Characteristics of flow behavior and heat transfer in the grooved channel for pulsatile flow with a reverse flow

International Journal of Heat and Mass Transfer 147 (2020) 118932

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Characteristics of flow behavior and heat transfer in the grooved channel for pulsatile flow with a reverse flow Junxiu Pan a, Yongning Bian a,⇑, Yang Liu a,⇑, Fengge Zhang c, Yunjie Yang a, Hirofumi Arima b a

State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China Institute of Ocean Energy, Saga University, Japan c Harbin Turbine Company Limited, China b

a r t i c l e

i n f o

Article history: Received 13 June 2019 Received in revised form 1 September 2019 Accepted 20 October 2019

Keywords: Pulsatile flow Grooved channel Heat transfer enhancement Oscillatory fraction Flow visualization

a b s t r a c t The present study investigates the flow behavior and heat transfer in the grooved channel for pulsatile flow with a reverse flow by experimental and numerical approaches at different Strouhal numbers (from 0 to 0.125) and different oscillatory fractions (from 0.6 to 1.4). The pulsatile flow patterns are visualized by the aluminum dust method, and the numerical model is validated by experimental results. It is observed that the flow is less stable in the reverse acceleration phase. At the same time, streamlines, temperature and vorticity in the upper and lower grooves are asymmetrical. Besides, this instability leads to a remarkable mix between the groove and the main flow and contributes to the enhancement of heat transfer. In addition, the onset of the unstable state gradually delays and the duration of the unstable state reduces with the increase of Strouhal number during the test range when the net Reynolds number is 375 and the oscillatory fraction is 1.4. Also, it demonstrates that the time-averaged Nusselt number increases with oscillatory fraction which suggests that the imposed reverse flow is capable of improving the heat transfer. It is further revealed that the maximum heat transfer enhancement factor is 2.74 when the oscillatory fraction is 1.4. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Ocean thermal energy conversion (OTEC) plays a pivotal role in the development and utilization of clean energy, which takes the advantage of the temperature difference between the deep sea (3–10 °C) and the surface sea (20–30 °C) [1]. However, the thermal efficiency of OTEC is relatively low even for the optimized Kalina cycle due to the limited temperature difference [2]. Therefore, it is crucial to improve the thermal efficiency of the heat exchangers adopted in the system. Many researchers are dedicated to explore the optimum geometry of the channels and tubes in the thermoplates and the effective techniques for the enhancement of the heat transfer [3–7]. Regarding the geometry of the channel, the contracted and expanded channels have been extensively taken into consideration. Bian et al. [8] conducted experimental researches about the characteristics in a wavy-walled tube with different dimensions for steady flow. Results show that the mass transfer can reach 1.3–4 times of that in the straight–walled tube. Mendes et al. [9] ⇑ Corresponding author at: State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China E-mail addresses: [email protected] (Y. Bian), [email protected] (Y. Liu). https://doi.org/10.1016/j.ijheatmasstransfer.2019.118932 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.

studied the correlation between the pressure drop and the heat transfer in the converging-diverging tubes experimentally. Their results suggested that the enhancement of heat transfer is accompanied by the increase of the pressure drop. Gutierrez et al. [10] added curved flow deflectors in a grooved channel, and they found that the deflectors contribute to the enhancement of the heat transfer in the grooved channel because the deflectors can enhance the fluid mix between the grooved area and the main flow. Similarly, Selimefendigil et al. [11] studied the effects of the location of the barriers added in the channel on the enhancement of the heat transfer, their results showed that when Re = 200, the maximum heat transfer enhancement can reach 228% in the presence of square obstacle compared to the no-obstacle case when the obstacle is located at a proper location. Wu et al. [12] numerically analyzed the heat transfer in a rectangular channel with longitudinal vortex generator (LVG). It was found that the essence of the heat transfer enhancement by LVG can also be explained by the field synergy principle. Herman et al. [13] carried out experiments to explore the characteristics of the local heat transfer by means of visualization of temperature fields under different Reynolds number in a grooved channel. They pointed out that the instability and self-sustained oscillation occurs when the Reynolds number exceeds a critical value due to the Tollmien-Schlichting (T-S) wave

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J. Pan et al. / International Journal of Heat and Mass Transfer 147 (2020) 118932

