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Copyright 0 1999 by ASME
1111111 111E110,111 11111 11
PREDICTION OF TRANSITIONAL HEAT TRANSFER CHARACTERISTICS
OF WAKE-AFFECTED BOUNDARY LAYERS
Kyounglin Kim and Michael E. Crawford
Mechanical Engineering Department
The University of Texas at Austin
Austin, Texas 78712
ABSTRACT
undisturbed steady freestream velocity at the boundary
The presence of wake-passing in the gas turbine environment
significantly modifies the heat transfer characteristics on the
downstream blade surface by causing wake-induced transition. In this
study, time-dependent boundary layer calculations were carried out
using a model for wake-induced transition based on a prescribed timedependent intermittent function. The model is determined from the
well-known turbulent spot propagation theory in a time-space diagram
and from experimental evidence in the ensemble-averaged sense.
Time-averaged heat transfer distributions are evaluated and compared
with experimental results for different flow and wake-generating
conditions over a flat plate. Comparison showed that the present timedependent calculations yield more accurate results than existing steady
superposition models.
NOMENCLATURE
a, b, w
a,
aLE , all
to the local freestream velocity
ratios of propagation velocity of leading and trailing
edges of turbulent strip to the local freestream velocity
wake-passing frequency
shape factor (= 8 . 10 )
characteristic length
Reynolds number
reduced wake-passing frequency (=JL 1 U.)
St
Stanton number
wake-passing period (=1/f)
time
turbulence level
Tu
,0
(=Ue (x) W(x,t))
inlet velocity
periodic fluctuation of ensemble-averaged velocity
( = 0-0 )
•
Ud
normalized velocity defect (= (U - Uk„, ) /C!, )
Unns
rms velocity of periodic fluctuation
W(x,r)
relative fluctuation function for freestream velocity
defect
streamwise distance
x,,,
constants for the model of fntestream velocity defect
ratio of traveling velocity of the center of turbulent strip
Re
layer edge
wake-disturbed unsteady freestream velocity
flint-mean and ensemble-averaged velocity
onset of wake-induced transition
normal distance from the wall
S.
rxx.0
V
displacement thickness
momentum thickness
time-dependent intermittent (or transitional) function
Stanton number based time-averaged intermittency
kinematic viscosity
duration of turbulent snip
Subscripts
L, lam
T. hub
Zn
101
freestream
laminar
turbulent
time-mean
steady
wake
total
Presented at the International Gas Turbine & Aeroengine Congress & Exhilsdion
Indianapolis, Indiana — June 7-June 10, 1999
This paper has been accepted for pubkation in the Transactions of the ASME
Discussion of a will be accepted at ASME Headquarters until September 30, 1999
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INTRODUCTION
Periodic wake-passing from upstream blade rows in the gas
turbine flowfield can strongly influence the boundary layer and heat
transfer characteristics on the surfaces of the downstream blades. This °
form of flow unsteadiness, which is caused by the relative motion of
adjacent blade rows, affects the unique transition process on the blade
surface, which is known as wake-induced transition. Since the optimal
blade design heavily relies on accurate heat transfer analysis of the
blade surface, it is important to predict the unsteady boundary layer
development due to the wake-induced transition.
A number of the measurements on wake-affected boundary layers
have been recently conducted in laboratory simulations using rotating
wake generators (Pfeil and Herbst, 1979; Pfeil et al., 1983; Dullenkopf
et al., 1991; Liu and Rodi, 1991; Orth, 1993; Chaldca and Schobeiri,
1997; Funazaki et al., 1997) and in a turbine-compressor environment
(Halstead et al., 1997). The experimental results showed that the
unsteady wakes produce an incident flow that has two distinctive
characteristics: a freestream velocity defect, and its associated high
turbulence during the wake-passing. Orth (1993) and Halstead et al.
(1997) concluded that the high turbulence of the incoming wakes,
rather than the level of freestream velocity defect, is primarily
responsible for initiating the wake-induced transition.
Figure 1 shows the general concept of wake-induced transition.
While the wakes pass over the surface approximately at the speed of
the freestrearn, high turbulence in the wakes penetrates into the
laminar boundary layer and initiates turbulent strips or spanwise
coalescence of turbulent spots. These turbulent strips convect
downstream and grow, due to different propagation speeds of their
leading and trailing edges. The passing wakes appear only to trigger or
initiate the starting location of the turbulent snips, and the wakes do
not affect the turbulent strips during their downstream development
This has been described by Orth (1993), and his experimental evidence
backs up the separate treatment of these two flow phenomena by
ignoring the interaction between the freestream velocity defect and the
turbulent strips, downstream of initiation. Thus, only the effect of
strong turbulence in the periodic wake passing is considered in the
model of wake-induced transition developed herein, while the effect of
periodic freestream velocity defect from the wakes is neglected.
The literature contains several different approaches for predicting
the transitional boundary layers due to wake-passing. For timeaveraged results, time-averaged intennittency models were proposed
high Tu
time
trailing edge
(- 0.SUJ
leading edge
(- 0.88l1.)
1
treestream
velocity detect
(- 1_1.)
