Characteristics of T-S wave in the grooved channels
with different groove length for steady flow
Yongning BIAN1, Hirofumi ARIMA2, Yasuyuki IKEGAMI2, Faming SUN1
1State Key Laboratory of Structural Analysis for Industrial Equipment
Dalian University of Technology, 116024, China
2
Institute of Ocean Energy, Saga University, 840-8502, Japan
Abstract
Pressure drop and time-averaged and time-various local pressure in the grooved channels with six kindsof groove length are measured with the differential transducer and the pressure sensor respectively, and theflow structures are also visualized by the aluminum dust method. The local pressure signal shows that T-S
wave appears in the first or second frequency, and the Strouhal number, based on the nature frequency of
the T-S wave, is almost equivalent for the 1st or 2nd frequency in the same channel. Meanwhile, theStrouhal number for each channel decreases monotonously with the groove length. Furthermore it is foundthat an operation of pressure increasing will result in higher amplitude of the T-S wave, this behavior isvery significant for the efficient heat exchanger in practical engineering.
Key Words: Grooved channel; T-S wave; 1st and 2nd frequency; Amplitude; Pressure increasing or decreasing
1. INTRODUCTION
Heat and mass transfer processes frequently limit thesize, performance and efficiency of practical engineering devices, such as the plate heat exchanger,
which is widely applied to the ocean thermal energy
conversion (OTEC) system. Many investigators devotetheir effort to finding the effective channel shape andthe technique to enhance the heat and mass transfer. As
a fundamental element of plate heat exchanger, twodimensional channels with various wall shapes are
considered to realize this purpose.
The earlier study was carried out in the planechannel. Nishioka et al.[1] conducted an experimental
investigation of the stability of plane Poiseuille flow.
They confirmed that when the Reynolds number
exceeds a critical value, the local velocity fluctuations
display the sinusoidal wave forms. As further increasing the Reynolds number, the wave form deviates
gradually from this regular shape to irregular one. The
above flow behavior is numerically proved by Ghaddar
et al.[2], they concluded that the flow instability in theplane channels displays the form of TollmienSchlichting wave, i.e. the T-S wave, and characterized by the wave number and frequency. Then,
Stephanoff[3] observed the self-sustained shear-layer
oscillation in a wavy channel. He found that when themean velocity through the channel exceeds a criticalvalue, this self-sustained shear-layer oscillation appears
as one, or sometimes two, selectively amplified frequencies. With the similar channel, Wang et al.[4]numerically studied the convective heat transfer. Theirresult shows that the T-S wave will lead to thedestabilization of the laminar thermal boundary layers,
and as a result of the unsteadiness, there is increased
mixing between the core and near-wall fluid, thereby
resulting in enhanced heat transfer rate. Furthermore,
the flow characteristics and the relationship between
fluid mixing and flow instability also have been investigated systematically in the grooved channel by
Nishimura et al.[5~7] and Greiner[8]. Both of theirexperimental and numerical results indicated that theexistence of T-S wave exists in all of two-dimensional
channels, and the nature of T-S wave has a closed
relationship with the heat and mass transfer rate.
Although many studies described the behavior of
T-S wave in the two-dimensional channels, the details
about the effect of the operation conditions on the T-S
wave is scarcely considered even though it is verysignificant for the heat transfer enhancement, it
prompts us to carry out the present experimental study.
In this study, the overall pressure drop and the local
pressure will be measured in the series grooved
channels with different groove length, and then theamplitude of T-S wave will be analyzed for the case ofincreasing or decreasing the channel pressure. Finally,combine with the flow visualization results, thecharacterristics of T-S wave in the grooved channels
will be clarified.
2. EXPERIMENTAL SETUP
The schematic of the experimental apparatus is depicted in Fig. 1. The flow is provided by a centrifugal
pump and the city water is used as the working fluid.
The flow rate is adjusted by the control valve andmeasured with an electromagnetic flowmeter; Q represents the flow rate. The overall pressure drop of thetest section is recorded by a differential transducer andfive pressure sensors are used to measure the timevarious pressure at eight different locations of the test
section. The valve-1 and valve-2, located at the outletof the test section and the downstream of the flowmeter
respectively, are used to obtain the decreasing or increasing pressure operating condition. The aluminum
dust method is used to realize the flow visualizationand the flow patterns are recorded by a high vision
digital video camera. The dimensions of the test section

