A solution for the quasi-one-dimensional linearised Euler equations with wall friction

The unsteady response of nozzles with wall friction forced by acoustic and/or entropy waves is modelled. The approach is based on the quasi-one-dimensional linearised Euler equations. The equations are cast in terms of three variables, namely the dimensionless mass, stagnation temperature and entropy fluctuations, which are invariants of the system at zero frequency and in the absence of wall friction. The resulting first-order system of differential equations is solved using the Magnus-expansion method, where the perturbation parameters are the normalised frequencies and Fanning friction factors. In this work, a measure of the flow non-isentropicity caused by wall friction is used for the first time as an expansion parameter. The solution method is applied to a converging–diverging nozzle with wall friction for both subsonic and supersonic flow cases, showing good agreement with numerical predictions. The nozzle flow with both wall friction and steady heat transfer is further explored. It is observed that the acoustic and entropy transfer functions of the nozzle strongly depend on the frequency, wall friction and heat transfer. Wall friction affects the unsteady response through changes in the mean flow and through coupling between acoustic, velocity and entropy fluctuations.