A computational study of three-dimensional turbulent flow and heat transfer was performed in four types of rotating channels.
The first type is a rotating rectangular channel with V-shaped ribs. The channel aspect ratio (AR) is 4:1, the rib height-to-hydraulic diameter ratio (e/Dh) is 0.078 and the rib pitch-to-height ratio (P/e) is 10. The rotation number and inlet coolant-to-wall density ratio were varied from 0.0 to 0.28 and from 0.122 to 0.40, respectively, while the
Reynolds number was varied from 10,000 to 500,000. Three channel orientations (90 degrees, −135 degrees, and 135 degrees from the rotation direction) were also investigated. The second type is a rotating rectangular channel with staggered arrays of pinfins.
The channel aspect ratio (AR) is 4:1, the pin length-to-diameter ratio is 2.0, and the pin spacing-to-diameter ratio is 2.0 in both the stream-wise and span-wise directions.
The rotation number and inlet coolant-to-wall density ratio varied from 0.0 to 0.28 and from 0.122 to 0.20, respectively, while the Reynolds number varied from 10,000 to iv 100,000. For the rotating cases, the rectangular channel was oriented at 150 degrees with respect to the plane of rotation.
In the rotating two-pass rectangular channel with 45-degree rib turbulators, three channels with different aspect ratios (AR=1:1; AR=1:2; AR=1:4) were investigated. Detailed predictions of mean velocity, mean temperature, and Nusselt number for two Reynolds numbers (Re=10,000 and Re=100,000) were carried out. The rib height is fixed as constant and the rib-pitch-to-height ratio (P/e) is 10, but the rib height-to-hydraulic diameter ratios (e/Dh) are 0.125, 0.094, and 0.078, for AR=1:1,
AR=1:2, and AR=1:4 channels, respectively. The channel orientations are set as 90 degrees, the rotation number and inlet coolant-to-wall density ratio varied from 0.0 to
0.28 and from 0.13 to 0.40, respectively.
The last type is the rotating two-pass smooth channel with three aspect ratios
(AR=1:1; AR=1:2; AR=1:4). Detailed predictions of mean velocity, mean temperature and Nusselt number for two Reynolds numbers (Re=10,000 and Re=100,000) were carried out. The rotation number and inlet coolant-to-wall density ratio varied from 0.0 to 0.28 and from 0.13 to 0.40, respectively.
A multi-block Reynolds-averaged Navier-Stokes (RANS) method was employed in conjunction with a near-wall second-moment turbulence closure.

A widely used method for cooling turbine blades is to bleed lowertemperature air from the compressor and circulate it within and around each blade. The coolant typically flows through a series of straight ducts connected by 180° turns and roughened with ribs or pin fins to enhance heat transfer.

They suggested that a low
Reynolds number turbulence model is necessary to simulate real gas turbine engine conditions and a Reynolds stress model is required to capture anisotropic effects

They concluded that V-shaped ribs pointing upstream performed superior to V-shaped ribs pointing downstream and transverse ribs

Han et al.20 found that the 60° and 45° Vshaped ribs perform better than the 60° and 45° parallel angled ribs in square channels.
Ekkad and Han11 also showed that the non-rotating square channel with 60° V-shaped rib
6
produced more heat transfer enhancement than 60° and 90° angled ribbed channels. Al-
Hadhrami et al.29, 30 studied the effect of rotation on heat transfer in rotating two-pass square and rectangular channels (AR = 2:1) with rib turbulators for five different channel orientations. They found that the parallel and V-shaped ribs produced better heat transfer enhancement than the crossed and inverted V-shaped ribs. Lee et al.31 studied the heat transfer in rotating rectangular channels (AR = 4:1) with V-shaped and angled rib turbulators with and without gaps. They concluded that the V-shaped ribs produced more heat transfer enhancement than the angled ribs for both the stationary and rotating cases, and the enhancement on both the leading and trailing surfaces increases with rotation. The experimental results of Lee et al.31 for narrow rectangular channels (AR =
4:1) with V-shaped ribs provided a validation database for the present work.
The above experimental studies show that V-shaped ribs produce overall better heat transfer enhancement than the angled ribs. However, the effects of rotation, channel orientation and large channel aspect ratio on the secondary flow and heat transfer calculations in rectangular channels with V-shaped ribs have not been systematically investigated in previous numerical studies.

