NOMENCLATURE
$Bo$
Buoyancy Number,
$\left(\frac{\Delta\rho}{\rho}\right)\left(\frac{R_x}{d_h}\right)Ro^{2}$
-
$d_h$
Channel Hydraulic Diameter
-
$F$
Secondary Coriolis Force
-
$h$
Heat Transfer Coefficient
-
$k$
Thermal Conductivity of Air
-
$L$
Total Length of the Channel
-
$Nu$
Nusselt Number
-
${Nu}_0$
Dittus-Boelter Correlation,
$0.023Re^{0.8}\ Pr^{0.4}$
-
$Re$
Reynolds Number,
$\frac{Vd_h}{v}$
-
$Ro$
Rotation Number,
$\frac{\Omega d_h}{V}$
-
$R_m$
Mean Rotating Radius
-
$T_b$
Bulk Fluid Temperature
-
$T_w$
Wall Temperature
-
$V$
Average Velocity of the Coolant
-
$V_y$
Velocity Component Normal to Main Flow
-
$X$
Distance from the Inlet of the Channel
Greek Symbols
-
$\rho$
Density of air
-
$v$
Kinematic viscosity
-
$\Omega$
Angular Velocity
Subscripts
-
$b$
Bulk Fluid
-
$w$
near wall fluid
- Dittus-Boelter Correlation
1.0 INTRODUCTION
The temperature of the hot gases exiting the combustor section can be as high as 1600
$^{\circ}\text{C} $
, which is significantly higher than the blade material withstanding temperature(1). Relatively colder air bled off from compressor discharge is fed to the internal serpentine cooling passages of gas turbine blades to cool its internal walls. However, increased usage of the coolant reduces the efficiency of gas turbines. Hence, it is imperative that coolant should be used judiciously. One other concern in turbine blade cooling is that the rotation significantly modifies the heat transfer on leading and trailing side internal walls due to influence of Coriolis force and centrifugal buoyancy forces. The heat transfer enhancement concepts developed under stationary conditions perform very differently when they are subjected to rotation. The non-uniform distribution of internal wall heat transfer coefficient leads to non-uniform metal temperature and hence increased thermal stresses.
There have been many experimental and computational studies on fluid flow and heat transfer enhancement mechanism in conventional ribbed multi-pass channels under stationary and rotating conditions. It is well documented that under rotation, for a radially outward flow, heat transfer enhances on the trailing side and reduces on the leading side. The behaviour is vice-versa for a radially inward flow. This phenomenon is the effect of Coriolis force due to rotation and centrifugal buoyancy which comes into play due to density difference between colder core fluid and hotter near-wall fluid(Reference Johnson, Wagner, Steuber and Yeh2–Reference Al-Hadhrami and Han11).
Relatively fewer studies have addressed the problem of non-uniformity in heat transfer behaviour of leading and trailing walls under rotation. Dutta and Han(Reference Dutta and Han12) investigated heat transfer behaviour under rotation for three different model orientations. Model B of their study showed that heat transfer augmentation on both leading and trailing sides of the two-pass ribbed duct. Recently, Singh et al.(Reference Singh and Ekkad13,Reference Singh, Li, Ekkad and Ren14) studied heat transfer enhancement mechanism for cooling feature based on Model B of(Reference Dutta and Han12) by means of transient liquid crystal experiments. The authors exhibited the advantages of the model orientation where rotation actually favoured the heat transfer enhancement on both the walls.
Above studies were carried out on orthogonal mode of rotation where the direction of fluid flow and rotation axis are perpendicular to each other. However, in parallel mode of rotation, where the direction of fluid flow and axis of rotation are parallel to each other, theoretically, the Coriolis force effect should be negated. In this study, this concept is utilised and the parallel rotation configuration investigated at different Rotation numbers is presented.
There have been few studies on the parallel mode rotation as well, e.g. Humphreys et al.(Reference Humphreys, Morris and Barrow15) studied convection heat transfer in the turbulent entry region of a circular tube rotating about an axis parallel to itself. The authors concluded that inlet swirl dominated the heat transfer enhancement in the entry region. Morris and Woods(Reference Morris and Woods17) studied the heat transfer behaviour in the entrance region of a circular tube rotating about a parallel axis in laminar and turbulent regimes. They witnessed Coriolis force effect in the developing region, which increased radial mixing and improved heat transfer. The authors also proposed correlations for Nusselt number in terms of Reynolds number and Rotational Reynolds number. Morris and Dias(Reference Morris and Dias18) studied the turbulent convective heat transfer in a square cross-sectioned tube in parallel axis rotation mode. They concluded that overall heat transfer increases with increasing rotational Reynolds number. They also demonstrated that heat transfer enhancement is less sensitive to the mean rotating radius/eccentricity parameter. Levy et al.(Reference Levy, Neti, Brown and Bayat19) studied heat transfer and pressure drop characteristics for a rectangular channel (Aspect ratio 2:1) in the laminar regime. The authors demonstrated that rotation increased heat transfer and pressure drop significantly for Grashof numbers greater than 1000.
