Use of Bayesian statistics to calculate transient heat fluxes on compressor disks

Future gas turbine engines require improved understanding of the heat transfer between compressor disks and air in compressor cavities under transient operating conditions. Calculation of transient heat fluxes from temperature measurements on compressor disks is a typical ill-posed inverse problem where small uncertainties of measurements can lead to large uncertainties of the calculated fluxes. This paper develops a Bayesian model for the heat flux to reduce the adverse nature of the problem by using a Gaussian prior distribution with Matérn covariance. To efficiently find the maximum a posterior, a neural network was used to solve the heat equation for compressor disks for any choice of parameters, allowing fast evaluation of the solution to the forward model for any heat flux of interest. The power of the Bayesian model is first demonstrated using numerically simulated data. Subsequently, the model is used to calculate heat fluxes from measurements of transient temperature collected from the Compressor Cavity Rig at the University of Bath. During these transient tests, the periphery of the rotating compressor disk was initially heated to a steady-state condition and then cooled rapidly by the ambient air. The fluxes for four transient cycles were calculated, with the operating range of $7.0×105<Reϕ<2.8×106, 0.0<βΔT<0.15$⁠, and $0.13<χ$⁠, where $Reϕ,βΔT$⁠, and χ are the rotational Reynolds number, the buoyancy parameter, and the compressibility parameter, respectively. The results show that, for all four cases, the flow and the heat transfer in the closed cavity were initially dominated by buoyancy effects, with heat transferred from the disk to the cavity air in the outer region and reversed in the inner region. The initial heat fluxes at $Reϕ=2.1×106$ were higher than those at $Reϕ=2.8×106$ owing to a compressibility effect. During the cooling transient, for cases with $Reϕ≤1.4×106$⁠, the magnitudes of the heat fluxes gradually decreased and eventually reached virtually zero. This indicated that the flow was first governed by the buoyancy effects and then became stratified. At $Reϕ=2.8×106$⁠, where the rotational speed was at its maximum, buoyancy-induced flow dominated the entirety of the transient process due to significant frictional heating at the periphery of the rig. The calculated fluxes present evidence for future theoretical and computational modeling of transient disk heat transfer, and the Bayesian model provides guidance for transient temperature data analysis.