The governing equations of the acoustic perturbations and the fluid flow are the same. The fluc-tuation data obtained using an unsteady compressible set of flow equations also contain the acoustic field induced by the flow. Flow noise prediction approaches making use of numeri-cal solvers which take into account this compressibility effects are named ‘direct approaches’.
The acoustic field is not explicitly modeled in direct approaches. The compressible equations inherently include a two-way coupling mechanism with the flow and the acoustic field. This allows the direct approaches to solve any aeroacoustic problem: tonal, broadband, combustion etc. noise. However, due to the disparity of scales and amplitudes between the hydrodynamic and acoustic perturbations, the accuracy of the numerical simulations are to be very high to be able the resolve the acoustic field properly. Such a demand for high accuracy results in ex-cessive computational costs making direct approaches infeasible for many engineering flows.
Hybrid approaches as an alternative can provide satisfactory noise predictions at significantly lower costs. The fundamental assumption of the hybrid approaches is that a one way coupling is considered between the flow and the acoustic fields. The prediction of the flow and the acoustic field are treated sequentially, ignoring the effect of the acoustic field on the flow. There are var-ious noise prediction methods counted as hybrid. Among them, aeroacoustic analogies take an important place for low Mach number noise prediction problems. The noise prediction strate-gies adopted in the present study are mainly based on the aeroacoustic analostrate-gies of Lighthill and Curle. Therefore, a detailed revisit is provided in the following subsections.
2.3.1 Lighthill’s aeroacoustic analogy
Lighthill [51] introduced the idea of reformulating the equations of fluid motion to allow a distinction between the sound generation and propagation parts, which are called the ‘source’
and ‘reference’ regions. Lighthill’s derivation of such a formulation starts with the conservation equations of mass and momentum. The conservative form of the momentum equation is given as;
where fis the density of the force field. Taking the time derivative of the continuity equation
and the divergence of Eq. (2.85) as;
∂
Eq. (2.87) is subtracted from Eq. (2.86) to obtain;
∂2ρ the wave propagation operator of d’Alembert on the left hand side as follows:
∂2ρ
Splitting the flow variables(ρ,p)into a uniform reference state (ρ0,p0)and a fluctuating part (ρ0,p0), assuming stagnant flow, and following an analysis similar to the one given in Sec-tion 2.1.2, Eq. (2.89) can be written in terms of the fluctuating variables to obtain Lighthill’s analogy:
withTi j, the so-called Lightill’s stress tensor defined as;
Ti j≡ρuiuj+(p0−c20ρ0)δi j−σi j. (2.91) An important notice to be made here is that Eq. (2.91) is the exact equation defining all the mechanisms generating acoustic waves in a uniform stagnant fluid. This implies that an attempt to predict the noise generated inside flow using Eq. (2.91) requires the knowledge of all the non-linear interactions among the flow variables, and therefore is not necessarily better than directly solving the compressible Navier-Stokes equation. On the other hand, it allows evaluating the relative importance of different noise generation mechanisms given a flow condition, so that simplifications are easy to make.
The term fi drops if there are no external forces exerted on the fluid elements. For typical industrial flow where high-Reynolds number assumption can be made, the contribution of the viscous term,σi jto noise generation with respect to theρuiujterm can be neglected. Moreover, for isentropic flows, the second term on the right hand side of Eq. (2.91) can also be dropped.
And finally, for low-Mach number flows, the compression effects can be considered to be very small so that the following simplification holds true:
ρuiuj 'ρ0uiuj. (2.92)
2.3.1.1 Integral formulation of Lighthill’s analogy
Assuming causality and initially silent medium, the integral formulation of Eq. (2.90) is given as; where x and y denote the listener and source positions, respectively. The volume integral on the right hand side of Eq. (2.93) correspond to the incident sound field generated by volu-metric sources, and the surface integral represents the scattering of the incident field over the boundaries, ∂V. This form of integral representation of Lighthill’s analogy is highly prone to numerical issues, as any error contained in the stress tensor data is severely amplified by double differentiation. Integration by parts can be applied two times on the volume integral in Eq. (2.93) to provide a more robust formulation as follows:
Z t
Substituting Eq. (2.94) in Eq. (2.93) yield;
ρ0(x,t)=
The double differentiation in the volume integral in Eq. (2.93) is now shifted to the Green’s function,Gfor which a more robust differentiation, if not analytical, can be achieved. Assuming free-field without any boundaries, the scattering terms vanish from Eq. (2.94). Note that, to be able to impose this condition, the Green’s function should satisfy the Sommerfeld boundary condition:
Lighthill’s analogy does not provide any solution to treat the scattering from the boundaries.
Therefore it is applicable only in free-field. Curle extended Lighthill’s analogy to take into
account the interaction of the incident field with steady surfaces. The analysis starts with finding
Thanks to the symmetry ofTi j, change of indices do yield the same result for the expression µ∂Ti j
Inserting the conservation of momentum equation (2.85) in the absence of external forces in Eq. (2.97) and using Eq. (2.98), the following expression is obtained:
∂Ti j
∂yi = −∂ρui
∂τ −c02∂ρ0
∂yi. (2.99)
Substituting Eq. (2.91) and Eq. (2.99) in Eq. (2.95) gives the following expression:
ρ0(x,t)= Note that there are terms canceling each other in Eq. (2.100). When all the cancellations are performed, the following expression is obtained:
ρ0(x,t)= There exists again a term in Eq. (2.101) where velocity is differentiated. Integrating this term by parts as follows; The first integral on the right hand side of Eq. (2.102) vanishes due to the virtue of causality, and the second integral vanishes for impermeable fixed boundaries due to the no-slip condition.
Eq. (2.101) then reduces to the following:
ρ0(x,t)= Eq. (2.103) is valid for any Green’s function. Curle’s analogy uses the free-field Green’s func-tion which is not discussed here since a tailored Green’s funcfunc-tion for cylindrical ducts is used instead in the this study. The discussion of implementation of a tailored Green’s function in Eq. (2.103) is left to the Section 7.2.
diaphragm flows
3.1 Introduction
The experimental investigation of the diaphragm flows include the determination of the con-figurations to be numerically investigated by means of some preliminary measurements, and providing reference flow and noise field data for the numerical analyses. All the experimental analyses were carried out using the test rig installed in the anechoic chamber of the von Karman Institute for Fluid Dynamics (VKI). Flow field measurements were conducted using hot-wire anemometry. The aeroacoustic measurements included various campaigns to measure the noise field inside and outside the duct, and to identify the active flow noise and the scattering charac-teristic of the diaphragms. This chapter is devotedi) to provide detailed information about the anechoic chamber and the test rig together with the instrumentation used, andii) to discuss the preliminary investigations conducted to determine the details of the test cases investigated in the study.