The present study is concerned specifically with the numerical prediction of flow and heat transfer in rectangular channels with V-shaped ribs.

Almost all of the studies have focused on square or close to square ducts and the previous numerical research focused on the validation of turbulence models and the physical behavior of the flow and heat transfer in the square duct (AR=1:1) or channels with aspect ratios greater than one (AR=2:1, and AR=4:1). However, confined by the blade configurations, the cooling passage should include not only these higher aspect ratio ducts, but also the lower aspect ratio ducts, such as the AR=1:2 and AR=1:4 channels considered in the present study.

The objective of this study is to use the second-moment RANS method of Chen et al.7, 8 (1) to predict the three-dimensional flow and heat transfer for rotating one-pass rectangular ducts (AR = 4:1) with V-shaped ribs and compare with the experimental data of Lee et al.31; (2)to evaluate the effects of lower aspect ratio rectangular smooth channels (AR = 1:1, AR = 1:2, and AR = 1:4) on heat transfer for both the non-rotating and rotating conditions; (3) to evaluate the effects of lower aspect ratio rectangular ribbed channels (AR = 1:1, AR = 1:2, and AR = 1:4) on heat transfer for both the nonrotating and rotating conditions; (4) to facilitate efficient simulation of detailed threedimensional flow and heat transfer in stationary and rotating pin-fin channels under high
Reynolds number and high rotation number conditions; (5) to investigate the effect of high rotation number, high Reynolds number, and high density ratio on the secondary flow field and the heat transfer characteristics in a rectangular duct with turbulators rotating at −135°, 90°, 135° and 150° orientations.

The Governing Equation and Chimera Method
For unsteady incompressible flow, the continuty equation and momentum equation were to be solved to get the velocity field,

The Reynolds stress tensor is the solution of the Reynolds stress transport equation :

The temperature T is obtained from the energy equation:

The present method solves the mean flow and turbulence quantities in arbitrary combination of embedded, overlapped, or matched grids using a chimera domain decomposition approach. In this approach, the solution domain is first decomposed into a number of smaller blocks which facilitate efficient adaption of different block geometries, flow solvers and boundary conditions for calculations involving complex configurations and flow conditions. Within each computational block, the finite-analytic numerical method was employed to solve the unsteady RANS equations.
3.1 Description of Problem
A schematic diagram of the geometry is shown in Figure 3.1. The channel has a rectangular cross section with a channel aspect ratio (AR) of 4:1. Of the four side walls, the two in the rotational direction are denoted as leading and trailing surfaces, while the other two are denoted as top and bottom surfaces. The channel hydraulic diameter, Dh, is 0.8in (2.03cm). The distance from the inlet of the channel to the axis of rotation is given by Rr/Dh = 20.0. The total length of the channel, L, equals to 22.5Dh. The channel consists of an unheated smooth starting section (L1/Dh = 9.92), a heated ribbed section
(L2/Dh = 7.58), and an unheated smooth exit section (L3/Dh = 5.00). In the ribbed section, as shown in Figure 3.1, the leading and trailing surfaces are roughened by nine equally-spaced 45° V-shaped ribs with square cross section. The in-line V-shaped ribs are parallel and point upstream. The rib height-to-hydraulic diameter ratio (e/Dh) is
0.078. The rib pitch-to-height ratio (P/e) is 10. Three channel orientations (β = 90°, β =
−135°, β = 135°) are studied, with β = 90° corresponding to the mid-portion and β =
−135° and β = 135° to the trailing edge region of a blade

