Heat transfer and flow structure in a detached latticework duct

Applied Thermal Engineering 155 (2019) 24–39

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Research Paper

Heat transfer and flow structure in a detached latticework duct a,b

Wei Du a b

a,⁎

a

b

, Lei Luo , Songtao Wang , Jian Liu , Bengt Sunden

b,⁎

T

School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China Division of Heat Transfer, Department of Energy Sciences, Lund University, Lund, Sweden

H I GH L IG H T S

clearance is shown to improve the thermal performance of the latticework duct. • The of Coriolis force and clearance on heat transfer in a detached latticework are revealed. • Effects • Impingement, turn, helical flows and leakage flow are found in the detached latticework duct.

A R T I C LE I N FO

A B S T R A C T

Keywords: Clearance Detached latticework Impingement Helical flow Leakage flow

A numerical method was used to study the effect of clearance and rotational number on the thermal performance in a detached latticework duct. The latticework duct, which had four-entry sub-channels, was located in a simplified rectangular channel. The crossing angle for each sub-channel was 45°. The numerical studies were conducted with various clearances (0–0.5) and various rotational numbers (0–0.5). The streamlines, wall shear stress and Nusselt number were analyzed. The results indicated that the detached latticework provided a small mechanical energy loss with considerable heat transfer enhancement. In the traditional latticework duct, the impingement, turn and helical flows dominated the flow structure. The impingement and helical flows brought high Nusselt numbers and high wall shear stress in the latticework duct. In the detached latticework, the leakage flow, which is induced by the clearance, also changed the flow structure and heat transfer significantly. As the clearance was increased, the leakage became strong while the impingement and helical flows became weak. In the rotational condition, the Coriolis force promoted the heat transfer on the suction side but weakened the heat transfer on the pressure side. In addition, the wall shear stress on the suction side and leakage flow were increased under high rotational number.

1. Introduction

the crossing angle (β ). Therefore, the internal passage within the blade is divided into some coplanar crossing channels or some sub-channels by repeated-parallel ribs. The ribs bridge the pressure side and suction side directly and promote the turbulence intensity simultaneously. Therefore, the latticework duct provides high structural strength and high heat transfer enhancement ratio compared to other internal cooling structures. Fig. 1 also shows the flow path of cooling air in the latticework duct. When the cooling air enters the latticework duct, it turns the flow direction at the end of the sub-channels (turning region) and impinges onto the beginning of the opposite sub-channel (impingement region). Then, the above flow pattern is repeated continuously until the cooling air leaves the latticework duct. However, studies of the latticework ducts are inadequate compared to traditional cooling structures (pin fins, ribs, dimples and impingements). Only a limited number of literatures about the latticework ducts have been

To protect a gas turbine blade, many internal cooling structures, i.e., pin fins [1], ribs [2], dimples [3] and impingement flow [4], are applied in gas turbines. These cooling structures have already been used widely and successfully in European and American gas turbines. However, Russian often use the latticework duct (also called vortex cooling structure or bounded vortical structure) to cool the turbine blades. The reliability and validity of the latticework ducts have already been proved by some Russian gas turbine designs, such as AL-31, GT25000. Fig. 1 shows a gas turbine blade, which uses the latticework duct. The internal space of the turbine blade is filled with latticework duct. The latticework duct consists of some repeated-parallel ribs. The ribs on the pressure side are in direct contact with the ribs on the suction side. The two sides of the ribs are oriented at a certain angle, which is named as

⁎

Corresponding authors. E-mail addresses: [email protected] (L. Luo), [email protected] (B. Sunden).

https://doi.org/10.1016/j.applthermaleng.2019.03.148 Received 7 January 2019; Received in revised form 27 March 2019; Accepted 28 March 2019 Available online 29 March 2019 1359-4311/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature

c Cp Dh f f0 H Ht L L1 L2 n Nu P Pin Pout Pr q R Re Ro Sh t

Tb Tin Tout Tw U Uin W Wt Y+

Clearance height (mm ) Pressure coefficient hydraulic diameter (mm) Friction factor Karman-Nikuradse equation Channel height (mm ) Sub-channel height (mm ) Latticework length (mm ) Inlet part length (mm ) Outlet part length (mm ) The distance to the wall (m ) Nusselt number Pressure (Pa ) Inlet pressure (Pa ) Outlet pressure (Pa ) Prandtl number Wall heat flux (W/m2 ) Rotation radial (mm ) Reynolds number Rotational number Sherwood number Rib width (mm )

Local bulk temperature (K ) Inlet temperature (K ) Outlet temperature (K ) Wall temperature (K ) Local relatively velocity(m/s) Inlet averaged relative velocity(m/s) Channel width (mm) Rib spacing (mm) Y plus

Greek symbols

ρ β μ λ τ Ω

Density (kg/m3) Rib crossing angle (°) Dynamic viscosity (Pa·s ) Thermal conductivity (W·m−1·K−1) Wall shear stress (Pa) Rotational speed (rad/s)

Abbreviations CFD TKE

Computational fluid dynamics Turbulent kinetic energy

Fig. 1. Latticework duct blade and flow network.

