49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition
4 - 7 January 2011, Orlando, Florida
AIAA 2011-682
Determining the Effect of Dimples on Drag in a Turbulent
Channel Flow
C. M. Tay1
National University of Singapore, Singapore, 119260, Singapore
An array of circular shallow dimples with a depth to diameter ratio of 5% located in a
channel was experimentally found to affect drag depending on the Reynolds number.
However, such effects on drag tend to be very small, typically less than 5% compared to a
smooth flat channel. A method relying on measuring the fully developed mean streamwise
pressure gradient of the flow can be used to determine such small changes in drag relatively
accurately. A long channel is required to allow the flow to become fully developed several
times as it flows over the different sections of the set-up. Dimple arrays with two different
coverage ratios, 40% and 90% are studied and their effects on drag are presented. When
compared to a flat channel, it is found that drag is only reduced beyond a certain Reynolds
number, depending on the coverage ratio. Below this critical Reynolds number, the drag
increases as the Reynolds number is reduced. However, the relationship of drag to Reynolds
number may not be monotonic. The results suggest that at the lower coverage ratio, drag
reduction occurs only for a small range of Reynolds numbers.
Nomenclature
D
d
h
LD
m
Pg
Ps
P1
P2
Re
V
x
z
β
∆d
∆p
δ
ρ
T
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
dimple diameter
dimple depth
channel height
length of dimple array
mean streamwise pressure gradient found by least squares fit
mean fully developed streamwise pressure gradient
local static pressure
value of intercept along static pressure axis without/upstream of dimples
value of intercept along static pressure axis downstream of dimples
Reynolds number
channel centerline velocity
streamwise coordinate measured from channel entrance
spanwise coordinate measured from channel centerline
distance between centers of adjacent dimples
change in drag due to dimple array compared to a flat smooth channel
change in static pressure due to dimple array normalized by the dynamic pressure
small change in quantity
density of air
I. Introduction
here has been much work carried out to study the effect of dimples on the flow over it. Most of the work carried
out in the past focused on the heat transfer aspect of the dimples. It was found that compared to conventional
1
Research Fellow, Mechanical Engineering, National University of Singapore, 21 Lower Kent Ridge Road,
Singapore 119077.
1
American Institute of Aeronautics and Astronautics
Copyright © 2011 by Jonathan Tay Chien Ming. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
intrusive flow turbulators, dimples can increase heat transfer at a smaller pressure drop penalty 1. The flow over the
dimples however, is affected by many dimple parameters. Dimple depth to diameter ratios 1, smooth or sharp edges,
dimple arrangement and area coverage ratios 2, as well as flow Reynolds numbers 1, 3-5 all have varying but significant
effects on the flow over it. Even the flow turbulence intensity can affect the flow over dimples 6. Both experimental7
as well as computational simulations8, 9 have been carried out in an effort to further understand such complicated
flows. As the focus of these studies were on enhancing heat transfer, the dimples studied were typically sharp edged,
deep dimples with depth to diameter ratios of 10% and above. These dimple characteristics typically result in
increased turbulent production and promote mixing in the flow.
Some recent studies have focused instead on the drag reduction aspect of dimples 10, 11. Investigators have
suggested the possibility of drag reduction with some dimple geometries 12. For these studies, smooth edged shallow
dimples with depth to diameter ratios of no more than 10% seem to be preferred. Lienhart et al.10 studied the dimples
both experimentally with a channel and an external flow boundary layer, as well as computationally with dimples in
a channel. The experiment with the channel was carried out at a Reynolds number between 10,000 and 70,000,
based on the bulk flow velocity and channel half height. The external flow experiment estimated the mean skin
friction by measuring the boundary layer profiles before and after a dimple array using a Pitot-tube rake. Both of
these experiments showed that the dimples did not affect the drag of the flow significantly. The measured difference
in drag from these experiments was deemed within the error band of the experiment. DNS was employed to simulate
a channel flow with dimples on just the lower wall of the channel as well as on both walls. In comparison with a flat
walled channel case, the simulations showed a slight increase in drag of about 2% for the dimples on the lower wall
only case, and an increase of about 4% for dimples on both walls.