Soong and Yan(Reference Soong and Yan20) performed numerical study on parallel rotation of a rectangular duct in the entrance region with uniform wall temperature and uniform heat flux thermal boundary conditions. The flow was laminar and aspect ratios were in the range of 0.2 to 5. The authors concluded that rotation had strongest effects for the unity aspect ratio. Authors also found that buoyancy induced vortices were stronger for iso-heat-flux case than for iso-thermal wall heating case. Mahadevappa et al.(Reference Mahadevappa, Rammohan Rao and Sastri21) presented a numerical study of fully developed laminar flow in rectangular and elliptical ducts rotating about a parallel axis for various aspect ratios and Prandtl numbers. They witnessed increase in heat transfer with increasing Prandtl number and Rotation number. They also stated that elliptical duct had superior thermal hydraulic performance. Sleiti and Kapat(Reference Sleiti and Kapat22) addressed the parallel rotation of square duct under constant heat flux case for high Rotation numbers. They concluded that Nusselt number increases on three surfaces and decreases on one surface with rotation. Recently, Pelle et al.(Reference Fasquelle, PellÉ, Harmand and Shevchuk23) carried out numerical study of convective heat transfer enhancement in a pipe rotating around a parallel axis for circular and elliptical geometries. They concluded that increased angle of attack opposite to pipe rotation increased heat transfer enhancement in the entrance region. They also reassured that elliptical configuration had superior thermal hydraulic performance than circular.
The present study addresses the parallel rotation of square duct in the fully developed turbulent regime under both constant temperature and constant heat flux cases at engine similar flow and rotation conditions. A detailed discussion on the effect of centrifugal buoyancy and secondary Coriolis force effect on heat transfer enhancement on each wall is presented. All the previous studies of parallel mode rotation treated its applicability only in electrical rotor machines. The novelty of this study resides in discussing the advantages of parallel mode rotation as applicable to gas turbine internal cooling channels, thereby eliminating non-uniformity in heat transfer enhancement levels.
Representation of orthogonal and parallel rotation configurations (not drawn to scale).
Hexahedral mesh (a) On the walls (b) At a cross section of the channel (flow into the plane).
2.0 TEST CONFIGURATION
Numerical study is carried out on a straight channel of length equal to 50 times the channel hydraulic diameter of square cross section. The mean rotating radius Rm was 49 times the hydraulic diameter for both orthogonal and parallel mode rotation as shown in Fig. 1. A hexahedral mesh as shown in Fig. 2 shows the mesh which was generated in ICEM software. The first cell distance from the wall was set such that the wall y+ was around unity, which is a requirement for the near-wall treatment used in this study. The cell count was approximately 14 million.
A grid independence study was carried out for parallel configuration with 10, 14 and 16 million cells at Reynolds number of 25000 and Rotation number of 0.1. Figure 3 shows the globally averaged Nusselt number in the fully developed region of last 25 hydraulic diameters on leading and trailing walls. The variation between two grid sizes was within 2%. Therefore, the solution was deemed independent of grid size, and further studies were carried with 14 M cells to balance the accuracy and computational cost.
Global averaged normalised Nusselt number variation with grid size.
3.0 NUMERICAL SETUP
3.1 Solver and turbulence model
Numerical simulations were carried out in commercial software package ANSYS FLUENT, which is a finite volume-based solver. Steady-state Reynolds-averaged Navier-Stokes equations were solved using second order discretization scheme. The working fluid was modeled as incompressible as the Mach number in the duct was less than 0.05 and hence a pressure-based solver was used. However, due to differences in the local fluid temperatures, the density was allowed to vary with local temperature. SIMPLE scheme is used for pressure-velocity coupling. The Reynolds number at inlet was 25000 and the Rotation numbers 0, 0.1, 0.15, 0.2, 0.25. To model the turbulent flow field, realizable version of k-
$\varepsilon$
turbulence model is used with enhanced wall treatment.