90-deg turned duct. The Nusselt number was measured by the liquid crystal thermochromic. Two different inlet channels were considered. The results illustrated that the 2-inlet channel provided high Nusselt numbers. However, the pressure drop was independent of the Reynolds number and different inlet channels. From the existing studies, the turning region near the side wall was responsible for the high Nusselt number in the latticework duct. Therefore, some researchers studied the heat transfer in a latticework duct with a different side wall. Deng et al. [9] used a slot between the rib and side wall to enhance the heat transfer. It was found that a proper slot could augment the heat transfer remarkably. Due to the limitation of the experimental methods, the previous studies mainly provide the heat transfer distribution in the latticework duct, and little flow structure information is reported. Tsuru et al. [10] used the nuclear magnetic resonance to show the flow structure in a latticework duct firstly. It was found that the helical flow was the main flow structure in the latticework duct. In addition, the

published. Gorelov et al. [5] was probably the earliest researchers to study the cooling efficiency in a gas turbine, which used a latticework duct in the turbine blade. The heat transfer coefficient was measured by calorimetry in melted Zn. Results showed that the maximum Nusselt number was found at β ≈ 90°. Gillespie et al. [6] applied a latticework in the trailing region of a gas turbine. To measure the heat transfer conveniently, the trailing region was simplified as a wedge duct. In addition, a single row of film holes was placed on the endwall. Results showed that the Nusselt number was decreased along the radial direction due to the lateral outflow. Bunker [7] measured the Nusselt number distributions in a simplified latticework duct by the liquid crystal and infrared thermography methods. Results showed that the latticework provided a uniform laterally averaged Nusselt number distribution. In addition, the cooling air near the turning region leads to the highest Nusselt number. Saha et al. [8] applied a latticework in a 25

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flow structure in each sub-channel could be divided into three parts. The turning effect dominated the flow near the side wall, the vortex was the main flow structure in the middle part of the sub-channel. The previous studies were carried out in a static duct. According to the existing literatures, the Coriolis force and centrifugal force alter the heat transfer and flow structure obviously in a rotating duct [11–13]. An investigation in a rotating duct was carried out by Acharya et al. [14] based on the work by Bunker [7]. It was found that the Nusselt number and friction factor were relatively independent of the rotational number (The rotational number, which accounts for the ratio between the Coriolis forces and the inertia of the fluid, was used to quantitatively evaluate the effect of the rotation) and density ratio at high

Reynolds number. Furthermore, the turning region also contributed to the high Nusselt number in the rotational duct. A few years later, Oh et al. [15] also experimentally studied the heat transfer in a latticework duct under rotational condition. The rotational number was in the range of 0–0.8. In the static duct, the Nusselt number distribution was similar to that of Bunker [7]. The impingement/turning/swirling affected the Nusselt number distributions primarily. In the rotating duct, the high rotational number brought high Nusselt numbers on the leading side (also called the suction side in a gas turbine), especially near the impingement region. These results were different compared to these of Acharya et al. [14]. For the trailing side (also called the pressure side in a gas turbine), the rotational number had a small influence

Fig. 2. Schematics of the numerical model. 26

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decrease the outlet boundary effect. The length of the outlet part is L 2 = 80 mm . The length of the latticework duct is L = 224 mm . In this study, the rotation direction points along the Y direction. Therefore, the direction of the Coriolis force points from the suction side to the pressure side. The rotation radius (R = 185.6 mm ) is the same as in Oh et al. [15]. The duct inlet cross section is W × H = 128 mm × 48 mm . According to the experimental results, there are four-entry sub-channels at the beginning of the latticework duct. The latticework is generated by the repeated-parallel ribs. The width of the rib is t = 8 mm and the distance between two adjacent ribs is Wt = 24 mm [15]. The rib crossing angle is β = 90 °. In each side, there are eleven sub-channels. For convenience of analysis, the sub-channels are numbered from S1 to S11 on the suction side and from P1 to P11 on the pressure side. The latticework duct includes the inlet region, middle region and outlet region on both the pressure side and suction side. Six different clearC ances ( H =0, 0.1, 0.2, 0.3, 0.4, 0.5) , corresponding to the Case 0, Case 1, t Case 2, Case 3, Case 4 and Case 5, are adopted to study the effect of the clearance. The rotational number covers realistic conditions and is ranging from 0 to 0.5.

on the Nusselt number. Due to the increased centrifugal force, the friction factor was decreased slightly as the rotational number increased. Carcasci et al. [16] also found that the friction factor was insensitive to the rotational number, similar to Acharya et al. [14]. In addition, the Nusselt number was higher in a rotating channel compared to a static channel. The earlier studies were mainly carried out by experimental methods. However, only a small amount of information can be obtained by the experimental methods. Due to the development of computational fluid dynamics (CFD), more abundant information of the flow structures and heat transfer characteristics are obtained by various CFD softwares [17–19]. Many researchers have already used the CFD to study the heat transfer in a latticework duct. Su et al. [20] used a latticework duct in the trailing region of a realistic gas turbine. Numerical results showed that the latticework duct made the temperature distributions more uniform in the middle part of the turbine blade. Hagari et al. [21] proved that the numerical method could predict the Nusselt number distributions reliably. The impingement and vortex formed at the turning region contributed to the high heat transfer enhancement ratio. Bu et al. [22] also found that the numerical method can obtain the acceptable and accurate results compared to experimental results, especially near higher Nusselt number region. Sun et al. [23] used a commercial software to predict the effect of the bleed hole on the heat transfer in a latticework duct. It was also found that the numerical method provided a good performance to predict the heat transfer characteristics in a latticework duct combined with the bleed hole. Through the previous studies, it has been shown that the latticework duct provides a desirable heat transfer coefficient. However, the high heat transfer performance is accompanied by a high friction factor for most of the passive heat transfer enhancement devices. In general, the friction factor in pin finned channels is ranging from 10 to 80, while in a ribbed channel it is ranging from 0 to 25 [24]. However, the friction factor in a latticework is more than 100 [9], and even it can reach up to 2500 [22]. The high friction factor in a latticework duct reduces the thermal performance and limits its further application in gas turbines. Therefore, it is necessary to decrease the friction factor in the latticework duct and keep the Nusselt number at a considerable level. Analogous to the detached pin fin, the detached latticework duct is adopted in this study to decrease the friction factor. According to the previous studies, a detached pin fin provides better thermal performance than the traditional pin fin in a pin finned duct [25,26]. The detached pin fin means that a clearance occurs between one side of the pin fin and the endwall surface. After adoption of a clearance in a pin finned duct, the Nusselt number is kept almost constant while the friction factor is decreased significantly. The low fluid flow block contributes to the low friction factor in a detached pin fin duct. It should be noted that the clearance could enhance the heat transfer in some situations. However, only a few papers have been published to combine the latticework duct with clearance. Therefore, it is important and interesting to study the heat transfer and fluid flow in a detached latticework. In a detached latticework, the ribs on the suction side do not contact the ribs on the pressure side. There is a clearance between the two opposite ribs in a detached latticework. In the detached pin fin duct, the value of the clearance dominates the heat transfer and friction factor. Thus, to find a suitable clearance in the detached latticework duct, six different clearances are investigated. In addition, six different rotational numbers are selected to find the relationship between the suitable clearance and different rotational number.