Nomenclature D
h D
p E f* h* L* L* Nu Nui Num Nup Nus P Pr p Dp Q
0 Q
s Q
i q
i Re Res Rec Rei St s* T
0

hydraulic diameter (mm) diameter of the piston pump (mm) heat transfer enhancement factor oscillatory frequency (Hz) representative height (mm) periodic length of the groove (mm) length of the expanded groove (mm) average local Nusselt number local Nusselt number time-averaged Nusselt number Nusselt number for the parallel plate channel Nusselt number for steady flow oscillatory fraction Prandtl number dimensionless pressure pressure drop (Pa) maximum flow rate of the oscillatory flow (m3s1) net volume flow rate (m3s1) instantaneous volume flow rate (m3s1) local heat flux (Wm2) Reynolds number net flow Reynolds number critical Reynolds number instantaneous Reynolds number Strouhal number stroke of the piston pump (mm) time period of pulsating cycle (s)

where the flow oscillations were first observed when Re = 1054. It is well known that the self-sustained oscillation can enhance the heat transfer and also lead to an increase in the pressure drop. However it is necessary to adopt a laminar flow rather than a flow with high Reynolds number to enhance the heat transfer from the point of energy conservation. As to the effective techniques, the imposed oscillatory flow is adopted to enhance heat transfer. Nishimura et al. [14] experimentally studied the effects of oscillatory frequency on the mass transfer enhancement. Their results indicated that the mass transfer enhancement in laminar flow is higher than that in turbulent flow by means of fluid oscillation. Bian et al. [15] analyzed the characteristics of the mass transfer in the wavy-walled tube for pulsatile flow. They suggested that there is an optimum Strouhal number which corresponds to the maximum mass transfer, and the highest enhancement occurs just before the transitional flow region. Jin et al. [16] adopted PIV method to observe the feature of the flow in a triangular grooved channel during a pulsating cycle for pulsatile flow. Their results indicated that the vortex grows, expands and detaches from the groove to the main flow during a pulsating cycle which ultimately improves the heat transfer. Aluminum dust method was adopted by Nishimura et al. [17,18] to analyze the flow pattern in a wavy-walled tube. In addition, Nishimura et al. [19] used numerical and experimental approaches to study the features of the mass transfer enhancement in the grooved channel for pulsatile flow. They found that the unsteady separation of the vortex leads to an effective mix between the groove area and main flow. Nishimura et al. [20] also concluded that the flow is more unstable during the decelerating phase for the pulsatile flow in the wavy-walled tube, and they pointed out the mass transfer is greatly enhanced during the unstable state. Besides, they proposed that the duration of the unstable state in an oscillatory cycle decreases with the increase of Strouhal number. Wang et al. [4] numerically investigated the effects of the pulsatile flow on the

T Tf Tw Tin t u
m u
s u
in u,v W* wz

dimensionless temperature dimensionless temperature of the local fluid dimensionless temperature of the wall dimensionless temperature of the inlet dimensionless time characteristic flow velocity (ms1) area-averaged flow velocity at the entrance for steady flow (ms1) inlet velocity (ms1) dimensionless velocity components in x- and ydirections, respectively width of the test zone (mm) dimensionless vortcity magnitude

Greek symbols a
i heat transfer coefficient (W(m2K)-1) l* viscosity of water (Pas) q
density of water (kgm3) thermal conductivity (W(mK)-1) k
Du phase lag (°) Subscripts c critical in inlet i instantaneous out outlet

convection in a rectangular channel containing a longitudinal vortex generator (LVG). Results suggested that the pulsatile flow in the channel with LVG can intensify the heat and mass transfer, and the heat transfer rate and friction factor along the flow direction increase gradually as the amplitude of the pulsatile flow increases. Selimefendigil et al. [21] numerically examined forced convection of pulsating nanofluid flow over a backward facing step with a corrugated bottom wall. They observed that the average Nusselt number distribution along the bottom wall downstream of the step enhances with the increase of the Reynolds number, length and height of the triangular wave. Later, Selimefendigil et al. [22] studied further about heat transfer in the forced convection of pulsating nanofluid flow over corrugated isothermal heaters in channel under the effects of an inclined magnetic field. Their results showed that heat transfer has been enhanced by 40.3% and 34% compared to the steady case in the absence and presence of the magnetic field respectively at a Hartmann number of 15 and pulsation amplitude of 0.9. Huang et al. [23] numerically and experimentally analyzed the heat transfer in the grooved channel for pulsatile flow with different groove lengths (l = 1.6, 2.4, 3.2, 4.0 and 4.8), they found that the maximum heat transfer coefficient occurs at l = 1.6. Besides, their results interestingly demonstrated the existence of the phase lag between the pressure drop and flow rate in the pulsatile flow. Their results are consistent with Wang et al. [24] whose results showed that the phase lag is related to the ratio of the acceleration pressure drop to total pressure drop. The brief discussion of the literature clearly shows that the characteristics of the pulsatile flow and the heat transfer in the contracted and expanded channels when the oscillatory fraction is lower than 1 (P < 1) has been fully studied. However, the features of the pulsatile flow with a reverse flow (P > 1) have not been clarified yet, which motivates the present investigation. Therefore, the characteristics of the heat transfer in the grooved channel for pulsatile flow with the consideration of a reverse flow are investigated

