Proper orthogonal decomposition of eigen modes in a gas affected by a mass force

TL;DR

This paper derives relations to decompose perturbations in a gas affected by a constant mass force into acoustic and entropy modes, enabling analysis of energy distribution at any instant.

Contribution

It introduces a method to uniquely decompose perturbations into acoustic and non-acoustic parts in a gas under a mass force, independent of time.

Findings

Relations enable decomposition of perturbations into modes at any time.

Method applies to gases with temperature varying along the force direction.

Examples demonstrate decomposition in temperature linearly dependent on space.

Abstract

The relations connecting perturbations in acoustic and entropy modes in a gas affected by a constant mass force, are derived. The background temperature of a gas may vary in the direction of an external mass force. The relations are independent on time. They make possible to decompose the total vector of perturbations into acoustic and non-acoustic parts uniquely at any instant. %The conclusions of distribution of the energy between sound and entropy modes may be also made at any instant. In order to do this, three quantities are required, according to the number of modes. In one dimension, the reference quantities may be total perturbations in entropy, pressure and velocity. The total energy of flow is determined. The examples of dismemberment of the total field into acoustic and entropy parts relate to the unperturbed temperature of a gas which linearly depends on the spacial co-ordinate.