StreeMwise distance
Figure 2. Convection of turbulent strips and freestream velocity
defect due to the wake-passing.
by Mayle and Dullenlcopf (1990, 1991), Hodson et al. (1992), and
Funazalci (1996). In contrast, time-accurate calculations have been
carried out by Iran and Taulbee (1992), Cho et al. (1993), and Fan and
Lalcshminarayana (1996) by solving the unsteady boundary layer
equations or the Navier-Stokes equations in conjunction with low
Reynolds number k-e models.
Recently, Kim and Crawford (1998) introduced a model for
wake-induced transition that reproduces the time-accurate
development of the wake-affected boundary layer. They used an
unsteady boundary layer scheme and a simple engineering model for
the wake-induced transition based on the theory of turbulent spots.
Their computational results were compared with various measured
time-resolved and time-averaged boundary layer profiles and
parameters. In the present paper, transition model is revised based on
the measurements of the ensemble-averaged intermittency by Chaldca
and Schobeiri (1997), and the transitional heat transfer characteristics
in the wake-affected boundary layers are predicted and compared with
the measured data and the predicted results of the time-averaged
superposition model.
MODEL FOR WAKE-INDUCED TRANSMON
To compute periodically unsteady transitional boundary layers, a
conventional eddy viscosity formulation (Hodson et al., 1992) is
adopted to estimate the effective viscosity,
wake passing - Uca
(x, y, =v
+ r(x,t)v
(x, y,t)
(I)
where vim, is the molecular viscosity, and v aa.b is the turbulent eddy
viscosity. The transitional function T(r,r) is considered in an
ensemble-averaged sense, and thus it is different from the
conventional steady state intermittency or instantaneous intermittency.
In the experimental work by Chakka and Schobeiri (1997), they
measured the ensemble-averaged intermittency as
formation of
turbulent strip
0.88U.
1 f
i
.17=—Nziu0i)
0.5U.
(2)
where the index i represents the phase-locked time (or fixed angle)
with respect to the wake-passing period, and N is the number of wake-
Figure 1. Development of turbulent strips on the wake-affected
surface.
2
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where du and arE are the ratios of propagation velocities for the
leading edge and tailing edge of the turbulent spot, respectively, to the
freestrearn velocity.
— present model
—Kim and Crawford (1998)
ULE = au Ue and UrE =
U,
(6)
F(x,r) becomes traveling function in the streamwise direction with a
center convecting velocity of ;De , which can be easily determined
0
from the propagation velocity of the leading (amU,) and trailing
leedip edge of
babulent strip
( ajEU, ) edges of the turbulent snip.
treble edge of
turbulent strip
For a given wake-passing frequency (or wake period), the
intermittent function for the trajectories of the turbulent strips that
successively move in the time-space domaimis prescribed using Figure
t/T
2 with propagation velocities set to be 88 percent ( au ) of local
Figure 3. Intermittent function for the transition model as a
function of time.
freestream velocity for the leading edge and 50 percent ( art ) for the
trailing edge in the computations. In reality, the propagation speed of
the turbulent spots or snip will be influenced by the existence of
pressure gradients. However, pressure dependence is not considered,
since the adverse pressure gradient is not important in the predicted
measurements considered herein. Figure 3 shows the present model of
the intermittent function in the form of a Gaussian distribution, along
with the older trapezoidal shaped model used by Kim and Crawford
(1998).
Determination of the onset for the transition presents a challenge,
because no reliable theory is available for the wake-disturbed
boundary layer. An existing empirical correlation for steady transition
developed by Abu-Ghannam and Shaw (1980) is utilized in the present
simulations. In their model, the momentum Reynolds number at the
start of transition is expressed as
passing period used in the evaluation. I is the instantaneous
intermittency which appears as the random step function with 0
(laminar) and 1 (turbulent). )7 is identical to the present transitional
function, f(x, r) . The first possibility for the determination of Iles)
can be derived from the propagation process of the turbulent strips as
shown in Figure 2, following the qualitative description of wakeinduced transition by Pfeil et al. (1983). The idea for the model is
simply to switch on and off the turbulent viscosity term in equation (1)
while following the locus of the turbulent strips. For example, at some
location x, the F(x,r) abruptly switches on with a periodicity of T
(Figure 2) and snitches off in a step function manner as depicted in
Figure 3. This switch is somewhat unrealistically abrupt at the
interface of the turbulent strip with the undisturbed laminar flow
because it implies a sudden variation of the effective viscosity.
Computationally, the switch tends to cause numerical difficulty at the
interface and, in turn, leads to an unrealistic time-resolved response of
the boundary layer.
For more realistic' prediction of turbulent snip behavior, the
intexmittent function flx.r) is determined using the observation of
experiments. Chaklca and Schobeiri (1997) measured the ensembleaveraged intermittency in a similar case of wake-passing on a curved
plate. Their results showed that the ensemble-averaged intermittency is
a Gaussian distribution in which a normalized shape is conserved
downstream. This idea is modeled as
F(x,r) = a 1{ ai
lt/T—x
/(atUeT)1)
iIT
Rem, =163+ exp(6.91— Tu)
where Tu is the freestream turbulence level in units of percent of
freestream velocity. Assuming that the snip formation occurs at the
point of maximum turbulence level in the wakes, and that its
maximum turbulence level is sufficiently high, the transition onset,
x„ can be estimated to be the leading term, Re e., =163 , even if the
turbulence level in the wakes is not precisely known. This is the
primary reason that the above correlation was selected over the one by
Mayle (1991), even though Mayle's correlation seems to be more
accurate in case the freestream turbulence level is known. We should
note that Chakka and Schobeiri (1997) found a dependence of the
transition onset on the wake-passing frequency.