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numerically and experimentally in this paper. Streamlines, temperature distribution and local Nusselt number at eight different times in a pulsating cycle are analyzed. Besides, the effects of the oscillatory fraction and Strouhal number on the characteristics of the flow behavior and heat transfer are discussed in detail. As a reminder of this paper, Section 2 presents the physical problem and the considered geometry. Section 3 describes numerical arrangement, followed by the experimental setup in Section 4. Section 5 and 6 present numerical validation, and results respectively. The conclusions of this study are summarized in Section 7. 2. Geometry and problem description The test zone consists of 55 modules and the dimensions of a typical module are shown in Fig. 1. The length, width, periodic length, height of the groove is 10 mm, 200 mm, 20 mm, 2.5 mm respectively and the relative uncertainty is 0:05 mm. Water is used as the working fluid and all the experiments are conducted at the room temperature. Previous studies showed that the calculation results obtained by a two-dimensional flow model compare well with experimental data [25,26]. The two-dimensional model is therefore adopted in this paper. The physical variables are made dimensionless using: L = L*/h*, l = l*/h*, where the superscript * denotes dimensional values. In this study, L = 8 and l = 4. To better understand the pulsatile flow, some parameters are defined as follows: the net flow Reynolds number (Res)

Res ¼ q
u
m h =l



ð1Þ

where q* and l* are the density and viscosity of water respectively. As suggested by Sun et al. [27], the characteristic velocity of the pulsatile flow um* is calculated by:

u
m ¼ 3=2
u
s

ð2Þ

where us* is the area-averaged velocity at the entrance and it is calculated as bellow:


u
s ¼ Q
s =ð2W
h Þ

ð3Þ

In the above equation, Qs* is the net volume flow rate, that is the steady one. The oscillatory fraction (P) of the flow rate is defined as:

P ¼ Q
0 =Q
s

ð4Þ

where Q0*is the maximum flow rate of the oscillatory flow and it is expressed as:

Q
0 ¼ 2pf s
ðpD
2 p =4Þ


ð5Þ

In Eq. (5), f* is the oscillatory frequency, s* is the stroke of the piston pump utilized to produce the pulsatile flow and Dp* = 50 mm is the diameter of the piston.

h* 2h* W* flow direction

The instantaneous volumetric flow rate of the pulsatile flow is given as:

Q
i ¼ Q
s þ Q
0 sinð2pf t
Þ ¼ Q
s ð1 þ Psinð2pf t
ÞÞ





ð6Þ

where t* is the dimensional flow time, and it is made dimensionless by t = t*f*. 3. Numerical arrangement The present study adopts the FVM (finite volume method) numerical approach to study the flow characteristics as well as the heat transfer properties in the grooved channel. The commercial software ANSYS Fluent is employed to account for the flow. It is worth noting that in the present study, the heat transfer is predominantly investigated by numerical means due to the limitation of the employed experimental instruments whilst the flow characteristics are studied both numerically and experimentally. 3.1. Geometry and mesh Previous study [15] showed that the instantaneous velocity fields repeat periodically in the fully developed region. The flow reaches fully developed state at approximately the 9th groove from the inlet. Considering the interest of the current study is also in this flow state, a model consisting of 25 typical modules is employed. A uniform mesh is constructed and utilized for the present study, and it is shown in Fig. 2. It needs to clarify that to minimize any side effect associated with the outlet boundary, especially the thermal condition, and an extended straight pipe is included at the exit of the grooved channel. The corresponding independency results are presented in Section 5.1. 3.2. Governing equations A dimensionless numerical approach is adopted for the present numerical study in the framework of ANSYS Fluent. The physical parameters are made dimensionless by:

x ¼ x
=h ; y ¼ y
=h ; u ¼ u
=u
m ; v ¼ v
=v
m ;









p ¼ p
=ðq
u
2 m Þ; T ¼ ðT  T in Þ=ðT w  T in Þ

ð7Þ

Then the dimensionless governing equations can be written as:

@u @ v þ ¼0 @x @y

ð8Þ

St

@u @u @u @p 1 @ 2 u @ 2 u þu þv ¼  þ ð 2 þ 2Þ @t @x @y @x Re @x @y

ð9Þ

St

@v @v @v @p 1 @ 2 v @ 2 v þu þv ¼  þ ð 2 þ 2Þ @x Re @x @t @x @y @y

ð10Þ

St

@T @T @T 1 @2T @2T þ Þ þu þv ¼ ð @t @x @y RePr @x2 @y2

ð11Þ

where St is the Strouhal number defined as St = f*h*/um*. 3.3. Boundary conditions and initial conditions

l* L*

(L*=20mm, l*=10mm, W*=200mm, h*=2.5mm) Fig. 1. Dimensions of the test zone.

