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Research Cite this article: Ghorbaniasl G, Siozos-Rousoulis L, Lacor C. 2016 A time-domain Kirchhoff formula for the convective acoustic wave equation. Proc. R. Soc. A 472: 20150689. http://dx.doi.org/10.1098/rspa.2015.0689 Received: 1 October 2015 Accepted: 18 February 2016
Subject Areas: mechanical engineering Keywords: aeroacoustics, Kirchhoff formula, moving medium, convective wave equation Author for correspondence: Ghader Ghorbaniasl e-mail: [email protected]
A time-domain Kirchhoff formula for the convective acoustic wave equation Ghader Ghorbaniasl1,2 , Leonidas Siozos-Rousoulis1 and Chris Lacor1 1 Vrije Universiteit Brussel (VUB), Pleinlaan 2, Brussels 1050, Belgium 2 University of Tehran, Tehran, Iran
Kirchhoff’s integral method allows propagated sound to be predicted, based on the pressure and its derivatives in time and space obtained on a data surface located in the linear flow region. Kirchhoff’s formula for noise prediction from high-speed rotors and propellers suffers from the limitation of the observer located in uniform flow, thus requiring an extension to arbitrarily moving media. This paper presents a Kirchhoff formulation for moving surfaces in a uniform moving medium of arbitrary configuration. First, the convective wave equation is derived in a moving frame, based on the generalized functions theory. The Kirchhoff formula is then obtained for moving surfaces in the time domain. The formula has a similar form to the Kirchhoff formulation for moving surfaces of Farassat and Myers, with the presence of additional terms owing to the moving medium effect. The equation explicitly accounts for the influence of mean flow and angle of attack on the radiated noise. The formula is verified by analytical cases of a monopole source located in a moving medium.
1. Introduction The Kirchhoff formula was suggested by Gustav Kirchhoff in 1883 [1] and was primarily used in the theory of light diffraction and other electromagnetic problems. It is one of the most useful results in wave propagation theory and particularly in acoustics. As an integral method applicable to noise prediction problems in modern computational aeroacoustics (CAA), the Kirchhoff method allows flow-generated noise to be propagated and predicted based on the pressure and its derivatives in time and space, on an arbitrary control surface (data surface). The linear wave equation 2016 The Author(s) Published by the Royal Society. All rights reserved.
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is assumed to be valid outside the data surface. The data surface should thus be placed in the nonlinear flow region, including all nonlinear flow effects. Acoustic pressures predicted by the Kirchhoff formula can be substantially in error if the Kirchhoff surface is located inside the nonlinear region. This is the main shortcoming of the method, compared with the Ffowcs Williams & Hawkings (FW-H) [2] equation, which may predict noise emissions fairly accurately, even when the integration surface is close to the noise-generating surface, thus neglecting external quadrupoles. The two methods are therefore equivalent when the Kirchhoff surface is placed in the linear flow region [3]. On the other hand, the FW-H equation is based on the conservation laws of fluid mechanics rather than the wave equation, which is the case for the Kirchhoff formula. Consequently, the FW-H equation is not appropriate for all types of wave propagation. In practical CAA problems, the Kirchhoff approach finds wide application to aerodynamic noise generation involving high-lift devices [4,5], rotors [3,6–9] and jet flows [10–14]. Initially, the Kirchhoff formula was limited to a stationary data surface [15]. In 1930, the original Kirchhoff formula was extended to moving surface applications by Morgans [16], using the Green’s function approach. Morino also used the Green’s function method and developed a time-domain and frequency-domain formulation for a uniformly translating surface [17,18], which he then extended to a rotating surface [19]. This approach however was significantly cumbersome and complicated. In order to achieve a simpler derivation of the Kirchhoff formula, Farassat & Myers in 1988 [20] suggested a fast and modern method to obtain Morgans’ result by using generalized functions theory in the time domain. They also provided an extension of this formula to moving surfaces, which has been widely used for high-speed helicopter rotor noise prediction. More recently, Farassat & Posey [21] presented a faster and simpler approach for the derivation of the aforementioned Kirchhoff formulation for moving surfaces [20]. An extensive review of the use of Kirchhoff’s method in CAA and its applications was provided by Lyrintzis [22]. In the presence of uniform constant flow, such as a typical wind tunnel case, all the aforementioned formulations require application of the moving observer method to implicitly include effects of uniform constant velocity on emitted noise. The terms associated with the mean flow velocity effects are not present in the formulations and, thus, the observer must be put in motion along with the data surface, with the velocity of the flow but in the opposite direction. The same problem, however, can be alternatively solved by the moving medium approach. The main advantage of moving medium methods is that terms related to incidence are explicitly included in the acoustic sources and acoustic propagation formulae, thus showing the effects of incidence on the Doppler amplification of radiated noise. In addition, it provides a way of examining the physics of the problem, while processing asymmetric inflow effects. When rotating noise sources such as propeller blades are concerned, the effects of mean flow incidence can be equivalent to an unsteady acoustic source, resulting in generation of more effective radiation modes [23,24]. Recently, Lee & Lee [25] proposed a convective Kirchhoff formula for boundary integral problems, by considering only streamwise flow motion. The Kirchhoff approach has thus not been implemented for aerodynamic noise prediction for sources in arbitrary motion located in a moving medium with an arbitrary orientation. The contribution of this paper is the development of a convected Kirchhoff formulation for moving surfaces that explicitly takes into account the effects of mean flow and angle of incidence on radiated noise. The formulation will be derived in the time domain for moving surfaces, with respect to a moving medium frame and will be an extension of Farassat & Myers’ formula [20] to moving medium problems. The derivation method of the formulation will follow the fast and modern procedure depicted by Farassat & Posey [21], by using generalized functions theory [26,27]. This formulation consists of the partial derivatives with respect to the source coordinates and time, thus making the formulation easy to use when studied numerically. The extension of the formula to moving medium problems will be realized according to the procedure described by Ghorbaniasl & Lacor [24], who suggested a moving medium solution of the FW-H equation [2]. A formula with the Doppler factor will be developed because it is more easily interpreted and is more helpful in
3
ˆn = —f
M• mean flow velocity
f (x, t) < 0 j˜ (x, t) = 0
v data surface velocity
f (x, t) > 0 j˜ (x, t) = j (x, t)
Figure 1. Schematic of the Kirchhoff data surface at time t.
Table 1. Nomenclature. ..........................................................................................................................................................................................................
∇
gradient operator
c0
speed of sound
f (x, t)
function describing the data surface
g
= τ − t + R/c0 , retarded time function
G
Green’s function
Mi = vi /c0 ,
local Mach number vector components
Mn = vn /c0 ,
source Mach number along data surface normal direction
MR
source Mach number in acoustic radius R-direction
M
source Mach number in acoustic radius R∗ -direction
M∞i = U∞i /c0 ,
local mean flow Mach number vector components
M∞n = Un /c0 ,
mean flow Mach number along data surface normal direction
M∞R
mean flow Mach number in acoustic radius R-direction
M
mean flow Mach number in acoustic radius R∗ -direction
.......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... ..........................................................................................................................................................................................................
R∗ .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... ..........................................................................................................................................................................................................
∞R∗ ..........................................................................................................................................................................................................
nˆ i
unit normal vector components
r
distance between observer position xi and source position yi
ˆri R˜ i , R˜i∗
= (xi − yi )/r
dS
element of data surface
t
observer time
.......................................................................................................................................................................................................... .......................................................................................................................................................................................................... ..........................................................................................................................................................................................................
gradients of acoustic radii R and R∗
.......................................................................................................................................................................................................... .......................................................................................................................................................................................................... ..........................................................................................................................................................................................................
δij
Kronecker delta
..........................................................................................................................................................................................................
θ, θ
∗
angles between normal direction and acoustic radii directions
..........................................................................................................................................................................................................
τ
retarded time or emission time
..........................................................................................................................................................................................................
Φ acoustic function outside data surface .......................................................................................................................................................................................................... Φ˜ acoustic function inside and outside data surface ..........................................................................................................................................................................................................
Φn
normal derivative of acoustic function, nˆ i ∂Φ/∂xi
Φt
observer time derivative of acoustic function, ∂Φ/∂t
Φ
= U ∂Φ/∂x
.......................................................................................................................................................................................................... ..........................................................................................................................................................................................................
U∞ ∞i i ..........................................................................................................................................................................................................
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a
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Kirchhoff data surface
body
(a) Derivation of the equation Let us assume that f (x, t) = 0 (table 1) is the moving Kirchhoff data surface with velocity v defined in such a way that the unit outward normal is nˆ = ∇f (figure 1). If this condition is not satisfied by the original definition of f , then one can redefine it as f /|∇f |, which does satisfy this condition on the surface. Furthermore, the definition of the unit normal in terms of the function defining the surface relies on a normalization of f , which is always possible. The aim is to show how to derive the Kirchhoff formula for acoustic wave prediction radiated from sources on the data surface in a moving frame, using generalized functions. Assume the medium moves in an arbitrary direction with a constant velocity of U ∞ . Then the convective wave equation can be expressed as 1 D2 Φ(x, t) − ∇ 2 Φ(x, t) = 0, Dt2 c20
(2.1)
where D/Dt = ∂/∂t + U∞i ∂/∂xi with xi being the ith component of x. Equation (2.1) is valid for the region outside the data surface f (x, t) = 0. In order to extend the domain of the problem to inside ˜ t) as follows: the surface, one defines the function Φ(x, Φ(x, t) f > 0 ˜ Φ(x, t) = (2.2) 0 f < 0. Because Kirchhoff’s formula [1] is generally valid for wave propagation in a homogeneous ˜ t) is used in this paper. However, when the suggested approach is applied medium, function Φ(x, ˜ t) will be equivalent to the acoustic pressure, p (x, t). to aeroacoustic problems, Φ(x, The extended governing equation is given by ˜ t) 1 D2 Φ(x, ˜ t) = 0. − ∇ 2 Φ(x, 2 2 Dt c0
(2.3)
˜ t) is discontinuous, the ordinary derivatives in equation (2.3) should be Because the function Φ(x, replaced by generalized derivatives [30]. In general and according to Farassat & Myers [27], if the function Φ˜ has a discontinuity across the surface f = 0, then the jump is defined as ˜ f = 0− ). ˜ f = 0+ ) − Φ( Φ˜ = Φ(
(2.4)
When Φ˜ is multiplied by the Dirac delta function δ( f ), by using the definition of equation (2.2), one has ˜ f = 0+ ) = Φ. Φ˜ = Φ(
(2.5)
According to Farassat [26], generalized spatial derivatives will be obtained by ∂ Φ˜ ∂f ∂ Φ˜ ∂¯ Φ˜ = + Φ˜ δ( f ) = + Φ nˆ i δ( f ), ∂xi ∂xi ∂xi ∂xi
(2.6)
...................................................