Even though the fretstream velocity defect in the wakes is not
considered a significant factor, and thus excluded from the transition
model, the freestrerun velocity defect is easily included in the
boundary conditions for a more realistic prediction. The freestream
velocity gradient is expressed using the unsteady Bernoulli equation as
(3)
where T is the wake-passing period, and r is the time duration for the
turbulent snip, as defined by
r=
F(x,r)dr
1 dP = aU„, n
„
+u
p dx
---
(4)
In reality, r in equation (3) can be approximated using the classical
theory of turbulent spot propagation,
( I
I
am oLE
U.
(7)
as
(8)
To include the freestream velocity defect in the time-resolved
calculations, the unsteady freestream velocity is modeled to be
U„(x,r)=L/ e (x) W(x,r), where Li e (x) is an undisturbed steady
freestrearn velocity and W(x.r) is a relative freestream time-dependent
part. 19(x,t) is modeled from fmestream experimenml data using a
Gaussian distribution,
(5)
•
3
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Complete results of the calculations with detailed discussion for
the other cases of the measurements are described in Kim and
Crawford (1998) for a slightly different intermittent function model.
However, the difference is found to be minor, when compared with the
present results.
cases without periodic wakes and the results agree well with the
measurements. Note that all predictions of no-wake transition were
performed using the length transition model of y, = I — exp(-517 3 )
where I? = (x — xs )/(xE —x5).
Prediction of the Heat Transfer Measurements
The effects of the periodic wake-passing on boundary layer
transition and heat transfer on a flat plate were studied experimentally
by Funazalzi et al. (1997). The test Reynolds number was 1.3x10 6,
based on the inlet flow velocity (20 m/s) and the length of a flat plate
(L = 1 m), and the inlet freestream turbulence level without wakepassing was about 0.5 percent. Using a flow accelerating device and
varying the inclination angle, four types of favorable pressure gradient
flow cases (types 1 to 4) were generated and used for heat transfer
measurements as well as the case of zero pressure gradient (type 0).
Figure 10 shows the velocity distributions for the cases with favorable
pressure gradient.
Figure II shows the steady boundary layer calculations for three
•
•
1Y118 1
type 2
type 3
type 4
The freestream turbulence is set to be 1.4 percent, which is higher
than the reported value of 0.5 percent. The experimental unheated
starting length of 45 mm from the leading edge was taken into account
for all the heat transfer predictions. Compared with the zero pressure
gradient case (TO), transition is delayed in the flow of type 1 (T1), and
is completely suppressed in the flow of type 2 (T2) over the entire
length of the measurement plate by the presence of the stronger
favorable pressure gradient
Wake-Passine Cases of Normal Rotation Funazaki et al.
(1997) used a wake generator of the spoke-wheel type to simulate
periodic wake-passing over the test plate. There is no effect of
secondary wakes with the spoke-wheel type wake-generator, but the
cylindrical bars on the wheel should be long enough to ensure the twodimensional flow over the test plate, in contrast to the squirrel-cage
type wake generator used by Liu and Rodi (1991). By changing the
rotation of the spoke-wheel, there were two types of wake-passing
created: (a) normal rotation (wake generating bars in front of the plate
move toward the measurement plate); and (b) reverse rotation (bars
move away from the measurement plate). In the measurements, timeaveraged heat transfer of the wake-disturbed plate for each type of the
flow was recorded for three cases of wake-passing by changing the
number of bars on the spoke-wheel. As a result, the corresponding
reduced wake-passing frequencies ( S = IL I U.) were 1.88, 2.83, and
•••
•
•
s ue.
••
1
occa
I .
6.11411 OBBogoecl000000000
0
4 188 80
15
0
0.2
0.4
(m)
0.8
The start (x5) and end (xE) of
transition are determined using equation (7) and the correlation of
ReeE = 2.667 Rees (Abu-Ghannam and Shaw, 1980).
5.65, based on the inlet flow Velocity and the plate length.
Predictions of the wake-affected heat transfer characteristics were
carried out using two approaches. The first approach used a steadyflow method involving superposition of a fully-laminar Stanton
0.8
number ( StL ) distribution and a fully turbulent Stanton number
Figure 10. Freestream velocity distributions from the
measurements by Funazaki et al. (1997).
( Sty. ) distribution, obtained without the presence of the wakes. For
this approach, superposition (Mayle, 1991) leads to
=StL, -117,„(St r —Sty)
where
(12)
is a superposition function that contains the wake-passing
effect.
From the theory of turbulent spot propagation and the time-space
diagram of Pfeil et al. (1983) in Figure 2, Funazaki (1996) proposed
.
1
)s
an ow
ix—x,l_i
L
Experimental values of am and
Cla
1
1 -x,„)
Ix
\au U„,T
(13)
from various measurements of
turbulent spots are about 0.5 and 0.88, respectively. This simple model
implies that the propagation and the growth of the turbulent ships are
independent of the movement of the wakes outside the boundary layer,
and that the time-averaged transition process can be estimated without
the detailed latowledge of the wake propagation.