(1) Inlet boundary conditions: the working fluid enters the grooved channel with an imposed pulsating velocity and a constant temperature, that is: uin ¼ 2ð1 þ P  sinð2ptÞÞ=3, v ¼ 0, T in ¼ 0 and the employed uin corresponds to Res = 250.

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J. Pan et al. / International Journal of Heat and Mass Transfer 147 (2020) 118932

Fig. 2. Schematic of the grid system in the grooved channel.

(2) Outlet boundary conditions: pressure-outlet boundary condition is applied. That is:

@T=@x ¼ 0; pout ¼ 0:01

(3) Wall boundary conditions: both the upper and the lower wall of the channel are kept at a constant temperature, and the no-slip boundary condition is applied. That is:

u ¼ 0; v ¼ 0; T w ¼ 1

v ¼ 0; T ¼ T in

The maximum instantaneous Reynolds number (Rei) in the numerical simulation is 600, lower than the critical Reynolds number (Rec = 604) [27], which means the flow keeps laminar during a pulsating cycle. Therefore, laminar model is selected in this simulation. The SIMPLE algorithm is employed for the coupling of pressure and velocity [6]. The solutions are considered to be converged when the normalized residual reaches 10-6 for the energy equation and 10-3 for continuity and velocity equations. 3.5. Parameter definition

(4) Initial conditions: the mean velocity and temperature at the inlet is taken as the initial velocity and temperature. Therefore, the initial conditions are:

u ¼ 2=3;

3.4. Numerical methods

The local Nusselt number Nui, the average local Nusselt number Nu and the time-averaged Nusselt number Num are calculated as bellow:

Nui ¼ a
i D
h =k


Fig. 3. Entire experiment setup.

ð12Þ

5

J. Pan et al. / International Journal of Heat and Mass Transfer 147 (2020) 118932

where a
i is the heat transfer coefficient, k
is the thermal conductivity of the fluid, Dh* is the hydraulic diameter and it is defined as Dh*=4h* [25]. The local heat flux is calculated as:

q
i ¼ k
ð@T
=@n
Þw;i ¼ a
i ðT
w  T
f Þ

ð13Þ

where ð@T
=@n
Þw;i is the local temperature gradient normal to the wall, T
f is the local fluid temperature. Circulating tank Centrifugal pump Control valve Piston pump Test zone

Differential transducer Flow-meter Overflow tank

Nu ¼

1 25L


Num ¼

Fig. 4. Diagram of the experimental system.

1 T
0

Z

25L


ð14Þ

Nui dx 0

Z

T
0

ð15Þ

Nudt

0

where T
0 is the time period of a pulsating cycle. Based on them, the heat transfer enhancement factor E is defined as:

rotation

E ¼ Num =Nus where Nus is the Nusselt number for steady flow. The dimensionless vorticity magnitude is defined as:

m 20m



translation piston

wz ¼

1  @v ð 2  @x

   @u Þ @y 

u
m =h

ð16Þ




4. Experiment setup Fig. 3 shows the entire experimental setup and a schematic of the experimental system can be seen in Fig. 4. Water is used as the working fluid and the flow is supplied to the test zone using a centrifugal pump and the flow rate is adjusted by the control valve. Feed water is stored at the circulating tank. The imposed oscillatory flow is obtained by the piston pump, which is driven by a variable speed motor through the Scotch-yoke mechanism to cover a range of pulsating flow parameters. After passing through the test zone, the flow rate is measured by the flowmeter and the relative uncertainty is 1 ml/s. The overall pressure drop of the test zone is recorded by the differential transducer and the relative uncertainty is 0:065 kPa. Additionally, the aluminum dust method is used to visualize the flow patterns. The aluminum dust method, i.e. perfusion with a suspension of aluminum particles which are about 40 lm in diameter, enables the observation of streamlines (i.e. the reflection of the aluminum dust) in the whole flow field. Since the instantaneous velocity fields repeat periodically in the fully developed area, the flow patterns of the 20th groove were taken as an example. A sheet light was employed to illuminate the 20th groove and the patterns were recorded by a

plate

Fig. 5. Diagram of the pulsating generator.

Fig. 6. Diagram of the photographing system.

