the velocity W variation around its pitchwise mass-averaged value WM
specific heat capacity at constant pressure, Rg the gas constant, f,
the total temperature, and p, the total pressure, are presented (Fig. W,„ = (WR - WRJeR + (We - WeJee (2)
8) at various axial (x = const) locations (Table 3). The flow that
is plotted at planes B, D, and G (Fig. 8). At plane B the leakage
leaks over the blade in the leading-edge (LE) neighborhood (plane vortex emanates from the gap. At plane D |his vortex has reached
B) mixes with the passage flow and creates a high-entropy region the shock-wave, which is identified in the Wm plots as a region of
that is convected toward the middle of the passage (plane c) until important radial flow migration. At plane o a large secondary flow
it reaches the shock-wave (plane D). Note that the shock-wave (sw) structure is observed, extending from the shock-wave to the
is clearly seen as an entropy jump (Fig. 8). Then this high-entropy pressure-side, corresponding to the second high entropy region
region traverses the shock-wave (plane F) and reaches the pressure- (Fig. 8). At plane i (Fig. 8) the high-entropy region is more diffuse,
side (PS) of the next blade (plane G). At this point two regions of with lower maxima as compared to upstream planes. This is
high entropy, one emanating from the tip-clearance gap and ex- mainly due to mixing by the secondary flows.
tending to midpassage (concentrated near the casing), and another An alternative sensor to entropy is turbulence kinetic energy k, that
from midpassage to the pressure-side (with greater spanwise pen- surprisingly has not yet been exploited by authors that solve an
etration) exist (Fig. 8). equation for k (Hah, 1986; Copenhaver et al., 1993; Jennions and
These high-entropy structures can be identified with secondary Turner, 1993; Turner and Jennions, 1993; Suder and Celestina, 1996;

Journal of Turbomachinery OCTOBER 1999, Vol. 121 / 755

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 sin a r j = •
+ v, yf^+vf

grid_D
gridx
grid-B
+ experiment

a: = 0.1016 m

95% span

-10 0 10 20 30 40 50 60 70 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70

!
' \
-~-zdr--
,»***«•"- — 70% span
- T ff|- |

c i i L
Ii I I
-10 0 10 20 30 40 50 60 70 10 20 30 40 50 60 70 80 00 0 10 20 30 40 50 60 70

I
****-< .*•
i i

50% span

31 E i
l.j
i

i
-10 0 10 20 30 40 50 60 70 JO 20 30 40 50 60 70 SO 90 0 10 20 30 40 50 SO 70

f
'

W
80 BO

-f
82

M ) > • >
82

84
IT1 30% span

86

1 ! 'I 88
88

•10 0 10 20 30 40 SO 60 70 10 20 30 40 50 60 70 80 90
1 t
0 10 20 30 40 50 60 70

Fig. 6 Comparison of computed and measured pitchwise distributions of ax0 for NASA_37 rotor (//TJTT.T = 9.85
kg s" 1 ; 7"„ = 3 percent; STc = 0.356 mm)

Copenhaver et al., 1996, 1997; Shabbir et al., 1997). Examination of the leading-edge) and J (near the trailing-edge) indicate the pres-
fc-levels at various axial planes (Fig. 9) shows that the same informa- ence of a highly three-dimensional vortical structure as the flow is
tion is contained in k as in s. The close-up offe-levelsnear the blade accelerated over the tip on the pressure-side (Fig. 10).
tip shows how k is produced by the mixing ofleakage flow emanating The use of a fine grid in the tip-clearance-gap (grid_D has 41
from the tip-clearance gap. Note that at plane i, near the trailing-edge radial surfaces, and a total of 140,097 points within the gap) makes
(TE) air comes over the tip from the green region of k, with almost it possible to resolve the boundary layers on the blade-tip and on
unchanged turbulence kinetic energy until it leaks on the suction-side the casing, and to plot velocity profiles inside the tip-clearance-gap
and mixes with the passage flow. The high values of k (Fig. 9) and s (Fig. 11). In the relative frame of reference the blade-tip is fixed
(Fig. 8) near the suction-side trailing-edge are indicative of boundary- (W = 0) and the casing rotates at We = —Q,R (in the present case
layer separation. O > 0 so that the relative velocity of the casing is in the negative
Plots of Mw in the blade-tip region (Fig. 10) indicate that for 6 direction). Radial profiles of the three components of the relative
most of the blade chordwise length (planes A-H) flow acceleration flow velocity, W„ WR, W0 are plotted for various positions from
from the pressure-side over the tip is supersonic through a Prandtl- the pressure-side (0 percent thickness) to the suction-side (100
Meyer expansion fan. The M„ contours clearly indicate the mixing percent thickness), at three axial planes A, G, and J (Fig. 11).
layer formed by the leakage flow emanating over the suction-side At plane A the flow is largely tangential (Fig. 11), with a
blade-tip. At plane G the leakage flow terminates in a low Mw turbulent Couette flow profile (Schlichting, 1979), between the
region, corresponding to the leakage-interaction region (Fig. 10). entraining casing and the fixed tip. At the pressure-side edge, near
The relative velocity vectors over the blade-tip at planes A (near the tip, there are high radial velocities, as the flow is radially