2. Governing equation in a moving medium
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examining the physics of systems. For subsonic cases, the present formulation explicitly shows the mean flow effect and its physical meaning. The formulation will be verified through two analytical subsonic test cases for stationary and moving Kirchhoff data surfaces. As already mentioned, supersonic jet flows is an area of significant application potential for the Kirchhoff approach. The presented technique may thus be used to derive a formula which contains no Doppler factor, enabling fast prediction of noise in supersonic cases, as depicted by Wells & Han [28]. Another approach without the Doppler factor and its mathematical explanation can be found in [29], leading to formulation 4 of Farassat.
ˆ In equation (2.6), a bar over the derivative operator is where nˆ i is the ith component of n. consistently used to denote generalized differentiation. From equation (2.6), one can also derive
where Φn = nˆ · ∇Φ. ¯ Similarly, generalized time derivatives D/Dt can be expressed in terms of ordinary time derivatives D/Dt as follows:
On the other hand
¯ Φ˜ DΦ˜ Df DΦ˜ Df D = + Φ˜ δ( f ) = +Φ δ( f ). Dt Dt Dt Dt Dt
(2.8)
∂f ∂f Df = + U∞i = −νn + U∞n , Dt ∂t ∂xi
(2.9)
where U∞n = nˆ · U ∞ and vn = nˆ · v. By substituting (2.9) into (2.8), one has ¯ Φ˜ DΦ˜ D = − Φ(νn − U∞n )δ( f ). Dt Dt Then ¯ 2 Φ˜ D D = Dt Dt2
¯ Φ˜ D Dt
−
DΦ (νn − U∞n )δ( f ). Dt
(2.10)
(2.11)
Substituting equation (2.10) into equation (2.11) yields ¯ 2 Φ˜ D2 Φ˜ D DΦ D [Φ(νn − U∞n )δ( f )] − (νn − U∞n )δ( f ). = − 2 2 Dt Dt Dt Dt
(2.12)
Using equations (2.7) and (2.12), one can form the following expression: ¯ 2 Φ˜ 1 D2 Φ˜ 1 D 1 D [Φ(Mn − M∞n )δ( f )] − ∇ 2 Φ˜ − Φn δ( f ) − ∇¯ 2 Φ˜ = 2 − 2 2 2 c0 Dt c0 Dt c0 Dt −
∂ Mn − M∞n DΦ δ( f ) − [Φ nˆ i δ( f )], c0 Dt ∂xi
(2.13)
where Mn = vn /c0 is the local normal Mach number of the surface, and M∞n = U∞n /c0 is the normal Mach number of the moving frame. Substituting equation (2.3) into (2.13) gives the Kirchhoff formula for bodies in motion in a moving medium as follows: ¯ 2 Φ˜ Mn − M∞n DΦ 1 D 2 ˜ ¯ δ( f ) − ∇ Φ = − Φn + c0 Dt c20 Dt2 −
∂ 1 D [Φ(Mn − M∞n )δ( f )] − [Φ nˆ i δ( f )]. c0 Dt ∂xi
(2.14)
Using the definition of D/Dt, one has ¯ 2 Φ˜ 1 D ¯ 2 Φ˜ = − Φn + Mn − M∞n (Φt + ΦU∞ ) δ( f ) − ∇ c0 c20 Dt2 −
∂ 1 ∂ [Φ nˆ i + Φ(Mn − M∞n )M∞i δ( f )], [Φ(Mn − M∞n )δ( f )] − c0 ∂t ∂xi
(2.15)
where Φt = ∂Φ/∂t and ΦU∞ = U ∞ · ∇Φ. Note that these variables will be functions of y and τ , i.e. Φt = Φt (y, τ ) and ΦU∞ = ΦU∞ (y, τ ), in the same manner as other terms inside the integrals. As will be shown in appendix A, the derived convective wave equation (2.15) is consistent with the convective FW-H equation [24], when the data surface is located in the linear propagation region (thus when the input data are compatible with the wave equation) [3].
...................................................
(2.7)
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∂ [Φ nˆ i δ( f )] + Φn δ( f ), ∇¯ 2 Φ˜ = ∇ 2 Φ˜ + ∂xi
5
(b) Solution of the equation
6
δ(g) 4π R∗
(2.16)
for τ ≤ t, where g = τ − t + R/c0 . The quantities R∗ and R are defined as r 1 + γ 2 M2∞r R∗ = γ
(2.17)
and R = γ 2 (R∗ − rM∞r ),
(2.18)
where γ 2 = 1/(1 − M2∞ ) and M∞ r = M ∞ · rˆ with rˆ being the unit vector of the distance between the source position y(τ ) and the observer position x(t), i.e. r = |x(t) − y(τ )|. Based on this Green’s function, the solution of equation (2.15) is given by +∞
Mn − M∞n δ(g) Φn + (Φt + ΦU∞ ) δ( f ) ∗ dy dτ c R 0 −∞ −∞ t +∞ 1 ∂ δ(g) − Φ(Mn − M∞n )δ( f ) ∗ dy dτ c0 ∂t −∞ −∞ R t +∞ ∂ δ(g) − [Φ nˆ i + Φ(Mn − M∞n )M∞i ]δ( f ) ∗ dy dτ . ∂xi −∞ −∞ R
˜ t, M ∞ ) = − 4π Φ(x,
t
(2.19)
Let S: f = 0 be defined in a reference frame, called the η-frame, fixed to the data surface, where the variable y is transformed to the variable η. For the sake of simplicity, we assume that the surface is non-deformable. It can, though, be easily extended to deformable surfaces. Therefore, equation (2.19) leads to t Mn − M∞n δ(g) ˜ Φn + (Φt + ΦU∞ ) dS dτ 4π Φ(x, t, M ∞ ) = − c R∗ 0 −∞ S 1 ∂ t δ(g) − Φ(Mn − M∞n ) ∗ dS dτ c0 ∂t −∞ S R t ∂ δ(g) − [Φ nˆ i + Φ(Mn − M∞n )M∞i ] ∗ dS dτ . (2.20) ∂xi −∞ S R Considering that x and y are independent variables, equation (2.20) can be rewritten in the following form: ˜ t, M ∞ ) = − 4π Φ(x,
t
−∞ S
∂ − ∂t t −
t
Φn +
−∞ S
−∞ S
Φ
Mn − M∞n δ(g) (Φt + ΦU∞ ) dS dτ c0 R∗
Mn − M∞n δ(g) dS dτ c0 R∗
[Φ nˆ i + Φ(Mn − M∞n )M∞i ]
∂ ∂xi
δ(g) dS dτ . R∗
(2.21)
In order to convert the space derivative to a time derivative, one can use the following relation [24]: ˜ ∗ δ(g) ˜ i δ(g) R δ(g) 1 ∂ R ∂ i = − (2.22) − ∂xi R∗ c0 ∂t R∗ R∗2
...................................................
G(x, t, y, τ ) =
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The governing wave equation obtained for the Kirchhoff formula is valid in the open space. We can now solve this equation using the Green’s function for the wave equation in the open threedimensional space. One may use the Green’s function for a moving medium approach [24], i.e.
with
Therefore, equation (2.21) becomes t δ(g) Mn − M∞n ˜ Φn + Φt + ΦU∞ dS dτ 4π Φ(x, t, M ∞ ) = − c0 R∗ −∞ S 1 ∂ t δ(g) + [Φ cosθ − (1 − M∞R )Φ(Mn − M∞n )] ∗ dS dτ c0 ∂t −∞ S R t δ(g) + [Φ(Mn − M∞n )M∞R∗ + Φ cosθ ∗ ] ∗2 dS dτ , R −∞ S
(2.23)
(2.24)
˜ cosθ ∗ = nˆ · R ˜ ∗ , M∞R = M ∞ · R ˜ and M∞R∗ = M ∞ · R ˜ ∗. where cosθ = nˆ · R, The integration over dτ can be performed by using the methods of generalized functions theory [27]. It can be seen that [24] t h(τ ) h(τ )δ(g) dτ = , (2.25) 1 − MR e −∞ ˜ The subscript e denotes that quantities inside the bracket are calculated in the where MR = M · R. emission time τe = t − R/c0 . This relation can be applied to equation (2.24), leading to Φn + ((Mn − M∞n )/c0 )(Φt + ΦU∞ ) ˜ t, M ∞ ) = − dS 4π Φ(x, R∗ (1 − MR ) S e cosθ − (1 − M∞R )(Mn − M∞n ) 1 ∂ Φ + dS c0 ∂t S R∗ (1 − MR ) e cos θ ∗ + M∞R∗ (Mn − M∞n ) Φ + dS. (2.26) R∗2 (1 − MR ) S e Equation (2.26) is the general Kirchhoff formula for moving surfaces, described in a moving medium. The derived Kirchhoff formula requires Φ and its time derivative Φt and space derivative ∇Φ as inputs for the wave propagation. It explicitly takes into account the convection effect on the noise prediction. The formulation is equivalent to the Kirchhoff formula derived by Farassat & Myers [20] if the medium is set to be at rest, i.e. M ∞ = 0, where θ ∗ = θ and R∗ = R = r. In general, the Kirchhoff data surface S can deform. However, for the sake of simplicity, the derivative in this paper used a non-deformable data surface. In addition, in many applications of the Kirchhoff formula in aeroacoustics, the data surface is assumed to be rigid. The derived formulation is based on the retarded time solution and its stability will depend on the relative position of the source and observer, the data surface velocity and the time step when it is studied numerically. The above equation is a noise prediction formula based on the classical retarded time algorithms. This formula explicitly takes into account the mean flow influence and the aerodynamic and acoustic effects of incidence and is easy to implement in noise prediction codes. This formulation is derived with the Doppler factor because it is more easily interpreted and is more helpful in examining the physics of systems.
(c) An alternative form of the solution Writing the formulation in terms of geometry and available kinematic quantities makes the formulation practically useful. In addition, in order to improve the accuracy of the results, it is also essential that the time derivative in the integrand of the second term on the right-hand side
...................................................
and
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⎫ r 2 ⎪ ⎬ (ˆ r + γ M M ) ∞r i ∞i γ 2 R∗ ⎪ ⎭ ˜ ∗ − M∞i ). R˜ i = γ 2 (R i
R˜ ∗i =
⎫ ∂R ∂R∗ ⎪ ⎪ = −c0 MR∗ , = −c0 MR , ⎪ ⎪ ∂τ ∂τ ⎪ ⎪ ⎪ ⎪ ∗ ∗ 2 ⎬ ˜ ˜ ∗ −c M + c γ (M − M M ) R ∂R 0 0 R ∞M i ∞i i i = , ⎪ ∂τ γ 2 R∗ ⎪ ⎪ ⎪ ⎪ ⎪ ∗ ˜ ˜ ⎪ ∂ Ri ⎪ 2 ∂ Ri ⎭ =γ , ∂τ ∂τ
(2.28)
where M∞M = M · M ∞ . One thus has ˙R c0 (γ 2 M2R∗ − M2 ) − c0 γ 2 M2∞M 1 M ∂ c0 MR∗ = + + , ∂τ R∗ (1 − MR ) R∗ (1 − MR )2 R∗2 (1 − MR ) R∗2 (1 − MR )2
(2.29)
˙ R=M ˙ · R. ˜ It should be noted that the dot over a quantity denotes the source time where M derivative of the quantity. Let Q = Φ cosθ − (1 − M∞R )(Mn − M∞n )Φ.