The present study focuses also on the drag reduction aspect of dimple flows. However, one common problem
with these studies is the very small changes to drag the shallow dimples usually cause. These very small changes
require very careful measurements to be made or they can easily fall within the error band of the experiment. For
this reason, the drag in the present study is determined experimentally in a channel flow by measuring the average
pressure drop over dimples after the flow is allowed to fully develop. A new method takes advantage of a relatively
long channel and dimpled section to quantify the effect of the dimples on drag very accurately. The method can in
fact be similarly applied to determine small changes in drag by other flow manipulators within a channel flow
environment.
II. Experimental details
Experiments were conducted in an air channel measuring 20mm in height and 400mm wide. A bell-mouth, four
screens, a honey comb and a 16:1 contraction preconditions the flow before it enters the channel. A 3.2m smooth
flat section immediately follows to allow the flow to fully develop before the test section where the dimples will be
located is reached 160 channel heights after entry. The test section measures 2.4m long and consist of a removable
dimpled upper wall which can be replaced with a smooth wall to allow comparison between the two. The dimple
array occupies the full 2.4m length of this section when the dimpled wall is in place. This is followed by another
2.4m of smooth flat channel before the end of the channel is reached to minimize any exit effects. A suction fan is
located at the downstream end of the channel to drive the flow. Static pressure taps line the top of the channel to
allow measurements at various streamwise locations. Several pressure taps are also made across the width of the
channel before and after the test section to ensure that the flow is two-dimensional within the channel.
Static pressure is measured using a multiplexer in conjunction with a Setra model 239 pressure transducer,
sampled at 1000Hz for about 32 seconds. The flow speed is controlled using a variable speed motor controller and is
determined using a total pressure tube in conjunction with a static pressure tapping at the total pressure tube inlet
location. This total pressure tube is located far downstream from the test section to minimize its effects on the flow.
The experiments are carried out over a range of Reynolds numbers of about 10,000 to 40,000 based on channel
height and centerline velocity.
The circular axisymmetric dimples studied are machined onto an aluminium test plate that forms one of the
removable channel walls. They have a diameter of 50mm and depth of 2.5mm, giving a depth to diameter ratio of
5%. The cross-section of each dimple along its diameter, illustrated in Figure 1 is made up of three circles joined
tangentially to one another and to the surrounding flat area. The edge radius of the dimples is 42.1mm and the base
radius is 84.2mm. The depth of the dimple in Figure 1 is greatly exaggerated for illustration. At a depth to diameter
ratio of only 5%, the dimples are very shallow in reality.
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D = 50mm
84.2mm
42.1mm
d = 0.05D
Figure 1. Dimple cross-section
The dimples are staggered as shown in Figure 2 and their centers form isosceles triangles with adjacent dimples.
Also shown in the figure are the axes used in the following discussion. The x axis lies along the channel centerline,
and the origin is located at the channel entrance. z is the spanwise coordinate measured from the channel centerline.
Two coverage ratios of 40% and 90% are studied for this dimple geometry and the coverage ratio determines the
distance β between the centers of adjacent dimples. The dimple are arranged so that their centers form isosceles
triangles. The dimple coverage ratio is defined as the total plan area occupied by the dimples divided by the total
plan area of the array.
SHAPE \* MERGEFORMAT
Flow direction
x
60o
z
β
β
Figure 2. Plan view of dimple arrangement in test section.
For a coverage ratio of 40%, β is about 75.3mm. At 90%, β is about 50.2mm, and the adjacent dimples edges
almost touch one another. This case represents the maximum limit of coverage ratio physically possible for
staggered dimples.
As mentioned before, the experimental set-up allows the dimpled plate to be replaced by a smooth flat plate.
Besides using this flat plate for comparison between the dimpled results with results for a channel with smooth flat
walls, the flat plate is also used for validation of the channel flow.