15.8

16.0

Num

Num

15.6 15.5

15.4

15.0 1x10 5

2x10 5

3x10 5

4x10 5

5x10 5

6x10 5

7x10 5

15.2

5.0x10 -4

1.0x10 -3

1.5x10 -3

nodes

time step

(a)

(b)

Fig. 7. Grid and time step independency tests results (a) grid (b) time step.

2.0x10 -3

6

J. Pan et al. / International Journal of Heat and Mass Transfer 147 (2020) 118932

Fig. 8. Comparison between numerical and experimental flow streamlines for Res = 250, P = 1.4 and St = 0.001575.

7

J. Pan et al. / International Journal of Heat and Mass Transfer 147 (2020) 118932

Fig. 8 (continued)

digital video camera with an exposure time of one fifteenth of a second. The pulsating generator is shown in Fig. 5, and it is composed of a plate on which a piston with a diameter of 0.05 m is connected by a metal stem. The stroke of the piston is controlled through the plate from 25 mm to 125 mm in step of 5 mm. The oscillatory frequency is controlled by the frequency control. Con-

sidering the photographing process, a special light-driven camera system is designed as shown in Fig. 6. In order to capture 8 typical moments in a pulsating cycle, the axis of the plate in the Scotchyoke mechanism is divided into 8 equal parts using special lines in the circumferential direction, and each line corresponds to a special moment in a pulsating cycle. At the same time a bolt is set in the axis and its position is adjusted according to the special

400 numerical result after linear translation experimental result numerical result

80 70

2

7 8

the 1st groove the 2nd groove the 3rd groove the 4th groove the 5th groove the 6th groove the 7th groove the 8th groove the 9th groove the 10th groove the 11th groove the 12th groove

10

50

100 0

40 30 20

-100 1.0

s 3 0

60

200

Nui

Δ p (Pa)

300

10

1.5

2.0

2.5

3.0

t Fig. 9. Comparison for numerical pressure drop and experimental pressure drop for Res = 250, P = 1.4 and St = 0.001575.

0

0

2

4

s

6

8

10

Fig. 10. Distribution of the local Nusselt number of different grooves at Re = 260.

8

J. Pan et al. / International Journal of Heat and Mass Transfer 147 (2020) 118932

lines. The shutter of the light-driven camera which is connected with a sensor will be pressed when the camera receives the reflected light signal and the flow pattern at this moment can be recorded.

the parallel plate channel. It can be seen that the numerical results are almost identical with the results obtained by Adachi et al. [25]. It is noted that the results of the Nusselt number along the inner surface of the groove are plotted in Figs. 10 and 11 to better illustrate the local heat transfer properties.

5. Numerical validation 6. Results 5.1. Validation of grid and time-step independency 6.1. Experimental results To check the grid independency, four sets of uniform meshes were constructed when Res = 250, P = 1.4 and St = 0.125 with a time step of 0.001 s. They are 166,470, 297,960, 467,450 and 647,940 nodes. The time-averaged Num of the four meshes are shown in Fig. 7(a). The time step independency test was performed by four time step sizes with grid nodes of 467,450: 0.0005 s, 0.001 s, 0.0015 s and 0.002 s under the same conditions as shown in Fig. 7(b). In order to keep the numerical accuracy, a grid with 467,450 nodes and a time step of 0.001 s were chosen. 5.2. Model validation In order to validate the present numerical results, the simulation is conducted under the same operating conditions as the experiment. Fig. 8 shows the comparison of the numerical and experimental streamlines when Res = 250, P = 1.4 and St = 0.001575. It is apparent that the numerical streamlines show good agreement with the experimental trajectories at different time during a pulsating cycle. At the same time, Fig. 9 shows comparison between the numerical pressure drop after linear translation and the experimental pressure drop for Res = 250, P = 1.4 and St = 0.001575. Due to the difference between the experimental and the numerical periodic number of the module, the numerical results have been linear transformed to make the comparison with the experimental results. However, the negative pressure cannot be measured in the experiment because of the limitation of the experimental equipment, so the measured pressure drop in the experiment fluctuates at 0 Pa at some time. Anyway, the numerical results after linear translation compare well with the experimental results when the pressure drop is positive. To further verify the correctness of the simulation, the numerical simulation is carried out with the same geometry and operating conditions (i.e. Re = 260 for steady flow) as presented in [25]. Fig. 10 presents the distributions of the local Nusselt number of different grooves. It illustrates that the periodic thermal condition is achieved after the 9th groove. The comparison results are shown in Fig. 11, where Nup = 7.54 [25] is the mean Nusselt number for