(2.30)
By using the following relations ⎫ n˙ R 1 ∂ cosθ −Mn + γ 2 (MR∗ cosθ ∗ − M∞M M∞n ) ⎪ ⎪ = + , ⎪ ⎪ ⎪ c0 ∂τ c0 R∗ ⎪ ⎪ ⎪ ⎪ ⎪ ˙ 1 ∂Mn Mn + n˙ M ⎪ ⎪ ⎪ = , ⎬ c ∂τ c 0
0
1 ∂M∞R −M∞M + γ 2 (MR∗ M∞R∗ − M∞M M2∞ ) , = c0 ∂τ R∗ 1 ∂M∞n n˙ M∞ = , c0 ∂τ c0
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(2.31)
one can have 1 ˙ Φ˙ Q = [cos θ − (1 − M∞R )(Mn − M∞n )] c0 c0 n˙ R −Mn + γ 2 (MR∗ cosθ ∗ − M∞n M∞M ) + +Φ c0 R∗ −
Φ ˙ n + n˙ M − n˙ M∞ ) (1 − M∞R )(M c0
+
Φ(Mn − M∞n ) [−M∞M + γ 2 (MR∗ M∞R∗ − M∞M M2∞ )] R∗
(2.32)
or Φ˙ 1 ˙ Q = [cos θ − (1 − M∞R )(Mn − M∞n )] c0 c0 +
Φ ˙ n + n˙ M − n˙ M∞ )] [n˙ R − (1 − M∞R )(M c0
+
Φ {MR∗ [γ 2 cosθ ∗ + γ 2 M∞R∗ (Mn − M∞n )] − Mn (1 + γ 2 M∞M )}, R∗
˙ n=M ˜ M ˙ · n, ˆ n˙ M = n˙ˆ · M and n˙ M∞ = n˙ˆ · M ∞ . where n˙ R = n˙ˆ · R,
(2.33)
...................................................
along with the following relations [24]:
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of equation (2.26) can be evaluated analytically. As demonstrated by Farassat & Succi [31], this can be realized by using the rule 1 ∂ ∂ = (2.27) ∂t x e 1 − MR ∂τ x e
Mn − M∞n (Φt + ΦU∞ ) E1 = − Φn + c0 +
˙ RΦ M [cos θ − (1 − M∞R )(Mn − M∞n )] c0 (1 − MR )2
+
1 {[cos θ − (1 − M∞R )(Mn − M∞n )]Φ˙ c0 (1 − MR )
˙ n + n˙ M − n˙ M )]Φ} + [n˙ R − (1 − M∞R )(M ∞
(2.35)
and E2 = cosθ ∗ + M∞R∗ (Mn − M∞n ) +
1 {MR∗ [cosθ + γ 2 cosθ ∗ − (1 − M∞R − γ 2 M∞R∗ )(Mn − M∞n )] 1 − MR
− Mn (1 + γ 2 M∞M )} +
γ 2 (M2R∗ − M2∞M ) − M2 (1 − MR )2
[cos θ − (1 − M∞R )(Mn − M∞n )].
(2.36)
This is the convected Kirchhoff formula for moving data surfaces. It should be noted that, as we have considered in the derivation procedure, Φ˜ = 0 inside the data surface. The derivation of this formula was based on generalized functions theory, which offers great advantages compared with the classical derivation. It can be checked that this formulation leads to the Kirchhoff formula for non-deformable moving surfaces derived by Farassat & Myers [20], under no mean flow influence.
(i) Simplified form for a stationary data surface The simple case of a stationary data surface is suitable for practical applications and is commonly used in the hybrid computational fluid dynamics and acoustic analogy approach. The derived formulation (2.34) along with (2.35) and (2.36) can be simplified for stationary data surfaces as follows: E1 E2 Φ ˜ dS + dS, (2.37) 4π Φ(x, t, M ∞ ) = ∗ ∗2 S R e S R e where
M∞n 1 (Φt + ΦU∞ ) + [cosθ + M∞n (1 − M∞R )]Φ˙ E1 = − Φn − c0 c0
(2.38)
E2 = cosθ ∗ − M∞R∗ M∞n .
(2.39)
and
In this case, the retarded time can be obtained by an explicit solution without need for any iterative solution. This is due to the radius r not being dependent on the retarded time.
(ii) Simplified form for a stationary data surface and a medium at rest As can be seen, when M ∞ = 0 (then cosθ ∗ = cosθ and R∗ = R = r), equation (2.37) represents the classical Kirchhoff formula derived by Lyrintzis [22], for stationary surfaces for an observer fixed to a medium at rest.
...................................................
where
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Substituting equations (2.27), (2.29) and (2.33) into equation (2.26) gives the following formulation: E1 E2 Φ ˜ t, M ∞ ) = dS + dS, (2.34) 4π Φ(x, ∗ ∗2 S R (1 − MR ) e S R (1 − MR ) e
3. Numerical results
10
In order to verify the present formulations, the analytical solution of a three-dimensional monopole source in uniform constant flow has been used, as described by Lyrintzis & Mankbadi [10]. This case is equivalent to the problem of a monopole in motion with a constant velocity in the x-direction, located in an ambient quiescent fluid. The analytical solution for the acoustic pressure at the data surface resulting from the monopole [10] is given by ps =
robs + M∞ (xs − x) , sinω t − robs c0 β 2
where β= and
robs =
1
1 − M2∞
(3.1)
⎫ ⎪ ⎬
⎪ (xs − x)2 + β 2 (ys − y)2 + β 2 (zs − z)2 .⎭
(3.2)
In equation (3.1), ω is the angular frequency of the source, whereas the monopole source is located at (xs , ys , zs ) and the observer at (x, y, z). In this study a spherical Kirchhoff surface of radius rs = 0.1 m is used as the integration (data) surface (a two-dimensional cross section of the domain is depicted in figure 2). The data surface is discretized into 4095 points, well distributed on the spherical surface, with angle increments equal to 4◦ . This resolution for the data surface was considered to be fine enough. The ambient speed of sound and the density of the undisturbed medium are chosen as c0 = 1 m s−1 and ρ0 = 1 kg m−3 , respectively. The monopole source is located at the origin (0, 0, 0) and its angular frequency ω is 1 rad s−1 . For discretization in time, 256 time steps are used. The inputs required for the Kirchhoff formula, namely the pressure and its derivatives in time and space, are calculated from equation (3.1). The acoustic pressure fluctuations p obtained by the developed Kirchhoff formulation are plotted versus the analytical results for different observer locations. Three observers are positioned in the x−y plane (θ = 0◦ ), on a circle of radius robs = 10 m and at angle increments of α1 = 0◦ , α2 = 90◦ and α3 = 180◦ , as defined in a spherical coordinate system by x = robs (cosα, sinα cosθ, sinα sinθ).
(3.3)
Additionally, the noise directivities of the RMS values of the acoustic pressure predicted by the derived Kirchhoff formulations are plotted against the analytical directivities, for 36 observers equally distributed in the x−y plane on a circle of radius robs = 10 m from the origin. Two cases corresponding to mean flow Mach numbers of M∞ = 0.2 and M∞ = 0.8 are considered. The time histories of the acoustic pressure fluctuations corresponding to the considered cases are depicted in figures 3 and 4, respectively, whereas the directivities are shown in figures 5 and 6, respectively. In both cases, it is obvious that the numerical method well duplicates the analytical solution for the observer locations chosen. The excellent agreement between the numerical and analytical results proves that the formulation predicts the acoustic pressure accurately.
...................................................
(a) TC1: monopole in uniform constant flow
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Here, the suggested convected Kirchhoff formulation for moving surfaces is validated through two analytical test cases. The first test case is a stationary monopole in mean flow with a stationary data surface, whereas the second case consists of a monopole translating along with the data surface, in a moving medium. Descriptions of the test cases and the calculated results are given in §3a,b.
y
11
M•
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rs
a
o (xs, ys, zs)
x
Figure 2. Schematic of the spherical Kirchhoff data surface in the x−y plane.
p¢ (Pa)
(a)
(b) 0.15
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0.10
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0
0
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analytic
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0
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1.0
–0.20
0
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(c)
0.6
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t/T 0.15 0.10
p¢ (Pa)
0.05 0 –0.05 –0.10 –0.15 –0.20
0
0.2
0.4
0.6
0.8
1.0
t/T
Figure 3. The calculated acoustic pressure is identical to the analytical solution for the three observer positions of (a) α1 , (b) α2 and (c) α3 , at a mean flow Mach number of M∞ = 0.2. (Online version in colour.)
(b) TC2: monopole in uniform constant flow with a moving data surface In order to achieve validation of the developed formulation for cases involving a data surface in motion, a second test case was considered. Because, to the best of our knowledge, an analytical test case of a moving data surface is not available in the literature, the aforementioned threedimensional monopole was set in translating motion along with the Kirchhoff data surface at a constant velocity of M = 0.2, while located in a uniform constant flow of M∞ = 0.2. The noise prediction was realized by the moving medium formulation developed here. The input for the formulation, and thus the sources on the data surface, was calculated for a mean flow of M∞ = 0.2, using equation (3.1). After the noise sources on the data surface were obtained, it was set in translating motion with velocity M = 0.2, towards the direction of the mean flow.
(a)
(b) 0.15
12
0.2
p¢ (Pa)
0 0 –0.05 –0.10
–0.1 present
–0.15
analytic
–0.20
0
0.2
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–0.2
1.0
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0
0.2
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(c)
0.6
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1.0
t/T 0.15 0.10
p¢ (Pa)
0.05 0 –0.05 –0.10 –0.15 –0.20
0
0.2
0.4
0.6
0.8
1.0
t/T
Figure 4. The calculated acoustic pressure is identical to the analytical solution for the three observer positions of (a) α1 , (b) α2 and (c) α3 , at a mean flow Mach number of M∞ = 0.8. (Online version in colour.)
120
90 0.1 0.05
150
60 30 0
180 210
330 240
present analytic
270
Figure 5. The calculated directivity of the acoustic pressure is identical to the analytical solution for 36 observers equally distributed on a circle of radius 10 m from the origin, at a mean flow Mach number of M∞ = 0.2. (Online version in colour.)
The reference for comparison was obtained by computing the emitted noise for the exact same case, using the Kirchhoff formula derived by Farassat for a medium at rest [20], applied with a moving observer approach. The inputs used for the reference solution were the noise sources calculated on the data surface, as described above. The reference solution was then obtained by imposing translational velocity, equal to M = 0.2, on the source, whereas the source is also translating along with the observer with velocity M∞ = 0.2. Therefore, translation in a moving medium is achieved, rendering the reference case equivalent to the moving medium case.
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0.1
0.05
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0.10
present analytic
90 0.15 60 0.1
180
0
210
330 240
300 270
Figure 6. The calculated directivity of the acoustic pressure is identical to the analytical solution for 36 observers equally distributed on a circle of radius of 10 m from the origin, at a mean flow Mach number of M∞ = 0.8. (Online version in colour.)