The effect of pressure gradient cannot be accounted for in the
simple model such as equation (13). Thus, a new formulation was
presented to include the streamwise variation of local freestream
velocity (Hodson et aL, 1992; Funazaki et aL, 1997).
Figure 11. Stanton number variations for the cases of no wakes:
symbols are the measurements by Funazaki et al. (1997); solid
lines are the steady boundary layer predictions.
7
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boundary layer is presented in Figure 4(a), where the time-mean
thickness is used for normalization. The predicted values are in
excellent agreement with the measurements both for the amplitude and
phase angle of the variation. In Figure 4(b), predicted friction
coefficients, normalized by their time-mean values, are compared with
measurements. Agreement is generally good, although the
computation overpredicted the amplitude of the lowest frequency case
of f = 0.01 Hz, and the phase angle is shifted. However, as the
frequency increases, the computation shows excellent agreement. In
addition, the comparison shows that the time-resolved surface
quantities such as friction coefficient are more difficult to predict than
the time-resolved boundary layer integral parameters.
Grid dependence of the computations in space and time were
thoroughly checked by performing calculations with grid systems of
much finer resolution, and the convergence of periodicity was checked
by comparing the calculations with the ones in which the wakepassing periods were doubled.
Figure 5. Modeled freestream velocity defects for case 3 using
Gaussian distribution: symbols are the measurements at y= 15
mm by Liu and Rodi (1991).
RESULTS AND DISCUSSION
Prediction of the Time-Resolved Boundary Layer
Development
Liu and Rodi (1991) conducted extensive hot-wire measurements
in the boundary layer developing along a plate which was subjected to
periodic wake-passing, and they reported a wide variety of timeresolved measurement data for the wake-affected boundary layer
development and transition process. Periodic wakes were created using
the squirrel-cage type wake generator in front of the test plate, and the
freestream velocity was almost constant at 15.5 tn/s. In the case of no
wake, the freestream turbulence level is so low (0.3 percent) that the
entire surface length of the plate (0.5 m) remained laminar. Liu and
Rodi (1991) created four cases of different wake-passing frequency:
case 2 (20 Hz) case 3 (40 Hz), case 4 (60 Hz), and case 5 (120 Hz).
However, only the calculations for case 3 are presented in this paper.
Transition onset was determined to be 0.075 m from the leading edge
by using the correlation by Abu-Ghatutam and Shaw (1980). As
described earlier, the effect of freestream velocity defect was included
in the calculation using the measurement data set at y = 15 mm, as
shown in Figure 5. Calculations were carried out with and without the
inclusion of freestream velocity defect in order to appreciate the effect
of freestream velocity defect on the boundary layer development.
Figure 6 presents the calculated time-resolved boundary layer
parameters at three streamwise locations, and the agreement with the
measurement data is good in all three locations. These results show the
temporal switching between the laminar and turbulent states during the
wake-passing and turbulent snip propagation. Note that that the
predicted results have the saint phase for all three locations, and this
trend shows that the convection path of the turbulent strips is different
from the one of the passing wakes.
Periodic fluctuations (i7) of the ensemble averaged boundary
layer velocity are shown in Figure 7 for three normal distances, and
the calculated ensemble-averaged velocity traces agree well with the
corresponding measurement data. Note that ri is defined as the
difference between the ensemble averaged velocity ((I ) and the timemean velocity (U). Comparison between the results of computation
with and without the freestream velocity defect shows the two
contributions to the ensemble-averaged velocity fluctuation, namely
the unsteady transition process and the freestream velocity defect
Predicted near-wall velocity traces (y = 0.3 mm) at x = 0.2 and 0.4 m
t/T
Figure 6. Time-resolved variation of boundary layer parameters
for case 3 of the measurements by Liu and Rodi (1991):
symbols are the measurements; solid lines are the predictions
with freestream velocity defect and dotted lines without
freestream velocity defect.
show significant discrepancy with the measurements, although the
predictions at x = 0.5 m show excellent agreement. It is not clear
whether that discrepancy is caused by the deeper penetration of wake
disturbances that move at the speed of freestream or the possible effect
of secondary wakes from the returning bars. Both effects are excluded
from the model in the present approach.
In Figure 8, the tins profiles of the periodic fluctuation
component ( ) are compared with the measurements. In the upstream
region closer to the transition onset, the rms profiles for both the
measurements and the calculations show the two local maxima near
the surface due to the periodic transition. As the transition proceeds,
the maxima grow continuously, and they decrease as the turbulent
strips merge.
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i.r.r
15
(a) casa 3 (f=40Hz): = o.20 m
çj
7CVPfl
0
x).3rn
24:12m
0.1
y = 6.0 mm
0 ia
0
10
y 3.0 mrn- 0.1
5' 0
,,0
-0.1
0.1
-0.1
0.5
1
t/T
1.5
Figure 8. Profiles of ms velocity of periodic fluctuation for case
3 of the measurements by Liu and Rodi (1991): symbols are the
measurements; solid lines are the predictions with freestream
velocity defect and dotted lines without freestream velocity
defect.