The theoretical instantaneous volume flow rate is shown in Fig. 12. Three kinds of flow patterns at P = 1.4, Res = 375 and St = 0.00757 are observed during a pulsating cycle which will be discussed as follows. As shown in Fig. 13, when t = 0, two stable symmetric vortexes form in the grooved region and the path lines of the main flow are parallel to each other and there is not any mix between the flow in the main flow and the groove. It is in a ‘‘stable” state and this state is marked by ‘‘+”. It is noticed that when t = 0.375, the trajectories are highly disturbed and there is a mix between the flow in the main flow and the groove. It indicates a chaotic motion and marked as ‘‘”. Besides when t = 0.75, it is observed that a diversion occurs at the location ‘‘A” and two vertexes form. It suggests that it is a reverse flow and marked as ‘‘r”. Furthermore, it is interesting to find that the flow is unstable in the deceleration phase for the forward flow (t = 0.375–0.627) and the transition phase between forward flow and reverse flow (around t = 0.627 and t = 0.873). A similar result has been obtained in a wavy-walled tube [24]. Besides, the flow remains stable when t = 0.25 though the instantaneous Reynolds number exceeds the critical Reynolds number (Re0.25 = 900 > 604). Further, during a pulsating cycle in the experiment when P > 1, it is observed that the vortices grow gradually in the groove at first. The vortices are flat initially and the centers of the vortices move downstream gradually. Besides, the upper vortex is counterclockwise, the lower vortex is clockwise. Vortices gradually fill the whole groove and extend to the mainstream. The two large vortices fill the whole module when the flow almost stops (i.e. the instantaneous flow rate is 0). Subsequently, the direction of the flow changes. The two large vortices gradually decrease, then the vortices begin to attach to the wall of the groove and gradually increase. The vortices fill the whole module again when the flow rate reaches 0, meanwhile the upper vortex is clockwise and the lower vortex is counterclockwise. Later, the flow returns to the forward flow and the vortices decrease gradually. However, the flow changes rapidly and not all of the abovementioned flow features can be captured due to the limitation of the experimental apparatus. Take Fig. 13 as an example, it is easy to notice that the vortices

Qi

4

Re=260

P=1.4

numerical results of the 20th groove Results of Adachi et al.[25]

Nui /Nup

3

s 3

2

2

0

7 8

Qs

10

t=0.627

1

t=0.873

0.0

0

0

2

4

s

6

8

10

Fig. 11. Comparison of the distribution of the local Nusselt number between the numerical results of the 20th groove and the results of Adachi et al. [25].

0.000

0.125

0.250

0.375

0.500

0.625

0.750

0.875

1.000

t Fig. 12. Instantaneous volume flow rate varies with time.

J. Pan et al. / International Journal of Heat and Mass Transfer 147 (2020) 118932

are flat at first and gradually fill the whole groove, and then extend to the main flow at t = 0.375, and there are two large vortices filling the whole module at t = 0.625 when the flow rate is close to 0 m3s1. The direction of the flow changes at t = 0.75, and the two vortices fill the whole module again at t = 0.875.

9

In order to further study the effects of Strouhal number on the characteristics of the pulsatile flow, a number of flow visualization results under different Strouhal numbers have been obtained in Table 1. For the forward flow, as the increase of the Strouhal number, the onset of the unstable state gradually delays and the duration of the unstable state reduces. For the reverse flow (i.e. at

Fig. 13. Flow patterns when P = 1.4, Res = 375 and St = 0.00757.

10

J. Pan et al. / International Journal of Heat and Mass Transfer 147 (2020) 118932 Table 1 Stable and unstable state of flow patterns during a pulsating cycle for different Strouhal numbers when P = 1.4, Res = 375.

