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We define the far field as being a distance, r, far enough away from the aerofoil so that the mean flow is approximately uniform, and amplitude terms of O(1/r) are negligible compared to terms retained in the asymptotic solution. Since there is no Rayleigh distance for this gust-aerofoil problem, we choose r = 25 in all following results to illustrate the far-field behaviour.

In Figure 3.2 we consider the effect of altering the strength of the mean shear, char-acterised by the parameter S = (M∞−M₀)/M∞, on the leading-edge directivity. Here we set = 0, so that the aerofoil reduces to a flat plate, and plot the quantity |D_l(θ)|

as defined in equation (3.4.0.11). When S = 0 the directivity pattern takes the famil-iar form cosθ/2, which is characteristic of low Mach number uniform flow gust-aerofoil interaction. Varying S away from zero has a significant effect; when the shear is jet-like (S > 0) the directivity is particularly reduced in the downstream direction, with little effect upstream, whereas for wake-like shear (S < 0) the directivity is reduced predominantly upstream. Mathematically, these directivity effects may be coming from two places; first, in (3.3.2.5) through the terms dependent on β₀; and second, from the ray-tube area scaling factors(r, θ) in (3.4.0.2). We have investigated the relative effects of both sets of terms, and have found that the directivity variations seen in Figure 3.2 are arising primarily from the second effect of the shear increasing the ray tube area (and therefore decreasing the pressure amplitude along the ray) in the downstream/upstream directions for S positive and negative respectively. See Figure 3.3.

In Figure 3.4 the effects of aerofoil thickness on the leading-edge directivity are con-sidered. In significant shear, S = 0.333, we see that increasing the thickness from zero changes the nature of the directivity. The leading-edge sound is made up of two contri-butions; the leading-order term corresponding to flat-plate scattering, see (3.3.2.5), and an additive correction term of relative sizeO(√

k)to account for the effects of thickness in the leading-edge region, see (3.3.2.17). The interference between these two sources in shear gives rise to the lobular directivity pattern seen in Figure 3.4. Note that the contribution from (3.3.2.17) takes the same form as the contribution from (3.3.2.5), but with a multiplicative correction factor which involves several effects (see the brief discus-sion following (3.3.2.17)). However, the variation with observer angle, θ, seen in Figure 3.4 can only arise from the term in this correction factor involving d(kβ₀, kψ), which in turn arises from the term on the right hand side of equation (3.1.0.9a), i.e. from the interaction between the leading-order scattered field and the non-uniformity of the mean shear flow. We stress that the interference within the leading-edge field is only present in shear flow.

We must now choose an upstream vorticity distribution in order to obtain quantitative results for the trailing-edge noise. We therefore choose Ω(η)˜ ∼ e^−kη2. In Figure 3.5 we see the relative effects of the two interactions generating sound at the trailing edge of the aerofoil. The sound generated by the rescattering of the leading-edge field by the trailing edge, given by k^−1/2D_t2 in (3.6.1.7), is familiar as it is the sole noise from the trailing edge in uniform flow gust-aerofoil interaction. The scattering of the pressure associated

-0.001 0.001 0.002 0.003 0.004 0.005 0.001

0.002 0.003

0.004 S=0.333

S=0.167 S=0.033 S=0.000

(a) PositiveS, withM∞= 0.3.

-0.001 0.001 0.002 0.003 0.004 0.005 0.001

0.002 0.003 0.004

S=-0.333 S=-0.167 S=-0.033 S=0.000

(b) NegativeS, withM0 = 0.3.

Figure 3.2: Leading-edge far-field acoustic pressure directivity for a flat plate, with k = 10 and varying strengths of shear, S= (M∞−M₀)/M∞.

-1.5 -1.0 -0.5 0.5 1.0 1.5

0.5 1.0 1.5

Figure 3.3: Magnitude of the scaling factor, s, for various shear flow profiles; dotted -M₀ = 0.3, M∞ = 0.2, solid - M₀ = M∞ = 0.3, dashed - M₀ = 0.2, M∞ = 0.3, at a distance r= 25.

-0.001 0.001 0.002 0.003 0.001

0.002 0.003 0.004

Ε=0.2 Ε=0.1 Ε=0

Figure 3.4: Leading-edge far-field acoustic pressure directivity with k = 10, M₀ = 0.2, M∞= 0.3, and S = 0.33, for Joukowski aerofoils of varying thickness, denoted by .

-0.0005 0.0005 0.0010

0.0002 0.0004 0.0006 0.0008 0.0010

Figure 3.5: Trailing-edge far-field pressure directivity with S = 0.333, k = 10, = 0.1.

The solid line denotes the total trailing-edge directivity. The dashed line denotes the contribution from the rescattering of the leading-edge field by the trailing edge, and the dotted line denotes scattering by the trailing-edge of pressure associated with the gust.

with the gust,MD_t1 from (3.6.1.7), is peculiar to shear flow interactions, and as we see in Figure 3.5 has a non-negligible effect on the total trailing-edge contribution upstream of the aerofoil. For our chosen vorticity distribution,MD_t1 is formallyO(k^−1/4)smaller than k^−1/2D_t2, which is most noticeable in the upstream region of Figure 3.5. Downstream of the aerofoil, the size difference is increased, since this scattering of pressure (the dotted line) takes a sinθ/2 type directivity pattern. At the trailing edge, as with the leading edge, there are two contributions to the acoustic solution; one from the rescattering of the leading-edge field, and one from the interaction of the unsteady self-noise with the perturbed steady flow. However, at the trailing edge, the perturbation to the steady mean flow is smaller than at the leading edge by a factor ofk^−1/2, hence the interaction of these two sources should be weaker than the interaction of the two leading-edge sources.

