Effect of heat flux on boundary layer flow under rotating conditions

International Journal of Heat and Fluid Flow 80 (2019) 108493

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Effect of heat flux on boundary layer flow under rotating conditions a,⁎

a

a

a

Ruquan You , Shengjun Zhou , Haiwang Li , Zhi Tao a

T

National Key Laboratory of Science and Technology on Aero-Engine Aero-thermodynamics, Beihang University Beijing 100191, China

ARTICLE INFO

ABSTRACT

Keywords: Gas turbine PIV Turbulence Boundary layer thickness Rotation number

The boundary layer flow behaviour in a smooth rotating channel with heated walls is measured by particle image velocimetry (PIV). To simulate the real operation environment of an internal coolant channel in a turbine blade, airflow is analysed in a rotating channel, whose four walls are uniformly heated by Indium Tin Oxide (ITO) glass. The flow is measured in the middle plane of the rotating channel with a Reynolds number equal to 10000 and rotation numbers ranging from 0 to 0.52. The results are presented for the boundary layer flow behaviour with and without heated thermal boundary conditions. The buoyancy force generated by the heated walls influences the flow behaviour under rotating conditions. Separated flow occurs, which substantially influences the turbulent flow behaviours. Sometimes, this buoyancy force can determine the flow behaviours. The results also showed that the displacement thickness and the momentum loss thickness present new changes at different radius positions due to the heated thermal boundary conditions. The displacement thicknesses of both the leading and trailing sides with heated walls are both thicker than those of the leading and trailing sides without heated walls. Then, the difference of the boundary layer thickness between these two cases increases with the increase of rotation number. For momentum loss thickness, a sharp drop happens when the rotation number increases to a certain value. At the large radius position, the drop in momentum loss thickness is much greater than that in the small radius position.

1. Introduction It is widely acknowledged that increasing the inlet gas temperature of a turbine is an important method for improving the turbine efficiency and increasing the thrust-weight ratio. Therefore, to achieve higher inlet temperatures—even beyond the melting point of materials—many kinds of cooling techniques have been investigated and applied. As a typical cooling technique, internal forced convection can maintain an adequate temperature for the blade material to resolve this conflict. Serpentine passages in the middle section of a blade are an effective and classical method of internal forced convection and have been researched, optimized and utilized in turbine blades for several decades. Han and Huh (2010) published a review paper, which reviewed a large number of studies related to the internal cooling of turbine blades. However, the majority of experiments focused on the heat transfer instead of the boundary layer flow and the velocity measurement, which is due to the intricate measurement technology at rotation conditions. To reveal the mechanism of heat transfer and flows in a rotating channel is a valuable reference for turbine blade designers. The connection between heat transfer and flow conditions in a rotating channel is complicated, especially when the channel walls are

⁎

heated. In this case, the Coriolis force, centrifugal force, and buoyancy force are three key forces that influence the heat transfer and flow conditions. Among them, the Coriolis force, which is induced by rotation, has been investigated in rotation channels by many researchers. The effect of the Coriolis force was numerically investigated in Ref. (Kheshgi H and Scriven L, 1985, Kristoffersen and Andersson H, 1993) and experimentally investigated in Ref. (Liou et al., 2003, Liou T et al., 1993, Liou et al., 2007, Liou T et al., 2001, Johnston J, 1998, Wagner J et al., 1991, Johnston J, 2006). All of these investigators reported that Coriolis force pushed the mainstream velocity to trailing side, and the Coriolis force caused the turbulent Reynolds stresses are asymmetric in the rotating channel. These authors also found that rotation stabilises the flow on the leading side and destabilises the flow on the trailing side. Moreover, Hart, (1971) found that on the stabilised layer, the turbulent stress decreases. Johnston J, (2006) found that the flow was stabilised on the leading side, but destabilised on the trailing side, which is due to the Coriolis force. Similar results were obtained via direct numerical simulation by Kristoffersen and Andersson H (1993). Hart (1971) also found stabilisation on the leading side, and destabilisation on the trailing sides. Moreover, they reported that the turbulent stress decreases on the stabilised layer. Cheah S et al. (1996) used

Corresponding Author. E-mail address: [email protected] (R. You).

https://doi.org/10.1016/j.ijheatfluidflow.2019.108493 Received 2 June 2019; Received in revised form 10 October 2019; Accepted 12 October 2019 Available online 31 October 2019 0142-727X/ © 2019 Elsevier Inc. All rights reserved.

