Sound generation by acceleration of density inhomogeneities and steady heat communication

This paper presents a theoretical and numerical study on the relative significance of two mechanisms of sound generation encountered in combusting flows. A non-diffusive, onedimensional, steady heat communicating flow, subjected to downstream travelling acoustic excitations is considered. It is first shown numerically and analytically that even without unsteadiness in heat communication the investigated flow can generate sound. Existing theories of acoustic energy reveal that such generation of sound can be due to either of acceleration of entropy disturbances or interactions between acoustics and steady heat communication. An analytical scaling analysis is then put forward to estimate the relative significance of these sound generating mechanisms. The results are validated against the numerical simulations and include both high and low frequency limits. In particular, it is shown that at high forcing frequencies the sound generation is dominated by the steady heat communication. Introduction It has been known since Rayleigh [1] that unsteadiness in heat communication can generate sound. Later, Ffowcs Williams and Howe [2, 3] showed that acceleration of density inhomogeneities can be also a sound source. Since combusting flows are normally accelerating and feature heat addition unsteadiness, these two mechanisms are sometimes considered as the main sources of combustion noise [4, 5]. Nonetheless, Bloxidge et al. [6] showed theoretically that interaction of acoustics with steady heat communication can be also a sound source. Recently Karimi et al. [7] investigated the acoustic and entropic fields of a simple one-dimensional heat communicating flow subjected to acoustic forcing. They showed analytically and numerically that even in the absence of unsteadiness in heat communication such a flow can generate sound for all forcing frequencies. However, the physical mechanisms responsible for this sound generation were not clear, since their investigated flow included both steady heat communication and convection of entropy disturbances by an accelerating mean flow. Importantly, the relative importance of each of these two mechanismss at varying forcing frequencies was unknown. This paper therefore aims at estimating the relative significance of the two mechanisms of acceleration of density disturbances and steady heat communication in simple steady heat communicating flows. Fig. 1 shows schematically the problem under investigation. A non-diffusive, steady heat communicating flow, is excited by downstream travelling acoustic waves. This inhomogeneous flow features a linear increase (or decrease) in stagnation temperature and semi-infinite homogeneous regions. Thus, there is no sound reflection from these homogeneous ends while the inhomogeneous region can be reflective. The system is excited by an incident harmonic acoustic wave I with reflected and transmitted counterparts R and T . Interaction of S Figure 1: Schematic of the problem configuration, one dimensional accelerating flow, l=1m, heating case: θt0=600 K, θtl=1200 or 1800 K, cooling case: θt0=1200 K, θtl=600 K where θt is the stagnation temperature. the acoustics with the heat communication flow also results in generation of an entropy wave S which convects downstream. Theoretical and numerical methods Equations of motion Consider the one-dimensional Euler equations applied to a calorifically perfect, ideal gas, ∂ ∂t (ρ)+ ∂ ∂x (ρu) = 0, (1) ∂ ∂t (ρu)+ ∂ ∂x ( p+ρu2 ) = 0, (2) ∂ ∂t ( p γ−1 +ρu 2 ) + ∂ ∂x ([ γp γ−1 +ρu 2 ] u ) = q, (3) where p (Pa), ρ (kg/m3), s (kJ/kg.K) and u (m/s) are respectively the static pressure, static density, entropy and flow velocity, q (W/m3) is the heat communication per unit volume and γ is the ratio of specific heats (γ = 1.4 ). Throughout this paper symbols (̄) and ()′ respectively refer to mean value and first order disturbance. Acoustic energy The acoustic energy balance for heat communicating flows was derived by Bloxsidge et al. [6] who extended Morfey’s earlier work [8]. ∂(eA) ∂t + ∂(EA) ∂x = D, (4) where e = p′2/2ρ̄c̄2 + ρ̄u′2/2+ ūp′u′/c̄2, (5) is the acoustic energy density (J/m3) and E = p′u′ + ρ̄ūu′2 + ū2 p′u′/c̄2 + ūp′2/ρ̄c̄2, (6) is the acoustic energy flux (W/m2). Bloxsidge et al. [6] expressed the acoustic energy source term D (W/m) as D = ( ρ̄ūu′ + p′ ) q̄(γ−1) γ p̄ ( q′ q̄ − p ′ p̄ − u ′ ū ) (7) + ( u′ + ūp ′ ρ̄c̄2 )[( ρ̄s′ cp − p ′ c̄2 ) ū dū dx − p ′ ρ̄ dρ̄ dx ] , which can be rearranged in the following way D = Duhc +Dshc +Dm fa, (8) where Duhc = q̄ (γ−1) γ p̄ ( ρ̄ūu′ + p′ ) q′ q̄ , (9) Dshc = − q̄ (γ−1) (γ) p̄ ( ρ̄ūu′ + p′ )( p′ p̄ + u′ ū ) (10) − 1 ρ̄ dρ̄ dx p′ ( u′ + ūp′ ρ̄c̄2 ) , Dm fa = −ū dū dx ( u′ + ūp ′ ρ̄c̄2 ) ρ′. (11) are respectively the sources terms due to unsteady heat communication, steady heat communication and mean flow acceleration. With the arrangement in Eqs. (8) to (11) we can consider two mechanisms which can generate sound in the absence of unsteady heat communication, i.e. Duhc = 0. First Dm fa describes sound generation by the acceleration of entropy inhomogeneities. The remaining terms then only depend on the steady heat communication and so are grouped as Dshc. It is not clear how significant Dshc is compared to the well-established mechanism of acceleration of entropy inhomogeneities [2]. Numerical solver The present work solves Eqs. (1) to (3) in conservation form by using the Dispersion-Relation-Preserving (DRP) scheme of Tam and Webb [9]. The specific DRP scheme chosen uses an optimised, four level, time marching scheme and seven point stencil for spatial differentiation. Further details and validation of the solver can be found in [7, 10]. Results and discussion Sound generation in steady heat communicating flow Consider the balance of acoustic energy as defined in Eqs. (4) to (6). The acoustic energy reflection and transmission coefficients are defined respectively as ΣR = (1− M̄0) (1+ M̄0) 2 | R I |2, ΣT = ρ̄0c̄0 ρ̄l c̄l (1+ M̄l ) 2 (1+ M̄0) 2 | T I |2. (12) Figs. 2 and 3 show the numerically calculated reflection and transmission coefficients at finite Mach numbers for heated and cooled flows respectively. These figures also include the analytically developed acoustic energy reflection and transmission coefficients for zero mean flow case [7]. As can be seen, there is a good agreement between the low Mach number numerical simulations and the analytical calculations. However, as the inlet Mach number increases there is a deviation from the zero mean flow coefficients. This results in a net generation or dissipation of acoustic energy in cooled and heated flows respectively. Further simulations show that generation (or dissipation) of acoustic energy becomes more significant as the inlet Mach number and the overall temperature ratio increase [10]. 0.01 0.02 0.03 0.04 0.05 a)