compared time-domain LEE and potential flow solutions for exhaust noise problems. They also used a vortex sheet, but enforced the continuities of normal displacement and of pressure instead.
In this chapter we revisit the models previously proposed in references [51, 95] and we address several outstanding issues: the application of a Kutta condition at the beginning of the vortex sheet, the discretisation of the vortex sheet, and a quantitative validation and comparison of these finite element models. In particular, we explicitly include the Kutta condition, an upwinding method is proposed to discretise the vortex sheet, and finally the different finite element formulations are validated against a semi- analytic model.
This chapter is structured as follows: we introduce the physical problem, by de- scribing the vortex sheet, the Kutta condition, and the Kelvin-Helmholtz instability. We review the governing equations, derive the potential theory model, and present the necessary continuity conditions. We derive the existing formulations, the proposed formulation, and we introduce a novel streamline upwind Petrov-Galerkin method. De- tails of the implementation of the method to a benchmark problem are given, and the results are discussed. Finally, a summary of the main findings is presented.
7.2
Description of the Problem
An example of the jet exhaust of a turbofan engine exhaust is illustrated in Figure 7.1. Coaxial jets issue from the bypass and core ducts, and shear layers develop at the interface between these jets, and between the bypass jet and the free stream flow. Broadband noise and tonal noise are generated upstream of the core and bypass ducts. This noise propagates along the ducts and radiates through the jets and into the far field. The sound waves will be diffracted by the trailing edges of the ducts, and refracted by the mean flow gradient as it propagates through the shear layers. It is essential to capture these effects to accurately predict the sound radiated to the far field.
7.2.1 Vortex Sheet
The shear layer of a jet is initially very thin close to the nozzle and gradually grows in the downstream direction. The convected wave equation (2.13) does not allow sheared flows, such as a mixing layer, to be included as a base flow. Instead we implement a
7. EXHAUST NOISE PREDICTION
Bypass jet Core jet
Free stream flow Free shear layers
Noise Noise
Figure 7.1: Diagram of the engine exhaust problem.
vortex sheet, which is essentially a surface of discontinuity in the mean flow representing an infinitely thin shear layer. This approximation is only valid when the acoustic wavelength is much larger than the shear layer thickness. This is the case either close to the nozzle and/or at low frequencies.
To fix ideas, let us consider two different flow regions Ω1 and Ω2 separated by a
vortex sheet Γ, as shown in Figure 7.2. The trailing edge of the nozzle, from which the vortex sheet originates, is labelled Γ0. The flow properties in each region are denoted
by subscripts 1 and 2. On Γ the mean flow is tangential: v01· n = v02· n = 0, where
n is the unit normal to Γ pointing into Ω2. The solutions on either side of the vortex
sheet are coupled by imposing kinematic and dynamic conditions [149]. For an inviscid fluid, the dynamic condition is the continuity of pressure:
p1+ ξ∂p01
∂n = p2+ ξ ∂p02
∂n , (7.1)
where ξ is the normal displacement of the vortex sheet. Note that the terms involving mean pressure gradients are present because the continuity of pressure for the total flow is imposed on a surface which oscillates with displacement ξ. Using Euler’s equation for the base flow we can write:
∂p0
7.2 Description of the Problem
Thus we see that the second terms on both sides of Equation (7.1) are related to the change in direction of the fluid elements along the vortex sheet.
The kinematic condition derived by Myers [110] is obtained by linearising the con- tinuity of total normal velocity along a moving interface:
v1· n = d01ξ
dt − ξn · [(n · ∇)v01] and v2· n = d02ξ
dt − ξn · [(n · ∇)v02] , (7.3) where the terms involving normal gradients account for the curvature of the vortex sheet. This condition can be understood to represent the continuity of the normal acoustic displacement across the vortex sheet [149].
As an alternative to (7.3), the continuity of normal acoustic velocity has sometimes been used. This can be written as:
v1· n = v2· n , or , ∂φ1 ∂n = ∂φ2 ∂n . (7.4) 7.2.2 Trailing Edge
The behaviour of the fluid at the trailing edge has a noticeable effect on the acoustic far field. The acoustic and hydrodynamic fields are coupled at the trailing edge, and, as a consequence, the amount of vorticity shed from the trailing edge and the behaviour of the sound field at that point are closely linked.
A basic physical property of the problem is that the streamline at the trailing edge is continuous, which amounts to requiring that the normal displacement of the vortex sheet at the trailing edge must be zero.
A stronger constraint is the Kutta condition, for which the normal acoustic velocity should also vanish at the trailing edge. Since we have v· n = d0ξ/dt the streamwise
gradient of the normal displacement at the trailing edge vanishes. The Kutta condition indicates that all the vorticity is shed from the trailing edge and, in that case, that the acoustic pressure vanishes with p∼√r near this point where r is the distance from the trailing edge. If the Kutta condition is not satisfied, the acoustic pressure is singular at the trailing edge and behaves like p∼ 1/√r.
This condition has been included in analytic models of exhaust duct problems. Rienstra [128] introduced a complex parameter which could be continuously varied to control the amount of vorticity shedding. Gabard & Astley [62] used this method when modelling the noise radiating from a jet pipe (the so-called Munt problem [109]).
7. EXHAUST NOISE PREDICTION y z θ r x n Ω1 Ω2 Γ v01 v02 Γ0
Figure 7.2: A schematic of a three-dimensional vortex sheet Γ emanating from the trailing
edge Γ0 of a nozzle.
They observed that vorticity shedding has no significant effect on the main lobe in far- field directivities, but did find that vorticity shedding noticeably affects the pressure amplitude in the rear arc.
An example of vortex shedding from a trailing edge, in the case of a uniform mean flow with an imposed Kutta condition, is given in Figure 7.3. Also included in the figure is a solution of the velocity potential field in the presence of non-uniform mean flow, but without any Kutta condition, note the refraction of the wave as it propagates the vortex sheet. This data has been obtained using the analytic solution which will be used to test the numerical predictions, yet to be presented.
7.2.3 Kelvin–Helmholtz Instability
The shear layer can exhibit Kelvin–Helmholtz instabilities. The onset of this hydrody- namic instability is characterized by the Strouhal number St = f δ/∆u, where f is the frequency, δ the shear layer thickness and ∆u is the velocity difference [89].
The instability wave will initially grow exponentially along the shear layer. If in- cluded, non-linear effects will saturate the growth of this instability. Even in a linear model, the gradual increase of the shear layer thickness will turn the instability wave into a decaying wave. When approximating the shear layer by a vortex sheet model, it is important to recognise that the vortex sheet model is unstable at any frequency since we take δ→ 0 [48]. Furthermore, since the growth of the shear layer thickness is