3.2 Boundary conditions
Mass flow rate at the inlet was specified based on the Reynolds number and channel hydraulic diameter. Also, uniform flux was of 5000 W/m2 was specified on all walls for constant flux case and uniform temperature of 425 K was imposed on all walls for constant temperature case. The fluid temperature at the inlet was specified 300 K. Turbulence intensity of 5% and turbulent length scale equal to hydraulic diameter was specified at the inlet. At the outlet, zero-gauge pressure was specified. The solution was assumed converged when the continuity, momentum and turbulence residuals reached 1e-6 and energy residual reached 1e-8. Temperature was monitored at multiple locations to ensure convergence. The simulations were carried out at different conditions listed in Table 1.
Test matrix
3.3 Data reduction
The local heat transfer coefficient at the walls subjected to constant heat flux is calculated from following equation.
\begin{equation} h = \frac{q^{\prime\prime}}{T_{w} - T_{m}}\end{equation}
The bulk fluid temperature was calculated by considering mass weighted average of the temperature at ten normal planes to the flow and was then linearly interpolated along the flow direction. The Nusselt number is then calculated as shown in Eq. 2.
\begin{equation} Nu = \frac{hd_{h}}{k}\end{equation}
The calculated Nusselt number was normalised with the Dittus-Boelter correlation for smooth circular tube given by Eq. 3.
$$N{u_0} = 0.023R{e^{0.8}}P{r^{0.4}}$$
Normalized Nusselt number 
$$(Nu/N{u_0})$$
contours in the last 
$${\rm{25}}{d_h}$$
region for orthogonal mode rotation at Re = 25000, Ro = 0.25 for constant flux wall condition 
$${(x/{D_h} = {\rm{25}}\;to\;{\rm{50}}}$$).
Normalized Nusselt number contours 
$$(Nu/N{u_{\bf{0}}})$$
in the last 25 dh region for parallel mode rotation at Re = 25000, Ro = 0.25 for constant flux wall condition (from x/d = 25 Dh to 50 Dh).
4.0 RESULTS AND DISCUSSION
4.1 Comparison of heat transfer enhancement between orthogonal and parallel rotation modes
Detailed normalised Nusselt numbers 
${(Nu/Nu_{0})}$
is shown in Figs. 4 and 5. The inlet effects were diminished by the time flow at 
$x/D_h\sim15$
. It can be seen that for the stationary case, the Nusselt number ratio was close to unity and invariant with the streamwise location as seen in Fig. 4, which indicates fully developed flow, both hydrodynamically and thermally. Such a configuration is chosen, so that the clear benefits of parallel rotation can be demonstrated. Rotational effects on orthogonal rotation mode can be seen in trailing and leading side heat transfer results. The Coriolis force pushes the coolant in the core towards the trailing side. Further, the coolant density being higher along the trailing side when coupled with centrifugal forces leads to fluid acceleration along that wall and hence in increased heat transfer. The presence of relatively warmer and low velocity coolant near the leading wall leads to reduced heat transfer. On the other hand, for parallel rotation (Fig. 5), the Nusselt number ratios for leading and trailing walls were near similar for the last 25 Dh of the straight duct. This trend, in turn, proves that the rotational effects (particularly Coriolis force) on heat transfer on leading and trailing walls can be negated when the coolant flow direction is same as the rotation vector. In the parallel rotation, although the direct effects of Coriolis force were negated, the centrifugal buoyancy forces still act on the fluid elements in blade root to tip direction. Centrifugal buoyancy forces push the coolant towards the radially outward wall of the channel, and hence the coolant mass flux along that wall will be relatively higher compared to the radially inward wall in order to satisfy the continuity equation.
To identify quantitative differences in heat transfer level, the regionally averaged heat transfer data has been presented in Fig. 6 for the highest Rotation number case of 0.25. For orthogonal mode rotation, the heat transfer enhances on trailing wall and reduces on leading side which is in agreement with previous studies. For parallel mode rotation, the Nusselt number on both leading and trailing walls are above stationary due to enhancement caused by centrifugal buoyancy, where heat transfer levels for leading and trailing walls were nearly similar, with trailing side heat transfer marginally higher than the leading side.
Span wise averaged normalized Nusselt number 
$$(Nu/N{u_{\bf{0}}})$$
distribution at Re = 25000, Ro = 0.25, for constant flux wall heating condition for (a) orthogonal rotation (b) parallel rotation.
