Theory of Non-Dimensional Groups in Film Effectiveness Studies

Abstract

The desire to improve gas turbines has led to a significant body of research concerning film cooling optimization. The open literature contains many studies considering the impact on film cooling performance of both geometrical factors (hole shape, hole separation, hole inclination, row separation, etc.) and physical influences (effect of density ratio (DR), momentum flux ratio, etc.). Film cooling performance (typically film effectiveness, under either adiabatic or diabatic conditions) is almost universally presented as a function of one or more of three commonly used non-dimensional groups: blowing—or local mass flux—ratio, density ratio, and momentum flux ratio. Despite the abundance of papers in this field, there is some confusion in the literature about the best way of presenting such data. Indeed, the very existence of a discussion on this topic points to lack of clarity. In fact, the three non-dimensional groups in common use (blowing ratio (BR), density ratio, and momentum flux ratio) are not entirely independent of each other making aspects of this discussion rather meaningless, and there is at least one further independent group of significance that is rarely discussed in the literature (specific heat capacity flux ratio). The purpose of this paper is to bring clarity to this issue of correct scaling of film cooling data. We show that the film effectiveness is a function of 11 (additional) non-dimensional groups. Of these, seven can be regarded as boundary conditions for the main flow path and should be matched where complete similarity is required. The remaining four non-dimensional groups relate specifically to the introduction of film cooling. These can be cast in numerous ways, but we show that the following forms allow clear physical interpretation: the momentum flux ratio, the blowing ratio, the temperature ratio (TR), and the heat capacity flux ratio. Two of these parameters are in common use, a third is rarely discussed, and the fourth is not discussed in the literature. To understand the physical mechanisms that lead to each of these groups being independently important for scaling, we isolate the contribution of each to the overall thermal field with a parametric numerical study using 3D Reynolds-averaged Navier–Stokes (RANS) and large eddy simulations (LES). The results and physical interpretation are discussed.

Method for Determining Adiabatic Film Effectiveness in Presence of Thermal Boundary Layer

“…Film cooling performance is often characterized using the adiabatic film effectiveness, η, a nondimensional temperature typically defined by η = T 01 − T w,ad T 01 − T 02 (1) where T w,ad is the adiabatic surface temperature at a point of interest, T 01 is the (typically hot) mainstream gas temperature, and T 02 is the (typically cold) coolant gas supply temperature. In an adiabatic system, the wall temperature can be thought of as being due to mixing of particular mass fractions of hot and cold gas at the surface.…”

“…In this sense, the adiabatic film effectiveness can be regarded as being directly related to the local mass fractions. More specifically, rearranging (1), we write T w,ad = ηT 02 + (1 − η)T 01 (2) where η can now be thought of as the mass fraction of cold gas and (1 − η) the mass fraction of hot gas. Strictly speaking, this formulation is only valid in the case of the mainstream and coolant stream having the same specific heat capacity.…”

“…In cases where the temperature ratio is not matched, and where air is used for both coolant flow and mainstream flow, the scaling can be quite complex (see, for example, Ref. [1]). The need to scale arises primarily from a mismatch in the specific heat capacity flux ratio.…”

“…This problem can be mitigated by using so-called foreign gas for one stream, typically the coolant stream, to bring the specific heat capacity flux ratio closer to the engine condition. Even if this approach is taken, second-order scaling issues would still need to be addressed [1].…”

“…More complex, second-order effects that lead to dissimilarity of the flow field are discussed in Ref. [1]. It is largely true to say that the second-order effects cannot be corrected for analytically, because they change the structure of the flow in a way that tends to be geometry specific.…”