International Journal of Heat and Fluid Flow 80 (2019) 108493

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Nomenclature

δ*Displacement thickness Displacement thickness(mm) * Sum of the displacement thicknesses of the leading side and trailing SUM side Sum of the displacement thicknesses of the leading side and trailing side(mm) ΩRotational speed Rotational speed(rpm) μMolecular dynamic viscosity Molecular dynamic viscosity(kg/(ms)) Viscous Viscous(-) ρDensity Density(kg/m3)

DHydraulic diameter Hydraulic diameter(mm) ITurbulence intensity Turbulence intensity(-) MMagnification factor Magnification factor(pixel/mm) RRotation radius Rotation radius(mm) RoRotation number Rotation numberΩD/U0(-) ReReynolds number Reynolds numberρvD/μ(-) qwHeat flux Heat flux(W/m2) uVelocity Velocity(m/s) XStreamwise direction Streamwise direction(-) YNormal direction Normal direction(-) ZSpanwise direction Spanwise direction(-)

Subscripts m e

Averaged value Environment

Greek symbols θMomentum loss thickness Momentum loss thickness(mm) another measurement method, laser Doppler velocimetry (LDV), to observe the impact of the Coriolis force on the mean flow at the symmetry plane in a U-duct under rotating conditions, and the same conclusions were drawn. Martensson et al. (2002) investigated the effect of rotation on the pressure drop in the rotating channel. They found that the friction coefficient increases with rotation numbers. Gallo and Astarita (2010) applied a PIV measurement system to obtain the velocity field in the rotating channel to investigate the rotation effects. Visscher and Andersson H (2011) using a PIV system to investigate the separated turbulent flows under rotation conditions. They found that the primary separation bubble length decreases monotonically with rotation numbers on leading side. In addition to the main flow, the secondary flow is also a focus of researchers. Gallo et al. (2012) and Elfert et al. (2012) measured the secondary flow to reveal the effect of

the Coriolis force in rotating channels. Comparing the flow data with the heat transfer data, Gallo found that the heat transfer increases with the increase in turbulent kinetic energy. Macfarlane and Joubert P (1998) carried out experiments on the effect of secondary flows on developing turbulence. There are few investigations on the boundary layer of rotating channels because the measurement of the boundary layer is a great challenge. Sante A and Den Braembussche (2010) using TR-PIV (time-resolved particle image velocimetry) system to measure the 3D visualization of the flow field in the rotating channels. They discussed the boundary layer thickness with rotation numbers. They found the boundary layer thickness decreases on the trailing side, but increases on the leading side. Then, the heated thermal boundary conditions were applied in the rotation channels to simulate the real gas turbine blade and to

Fig. 1. The rotating facility and PIV set up 2

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commercial CFD software.