(a)
(b) 0.15 0.10
p¢ (Pa)
0.15
present Farassat and Myers
0.10
0.05
0.05
0
0
–0.05
–0.05
–0.10
–0.10 0
0.2
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0
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t/T
(c)
0.6
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0.15
p¢ (Pa)
0.10 0.05 0 –0.05 –0.10
0
0.2
0.4
0.6
0.8
1.0
t/T
Figure 7. The calculated acoustic pressure is identical to one obtained from Farassat and Myers’ formula [20] with the moving observer method for the three observer positions of (a) α1 , (b) α2 and (c) α3 . The data surface is in motion at M = (0.2, 0, 0) and is located in a mean flow of M∞ = 0.2. (Online version in colour.)
The acoustic pressure time history obtained by the developed Kirchhoff formulation is plotted versus the results from the moving observer method for the three aforementioned observer locations. Moreover, noise directivity obtained by the moving medium and moving observer approaches were calculated and plotted for 36 observers equally distributed in the x−y plane on a circle of radius robs = 10 m from the origin. All required parameters, as well as the time and space discretization, are defined as described in §3a.
...................................................
30
0.05
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150
13
90 120
30
0
330 300
Figure 8. The calculated directivity of the acoustic pressure is identical to the solution obtained from Farassat and Myers’ formula [20] for 36 observers equally distributed on a circle of radius 10 m from the origin. The data surface is in motion at M = (0.2, 0, 0) and is located in a mean flow of M∞ = 0.2. (Online version in colour.)
The outcome of the comparison of the present formulation against the moving observer method is depicted in figures 7 and 8, for the acoustic pressure time history and directivity, respectively. Apparently, the results of the two methods are in excellent agreement. In this case, more significant than the actual input to the methods is the final outcome, which proves that both methods provide identical results, thus validating the developed formulation for a moving Kirchhoff data surface.
4. Conclusion A convected Kirchhoff formulation was suggested for arbitrarily moving surfaces in a uniform moving medium of arbitrary orientation. The convective wave equation was obtained using generalized functions theory, and the Kirchhoff formula was derived in the time domain with respect to a moving medium frame. The equation explicitly takes into account mean flow and incidence effects on the radiated noise and is easy to implement in noise prediction codes. It was shown that the suggested formula can be regarded as an extension of Farassat and Myers’ formula [20] to moving medium problems. The derived formula consisted of the Doppler factor because it is more easily interpreted and is more helpful in examining the physics of systems. The same methodology can be used to obtain a formula which contains no Doppler factor, thus facilitating noise prediction for supersonic cases. A verification study was conducted based on two test cases involving a stationary and a moving data surface for an observer located in a moving medium. In the former case, the calculated results were compared with an analytical solution, whereas for the latter case results from the moving observer approach served as a reference. The comparisons verified the accuracy of the developed formula. The suggested formulations may be applied to codes based on the Kirchhoff method for prediction of aerodynamic noise. Specific cases include jet noise, rotors and high-lift devices and in general any problems where the influence of mean flow and angle of attack is significant and should be taken into account. Data accessibility. This work does not have any experimental data. Authors’ contributions. G.G. conceived the formulations and the required mathematical methodology, then carried out and drafted the derivation procedure; L.S.-R. validated the formulations through the analytical test cases and drafted the manuscript; C.L. coordinated the research and helped draft the manuscript. All authors gave final approval for publication. Competing interests. We have no competing interests.
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180
210 present Farassat and Myers 270
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0.1
Funding. This research is funded by the Research Foundation - Flanders (FWO) and a PhD grant from the
It has already been mentioned that the Kirchhoff formulation is consistent with the FW-H equation, when the data surface is located in the linear propagation region (thus when the input data are compatible with the wave equation) [3]. We will show here the equivalence of the moving medium FW-H equation and the moving medium Kirchhoff formula, equation (2.14), in the linear region, as shown by Brentner & Farassat [3] for a medium at rest. The parameter under consideration here is acoustic pressure, because this equivalence is of interest for aeroacoustic applications. The FW-H equation in a moving medium can be given as [24] ∂2 ∂ D 1 D2 2 [Qδ( f )] (A 1) − ∇ {p (x, t)H( f )} = [Tij H( f )] − [Li δ( f )] + 2 2 ∂xi ∂xj ∂xi Dt c0 Dt with
⎫ Tij = ρui uj + [(p − p0 ) − c20 (ρ − ρ0 )]δij − σij ,⎪ ⎪ ⎪ ⎬ Li = ρui [un − (vn − U∞n )] + Pij nˆ j , ⎪ ⎪ ⎪ Q = ρ[un − (vn − U∞n )] + ρ0 (vn − U∞n ), ⎭
(A 2)
where the flow quantities are given in coordinates fixed to the undisturbed medium. From (A 1) and (A 2), one can obtain the following equation: 1 D2 ∂2 2 − ∇ (x, t)H( f )} = [Tij H( f )] {p ∂xi ∂xj c20 Dt2 −
∂ {[ρui [un − (vn − U∞n )] + Pij nˆ j ]δ( f )} ∂xi
+
D {[ρ[un − (vn − U∞n )] + ρ0 (vn − U∞n )]δ( f )}. Dt
(A 3)
In order to show the relation between equations (2.14) and (A 3), let the Kirchhoff source terms in equation (2.14) be denoted by QKir , where ϕ corresponds to the pressure fluctuation p . Subsequently, the Kirchhoff source terms are added to and subtracted from equation (A 3), resulting in 2 p (x, t) = QKir + + −
∂2 [Tij H( f )] ∂xi ∂xj
(Mn − M∞n ) Dp D (Mn − M∞n ) ∂p + δ( f ) + (p − c20 ρ ) δ( f ) ∂n c0 Dt Dt c0
D ∂ [ρun δ( f )], {ρui [un − (vn − U∞n )]δ( f )} + ∂xi Dt
(A 4)
where 2 is the wave or D’Alembertian operator in three-dimensional space. Having D/Dt = ∂/∂t + U ∞ · ∇ and the relation ∂ ∂ [nˆ i δ( f )] = − [vn δ( f )] ∂t ∂xi
(A 5)
...................................................
Appendix A. Comparison of the Kirchhoff and FW-H equations in the moving medium
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Agency for Innovation by Science and Technology in Flanders (IWT). Acknowledgements. This research is supported by the Research Foundation - Flanders (FWO) and the Agency for Innovation by Science and Technology in Flanders (IWT). The authors gratefully acknowledge this support.
one can obtain
16
∂ [(vn − U∞n )δ( f )]. ∂xi
(A 6)
By using the continuity and momentum equations [24], as well as equation (A 6), one can rewrite equation (A 4) as follows: 2 p (x, t) = QKir +
∂2 [Tij H( f )] ∂xi ∂xj
+
Mn − M∞n D Mn − M∞n D [p − ρ c20 ] (p − ρ c20 ) δ( f ) + δ( f ) Dt c0 Dt c0
−
∂ ∂ [ρui uj ]nˆ i δ( f ) − [ρui un δ( f )]. ∂xj ∂xi
(A 7)
We can finally conclude that in the linear flow region, where p = ρ c20 , equation (A 7) becomes 2 p (x, t) = QKir +
∂ 2 (ρui uj ) ∂xi ∂xj
H( f ).
(A 8)
Evidently, the only remaining source term in equation (A 8) is second order in the perturbation quantity ui and would be negligible in the derivation of the wave equation from the fluid conservation laws. Therefore, it is shown that the Kirchhoff and the FW-H formulations for a moving medium are equivalent, when the data surface is placed in the linear propagation region.
References 1. Kirchhoff GR. 1883 Zur Theorie der Lichtstrahlen. Ann. Phys. Chem. 18, 663–695. (doi:10.1002/ andp.18832540409) 2. Ffowcs Williams JE, Hawkings DL. 1969 Sound generation by turbulence and surfaces in arbitrary motion. Phil. Trans. R. Soc. Lond. A 264, 321–342. (doi:10.1098/rsta.1969.0031) 3. Brentner KS, Farassat F. 1998 Analytical comparison of the acoustic analogy and Kirchhoff formulation for moving surfaces. AIAA J. 36, 1379–1386. (doi:10.2514/2.558) 4. Yao HD, Eriksson LE, Davidson L, Grundestam O, Peng SH, Eliasson PE. 2012 Aeroacoustic assessment of conceptual low-noise high-lift wing configurations. In AIAA 2012-0383, Proc. 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Nashville, TN, 9–12 January 2012. Reston, VA: American Institute of Aeronautics and Astronautics. 5. Nebenführ B, Yao H, Peng SH, Davidson L. 2013 Hybrid RANS/LES simulations for aerodynamic and aeroacoustic analysis of a multi-element airfoil. In Proc. 19th AIAA/CEAS Aeroacoustics Conference, Berlin, Germany, 27–29 May 2013. Reston, VA: American Institute of Aeronautics and Astronautics. 6. di Francescantonio P. 1997 A new Kirchhoff formulation for transonic rotor noise. J. Sound Vib. 202, 491–509. (doi:10.1006/jsvi.1996.0843) 7. Brentner KS, Farassat F. 2003 Modeling aerodynamically generated sound of helicopter rotors. Prog. Aerosp. Sci. 39, 83–120. (doi:10.1016/S0376-0421(02)00068-4) 8. Strawn RC, Biswas R, Lyrintzis AS. 1996 Helicopter noise predictions using Kirchhoff methods. J. Comput. Acoust. 4, 321–339. (doi:10.1142/S0218396X96000106) 9. Chae S, Yee K, Yang C, Aoyama T, Jeong S, Obayashi S. 2010 Helicopter rotor optimization for the improvement of aeroacoustic performance in hover. J. Aircraft 47, 1770–1783. (doi:10. 2514/1.C000283) 10. Lyrintzis AS, Mankbadi R. 1996 Prediction of the far-field jet noise using Kirchhoff’s formulation. AIAA J. 34, 413–416. (doi:10.2514/3.13079) 11. Uzun A, Lyrintzis AS, Blaisdell GA. 2004 Coupling of integral acoustic methods with LES for jet noise prediction. Int. J. Aeroacoust. 3, 297–346. (doi:10.1260/1475472043499290)
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=−
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D ∂ ∂ [nˆ i δ( f )] = [nˆ i δ( f )] + U∞i [nˆ i δ( f )] Dt ∂t ∂xi
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12. Pan FL, Uzun A, Lyrintzis AS. 2008 Surface integral methods in jet aeroacoustics: refraction corrections. J. Aircraft 45, 381–387. (doi:10.2514/1.23513) 13. Bodony DJ, Lele SK. 2008 Current status of jet noise predictions using large-eddy simulation. AIAA J. 46, 364–380. (doi:10.2514/1.24475) 14. Spalart PR, Shur ML. 2009 Variants of the Ffowcs Williams–Hawkings equation and their coupling with simulations of hot jets. Int. J. Aeroacoust. 8, 477–492. (doi:10.1260/14754 7209788549280) 15. Pierce AD. 1981 Acoustics: an introduction to its physical principles and applications. New York, NY: McGraw-Hill. 16. Morgans RP. 1930 The Kirchhoff formula extended to a moving surface. Lond. Edinb. Dublin Philos. Mag., J. Sci. Ser. 7 9, 141–161. (doi:10.1080/14786443008564988) 17. Morino L. 1974 A general theory of unsteady compressible potential aerodynamics. NASA Contractor Report no. CR-2464. See http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa. gov/19750004821.pdf. 18. Morino L. 1985 Mathematical foundations of integral methods. In Computational methods in potential aerodynamics. New York, NY: Springer. 19. Morino L, Freedman MI, Deutsch DJ, Sipcic SR. 1985 An integral equation method for compressible potential flows in an arbitrary frame of reference. In Computational methods in potential aerodynamics. Southampton, UK: Computational Mechanics Publications. 20. Farassat F, Myers MK. 1988 Extension of Kirchhoff’s formula to radiation from moving surfaces. J. Sound Vib. 123, 451–460. (doi:10.1016/S0022-460X(88)80162-7) 21. Farassat F, Posey J. 2007 A fast method of deriving the Kirchhoff formula for moving surfaces. J. Acoust. Soc. Am. 122, 2965. (doi:10.1121/1.2942569) 22. Lyrintzis AS. 1994 Review: the use of Kirchhoff’s method in computational aeroacoustics. J. Fluids Eng. 116, 665. (doi:10.1115/1.2911834) 23. Mani R. 1990 The radiation of sound from a propeller at angle of attack. NASA Contractor Report no. CR-4262. See http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/ 19900012286.pdf. 24. Ghorbaniasl G, Lacor C. 2012 A moving medium formulation for prediction of propeller noise at incidence. J. Sound Vib. 331, 117–137. (doi:10.1016/j.jsv.2011.08.018) 25. Lee YW, Lee DJ. 2014 Derivation and implementation of the boundary integral formula for the convective acoustic wave equation in the time domain. J. Acoust. Soc. Am. 136, 2959. (doi:10.1121/1.4898427) 26. Farassat F. 1996 The Kirchhoff formulas for moving surfaces in aeroacoustics— the subsonic and supersonic cases. NASA Technical Memorandum no. 110285. See http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19970001733.pdf. 27. Farassat F, Myers MK. 2011 Multidimensional generalized functions in aeroacoustics and fluid mechanics—part 1: basic concepts and operations. Int. J. Aeroacoust. 10, 161–200. (doi:10.1260/1475-472X.10.2-3.161) 28. Wells VL, Han AY. 1995 Acoustics of a moving source in a moving medium with application to propeller noise. J. Sound Vib. 184, 651–663. (doi:10.1006/jsvi.1995.0339) 29. Farassat F, Brentner KS, Dun MH. 1998 A study of supersonic surface sources - the Ffowcs Williams–Hawkings equation and the Kirchhoff formula. In AIAA 98-2375, Proc. 4th AIAA/CEAS Aeroacoustics Conference, Toulouse, France, 2–4 June 1998. Reston, VA: American Institute of Aeronautics and Astronautics. 30. Gelfand IM, Shilov GE. 1964 Generalized functions, volume 1: properties and operations. San Diego, CA: Academic Press. 31. Farassat F, Succi GP. 1983 The prediction of helicopter discrete frequency noise. Vertica 7, 309–320.