This minimal difference may be attributed to the Coriolis force effect due to fluid velocity component normal to the bulk fluid flow. Due to centrifugal buoyancy, normal velocity to the bulk flow is relatively stronger. This component perpendicular to the axis of rotation induces minor Coriolis effect. This effect pushes the colder core fluid towards the trailing wall, resulting in marginally higher heat transfer at that wall. This phenomenon is observed for other Rotation numbers as well. The magnitude of secondary Coriolis effect at different planar locations normal to bulk fluid flow is shown in Fig. 7. It is calculated as
\begin{equation}F=2\rho V_y \omega \end{equation}
It indicates that the secondary Coriolis effect is stronger towards the trailing side than the leading side, which supports the trend of higher heat transfer due to secondary Coriolis effect put forth earlier.
4.2 Variation of normalised Nusselt number
$$(Nu/N{u_{\bf{0}}})$$
with Rotation number
The variation of normalised Nusselt number with Rotation number for orthogonal and parallel rotation modes is shown in Fig. 9. For orthogonal rotation case, heat transfer increases on trailing wall with increasing Rotation number and similar trend was observed on leading side, where at the same time, leading side heat transfer was consistently lower than the stationary case. On the leading wall, heat transfer first decreases and then increases farther from the inlet with Rotation number. The heat transfer on leading wall starts increasing along the flow when the centrifugal buoyancy effects are significant, and this trend was observed for all investigated Rotation numbers. The point of increase moves closer to inlet with increase in Rotation number. This behaviour is in good agreement with the earlier studies on orthogonal rotation(Reference Johnson, Wagner, Steuber and Yeh2–Reference Chang and Morris9).
Secondary Coriolis force on the planes normal to bulk flow at various axial locations.
Comparison of leading and trailing wall normalised Nusselt numbers for all Rotation numbers with constant flux case for (a) Orthogonal mode rotation (b) Parallel mode rotation.
Variation of Span-wise averaged normalised Nusselt number along axial location parallel mode rotation on (a) Trailing wall (b) Leading wall (c) Top wall (d) Bottom wall with constant flux heating condition.
For parallel rotation, the heat transfer almost remains constant until the flow is developed and increases at a later stage with Rotation number. On the trailing wall, it increases with Rotation number. The increase of heat transfer on leading and trailing wall with increase in Rotation number can be attributed to the transport assisted by centrifugal buoyancy 
$Bo-(\frac{\Delta \rho}{\rho})(\frac{R_{x}}{d_h})Ro^2$
, the strength of which increases with increasing Rotation number.
From Fig. 8, it is clear that the parallel mode rotation has two benefits over the conventional orthogonal mode rotation: (1) the heat transfer on leading and trailing side heat transfer is near similar, leading to uniformity in heat transfer, and (2) rotational energy (combined with density ratio) being used in favour of heat transfer enhancement on both leading and trailing walls.
The normalised Nusselt number for parallel rotation case at the four different walls is shown separately in Fig. 9. Heat transfer increases with Rotation number for leading, trailing and top walls where buoyancy contributes to enhancement, whereas the bottom wall heat transfer was mostly invariant with rotation. For trailing wall, the Nusselt number decreases initially as the flow develops and increases later once the secondary flow is established. Also, with increase in Rotation number, the secondary flow develops closer to inlet due to increase in buoyancy effects. This impact of secondary flow development on heat transfer enhancement is also visible on leading wall and top wall.
For parallel rotation case, the variation of globally averaged normalised Nusselt number (in the last 25 dh) with Rotation number is shown in Fig. 10 for leading and trailing walls, for two different thermal boundary conditions. The heat transfer enhancement increases with Rotation number under both constant heat flux and constant temperature wall heating conditions, indicating the increased effect of centrifugal buoyancy with Rotation number. However, the rate of increase is different for constant heat flux and constant temperature wall heating condition, which is explained in detail in the subsequent sections.
Variation of global averaged normalised Nusselt number in the last 25 dh with Rotation number for parallel mode rotation (a) constant flux wall heating condition (b) Constant temperature wall heating condition.