investigate the connection between heat transfer and flow behaviours. The buoyancy force is induced by the temperature gradient near the channel wall; this is another force that has an important influence on the characteristics of flow and heat transfer. Siegel (1985) studied influence of heated walls on the heat transfer of developed laminar flow by CFD. They also found that the buoyancy force improves the heat transfer in the inflow passage and reduces the heat transfer in the outflow passage. Han and Zhang Y (1992) investigated the effect of thermal boundary conditions on local heat transfer in rotating channels. Three cases are discussed: (A) four walls with uniform temperature, (B) four walls with uniform heat flux and (C) leading and trailing walls hot and two side walls cold. They compared the value of on the leading side, and found that case B is larger than that of case C and case A. On the trailing side, the result is case A > case C > case B. However, on the trailing side, the differences of among three cases is less than that on leading side. The results suggest that it is the local buoyancy forces changing the influence of rotation, which changing the local heat transfer coefficients on both leading and trailing side. The results revealed that the non-uniform wall temperature generates a local buoyancy force, which has an important role on the impact of rotation. Dutta et al. (1996) predicted the flow field by a two-equation turbulent model in a rotating channel, and they discovered flow separation, which appeared at the leading side. Viscous force is another force in the boundary layer. Separated flow happens when the buoyancy force is larger than the viscous force. Li, You and Deng (Li et al., 2016, Li et al., 2017] investigated the heat transfer and flow behaviour in a rotating channel. They found the Coriolis force and buoyancy force play an important effect on the heat transfer on both leading and trailing surface. Shen et al. (2015) investigate the heat transfer and flow field in a U-shaped channel with different kinds of ribs numerically. They found that the ribs can enhanc the heat transfer in the channel comparing with the smooth channel. Mayo et al. (2015) and Coletti et al. (2014) using TR-PIV to measure the flow field in a rotating channel. They found that buoyancy force is an important factor on the distribution of the vortices near the leading side. Bons (1997) also investigated the effect of the Coriolis force and buoyancy force on the flow field in a rotating channel by PIV. However, he haven't obtained velocity data in the boundary layer because the laser reflection near the channel wall. As mentioned above, previous research covers many aspects of the flow behaviours in rotating channels. However, to best of our knowledge, there are few measurements of boundary layer flow in rotating channels with heated thermal boundary conditions, which can be used to investigate the effect of thermal boundary conditions on boundary layer flow. The current work focuses on the boundary layer flow, especially the boundary layer thickness, under the effect of thermal boundary conditions. The four channel walls are simultaneously heated by ITO glass to simulate thermal boundary conditions that are similar to those of real turbine blades. The boundary layer flow field experimental data at rotating conditions can be used for validation on the accuracy of

2. Experimental set up and measurement system To achieve the aim of this experimental investigation, a set of experimental apparatuses and a measurement system have been established, which are shown in Fig. 1. The four main parts are described below. The rotating facility The rotating facility used in current work is presented in Fig. 1, which was described by You et al. (2017). A rotating disc and the extensions arms are connected to a vertical hollow shaft, which is the rotatory axis. These devices are driven by a DC motor. All of the data acquisition modules are putted on the rotating disc. A pair of transparent test channels are fixed on the rotating extensions arms. Fig. 2 presents the experimental system of coolant air in current work. The compressed air produced by the blower passes through a cooling system, which keeps the air temperature constant. A Laskinnozzle-based particle generator is used to produce suitable seeding particles. Before this cool air enters the flow meter, the seeding particles are evenly produced and proportionally mixed with the seeding particles for better capture by the PIV system. Then, the mixed flow of air and seeding particles passes through the flow meter. The rotary joint enables the transition between the static and rotating conditions of the flow. Finally, the air enters the rotating transparent channel after stabilisation in the settling chamber. A thin indium-tin-oxide (ITO) heater glass with a thickness of 1 mm is pasted onto each inner side of the channel walls. ITO glass is a special type of glass that has excellent electrical conductivity and appropriate electric resistance so that uniform heat flux can be provided when the ITO glass is subjected to an electric charged. The ITO glass is transparent, which ensures that the PIV laser completely covers the test area. The temperature on the rotating frame can be measured by the thermocouples, whose cold ends are fixed on the disc. In the rotating conditions, the analogue signals of temperature are transmitted to the static facilities by signal slip rings, which provide the digital signals. Direct current is conveyed by a slip ring for heating the channel. To maintain the balance of the disc and extension, an extra counterweight is fixed on the opposite extension arm. Test section and conditions As shown in Fig. 3, the cross-section of the test channel is an 80 mm × 80 mm square. X is along the direction of the mean flow, and X/ D=0 represents the entrance of the channel. This entrance is at the rotating radius of 281 mm, where the heat flux starts to be provided. In the current work, the region of main flow that is measured and analysed is 3.5
Fig. 2. Experimental system 3

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Fig. 3. Test section

the rotation number Ro, which is calculated as follows:

Ro =

D U0

validation algorithms when the vector field is completely calculated. Regarding the peak ratio value of 2 as the post-processing standard, the inferior vectors are eliminated, and then the empty area is filled by subpixel interpolation. FPGA-based trigger signal generator As mentioned above, the PIV system is in a static condition outside the rotating facility. The PIV system ought to be precisely synchronized to a specified circumferential position of the test plane. To obtain the accurate relative velocity of the air in the channel, we can measure the absolute velocity by the static PIV system and obtain an accurate peripheral velocity at each position of the test section. Hence, a trigger signal generator based on field-programmable gate array (FPGA) is applied to obtain the peripheral velocity with a short delay time by a phase-locked measurement. This trigger signal generator consists of an FPGA module, a high-precision incremental encoder, a level comparison and conversion module, a serial communication module and a transistor-transistor logic (TTL) trigger pulse output module. As the trigger signal generator operates, the encoder resolves the circumferential position of the rotor with an accuracy of 0.0038 degrees and generates the pulse signal. The level comparison and conversion module converts the signal to the form of the FPGA module. Comparing the pulse signal count with the predetermined threshold, the TTL trigger pulse output module activates the PIV system. In addition, the instantaneous rotational velocity is calculated by the FPGA module with an accuracy of 0.0127 rpm. Then, the velocity is output to the host computer via the communication module. Therefore, the peripheral velocity at each position of the plane can be calculated from the rotating radius and the obtained instantaneous rotational speed.

(1)

where Ω is the angular velocity of the rotating channel, UO is the relative bulk flow velocity, and D is the characteristic length (i.e., the channel hydraulic diameter of 80 mm). The thermal boundary condition is in a steady state. A thin film ITO glass heater with a 1 mm thickness is pasted onto each wall to generate uniform heat flux (qw = 380W/m2 ) from the four walls. The experimental turntable rotates at various circumferential velocities, and the rotation numbers are 0, 0.13, 0.26, 0.39, and 0.52 at each of the tested velocities. In this case, the operational environment of the turbine blades has been simulated, and the flow behaviours can be compared with those without heated walls to determine the effect of the thermal conditions on the flow in rotating channels. The mean velocity measured by PIV is validated by the data obtained with a 1D hot-wire measurement system [29], which is presented in Fig. 4. The measurement result of the PIV system is in accordance with the result of the 1D hot-wire measurement system. The turbulence intensity is approximately 4% in the mean flow. PIV setup and methodology The CCD camera, laser module and seeding generator are three key components of the PIV setup used in this study. The schematic of the rotating facility and the layout of PIV can be seen in Fig. 1. In the current work, the CCD camera and the laser module are located outside of the rotating disc. The CCD camera is fixed on static ground, and it is positioned below the rotating channel. The laser module is fixed on static ground, and it is positioned on one side of the measured channel. Because the PIV system is outside of the rotation, the camera and laser module are synchronized by a high accuracy encoder to capture pairs of images at appropriate moments. The light pulses are generated by a dual-cavity 135 mJ Nd:YAG pulse laser device and guided via a lightguiding arm to the test plane with a thickness less than 1 mm. The seeding is provided by a Laskin-nozzle-based seeding generator with dioctyl sebacate (DEHS), which ensures that the particles have an average diameter of 1 μm. A CCD camera is used to capture images. Although the CCD camera has a resolution of 2048 × 2048 pixel2, the resolution of the active pixels that cover the 80 mm × 80 mm region of each plane are 1778 × 1778 pixel2 for . In that case, a Nikon 105 mm macro lens for planes is used, and the magnification factor is 22.225 pixels/mm. In the experiments, rotational speed and delay time result in displacement between image pairs, which causes particle displacements that are larger than a fourth of the interrogation region dimension. An initial window shift is used to reconstruct the image so that the displacement of the particle is approximately 8-10 pixels. In that case, the software Flow Master can be used to analyse the collected image pairs. The spurious vectors can be recognized and eliminated by vector

3. Analysis of uncertainties The displacement thickness and momentum loss thickness of the boundary layer are indirectly measured. We can analyse the uncertainties of these characteristics with the error transfer function; this

Fig. 4. Comparison between the PIV measurements and hot-wire measurements [You et al., 2018] 4

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process is described below. According to the formula of displacement thickness u ( * = 0 (1 U ) dy ) and the velocity profile of boundary layer pree sented by You et al. (2017), we can use a method called the discrete area formula to obtain the magnitude of the displacement thickness. After derivation, the calculation formula is obtained as shown below. n

*=

0

1 * (1 2

u )1 + (1 Ue

u ) 0 *(y1 Ue

y0 )

Table 1 Uncertainty of the parameters.