Research Cite this article: Ghorbaniasl G, Siozos-Rousoulis L, Lacor C. 2016 A time-domain Kirchhoff formula for the convective acoustic wave equation. Proc. R. Soc. A 472: 20150689. http://dx.doi.org/10.1098/rspa.2015.0689 Received: 1 October 2015 Accepted: 18 February 2016
Subject Areas: mechanical engineering Keywords: aeroacoustics, Kirchhoff formula, moving medium, convective wave equation Author for correspondence: Ghader Ghorbaniasl e-mail: [email protected]
A time-domain Kirchhoff formula for the convective acoustic wave equation Ghader Ghorbaniasl1,2 , Leonidas Siozos-Rousoulis1 and Chris Lacor1 1 Vrije Universiteit Brussel (VUB), Pleinlaan 2, Brussels 1050, Belgium 2 University of Tehran, Tehran, Iran
Kirchhoff’s integral method allows propagated sound to be predicted, based on the pressure and its derivatives in time and space obtained on a data surface located in the linear flow region. Kirchhoff’s formula for noise prediction from high-speed rotors and propellers suffers from the limitation of the observer located in uniform flow, thus requiring an extension to arbitrarily moving media. This paper presents a Kirchhoff formulation for moving surfaces in a uniform moving medium of arbitrary configuration. First, the convective wave equation is derived in a moving frame, based on the generalized functions theory. The Kirchhoff formula is then obtained for moving surfaces in the time domain. The formula has a similar form to the Kirchhoff formulation for moving surfaces of Farassat and Myers, with the presence of additional terms owing to the moving medium effect. The equation explicitly accounts for the influence of mean flow and angle of attack on the radiated noise. The formula is verified by analytical cases of a monopole source located in a moving medium.
1. Introduction The Kirchhoff formula was suggested by Gustav Kirchhoff in 1883 [1] and was primarily used in the theory of light diffraction and other electromagnetic problems. It is one of the most useful results in wave propagation theory and particularly in acoustics. As an integral method applicable to noise prediction problems in modern computational aeroacoustics (CAA), the Kirchhoff method allows flow-generated noise to be propagated and predicted based on the pressure and its derivatives in time and space, on an arbitrary control surface (data surface). The linear wave equation 2016 The Author(s) Published by the Royal Society. All rights reserved.
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is assumed to be valid outside the data surface. The data surface should thus be placed in the nonlinear flow region, including all nonlinear flow effects. Acoustic pressures predicted by the Kirchhoff formula can be substantially in error if the Kirchhoff surface is located inside the nonlinear region. This is the main shortcoming of the method, compared with the Ffowcs Williams & Hawkings (FW-H) [2] equation, which may predict noise emissions fairly accurately, even when the integration surface is close to the noise-generating surface, thus neglecting external quadrupoles. The two methods are therefore equivalent when the Kirchhoff surface is placed in the linear flow region [3]. On the other hand, the FW-H equation is based on the conservation laws of fluid mechanics rather than the wave equation, which is the case for the Kirchhoff formula. Consequently, the FW-H equation is not appropriate for all types of wave propagation. In practical CAA problems, the Kirchhoff approach finds wide application to aerodynamic noise generation involving high-lift devices [4,5], rotors [3,6–9] and jet flows [10–14]. Initially, the Kirchhoff formula was limited to a stationary data surface [15]. In 1930, the original Kirchhoff formula was extended to moving surface applications by Morgans [16], using the Green’s function approach. Morino also used the Green’s function method and developed a time-domain and frequency-domain formulation for a uniformly translating surface [17,18], which he then extended to a rotating surface [19]. This approach however was significantly cumbersome and complicated. In order to achieve a simpler derivation of the Kirchhoff formula, Farassat & Myers in 1988 [20] suggested a fast and modern method to obtain Morgans’ result by using generalized functions theory in the time domain. They also provided an extension of this formula to moving surfaces, which has been widely used for high-speed helicopter rotor noise prediction. More recently, Farassat & Posey [21] presented a faster and simpler approach for the derivation of the aforementioned Kirchhoff formulation for moving surfaces [20]. An extensive review of the use of Kirchhoff’s method in CAA and its applications was provided by Lyrintzis [22]. In the presence of uniform constant flow, such as a typical wind tunnel case, all the aforementioned formulations require application of the moving observer method to implicitly include effects of uniform constant velocity on emitted noise. The terms associated with the mean flow velocity effects are not present in the formulations and, thus, the observer must be put in motion along with the data surface, with the velocity of the flow but in the opposite direction. The same problem, however, can be alternatively solved by the moving medium approach. The main advantage of moving medium methods is that terms related to incidence are explicitly included in the acoustic sources and acoustic propagation formulae, thus showing the effects of incidence on the Doppler amplification of radiated noise. In addition, it provides a way of examining the physics of the problem, while processing asymmetric inflow effects. When rotating noise sources such as propeller blades are concerned, the effects of mean flow incidence can be equivalent to an unsteady acoustic source, resulting in generation of more effective radiation modes [23,24]. Recently, Lee & Lee [25] proposed a convective Kirchhoff formula for boundary integral problems, by considering only streamwise flow motion. The Kirchhoff approach has thus not been implemented for aerodynamic noise prediction for sources in arbitrary motion located in a moving medium with an arbitrary orientation. The contribution of this paper is the development of a convected Kirchhoff formulation for moving surfaces that explicitly takes into account the effects of mean flow and angle of incidence on radiated noise. The formulation will be derived in the time domain for moving surfaces, with respect to a moving medium frame and will be an extension of Farassat & Myers’ formula [20] to moving medium problems. The derivation method of the formulation will follow the fast and modern procedure depicted by Farassat & Posey [21], by using generalized functions theory [26,27]. This formulation consists of the partial derivatives with respect to the source coordinates and time, thus making the formulation easy to use when studied numerically. The extension of the formula to moving medium problems will be realized according to the procedure described by Ghorbaniasl & Lacor [24], who suggested a moving medium solution of the FW-H equation [2]. A formula with the Doppler factor will be developed because it is more easily interpreted and is more helpful in
3
ˆn = —f
M• mean flow velocity
f (x, t) < 0 j˜ (x, t) = 0
v data surface velocity
f (x, t) > 0 j˜ (x, t) = j (x, t)
Figure 1. Schematic of the Kirchhoff data surface at time t.
Table 1. Nomenclature. ..........................................................................................................................................................................................................
∇
gradient operator
c0
speed of sound
f (x, t)
function describing the data surface
g
= τ − t + R/c0 , retarded time function
G
Green’s function
Mi = vi /c0 ,
local Mach number vector components
Mn = vn /c0 ,
source Mach number along data surface normal direction
MR
source Mach number in acoustic radius R-direction
M
source Mach number in acoustic radius R∗ -direction
M∞i = U∞i /c0 ,
local mean flow Mach number vector components
M∞n = Un /c0 ,
mean flow Mach number along data surface normal direction
M∞R
mean flow Mach number in acoustic radius R-direction
M
mean flow Mach number in acoustic radius R∗ -direction
.......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... ..........................................................................................................................................................................................................
R∗ .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... ..........................................................................................................................................................................................................
∞R∗ ..........................................................................................................................................................................................................
nˆ i
unit normal vector components
r
distance between observer position xi and source position yi
ˆri R˜ i , R˜i∗
= (xi − yi )/r
dS
element of data surface
t
observer time
.......................................................................................................................................................................................................... .......................................................................................................................................................................................................... ..........................................................................................................................................................................................................
gradients of acoustic radii R and R∗
.......................................................................................................................................................................................................... .......................................................................................................................................................................................................... ..........................................................................................................................................................................................................
δij
Kronecker delta
..........................................................................................................................................................................................................