III. Results and discussion
Before any measurements with the dimples were carried out, static pressure measurements were made with the
smooth plate to ensure that the flow is fully developed before it reaches the test section. Checks were also made to
ensure that the flow is 2-dimensional within the channel, and that the streamwise pressure gradients are comparable
with those found in the literature.
The dimple array will be located between x = 3200mm and x = 5600mm from the entrance of the channel when
in place. Figure 3 shows the spanwise variation of the static pressure at various flow Reynolds numbers at x =
2950mm. These measurements are carried out with the smooth flat plate and at a location slightly upstream of the
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intended dimple array. Similarly, Figure 4 shows the spanwise pressure variation for the smooth wall case at x =
5950mm, just downstream of the intended dimple array.
Figure 3. Spanwise variation of static pressures at x = 2950mm
Figure 4. Spanwise variation of static pressures at x = 5950mm
The static pressure is practically constant across the width of the central portion of the channel at each Reynolds
number. In fact, as a percentage of their respective dynamic pressures, the static pressures vary by no more than 1%
from their mean value at all the locations shown, suggesting reasonable 2-dimensionality of the flow at least within
the test section of the channel.
Streamwise pressure measurements show that the streamwise variation of the static pressure appears linear
before the test section at x = 3200mm is reached. Static pressure measurements made from x = 1950mm (x/h = 97.5)
onwards show a linear variation in static pressure with distance for the range of Reynolds numbers shown in Figure
5, suggesting fully developed flows. No static tappings were located within the test section from x = 3200mm to x =
5600mm to avoid any possibility of flow perturbations due to the presence of the static taps.
The variation of the streamwise pressure gradient with Reynolds number also compares favorably with that
predicted by the Colebrook-White equation for fully developed flows in conduits 13 as Figure 6 shows. It might be
worth nothing that while the Reynolds number used in this study is based on the channel height and centerline
velocity, the Reynolds number used in the Colebrook-White equation is based on the hydraulic diameter and bulk
flow velocity. A conversion was done for both these quantities, with the bulk flow velocity approximated to 87%
that of the centerline velocity14, to allow a valid comparison with the present results.
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Figure 5. Streamwise variation of static pressures at various Reynolds numbers
Figure 6. Variation of average streamwise pressure gradient with Reynolds numbers. Diamonds:
measurements. Solid line: Colebroook-White equation
After verifying that the flow within the channel was 2-dimensional and fully developed by the time it reaches the
test section, the dimple section was installed for comparison with the flat smooth channel results. Figure 7 shows the
streamwise variation of two cases at comparable Reynolds numbers, one with dimples present and another with
dimples absent. The solid line is fitted to the flat channel data using a least squares fit. The difference due to the
dimples on the streamwise pressure variation is barely noticeable from the figure. This very small difference similar
to Lienhart et al.10 results and is typical for the various Reynolds numbers presently studied.
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Figure 7. Effect of dimples on the streamwise pressure distribution.
During the experiment, every time the channel configuration was changed from a dimpled one to one without
dimples, the fan powering the channel had to be shut down and then turned on again when the reconfiguration was
complete. The very low pressure present within the channel makes removal of the test plates particularly at high
flow speeds very difficult. The finite accuracy of the motor speed controller resulted in a very small change in the
fan speed even between consecutive runs, resulting in a small but definite change in the Reynolds number such as
that between the two runs in Figure 7. Using the Colebrook-White equation13 of Figure 6, a simple analysis was
carried out to estimate the significance of this change in the Reynolds number. Simply stated, the mean streamwise
pressure gradient Pg is a function of Re, the Reynolds number of the flow.