Computational grid details
Figure 3.2(a) shows the computational grid around the ribs for the ribbed duct. The number of grid points in the streamwise direction is 402 (with 373 in the ribbed section), and the number of grid points in the cross-stream plane is 43×75 for Re = 10,000 cases.
For the high Reynolds number (Re = 100,000 and 500,000) cases, the number of grid points in the cross-stream plane was increased from 33×75 to 43×85 while the number of grid points in the streamwise direction was maintained at 373 for the ribbed section. It is clearly seen from the grid refinement studies shown in Figures 3.2(b) and 3.2(c) that these grid distributions produced nearly grid-independent solutions for the Re =10,000 and 500,000 cases, respectively.
3.2 Velocity and Temperature Fields
As summarized in Table 3.1, computations were performed with Reynolds number
(Re) ranging from 104 to 5×105, rotation number (Ro) from 0 to 0.28, inlet coolant-to wall density ratio (Δρ/ρ) from 0.122 to 0.4, and channel orientations from −135° to 90° to 135°. The Nusselt numbers presented here are normalized with a smooth tube correlation by Dittus-Boelter for fully developed turbulent non-rotating tube flow:
Nuo = 0.023 Re0.8 Pr0.4 (3-1)
Fig. 3.3 shows the velocity vector field and dimensionless temperature contour (θ) near the ribbed surface for the non-rotating case (case 1) listed in Table 3.1. As we can see from the figure, since the ribs are oriented at a 45° angle pointing upstream, the fluid

In this chapter, we perform calculations for rotating/non-rotating two pass rectangular channels (AR=4:1) with pin-fins as tested by Wright et al.48 using the near wall second-order Reynolds stress closure model.
4.1 Description of the Problem
A schematic diagram of the pin-fin configuration used in the present study is shown in Fig. 4.1. This test section shown in Fig. 4.1(a) has 12 rows of pins in the flow direction and is identical to that used by Wright et al. in their experimental investigations. For the non-rotating case, the flow is symmetric with respect to both the y and z coordinates at each longitudinal cross-section. Therefore, it is possible to simulate only one-quarter of the channel cross-section but include all 12 rows of pinfins.
Under rotating conditions, however, the flow is no longer symmetric due to the presence of centrifugal and Coriolis forces induced by the channel rotation.
Consequently, it will be necessary to include the full channel cross-section for the rotating pin-fin channels. Due to the limitation in available computer resources, the present rotating channel flow simulations were performed for the full channel crosssection but with only 6 rows of pin-fins, as shown in Fig. 4.1(b). The channel has a rectangular cross section with a channel aspect ratio (AR) of 4:1. Of the four sidewalls, the two in the rotational direction are denoted as the leading and trailing surfaces, while the other two are denoted as the inner and outer surfaces. The channel hydraulic diameter, Dh, is 0.8in (2.03cm). The distance from the inlet of the channel to the axis of rotation is given by Rr/Dh = 20.0. The channel consists of an unheated smooth starting section (L1/Dh = 10.3), a heated section with pins (L2/Dh = 7.5 for channel A and L2/Dh =
3.75 for channel B in Fig.4.1), and an unheated smooth exit section (L3/Dh = 5.00). A summary of all six cases studied is given in Table 4.1.
Computational grid details
Fig. 4.2(a) shows the computational grid around the pins which model channel A with the AR=4:1 in Fig. 4.1(a). In this multi-block chimera grid system, the computational domain was divided into 25 overlapping and embedding chimera grid blocks (one block is for the rectangular duct and the other 24 blocks are for the pins) to simplify the grid generation process. Fig. 4.2(b) shows the computational grid around the pins which models channel B (Fig. 4.1(b) ) which has a total of 6 rows of pins. For numerical grid (B), the number of grid points in the rectangular duct is 296 × 171 × 33 and the number of grid points for each pin is 62× 31 × 33, the identical grid distribution was applied in numerical grid (A). The total number of grid points is just over 3 million.
Fig. 4.3(a) and 4.3(b) present grid refinement studies for Re =10,000 and 100,000, respectively, which indicates the present simulation results are nearly grid-independent.
4.2 Velocity and Temperature Fields
As summarized in Table 4.1, computations were performed with the Reynolds number (Re) ranging from 104 to 105, rotation number (Ro) from 0 to 0.28, inlet coolant to wall density ratio (Δρ/ρ) from 0.122 to 0.2, and for the rotating cases, the channel orientation was fixed at 150o.