Furthermore, by considering Figure 3.5, we see that the contribution to the trailing-edge field due to the scattering of gust self-noise is much smaller than the contribution due to the rescattering of the leading-edge field. We therefore do not observe a modulated trailing-edge field despite (as with the leading-edge field) there being multiple types of interaction occurring.

We now consider the total scattered acoustic pressure as the sum of leading- and trailing-edge fields. In Figure 3.6 we consider the far-field pressure directivity in the two very low shear cases S =±0.03 for the flat plate, = 0. The significant modulation of the directivity is now caused by the interference between the leading- and trailing-edge fields, and is of course absent in the comparable plots of just the leading-edge flat-plate field (see Figure 3.2). We repeat these flat-plate calculations in Figure 3.7, but now with significant shear, and similar directivity patterns are again observed. Note that positive shear significantly increases/decreases the sound level in the upstream/downstream di-rections respectively, and vice versa for negative shear. This effect cannot be explained by simple ray tracing arguments, which would suggest that rays in positive/negative shear flow would tend to bend in the direction of decreasing/increasing θ (see Amiet (1978)). Rather, the refraction experienced by the leading- and trailing-edge fields is the same as they propagate through the shear, therefore the pattern seen in Figure 3.7 is influenced by the interaction of the two fields via their phase shift, %_s, and their am-plitudes (which both contain the scaling term, s, yielding the patterns shown in Figure 3.3). The refraction experienced by rays from the leading and trailing edges is contained within the overall phase function multiplying the total far field solution, (3.7.0.2), thus a directivity pattern does not pick out these effects.

In Figure 3.8 we introduce non-zero thickness to the finite-chord total far-field pres-sure results. We observe that increasing the thickness predominantly alters the upstream

-0.001 0.001 0.002 0.003 0.004 0.005 0.001

0.002 0.003

0.004 S=0.033

S=-0.034

Figure 3.6: Far-field pressure directivity at r = 25 in almost uniform Gaussian shear flow withM ≈0.3,k = 10, and = 0.

-0.001 0.001 0.002 0.003 0.004 0.001

0.002 0.003

0.004 S=0.333

S=-0.500

Figure 3.7: Far-field pressure directivity at r = 25 for jet-like and wake-like shear flows, with k= 10, and = 0.

-0.002 -0.001 0.001 0.002 0.003 0.004

0.001 0.002 0.003 0.004

Ε=0.2 Ε=0

A B

C D

Figure 3.8: Far-field pressure directivity at r = 25 with M₀ = 0.2, M_∞ = 0.3, k = 10, and varying thickness.

-0.001 0.001 0.002 0.003 0.004 0.001

0.002 0.003 0.004

S=0.000 S=0.200 S=0.333

Figure 3.9: Far-field pressure directivity at r = 25 with M₀ = 0.2, k = 10, = 0.1 and varying S.

far-field pressure directivity, and one of these effects is to make the lobes more uneven.

For non-zero thickness we see that the lobes labelled by A, B, C, and D have markedly different amplitudes (notably the lobes A and C are very large but B and D are less pro-nounced). For zero thickness, each lobe has a roughly equal amplitude in the upstream region. This uneven lobe pattern arises because of the leading-edge interaction seen in Figure 3.4 being superimposed on the interaction between the leading- and trailing-edge ray fields. Even-sized lobes are produced by leading-trailing interaction, as seen in Figure 3.7, but on introducing leading-edge interaction by allowing for non-zero thickness, the overall pattern of the total directivity field becomes modulated by an oscillating leading-edge field, thus allowing the uneven lobes to appear. This characteristic of the far-field pressure directivity for a finite length aerofoil of non-zero thickness is peculiar to shear flow interactions, as it is the interaction of the two leading-edge sources that creates it. Another upstream effect we see in Figure 3.8 due to variations in thickness is the positioning of the lobes and overall magnitude of the pressure. We see little difference in the field downstream of the aerofoil. We expect to see the most difference to the far-field pressure in the upstream region as we vary thickness because it is the blocking of the horizontal gust velocities, and scattering of gust self-noise by the nose of the aerofoil that is a new important source of noise for aerofoils of non-zero thickness.

In Figure 3.9 we show the effects of altering the shear strength,S, whilst maintaining a fixed value of M₀. It is important we mention that for this figure, M₀ is fixed, because

-0.002 0.002 0.004 0.006 0.008 0.010 0.002

0.004 0.006 0.008 0.010

Figure 3.10: Far-field pressure directivity for gust-aerofoil interaction in uniform flow withM = 0.6around a Joukowski aerofoil of thickness0.1, and frequency (as defined in Chapter 2)10.

we then know to attribute any variation in lobe position to the variation inM∞. This is sensible, as it is M∞ that has a dominant effect in the phase shift between the leading-and trailing-edge ray fields which therefore governs the leading-trailing interaction in the far field hence the positioning of the lobes. We see that as the strength of the shear is decreased towards zero, the overall shape of the directivity pattern reduces to a familiar pattern seen in uniform flow for purely transverse gusts, as illustrated by Figures 3.10, and 2.12 from Chapter 2. AsS increases away from0, the overall shape of the directivity pattern distorts in accordance to the ray tube scaling term,s, as seen in Figure 3.3. Once again, for significant shear and non-zero thickness, we see uneven lobes in Figure 3.9 due to the leading-edge interaction combining with the leading-trailing interaction.