Elsevier

Journal of Sound and Vibration

Volume 329, Issue 22, 25 October 2010, Pages 4705-4718
Journal of Sound and Vibration

On the interaction of sound with steady heat communicating flows

https://doi.org/10.1016/j.jsv.2010.05.009Get rights and content

Abstract

This paper presents a theoretical and numerical investigation of the interaction between sound waves and non-diffusive, quasi-one-dimensional, subsonic flows with only steady heat communication. It is first shown that a steady heat communicating flow can attenuate or generate sound even in the absence of mean flow acceleration. The relative significance of sound generation by the heat addition induced acceleration of density or entropy inhomogeneities and the effect of steady heat communication on the incident acoustic wave is then found. It is shown through scaling arguments that at high frequencies mean flow acceleration effects are negligible and the only significant sound generating mechanism involves steady heat communication. At low frequencies, however, the two mechanisms are more comparable.

Introduction

Sound production in fluid flows with simultaneous heat communication and acceleration has important application in numerous engineering devices. These include rockets, gas turbines, heat exchangers and refrigerators, to name a few. In these cases, sound often coexists with unsteady heat addition or removal, as well as hydrodynamic and temperature fluctuations.

It has been known since Rayleigh [1] that unsteadiness in the communication of heat to or from a flow can generate or attenuate sound. Ffowcs Williams and Howe [2] and Howe [3], [4] also showed that the acceleration of density and entropy inhomogeneities can be a sound source. As heat communicating flows are normally accelerating or decelerating, these mechanisms are sometimes considered to be the main sound generating mechanisms in combusting flows [5], [6], [7], [8], [9]. There are other sources of noise in combusting flows, such as those rising from viscous and diffusion effects [10], [11], chemical inhomogeneities [12] and flame front interactions [6].

Recently, Karimi et al. [13] considered a non-reacting, inviscid and non-heat conducting flow, and showed theoretically and numerically that steady heat communication can generate sound. However, the origin of this sound generation was not clear since their test cases included flows with both steady heat communication and steady flow acceleration. It is well known that the interaction of sound with heat communicating flows can generate entropy disturbances [13], [14], [15], [16], [17]. Following Howe [3], these disturbances can then generate sound as they are convected by the accelerating mean flow due to the mean temperature gradient. Therefore, the sound generation in studies such as Karimi et al. [13] can either be by the acceleration of these entropic disturbances or potentially involve additional sources.

Two duct flows are considered in this paper (Fig. 1). In each group both flows have identical upstream mean Mach number M¯0, static temperature θ¯0 and pressure p¯0. Two homogeneous, semi-infinite regions are located upstream and downstream of a region of steady heat communication, which have constant cross-sectional areas. The steady heat communication is through external means and results in a linear increase or decrease in the mean stagnation temperature from θ¯t0 at x=0 to θ¯tl at x=l. Thus, the mean heat communication to the two flows is the same. In all cases studied throughout this paper the unsteadiness in heat communication is zero. Further, as will be detailed in Section 3.1, the cross-sectional area in Fig. 1b changes in such a way that the mean flow does not accelerate. The system is excited by a downstream travelling acoustic wave I and the response is characterised by the reflected R and transmitted T acoustic waves as well as an outgoing entropy disturbance Sl. These acoustic disturbances are given by p(t,0)=Re[exp(iωt)(I+R)],u(t,0)=Re[exp(iωt)(IR)1/ρ¯0c¯0],and p(t,l)=Re[exp(iωt)(Texp(iωl/c))],where Re symbol indicates the real part.

In these equations the terms p (Pa), ρ (kg/m3), c (m/s), ω (rad/s) and u (m/s) are respectively the static pressure, static density, isentropic sound speed, frequency and flow velocity. In general any given property g may be split into a steady g¯ and disturbance quantity g′ such that g=g¯+g. Unless otherwise stated, throughout this paper the inlet and exhaust stagnation temperature and length of the inhomogeneous region (l) are those specified in Fig. 1. Further, the configurations in Fig. 1a and b are referred to as cases A and B respectively.

It should be noted that the flows shown in Fig. 1 are not intended to be closely representative of real flows. In particular, the flow within a gas turbine combustor features unsteady heat release, strong turbulence and non-equilibrium chemistry, to name a few. None of these are considered in this paper. In particular, unsteady heat release is a well-known and significant sound source in combustion [5], [6]. The relative magnitude of sound generation by steady versus unsteady heat addition depends on the relative magnitude of the mean and unsteady sources, which is not the subject of this paper. Rather, sound sources due only to steady heat communication and the acceleration that it can induce are examined over temperature ratios that are representative of some devices.

Section snippets

Equations of motion

Consider the quasi-one-dimensional Euler equations applied to a calorifically perfect and ideal gast(ρA)+x(ρuA)=0,t(ρuA)+x([p+ρu2]A)=pdAdx,tpγ1+12ρu2A+xγpγ1+12ρu2uA=Q,where Q (W/m) is the heat communication per unit length and γ=1.4 throughout this paper. The terms q (W/m3), γ, cp (J/kg K), s (J/kg K), and A (m2) are respectively the heat communication per unit volume, specific heat ratio, specific heat capacity, entropy and the cross-sectional area.

It is important to note that

Zero and low mean flow velocity

The cross-sectional area variation for the non-accelerating mean flow, case B, is found from the equations of motion. It follows from Eqs. (1), (2) that for a quasi-one-dimensional, constant mean velocity flow the mean pressure must remain constant. The equation of state for an ideal gas then requires that θ¯dθ¯/dx=ρ¯dρ¯/dx, which upon substitution into Eq. (1) results in1AdAdx=1θ¯dθ¯dx.Here, θ¯t(x)=θ¯t0+mx in which m (K/m) is a constant, is considered along the inhomogeneous region. Noting

Conclusions

The generation of sound due to the interaction of incident acoustics and entropic disturbances with steady heat communication was studied numerically and theoretically. Two sets of flows were considered with identical upstream conditions and steady heat addition or heat removal. One of these flows was one-dimensional, and therefore accelerating, whilst the other featured a varying cross-sectional area such that the mean flow velocity was constant. The numerical simulation of these two flows was

Acknowledgement

The authors would like to thank Prof. Nigel Peake of the University of Cambridge for thoughtful discussions on earlier drafts of this work.

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