FLOW AND HEAT TRANSFER IN ROTATING TWO-PASS RECTANGULAR
CHANNELS (AR=1:1, 1:2, AND 1:4) WITH SMOOTH WALLS
In this chapter, we perform calculations for rotating/non-rotating two pass rectangular channels (AR=1:1, 1:2, and 1:4) with smooth wall as tested by Fu et al.63 using the near wall second-order Reynolds stress closure model.
5.1 Description of Problem
A schematic diagram of the geometry for AR=1:4 duct is shown in Fig. 5.1. As mentioned earlier, this investigation covers three channels with different aspect ratios
(AR = 1:1, AR = 1:2 and AR = 1:4). For the sake of brevity, however, only the geometry for the AR = 1:4 channel is given in Fig. 5.1. All three ducts have the same length, same bend radius, and the same width between the inner and outer wall. Of the four side walls, the two in the rotational direction are denoted as leading and trailing surfaces, while the other two are denoted as inner and outer surfaces. The total length of the channel, L, equals to 37.47cm (14.75 in), the distance from the inlet of the channel to the axis of rotation is given by Rr = 40.64cm (16 in). The channel consists of an unheated smooth starting section (L1 = 22.23cm (8.75 in)) and a heated section (L2 = 15.24cm (6 in)). The radius of curvature of the 180° sharp turn is ri = 0.635 cm (0.25 in) and the minimum gap G between the inner and outer surfaces of the bend is 1.27 cm (0.5 in). The channel width W is kept the same at 1.27 cm (0.5 in) for all three different aspect ratio channels.
The channel height is increased from 1.27 cm (0.5 in) for square duct to 5.08 cm (2.0 in)
72
Fig. 5.1 Geometry and Conceptual View of Rotating Channel Orientation for
Rectangular Duct (AR=1:4) and Numerical Grid
73
for AR =W/H = 1:4 channel. It should be noted that the hydraulic diameters Dh are different for different aspect ratio channels with Dh = 1.27 cm, 1.69 cm, and 2.03 cm for
AR = 1:1, 2:1, and 4:1 channels, respectively. For the rotating cases, the channel orientations is given as β = 90°. A summary of all fifteen (15) cases studied is given in
Table 5.1. Because the hydraulic radius are different for different aspect ratio ducts, it is necessary to adjust the channel inlet velocity and rotating speed in order to maintain the same Reynolds number and the same rotation number in the present numerical simulations. Computational grid details
Fig. 5.1(b) shows the computational grid for the smooth rectangular duct with AR
= 1:4. For the sake of brevity, the numerical grids for AR = 1:1 and AR = 1:2 channel are not included here. A systematic grid refinement study was performed for all three different aspect ratio ducts. The grid refinement in the axial direction at the bend region has produced only minor changes of Nusselt number ratio. The refinement of cross sectional grids resulted in a maximum improvement of 3.5% in Nusselt number ratios when the grid number increase about 50 %. The minimum grid spacing for the Re =
100,000 cases is maintained at 2×10−5 of the hydraulic diameter which corresponds to wall coordinate y+ on the order of 0.1 also. The refinement in the axial direction at the bend region produced only minor changes of Nusselt number ratio, while the refinement of cross sectional grids caused a maximum of 4.1% improvement at the bend region in
Nusselt number ratios when the grid number increase about 50 %.