t = 0.75), it gradually transforms to unstable state as the Strouhal number increases. 6.2. Numerical results 6.2.1. Heat transfer Fig. 14 shows the local Nusselt number along the surface of the 20th groove at eight different times in a pulsating cycle for P = 1.4, Res = 250 and St = 0.0025. It is interesting to notice that the local Nusselt numbers of the upper and lower walls are different from t = 0.375 to t = 0.75, which is different with the steady flow. At the same time the streamlines, the dimensionless vorticity magnitude (wz) and temperature of the upper and lower grooves are asymmetrical during t = 0.375–0.75 as shown in Figs. 15 and 16. It suggests that the flow is more unstable and asymmetrical during the reverse acceleration phase. From Fig. 14, at t = 0 Nui reaches the maximum value when s = 8 due to the redevelopment of the thermal boundary layer. Meanwhile the vorticity magnitude is the largest and the temperature boundary layer is the thinnest at s = 8 (i.e. x = 6 and y = 3) as illustrated in Figs. 15 and 16. Besides, Nui reaches the minimum value when s = 3 and s = 7 (the corner of the groove). Simultaneously, its corresponding temperature boundary layer is the thickest and the vorticity magnitude is the smallest. However, at t = 0.75 (i.e. reverse flow) Nui reaches the maximum value when s = 2. Still Nui reaches the minimum value when s = 3 and s = 7 at the moment. In a word, the maximum Nui occurs at the edge point and the minimum Nui occurs at the corner of the groove. In order to easily understand the interaction between the thermal and flow, the isotherms contour and some streamlines are presented in Fig. 16. First, the vortices are flat and there is a mix between the main flow and the groove at the downstream of the groove as shown in Fig. 16(a). Then, from t = 0 to t = 0.25, the vortices in the groove grow and fill the whole groove gradually, and the warm fluid in the groove is carried out away and ejected into the main flow by the vortex formed in the groove. Later, at t = 0.375, the vortex decomposes into two small vortices and the streamlines of the upper and lower grooves are asymmetrical which increases the instability of the flow. At the same time, some cold fluid penetrates into the groove and promotes the mix between the hot and cold fluid. Next at t = 0.5 the small vortices grow gradually and the thermal boundary layer on the horizontal groove is compressed and there is still some cold fluid penetrating into the groove. Two large asymmetrical vortices and some small vortices fill the groove when t = 0.625 which promote the heat exchange and increase the temperature of the main flow. When

t = 0.75, the direction of the flow changes and the thermal boundary layer is very thin at the downstream. When t = 0.875 two large symmetrical vortices form in the groove, and the flow is almost motionless. The relationship between the heat transfer enhancement factor E and the Strouhal number under different P is illustrated in Fig. 17 when Res = 250. E increases with St first and then decreases gradually though there is a little fluctuation at St = 0.08 when P = 0.6. Besides, the enhancement factor increases with the oscillatory fraction and the maximum heat transfer enhancement factor is 2.74 when P = 1.4. These suggest that the pulsatile flow with a reverse flow can enhance the heat transfer. Fig. 18 depicts the variation of Nu with different Strouhal numbers in a pulsating cycle for Res = 250 and P = 1.4. It can be seen that Nu is relatively higher during t = 0.375–0.625 for St = 0.0025, and the flow is unstable during that period as mentioned above. Thus it can be concluded that the unstable state in the reverse acceleration phase increases heat exchange between the main area and the grooved area and contributes to the enhancement of the heat transfer. Additionally, a secondary heat transfer enhancement occurs at about t = 0.75 under different Strouhal numbers. Fig. 19 illustrates the variation of Nu with different P in a pulsating cycle for Res = 250 and St = 0.05. A secondary heat transfer enhancement occurs at about t = 0.75 for P = 1 and P = 1.4. Besides it is apparent that the maximum Nu can be achieved around T0/4 (i.e. t = 0.25) for different P. While the time corresponding to the best heat transfer in a pulsating cycle is different under different Strouhal numbers as shown in Fig. 18. These indicate that the onset of the unstable state is different for different Strouhal numbers, which agrees with the experimental results mentioned above. 6.2.2. Phase lag Fig. 20 shows the numerical results of the overall pressure drop and the volume flow rate. It is interesting to notice that the volume flow rate is positive when the pressure drop reaches 0 Pa. This could be attributed to the following reasons. Based on the researches of Wang et al. [24] and Moschandreou et al. [28], the total pressure drop of the pulsating flow is consisted of friction pressure drop and acceleration pressure drop. The total pressure drop is negative when the reverse acceleration pressure drop is larger than the friction pressure drop. Therefore there exists a phase lag between the pressure drop and volume flow rate, and the same phenomenon has been found in the experiment as shown in Fig. 21. In addition, the phase lag increases with Strouhal number at first and then almost keeps at a stable value of 86° as shown in Fig. 22.

11

J. Pan et al. / International Journal of Heat and Mass Transfer 147 (2020) 118932 40 30 0

2

15

0

2 s 3

8 7

20

10

Nui

20

7 8

s 3

10

0

2

7 8

0

2 s 3

8 7

s 3

15 10

10

t=0.125

lower wall upper wall

30 25

25

Nui

35

t=0

lower wall upper wall

35

10 10

5

5 0

0

-5

-5

0

2

4

6

s

8

10

0

2

4

s

(a)

10

80

t=0.25

lower wall upper wall

20 0

2

7 8

0

2 s 3

8 7

s 3

15 10

70

50 10 10

t=0.375

lower wall upper wall

60

Nui

25

Nui

8

(b)

30

40

0

2

7 8

30

0

2 s 3

8 7

s 3

10 10

20

5

10 0

0 0

2

4

6

s

8

-10

10

0

2

4

s

(c) 100

2

7 8

0

2 s 3

8 7

40

80

t=0.5

8

lower wall upper wall

60 10

0

2

7 8

0

2 s 3

8 7

s 3

10

Nui

0

s 3 60

6

10

(d)

lower wall upper wall

80

Nui

6

40

t=0.625

10 10

20

20

0

0 0

2

4

6

s

8

10

0

2

4

s

(e)

6

8

10

(f)

40

lower wall upper wall

30

Nui

25 0

2

7 8

0

2 s 3

8 7

s 3

20 15

20

t=0.75

lower wall upper wall

15 10

10

0

2

7 8

0

2 s 3

8 7

s 3

10

Nui

35

10

t=0.875

10 10

5

5 0 -5

0

0

2

4

s

6

8

10

0

2

4

(g) Fig. 14. Variation of Nui in a pulsating cycle for P = 1.4, Res = 250 and St = 0.0025.

s

(h)

6

8

10

12

J. Pan et al. / International Journal of Heat and Mass Transfer 147 (2020) 118932

Fig. 15. The distribution of the instantaneous dimensionless vorticity magnitude in a pulsating cycle for P = 1.4, Res = 250 and St = 0.0025.

J. Pan et al. / International Journal of Heat and Mass Transfer 147 (2020) 118932

Fig. 16. Instantaneous streamlines and the distribution of the dimensionless temperature in a pulsating cycle for P = 1.4, Res = 250 and St = 0.0025.

13

J. Pan et al. / International Journal of Heat and Mass Transfer 147 (2020) 118932 3.0

E

2.5

2.0

150

Q i ×10 6(m 3⋅s -1 )

P=1.4 P=1.0 P=0.6

Re s =250

250

Qi

200

Δp 100

150

Δϕ

100

50

50 0

0

1.5

0.02

0.04

0.06

0.08

0.10

0.0

0.12

0.5

1.0

Fig. 21. Experimental results of the overall pressure drop and volume flow rate when P = 1.4, Res = 250 and St = 0.002365.

Fig. 17. Effect of Strouhal number on E with different P for Res = 250.

90

20

86°

Re s=250, P=1.4

85

Δϕ (°)

15

Nu

St=0.0025 St=0.05 St=0.08125

10

80

75

5 0.000

0.125

0.250

0.375

0.500

0.625

0.750

0.875

1.000

70 0.00

0.05

Fig. 18. Variation of Nu with different Strouhal numbers in a pulsating cycle for Res = 250 and P = 1.4.

P=1.4 P=1.0 P=0.6

Re s=250, St=0.05

16 14 12 10 0.000

0.125

0.250

0.375

0.500

0.625

0.750

0.875

1.000

t Fig. 19. Variation of Nu with different P in a pulsating cycle for Res = 250 and St = 0.05.

200 Δp

100

100

50

50

-50 0.0

0

Δϕ

0

Δ p (Pa)

Qi

150

-50 0.5

1.0

1.5

0.15

0.20

0.25

0.30

0.35

0.40

Fig. 22. Variation of phase lag (Du) with St for P = 1.4.

7. Conclusions In this paper, experimental and numerical researches are conducted to explore the characteristics of the pulsatile flow with a reverse flow in the grooved channel. The main conclusions are obtained as follows:

20 18

0.10

St

t

Nu

-50 2.0

1.5

t

St

Q i ×10 6 (m 3⋅s -1 )

Δ p (Pa)

14

2.0

t Fig. 20. Numerical results of the overall pressure drop and volume flow rate when P = 1.4, Res = 250 and St = 0.002365.

(1) Flow is less stable during the reverse acceleration phase. Streamlines, temperature and vorticity of the upper and lower grooves are asymmetrical during the unstable state. Further, the instability contributes to the enhancement of heat transfer. (2) The onset and the duration of the unstable state are different for different St. The onset of the unstable state gradually delays and the duration of the unstable state reduces with the increase of St during the test range when Res = 375 and P = 1.4. (3) The heat transfer enhancement factor increases with St at first and then decreases, and the maximum heat transfer enhancement factor is 2.74 when P = 1.4 in the numerical range. (4) The time-averaged Nusselt number almost increases with P and a secondary heat transfer enhancement occurs when P > 1 and the heat transfer enhancement factor increases with P. These suggest that the pulsatile flow with a reverse flow can enhance the heat transfer. The above results can be used as a useful reference for the design and operation of high efficiency heat exchangers. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

J. Pan et al. / International Journal of Heat and Mass Transfer 147 (2020) 118932

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