3. Computational details 3.1. Parameters definitions To decide the geometry and working conditions, the inlet hydraulic diameter is defined as:

Dh =

4×W×H 2(W + H )

(1)

where W is the width of the duct inlet, H is the height of the duct inlet. To define the inlet relative velocity and nondimensionalized Nusselt number, the Reynolds number is defined as:

Re =

ρUin Dh μ

(2)

where ρ is the density of the inlet air, μ is the dynamic viscosity of the inlet air, Uin is the inlet averaged relative velocity. The rotational number, which accounts for the ratio between the Coriolis forces and inertia force, is defined as:

Ro =

ΩDh Uin

(3)

where Ω is the rotational speed. The local bulk temperature is used as the reference temperature. It is defined as [21]:

Tb = Tw − (Tw − Tin )(

Tw − Tout y )L Tw − Tin

(4)

where Tin is the inlet temperature of the air, Tout is the outlet temperature of the air, Tw is the local wall temperature, y is the local coordinate along the Y direction, L is the length of the heated wall. The Nusselt number is defined as:

Nu =

q D · h Tw − Tb λ

(5)

where λ is the thermal conductivity of the inlet air, q is the wall heat flux calculated by ANSYS FLUENT. The Nusselt number is nondimensionalized by the Gnielinski modification of the Popov- Pehtukhov formula, which is written as [27]:

2. Geometry A simplified geometry is applied to study the heat transfer and flow structure in a detached latticework duct under static and rotational conditions. The geometry is divided into three parts, see Fig. 2. The inlet part is used to provide a fully development turbulent fluid flow. The length of the inlet part is L1 = 100 mm . The outlet part is used to

Nu 0 =

f (Re 8

− 1000) Pr

1 + 12.7.

f 8

2

(Pr 3 − 1)

The pressure coefficient in a static duct is defined as: 27

(6)

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Cp =

P − Pout 0.5 × ρ × Uin2

this study. To increase the mesh quality near the side wall, the Y-Block is used to split the flow domain, as shown in Fig. 3(a). In order to capture the boundary layer flow by the enhanced wall functions, the thickness of the first mesh element is about 0.01 mm, corresponding to Y + ≤ 1. Fig. 3(b) shows the Y + distribution on the endwall surface. In addition, the growth ratio of the grid height is set as 1.1 near the endwall. A mesh independence study is also carried out. Three different mesh numbers are selected (3.5 million, 5.7 million, 8.5 million). The laterally-averaged Nusselt number in S5 and P6 are used to judge the mesh independence. Fig. 3(c) shows the comparison of the laterallyaveraged Nusselt numbers between the different mesh numbers at Ro = 0.5 for the Case 0. It is found that the 5.7 million mesh number is sufficient to capture the Nusselt number distributions on both the pressure side and suction side. Therefore, the 5.7 million mesh number is used for all cases.

(7)

where P is the local static pressure in the latticework duct, Pout is the outlet pressure of the latticework duct. To evaluate the mechanical energy loss in a static duct, the friction factor is defined as [28]:

f=

Pin − Pout D · h 0.5 × ρ × Uin2 4L

(8)

where Pin is the inlet pressure of the latticework duct. The Karman-Nikuradse equation is used to nondimensionalize the friction factor. It is defined as [29]:

f0 = 0.046Re−0.2

(9)

The wall shear stress is defined as: 3.3. Calculation settings

∂U ⎞ τ = μ⎛ ⎝ ∂n ⎠n = 0

(10) The CFD software ANSYS FLUENT is used to process this numerical study. The air as an ideal gas with constant thermal conductivity and specific heat is used as the cooling medium. The Reynolds number is fixed at 44,000 for all cases [15]. The Ro number, which contains the actual working condition, is ranging from 0 to 0.5. For the inlet, a uniform velocity profile and temperature profile (Tin = 20 °C) are used

where U is local relatively velocity, n is the distance to the wall. 3.2. Mesh generated A structured mesh, which is generated by ANSYS ICEM, is used in

(a) Mesh topology and node distribution

(b)

(c) Nusselt number distributions with different mesh numbers

distribution on the latticework duct

Fig. 3. Mesh information. 28

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numerical method is compared with Sh obtained by the experimental Sh 0 results in this study. The Reynolds number Re = 44,000 and rotational number Ro = 0.4 are selected in the numerical validation. Fig. 4 shows the comparison of Nusselt numbers between the numerical results and experimental results. Fig. 4(a) shows the local Nusselt number distributions on both the pressure side and suction side. It is found that the realizable k − ε turbulence model can predict the Nusselt number near the impingement region and turning region precisely. The impingement region has high Nusselt numbers while the turning region has low Nusselt numbers. The realizable k − ε turbulence model also captures the high Nusselt number near the windward side in each sub-channel. Fig. 4(b) shows the laterally averaged Nusselt number distributions. It is found that the Nusselt number is decreased from the impingement region to the turning region. In addition, the leading side has higher Nusselt number than the trailing side. This distribution trend obtained by the realizable k − ε turbulence model agrees well with the experimental results. Therefore, the realizable k − ε turbulence model is suitable in this study to predict the heat transfer and fluid flow in the latticework duct.

