Film-Cooling Jets Analyzed with Proper Orthogonal Decomposition and Dynamic Mode Decomposition

Abstract

Film cooling jets were simulated using Large Eddy Simulation (LES) at blowing ratios of 0.150, 0.50 and 1.0. The results were analyzed using Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) as a statistical analysis tool and the first step toward reduced order modeling. The results of both decompositions are presented and put in perspective with dominant vortical structures and heat transfer mechanisms observed in the instantaneous velocity and temperature fields. Assessment of the performance of the decompositions is provided in terms of energy content and rendered dominant frequencies. Evolution of the dominant modes and decomposition metrics are discussed as the blowing ratio is increased. Nomenclature BR = Blowing ratio D j = Jet diameter, mm Re δ = Boundary layer Reynolds number (U ∞ .δ/ν) St = Strouhal number (f D j /U ∞ ) <T> = Time averaged temperature, K ΔT = Normalized temperature fluctuation (T-<T>)/(T j -T ∞ ) (T j , T ∞ jet and cross-flow temperatures resp.) U ∞= Average free-stream velocity, m.s -1 X j = Normalized streamwise coordinate (x/D j ) Y j = Normalized spanwise coordinate (y/D j ) Z j = Normalized vertical coordinate (z/D j ) δ = 99% Boundary layer thickness, mm ν = Air kinematic viscosity, m 2 .s -1 φ i , a i = Velocity POD shape function and temporal coefficient ψ i , b i = Temperature POD shape function and temporal coefficient Φ i , λ i = Velocity DMD shape function and eigenvalue Ψ i , μ i = Temperature DMD shape function and eigenvalue Λ 2 = Second invariant of the tensor S 2 +Ω 2 (S -stress tensor, Ω -rate of rotation tensor)

Structure analysis of adiabatic film cooling effectiveness in the near field of a single inclined jet: Measurement using fast-response pressure-sensitive paint

“…To further resolve the dominant signatures of the flow structures on the wall, a snapshot POD analysis was conducted on a set of 300 instantaneous fluctuating terms of effectiveness fields. Throughout the entire coolant region, the strongest modes are located far from the jet exit hole [37]. Therefore, the near-hole area x/D < 2.5 is used for calculations to exclude the far downstream behavior of strong unsteady structures.…”

“…As expected, the first two modes are oscillating together with a phase delay, but CRVP structures reflect a quasi-periodic pattern on the wall. The upward and downward motions of the side vorticities influence the wall behavior [37]. These motions entrain cross-flow fluid near the wall.…”