A Revised Equation for Heat Flux Reduction in Film Cooling Studies and Discussion of Its Applications

In film cooling experimental studies, due to the difficulty in measuring the surface heat flux variation, a Heat Flux Ratio (HFR) equation originally derived by Mick and Mayle [1], has been widely employed to calculate the surface heat flux distribution using the measured adiabatic film effectiveness and surface temperature. A close examination of the derivation process and applications of the HFR equation reveals two issues of concern. First, an implicit assumption was introduced by letting the wall surface temperature of the system withoutfilm be the same as that which would occur with a film-cooled condition. A revised equation is then derived by removing this implicit assumption and incorporating the wall temperature change due to film cooling Secondly, a uniform value of the non-dimensional metal temperature φ (or film cooling effectiveness) has been used in all the previous applications of the HFR equation. This practice implicitly implies that a uniform wall temperature is distributed throughout the entire surface under film cooling, which is usually not the case in real conditions. A series of computational experiments are conducted to verify the revised HFR equation under different conditions as well as examine the validity of using a constant surface temperature in the HFR equation. Results reveal that using a constant value of φ (0.5 ~ 0.7) to calculate surface heat flux may result in a negative HFR in some simulated cases showing the commonly adopted value φ=0.5~0.7. This could induce errors and give false HFR. The error is reduced in 3D cases because the streamwise wall temperature becomes more uniform than 2D cases. The difference between the old and new equations can reach about 20%. A conjugate wall cooling simulation shows negative HFR is possible in the region close to the film hole due to the heat conduction from the downstream hotter region into the cooler region near the film hole. Using the actual wall temperature as the φ-value, the newly revised HFR equation produces the exact heat flux as calculated by CFD including the correct calculation of negative heat flux caused by the conjugate wall. NOMENCLATURE b coolant injection slot width (mm) haf adiabatic film heat transfer coefficient (haf = q" / (TawTw)) (W/mK) HFR heat flux ratio (q" / q"o) k turbulence kinetic energy (m/s) l chord length (mm) M blowing ratio, (ρu)j/(ρu)g Nux Nusselt number, hx/λ, x is the distance from the injection hole in streamwise direction NHFR net heat flux reduction (1q" / q"o) Pr Prandtl number (ν/α) q" heat flux (W/m), positive value for heat flowing from gas into the wall r recovery factor Re l Reynolds number based on chord length, ul/ν Taw adiabatic wall temperature (K) Tw wall surface temperature in contact with gas (K) Tg main gas flow temperature (K) Tj coolant temperature at the cooling jet hole exit (K) Tci internal coolant temperature (K) Tr recovery temperature (K) Tu turbulence intensity Greek Letters α thermal diffusivity (m/s) ε turbulence dissipation rate (m/s) η adiabatic film cooling effectiveness, (Tg-Taw)/(Tg-Tj) λ heat conductivity (W/mK) ν kinematic viscosity (m/s) ' v ' u ρ Reynolds stress φ film cooling effectiveness, φ = (Tg-Tw) / (Tg-Tj) (or non-dimensional metal temperature, overall cooling effectiveness) Subscripts aw adiabatic wall ci internal cooling conj conjugate blade f with film cooling g main flow of hot gas/air j coolant or jet flow o without film w wall