[21]. In addition, the turbulence intensity and turbulent viscosity ratios are assumed as 5% and 10, respectively. The rib surface and endwall have the constant temperature (Tw = 50 °C) [21]. The average static pressure is applied at the outlet. The commercial software ANSYS FLUENT is used to calculate the incompressible-steady flow and heat transfer. The pressure and velocity coupling is handled by the SIMPLE algorithm. The computational domain is discretized by second-order differences.

3.4. Validation of the numerical approach The realizable k − ε turbulence model is selected to calculate the heat transfer and fluid flow in the latticework duct. The experimental results in [15] are used to validate the accuracy of the numerical method. The experimental results in [15] are obtained by the naphthalene sublimation method, which is a convenient mass transfer method. In heat transfer experiments, it is often difficult to measure local heat transfer coefficients in detail when the temperature changes rapidly over a small region. Such measurements include large wallconduction errors resulting from steep gradients in the transfer rates across the region. However, with mass transfer techniques one can readily measure mass transfer coefficients in the analogous situations, even near a corner or edge, i.e., a singular point, with a large convective coefficient gradient. The mass transfer results (Sh number) can be converted to heat transfer results (Nu number) by a heat/mass transfer analogy [30]. As Sh number and Nu number are dimensionless, the Sh is the same as

Nu Nu0

in many situations. Therefore,

Nu Nu0

4. Results and discussions 4.1. Overall performance To evaluate the performance of the detached latticework, the averaged Nusselt number and wall shear stress distribution for different Ro numbers and clearances are displayed in Fig. 5. Fig. 5(a) shows that the Nu/Nu0 in the static traditional latticework (Case 0) can reach the

Sh 0

obtained by the

(a) Local Nusselt number distributions

(b) Laterally averaged Nusselt number distributions Fig. 4. A comparison the Nusselt number distributions between the numerical results and experimental results [15]. 29

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(a) Nu/Nu0

(b) Wall shear stress

Fig. 5. Comparison of the Nu/Nu0 and wall shear stress for various rotational numbers and clearances.

Fig. 6. Velocity vector distributions colored by the velocity in z direction on the middle plane with different clearance.

latticework duct. According to the previous studies, the pressure side (trailing side) has higher Nusselt number than the suction side (leading side) in the pin finned duct [12,31]. However, the suction side provides higher Nusselt number than the pressure side in the rotating latticework duct. In addition, the pressure side and suction side have different variation trend as the Ro number is increased. For the suction side, high Ro number means high Nusselt number for all clearances. However, the

value 4.21. This value is far above the heat transfer enhancement ratio in the pin finned duct and ribbed duct. However, the Nu/Nu0 is decreased moderately after adoption of the clearance in the latticework duct. The Nusselt number becomes smaller when the clearance is increased. The Nu/Nu0 in the static duct is 3.99, 3.53, 3.13, 2.88 and 2.56 for Case 1, Case 2, Case 3, Case 4 and Case 5, respectively. Fig. 5(a) also shows that the Ro number has an effect on the Nusselt number in the 30

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Fig. 7. 3D streamline colored by the velocity with various clearance in the static condition.

Fig. 8. Averaged Nusselt number in each sub-channel with different clearance on the pressure side.

Fig. 9. A comparison of the laterally-averaged Nusselt number in P6 along the Y direction between the different cases.

high Ro number leads to low Nusselt number on the pressure side for Case 0 and Case 1. As the clearance is greater than 0.3 (Case 3, Case 4 and Case 5), the Nusselt number is decreased at small Ro number and increased at high Ro number. Furthermore, the Nusselt number difference between the pressure side and suction side becomes larger as

the Ro number is increased. In heat transfer devices, the mechanical energy loss is very important for evaluation of the thermal performance. In a static duct, only the pressure drop drives the cooling air. However, as the rotational 31

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Fig. 10. Local Nusselt number distributions on the pressure side with different clearance.

Fig. 11. Pressure coefficient distributions along the S2, P6, S10.

number is increased, the centrifugal force and pressure drop push the cooling air together. As the Ro number is above a critical value, the centrifugal force is sufficient to push the cooling air. Therefore, the pressure drop is not suitable to evaluate the mechanical energy loss in the rotating channel. The wall shear stresses on the pressure side and suction side surface are used to illustrate the mechanical energy loss in this study [12]. A small wall shear stress means less mechanical energy

Table 1 Relative friction factors for different clearances at the static condition. Case

Case 0

Case 1

Case 2

Case 3

Case 4

Case 5

f / f0

381.40

204.20

138.60

82.67

56.71

37.13

32

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(a) Suction Side

(b) Pressure side Fig. 12. Local Nusselt number distributions in 6th sub-channel between the suction and pressure side.

sub-channel with different clearances. The velocity vector is colored by the velocity in the Z - direction. The velocity vector distributions in Case 0 is similar to the previous studies [32,33]. The cooling air, which comes from the end of the sub-channel S2 and S3, impinges onto the pressure side (sub-channel P6). The impingement region has a high velocity in the Z - direction. After impingement, the cooling air turns the flow direction from the Z - direction to the Y- direction. Due to the shear layer between the pressure side and suction side, a large vortex is formed in the sub-channel S3 to S9. This vortex has an axial velocity component and a swirl velocity component [33]. The swirl velocity component changes the flow direction of the cooling air on the pressure side and transports parts of the cooling air from the pressure side to the suction side. Furthermore, the axial velocity component generates a helical flow structure in the sub-channel on the suction side. In other words, the cooling air on the pressure side can pass the interaction between the pressure side and suction side and enhance the helical flow on the suction side. However, these flow behaviors are weak at the beginning of the P6 sub-channel and are augmented gradually along the flow direction. Therefore, the velocity in the Z - direction is increased along the flow direction. At the end of P6, the cooling air has to turn direction from the pressure side to the suction side. Therefore, this region is named as the turning region. In the traditional latticework duct, the repeated impingement, helical flow and turning are the main flow structures. By using the clearance, the impingement, vortex and turning

loss in the rotational latticework duct. Fig. 5(b) shows the wall shear stress distribution in the latticework duct with various Ro numbers and clearances. It is clear that the clearance offers massive benefits for the wall shear stress in the latticework duct both for static and rotational conditions. In the static duct, the wall shear stress is dropped from 0.0724 (Case 0) to 0.0516 (Case 1), 0.045 (Case 2), 0.0314 (Case 3), 0.025 (Case 4) and 0.0195 (Case 5). This means that the mechanical energy loss will be decreased significantly by using the clearance. Fig. 5(b) also shows that the wall shear stress is increased on the suction side while it is decreased on the pressure side as the Ro number is increased. Therefore, the wall shear stress difference between the pressure side and suction side becomes larger as the Ro is increased. In addition, the effect of the clearance on the wall shear stress is decreased as the clearance is increased. It can be inferred that a continuous increase of the clearance will cause less wall shear stress drop. To explain the averaged Nusselt number and wall shear stress distribution in the detached latticework duct, the flow structure, local wall shear stress and local Nusselt number distributions with various different clearances and rotational numbers are analyzed, respectively. 4.2. Static duct In the static condition, the flow structure is analyzed firstly. Fig. 6 shows the velocity vector distributions on the middle plane of the P6 33

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(a) Suction side

(b) Pressure side Fig. 13. Limiting streamline colored by the wall shear stress in the 6th sub-channel between the suction and pressure side.

clearance acts as a crossflow for the uptrend spiral flow and helical flow. This flow behavior is similar to the crossflow in jet/impingement [34]. In general, the crossing flow reduces the heat transfer enhancement. After presentation the flow structures, the averaged Nusselt numbers in each sub-channel with different clearances on the pressure side are shown in Fig. 8. It is found that the clearance counteracts the heat transfer enhancement in the latticework duct. As the clearance is increased from 0 to 0.5, the Nusselt number in the middle region and outlet region is decreased obviously. However, the clearance promotes the Nusselt number in the inlet region. The Case 0 without any clearance has the lowest Nusselt number while the Case 1 with the smallest clearance has the highest Nusselt number in the inlet region. As the clearance is increased, the Nusselt number is reduced gradually in the inlet region. The clearance also changes the Nusselt number distribution characteristics along the sub-channel. For the traditional latticework duct, the Nusselt number is increased from the first sub-channel to the third sub-channel significantly. From the third sub-channel to the forth sub-channel, the Nusselt number remains roughly constant. Then the Nusselt number maintains a sustained growth from the 4th subchannel to the 7th sub-channel. From the 7th to the 9th sub-channel, the Nusselt number shows a slight drop. In the outlet region, the Nusselt number reaches the highest value. After adoption of the clearance in the latticework duct, the Nusselt number is decreased from the third to fourth sub-channel for Case 1 to Case 5 while it is kept constant for Case 0. For the middle region, the Case 1 and Case 2 have similar distribution trends compared to Case 0. The maximum Nusselt number in the middle part is found at the 7th sub-channel. The Nusselt number fluctuates near 2.8 for Case 3 and Case 4 in the middle region. As the

region also dominate the flow structures in the detached latticework. However, the flow structure shows some changes as the clearance is increased from 0 to 0.5. Firstly, the impingement region becomes weak as the clearance is increased. In Case 1 and Case 2, impingement flow is found on S2 and S3. However, the impingement flow on S3 is replaced by the helical flow for Case 3, Case 4 and Case 5. Secondary, the size of the vortex is decreased as the clearance is increased. This means that less and less fluid in the P6 is transported to the suction side due to the small velocity in the Z - direction. To show more information about the flow structures, 3D streamlines colored by the velocity with various clearances are displayed in Fig. 7. The streamlines origin from S2, then impinge onto P6 and finally exit from S10. For Case 0, the cooling air encounters obstruction at the end of S2 and generates a low-speed recirculation near the side wall. Then the cooling air experiences an upward spiral flow from S2 to P6. In the sub-channel P6, the helical flow replaces the upward spiral flow and is enhanced by the vortex on the suction side, which has already been discussed in Fig. 6. Near the end of P6, the cooling air turns flow direction again. However, most of the cooling air exits from S8 and S9 and only a small part of the cooling air exits from S10 due to the small flow passage area. In Case 1, the clearance changes the streamlines obviously. The leakage flow, which exits from the clearance, is found near P6. The leakage flow becomes strong along the flow direction. Near the turning region, more and more fluid belonging to P6 exits from the clearance. The helical flow in P6 also becomes weak because of the leakage flow. Therefore, the flow resistance and mechanical energy loss can be decreased in the latticework. As the clearance is increased, the uptrend spiral flow and helical flow becomes weaker because of the stronger leakage flow. The reason is that the leakage flow in the 34

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Fig. 14. Streamline and TKE distributions at Y = −0.062 m.

compared to Case 0 and Case 1. The impingement region has a remarkable contraction from Case 2 to Case 5. Thus, the location of the peak value of the Nusselt number shifts to the side wall. In addition, the decrease rate of the Nusselt number in the middle part for Case 2 to Case 5 becomes slow along the Y - direction compared to Case 0 and Case 1. The laterally-averaged Nusselt number distributions in the turning region for Case 2, Case 3, Case 4 and Case 5 are similar to Case 0 and Case 1. This indicates that the clearance has a significant impact on the impingement region while it has little effect on the turning region. Fig. 10 shows the local Nusselt number distributions on the pressure side for different clearances in the static duct. It is found that the Nusselt number distribution is closely related to the clearance. For the traditional latticework without the clearance (Case 0), the inlet region has low Nusselt number due to the small turbulence intensity. However, a relatively high Nusselt number can be found at the windward side of the sub-channel at the inlet region. After the inlet region, the middle region and outlet region provide a considerable heat transfer enhancement. For these locations, the impingement region has the highest Nusselt number, which agrees well with Fig. 9. As the cooling air flows

clearance is equal to 0.5, the Nusselt number is decreased along the flow direction in the middle region. The clearance also affects the Nusselt number in the outlet region remarkably. For Case 0, Case 1, Case 2, the outlet region has higher Nusselt number than the inlet region while the inlet region has higher Nusselt number than the outlet region in Case 5. Fig. 9 shows a comparison of the laterally-averaged Nusselt number along the Y - direction in P6 between the different cases. It is noted that the Nusselt number from the impingement region to the turning region has noticeable changes. For Case 0, the Nusselt number has a sharp growth near the impingement region, peaking at Y = −0.035 m. Then, the Nusselt number continues to decline throughout the middle region until it reaches the lowest value near Y = 0.06. Finally, the Nusselt number has a slight increase near the side wall at the turning region. This trend is consistent with the results obtained by [15]. The Nusselt number distribution in Case 1 is similar to Case 0. However, Case 1 has lower Nusselt number than Case 0. In addition, the position of the lowest Nusselt number for Case 1 is far away from the side wall in contrast to Case 0. The distributions of the laterally-averaged Nusselt number in Case 2, Case 3, Case 4 and Case 5 are completely different

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Fig. 15. Streamline and TKE distributions at Y = 0.062 m.

the S2, P6, S10 sub-channel are shown in Fig. 11. The S2 sub-channel belongs to the inlet region, P6 belongs to the middle region and the S10 belongs to the outlet region. In general, a high pressure coefficient means that more differential pressure is required to force the cooling air through the latticework duct. Fig. 11 shows that the Case 0 has the largest pressure coefficient throughout the latticework duct. At the junction between S2 and P6, the pressure coefficient experiences a rapid decline. In the P6 region, the pressure coefficient is decreased slowly because of the regular fluid flow. Another rapid decline of the pressure coefficient occurs at the junction between P6 and S10. Therefore, it can be inferred that the impingement and turning region are responsible for the high mechanical energy loss in the latticework duct. As the clearance is ranging from 0.1 to 0.2, the pressure coefficient shows an obvious declining trend. The trend of the pressure coefficient is similar to that of Case 0. Once the clearance is above 0.2, the pressure coefficient trend is quite different from Case 0. The pressure coefficient only has a very slow declining trend for Case 3 to Case 5. The reason is that the large clearance reduces the influence of impingement and turning region, which was already discussed in Fig. 7. In addition, the Case 3, Case 4 and Case 5 have similar values of the

from the impingement region to the turning region, the Nusselt number is diminishing gradually. However, the Nusselt number has a modest rebound near the turning region. This means that the lowest Nusselt number is found upstream of the turning region. By using a small clearance in the latticework duct (Case 1), the Nusselt number near the turning region is unchanged almost while it is decreased significantly near the impingement region. This implies that the small clearance is a disadvantage of the heat transfer enhancement near the impingement region. As the clearance is more than 0.2, the Nusselt number near the windward side of each sub-channel declines remarkably. In addition, the Nusselt number near the impingement region also has a remarkable drop. It should be noted that the difference of the heat transfer enhancement between the inlet region and middle region is decreased as the clearance is increased. This means that the Nusselt number distribution on each sub-channel is more uniform as the clearance is increased. The purpose of this study is to decrease the mechanical energy loss in the latticework duct. From Fig. 5, it is found that the wall shear stress is decreased significantly by using the clearance. To analyze the mechanical energy loss in detail, pressure coefficient distributions along 36

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(a) Nu/ Nu0

(b) Wall shear stress

Fig. 16. The laterally averaged wall shear stress and Nusselt number distributions along the Y direction with different Ro number.

small part of the fluid generates a low-speed recirculation with low pressure which was already shown in Fig. 11. Another part of the cooling air turns into the helical flow. The impingement and helical flow are responsible for the high wall shear stress in the sub-channel. Due to viscous dissipation, the helical flow becomes weak and leads to the low wall shear stress. At the end of the sub-channel, the turning region is characterized by the spiral point. The turning region results in a small wall shear stress. Fig. 13(a) also shows that the separation line is insensitive to the Ro number while the wall shear stress near the helical flow is sensitive to the Ro number. The high Ro number promotes the wall shear stress at the middle part of the sub-channel on the suction side. In addition, the turning region becomes weak as the Ro increased. This means that the leakage flow is enhanced at the high Ro number condition. The limiting streamline and wall shear stress distributions on the pressure side are very similar to those on the suction side, as shown in Fig. 13(b). However, the wall shear stress on the pressure side is kept almost constant as the Ro number is increased from 0 to 0.5. Near the turning region, the spiral point is invisible at high Ro number. This means that only a small part of the cooling air will change flow direction near the turning region and most of the cooling air is leaving by the leakage flow. It can be inferred that the high Ro number is in favor of the leakage flow. Figs. 12 and 13 show that the Ro number has a significant impact on the Nusselt number and wall shear stress near the impingement region and turning region. Therefore, Fig. 14 shows the streamlines and turbulent kinetic energy (TKE) distributions near the connection between S2 and P6 on the Y = −0.062 m. It is found that the upward spiral flow, which occupies most of the space, is insensitive to the Ro number. Near the windward side of the S2 sub-channel, a large vortex is induced by the upward spiral flow. As the Ro number is increased, this vortex is decreased. At the P6 sub-channel, a low-speed recirculation with a low TKE is found near the core region of the leeward side. As the Ro number is increased, the low-speed recirculation becomes larger and the impingement region becomes smaller. The small impingement region leads to low Nusselt numbers on the pressure side, as shown in Fig. 14. The Ro number also has an influence on the leakage flow. For Ro = 0, some leakage flow is sucked by the upward spiral flow. However, a reversed flow is generated gradually as the Ro number is increased at the clearance. The revised flow restricts the suction effect. As the Ro number is higher than 0.3, some part of the upward spiral flow is pushed into the clearance and merged with the leakage flow. Therefore, the leakage flow is enhanced as the Ro number is increased. The TKE

pressure coefficient. This means that the influence of the clearance on the pressure coefficient has been weakened as the clearance is above 0.2. The relative friction factors in the detached latticework duct for different clearances at the static condition are shown in Table 1. It is found that the friction factor is decreased amazingly. As the clearance is increased from 0 to 0.5, the relative friction factor is decreased from 381.4 to 37.13. However, the Nusselt number is only decreased from the 3.864 to 2.353. Therefore, the thermal performance in the detached latticework is improved. The low friction factor with high Nusselt number proves the effectiveness of the detached latticework duct. 4.3. Rotational duct Case 2 with different clearances is used to analyze the Ro number effect on the flow structure and heat transfer in a detached latticework duct. Fig. 12 shows the local Nusselt number distribution in 6th subchannel both on the pressure side and suction side. In the rotational latticework duct, the suction side provides higher Nusselt number than the pressure side for all Ro number. Fig. 12(a) shows that the Ro number has a remarkable effect on the Nusselt number distribution near the impingement region while it has a small effect near the turning region on the suction side. As the Ro number is increased, the Nusselt number is increased near the impingement region but is kept almost constant near the turning region. In addition, the relatively high Nusselt number region at the middle part of the 6th sub-channel experiences a small growth. Therefore, the averaged Nusselt number in 6th subchannel is increased as the Ro number is ranging from 0 to 0.5 for the suction side. Fig. 12(b) shows the Nusselt number distribution on the pressure side. The variation of the Nusselt number with Ro number on the pressure side is different than that on the suction side. The low Ro number is favorable for the heat transfer enhancement near the impingement region on the pressure side. However, the Nusselt number near the turning region is independent of the Ro number on the pressure side. To explain the relationship between the wall shear stress and Ro number, the limiting streamlines colored by the wall shear stress in the 6th sub-channel both on the pressure side and suction side are shown in Fig. 13. It is found that the limiting streamline and wall shear stress are sensitive to the Ro number on both the pressure side and suction side. For the suction side, a separation line induced by the impingement is visible at the beginning of the 6th sub-channel. After impingement, a 37

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was sensitive to the rotational number. (4) The impingement region contributed to the large mechanical energy loss in the latticework duct. As the clearance was increased, the impingement became weak and the mechanical energy loss was decreased. In the rotational duct, the high rotational number made the wall shear stress to increase on the suction side while it decreased on the pressure side.

distribution also changes with different Ro number. The TKE is decreased near the leakage flow and low-speed recirculation as the Ro number is increased. Fig. 15 shows the streamline distributions near the connection between the P6 and S10 on the Y = 0.062 plane. The flow structure in Fig. 15 is similar to that in Fig. 14. The upward spiral flow, vortex and low-speed recirculation are also observed in Fig. 16. However, the Ro number has a different influence on the flow structure due to the different direction of the Coriolis force. As the Ro number is increased, the TKE in the low-speed recirculation region is augmented. The high TKE is beneficial for the heat transfer enhancement on the suction side. Therefore, the Nusselt number near the impingement region on the suction side is increased as the Ro number is ranging from 0 to 0.5, as shown in Fig. 12. It is also found that the Ro number has a small effect on the leakage flow and vortex near the top of the P6. Thus, the Nusselt number near the turning region has minor changes for different Ro number. To quantitatively describe the Ro number effect on the Nusselt number and wall shear distributions, Fig. 16 shows the laterally averaged Nusselt number and wall shear stress distributions along the Y direction in P6 and S6 sub-channel. Fig. 16(a) shows the Nusselt number distribution. It is found that the suction side has higher Nusselt number than the pressure side. However, the pressure side provides more uniform Nusselt number distributions in the sub-channel. In addition, the pressure side and suction side have similar values of the Nusselt number in the turning region. For the suction side, the high Ro number induces high Nusselt number near the impingement region and middle part of the sub-channel. For the pressure side, the high Ro number is in favor of the heat transfer near the turning region while it counteracts the heat transfer near the impingement region. Fig. 16(b) shows the wall shear stress distribution. The results agree well with Fig. 13. From the impingement region to the turning region, the wall shear stress is decreased both on the pressure side and suction side. However, the suction side has higher descent rate compared to the pressure side. It is also found that the wall shear stress near the impingement region is more sensitive to the Ro number on both the pressure side and suction side. For the impingement region, the high Ro promotes the wall shear stress on the suction while it suppresses wall shear stress on the pressure side.