Fig. 1

Schematic diagram of the experimental
systemFig. 2Dimensions of the test section

Fig. 3are shown in Fig. 2. Total six kinds of grooved
channels are examined in this study. Every grooved
period length is L=20mm, W=200mm, h=2.5mm andthe grooved lengths are l=4, 6, 8, 10, 12, 14mm,
respectively. The local pressures are measured at eight
different locations of the grooved channel, as shown in
Fig. 3. The Reynolds number Re, Strouhal number St
and friction factor f are calculated according to thefollowing definitions:(1)
Re = ρ uh / µ
St = fT h / u
(2)
2
(3)
f = h∆P /(2 LP ρ u )where, fT is the frequency of T-S wave, ρ and µ are thedensity and viscosity of water, respectively. u=(3/2)um
is the characteristic flow velocity, here, um=Q/(2Wh).
∆P is the overall pressure drop. LP=1295mm, is the
distance of the taps between the inlet and outlet. All theexperiments are conducted at the room temperature.
3. RESULTS AND DISCUSSION
3.1 Effect of the groove length on the critical Reynolds number
The friction factor, obtained by the pressure drop, is
one of the parameters reflecting fluid mechanicalcharacteristics in channels. The relationships between
the overall friction factor and the Reynolds number inthe six kinds of grooved channels are shown in Fig. 4.
It is seen that the lines of f-Re for all channels almost
have the same slop in laminar flow regime, but theirdevelopment becomes a little different when the Reynolds numbers Re exceed their critical value Rec, i.e.,the Rec will increase with the decreasing of the groove
length, meanwhile, the friction factor increases gradually. Figure 5 shows the effect of the groove length
on the Rec. For the series grooved channels used in thisstudy, the Rec approximately decreases in linear with
the increment of the groove length. According to thisresult, the T-S wave only occurs in the upper area ofthe dashed line.
3.2 Local pressure in different channels
The local pressures in the eight different locations
are measured for all the grooved channels. Figure 6
shows the representative result obtained at P3, P4 andLocations of the local pressure measurement

100

P7 from the channel with l=8mm. Obviously, the local
pressure is nearly same when Re<1500, and the P7
decreases a little with further increasing the Re, means
the pressure loss is increase contrast to the P3 and P4
and the power consumption must be considered fully
when the flow speed is high enough. The comparisonof local pressure for all channels at the same P7 isdisplayed in Fig. 7. All of the curves close each other,indicating that that all of the grooved channels have thesame characteristics on the flow resistance.

10-1

f [-]

10-2

10-3

10-4

l=4, Re c=1127
l=6, Re c=1049
l=8, Re c=807
l=10, Rec=604
l=12, Rec=532
l=14, Rec=379

10-5 0
10

101

102

103

104

Re [-]

Fig. 4

f vs.Re in all channels

1500

Rec [-]

1200

T-S wave

900
600

no T-S wave

300
0
0

2

4

6

8

10

12

14

16

18

Groove length l [mm]Relationship between Rec and groove length30

Local Pressure P [kPa]

25
20
15
10
P3
P4
P7

5
0

0

500

1000

1500

2000

2500

3000

Re [-]

Fig. 6

P vs Re in the channel l=8 mm

Fig. 8T-S wave in the channel l=8 mm

Fig. 9T-S wave in the channel l=10 mm

30
25Local Pressure P7 [kPa]

Fig. 5

3.3 Tim-variation of the local pressure
To check the relationship between the groove lengthand the nature frequency of the T-S wave, the timevariation local pressures are measured in different
location for all channels. The T-S waves have been
captured in three channels, that is l=8,10,12mm. In theother three channels, the T-S wave could not berecorded. The reason for this is considered to the limitof the maximum flow rate and the sensitivity of thepressure sensor. Figure 8~10 show the measurement
results at P3 in the three kinds of grooved channels. It
is found that the oscillations in the 1st and the 2nd

20
15
l=4
l=6
l=8
l=10
l=12
l=14

10
5
0

0

500

1000

1500

2000

2500

3000

Re [-]

Fig. 7

P vs Re in all grooved channels

quency is obtained, the nature frequency of the T-S
wave at any Reynolds number could be calculated.
Based on the averaged St for each used grooved
channel in this study, a simple relationship could beconcluded, that is, the Strouhal number decreases
monotonously with the groove length, as shown in
Fig.11. According to this conclusion, the St could be
calculated conveniently for the channels with l=8~12mm.