TRANSFER IN TWO-PASS ROTATING RECTANGULAR CHANNELS
(AR=1:1, AR=1:2, AR=1:4) WITH 45-DEG ANGLED RIBS
In this chapter, we present calculations for rotating/non-rotating two pass rectangular channels (AR=1:1, 1:2, and 1:4) with ribbed wall using the near wall second-order Reynolds stress closure model.
6.1 Description of Problem
A schematic diagram of the geometry for AR=1:4 duct is shown in Fig. 6.1. As mentioned earlier, this investigation covers three channels with different aspect ratios
(AR = 1:1, AR = 1:2 and AR = 1:4). For the sake of brevity, however, only the geometry for the AR = 1:4 channel is given in Fig. 6.1. All three ducts have the same length, same bend radius, and the same width between the inner and outer wall. Of the four side walls, the two in the rotational direction are denoted as leading and trailing surfaces, while the other two are denoted as inner and outer surfaces. The total length of the channel, L, equals to 37.47cm (14.75 in), the distance from the inlet of the channel to the axis of rotation is given by Rr = 40.64cm (16 in). The channel consists of an unheated smooth starting section (L1 = 22.23cm (8.75 in)) and a heated section (15.24cm
(6 in)). The radius of curvature of the 180° sharp turn is ri = 0.635 cm (0.25 in) and the minimum gap G between the inner and outer surfaces of the bend is 1.27 cm (0.5 in).
The channel width W is kept the same at 1.27 cm (0.5 in) for all three different aspect ratio channels. The channel height is increased from 1.27 cm (0.5 in) for square duct to
101
Fig. 6.1 Geometry and Conceptual View of Rotating Channel Orientation for Rectangular Duct (AR=1:4) with Angle Ribs and Numerical Grid
102
5.08 cm (2.0 in) for AR =W/H = 1:4 channel. It should be noted that the hydraulic diameters Dh are different for different aspect ratio channels with Dh = 1.27 cm, 1.69 cm, and 2.03 cm for AR = 1:1, 2:1, and 4:1 channels, respectively. The leading and trailing surfaces are roughened by nine equally-spaced 45° ribs with square cross section. The ribs are parallel (inline on leading and trailing surfaces) and point upstream. The rib height is fixed at 0.158 cm (0.0625 in) and the rib-pitch-to-height ratio (P/e) is 10, but the rib height-to-hydraulic diameter ratios (e/Dh) are 0.125, 0.094, and 0.078, for
AR=1:1, AR=1:2, and AR=1:4 channel, respectively. For the rotating cases, the channel orientations is given as β = 90°. A summary of all fifteen (15) cases studied is given in
Table 6.1. It should be remarked here that the square duct has the highest inlet velocity and highest rotating speed under the same Reynolds number and same rotation number conditions since it has the smallest hydraulic diameter among the three ducts considered.
Computational grid details
Fig. 6.1(b) shows the computational grid for the rectangular duct with AR = 1:4.
For the sake of brevity, the numerical grids for AR = 1:1 and AR = 1:2 channel are not included here. Systematic grid-refinement studies were performed for all three channels for Re = 10,000 cases. The grid refinement in the axial direction at the bend region has produced only minor changes of Nusselt number ratio. The refinement of cross sectional grids resulted in a maximum improvement of 3.1% in Nusselt number ratios. The minimum grid spacing for the Re = 100,000 cases is maintained at 2×10−5 of the hydraulic diameter which corresponds to wall coordinate y+ on the order of 0.1 also. The grid-refinement study for all three channels at this Reynolds number was also performed.