Pg = f (Re)
(1)
Differentiating with respect to the Reynolds number gives:
δ Pg
= f ' (Re)
δ Re
(2)
For a simple estimation of small changes in the Reynolds number,
δ Pg f ' (Re)δ Re
=
Pg
Pg
(3)
Figure 8 shows the percentage change in the mean pressure gradient when the Reynolds number changes by 400,
ie. δRe = 400, which is approximately the difference between the two cases presented in Figure 7. The effect varies
with the Reynolds number and is as high as 14% at a Reynolds number of 5000, and decreases to 3.6% at a
Reynolds number of 20,000. At a Reynolds number of 40,000, the change in mean pressure gradient drops further to
about 1.8%. These percentage changes are deemed to be significantly large as the expected changes in drag due to
the dimples are very small10. Thus a simple comparison of the results by changing the test plates of the set-up is not
possible due to the accuracy required and another method of estimating the effect on the dimples on drag was
sought.
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Figure 8. Effect on mean pressure gradient with a Reynolds number change of 400.
It is observed in Figure 7 that the mean pressure gradient in the flat portions of the channel both upstream and
downstream of the dimples is similar. Particularly, the value of the mean pressure gradient downstream of the
dimple array is not significantly affected by the dimples but is of similar value to the fully developed flow in the flat
section upstream of the dimple. This may be expected since as long as the channel is sufficiently long, the flow is
expected to revert to its fully developed state regardless of its initial condition . It is known that even before the flow
reaches its fully developed state, the streamwise pressure gradient will reach its fully developed value first 15. Barbin
and Jones16 suggests that for a circular pipe flow, the mean pressure gradient takes only about 15 pipe diameters for
it to reach its fully developed value. Given that in the present case, the velocity profile upon leaving the dimple
section is already somewhat parabolic and not the top hat profile commonly used for the pipe entrance profile in
such studies, it is likely that the flow would take much less than this length for the pressure gradient to reach its fully
developed value. In any case, Figure 7 shows that the static pressure varies linearly with distance over all the
measured locations downstream of the dimples.
Consider the hypothetical cases shown in Figure 9. The crosses show a linear pressure variation for the smooth
flat channel case. The diamonds show the case where the presence of dimples causes a vertical shift in the static
pressures at those points downstream of the dimple array. This vertical shift upwards thus means a shallower
pressure gradient over the dimple array, and thus means a drag reduction by the dimples. If a vertical shift
downwards is observed, it means the dimples cause a drag increase. If the magnitude of this vertical shift is known,
whether upwards or downwards, the effect on drag by the dimples can be calculated.
Figure 9. Hypothetical static pressure distributions. Diamonds: Hypothetical case with dimples. Crosses:
Hypothetical case without dimples.
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It was found that a least squares fit can be applied to the data and the vertical shift deduced from the coefficients.
Considering only the data set for the case when the dimples are present, equation 4 can be used to fit the data points
located upstream of the dimples. For a particular Reynolds number, the static pressures at these locations are not
significantly affected by the presence of the dimples due to their upstream location (see Figure 7). The static
pressures downstream of the dimples are more significantly affected by the dimples, and equation 5 may be used to
fit these points. Note that only the data set for the case with the dimples present is required for fitting, the case with
the flat walls without the dimples is not required at all. Since these two equations share the same gradient m, they
must be solved simultaneously for a given data set. The amount of vertical shift caused by the dimples is simply
given by (P2 – P1).
Ps = mx + P1
Ps = mx + P2
(4)
(5)
To further identify any possible errors given the very small changes to the expected drag, the variation in the
static pressures was examined in greater detail. For the flat channel cases with no dimples, the difference between
the actual measured static pressure and their respective fitted line is plotted for the various distances along the
channel. This difference would correspond to the vertical distance between each data point and the solid line in
Figure 7. When this difference is normalized by the flow Reynolds number, the streamwise variation of this
difference appears to become independent of the Reynolds number. Figure 10 shows some examples of this
variation when normalized using their respective dynamic pressures, and their relative independence of the Reynolds
number.
Figure 10. Variation of difference in measured static pressures from least squares fit.
When the difference between the measured pressures and their respective least squares fit given by equation 4 is
plotted for the dimple cases, similar trends in their variations result, with the exception of a noticeable vertical shift
for those points that lie downstream of the dimples. An example of this is shown in Figure 11. The diamonds
correspond to data obtained at a Reynolds number of about 30,000 with dimples present, and the crosses represent
data at a similar Reynolds number without dimples. The value on the vertical axis refer to the amount of difference
between the measured static pressure and the line obtained by equation 4 above, expressed as a percentage of the
flow dynamic pressure. Note that for the smooth channel case, equation 5 does not apply as the variation in static
pressure is completely linear through all points, and P1 = P2.
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Figure 11. Variation in difference from line from least squares fit.
Besides the obvious vertical shift of the diamonds in Figure 11 at locations downstream of the dimples, the
vertical magnitude of the shift is similar for all points downstream of the dimples. The trend in the variation remains
the same even with the dimples present, further showing that the dimples do not affect the flow significantly, except
by raising or lowering the absolute static pressure through its effect on drag and its associated pressure drop over the
dimple array. The static pressures upstream of the dimples are not affected by the dimples, and the variation is
similar whether dimples are present or not. This consistency of the results is good news since an objective method
using least squares can be used to fit the lines. It may additionally be noted that a comparison with data obtained
from a flat channel case at comparable Reynolds number is not required, since the variation of the difference in
static pressures from the least squares fit is independent of Reynolds number as Figure 10 shows, at least for the
range encountered in the present study.
From the values of P1 and P2 obtained from the data set corresponding to the diamonds in Figure 11, the vertical
shift can be calculated, and the drag reduction for this case is determined to be about 0.5%. It is also worth noting
that Figure 11 shows data from three consecutive runs, and very limited scatter is observed within the
measurements, showing the consistency of the results. The same can also be said of the measurements presented in
Figure 10, also with three consecutive runs shown for each Reynolds number. Although the change in drag is only
0.5% for the case presented in Figure 11, the observed vertical shift is substantial compared to the small scatter
within the measurements, giving increased confidence of the results. The magnitude of this shift ∆p is largely
affected by the length of the dimple array LD, the change in drag due to the dimple array ∆d and the mean streamwise
pressure gradient Pg according to the following equation:
∆ d LD Pg
1
ρV2
2
(6)
( P1 − P2 ) ( P1 − P2 )
=
LD Pg
LD m
(7)
∆
p
=
where ½ρV2 is the dynamic pressure. From the Colebrook-White equation in Figure 6, it can be seen that the
streamwise pressure gradient Pg is essentially related to the dynamic pressure if the density ρ remains constant.
Increasing the length of the dimple array LD will lead to an increase in the magnitude of the vertical shift observed,
thus increasing the sensitivity of experiment and allowing even very small changes in drag to be measured
confidently. The relatively long dimple section at 2.4 meters in the present study helps in this respect. It may be
further verified that the change in drag ∆d is easily determined by the equation:
∆
d
=
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where m according to equations 4 and 5 represents the mean pressure gradient Pg. A positive value means a drag
increase, while a negative value means a drag reduction. This method is applicable not just to study dimples but also
to any other studies involving the effect of a geometry or device on drag in a channel flow.
Using this method, the effect of the dimples on drag may be determined for various Reynolds numbers. Figure
12 shows the result when this method is employed to determine the effect of the dimples on drag when the coverage
ratio is 40%. The percentage change in drag refers to the change when compared to the flat smooth channel. A
positive value means that the drag with the dimples is increased, and a negative value means the drag is reduced
compared to a flat smooth channel at the same Reynolds number. The effect clearly depends on the Reynolds
number of the flow. Below a Reynolds number of about 10,000, transition effects in the channel render the
measurements inaccurate. Drag reduction is observed only for Reynolds numbers above about 25,000. Below that
drag increases rapidly as the Reynolds number is reduced. Maximum drag reduction appears to occur at a Reynolds
number of about 30,000, with a drag reduction of about 0.7%. Beyond that, the drag reduction decreases back to
about zero at a Reynolds number of about 36,000. The scatter within the data is also relatively small with the use of
the least squares fit, showing again the consistency of the measurements.
