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We now consider the effect of homogeneous weak turbulence from upstream incident on the stagnation point of the ellipse (see Goldstein (1978b) and Hunt (1973) for early work on this topic). We assume the three dimensional upstream turbulent spectrum, Φ^(∞)_i,j (k₁, k₂, k₃), is known. The one dimensional non-dimensionalised “ν, µ” turbulent

spectrum is defined by

Θ_ν,µ(x, k₁) = 1 2π

Z ∞

−∞

R_ν,µ(x, τ)e^ik1τdτ, (4.3.0.1)

whereν, µ ={r, θ, z} and

R_ν,µ(x, τ) = u_ν(x, t)u_µ(x, t+τ) (4.3.0.2) is the one-point turbulent velocity correlation tensor, where the long over-bar denotes time averaging. Θ_ν,µ relates to the known upstream turbulent spectrum via

Θ_ν,µ(x, k₁) = Z ∞

−∞

Z ∞

−∞

M¯_ν,jM_µ,nΦ^(∞)_j,n (k)dk₂dk₃, (4.3.0.3)

where here the short over-bar denotes complex conjugation, and the M_ν,j are given by u_ν =A_jM_ν,j(r, θ)e^{ik1(∆−t)+ik2Ψ+ik3z} for ν =r, θ, z. (4.3.0.4) We assume the turbulence is isotropic and homogeneous upstream, so choose the von Kármán spectrum, as done by Goldstein (1978b),

Φ^(∞)_j,n =α (k²₁+k₂²+k²₃)δjn−kjkn

(g₂/l²+k₁²+k₂²+k₃²)^17/6, (4.3.0.5) where

α= 55g1u²_∞

36πl^2/3, (4.3.0.6)

theg_i are constants to be determined from experimental data, andl denotes the integral lengthscale of the turbulence. The far upstream magnitude of the vortical perturba-tion is given by u_∞. This von Kármán spectrum is chosen because it agrees with the Kolmogoroff −5/3 law (Kolmogorov, 1941) which states that E(k) ∼ e^2/3k^−5/3, where E(k) = 2πΦ_ii(k)k² is the energy density (per unit mass) of the turbulence, and e is the rate of energy dissipation per unit mass. Commonly this law is just quoted as E(k) ∼ k^−5/3 as the dissipation is either unknown or not of interest. See Durbin &

Pettersson Reif (2001) for further details.

We define the one-dimensional turbulent pressure spectrum as Θ_pp(x, k₁) =

Z ∞

−∞

Z ∞

−∞

N¯_iN_jΦ^(∞)_i,j (k)dk₂dk₃, (4.3.0.7)

where the N_i are defined by p

ρ =N_j(r, θ)A_je^{ik1(∆−t)+ik2Ψ+ik3z}, (4.3.0.8) and the unsteady pressure can be found using

p=−ρ₀D₀φ

Dt . (4.3.0.9)

We present expressions for Θ_pp in Appendix B.

4.3.1 High-Frequency Pressure Spectra in the Far Field

We can extend our solution for Θ_pp into the far field, for θ = O(1), using (4.2.1.14).

For the ellipse, in the far field, the homogeneous solution is negligible, hence n_i → 0 for i = 2,3,4,5 (where the n_i are defined in Appendix B). Also φ^p = O(_k3/2¹√

r), so C =O(^√1_r)(Appendix B). Substituting this into (B.1) gives

Θ_pp∼ 2g₁u²_∞√ π 3r

Γ ¹₃

Γ ⁵₆cosθsin⁴θl²M

( g₂^−4/3 k₁l1

(k₁l)^−8/3 k₁l1 . (4.3.1.1) Hence for k₁l 1, Θ_pp =O(l²M r⁻¹) and for k₁l1, Θ_pp=O(k^−8/3₁ l^−2/3M r⁻¹).

We construct a composite function to combine the high and low reduced frequency results for Θ_pp given by (4.3.1.1). This yields

Θ_pp ∼ 2u²_∞√

πg₁l²MΓ ¹₃ 3rΓ ⁵₆

g₂^−4/3cosθsin⁴θ

1 + ^(k_g¹4^l)8 2

1/3 = const

r h(θ, k₁l). (4.3.1.2) We plot the functional behaviour of the far-field turbulent pressure spectrum in Figure 4.3, which we discuss later in Section 4.4.

4.3.2 High- and Low-Frequency Surface Pressure Spectrum

Low-Frequency

In the low-frequency regime, k 1, the scattered pressure can be written as p=−ρ0e^ik1t+ik3z

−ik1φ¯+Ur

∂φ¯

∂r + U_θ r

∂φ¯

∂θ

, (4.3.2.1)

whereφ¯= ¯φ^h+ ¯φ^p. Evaluating each term in the small−θ approximation gives

p^p =O(1−b²), p^h =O(1), (4.3.2.2) where p^p,h are the contributions to the pressure due to the particular and homogeneous solutions respectively. Thus, for1−b 1, the homogeneous solution dominates the pres-sure on the surface of the ellipse near the nose. This agrees with the high-frequency limit, where the same result was found. This is a result of both high- and low-frequency theo-ries because the particular solutions arise from source terms in the governing equations, which are generated by significant changes in momentum associated with the convected disturbance. It is sensible to assume that close to the boundary, in a thin wedge around the nose of the ellipse, the variation in momentum gradients is small, hence the volume sources make little contribution to the overall sound generated. The direct interaction between the boundary and the gust therefore generates the majority of the sound.