θ, θ
∗
angles between normal direction and acoustic radii directions
..........................................................................................................................................................................................................
τ
retarded time or emission time
..........................................................................................................................................................................................................
Φ acoustic function outside data surface .......................................................................................................................................................................................................... Φ˜ acoustic function inside and outside data surface ..........................................................................................................................................................................................................
Φn
normal derivative of acoustic function, nˆ i ∂Φ/∂xi
Φt
observer time derivative of acoustic function, ∂Φ/∂t
Φ
= U ∂Φ/∂x
.......................................................................................................................................................................................................... ..........................................................................................................................................................................................................
U∞ ∞i i ..........................................................................................................................................................................................................
...................................................
a
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Kirchhoff data surface
body
(a) Derivation of the equation Let us assume that f (x, t) = 0 (table 1) is the moving Kirchhoff data surface with velocity v defined in such a way that the unit outward normal is nˆ = ∇f (figure 1). If this condition is not satisfied by the original definition of f , then one can redefine it as f /|∇f |, which does satisfy this condition on the surface. Furthermore, the definition of the unit normal in terms of the function defining the surface relies on a normalization of f , which is always possible. The aim is to show how to derive the Kirchhoff formula for acoustic wave prediction radiated from sources on the data surface in a moving frame, using generalized functions. Assume the medium moves in an arbitrary direction with a constant velocity of U ∞ . Then the convective wave equation can be expressed as 1 D2 Φ(x, t) − ∇ 2 Φ(x, t) = 0, Dt2 c20
(2.1)
where D/Dt = ∂/∂t + U∞i ∂/∂xi with xi being the ith component of x. Equation (2.1) is valid for the region outside the data surface f (x, t) = 0. In order to extend the domain of the problem to inside ˜ t) as follows: the surface, one defines the function Φ(x, Φ(x, t) f > 0 ˜ Φ(x, t) = (2.2) 0 f < 0. Because Kirchhoff’s formula [1] is generally valid for wave propagation in a homogeneous ˜ t) is used in this paper. However, when the suggested approach is applied medium, function Φ(x, ˜ t) will be equivalent to the acoustic pressure, p (x, t). to aeroacoustic problems, Φ(x, The extended governing equation is given by ˜ t) 1 D2 Φ(x, ˜ t) = 0. − ∇ 2 Φ(x, 2 2 Dt c0
(2.3)
˜ t) is discontinuous, the ordinary derivatives in equation (2.3) should be Because the function Φ(x, replaced by generalized derivatives [30]. In general and according to Farassat & Myers [27], if the function Φ˜ has a discontinuity across the surface f = 0, then the jump is defined as ˜ f = 0− ). ˜ f = 0+ ) − Φ( Φ˜ = Φ(
(2.4)
When Φ˜ is multiplied by the Dirac delta function δ( f ), by using the definition of equation (2.2), one has ˜ f = 0+ ) = Φ. Φ˜ = Φ(
(2.5)
According to Farassat [26], generalized spatial derivatives will be obtained by ∂ Φ˜ ∂f ∂ Φ˜ ∂¯ Φ˜ = + Φ˜ δ( f ) = + Φ nˆ i δ( f ), ∂xi ∂xi ∂xi ∂xi
(2.6)
...................................................
2. Governing equation in a moving medium
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examining the physics of systems. For subsonic cases, the present formulation explicitly shows the mean flow effect and its physical meaning. The formulation will be verified through two analytical subsonic test cases for stationary and moving Kirchhoff data surfaces. As already mentioned, supersonic jet flows is an area of significant application potential for the Kirchhoff approach. The presented technique may thus be used to derive a formula which contains no Doppler factor, enabling fast prediction of noise in supersonic cases, as depicted by Wells & Han [28]. Another approach without the Doppler factor and its mathematical explanation can be found in [29], leading to formulation 4 of Farassat.
ˆ In equation (2.6), a bar over the derivative operator is where nˆ i is the ith component of n. consistently used to denote generalized differentiation. From equation (2.6), one can also derive
where Φn = nˆ · ∇Φ. ¯ Similarly, generalized time derivatives D/Dt can be expressed in terms of ordinary time derivatives D/Dt as follows:
On the other hand
¯ Φ˜ DΦ˜ Df DΦ˜ Df D = + Φ˜ δ( f ) = +Φ δ( f ). Dt Dt Dt Dt Dt
(2.8)
∂f ∂f Df = + U∞i = −νn + U∞n , Dt ∂t ∂xi
(2.9)
where U∞n = nˆ · U ∞ and vn = nˆ · v. By substituting (2.9) into (2.8), one has ¯ Φ˜ DΦ˜ D = − Φ(νn − U∞n )δ( f ). Dt Dt Then ¯ 2 Φ˜ D D = Dt Dt2
¯ Φ˜ D Dt
−
DΦ (νn − U∞n )δ( f ). Dt
(2.10)
(2.11)
Substituting equation (2.10) into equation (2.11) yields ¯ 2 Φ˜ D2 Φ˜ D DΦ D [Φ(νn − U∞n )δ( f )] − (νn − U∞n )δ( f ). = − 2 2 Dt Dt Dt Dt
(2.12)
Using equations (2.7) and (2.12), one can form the following expression: ¯ 2 Φ˜ 1 D2 Φ˜ 1 D 1 D [Φ(Mn − M∞n )δ( f )] − ∇ 2 Φ˜ − Φn δ( f ) − ∇¯ 2 Φ˜ = 2 − 2 2 2 c0 Dt c0 Dt c0 Dt −
∂ Mn − M∞n DΦ δ( f ) − [Φ nˆ i δ( f )], c0 Dt ∂xi
(2.13)
where Mn = vn /c0 is the local normal Mach number of the surface, and M∞n = U∞n /c0 is the normal Mach number of the moving frame. Substituting equation (2.3) into (2.13) gives the Kirchhoff formula for bodies in motion in a moving medium as follows: ¯ 2 Φ˜ Mn − M∞n DΦ 1 D 2 ˜ ¯ δ( f ) − ∇ Φ = − Φn + c0 Dt c20 Dt2 −
∂ 1 D [Φ(Mn − M∞n )δ( f )] − [Φ nˆ i δ( f )]. c0 Dt ∂xi
(2.14)
Using the definition of D/Dt, one has ¯ 2 Φ˜ 1 D ¯ 2 Φ˜ = − Φn + Mn − M∞n (Φt + ΦU∞ ) δ( f ) − ∇ c0 c20 Dt2 −
∂ 1 ∂ [Φ nˆ i + Φ(Mn − M∞n )M∞i δ( f )], [Φ(Mn − M∞n )δ( f )] − c0 ∂t ∂xi
(2.15)
where Φt = ∂Φ/∂t and ΦU∞ = U ∞ · ∇Φ. Note that these variables will be functions of y and τ , i.e. Φt = Φt (y, τ ) and ΦU∞ = ΦU∞ (y, τ ), in the same manner as other terms inside the integrals. As will be shown in appendix A, the derived convective wave equation (2.15) is consistent with the convective FW-H equation [24], when the data surface is located in the linear propagation region (thus when the input data are compatible with the wave equation) [3].
...................................................
(2.7)
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∂ [Φ nˆ i δ( f )] + Φn δ( f ), ∇¯ 2 Φ˜ = ∇ 2 Φ˜ + ∂xi
5
(b) Solution of the equation
6
δ(g) 4π R∗
(2.16)
for τ ≤ t, where g = τ − t + R/c0 . The quantities R∗ and R are defined as r 1 + γ 2 M2∞r R∗ = γ
(2.17)
and R = γ 2 (R∗ − rM∞r ),
(2.18)
where γ 2 = 1/(1 − M2∞ ) and M∞ r = M ∞ · rˆ with rˆ being the unit vector of the distance between the source position y(τ ) and the observer position x(t), i.e. r = |x(t) − y(τ )|. Based on this Green’s function, the solution of equation (2.15) is given by +∞
Mn − M∞n δ(g) Φn + (Φt + ΦU∞ ) δ( f ) ∗ dy dτ c R 0 −∞ −∞ t +∞ 1 ∂ δ(g) − Φ(Mn − M∞n )δ( f ) ∗ dy dτ c0 ∂t −∞ −∞ R t +∞ ∂ δ(g) − [Φ nˆ i + Φ(Mn − M∞n )M∞i ]δ( f ) ∗ dy dτ . ∂xi −∞ −∞ R
˜ t, M ∞ ) = − 4π Φ(x,
t
(2.19)
Let S: f = 0 be defined in a reference frame, called the η-frame, fixed to the data surface, where the variable y is transformed to the variable η. For the sake of simplicity, we assume that the surface is non-deformable. It can, though, be easily extended to deformable surfaces. Therefore, equation (2.19) leads to t Mn − M∞n δ(g) ˜ Φn + (Φt + ΦU∞ ) dS dτ 4π Φ(x, t, M ∞ ) = − c R∗ 0 −∞ S 1 ∂ t δ(g) − Φ(Mn − M∞n ) ∗ dS dτ c0 ∂t −∞ S R t ∂ δ(g) − [Φ nˆ i + Φ(Mn − M∞n )M∞i ] ∗ dS dτ . (2.20) ∂xi −∞ S R Considering that x and y are independent variables, equation (2.20) can be rewritten in the following form: ˜ t, M ∞ ) = − 4π Φ(x,
t
−∞ S
∂ − ∂t t −
t
Φn +
−∞ S
−∞ S
Φ
Mn − M∞n δ(g) (Φt + ΦU∞ ) dS dτ c0 R∗
Mn − M∞n δ(g) dS dτ c0 R∗
[Φ nˆ i + Φ(Mn − M∞n )M∞i ]
∂ ∂xi
δ(g) dS dτ . R∗
(2.21)
In order to convert the space derivative to a time derivative, one can use the following relation [24]: ˜ ∗ δ(g) ˜ i δ(g) R δ(g) 1 ∂ R ∂ i = − (2.22) − ∂xi R∗ c0 ∂t R∗ R∗2
...................................................
