experimental study on momentum transfer of surface texture
TRANSCRIPT
ORIGINAL ARTICLE
Experimental Study on Momentum Transfer of Surface Texturein Taylor-Couette Flow
Yabo XUE1,2• Zhenqiang YAO1,2
• De CHENG1,2
Received: 21 November 2016 / Revised: 10 January 2017 / Accepted: 9 February 2017 / Published online: 21 March 2017
� Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2017
Abstract The behavior of Taylor-Couette (TC) flow has
been extensively studied. However, no suitable torque
prediction models exist for high-capacity fluid machinery.
The Eckhardt-Grossmann-Lohse (EGL) theory, derived
based on the Navier–Stokes equations, is proposed to
model torque behavior. This theory suggests that surfaces
are the significant energy transfer interfaces between
cylinders and annular flow. This study mainly focuses on
the effects of surface texture on momentum transfer
behavior through global torque measurement. First, a
power-law torque behavior model is built to reveal the
relationship between dimensionless torque and the Taylor
number based on the EGL theory. Second, TC flow appa-
ratus is designed and built based on the CNC machine tool
to verify the torque behavior model. Third, four surface
texture films are tested to check the effects of surface
texture on momentum transfer. A stereo microscope and
three-dimensional topography instrument are employed to
analyze surface morphology. Global torque behavior is
measured by rotating a multi component dynamometer, and
the effects of surface texture on the annular flow behavior
are observed via images obtained using a high-speed
camera. Finally, torque behaviors under four different
surface conditions are fitted and compared. The experi-
mental results indicate that surface textures have a
remarkable influence on torque behavior, and that the peak
roughness of surface texture enhances the momentum
transfer by strengthening the fluctuation in the TC flow.
Keywords Torque behavior � Momentum transfer
behavior � Surface texture � Experimental study � Taylor-Couette flow (TC flow)
1 Introduction
Flow behavior is an antiquated but vigorously researched
topic. Researchers from various fields, including physics,
mechanics, and engineering, are greatly interested in this
topic. After Taylor’s pioneering exploration [1–3], Taylor-
Couette (TC) flow received much attention. The TC flow
consists of two concentric rotating cylinders, as shown in
Fig. 1. In this figure, line O1O2 is the corotating axis of the
inner and outer cylinders, and R1 and R2 are the inner and
outer radii, respectively. The inner cylinder rotates with an
angular speed of x. The fluid is confined in the annular gap
between the two cylinders, and because of the viscosity of
fluid, it rotates with the cylinders. This system is usually
employed for measuring viscosity, verifying flow stability
theory [4–6], investigating momentum transfer behavior
[7–9], and measuring torque behavior [10–12]. Among
these topics, torque behavior has been discussed for many
years, and numerous models were proposed to understand
the momentum transfer mechanism.
The flow between rotating plates and TC flow have been
investigated as typical rotating flow systems by many
researchers. Unlike the TC flow, the flow between rotating
plates has been studied well in the case of the wet clutch
Supported by National Programs for Fundamental Research and
Development of China (Grant Nos. 2009CB724308, 2015CB057302),
and National Science and Technology Major Project of China (Grant
No. 2013ZX06002002-017).
& Zhenqiang YAO
1 State Key Laboratory of Mechanical System and Vibration,
Shanghai Jiao Tong University, Shanghai 200240, China
2 School of Mechanical Engineering, Shanghai Jiao Tong
University, Shanghai 200240, China
123
Chin. J. Mech. Eng. (2017) 30:754–761
DOI 10.1007/s10033-017-0094-4
[13–15]. However, torque behavior is much more compli-
cated in a TC flow. In the 1990s, LATHROP, et al [16],
focused on this topic and argued that the flow should have
an upper bound based on Kolmogorov’s turbulence theory.
Further, flow transition was found to occur at a Reynolds
number of 1.3 9 104. However, the upper bound model
greatly deviated from the measured data. Based on the
observation of flow state transition, LATHROP, et al [17],
LEWIS, et al [18], and DUBRULLE, et al [8], separately
proposed composite models to describe the torque behav-
ior. Although a great deal of work was conducted in
Lohse’s lab, experimental results indicated that there
existed no universal scaling law based on the Reynolds
analogy [20, 22, 25].
High-capacity fluid machinery poses the challenge of
torque prediction in turbulent TC flow. In some cases, the
torque behavior models fail. Meanwhile, a few researchers
dealt with small gap TC flows [21]. New models have been
explored for such flows [22–25]. Refs. [22, 24] discuss the
different roles of boundary layer and turbulent bulk flows.
Based on these discussions, the bulk flow theory was
proposed to reveal the roles of the turbulent core regime
and boundary layer. Correspondingly, a linear weighted
model was proposed to describe torque behavior [25].
Recently, the Eckhardt-Grossmann-Lohse (EGL) theory
was proposed to gain a better understanding of momentum
transfer behavior [7, 8, 23]. They argued that the TC flow
system and Rayleigh-Bernard (RB) convection system
should have identical dynamic behavior [23]. Therefore,
the flux of angular velocity is derived from the Navier–
Stokes equations, in analogy to RB convection [23].
Meanwhile, they also argued that the radial flux of angular
velocity remains constant in the narrow gap between the
two cylinders. Many researchers have since worked on the
EGL theory, leading to some excellent results. The
relationship between the flux of angular velocity and
Taylor number was proposed in recent study to build a new
torque behavior model. However, van den Berg’s work
indicates that smooth and rough surfaces exhibit different
torque behavior in a TC flow [25]. WU’s work indicates
that smooth and non-smooth surface perform a different
lubrication behavior [26]. Thus, how does the surface
texture affect torque behavior is a topic to be explored.
In this study, the effect of surface texture on momentum
transfer behavior was investigated through torque mea-
surement and high-speed photography. This paper is
organized as follows. First, a momentum transfer behavior
model is introduced to describe the mechanism of torque
behavior. Second, the TC apparatus built in our lab and the
surface-testing instruments are described. Third, analysis
and comparison of the surface textures, which were
observed through a stereo microscope and a three-dimen-
sional topography instrument, are presented. Thereafter,
the experimental work and results are described. Finally,
the conclusions are drawn based on a discussion of the
experimental results.
2 Momentum Transfer in Taylor-Couette Flow
Momentum transfer behavior was widely investigated in
previous works. However, experimental results indicate
that there is no uniform scaling law based on the Reynolds
analogy. The similarity in the dynamic behavior of TC flow
and RB convection throws light upon this problem. The
corresponding parameters of both systems were compared
[23]. The EGL theory was proposed based on collaborating
works, and the Taylor number is defined as shown in
Eqs. (1) and (2), in analogy to the Rayleigh number [19], to
characterize the flow state:
Ta ¼ 1
4rðR2 � R1Þ2ðR1 þ R2Þ2x2
m2; ð1Þ
r ¼ ð1� gÞ=2ffiffiffi
gp
� �4
; ð2Þ
where r is a geometric parameter, and m represents the
kinematic viscosity of fluid.
As known, the momentum transfer behavior in the TC
flow varies with changes in the flow state. When Ta\ Tac,
the flow is laminar, and the flux of the angular velocity can
be derived from the velocity profile in the TC flow, as
shown in Eq. (3):
Jx;lam ¼ 1
4mR21R
22ðR1 þ R2Þ2xðR2 � R1Þ2
; ð3Þ
where Jx,lam is the laminar flux of angular velocity [19].
Fig. 1 Schematic diagram of Taylor-Couette flow
Experimental Study on Momentum Transfer of Surface Texture in Taylor-Couette Flow 755
123
When Ta[Tac, vortexes appear, and the flow becomes
turbulent. Because of this phenomenon, the flux of angular
velocity cannot be derived from the velocity profile of the
TC flow. Therefore, the momentum transfer behavior has
to be modeled in a different manner. Direct numerical
simulation shows that convection transfer dominates the
turbulent bulk flow, and molecular viscosity transfer
dominates the boundary layer flow [28]. As shown by
Eq. (4), the flux of angular velocity is the sum of the
convection term and molecular viscosity term:
Jx ¼ r3ð\urus [ A;t � mor\x[ A;tÞ: ð4Þ
Although the flux of angular velocity reveals the
momentum transfer behavior, it cannot be directly used to
model the torque behavior. In analogy to RB convection,
the turbulent flux of angular velocity is made dimension-
less by dividing it by the laminar flux of angular velocity
[19]; here, the dimensionless flux of angular velocity, Nux,
is defined as in Eq. (5):
Nux ¼ Jx
Jx;lam: ð5Þ
This theory makes it possible to build a torque behavior
model with a broad scope. However, such a model should
be built based on experiments. Based on previous torque
behavior modeling methods, the torque was made dimen-
sionless via Eq. (6) [28], and the dimensionless torque was
calculated as shown in Eq. (7) [19]:
G ¼ T
2pHqm2; ð6Þ
G ¼ NuxJx;lamm2: ð7Þ
The dimensionless flux of angular velocity, Nux, can be
described by the power law [29], as shown in Eq. (8):
Nux / Taa: ð8Þ
Based on Eq. (8), the relationship between the dimen-
sionless torque G and Taylor number Ta is defined as in
Eq. (9):
G ¼ cTaa; ð9Þ
where c is a constant and a is the exponent.
HUISMAN’s work indicated that the exponent remains
constant [29]. This theory seems suitable for modeling
torque behavior. However, VAN DEN BERG’s work
showed that the surface of between fluid and solid has a
remarkable influence on torque behavior [25]; further,
Eq. (4) shows that the boundary layer affects the flux of
angular velocity. It is known that momentum is transferred
from the boundary layer of the inner cylinder to the tur-
bulent core region. If surface texture disturbs the boundary
layer of the inner cylinder, and the laminar sub-layer is
destroyed, how does is the momentum transfer behavior
affected? This effects are mainly investigated in the fol-
lowing parts.
3 Experimental Apparatus and Test Instruments
3.1 Experimental Apparatus
The configuration of the TC flow system proposed in this
study is based on the CNC machine tool, with reference to
the T3C flow system in Twente University [27]. As Fig. 2
shows, the system mainly comprises a machine tool, rotor,
stator, dynamometer, camera, and data acquisition system.
The stator is fixed on the table, and the rotor is installed on
the spindle through a dynamometer. The system is
assembled using a chuck, and the rotor can be easily dis-
assembled to cover it by surface films.
As shown in Figs. 2 and 3, the TC flow apparatus was
built. The inner cylinder is made of an aluminum alloy, and
the outer cylinder is made of acrylic glass for observing the
flow state. The narrow gap between the inner and outer
cylinders is filled with water. The radii of inner and outer
cylinders are R1 = 65.0 mm and R2 = 69.0 mm, respec-
tively, and the rotor length is H = 385.0 mm for raising
the resonance frequency. Thus, a radius ratio of g = 0.942
and aspect ratio of C = 96.25 are attained. The detailed
parameters are listed in Table 1. The inner cylinder is
controlled by a spindle, and it is driven up to a rotation rate
of f1 = 133.3 Hz and the outer cylinder remains stationary.
As shown in Table 2, water with a kinematic viscosity of
m = 9.12 9 10-7 m2/s and density of q = 995.1 kg/m3 (at
a room temperature of 24 �C) is used as the working fluid.
Fig. 2 Schematic diagram of the experimental apparatus
756 Yabo XUE et al.
123
When the spindle is rotated, the driving torque is measured
by a dynamometer. Meanwhile, the dynamic behavior of
the annular flow is captured using a high-speed camera, as
shown in Fig. 3.
During the experiment, four different surface films were
tested. Only one rotor was used to reduce the manufac-
turing errors and to make the results comparable. Figure 4
shows the rotor covered with one of the selected surface
films. The rotor surface was covered with different surface
texture films separately to study their individual effects.
When the effects of different films were tested, the rotor
was removed from the spindle.
3.2 Surface Topography Measurement Instruments
The effects of surface texture on momentum transfer
behavior are discussed in this paper. A stereo microscope
and three-dimensional topography instrument were
employed to analyze the surface textures. Figure 5 shows
the stereo microscope, and Fig. 6 shows the three-dimen-
sional topography instrument. The indexes of the stereo
microscope are listed in Table 3. For a table size is
180 mm 9 180 mm, the scan area was
100 mm 9 100 mm; the maximum scan speed was
150 mm/s; the minimum resolution was 0.1 lm; the
repeatability precision was ± 0.5 lm; and the absolute
precision was 4 lm. The indexes of the three-dimensional
topography instrument are listed in Table 4; the minimum
magnification was 6.59, and the maximum magnification
was 509. Further, the working distance was 90 mm, and
the largest distance of the sample was 35 mm.
4 Surface Topography Measurement
The EGL theory indicates that surface texture affects the
flux of angular velocity and enhances momentum transfer.