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 Leakage Flow Structure
Several authors (Adamczyk et aI., 1993; Suder and Celestina,
1996; Copenhaver et aI., 1996; Puterbaugh and Brendel, 1997)
have studied the structure of leakage flow in transonic compres-
sors, and attributed the growth of the vortex an(l the creation of the
high-entropy, low-speed region to the interaction of the vortex
with the shock-wave. The interaction of the streamwise vorticity
with the shock-wave creates an axial velocity deficit in the vortex
core (Smart and Kalkhoran, 1997). However, examination of the
flow structure (Fig. 8) raises the question whether this effect is not
simply an additign to the well-known subsonic compressor
leakage-interaction region. In this case Lakshminarayana et al.
(1995) have suggested that the interaction of the leakage vortex
with the passage secondary flows creates a region of low velocity
and high entropy, which they called leakage-interaction region. In
the present transonic flow, ii'om the leading-edge to plane D, the
shock-wave is seen to separate the leakage-flow from the passage
secondary flow occurring in the region between the shock-wave
and the pressure-side (Fig. 8). When the leakage flow crosses the
shock-wave, it interacts with the passage secondary flow, creating
the leakage-interaction region. It is plausible to assume that there
is a direct analogy between the subsonic and transonic case, and
that the shock-wave acts as a barrier between the pressure-side
secondary flows and the leakage flow. When this barrier is crossed,
the leakage-interaction region is created in much the same way as
in the subsonic flow case (Lakshminarayana et aI., 1995), with the
additional effect of vortex growth and axial velocity deficit due to
the interaction with the shockwave.
In order to understand the leakage flow structure better, stream-

?
P8% span
lines near the blade tip are plotted (Fig. 12). Examination of the
relative flow streamlines cOIning from upstream and going into the
passage at 95 percent span (green streamlines) and at 98 percent
!
. ~ ... ......... """""
",..
span (red streamlines) shows clearly the secondary flows near the
pressure-side in the leading-edge neighborhood. Because of the
three-dimensional structure of the casing boundary layer (and the
relative motion between the casing and the blades) the incOlning
flow direction differs greatly between 95 percent span and 98
percent span. At 95 percent span, as the flow goes into the passage
it is deflected by the blades (green streamlines). At 98 percent
span, the streamlines near the pressure-side of the blade go straight
(and slightly upward) into the clearance gap, feeding the leakage
flow of the next channel for the first half of the blade (red
streamlines). This fluid crosses the gap and reappears on the
suction-side of the next blade creating the leakage vortex in the
leading-edge neighborhood (blue streamlines near the LE). It is
noticeable that this vortex continues in much the same direction as
the incOlning flow feeding the gap (blue streamlines forming the
vortex are almost parallel to the incOlning red streamlines). The
vortex is formed both from leakage fluid and from fluid entrained
Fig. 7 Computed M,., at various spanwise stations near the blade-tip for from the suction-side boundary-layer near the tip. The vortex
= =
NASA_37 rotor (rilrrT'~ 9.85 kg S-1; T u 3 percent; I)TC 0.356 mm) = moves forward in this direction (Fig. 12) until it meets the
pressure-side flow, that has been deflected by the blades. At this
point the vortex mixes rapidly with the pressure-side flow, turns
accelerated over the tip. Inside the gap, at station A radial velocities sharply in its direction, and grows substantially. This is the
are negligible, while axial velocities change sign from negative leakage-interaction-region (Lakshminarayana et aI., 1995) corre-
sponding to low velocity and high loss. It is important to note how
near the tip at the pressure-side to positive further inside the gap.
the vortex turns sharply to follow the direction of the blades. This
The blade-tip boundary-layer thickens substantially inside the gap.
sharp turning of the vortex trcYectory is not due to the shock-wave,
The vortical flow structure observed at plane A (Fig. 10) is asso-
which is almost normal to the flow for the present configuration. It
ciated with the variations of lV, and lVR' At plane G, near mid- occurs after the vortex has crossed the shock-wave. It is noted that
chord, the lVo flow structure is again a Couette-flow with pressure- the vortex trajectory calculated by Suder and Celestina (1996) for
gradient, while there are important variations of lV, (Fig. II). Note the same configuration is quite the same as the one computed in the
that these planes are x = const planes, while the flow across the tip present work (although these authors credited the shock-wave/
makes an angle of about -110 to -120 deg in the x-I.l plane with vortex interaction theory with explaining the important radial
the x direction. Again the important variation of lV., across the gap penetration of the leakage flow). In the present computations we
near the tip is associated with the three-dimensional flow structure. did not remark any noticeable modification of the vortex size or
At plane J the lVe and lV, profiles indicate that the tip boundary- direction when it interacts with the shock-wave.
layer undergoes three-dimensional separation and reattachment Examination of the flow after the sharp turning of the vortex
(Fig. II). near the pressure-side shows that for the second half of the blade,

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 -.:O=- --,B

-.:D=- -I-----, D

shock wave

~G~_ _____lf-+-___,G

D
G

Fig. 8 Entropy levels and secondary flows In the interblade passage of NASA_37 rotor (rim;.} = 9.85 kg s -';
Tu = 3 percent; liTe = 0.356 mm; grid_D)

it is the leakage vortex of the previous blade that feeds the leakage The leakage flow structure between these two vortices is im-
flow through the gap (blue streamlines going into the gap). The portant because it presumably explains the presence in the entropy
flow leaking out of the gap on the suction-side near the trailing- contours (Fig. 8) of a high-entropy region concentrated near the
edge is almost parallel to the leakage vortex direction near the tip, between midpassage and the suction-side of the blades (plane
pressure-side (after the leakage vortex has turned). This flow, G). In this region the leakage flow and the suction-side boundary-
together with the suction-side boundary-layer that it entrains near layer fluid it entrains move almost tangentially toward the neigh-
the blade-tip, forms a second trailing-edge tip-vortex, clearly seen boring blade (Fig. 12). The high-entropy fluid of the suction
in the plots (Fig. 12). Note that Suder and Celestina (1996) boundary-layer mixes with the flow leaking from the gap to form
observed such a second vortex in their calculations of the NASA_37 a high entropy region (Fig. 8). The streamlines, although this is not
rotor at 60 percent design speed, but did not report its presence also seen in the plot, have at this station (plane G) a marked tendancy
at design speed. In our case however, this second TE-tip-vortex is to move radially upward, thus concentrating the high-entropy fluid
formed both by leakage fluid and by fluid moving radially upward near the casing (Fig. 8). This movement might be enhanced by the
in the suction-side boundary-layer. interaction between the shock-wave and the fluid near the casing.
Note, however, that the casing is moving in the relative frame of
Table 3 Axial planes location reference, in the same direction as the leakage flow, and as a
planeABCDEFGH I J KL consequence this is neither a classical shock-wave/boundary-layer
x (m) 0.007 0.008 0.010 0.012 0.013 0.015 0.020 0.025 0.030 0.033 0.040 0.042
interaction on a fixed wall, neither a shock-wave/vortex interac-

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 LE two axial chords downstream of the trailing-edge. This renders the
tip-clearance zoom test case more difficult, since the complex mixing downstream of
the rotor directly influences the comparison. This is a flow relaxing
after having sustained the pressure gradients within the rotor,
precisely the type of flow where the k-E model is at its worse
c (Gerolymos, 1990; Gerolymos and Vallet, 1996, 1997). At plane L,
which is situated just before the trailing-edge near the hub, and just
F after the trailing-edge for the upper 60 percent span, plots of !VI"
and k (Fig. 13) indicate both the downstream evolution of the
G tip-clearance flow and the important boundary-layer separation on
the suction-side between 40 and 80 percent span. On the contrary,
the present computations underestimate the corner stall computed
H by Hah and Loellbach (1999) near the hub suction-side trailing-
edge. This is presumably a deficiency of the Launder-Sharma
(1974) k-E model, which seems to underestimate flow separation
compared to the Chien (1982) model used by Hah and Loellbach
(1999).
There is important mixing of the flow downstream toward
station 4 (Fig. 14). Spanwise mixing of the blades' wakes and the
D tip-clearance flow form a large secondary flow region that fills the
C upper 30-40 percent of the flowpath. The flow mixing is compli-
D cated by the important acceleration near the hub due to the flow-
F
path convergence (Fig. 1).
G
Conclusions
A three-dimensional compressible Navier-Stokes solver with
near-wall low-turbulence-Reynolds-number closure was used to
study the flow with tip-clearance in a transonic compressor rotor
(NASA_37). Comparison of computational results with measure-
ments is state of the art, with the discrepancy in the P 1M distribution
near the hub that is observed in most computations of this test case.
This discrepancy is attributed to underestimation of hub corner
Fig. 9 Turbulence kinetic energy in the interblade passage of NASA_37
rotor (riIlTi'.\ = 9.85 kg 5-'; Tu = 3 percent; lite = 0.356 mm; grid_D)
stall by the turbulence model, and perhaps leakage problems in the
experimental setup. Comparison of computations and measure-
ments near the tip is satisfactory. Results using three different grids
tion. It is believed that this second high-entropy region has not an of 10 6 , 2 X 10 6 , and 3 X 10 6 points, with accurate resolution of the
important effect on the general leakage flow structure. flow within the tip-clearance-gap, demonstrate that the results are
grid independent everywhere, with the exception of the T'M distri-
bution near the tip, where grid refinement substantially improves
Downstream Flow Mixing the agreement between computations and measurements.
The rake measurements used for comparison with computations Investigation of the flowfield inside the interblade-passage, near
for the NASA_37 rotor were obtained at station 4, located more than the tip, leads to suggesting an alternative scenario to the leakage

casing
I
....... '."."..~ ....... ~'_.-_--:..---~~

pressurc-.ide

Fig. 10 Relative Mach number levels and clearance-gap relative flow in the tip-region of NASA_37 rotor (riIni'J =
9.85 kg 5-'; Tu = 3 percent; liTC = 0.356 mm; grid_D)

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 Bp - B 0.0004
0.0004 c:: )
Bs - Bp i !
0.0003 0.0003
0%
____. 30% 0.0002
50%
100% 0.0001
\ 0.0001 r·!.·,!,.!1·\ I.···\···········\"· -t
f'..,
c"'\ ~-- '" . ",.""."
,so1:-:0-.4~00::="".:.30:':'0~.20~O:""'.~"~O'""'"'"!

,
o
-200 -150 -100 .50 D· 50 100 150 200 ·100 -50 0 60 100 150 200 250 300 350

0.0004

0,0003 ....

0.0002

0.00(11

f,
o
-400 -300 ·200 -100
'" -500

0.0004 , - - - - , - - - , - - - - , 0.0004 , - - , - - . , - - , - - - , 0._ ,----r-..,-,--,--,

0.0003 ... -_.....

t 0.0003

R- R'ip 0.0002 0.0002 0.0002 r j !\\ .. · .. i· .. j -t

(m)
0.0001 0.0001

A--------
O"''::--d=:::;.......=----:-!'OO
..
W. (m 8,,1)
Fig. 11 Relative velocities inside the tip-clearance gap for NASA_37 rotor (rilTl"T.} = 9.85 kg S-1; Tu =3
percent; Ilrc = 0.356 mm; grid_D)

flow interaction with the passage shock-wave. It is believed that in downstream, the leakage flow crosses the shock-wave, it interacts
the fore part of the blade the shock-wave surface separates the both with the shock-wave and with the pressure-side secondary
leakage flow from the secondary flows between the shock-wave flows generating a leakage-interaction region of low speed, high
and the pressure-side of the blade. When, as the flow progresses entropy, and high turbulence, in a way analogous to the subsonic

TE
j

TE-tip-vortex

TE- ti p- vortex

Fig. 12 Computed relative flow streamlines near the blade·tip for NASA_37 rotor (blue streamlines coming
from the lip-clearance gap and the suction-side near-tip boundary-layer; red streamlines coming from
upstream at 98 percent span; green streamlines coming from upstream at 95 percent span; ril1fT.} 9.85 kg =
s-'; Tu = 3 percent; Ilrc = 0.356 mm; grid_D)

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 Detailed velocity profiles inside the tip-clearanCe-gap are ob-
tained (to the authors' knowledge for the first time in a transonic
compressor) with sufficient resolution of blade-tip and casing
boundary-layers. The flow is a Couette-type flow, between the tip
and the casing. Flow acceleration over the blade-tip is associated
with a three-dimensional vortical structure, while important tip
boundary-layer thickening occurs within the gap. Both these ef-
fects are associated with flow blockage inside the tip-clearance
gap. It is believed that such computations will improve understand-
ing of the flow inside the gap, and as consequence will help in the
development of simple models to compute leakage massflow and
losses.
For the particular test case studied (NASA_37 rotor) rake mea-
surements were taken far downstream of the trailing-edge. The
region of important flow mixing between the blades' trailing-edge
and the measurement station is particularly difficult to compute
using classical turbulence models, which invariably fail to cor-
rectly reproduce turbulent flow relaxation after separation. This
may also partially explain why this test case is so difficult to
predict accurately.

Acknowledgments
The computations presented were run at the Institut pour Ie
Developpement des Ressources en Informatique Scientifique (!D-
RIS), where computer resources were made available by the Comite
Scientifique. This work is part of the turbo_3D project of the
Laboratoire d'Energetique, Universite Pierre-et-Marie-Curie. The
support of the Region Ile-de-France through a SESAME Grant is
acknowledged. Authors are listed alphabetically.

Fig. 13 Flow structure near the trailing-edge of NASA_37 rotor (mnY.~ =

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