G(x, t, y, τ ) =
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The governing wave equation obtained for the Kirchhoff formula is valid in the open space. We can now solve this equation using the Green’s function for the wave equation in the open threedimensional space. One may use the Green’s function for a moving medium approach [24], i.e.
with
Therefore, equation (2.21) becomes t δ(g) Mn − M∞n ˜ Φn + Φt + ΦU∞ dS dτ 4π Φ(x, t, M ∞ ) = − c0 R∗ −∞ S 1 ∂ t δ(g) + [Φ cosθ − (1 − M∞R )Φ(Mn − M∞n )] ∗ dS dτ c0 ∂t −∞ S R t δ(g) + [Φ(Mn − M∞n )M∞R∗ + Φ cosθ ∗ ] ∗2 dS dτ , R −∞ S
(2.23)
(2.24)
˜ cosθ ∗ = nˆ · R ˜ ∗ , M∞R = M ∞ · R ˜ and M∞R∗ = M ∞ · R ˜ ∗. where cosθ = nˆ · R, The integration over dτ can be performed by using the methods of generalized functions theory [27]. It can be seen that [24] t h(τ ) h(τ )δ(g) dτ = , (2.25) 1 − MR e −∞ ˜ The subscript e denotes that quantities inside the bracket are calculated in the where MR = M · R. emission time τe = t − R/c0 . This relation can be applied to equation (2.24), leading to Φn + ((Mn − M∞n )/c0 )(Φt + ΦU∞ ) ˜ t, M ∞ ) = − dS 4π Φ(x, R∗ (1 − MR ) S e cosθ − (1 − M∞R )(Mn − M∞n ) 1 ∂ Φ + dS c0 ∂t S R∗ (1 − MR ) e cos θ ∗ + M∞R∗ (Mn − M∞n ) Φ + dS. (2.26) R∗2 (1 − MR ) S e Equation (2.26) is the general Kirchhoff formula for moving surfaces, described in a moving medium. The derived Kirchhoff formula requires Φ and its time derivative Φt and space derivative ∇Φ as inputs for the wave propagation. It explicitly takes into account the convection effect on the noise prediction. The formulation is equivalent to the Kirchhoff formula derived by Farassat & Myers [20] if the medium is set to be at rest, i.e. M ∞ = 0, where θ ∗ = θ and R∗ = R = r. In general, the Kirchhoff data surface S can deform. However, for the sake of simplicity, the derivative in this paper used a non-deformable data surface. In addition, in many applications of the Kirchhoff formula in aeroacoustics, the data surface is assumed to be rigid. The derived formulation is based on the retarded time solution and its stability will depend on the relative position of the source and observer, the data surface velocity and the time step when it is studied numerically. The above equation is a noise prediction formula based on the classical retarded time algorithms. This formula explicitly takes into account the mean flow influence and the aerodynamic and acoustic effects of incidence and is easy to implement in noise prediction codes. This formulation is derived with the Doppler factor because it is more easily interpreted and is more helpful in examining the physics of systems.
(c) An alternative form of the solution Writing the formulation in terms of geometry and available kinematic quantities makes the formulation practically useful. In addition, in order to improve the accuracy of the results, it is also essential that the time derivative in the integrand of the second term on the right-hand side
...................................................
and
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⎫ r 2 ⎪ ⎬ (ˆ r + γ M M ) ∞r i ∞i γ 2 R∗ ⎪ ⎭ ˜ ∗ − M∞i ). R˜ i = γ 2 (R i
R˜ ∗i =
⎫ ∂R ∂R∗ ⎪ ⎪ = −c0 MR∗ , = −c0 MR , ⎪ ⎪ ∂τ ∂τ ⎪ ⎪ ⎪ ⎪ ∗ ∗ 2 ⎬ ˜ ˜ ∗ −c M + c γ (M − M M ) R ∂R 0 0 R ∞M i ∞i i i = , ⎪ ∂τ γ 2 R∗ ⎪ ⎪ ⎪ ⎪ ⎪ ∗ ˜ ˜ ⎪ ∂ Ri ⎪ 2 ∂ Ri ⎭ =γ , ∂τ ∂τ
(2.28)
where M∞M = M · M ∞ . One thus has ˙R c0 (γ 2 M2R∗ − M2 ) − c0 γ 2 M2∞M 1 M ∂ c0 MR∗ = + + , ∂τ R∗ (1 − MR ) R∗ (1 − MR )2 R∗2 (1 − MR ) R∗2 (1 − MR )2
(2.29)
˙ R=M ˙ · R. ˜ It should be noted that the dot over a quantity denotes the source time where M derivative of the quantity. Let Q = Φ cosθ − (1 − M∞R )(Mn − M∞n )Φ.
(2.30)
By using the following relations ⎫ n˙ R 1 ∂ cosθ −Mn + γ 2 (MR∗ cosθ ∗ − M∞M M∞n ) ⎪ ⎪ = + , ⎪ ⎪ ⎪ c0 ∂τ c0 R∗ ⎪ ⎪ ⎪ ⎪ ⎪ ˙ 1 ∂Mn Mn + n˙ M ⎪ ⎪ ⎪ = , ⎬ c ∂τ c 0
0
1 ∂M∞R −M∞M + γ 2 (MR∗ M∞R∗ − M∞M M2∞ ) , = c0 ∂τ R∗ 1 ∂M∞n n˙ M∞ = , c0 ∂τ c0
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(2.31)
one can have 1 ˙ Φ˙ Q = [cos θ − (1 − M∞R )(Mn − M∞n )] c0 c0 n˙ R −Mn + γ 2 (MR∗ cosθ ∗ − M∞n M∞M ) + +Φ c0 R∗ −
Φ ˙ n + n˙ M − n˙ M∞ ) (1 − M∞R )(M c0
+
Φ(Mn − M∞n ) [−M∞M + γ 2 (MR∗ M∞R∗ − M∞M M2∞ )] R∗
(2.32)
or Φ˙ 1 ˙ Q = [cos θ − (1 − M∞R )(Mn − M∞n )] c0 c0 +
Φ ˙ n + n˙ M − n˙ M∞ )] [n˙ R − (1 − M∞R )(M c0
+
Φ {MR∗ [γ 2 cosθ ∗ + γ 2 M∞R∗ (Mn − M∞n )] − Mn (1 + γ 2 M∞M )}, R∗
˙ n=M ˜ M ˙ · n, ˆ n˙ M = n˙ˆ · M and n˙ M∞ = n˙ˆ · M ∞ . where n˙ R = n˙ˆ · R,
(2.33)
...................................................
along with the following relations [24]:
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of equation (2.26) can be evaluated analytically. As demonstrated by Farassat & Succi [31], this can be realized by using the rule 1 ∂ ∂ = (2.27) ∂t x e 1 − MR ∂τ x e
Mn − M∞n (Φt + ΦU∞ ) E1 = − Φn + c0 +
˙ RΦ M [cos θ − (1 − M∞R )(Mn − M∞n )] c0 (1 − MR )2
+
1 {[cos θ − (1 − M∞R )(Mn − M∞n )]Φ˙ c0 (1 − MR )
˙ n + n˙ M − n˙ M )]Φ} + [n˙ R − (1 − M∞R )(M ∞
(2.35)
and E2 = cosθ ∗ + M∞R∗ (Mn − M∞n ) +
1 {MR∗ [cosθ + γ 2 cosθ ∗ − (1 − M∞R − γ 2 M∞R∗ )(Mn − M∞n )] 1 − MR
− Mn (1 + γ 2 M∞M )} +
γ 2 (M2R∗ − M2∞M ) − M2 (1 − MR )2
[cos θ − (1 − M∞R )(Mn − M∞n )].
(2.36)
This is the convected Kirchhoff formula for moving data surfaces. It should be noted that, as we have considered in the derivation procedure, Φ˜ = 0 inside the data surface. The derivation of this formula was based on generalized functions theory, which offers great advantages compared with the classical derivation. It can be checked that this formulation leads to the Kirchhoff formula for non-deformable moving surfaces derived by Farassat & Myers [20], under no mean flow influence.
(i) Simplified form for a stationary data surface The simple case of a stationary data surface is suitable for practical applications and is commonly used in the hybrid computational fluid dynamics and acoustic analogy approach. The derived formulation (2.34) along with (2.35) and (2.36) can be simplified for stationary data surfaces as follows: E1 E2 Φ ˜ dS + dS, (2.37) 4π Φ(x, t, M ∞ ) = ∗ ∗2 S R e S R e where
M∞n 1 (Φt + ΦU∞ ) + [cosθ + M∞n (1 − M∞R )]Φ˙ E1 = − Φn − c0 c0
(2.38)
E2 = cosθ ∗ − M∞R∗ M∞n .
(2.39)
and
In this case, the retarded time can be obtained by an explicit solution without need for any iterative solution. This is due to the radius r not being dependent on the retarded time.
(ii) Simplified form for a stationary data surface and a medium at rest As can be seen, when M ∞ = 0 (then cosθ ∗ = cosθ and R∗ = R = r), equation (2.37) represents the classical Kirchhoff formula derived by Lyrintzis [22], for stationary surfaces for an observer fixed to a medium at rest.
...................................................
where
9
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Substituting equations (2.27), (2.29) and (2.33) into equation (2.26) gives the following formulation: E1 E2 Φ ˜ t, M ∞ ) = dS + dS, (2.34) 4π Φ(x, ∗ ∗2 S R (1 − MR ) e S R (1 − MR ) e
3. Numerical results
10
In order to verify the present formulations, the analytical solution of a three-dimensional monopole source in uniform constant flow has been used, as described by Lyrintzis & Mankbadi [10]. This case is equivalent to the problem of a monopole in motion with a constant velocity in the x-direction, located in an ambient quiescent fluid. The analytical solution for the acoustic pressure at the data surface resulting from the monopole [10] is given by ps =
robs + M∞ (xs − x) , sinω t − robs c0 β 2
where β= and
robs =
1
1 − M2∞
(3.1)
⎫ ⎪ ⎬
⎪ (xs − x)2 + β 2 (ys − y)2 + β 2 (zs − z)2 .⎭
(3.2)
In equation (3.1), ω is the angular frequency of the source, whereas the monopole source is located at (xs , ys , zs ) and the observer at (x, y, z). In this study a spherical Kirchhoff surface of radius rs = 0.1 m is used as the integration (data) surface (a two-dimensional cross section of the domain is depicted in figure 2). The data surface is discretized into 4095 points, well distributed on the spherical surface, with angle increments equal to 4◦ . This resolution for the data surface was considered to be fine enough. The ambient speed of sound and the density of the undisturbed medium are chosen as c0 = 1 m s−1 and ρ0 = 1 kg m−3 , respectively. The monopole source is located at the origin (0, 0, 0) and its angular frequency ω is 1 rad s−1 . For discretization in time, 256 time steps are used. The inputs required for the Kirchhoff formula, namely the pressure and its derivatives in time and space, are calculated from equation (3.1). The acoustic pressure fluctuations p obtained by the developed Kirchhoff formulation are plotted versus the analytical results for different observer locations. Three observers are positioned in the x−y plane (θ = 0◦ ), on a circle of radius robs = 10 m and at angle increments of α1 = 0◦ , α2 = 90◦ and α3 = 180◦ , as defined in a spherical coordinate system by x = robs (cosα, sinα cosθ, sinα sinθ).
(3.3)
Additionally, the noise directivities of the RMS values of the acoustic pressure predicted by the derived Kirchhoff formulations are plotted against the analytical directivities, for 36 observers equally distributed in the x−y plane on a circle of radius robs = 10 m from the origin. Two cases corresponding to mean flow Mach numbers of M∞ = 0.2 and M∞ = 0.8 are considered. The time histories of the acoustic pressure fluctuations corresponding to the considered cases are depicted in figures 3 and 4, respectively, whereas the directivities are shown in figures 5 and 6, respectively. In both cases, it is obvious that the numerical method well duplicates the analytical solution for the observer locations chosen. The excellent agreement between the numerical and analytical results proves that the formulation predicts the acoustic pressure accurately.
...................................................