Fig.10T-S wave in the channel l=12 mmTable 1 Nature frequency of T-S wave in different
channels
1st frequency

l
(mm)

Re

fT(Hz)

St

8

1052

18.5

0.1099

1239

18.8

1495
1914
2063

19.3

0.058472245

200.0556810

12

2nd frequency
Re

fT(Hz)

St

1586

30

0.11821760

31

0.1101

0.094832432

330.0848120

0.08361

2607

33.5

0.0803118.8

0.06139

3.4 Amplitude of T-S wave at in different locations
To have a insight to the amplitude of T-S wave in thechannels, the T-S wave are captured at different locations for the three kinds of channels, and then all of theexperimental data are processed with a program filter
and the amplitude variation of the T-S wave with theflow rate are obtained as shown in Fig. 12~14. As
described in the former, it is clear to see that the T-S
wave in the 1st and the 2nd frequency exist in thechannels with l=8 and 10, but only the 2nd frequency
exists in the channel wit l=12. Moreover, the amplitude
of T-S wave in the channel with l=10 is nearly twotimes higher than it in the other two channels, whateverfor the 1st or the 2nd frequency. Another characteristic

0.15

St [-]

0.1

0.05
St=-0.013548 l+0.22118

0
6

8

10

12

14

Groove length l [mm]Fig. 11

Relationship between St and groove length

frequency appear in Fig.8 and 9, but only the 2nd frequency is found in Fig. 10. From these three figures,the frequency of the T-S wave in the 1st frequency andthe 2nd frequency could be read easily. Based on theformula (1) and (2), the Strouhal numbers are calculated and shown in Table 1. From this table it is seenthat for the same channel, whatever the 1st or the 2nd
frequency, the value of the Strouhal numbers are very
closed to each other, and the maximum error does notexceed 18%. This result agrees well with Stephanoff’s
study[3]. This fact is very important for the practicalengineering; it means that once the 1st or the 2nd fre-Fig. 12

Amplitude in the channel l=8 mm

Fig. 13

Amplitude in the channel l=10 mm

Fig. 14

Amplitude in the channel l=12 mm

Fig. 16

Effects of pressure on amplitude in thechannel l=10 mm at P4

Fig. 17

Effects of pressure on amplitude in thechannel l=12 mm at P2of T-S wave, as shown in Fig. 12, could be found thatthe higher of the amplitude in the 1st frequency, thelower of it in the 2nd frequency, the same tendency isalso seen in Fig. 13.
3.5 Effects of pressure increasing or decreasing on
the amplitude of T-S wave
To understand the amplitude variation of the T-S
wave, the effect of pressure on it is explored. The
condition of pressure increasing or decreasing could beobtained by adjust the valve-2 or valve-1, and thevariation of the pressure is controlled withinΔQ=±
50ml/s. Some experimental results are shown in Fig.15~17. For two locations in the same channel l=10, increasing pressure could make the amplitude of the T-S
wave higher than no pressure change, and in the otherside, decreasing pressure could make the amplitude of
T-S wave lower than no pressure change, as shown in
Fig. 15 and 16. For the other channel l=12, as shown in
Fig. 17, the similar phenomenon is displayed. This
result is very significant for the relative operation of
the higher efficient heat exchanger in practical engineering. Generally, there has an optimum value ofFig. 15

Effects of pressure on amplitude in thechannel l=10 mm at P7

pressure increasing, in which, the maximum amplitude
of T-S wave with minimum power consumption should
be attained. It is an important work in the future.
3.6 Characteristics of the flow structure related to
the T-S wave
It is known that the efficient heat and mass transfer
must accompany the fluid mixing. To examine therelationship between the flow structure and the T-S
wave, flow visualization is finished in the channel
l=12mm, and some representative flow patterns areshown in Fig. 18. Obviously, in the laminar flow
regime at Re=188, two stable large vortex form in thegroove and the streamlines are arrayed in parallel. But
after the flow enters transitional flow regime, the flow
oscillation could be recognized at Re=532 and 600,
although the weak oscillation could not be captured by
the pressure sensor. It is seen that the fluid mixing
occurs between the groove and the mainstream since
the flow oscillation and hence destroys the vortex. As
the flow enters turbulent flow regime at Re=1875, theflow oscillation in higher frequency appears and theflow mixing is further enhanced.
Laminar flow patterns
Re=188Transitional flow patterns
Re=532It means that once the 1st or the 2nd frequency is
obtained, the nature frequency of the T-S wave at
any Reynolds number could be calculated for thegrooved channel with a constant period length.
(2) The operating condition of pressure increasing or
decreasing strongly affects the amplitude of the T-S
wave. Increasing the pressure of the channel will
make the amplitude higher, and there must exist an
optimum value of pressure increasing. This conclusion is very significant for the operation of thehigher efficient heat exchanger. Confirming thisoptimum value is an important work in the future.
REFERENCES

Re=600Turbulent flow patterns
Re=1875

Fig. 18

Flow patterns in different flow regime
in the channel l=12 mm

4. CONCLUSIONSIn the present study, the characteristics of T-S wave
in the grooved channels with different groove length
are explored experimentally. The most important results are concluded as follows:(1) The T-S wave could be captured by analyzing thetime-variation local pressure signal. The Strouhal
number, based on the 1st or the 2nd frequency of
the T-S wave, is almost equivalent for the samechannel. Furthermore, the Strouhal number is foundto decrease monotonously with the groove length.

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