2
x 0.2 m
1_o.25
2 -020
3 -0.16
- • -0.10
6 -0.06
6 -0.02
7 +0.02
6 -006
9 +310
10 +0.16
11 +020
12+025
0.1
-0.1
1 .0
t/T
2.0
(a) case 3 at x = 0.2 m
0.5
1
t/T
1.5
2
2.05m
I -023
3-an
340.I5
---- • -0.10
5 -0.06
6 -002
7 +OM
6 +0.06
9+0.10
10 +0.15
II +020
12 +623
(c) caw 3 (f=40Hz): x=0.50 m
0.1
y = 6.0 mm
y= 3.0 mm- 0.1
1.0
20
1/1
(b) case 3 at x = 0.8 m
0.1
-0.1
Figure 9. Predicted velocity defect contours for case 3 of the
measurements by Liu and Rodi (1991).
y = 0.3 mm
-0.1
defined as .; .(0-u,)/u e , where U1
0.5
1
t/T
1.5
from the undisturbed
laminar profile for the case of no wake-passing. These contours
represent the time history of the disturbed flow at a fixed streamwise
location. In Figure 9(a), the negative contours of round-edged
triangular shape (t/T = 0.1 - 0.4) show the duration of the turbulent
strip and the temporal transition to turbulent status. After the turbulent
strip passes, the negative contours immediately disappear, but the
positive contours, which are confined to the near-wall region during
the convection of the strips, last longer. This time period (UT = 0.4 0.9) can be interpreted as the existence of the becalmed region behind
the turbulent strips. As the strips grow while convecting downstream,
they terminate the becalmed region of the preceding strips and begin to
merge with each other, as shown in Figure 9(b).
2
Figure 7. Periodic fluctuation of ensemble-averaged boundary
layer velocity for case 3 of the measurements by Liu and Rodi
(1991): symbols are the measurements; solid lines are the
predictions with freestream velocity defect and dotted lines
without freestream velocity defect.
Figure 9 presents the contours of the velocity defect at the
streamwise location of x = 0.2 and 0.5 m. Note that the velocity defect
represents the level of disturbed boundary layer velocity, and it is
6
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Complete results of the calculations with detailed discussion for
the other cases of the measurements are described in Kim and
Crawford (1998) for a slightly different intermittent function model.
However, the difference is found to be minor, when compared with the
present results.
cases without periodic wakes and the results agree well with the
measurements. Note that all predictions of no-wake transition were
performed using the length transition model of y, = I — exp(-517 3 )
where I? = (x — xs )/(xE —x5).
Prediction of the Heat Transfer Measurements
The effects of the periodic wake-passing on boundary layer
transition and heat transfer on a flat plate were studied experimentally
by Funazalzi et al. (1997). The test Reynolds number was 1.3x10 6,
based on the inlet flow velocity (20 m/s) and the length of a flat plate
(L = 1 m), and the inlet freestream turbulence level without wakepassing was about 0.5 percent. Using a flow accelerating device and
varying the inclination angle, four types of favorable pressure gradient
flow cases (types 1 to 4) were generated and used for heat transfer
measurements as well as the case of zero pressure gradient (type 0).
Figure 10 shows the velocity distributions for the cases with favorable
pressure gradient.
Figure II shows the steady boundary layer calculations for three
•
•
1Y118 1
type 2
type 3
type 4
The freestream turbulence is set to be 1.4 percent, which is higher
than the reported value of 0.5 percent. The experimental unheated
starting length of 45 mm from the leading edge was taken into account
for all the heat transfer predictions. Compared with the zero pressure
gradient case (TO), transition is delayed in the flow of type 1 (T1), and
is completely suppressed in the flow of type 2 (T2) over the entire
length of the measurement plate by the presence of the stronger
favorable pressure gradient
Wake-Passine Cases of Normal Rotation Funazaki et al.
(1997) used a wake generator of the spoke-wheel type to simulate
periodic wake-passing over the test plate. There is no effect of
secondary wakes with the spoke-wheel type wake-generator, but the
cylindrical bars on the wheel should be long enough to ensure the twodimensional flow over the test plate, in contrast to the squirrel-cage
type wake generator used by Liu and Rodi (1991). By changing the
rotation of the spoke-wheel, there were two types of wake-passing
created: (a) normal rotation (wake generating bars in front of the plate
move toward the measurement plate); and (b) reverse rotation (bars
move away from the measurement plate). In the measurements, timeaveraged heat transfer of the wake-disturbed plate for each type of the
flow was recorded for three cases of wake-passing by changing the
number of bars on the spoke-wheel. As a result, the corresponding
reduced wake-passing frequencies ( S = IL I U.) were 1.88, 2.83, and
•••
•
•
s ue.
••
1
occa
I .
6.11411 OBBogoecl000000000
0
4 188 80
15
0
0.2
0.4
(m)
0.8
The start (x5) and end (xE) of
transition are determined using equation (7) and the correlation of
ReeE = 2.667 Rees (Abu-Ghannam and Shaw, 1980).
5.65, based on the inlet flow Velocity and the plate length.
Predictions of the wake-affected heat transfer characteristics were
carried out using two approaches. The first approach used a steadyflow method involving superposition of a fully-laminar Stanton
0.8
number ( StL ) distribution and a fully turbulent Stanton number
Figure 10. Freestream velocity distributions from the
measurements by Funazaki et al. (1997).
( Sty. ) distribution, obtained without the presence of the wakes. For
this approach, superposition (Mayle, 1991) leads to
=StL, -117,„(St r —Sty)
where
(12)
is a superposition function that contains the wake-passing
effect.
From the theory of turbulent spot propagation and the time-space
diagram of Pfeil et al. (1983) in Figure 2, Funazaki (1996) proposed
.
1
)s
an ow
ix—x,l_i
L
Experimental values of am and
Cla
1
1 -x,„)
Ix
\au U„,T
(13)
from various measurements of
turbulent spots are about 0.5 and 0.88, respectively. This simple model
implies that the propagation and the growth of the turbulent ships are
independent of the movement of the wakes outside the boundary layer,
and that the time-averaged transition process can be estimated without
the detailed latowledge of the wake propagation.
The effect of pressure gradient cannot be accounted for in the
simple model such as equation (13). Thus, a new formulation was
presented to include the streamwise variation of local freestream
velocity (Hodson et aL, 1992; Funazaki et aL, 1997).
Figure 11. Stanton number variations for the cases of no wakes:
symbols are the measurements by Funazaki et al. (1997); solid
lines are the steady boundary layer predictions.
7
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If
art
au x„ tie
aLE
r
xi,
For type 0 and I flows, steady transition due to the background
turbulence occurs in the case of no wake-passing, and the slight
(14)
Us T
wheref is the wake-passing frequency (1= 11T).
For the second approach, time-resolved boundary layer
calculations were performed for the periodic boundary layer
development due to wake-passing for each type of the flow. Timedependent variation of the intermittent function (not to be confused
with the superposition intermittency function in the first approach) was
prescribed in the formulation of the turbulent viscosity, similar to that
carried out for the measurements of Liu and Rodi (1991). The wake
effect on the freestream velocity was neglected in the present
calculations, because the measurements did not provide sufficient
information. However, this would not significantly affect the timeaveraged results, since freestream fluctuation is only a minor factor for
the transition process. For all the predictions, transition onset was
determined from the correlation of starting location by Abu-Ghannam
and Shaw (1980), and the resulting starting locations of transition from
the leading edge of the test plate in meters are 0.0492 (type 0), 0.0498
(type 1), 0.0524 (type 2), 0.0519 (type 3), and 0.0537 (type 4). Steady
transition due to the background turbulence was ignored, and the
related issue on multimode transition will be discussed later.
The heat transfer results from the predictions using the timeaveraged steady superposition model (equation 13) and the timeaveraged results of the time-resolved computation are presented in
Figure 12 for three cases at three reduced wake-passing frequencies
and compared with the measurements. When the pressure gradient is
zero or mildly favorable, the steady model from equation (13) shows
reasonable prediction of time-averaged heat transfer, but the prediction
using the steady model starts to deviate from the measurements when
the favorable pressure gradient becomes significant In contrast, the
time-averaged results from the time-resolved calculation provide good
agreement for all three types of flow. However, significant
undaprediction occurs at S = 5.65 for the flow of type 0, and this
indicates that the transition onset is earlier than predicted by the
correlation. Assigning an earlier onset will yield better results.
Generally speaking, computational results for the time-avenged heat
transfer on the surface show that the prediction is less favorable in the
early region around Rex = 2x105 where the turbulent strips start to
develop. A possible explanation is that the interaction between the
newly formed turbulent strips and the passing wakes containing high
disturbance is intense, unlike the present assumption of no interaction
between them.
Funazaki et al. (1997) reported only the time-averaged
intermittency distribution without providing the corresponding heat
transfer results for their higher acceleration cases: types 3 and 4, as
shown in Figure 13. To compare the experimental results with the
time-dependent calculations, the time-resolved heat transfer is
avenged and inserted into the rearranged superposition equation (12)
to yield a calculating equation for the time-averaged intennittency
distribution.
St—St L
St r
(a) type 0 (norrn)
4
= 0.0492 m
3
2
1
•
0
210'
4 10'
Re*
L
610
a10'
1 e
(e) type 2 (norm)
ew e 0.0524 in
*********
•
21e
(15)
••
■*****
410' 6 10' B 11:51 i10' 1.2 1CP 1.4 Hi
Re
Figure 12. Time-averaged Stanton number diltributions for the
cases of normal rotation: symbols are the measurements of
Funazaki et al. (1997) (•: no wake, rD : S= 1.88, a : S= 2.83,
0: S= 5.65, and • : fully turbulent); solid lines are the
corresponding time-resolved predictions for the cases of wakepassing; dotted lines are the predictions of the steady
superposition model for the corresponding wake-passing cases
These results are plotted in Figure 13, and they compare very
accurately with the measurements. The results from the steady
superposition models are also plotted in Figure 13, showing a
significant disagreement for large x.
(equation 13).
8
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and proposed the following superposition for the multimode transition,
underprediction of the time-resolved computations near the trailing
edge of the plate at S = 1.88 could be attributed to a multimode
transition which is associated with the steady transition between the
wakes. Mayle and Dullenkopf (1990) assumed that the production of
steady and wake-induced transitions are independent of each other,
absent in steady boundary layers. Preliminary testing of equation (16)
with bath of the time-averaged intermittency models produced a
significant overprediction (not presented in the papa). First of all, the
presence of wake-passing modifies the flow before the region of
steady transition and forces the wake-affected flow to have a flow
history which is different from the no wake-passing case. Thus, even if
the assumption that permits independent existence of two modes of
transition is valid, superposition in equation (16) using information
from the no wake-passing case would lead to incorrect results.
Secondly, the two modes of transition are not independent, and thus
they should influence each other. The activity of the becalmed region,
which is excluded in the analytic models, may play the role of
stabilizing the surrounding flow and suppressing the effect of
background turbulence to trigger the transition between the turbulent
strips, until the becalmed region is terminated by the following wakes
or the surrounding turbulent flow. For the flows of types 2, 3, and 4,
steady transition does not appear because of the large favorable
pressure gradient, eliminating the opportunity of multimode transition.
(a) type 2
(S 02.83• x_ =0.0524 m)
0.8
0.4
w-
O
0.2
o
-
o
0. 1
0.2
0.3
0.4
x (m)
measurement
unsteady code
steady model 1
steady model 2
0.5
0.6
07
Wake-Passinq Cases of Reverse Rotation Funazaki et al.
(1997) also conducted heat transfer measureinents for reverie rotation
of the wake-generating bars, with the rest of flow conditions remaining
the same. Compared with the results from normal rotation of the bars,
time-averaged heat transfer was significantly reduced in the cases of
the reverse rotation on the same test surface,. according to the
measurements. It indicates that the flow mechanism associated with
the influence of wake-passing is quite different from the normal
rotation of moving bars. Figure 14 illustrates the flowfield created by
the different rotation of wake-generating bars. The flow pattern for the
normal rotation case is on the upper surface, and the reverse rotation
case can be considered as the flow on the lower surface or the pressure
side of a turbine blade. After the cutting of wakes by the test plate, a
flow toward the upper or suction surface (test surface with normal
rotation) results in accumulation of flow inside the wake. In contrast,
flow leaving the lower or pressure surface (test surface with reverse
rotation) causes the wake to be diminished (Binder et al., 1985).
Funazaki and Kitazawa (1997) also state that the wake-induced
1
(B) type 3
(3=2.83, x =0.0519 m)
0.8
o.s
0.4
w-
O
0.2
-
0. 1
0.2
0.3
0.4
x (m)
measirement
unsteady code
steady model 1
Pearly model 2
0.5
0.6
(16)
where y, is the intermittency distribution when the wake-passing is
•
,,,,,,
os
feat =1-0-FJ(l-y,)
07
1
(c) type 4
(S02.83, 0.00.0537 m)
0.8
0.8
measurement
unsteady code
- steady model 1
steady model 2
▪
0.2
o
o
0.1
0.2
0.3
0.4
x (m)
0.5
0.6
07
Figure 13. Comparison of predicted time-averaged intermittency
factor with the measurements (Funazaki et al., 1997) for high
acceleration cases.
Figure 14. Wake and surface interaction for normal and reverse
rotations of wake-passing (adapted from Binder et al., 1985).
9
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turbulent region in the reverse rotation case was reduced by the effect
of the negative jet away from the plate surface.
Funazaki and Kitazawa (1997) measured the time-resolved
turbulent intensity for the cases of zero-pressure gradient and S = 2.83
using both rotational directions. From the observation of their
measured turbulence intensity at y = 0.2 mm, reverse rotation produces
a much weaker effect on the boundary layer than normal rotation in
both strength and duration of turbulent fluctuations. Quantitatively
comparing the two experimental cases the duration of turbulent
fluctuation near the surface was reduced by approximately 25 percent
for the reverse rotation case. Therefore, the intermittent function for
reverse rotation was modified, yielding the duration of turbulent strip,
2, in the intermittent function to be 75 percent of that for normal
rotation case.
(17)
if L
2W
4W
6W
e105
ie
= 0.75 11
T Imo/
In the time-resolved prediction for the reverse rotation case, a
modified distribution of the intermittent function was used, but with
the same location of transition onset for each type of flows. The timemean results from the predictions are shown in Figure 15, along with
the measured data. Although the intermittent function for the reverse
rotation case is rather arbitrarily determined and calibrated using the
experimental observation, predicted time-mean heat transfer results
agree well with the experimental data. There is a slight overprediction
in the flow of type 2 in Figure 15(c), but the prediction can be
improved by shifting forward the transition onset.
Time-resolved variations of the boundary layer parameters from
unsteady boundary layer computations are compared with the
measurements by Funazaki and Kitazawa (1997) in Figure 16. Despite
some discrepancies with the measurements, boundary layer parameters
are well predicted for cases of either normal or reverse rotation. Wakeaffected variations of displacement and momentum thicknesses show
the distinct characteristics of these cases In normal rotation, both
1 01
Rex
(a) type 0 (normal rotation) :5=2.83. x/1. =0.3
los
2W 4W 6106 8 105
Re,
(c) type 2 (rev)
1.2W
-
xs, = 0.0524 m
1/4
•
U
M Dam pac
(b) type 0 (reverse rotation): S=2.83.
Trro—nr•
: 00000 00000000000
=0.3
3
0 0 0
........
......
..
0
.....
O..
0,
2
•
1
.
2 105 4 105 8 105 8 105 i106 1.2 103 1.4 1CP
Re x
0
Figure 15. Time-averaged Stanton number distributions for the
cases of reverse rotation: symbols are the measurements of
Funazaki et al. (1997) (41 : no wake, '0 : S = 1.88, a. : S = 2.83,
0 : 5= 5.65, and • : fully turbulent); solid lines are the
corresponding time-resolved predictions for the cases of wakepassing.
0
0.01
0.02
t (wc.)
0.03
0.04
Figure 16. Time-resolved variations of the boundary layer
parameters: (a) normal rotation and (b) reverse rotation;
symbols are the measurements by Funazaki and Kitazawa
(1997); solid lines are the time-resolved predictions.
10
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thicknesses increase rapidly with the arrival of the turbulent snip and
decrease gradually toward the trailing edge of the strip. However, the
reverse rotation ease shows an almost symmetric increase and decrease
during the passage of the turbulent snips, but with narrower duration.
However, both cases does not show major difference in the variation
of boundary layer integral parameters, although the wake-passing due
to normal rotation has a larger impact on the laminar boundary layer,
as shown in the comparison of the time-averaged heat transfer of
Figures 12 and 15.
Layers in Cascades: Part I -Description of the Approach and
Validation," ASME Journal of Turbomachinery, Vol. 118, pp. 96-108.
Funazaki, K., 1996, "Unsteady Boundary Layers on a Flat Plate
Disturbed by Periodic Wakes: Part 1-Measurement of Wake-Affected
Heat Transfer and Wake-Induced Transition Model," ASME Journal of
Turbomachinety, Vol. 118, pp. 327-336.
Funazidci, K., and Kitazawa, T., 1997, "Boundary Layers
Transition Induced by Periodic Wake Passage (Measurements of the
Boundary Layer by Hot-Wire Anemometry)," Bulletin of GTSJ, p. 26,
also private communication with K. Funazaki.
CONCLUSIONS
A simple model for the evolution of wake-induced transition is
proposed to simulate the time-resolved variation of wake-affected
boundary layer development. The model is based on the classical
theory of the turbulent spot propagation and experimental observation.
In the process of transition, it is assumed that there is no interaction
between the turbulent strips and the wakes after the high turbulence in
the wakes themselves initiates the turbulent snip at an earlier location.
The ensemble-averaged intermittent function was modeled using
Gaussian distribution and the linear propagation theory of turbulent
spots. Comparison with the measurement data showed the capability
of model to capture the details of ensemble-averaged variation of
wake-affected boundary layer flow.
Heat transfer measurements under zero and favorable pressure
gradients were calculated at several reduced wake-passing frequencies,
and the estimated time-averaged results show that the present method
provides more accurate predictions than a time-averaged superposition
model, especially when the effect of pressure gradient is significant.
The reverse rotation cases of wake-generating bars were also
considered. The duration time of the intermittent function was set to be
75 percent of the one for the normal rotation, yielding good agreement
with the measurements.
Funazalci, IC, Kitazawa, T., Koizumi, K., and Tadashi, T., 1997,
"Studies on Wake-Disturbed Boundary Layers Under the Influences of
Favorable Pressure Gradient and Free-Stream Turbulence Part I:
Experimental Setup and Discussions on Transition Model," ASME
Paper No. 97-OT-45 I.
Halstead, D. E., Wisler, D. C, Oldishi, T. H., Walker, G. J.,
Hodson, H. P., and Shin, H.-W., 1997, "Boundary Layer Development
in Axial Compressors and Turbines Part I of 4: Composite Picture,"
ASME Journal of Turbomachinety, Vol. 119, pp. 114-127.
Hodson, H. P., Addison, J. S., and Shepherdson, C. A., 1992,
"Models for Unsteady Wake-Induced Transition in
Axial
Turbormichines," Journal de Physique lll, Vol. 2, pp. 545-574.
Liu, X., and Rodi, W., 1991, "Experiments on Transitional
Boundary Layers with Wake-Induced Unsteadiness," Journal of Fluid
Mechanics, Vol. 231, pp. 229-256.
Kim, K., 1998, "Computation of Wake-Passing Effects on
Turbine Blade Boundary Layers," Ph.D. Dissertation, The University
of Texas at Austin, Austin, TX.
Kim, K., and Crawford, M. E., 1998, "Prediction of Unsteady
Wake-Passing Effects on Boundary Layer Development," presented at
the 1998 ASME International Mechanical Engineering Congress and
Exposition, Anaheim, CA, Nov. 15-20.
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Mayle, It E., 1991, "The Role of Laminar-Turbulent Transition
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Pfeil, H., Herbst, It, and Schroder, T., 1983, "Investigation of the
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Dullenkopf, K., Schulz, A., and Wittig, S., 1991, "The Effect of
Incident Wake Conditions on the Mean Heat Transfer of an Airfoil,"
ASME Journal of Turbomachinery, Vol. 113, pp. 412-418.
Iran, L. T., and Taulbee, D. B., 1992, "Prediction of Unsteady
Rotor-Surface Pressure and Heat Transfer From Wake Passings,"
ASME Journal of Turbomachinery, Vol. 114, pp. 807-817.
Fan, S., and Lalcshminarayana, B., 1996, "Computation and
Simulation of Wake-Generated Unsteady Pressure and Boundary
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