We write

p≈A₁N₁e^ik1t+ik3z, (4.3.2.3) where N₁ is independent of k_2,3 and is O(1) in the low-frequency regime. Hence using (4.3.0.7) we find that

Θ_pp∼ |N₁|²u²_∞g₁g^−5/6₂ , (4.3.2.4) which is independent of k₁, and is O(1) for θ,(1−b²) = O(k^1/2) (these scalings of the small angle and thickness are consistent with the asymptotic analysis throughout when k 1). Since the solution does not depend on the thickness of the ellipse at leading order then we are safe to indeed assume that the wavelength of the gust is so large that is does not see an elliptic body in the steady flow, but just a flat plate, and the pressure spectrum is dominated by vertical blocking of the gust velocity by the solid surface. To compute the coefficient N₁ we therefore look to Mish (2001), who considers spectra for a NACA 0015 aerofoil, based on the work by Amiet (1975) for a flat plate, and writes the pressure jump across a flat plate as

∆p^∗(x, y, t) = 2πρ^∗₀U^∗b^∗w^∗₀g(x, k₁, k₃)e^{ik∗3z∗−k∗1U∗t∗} (4.3.2.5) in dimensional form. Here g is the transfer function between turbulent velocity and aerofoil pressure jump, and w₀^∗ is the magnitude of the gust velocity. We can therefore

write that the cross power spectral density at a given point is S_qq(x, x,0, ω^∗) = 4(2πρ^∗₀b^∗)²

Z ∞

−∞

Z ∞

−∞

|g(x,−ω/U, k₃)|²Φ^(∞)_1,1 (−ω^∗/U^∗, k₂^∗, k^∗₃)dk^∗₂dk^∗₃, (4.3.2.6) where ω^∗ is the circular frequency, U^∗ is the steady free-stream velocity, and b^∗ is the semi-chord length of the flat plate. Mish (2001) provides us with the transfer function for a thin aerofoil, which, in our limit of small k₁^∗ and low Mach number, is

g(x, k₁, k₃) = −

1− rx

2

1−erfhp

2(2−x)k₃i

e^−ik3x π^3/2√

x√

k₃+ik₁. (4.3.2.7) Equation (4.3.2.6) can then be evaluated numerically. We see that (4.3.2.7) is dependent on k₁ thus we would naively expect our expression given in (4.3.2.4) to be incorrect, and Θ_pp should be dependent on k₁ in the low-frequency limit. However, we must now recall that whilst the overall scaling on our asymptotic estimate of Θpp is independent explicitly ofk₁, this only arises through cancellation of orders of magnitude ofk₁ with the thickness factor (1−b²), thus to compute N₁ to agree with experimental data, we must evaluate (4.3.2.6) for the value of small k₁ associated to the thickness of the aerofoil in the experiment. This then gives us a low-frequency asymptotic estimate for the turbulent pressure spectrum.

We evaluate (4.3.2.6) close to the nose of the aerofoil, at x = 0.01 say, and for a circular frequency of 2.31, steady free-stream velocity of 30, semi-chord length of 0.3, and integral lengthscale of turbulence (measured close to the surface) 0.0078, to find

S_qq ≈2.06. (4.3.2.8)

We choose the positionx, integral lengthscale,l, and free-stream velocityU in this way to agree with the experimental set up of Mish & Devenport (2003). The circular frequency is chosen to concur with our relation between thickness of the aerofoil and frequency of the gust (which in the low-frequency limit is (1−b²) =O(k^1/2)). We do not consider a high reduced frequency limit in this case because it would be unphysical for turbulence to have such a large integral lengthscale to allow k₁ 1 but k₁l 1.

High-Frequency

For small θ and high frequency, k₁ 1, we consider the turbulent spectrum on the nose of the ellipse by considering the limiting behaviour of (B.1) in Appendix B. It is expected that this depends on the exact location of the stagnation point, and hence

0.2 0.5 1.0 2.0 5.0 10.0k₁l 10^-6

10^-5 10^-4 0.001 0.01

hHΘ,k₁lL

Θ=А30 Θ=А9 Θ=А8

Figure 4.3: Plot of h(θ, k₁l) governing the turbulent pressure spectrum far from the ellipse, for varying values of θ. The constant,g₂ is chosen to be 0.6.

depends explicitly on the thickness parameterb, unlike in the far field where the solution decays exponentially and this dependence is negligible (as previously seen in the high-frequency limit in the far field). At intermediate distances, solutions for both small and O(1)values ofθ depend on thickness for the high- and low-frequency limits. Indeed from Appendix B we find in the high-frequency limit,

Θ_pp ∼ g₁u²_∞√ πΓ ¹₃

l²M 3Γ ⁵₆

(1−b²)²

( g₂^−4/3 k₁l1

(k₁l)^−8/3 k₁l1 , (4.3.2.9) soΘ_pp=O(k₁l²M)fork₁l 1and=O(k^−5/3₁ l^−2/3M)fork₁l1when we take account of the scaling of the thickness. A composite function can be constructed similarly for the far-field high-frequency solution.