Uncoupling Coriolis Force and Rotating Buoyancy Effects on Full-Field Heat Transfer Properties of a Rotating Channel

While thermodynamic laws dictate the improved specific power and thermal efficiency of a gas turbine engine by elevating the turbine entry temperature, several hot engine components, such as turbine blades, are prone to thermal damage. Internal cooling of a gas turbine rotor blade permits a turbine entry temperature in excess of the temperature limits of the creep resistance of the blade material. However, the configurations of the internal cooling channels must comply with the blade profile. In particular, the coolant rotates within the rotor blade. With such harsh thermal conditions for a running gas turbine rotor blade, an effective blade cooling scheme is crucial to ensure the structure's integrity. Thus, the local heat transfer properties for a rotating channel are important for the efficient usage of the limited coolant flow available. The acquisition of useful heat transfer data that are applicable to the design of the internal coolant passages at realistic engine conditions is of primary importance when an experimental method is developed for measuring the heat transfer properties of a simulated cooling passage inside a gas turbine rotor blade.

Rotation at a speed above 10,000 rpm considerably alters the cooling performance of a rotating channel inside a gas turbine rotor blade. The identification of engine conditions for such a rotating channel is permissible using the similarity law. With rotation, the dimensionless groups that control the transport phenomena inside a radially rotating channel can be revealed by deriving the flow equations relative to a rotating frame of reference. Morris¹ has derived the momentum conservation equation of flow relative to a rotating frame of reference as:

      (1)

In equation (1), the local fluid velocity, v̄, with the position vector, r̄, relative to a frame of reference rotating at the angular velocity, ω, is affected by the Coriolis acceleration in terms of 2(ω×v̄), the decoupled centripetal buoyancy force, β(T–T_ref)(ω×ω×r̄), the driven piezo-metric pressure gradient, , and the fluid dynamic viscosity, ν. The referenced fluid density, ρ_ref, is referred to a pre-defined fluid reference temperature T_ref, which is typical of the local fluid bulk temperature for experiments. If the irreversible conversion of mechanical energy into thermal energy is negligible, the energy conservation equation is reduced to:

      (2)

The first term of equation (2) is obtained by treating the specific enthalpy to be directly related to the local fluid temperature, T, via the constant specific heat, C_p. As the perturbation of fluid density caused by the variation of fluid temperature in a heated rotating channel provides considerable influence on the motion of fluids when it links with the centripetal acceleration in equation (1), the fluid velocity and temperature fields in an axially rotating channel are coupled. Also, both Coriolis and centripetal accelerations vary simultaneously as the rotating speed is adjusted. Thus, the effects of Coriolis force and rotating buoyancy on the fields of fluid velocity and temperature are naturally coupled.

Equations (1) and (2) in the dimensionless forms disclose the flow parameters that govern the heat convection in a rotating channel. With a basically uniform heat flux imposed on a rotating channel, the local fluid bulk temperature, T_b, increases linearly in the streamwise direction, s, from the reference inlet level, T_ref. The local fluid bulk temperature is determined as T_ref + τs, where τ is the gradient of the fluid bulk temperature in the direction of flow. Substitutions of the following dimensionless parameters of:

      (3)

      (4)

      (5)

      (6)

      (7)

into equations (1) and (2), where V_mean, N and d respectively stand for the mean flow through velocity, rotating velocity and channel hydraulic diameter, the dimensionless flow momentum and energy equations are derived as equations (8) and (9) respectively.

      (8)

      (9)

Evidently, η in equation (9) is a function of Re, Ro, and Bu = Ro²βτdR, which are respectively referred to as Reynolds, rotation and buoyancy numbers. The Rossby number that quantifies the ratio between inertial and Coriolis forces is equivalent to the inverse rotation number in equation (8).

When T_b is calculated as T_ref + τs in a rotating channel subject to a uniform heat flux, the τ value can be alternatively evaluated as Q_f/(mC_pL) in which Q_f, m and L are the convective heating power, coolant mass flow rate and channel length, respectively. Thus, the dimensionless local fluid bulk temperature, η_b, is equal to s/d and the dimensionless temperature at channel wall, η_w, yields [(T_w–T_b)/Q_f][mC_p][L/d]+s/d. With the convective heat transfer rate defined as Q_f/(T_w–T_b), the dimensionless wall-to-fluid temperature difference, η_w–η_b, is convertible into the local Nusselt number via equation (10) in which ζ is the dimensionless shape function of heating area and channel sectional area.

      (10)

With a set of predefined geometries and the hydrodynamic and thermal boundary conditions, the dimensionless groups controlling the local Nusselt number of a rotating channel are identified as:

      (11)

      (12)

      (13)

With experimental tests, the adjustment of rotating speed, N, for varying Ro to generate the heat transfer data at different strengths of Coriolis forces inevitably changes the centripetal acceleration, and thus, the relative strength of rotating buoyancy. Moreover, a set of heat transfer data collected from a rotating channel is always subject to a finite degree of rotating buoyancy effect. To disclose the individual effects of Coriolis-force and buoyancy on the heat transfer performance of a rotating channel requires the uncoupling of the Ro and Bu effects on Nu properties through the post data processing procedure that is inclusive in the present experimental method.