(a) TC1: monopole in uniform constant flow
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Here, the suggested convected Kirchhoff formulation for moving surfaces is validated through two analytical test cases. The first test case is a stationary monopole in mean flow with a stationary data surface, whereas the second case consists of a monopole translating along with the data surface, in a moving medium. Descriptions of the test cases and the calculated results are given in §3a,b.
y
11
M•
robs ...................................................
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rs
a
o (xs, ys, zs)
x
Figure 2. Schematic of the spherical Kirchhoff data surface in the x−y plane.
p¢ (Pa)
(a)
(b) 0.15
0.15
0.10
0.10
0.05
0.05
0
0
–0.05
–0.05
–0.10
–0.10 present
–0.15
–0.15
analytic
–0.20
0
0.2
0.4
0.6
0.8
1.0
–0.20
0
0.2
0.4
(c)
0.6
0.8
1.0
t/T
t/T 0.15 0.10
p¢ (Pa)
0.05 0 –0.05 –0.10 –0.15 –0.20
0
0.2
0.4
0.6
0.8
1.0
t/T
Figure 3. The calculated acoustic pressure is identical to the analytical solution for the three observer positions of (a) α1 , (b) α2 and (c) α3 , at a mean flow Mach number of M∞ = 0.2. (Online version in colour.)
(b) TC2: monopole in uniform constant flow with a moving data surface In order to achieve validation of the developed formulation for cases involving a data surface in motion, a second test case was considered. Because, to the best of our knowledge, an analytical test case of a moving data surface is not available in the literature, the aforementioned threedimensional monopole was set in translating motion along with the Kirchhoff data surface at a constant velocity of M = 0.2, while located in a uniform constant flow of M∞ = 0.2. The noise prediction was realized by the moving medium formulation developed here. The input for the formulation, and thus the sources on the data surface, was calculated for a mean flow of M∞ = 0.2, using equation (3.1). After the noise sources on the data surface were obtained, it was set in translating motion with velocity M = 0.2, towards the direction of the mean flow.
(a)
(b) 0.15
12
0.2
p¢ (Pa)
0 0 –0.05 –0.10
–0.1 present
–0.15
analytic
–0.20
0
0.2
0.4
0.6
–0.2
1.0
0.8
0
0.2
0.4
t/T
(c)
0.6
0.8
1.0
t/T 0.15 0.10
p¢ (Pa)
0.05 0 –0.05 –0.10 –0.15 –0.20
0
0.2
0.4
0.6
0.8
1.0
t/T
Figure 4. The calculated acoustic pressure is identical to the analytical solution for the three observer positions of (a) α1 , (b) α2 and (c) α3 , at a mean flow Mach number of M∞ = 0.8. (Online version in colour.)
120
90 0.1 0.05
150
60 30 0
180 210
330 240
present analytic
270
Figure 5. The calculated directivity of the acoustic pressure is identical to the analytical solution for 36 observers equally distributed on a circle of radius 10 m from the origin, at a mean flow Mach number of M∞ = 0.2. (Online version in colour.)
The reference for comparison was obtained by computing the emitted noise for the exact same case, using the Kirchhoff formula derived by Farassat for a medium at rest [20], applied with a moving observer approach. The inputs used for the reference solution were the noise sources calculated on the data surface, as described above. The reference solution was then obtained by imposing translational velocity, equal to M = 0.2, on the source, whereas the source is also translating along with the observer with velocity M∞ = 0.2. Therefore, translation in a moving medium is achieved, rendering the reference case equivalent to the moving medium case.
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Figure 6. The calculated directivity of the acoustic pressure is identical to the analytical solution for 36 observers equally distributed on a circle of radius of 10 m from the origin, at a mean flow Mach number of M∞ = 0.8. (Online version in colour.)
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Figure 7. The calculated acoustic pressure is identical to one obtained from Farassat and Myers’ formula [20] with the moving observer method for the three observer positions of (a) α1 , (b) α2 and (c) α3 . The data surface is in motion at M = (0.2, 0, 0) and is located in a mean flow of M∞ = 0.2. (Online version in colour.)
The acoustic pressure time history obtained by the developed Kirchhoff formulation is plotted versus the results from the moving observer method for the three aforementioned observer locations. Moreover, noise directivity obtained by the moving medium and moving observer approaches were calculated and plotted for 36 observers equally distributed in the x−y plane on a circle of radius robs = 10 m from the origin. All required parameters, as well as the time and space discretization, are defined as described in §3a.
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Figure 8. The calculated directivity of the acoustic pressure is identical to the solution obtained from Farassat and Myers’ formula [20] for 36 observers equally distributed on a circle of radius 10 m from the origin. The data surface is in motion at M = (0.2, 0, 0) and is located in a mean flow of M∞ = 0.2. (Online version in colour.)
The outcome of the comparison of the present formulation against the moving observer method is depicted in figures 7 and 8, for the acoustic pressure time history and directivity, respectively. Apparently, the results of the two methods are in excellent agreement. In this case, more significant than the actual input to the methods is the final outcome, which proves that both methods provide identical results, thus validating the developed formulation for a moving Kirchhoff data surface.
4. Conclusion A convected Kirchhoff formulation was suggested for arbitrarily moving surfaces in a uniform moving medium of arbitrary orientation. The convective wave equation was obtained using generalized functions theory, and the Kirchhoff formula was derived in the time domain with respect to a moving medium frame. The equation explicitly takes into account mean flow and incidence effects on the radiated noise and is easy to implement in noise prediction codes. It was shown that the suggested formula can be regarded as an extension of Farassat and Myers’ formula [20] to moving medium problems. The derived formula consisted of the Doppler factor because it is more easily interpreted and is more helpful in examining the physics of systems. The same methodology can be used to obtain a formula which contains no Doppler factor, thus facilitating noise prediction for supersonic cases. A verification study was conducted based on two test cases involving a stationary and a moving data surface for an observer located in a moving medium. In the former case, the calculated results were compared with an analytical solution, whereas for the latter case results from the moving observer approach served as a reference. The comparisons verified the accuracy of the developed formula. The suggested formulations may be applied to codes based on the Kirchhoff method for prediction of aerodynamic noise. Specific cases include jet noise, rotors and high-lift devices and in general any problems where the influence of mean flow and angle of attack is significant and should be taken into account. Data accessibility. This work does not have any experimental data. Authors’ contributions. G.G. conceived the formulations and the required mathematical methodology, then carried out and drafted the derivation procedure; L.S.-R. validated the formulations through the analytical test cases and drafted the manuscript; C.L. coordinated the research and helped draft the manuscript. All authors gave final approval for publication. Competing interests. We have no competing interests.
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Funding. This research is funded by the Research Foundation - Flanders (FWO) and a PhD grant from the
It has already been mentioned that the Kirchhoff formulation is consistent with the FW-H equation, when the data surface is located in the linear propagation region (thus when the input data are compatible with the wave equation) [3]. We will show here the equivalence of the moving medium FW-H equation and the moving medium Kirchhoff formula, equation (2.14), in the linear region, as shown by Brentner & Farassat [3] for a medium at rest. The parameter under consideration here is acoustic pressure, because this equivalence is of interest for aeroacoustic applications. The FW-H equation in a moving medium can be given as [24] ∂2 ∂ D 1 D2 2 [Qδ( f )] (A 1) − ∇ {p (x, t)H( f )} = [Tij H( f )] − [Li δ( f )] + 2 2 ∂xi ∂xj ∂xi Dt c0 Dt with
⎫ Tij = ρui uj + [(p − p0 ) − c20 (ρ − ρ0 )]δij − σij ,⎪ ⎪ ⎪ ⎬ Li = ρui [un − (vn − U∞n )] + Pij nˆ j , ⎪ ⎪ ⎪ Q = ρ[un − (vn − U∞n )] + ρ0 (vn − U∞n ), ⎭
(A 2)
where the flow quantities are given in coordinates fixed to the undisturbed medium. From (A 1) and (A 2), one can obtain the following equation: 1 D2 ∂2 2 − ∇ (x, t)H( f )} = [Tij H( f )] {p ∂xi ∂xj c20 Dt2 −
∂ {[ρui [un − (vn − U∞n )] + Pij nˆ j ]δ( f )} ∂xi
+
D {[ρ[un − (vn − U∞n )] + ρ0 (vn − U∞n )]δ( f )}. Dt
(A 3)
In order to show the relation between equations (2.14) and (A 3), let the Kirchhoff source terms in equation (2.14) be denoted by QKir , where ϕ corresponds to the pressure fluctuation p . Subsequently, the Kirchhoff source terms are added to and subtracted from equation (A 3), resulting in 2 p (x, t) = QKir + + −
∂2 [Tij H( f )] ∂xi ∂xj
(Mn − M∞n ) Dp D (Mn − M∞n ) ∂p + δ( f ) + (p − c20 ρ ) δ( f ) ∂n c0 Dt Dt c0
D ∂ [ρun δ( f )], {ρui [un − (vn − U∞n )]δ( f )} + ∂xi Dt
(A 4)
where 2 is the wave or D’Alembertian operator in three-dimensional space. Having D/Dt = ∂/∂t + U ∞ · ∇ and the relation ∂ ∂ [nˆ i δ( f )] = − [vn δ( f )] ∂t ∂xi
(A 5)
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Appendix A. Comparison of the Kirchhoff and FW-H equations in the moving medium
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Agency for Innovation by Science and Technology in Flanders (IWT). Acknowledgements. This research is supported by the Research Foundation - Flanders (FWO) and the Agency for Innovation by Science and Technology in Flanders (IWT). The authors gratefully acknowledge this support.
one can obtain
16
∂ [(vn − U∞n )δ( f )]. ∂xi
(A 6)
By using the continuity and momentum equations [24], as well as equation (A 6), one can rewrite equation (A 4) as follows: 2 p (x, t) = QKir +
∂2 [Tij H( f )] ∂xi ∂xj
+
Mn − M∞n D Mn − M∞n D [p − ρ c20 ] (p − ρ c20 ) δ( f ) + δ( f ) Dt c0 Dt c0
−
∂ ∂ [ρui uj ]nˆ i δ( f ) − [ρui un δ( f )]. ∂xj ∂xi
(A 7)
We can finally conclude that in the linear flow region, where p = ρ c20 , equation (A 7) becomes 2 p (x, t) = QKir +
∂ 2 (ρui uj ) ∂xi ∂xj
H( f ).
(A 8)
Evidently, the only remaining source term in equation (A 8) is second order in the perturbation quantity ui and would be negligible in the derivation of the wave equation from the fluid conservation laws. Therefore, it is shown that the Kirchhoff and the FW-H formulations for a moving medium are equivalent, when the data surface is placed in the linear propagation region.
References 1. Kirchhoff GR. 1883 Zur Theorie der Lichtstrahlen. Ann. Phys. Chem. 18, 663–695. (doi:10.1002/ andp.18832540409) 2. Ffowcs Williams JE, Hawkings DL. 1969 Sound generation by turbulence and surfaces in arbitrary motion. Phil. Trans. R. Soc. Lond. A 264, 321–342. (doi:10.1098/rsta.1969.0031) 3. