A solution for the quasi-one-dimensional linearised Euler equations with heat transfer

TL;DR

This paper develops an analytical solution for the unsteady response of nozzles with steady heat transfer using the quasi-one-dimensional linearised Euler equations, employing the Magnus expansion method to account for heat transfer effects.

Contribution

It introduces a novel approach using the Magnus expansion to solve for heat transfer effects in quasi-one-dimensional Euler equations, considering non-isentropic flow as an expansion parameter.

Findings

Good agreement with numerical predictions for converging-diverging nozzles.

Flow response depends strongly on frequency and heat transfer.

Method applicable to both sub-critical and super-critical flows.

Abstract

The unsteady response of nozzles with steady heat transfer 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 with no heat transfer. The resulting first-order system of differential equations is then solved using the Magnus expansion method, where the perturbation parameters are the normalised frequency and the volumetric heat transfer. In this work, a measure of the flow non-isentropicity (in this case the steady heat transfer) is used for the first time as an expansion parameter. The solution method was applied to a converging-diverging nozzle with constant heat transfer for both sub-critical and super-critical flow cases, showing good agreement with numerical predictions. It was observed that the acoustic and entropy transfer functions of the nozzle strongly depend on the frequency and heat transfer.