The engine and laboratory flow conditions for a rotating channel inside a gas turbine rotor blade can be specified by the ranges of Re, Ro and Bu. The typical engine conditions for the coolant flow through a gas turbine rotor blade, as well as the construction and commissioning of the rotating test facility that allowed experiments to be performed near the actual engine conditions was reported by Morris². Based on the realistic engine conditions summarized by Morris², Figure 1 constructs the realistic operating conditions in terms of Re, Ro and Bu ranges for a rotating coolant channel in a gas turbine rotor blade. In Figure 1, the indication of an engine's worst condition is referred to as the engine running condition at the highest rotor speed and the highest density ratio. In Figure 1, the lower limit and worst engine operating conditions respectively emerge at the lowest and highest engine speeds. It is extremely difficult to measure the full-field Nu distribution of a rotating channel running at a real engine speed between 5000 and 20,000 rpm. However, based on the similarity law, laboratory-scale tests have been conducted at reduced rotating speeds but with several attempts to provide a full coverage of the real-engine Re, Ro and Bu ranges. As an innovative experimental method, the NASA HOST program^3,4,5,6 adopted the high-pressure tests for increasing the fluid densities at the predefined Re in order to extend the Ro range by reducing the mean fluid velocity. In this regard, the specific relationships between Re, Ro and Bu for an ideal gas with a gas constant, R_c, and viscosity, μ, are related as:

      (14)

      (15)

To bring the laboratory conditions into the nominal correspondence with engine conditions seen in Figure 1, the rotating speed, N, coolant pressure, P, channel hydraulic diameter, d, rotating radius, R, and wall-to-fluid temperature difference, T_w–T_b, need to be controlled for matching the realistic Re, Ro and Bu ranges. Clearly, one of the most effective approaches to extend the Ro range is to increase channel hydraulic diameter, as Ro is proportional to d². As the laboratory heat transfer test at realistic N is extremely difficult, the coolant pressure, P, is technically easier to be raised for extending Ro range; even if Ro is only proportional to P. Based on this theoretical background, the design philosophy of the present experimental method is to increase Ro by pressurizing the rotating test channel using the maximum channel hydraulic diameter allowed to fit into the rotating rig. Having increased the Ro range, the range of Bu is accordingly extended as Bu is proportional to Ro². In Figure 1, the laboratory test conditions adopted to generate the heat transfer data of rotating channels are also included^{3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29}. As indicated in Figure 1, the coverage of realistic engine conditions by the available heat transfer data is still limited, especially for the required Bu range. The open and the colored solid symbols depicted in Figure 1 are the pointed and full-field heat transfer experiments, respectively. As collected in Figure 1, most of the heat transfer data with cooling applications to gas turbine rotor blades^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,20,21,22,23,24,25,26} are point measurements using the thermocouple method. The wall conduction effects on measuring the wall conductive heat flux and the temperatures at fluid-wall interfaces undermine the quality of heat transfer data converted from the thermocouple measurements. Also, the heat transfer measurements^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,20,21,22,23,24,25,26} using the thermocouple method cannot detect the two-dimensional heat transfer variations over a rotating surface. With the present experimental method^29,30,31,32, the detection of full-field Nusselt number distributions over the rotating channel wall is permissible. The minimization of wall conduction effect using 0.1 mm thick stainless-steel foils with Biot numbers >>1 to generate the heating power by the present experimental method permits the one-dimensional heat conduction from the heating foil to the coolant flow. In particular, the acquisition of full-field heat transfer data involving both Ro and Bu effects is not permissible using the transient liquid crystal technique and the thermocouple method. With the current steady-state liquid crystal thermography method¹⁹, the detectable temperature range of 35-55 °C disables the generation of heat transfer data with realistic density ratios.

Using the flow parameters governing the heat convection in a rotating channel to demonstrate that the full coverage of realistic engine conditions seen in Figure 1 has not yet been achieved, so the need for acquiring the full-field heat transfer data at realistic engine conditions has been continuously urged. The present experimental method enables the generation of full-field heat transfer with both Coriolis-force and rotating-buoyancy effects detected. The protocols are aimed at assisting the investigators to devise an experimental strategy relevant to the realistic full-field heat transfer measurement of a rotating channel. Along with the method of parametric analysis that is unique to the present experimental method, the generation of heat transfer correlation for assessing the isolated and interdependent Ro and Bu effects on Nu is permitted.

The article illustrates an experimental method aimed at generating the two-dimensional heat transfer data of a rotating channel with flow conditions similar to the realistic gas turbine engine conditions but operating at much lower rotating speeds in the laboratories. The method developed to select the rotating speed, the hydraulic diameter of test channel and the range of wall-to-fluid temperature differences for acquiring the heat transfer data at realistic engine conditions is illustrated in the introduction. The calibration tests for the infrared thermography system, the heat loss calibration tests and the operation of the rotating heat transfer test rig are shown. The factors causing the significant uncertainties for heat transfer measurements and the procedures for decoupling the Coriolis-force and buoyancy effects on the heat transfer properties of a rotating channel are described in the article with the selective results to demonstrate the present experimental method.