(2)

In the formula above, n is the number of velocity points from the wall to the outlet of the boundary layer. The error of the displacement thickness can be calculated by the following formula.

( )( * 2 u1

n 0

*=

u1)2 +

( ue 0 )2 + n 0

*=

2*(

( )( * 2 u0

( )( * y1

2

y1) 2 +

(

y1 y0 2 ) ( 2Ue

( )( * 2 ue1

u 0 )2 +

u )2 + 2*

( ) ( y) 2

* y0

u * (y1

0

2Ue2

y0 )

ue1) 2 +

* 2 u e0

2

2

e

2

2

Some characteristics will be explained hereafter. First, y is the distance from the velocity point to the channel wall. The error of y can be ignored in the current work, as Δy ≈ 0. The velocity outside of the boundary layer ΔUe has the same error as the velocity in the boundary layer, so we can consider that u = Ue . Finally, the uncertainty of δ*can be calculated by Eq. (4). n

( 0

u*(y1 y0 ) y1 y0 2 ) ( u) 2 + 2Ue 2Ue2

The boundary layer thickness is discussed in this part by measuring the boundary layer flow with different rotation numbers and different radius positions without wall heated. u Displacement thickness ( * = 0 (1 U ) dy ) is the thickness of e mass loss induced by the inviscid fluid flowing through the channel. Fig. 5 presents the comparison of displacement thickness between the leading side and trailing side. Due to Fig. 5, the displacement thickness on leading side is larger than that on trailing side. This is due to the Coriolis force pointing to the trailing side, making the boundary layer is compressed near the trailing side, then the boundary layer thickness becoming thinner on trailing side. With the increase of rotation number, the difference between the two sides is enlarged. This is due to the Coriolis force in enhanced with the increase of rotation numbers. u u Momentum loss thickness ( = 0 U (1 U ) dy ) is another e e boundary layer thickness that represents momentum loss induced by inviscid in the boundary layer. Fig. 6 presents the same distribution of the momentum loss thickness with that of displacement thickness. The momentum loss thickness of the leading side is thicker than that of the trailing side. With the increase of the rotation numbers, the momentum loss thickness of the leading side becomes thicker. However, at the large radius position (X/D=5, 6), the momentum loss thickness of the leading side first increases and then decreases with increases in rotation number. In contrast, the momentum loss thickness of the trailing side first decreases and then increases. Some theoretical analyses can account for the above-mentioned evolution. The Coriolis force and pressure gradient component are two main forces that act on the cross-section of the leading side of the rotating channel without heated thermal boundary conditions. Fig. 7 reveals the relationship of these two main forces along the flow direction. Cooling air flows through the test channel in the streamwise direction at a constant speed; therefore, the tangential Coriolis force maintains its magnitude along the stream and its direction points towards the trailing side at a certain rotation number. In addition, the pressure gradient component points to the leading side, which is contrary to the direction of the tangential Coriolis force; the pressure gradient also increases along the streamwise direction. When the Coriolis force is larger than pressure gradient component, the main flow is pushed to the trailing side by Coriolis force, and both displacement thickness and momentum loss thickness is larger at the leading side than that of the trailing side. With the increase of rotation number, the difference of the boundary layer thickness between the leading and trailing side becoming larger. In a certain rotation number, the pressure gradient component

(4)

Twenty velocity points were used to calculate the displacement thickness of the boundary layer; therefore, we can consider that n is 20 and y1 y2 = 20 . u Ue

For the momentum loss thickness, ( = 0 method has been applied, as shown hereafter.