CHAPTER VII
SUMMARY AND CONCLUSIONS
A multi-block RANS method was employed to predict three-dimensional flow and heat transfer in rotating rectangular channel with several turbulators. It predicted fairly well the complex three-dimensional flow and heat transfer characteristics resulting from the different channel aspect ratios, rotation, centrifugal buoyancy forces, and channel orientation. The main findings from this study are summarized as follows.
For channel with V-shaped ribs:
1. The Nusselt number ratios predicted by the present near-wall second-moment closure model are in very good agreement with the experimental data for both the non-rotating and rotating cases.
2. The V-shaped ribs induce four counter-rotating vortices that oscillate in size along the streamwise direction. For the non-rotating case, the secondary flow results in steep temperature gradients and high heat transfer coefficients on the surfaces with
V-shaped ribs as well as side walls.
3. For rotating cases, the rotation-induced cross-stream secondary flow distorts the Vshaped rib-induced vortices and affects the heat transfer on both the leading and trailing surfaces.
4. The V-shaped ribs create a symmetric heat transfer enhancement from the channel centerline towards the sidewalls. The rotation increases heat transfer enhancement
128
on the trailing surface and decreases heat transfer enhancement on the leading surface. 5. High Reynolds numbers tend to weaken the heat transfer enhancement effect of the
V-shaped rib-induced secondary flow.
6. For high Reynolds number cases, an increase in rotation number and density ratio leads the Nusselt number ratios on the leading surface to further decrease, and the
Nusselt number ratios on the side walls to further increase.
For channel with pin-fins:
1. The Nusselt number ratios predicted by the second moment closure model for both the non-rotating and rotating cases are in good agreement with the experimental data.
2. The pin-fins are very effective in heat transfer enhancement due to turbulent mixing caused by flow separation around the pin-fins and the formation of horseshoe vortices in the junction between the pin-fins and channel walls.
3. The Nusselt number ratio reaches a maximum value around the third row and decreases slightly towards the channel exit.
4. Both the rotation and density ratio have only minor effects on the heat transfer enhancement. 5. High Reynolds numbers tend to reduce the heat transfer enhancement effect of the pin-fins. For smooth channels with different aspect ratios:
1. For non-rotating ducts, a pair of symmetric counter-rotating vortices was generated in the bend region. The highest heat transfer enhancement was observed in AR = 1:2
129
duct when the counter-rotating vortices grow to full strength and occupy the entire channel. As the aspect ratio reduces to AR = 1:4, the vortices and the heat transfer enhancements are confined to the leading and trailing surfaces.
2. For all three rotating ducts at β = 90° channel orientation, the Coriolis force produce a pair of counter-rotating vortices perpendicular to the bend-induced vortices. The combined effects of rotation and turn lead to the growth of the vortex near the leading surface. The leading surface vortex continue to expand in the vertical direction as the channel aspect ratio was reduced from AR = 1:1 to AR = 1:4.
3. For all three different aspect ratio ducts, the Nusselt number ratios decrease with increasing Reynolds numbers. The friction factor ratios also decrease when the
Reynolds number was increased from 10,000 to 100,000.
4. For non-rotating ducts, the channel aspect ratio has a small effect on spanwiseaveraged
Nusselt number ratio and friction factor ratio except for the turn region.
5. The channel rotation leads to higher heat transfer on first pass trailing surface and lower heat transfer on first pass trailing surface. This trend is reversed in the second pass. The effects of rotation on Nusselt number ratio become much more pronounced for low aspect ratio (AR = 1:2 and 1:4) rectangular channels.
For rib-roughed channels with different aspect ratios:
1. The calculations for three different aspect ratio channels were performed using the same channel width, the same rib height and rib pitch, but different channel heights.
The square duct has the highest inlet velocity and highest rotating speed under the
130
same Reynolds number and same rotation number conditions since it has the smallest hydraulic diameter and largest e/Dh.
2. In the non-rotating channels, the size and strength of the secondary flow vary with the channel aspect ratio. In the first passage, the square duct (AR = 1:1) produced the highest heat transfer enhancement due to the relatively higher rib roughness, higher inlet velocity, and the strong mixing of the rib-induced secondary flows in the core region. In the second passage, however, the highest heat transfer enhancement were observed in the AR = 1:2 duct since the bend-induced secondary flow reaches maximum strength when the vortices are nearly circular and occupy the entire channel cross section.
3. For all three rotating ducts at Re = 10,000 and Ro = 0.14, the square duct again produced the highest heat transfer enhancement on both the leading and trailing surfaces of the first passage. High Nusselt number ratios were also obtained in the second passage far downstream away from the bend. In the 180° bend region, however, the AR = 1:2 channel still produces the highest Nusselt number ratios due to the presence of strong turn-induced vortices.
4. For the low-Ro and low-Re cases, the rotation effect on the Nusselt number is more significant in the AR = 1:2 channel than those observed in either the square or AR =
1:4 ducts due to the presence of strong turn-induced vortices. For the high-Ro and high-Re cases, however, the rotation effect decreases continuously when the channel aspect ratio was changed from AR = 1:1 to 1:2 and 1:4 since the bend-induced vortices are less important under the high rotation number conditions.
131
5. For all three different aspect ratio ducts, the heat transfer enhancement decreases when the Reynolds number was increased from 10,000 to 100,000.
6. At higher rotation number and higher Reynolds number, the density ratio increase has a dramatic effect on reduced heat transfer in the second passage of the lower aspect ratio duct. In general, the Nusselt number ratio decreases with increasing density ratios.