## 1 Introduction

1.1 Motivation-. To improve thermal efficiency, gas-turbine stages are being designed to operate at increasingly higher inlet temperatures. A widely used method for cooling turbine blades is to bleed lower-temperature gas from the compressor and circulate it within and around each blade. The coolant typically flows through a series of straight ducts connected by 180° bends and roughened with ribs or pin fins to enhance heat transfer. These cooling ducts may not only be square in cross section or normal to the rotational direction of the blade. In fact, the aerodynamic shape of the turbine blade dictates the us e of cooling channels that are rectangular in cross section (with different aspect ratios) and are at an angle, b, from the direction of rotation. Rotation of the turbine blade cooling passages adds another complexity to the problem. It gives rise to Coriolis and buoyancy forces that can significantly alter the local heat transfer in the internal coolant passages from the non-rotating channels. The presence of rib turbulators adds a further complexity since these ribs produce complex flow fields such as flow separation, reattachment and secondary flow between the ribs, which produce a high turbulence level that leads to high heat transfer coefficients.

1.2 Literature Review: Experimental Studies. The complex coupling of the Coriolis and buoyancy forces with flow separation/reattachment by ribs has prompted many investigators to study the flow and temperature fields generated in heated, rotating ribbed wall passages. Most experimental studies on internal cooling passages have focused on non-rotating ducts. See, for example, Han and Park [1], Han et al. [2] Ekkard and Han [3] and Liou et al. [4] and the references cited there. Experimental studies on rotating ducts have been less numerous. Wagner et al. [5], Dutta and Han [6], Soong et al. [7] and Azad et al. [8] investigated rotating ducts with smooth walls. Wagner et al. [9], Johnson et al. [10 and 11], Parsons et al. 12] and Zhang et al.[13] reported studies on rotating square channels with normal and angled ribs. Azad et al. [8] also investigated the effect of channel orientation on rotating ribbed two pass rectangular channel. Griffith et al. [14] studied the effect of channel orientation on rotating smooth and ribbed rectangular channels with channel aspect ratio of 4:1. They investigated a broad range of flow parameters including Reynolds number (Re = 5000-40000), rotation number (Ro = 0.040.3) and coolant to wall density ratio (Ap/p = 0.122). Their experimental results provided a database for the present work.

1.3 Literature Review: Numerical Studies

1.3.1 Smooth Surfaces: In addition to the experimental studies mentioned above, several studies have been made to predict numerically the flow and heat transfer in radially rotating smooth and ribbed ducts. Iacovides and Launder [15], Prakash and Zerkle [16], Dutta et al. [17] and Bo et al. [18] studied one passage smooth ducts with normal channel orientation from the direction of rotation i.e., (5 = 90°. Sathyamurthy et al. [19], Stephens et al. [20], Iacovides et al. [21] and Bonhoff et al. [22] reported numerical predictions for rotating smooth two passage ducts and (5 = 90°. The differential Reynolds stress model (RSM) with wall function in FLUENT code was used in the calculation of Bonhoff et al. [22]. Chen et al. [23, 24] predicted the flow and heat transfer in a rotating smooth two-pass square channel which is the first two passages of the four-pass serpentine passage that was experimentally investigated by Wagner et al. [5]. They used two turbulence models: a two-layer k-e isotropic eddy viscosity model and a near-wall second-moment closure model. The near-wall second-moment closure model accurately predicted the complex three-dimensional flow and heat transfer characteristics resulting from the rotation and strong wall curvature. They provided the most reliable predictions in comparison with the data of Wagner et al. [5]. Al-Qahtani et al. [25] predicted the flow and heat transfer in a rotating smooth two-pass rectangular channel with a 180° sharp turn and an aspect ratio of 2:1 which was also experimentally investigated by Azad et al. [8]. Two channel orientations were studied: b = 90° and 135°. They also investigated the effect of the rotation number, Ro, and inlet coolant-to-wall density ratio Ap/p

1.3.2 Ribbed Surfaces : Stephens et al. [26, 27] studied inclined ribs in a straight non-rotating square duct. Stephens and Shih [28] investigated the effect of angled ribs on the heat transfer coefficients in a rotating two-passage duct using a low-Re number k-w turbulence model. They studied the effects of Reynolds numbers, rotation numbers, and buoyancy parameters. Prakash and Zerkle [29], employing a high Reynold s number k-e turbulence model with wall function, performed a numerical prediction of flow and heat transfer in a ribbed rectangular duct (90° rib) with and without rotation. However, their calculations used periodicity and neglected buoyancy effects. 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. Bonhoff et al. [22] calculated the heat transfer coefficients and flow fields for rotating U-shaped coolant channels with angled ribs (45° ). They used a Reynolds stress turbulence model with wall functions in the FLUENT CFD code. Using the periodicity of the flow, Iacovides [30] computed flow and temperature fields in a rotating straight duct with 90° ribs. Two zonal models of turbulence were tested: a k-e with a one-equation model of k transport across the near-wall region and a low-Re differential stress model. He concluded that the differential stress model thermal computations were clearly superior to those of the k-e/one-equation model.

Using the same model and method of Chen et al. [23, 24], Jang et al. [31, 32] studied flow and heat transfer behavior in a non-rotating two-pass square channels with 60° and 90° ribs, respectively. Their results were in good agreement with Ekkad and Han's [3] detailed heat transfer data which validated their code and demonstrated the second-moment closure model superiority in predicting flow and heat transfer characteristics in the ribbed duct. In a later study, Jang et al. [33] predicted flow and heat transfer in a rotating square channel with 45° angled ribs by the same second-moment closure model. Heat transfer coefficient prediction was well matched with Johnson et al. [11] data for both stationary and rotating cases. Al-Qahtani et al. [34] predicted flow and heat transfer in a rotating two-pass rectangular channel with 45° angled ribs by the same second-moment closure model of Chen et al. [23, 24]. Heat transfer coefficient prediction was compared with the data of Azad et al. [8] for both stationary and rotating cases. It predicted fairly well the complex three-dimensional flow and heat transfer characteristics resulting from the angled ribs, sharp 180° turn, rotation, centrifugal buoyancy forces and channel orientation.

This affirmed the superiority of the second-moment closure model compared to simpler isotropic eddy viscosity turbulence models. This model solves each individual Reynolds stress component directly from the ir respective transport equations. The primary advantage of this model is that it resolves the near-wall flow all the way to the solid wall rather than using log-law assumption in the viscous sublayer. With this near-wall closure, surface data like heat transfer coefficients and friction coefficients can be evaluated directly from velocity and temperature gradients on the solid wall.

In practice, the aerodynamic shape of the turbine blade dictates the use of cooling channels that are rectangular in cross section and are at an angle (5 from the direction of rotation. The effect of rotation, channel orientation and large channel aspect ratio on the secondary flow and heat transfer in rectangular channels may vary from the square channels. None of the previous studies predicted the characteristics of fluid flow and heat transfer in rotating rectangular channels that have an aspect ratio, AR, of 4:1 whether perpendicular or at an angle from the direction of rotation.

The objective of this study is to use the second moment RANS method of Chen et al. [23, 24] to (1) predict the three-dimensional flow and heat transfer for rotating smooth and ribbed one-pass rectangular ducts (AR = 4:1) and compare with the experimental data of Griffith et al. [14] and (2) to investigate the effect of high rotation and high density ratios on the secondary flow field and the heat transfer characteristics in a ribbed duct at 135° orientation.

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