Enhancement of momentum transfer will affect the torque
Fig. 3 Configuration of the apparatus for studying Taylor-Couette
flow
Table 1 Parameters of Taylor-Couette flow system
Parameter Value
Radius of the inner cylinder R1/mm 65.0
Radius of the outer cylinder R2/mm 69.0
Width of the annular gap d/mm 4.0
Height of the cylinders H/mm 385.0
Gap ratio g 0.94
Aspect ratio C 96.25
Table 2 Physical parameters of working fluid (24 �C)
Fluid Kinematic viscosity m/(m2�s-1) Density q/(kg�m-3)
Water 9.12 9 10-7 995.1
Fig. 4 Test rotor covered by a smooth surface film
Fig. 5 Stereo microscope used in the study
Experimental Study on Momentum Transfer of Surface Texture in Taylor-Couette Flow 757
123
behavior. To verify the hypothesis that surface texture can
lead to enhancement of momentum transfer, four different
surface films were chosen, as shown in Fig. 7. Figure 7(a),
(b), (c), and (d) show a smooth surface, a rough surface
with small pits, a rough surface with cross ribs, and a rough
surface with sharp angles, respectively.
To investigate the effects of surface texture on
momentum transfer, the surfaces were observed using the
stereo microscope; Fig. 8 shows the photographs of the
surfaces obtained using the microscope. Figure 8(a), (b),
(c), and (d) show the smooth surface film, rough surface
film with small pits, rough surface film with cross ribs, and
rough surface film with sharp angles, respectively. The
surfaces become rougher in the order shown in the figure,
and they represent four different manufacturing surfaces.
To further compare the surface textures, 3D photographs
(Fig. 9) of these four surface textures were obtained using
the three-dimensional topography instrument. Figure 9(a),
(b), (c), and (d) show the smooth surface film, rough sur-
face film with small pits, rough surface film with cross ribs,
and rough surface film with sharp angles, respectively.
Table 5 lists the surface roughness of these four films. The
average roughness of the surface films ranges from 1 lm to
25 lm, and the peak roughness varies from 4.5 lm to
72.7 lm.
Fig. 6 Three-dimensional topography instrument
Table 3 Parameters of stereo microscope system
Parameter Value
Table size A/mm2 180 9 180
XY scan area A1/mm2 100 9 100
Max. speed U/(mm�s-1) 150
Min. resolution Pm/lm 0.1
Repeatability precision PR/lm ±0.5
Absolute precision PA/lm 4
Table 4 Parameters of three-dimensional topography instrument
system
Parameter Value
Min. magnification Kmin 6.59
Max. magnification Kmax 509
Working distance S/mm 90
Largest distance of sample S1/mm 35
Fig. 7 Films with different surface textures
Fig. 8 Stereo microscope photographs showing the surface textures
of the four films
758 Yabo XUE et al.
123
5 Flow Behavior: Experiments and Results
The EGL theory indicates that the boundary layer plays a
significant role in momentum transfer and leads to differ-
ences in the global torque behavior. Based on this idea, the
global torque behavior was tested using our TC flow
apparatus to study the effect of surface texture on
momentum transfer behavior. During the experiment, tor-
que data were acquired in every flow state at intervals of
100 r/min, from 100 r/min to 2000 r/min. The EGL model
indicates that the molecular transport is affected by these
four surface textures. In order to gain comprehensive
understanding of the dynamic behavior in annular flow, a
high-speed camera was used to capture the flow states in
the narrow gap for rotational speeds varying from 800
r/min to 2000 r/min.
5.1 Fluctuation of Torque Behavior
Through the momentum transfer behavior experiments, the
torque values for the four surface textures were measured
by a dynamometer. Figure 10 shows the original torque
signals of the four samples. As shown, the fluctuation in the
torque increases with speed. Meanwhile, the fluctuation
seems more intensive for rougher surfaces. The more
fluctuation in the flow, the more is the energy consumed by
the system. The fluctuation in the annular flow enhances
momentum transfer.
5.2 Fluctuation of the Flow State
To observe the annular flow states of these four films, a
high-speed camera was used in the experiment. Figure 11
shows the changes in the flow state under these four surface
conditions; the variation in flow states at typical speeds of
800 r/min, 1400 r/min, and 2000 r/min are shown. The flow
states show no differences at a low speed; however, the
differences become apparent at high speeds. In these
experiments, several girdle-like flows appeared, and they
became more apparent as the angular speed increased. The
transition appeared earlier in the case of the rough surface
than in the case of the smooth surface. When the surface
was very rough, girdle-like flows vanished early as seen
from the flow state of sample 4 in Fig. 11.
5.3 Effects of Surface Texture on Torque Behavior
The torque data for the four surface conditions were made
dimensionless by using Eq. (6). The Taylor number for
each speed was calculated using Eq. (1). Figure 12 shows
that the torque varies with the Taylor number under dif-
ferent surface conditions. The blue, green, red, and brown
lines represent the measured data for the smooth surface,
rough surface with small pits, rough surface with cross ribs,
and rough surface with sharp angles, respectively. As seen
from the figure, torque clearly increased with the Taylor
number, and the torque behavior showed big differences
among the four surfaces as the rotating speed increased.
The torque behavior under all the surface conditions
remained consistent at low speed. However, they become
remarkably distinct when the speed exceeded 500 r/min. In
addition, the difference in torque behavior increased with
speed. The torque increased by more than 50% when the
roughness of the test surfaces increased from 4.5 lm to
72 lm.
The flux of angular velocity in the laminar flow
remained consistent as suggested by Eq. (3). However,
torque behavior showed great differences at speeds[ 500
r/min. The torque data at high speeds ([ 500 r/min) were
fitted using Eq. (9). During data fitting, the relationship,
log G = log c ? a log Ta, was used instead of Eq. (9) to
fit the data. As shown in Fig. 13, the blue, green, red, and
brown circles represent the measured data for the smooth
surface, rough surface with small pits, rough surface with
cross ribs, and rough surface with sharp angles,
Fig. 9 Surface topography images obtained using the three-dimen-
sional topography instrument
Table 5 Roughness of the surface films
Sample Average roughness
Ra/lmPeak roughness
Rp/lm
1# (Smooth surface) 1.002 4.508
2# (Rough surface with
small pits)
5.176 9.108
3# (Rough surface with
cross ribs)
4.652 17.485
4# (Rough surface with
sharp angles)
24.535 72.719
Experimental Study on Momentum Transfer of Surface Texture in Taylor-Couette Flow 759
123
respectively. Correspondingly, the lines of the same color
were fitted using the measured data. Here, ci and ai are thefitting constants of sample i (i = 1, 2, 3, and 4). As shown,
the fitting lines and measured data keep consistent well
with speed varying from 500 r/min to 2000 r/min. How-
ever, the torque behaves differently under the four surface
textures. The torque of the smooth surface increases more
slowly than that of the rough surfaces, and the exponent
increases from 0.91 to 1.03.
6 Discussion and Conclusions
(1) In this paper, a power-law model of torque behavior
based on EGL theory is proposed to associate the
torque behavior with the Taylor number. Following
this work, the experimental apparatus was designed
and built in our lab based on the CNC machine tool to
verify the torque behavior. The results indicate that the
power-law model fits the experimental data well.
(2) The EGL theory indicates that boundary layer affects
the flux of angular velocity and the torque behavior.
Based on this idea, experiments employing our TCflow
apparatus were designed and conducted to investigate
the surface texture effect on momentum transfer
behavior. Different films were analyzed and compared
using a stereo microscope and three-dimensional
topography instrument. Further, torque behavior was
measured, and flow states were observed under these
four different surface texture conditions. The torque
behavior obeyed the power law well. However, the
exponent variedunder different surface conditions.The
experimental results prove that the surface texture has a
remarkable influence on torque behavior.
(3) The experimental results indicated that the torque
can be increased by more than 50% by changing the
Fig. 10 Torque signals in the time domain under different surface
conditions
Fig. 11 Dynamic behavior of annular flow as captured by a high-
speed camera
Fig. 12 Torque behavior in TC flow Fig. 13 Data fitting of torque behavior in TC flow
760 Yabo XUE et al.
123
surface texture. Moreover, comparison of the surface
roughness and torque behavior of samples 3 and 4
show that the peak roughness affects momentum
transfer to a greater extent than does the average
roughness. Finally, it can be concluded that the
convex ribs destroy the boundary layer more easily
than do the concave pits.
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Yabo XUE, born in 1986, is currently a PhD candidate at School of
Mechanical Engineering, Shanghai Jiao Tong University, China.
E-mail: [email protected]
Zhenqiang YAO, born in 1962, is currently a professor at School of
Mechanical Engineering, Shanghai Jiao Tong University, China. He
received his PhD degree on mechanical engineering from Shanghai
Jiao Tong University, China, in 1982. Tel: ?86-21-34206583;
E-mail: [email protected]
De CHENG, born in 1988, is currently a PhD candidate at School of
Mechanical Engineering, Shanghai Jiao Tong University, China.
E-mail: [email protected]
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