Plume Model for Buoyancy-Induced Flow and Heat Transfer in Closed Rotating Cavities

The radial clearance between the rotating blades and stationary casing of a gas-turbine compressor depends on the radial growth of the rotating discs to which the blades are attached, and this growth depends on the buoyancy-induced flow and heat transfer in the air-filled cavities between adjacent discs. In some engines, the cavities are sealed, which creates a closed rotating cavity. A theoretical model has been developed to calculate the radial distribution of the temperature of the disc in a closed rotating cavity. The principal assumptions are that the convective heat transfer from the hot shroud to the cold hub of the cavity is via plumes of fluid in which the cold fluid moves radially outward, and the hot fluid inward, inside an inviscid quasi-axisymmetric core of rotating fluid. The fluid core is surrounded by boundary layers on all rotating surfaces, with free-convection layers on the surfaces of the shroud and hub and laminar Ekman layers on the surface of the discs. In addition to the convection, heat is transferred by one-dimensional radial conduction in the rotating discs. Using the model, equations have been derived to calculate the radial distribution of temperature in the discs and fluid core. These equations reveal that the non-dimensional temperatures for the disc and core, θ_d and θ_c, are controlled by three non-dimensional parameters: $Reϕ$⁠, βΔT, and χ, the rotational Reynolds number, the buoyancy parameter, and the compressibility parameter, respectively. The compressibility parameter is defined as $χ=defMa2/βΔT$ where Ma is a Mach number, and χ is shown to strongly affect the radial distribution of the core temperatures. There will be a critical value of χ at which the core temperature equals that of the heated shroud. For a closed cavity with adiabatic discs and x_a = 0.5, the critical value is 6.7. Above this critical value, stratification is expected to occur and heat transfer from the shroud to the core will be by conduction rather than by convection. The theoretical model predicts radial distributions of temperatures in the discs and fluid core that are in good agreement with the experimentally derived values in a companion paper.