1 * 2

n

=

0

u 1 Ue

u Ue

u 1 Ue

+ 1

u Ue

(1

*(y1

u ) dy ), Ue

a similar

y0 )

(5)

0

The equation above is derived with the discrete area formula, where n is the number of velocity points from the wall to the boundary. The error of the momentum loss thickness can be calculated by the following formula.

=

n 0

( )( 2

u1)2 +

u1

( ue0 ) 2 +

( )( 2

u0

( )( 2

u 0 )2 +

y1)2 +

y1

( )( 2

ue1

( )( 2

y0

ue1) 2 +

( )

2

u e0

y0 )2

= n 0

2* (y1

y0 )

(

1 2Ue

u Ue2

)

2

( u )2 + 2* (y1

y0 )

(

1 2Ue2

+

u2 Ue3

)

2

( Ue )2 + 2*( y )2 (6) As described above, we can consider that uncertainty of θ can be calculated by Eq. (7).

u = Ue . Finally, the

= n 0

2* (y1

y0 )

1 2Ue

u Ue2

2

+ 2* (y1

y0 )

u u2 + 3 2 2Ue Ue

± 0.5 m3/h ± 1°C ± 0.044 m/s ± 0.19 mm ± 0.14 mm

4.1. Unheated thermal boundary conditions

2

( u) 2

Flow meter Temperature (T) Velocity (u) Displacement thickness (δ*) Momentum loss thickness (θ)

4. Results and discussion

(3)

*=

Uncertainty

Twenty velocity points are taken within the boundary layer; therefore, we can consider that n = 20 and y1 y2 = 20 . In this experiment, all the uncertainty can be seen in Table 1. The uncertainty of the flow meter is ± 0.5m3/h, and the uncertainty of the temperature measurement is ± 1∘C. The PIV system gives an uncertainty of velocity of ± 0.044 m/s. Therefore, the maximum uncertainty of the displacement thickness and momentum loss thickness is ± 0.19 mm and ± 0.14 mm, respectively.

( )

) ( U ) + 2*( y)

Parameters

2

( u)2 (7) 5

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Fig. 5. Displacement thickness of the leading side and trailing side without heated walls at different rotation numbers.

increases along the stream wise directions. When the pressure gradient component and the Coriolis force and have the same magnitude in the cross-section of the channel, both displacement thickness and momentum loss thickness near the leading side reach maximum values. This maximisation indicates the critical point of the boundary layer thickness evolution. Beyond this critical point, the tangential Coriolis force continues to increase and surpasses the pressure gradient component as the rotation number increases and the radius position changes. The net force for the tangential Coriolis force and pressure gradient points to the leading side. At the large radius position (X/ D=6), the displacement thickness and momentum loss thickness of the leading side start to gradually decrease. In addition, both the displacement thickness and the momentum loss thickness of the trailing side start to decrease.

displacement thickness of the leading side becomes thicker than that of the trailing side, and the gap between the thicknesses of the trailing side and leading side increases as the radius position increases. When the test channel is heated by the ITO glass, both the displacement thicknesses of the leading side and trailing side are thicker than those without heated thermal boundary conditions. The momentum loss thickness exhibits a similar trend at the small radius position without heated thermal boundary conditions. However, a special case happens in the right series of figures. The momentum loss thickness starts to decrease at the large radius position with heated thermal boundary conditions. It is very obvious that the momentum loss thickness decreases sharply when the rotation number is increased to 0.52 in the last figure. This sharp decrease in the momentum loss thickness is caused by the flow separation, which will be described in the following chapters. The sum of the displacement thicknesses of the leading side and * , which represents the thickness of mass trailing side is defined as SUM loss induced by the inviscid fluid flowing through the channel. Fig. 9 * with or without heated thermal shows the trend and comparison of SUM boundary conditions. When the test channel is rotating at a low speed (Ro= 0 or 0.13), the difference between the heated and unheated conditions is negligible. The heated thermal boundary conditions had little effect a low rotating speed. Then, the sum of the displacement thicknesses of the leading side and trailing side increases when the rotating channel is heated by the ITO glass and rotated at a relatively * increases, high speed (Ro= 0.26, 0.39, 0.52). When the value of SUM the mainstream thickness decreases. In this case, the mainstream speed increases because of the constant mass flow, as shown in Fig. 12. Heated thermal boundary conditions lead to a density gradient in the airflow. As a result, the buoyancy force, whose direction is opposite to the flow, acts on the airflow near the channel walls. The airflow is pressed to the middle of the channel from the trailing side and leading