4.3 Secondary flow due to centrifugal buoyancy in parallel mode rotation
In parallel mode rotation, the heat transfer is highest on the top wall, which is radially outward and lowest on the bottom wall, which is radially inward as witnessed in Fig. 6(b). This is the effect of centrifugal buoyancy, which always pushes the fluid away from the axis of rotation. This induces secondary flow, creating a pair of counter rotating vortices as shown in Fig. 11. These vortices contribute to heat transfer augmentation on three walls of the channel compared to stationary. Centrifugal buoyancy decreases the heat transfer on bottom wall since the fluid is always pushed away from it. As the Rotation number increases, the heat transfer increases due to generation of stronger counter rotating pair of vortices that pushes the fluid towards the walls with a greater velocity.
Secondary flow due to centrifugal buoyancy and normalised bulk fluid temperatures on the planes normal to bulk flow at various axial locations (a) constant heat flux (b) constant wall temperature.
Normalised turbulent kinetic energy contours in the last 25 dh region for orthogonal mode rotation at Re = 25000, Ro = 0.25 for constant flux wall condition (from x/d = 25 Dh to 50 Dh).
Normalised turbulent kinetic energy contours in the last 25 dh region for parallel mode rotation at Re = 25000, Ro = 0.25 for constant flux wall condition (from x/d = 25 Dh to 50 Dh).
Comparison of normalised Nusselt number between constant flux and constant temperature conditions at Rotation numbers (a) Ro 0.1 (b) Ro 0.15 (c) Ro 0.2 (d) Ro 0.25.
4.4 Turbulent kinetic energy on leading and trailing walls
Detailed normalised turbulent kinetic energy along the flow on planes very close to leading and trailing walls is shown in Fig. 12 for orthogonal rotation. It can be seen that trailing wall had higher turbulent kinetic energy than the leading wall due to the Coriolis force which destabilises the flow near trailing wall and stabilises the flow near leading wall. However, this difference in turbulent kinetic energy is not observed in parallel rotation as shown in Fig. 13. The absence of Coriolis effect ensures near similar turbulent energy on both the walls, aiding to similar heat transfer enhancement levels, which may have been different if the turbulent kinetic energy would be different on the two walls. Turbulent kinetic energy enhancement is also one of the mechanisms which leads to increased transport of energy from near wall to the channel core.
4.5 Effect of thermal boundary conditions on Nusselt number ratio
$$(Nu/N{u_{\bf{0}}})$$
The comparison of normalised Nusselt numbers for leading and trailing walls between two wall heating conditions is shown in Fig. 14 for all Rotation numbers. It can be seen that constant heat flux shows slightly higher enhancement than constant wall temperature case. This indicates that effect of centrifugal buoyancy is greater in constant heat flux case than constant temperature case, which was also witnessed for laminar region by Soong and Yan(Reference Soong and Yan20).
The variation of buoyancy number along the axial location is shown in Fig. 15. The buoyancy number is greater in constant flux case than constant temperature case. This can be attributed to the difference in nature of variation of temperature under these two wall heating conditions. The wall temperature variation affects the magnitude of centrifugal buoyancy forces since the fluid density is dependent on temperature for incompressible flow.
Comparison of buoyancy number between constant flux and constant temperature conditions, Ro = 0.25.
5.0 CONCLUSIONS
Detailed heat transfer studies are reported for a smooth, straight square channel of length 50 dh under orthogonal and parallel mode rotation, assuming steady state. Numerical simulations were carried out at Reynolds number of 25000 and Rotation numbers in the range of 0 to 0.25 for both the configurations. Two wall heating conditions of constant heat flux and constant temperature are considered to verify that the heat transfer behaviour is similar for both the conditions. Some of the conclusions from the present study are:
(1) For orthogonal rotation, heat transfer increases on trailing wall and decreases on leading wall when compared to stationary. The difference in heat transfer enhancement levels between leading and trailing walls is due to the presence of Coriolis effect. The heat transfer on trailing wall increases with Rotation number. On the leading wall, heat transfer first decreases and increases when buoyancy effects are significant, with Rotation number.
(2) For parallel rotation, heat transfer increases on both leading and trailing walls when compared to stationary. The heat transfer enhancement levels are similar on both leading and trailing walls. This indicates the absence of Coriolis effect in parallel mode rotation. The heat transfer on both the walls increases with increase in Rotation number.
(3) In parallel rotation, centrifugal buoyancy contributes to heat transfer augmentation on leading, trailing and top (radially outward) walls. The heat transfer on the bottom (radially inward) wall decreases due to buoyancy.
(4) Higher heat transfer is seen for constant flux case than constant temperature case due to variation in centrifugal buoyancy effects. This variation is due to the inherent difference in variation of wall temperature for different wall heating conditions.
