Reynolds analogy and turbulent heat transfer on rotating curved surfaces

Abstract

To evaluate turbulent heat transfer on rotating turbine blades, knowledge of the Reynolds stress and heat conduction moments on the blades is required. This involves solving the mean flow equations of momentum and thermal energy on the rotating blades. In this paper, the turbine blades are represented by rotating curved surfaces, and turbulent heat transfer calculations are carried out by invoking a valid Reynolds analogy because of its simplicity. The classic Reynolds analogy is defined as the ratio between the eddy and thermal diffusivity in a stationary plane flow; however, its counterpart for rotating curved flows is still not known. To derive a Reynolds analogy for rotating curved flows, appropriate turbulence models to close the Reynolds transport equations for momentum and thermal energy are required. Assuming that, in the constant flux region, advection and diffusion of Reynolds stress and heat conduction moments are negligible compared to the production of turbulent energy, the Reynolds shear stresses and heat conduction moments can be solved in terms of the mean flow gradients, a turbulent Prandtl number (Prt)rc and a gradient Richardson number for rotating curved flows Rirc. These dimensionless numbers are dependent on mean flow properties, surface curvature, and system rotational speed. Thus derived, the classic Reynolds analogy can be recovered only when Rirc << 1 and (Prt)rc = 1. For turbine blade heat transfer, Rirc is not necessary very small and (Prt)rc differs from unity in the whole flow field. Therefore, using the classic Reynolds analogy to estimate turbine blade heat transfer will lead to incorrect result. The new Reynolds analogy, formulated for turbulent thermal flows on rotating curved surfaces, takes all pertinent external body force effects into account; thus, turbulent heat transfer evaluation on turbine blade surfaces can be substantially improved.