Acknowledgment The authors acknowledge the financial support provided by China Scholarship Council (CSC), Natural Science Foundation of China (No. 51706051), China postdoctoral science foundation funded project (No. 2017M620116), Heilongjiang Postdoctoral Fund (No. LBH-Z17066), the General and Special Program of the Postdoctoral Science Foundation of China (No. 2018T110296) and the Fundamental Research Funds for the Central Universities (Grant No. HIT.NSRIF.2019061). References [1] W. Du, L. Luo, S. Wang, X. Zhang, Flow structure and heat transfer characteristics in a 90-deg turned pin fined duct with different dimple/protrusion depths, Appl. Therm. Eng. 146 (2019) 862 862-842. [2] P. Singh, J. Pandit, S.V. Ekkad, Characterization of heat transfer enhancement and frictional losses in a two-pass square duct featuring unique combinations of rib turbulators and cylindrical dimples, Int. J. Heat Mass Transf. 106 (2017) 629–647. [3] S. Wang, W. Du, L. Luo, D. Qiu, X. Zhang, S. Li, Flow structure and heat transfer characteristics of a dimpled wedge channel with a bleed hole in dimple at different orientations and locations, Int. J. Heat Mass Transf. 117 (2018) 1216–1230. [4] F. Xue, M.E. Taslim, Detailed flow and heat transfer analyses in a rib-roughened trailing-edge cooling cavity with impingement, J. Turbomach. 141 (2019) 051003. [5] V. Gorelov, M. Goikhenberg, V. Malkov, The investigation of heat transfer in cooled blades of gas turbines, 26th J. Propuls. Conf. (1990) AIAA-90-2144. [6] D.R.H. Gillespie, P.T. Ireland, G.M. Dailey, R. Royce, Detailed flow and heat transfer coefficient measurements in a model of an internal cooling geometry employing orthogonal intersecting channels, ASME Paper No. 2000-GT-0653. [7] R.S. Bunker, Latticework (Vortex) Cooling Effectiveness: Part 1—Stationary Channel Experiments, ASME Paper No. GT2004-54157. [8] K. Saha, S. Acharya, C. Nakamata, Heat transfer enhancement and thermal performance of lattice structures for internal cooling of airfoil trailing edges, J. Therm. Sci. Eng. Appl. 5 (2013) 011001. [9] H. Deng, K. Wang, J. Zhu, W. Pan, Experimental study on heat transfer and flow resistance in improved latticework cooling channels, J. Therm. Sci. 22 (2013) 250–256. [10] T. Tsuru, K. Ishida, J. Fujita, K. Takeishi, Three-dimensional visualization of flow characteristics using a magnetic resonance imaging (MRI) in a lattice cooling channel, J. Turbomach. 141 (2019) 061003. [11] H. Deng, L. Li, J. Zhu, Z. Tao, S. Tian, Z. Yang, Heat transfer of a rotating two-inlet trailing edge channel with lateral fluid extractions, Int. J. Therm. Sci. 125 (2018) 313–323. [12] W. Du, L. Luo, S. Wang, X. Zhang, Effect of the dimple location and rotating number on the heat transfer and flow structure in a pin finned channel, Int. J. Heat Mass Transf. 127 (2018) 111–129. [13] S.M. Hosseinalipour, H.R. Shahbazian, B. Sunden, Experimental investigations and correlation development of convective heat transfer in a rotating smooth channel, Exp. Therm. Fluid Sci. 94 (2018) 316–328. [14] S. Acharya, F. Zhou, J. Lagrone, G. Mahmood, R.S. Bunker, Latticework (Vortex) cooling effectiveness: rotating channel experiments, J. Turbomach. 127 (2005) 471–478. [15] I.T. Oh, K.M. Kim, D.H. Lee, J.S. Park, H.H. Cho, Local heat/mass transfer and friction loss measurement in a rotating matrix cooling channel, J. Heat Transf. 134 (2012) 011901. [16] C. Carcasci, B. Facchini, M. Pievaroli, L. Tarchi, A. Ceccherini, L. Innocenti, Heat transfer and pressure drop measurements on rotating matrix cooling geometries for airfoil trailing edges, ASME Paper No. GT2015-42594. [17] L. Luo, Z. Zhao, X. Kan, D. Qiu, S. Wang, Z. Wang, On the heat transfer and flow structures characteristics of turbine blade tip underside with dirt purge holes at different locations by using topological analysis, J. Turbomach. 141 (2019) 071004. [18] L. Luo, W. Du, S. Wang, W. Wu, X. Zhang, Multi-objective optimization of the dimple/protrusion channel with pin fins for heat transfer enhancement, Int. J. Numer. Methods Heat Fluid Flow 29 (2019) 790–813. [19] R.J. Simoneau, F.F. Simon, Progress towards understanding and predicting heat transfer in the turbine gas path, Int. J. Heat Fluid Flow 14 (1993) 106–128. [20] S. Su, J.-J. Liu, J.-L. Fu, J. Hu, B.-T. An, numerical investigation of fluid flow and heat transfer in a turbine blade with serpentine passage and latticework cooling, ASME Paper No. GT2008-50392. [21] T. Hagari, K. Ishida, Numerical investigation on flow and heat transfer in a lattice (matrix) cooling channel, ASME Paper No. GT2013-95412.

5. Conclusions A numerical investigation was carried out to study the effect of the clearance and Ro number on the heat transfer and flow structure in a detached latticework duct. The main conclusions can be summarized as follows: (1) After adoption of the clearance, the mechanical energy loss was significantly reduced while the Nusselt number was only decreased slightly both in the static duct and rotational duct. In the static duct, the wall shear stress decreased from 0.0724 to 0.0195 while the Nusselt number only decreased from 4.21 to 2.56. (2) The upward spiral flow near the side wall, helical flow in each subchannel, and the leakage flow at the clearance were the main flow structures in the detached latticework duct. As the clearance was increased, the leakage flow was augmented while the upward spiral flow and helical flow were suppressed. Under the rotating condition, a high rotational number also enhanced the leakage flow at the clearance. However, the high rotational number was against the helical flow and upward spiral flow. (3) The Nusselt number in each sub-channel was decreased from the impingement region to the turning region. The large clearance was a disadvantage for the heat transfer enhancement in the latticework duct. In the rotational duct, the Nusselt number was enhanced on the suction side while it was decreased on the pressure side. In addition, the high Nusselt number near the impingement region 38

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