A plate with a higher coverage ratio was also made and experimental runs with it yielded the result shown in
Figure 13. At this higher coverage ratio, the effect on drag was also Reynolds number dependent. Most significantly,
no minimum is observed within the measured range of Reynolds numbers this time. The drag steadily decreases as
Reynolds number increases, with a drag reduction of about 2% at a Reynolds number of about 40,000. It is likely
that increasing the Reynolds number still further may yield a greater drag reduction, but unfortunately, the present
facility does not allow this. At low Reynolds numbers, the dimples causes drag to increase at a higher rate than when
the coverage ratio is 40%. The cross-over point when drag increases become drag reductions occur at a Reynolds
number between 25,000 and 30,000, similar to that at the lower coverage ratio of 40%.
Figure 12. Effect of dimple array with area coverage ratio = 40% on drag.
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Figure 13. Effect of dimple array with area coverage ratio = 90% on drag.
IV. Conclusion
The results presented show simply the effect of dimples on drag over a range of Reynolds numbers. Although the
results are presently limited, they show significantly that dimple arrays have the potential to reduce turbulent drag in
a channel flow. The effect on drag is dependent on both the Reynolds numbers and the coverage ratio of the dimple
array. A greater coverage ratio tends to magnify the effect the dimples have on drag, so that at low Reynolds
numbers, a 90% coverage ratio will yield a greater amount of drag increase compared to the case with a coverage
ratio is 40%. Similarly, the potential for drag reduction is also increased for the higher coverage ratio at Reynolds
numbers where drag reduction occurs. Comparison with results in the literature is difficult as much depends on the
detailed parameters of the dimple array such as dimple geometry and area coverage ratio. Unfortunately, much of
such details are unreported in the relatively few studies involving shallow dimples for drag reduction, making proper
comparisons difficult. More work is planned and remains to be done to understand what is happening to the flow. Of
great interest is the extension of the Reynolds number range to observe what happens to the drag when the Reynolds
number is increased further, particularly for the case with a coverage ratio of 90%.
The present method of drag determination is useful for quantifying small changes in drag caused by changes in
geometry such as the shallow dimples studied here. The effects of drag altering devices in channel flows may also
be quantified using this method. This method though accurate, requires several practical considerations to be made.
It requires a relatively long channel to allow the flow to reach its fully developed state three times along the channel.
In the case of the present study, the flow has to become fully developed before the dimple array and provide
sufficient distance to allow the accurate determination of the streamwise pressure gradient. The flow then has to be
allowed to “fully develop” again and reach some kind of equilibrium over most of the dimple array. This
equilibrium will likely be cyclical over the wavelength of the dimple array for this present study. Finally, the flow
will need to again fully develop over the downstream flat section and allow the streamwise pressure gradient to
again reach its fully developed value. The distance for a flow to become fully developed is suggested to be as short
as 15 pipe diameters in a pipe flow 16. However, such studies usually measure entrance lengths where the inlet
velocity profile is a top hat profile. For a flow such as that in the present study, where the velocity profile of the flow
going into and out of the dimple section is already somewhat parabolic, the flow may take even shorter lengths than
this to become fully develop again as Figure 9 shows. The channel used in the present study has a total length of 400
channel heights to allow proper flow development to take place. Other studies involving geometries or devices that
cause greater perturbation to the mean velocity profile may require longer distances for the streamwise pressure
gradient to reach its fully developed value.
To further increase the sensitivity and accuracy of the experiment, the dimple array or test section may be made
relatively long. A test section that is too short, besides decreasing the sensitivity of the set-up, will result in
underestimating the changes in drag as the flow does not have time to reach a new equilibrium over the test section.
Due to the relatively long channel length, an accurate pressure transducer has to be used to measure both the small
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static pressures at the upstream end as well as the large static pressures at the downstream end satisfactorily. The
accuracy of pressure transducers are often rated in terms of full scale pressures, and one with a low accuracy may
not be able measure the smaller upstream pressures accurately while at the same time measure the much larger
pressures at the downstream end of the channel.
Acknowledgments
This work was supported by a research contract from Airbus Operations Ltd.
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