4.2. Heated thermal boundary conditions As mentioned above, the rotating channel can be heated evenly from each of the four walls by thin-film ITO glass heaters with a constant heat flux (qw = 380w/m2 ), which can keep the density ratio in current work up to 0.1 according to our previous work (You et al., 2017]. Under these heated thermal boundary conditions, we can obtain the new trend in displacement thickness and momentum loss thickness of the boundary layer at various peripheral velocities along the direction of the flow, as shown in Fig. 8. The series of images shown in Figs. 8(a) and (b) illustrate the change in displacement thickness and momentum loss thickness at different circumferential positions, respectively. First, some similar trends can be found between these two series of figures. When the test channel is under static conditions, these two kinds of thicknesses at the trailing side and leading side are almost the same, regardless of whether the test channel is heated. Once the test channel starts to rotate, the

Fig. 6. Momentum loss thickness of the leading side and trailing side without heated walls at different rotation numbers. 6

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Fig. 7. Pressure gradient component and tangential Coriolis force of the leading side without heated thermal boundary conditions

side. The boundary layer becomes thicker and the mainstream speed becomes faster than that of the unheated case. The comparison between the heated unheated thermal boundary conditions is shown in Fig. 10. First, the displacement thicknesses of the leading side and trailing side with heated walls are thicker than those without heated walls. The gap in the displacement thicknesses between these cases increases as the rotation number increases. Some theoretical analyses can account for this increase. The buoyancy force is induced by the density gradient in the airflow within the rotating heated channel. Since the direction of the buoyancy force is opposite to the flow, the boundary layer is restrained and becomes thicker than that in the unheated case. For the momentum loss thickness of the boundary layer, the trend is roughly the same as that of the displacement thickness. When the rotation number increases to a certain value, the momentum loss thickness decreases sharply, as shown in Fig. 11. At the large radius position, the drop in momentum loss thickness is substantially increased. The momentum loss thickness represents the thickness of momentum loss induced by the inviscid fluid flowing through the channel. In other words, the momentum thickness loss is the product of the mass and vector velocity. Separated flow was first obtained on the leading side with heated walls by You et al. (2017). The temperature gradient induced by the heated walls causes the density difference and the buoyancy force. Fig. 12 shows the velocity profile of the mean flow when the rotation number equals 0.52 with heated and unheated thermal boundary conditions. The direction of the leading side velocity becomes opposite to its original direction, especially at a large rotation radius X/ D=6. As shown in Fig. 12(c), the buoyancy force and inertial force are a pair of interactions. In the experiment, the velocity profile is pushed towards the trailing side owing to the Coriolis force, which decreases the magnitude of the inertial force. At the same time, the increasing buoyancy force becomes greater than the inertia force, and then the flow separates at the leading side. When the separated flow happens at the leading side, the stream direction becomes opposite to its original direction. As a result, the momentum loss thickness starts to decrease sharply.

5. Conclusions The effect of the heated thermal boundary conditions on the flow behaviours in the boundary layer has been investigated by means of PIV measurements. Experiments were carried out with a constant Reynolds number of 10000, and a series of rotation numbers, ranging from 0 to 0.52 with an interval of 0.13. The boundary layer flow is measured in the middle plane of the channel in the position of 3.5
Fig. 8. Displacement thickness and momentum loss thickness at different circumferential positions 7

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Fig. 9. The sum of the displacement thicknesses of the leading side and trailing side with and without heated walls at different rotation numbers.

Fig. 10. Displacement thickness of the leading side and trailing side with and without heated walls at different rotation numbers.

Fig. 11. Momentum loss thickness of the leading side and trailing side with and without heated walls at different rotation numbers. 8

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Fig. 12. Velocity profile of the mean flow with Ro=0.52.

Acknowledgements

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