# Analysis of heat-driven oscillations of gas flows

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## Summary

It has been known for a long time that gas flows through systems containing heat sources can produce acoustic oscillations which are coupled with a periodic heat release of the heat source (heat-driven oscillations). This phenomenon may at times occur in industrial combustion systems and in jet engines, and can under certain circumstances be very awkward owing to the intensity and shrillness of the noise, sometimes also owing to the actual damage caused by the pressure fluctuations, and to the reduced performance of the combustor or jet engine. In this paper an effort is made to create a basis for the theoretical treatment of heat-driven oscillations. The fundamental ideas underlying the theory are discussed with reference to a simple model consisting of a tube containing a disk-shaped heat source perpendicular to the axis of the tube. Upstream of the heat source the oscillations are mainly acoustic, but downstream of the heat source there is not only an acoustic mode but also an entropy mode of the oscillations. It is proved that the existence of the entropy mode can be neglected if the effects of the viscosity and Mach number on the oscillations are slight, so that quadratic and higher terms in the viscosity and Mach number can be neglected. Under these circumstances the characteristic equation governing the occurrence of heat-driven oscillations has to be derived from the laws of conservation of momentum and energy applied across the heat source, since it appears that these laws contain only acoustic quantities. In order to obtain the characteristic equation in a general form, transfer functions of the heat source and acoustic admittances and impedances are introduced. The transfer functions of the heat source describe the response of the heat release to fluctuations in the velocity and thermodynamic variables of the gas flow, and in some forthcoming papers will be considered in more detail for hot wire gauzes and flames of premixed gases.

## Keywords

Heat Source Mach Number Heat Release Characteristic Equation Acoustic Mode## List of Symbols

*A*area of cross-section of the tube

*a*velocity of sound propagation

*B*\(c_{p,2} \bar T_2 /(c_{p,1} \bar T_1 )\)

*c*_{p}heat capacity per unit mass for constant pressure

*c*_{v}heat capacity per unit mass for constant volume

*E*_{q,k}real part of transfer function (

*k=u, p*)- e
_{x} unit vector in axial direction

*F*_{q,k}imaginary part of transfer function (

*k=u, p*)*f*frequency

*G*_{i}conductance (

*i*=1, 2)*G*_{M}conductance induced by the mean flow (\(( = A\bar U_1 /\bar P_1 )\))

*g*dimensionless number (=νω/a

^{2})*I*_{i}acoustic intensity (

*i*=1, 2)*I*_{M}acoustic intensity induced by mean flow

*I*_{q}acoustic energy produced by heat source

- j
\(\sqrt { - 1} \)

*K*_{i}susceptance (

*i*=1, 2)*L*_{K}differential operators defined by (1, 19) (

*k=a, e*)*M*Mach number (\((\bar U/a)\))

*P*pressure

*p*reduced amplitude of the fluctuations in pressure

*P*_{r}Prandtl number

*Q*heat released by the heat source per unit time

*q*reduced amplitude of the fluctuations in heat release

*T*temperature

*t*time

*Tr*_{q,k}transfer function of t he heat source (

*k=u, p*)*U*axial velocity of gas flow

*u*reduced amplitude of the fluctuations in axial velocity

- V
vector velocity of gas flow

*x*axial co-ordinate

*Y*_{i}admittance (

*i*=1, 2)*Z*_{i}impedance

- γ
ratio of heat capacities (

*c*_{ p }/*c*_{ υ })- Γ
dimensionless number defined by (1,39)

- ϑ
reduced amplitude of fluctuations in temperature

- μ
dynamic viscosity of gas

- ν
kinematic viscosity of gas (μ/ϱ)

- ϱ
density

- σ
reduced amplitude of fluctuations in density

- ω
circular frequency

## Subscripts

*a*fluctuating properties of acoustic mode

*e*fluctuating properties of entropy mode

*p*fluctuations in pressure

*q*fluctuations in heat release

*u*fluctuations in velocity

*v*fluctuating properties of vorticity mode

- 1
cold gas upstream of the heat source

- 2
hot gas downstream of the heat source

*L*wave propagating to the left

*R*wave propagating to the right

## Superscripts

- w
^{*} complex conjugate of quantity

*w*- w′
amplitude of fluctuations in quantity

*w*- \(\bar w\)
steady-state value of quantity

*w*

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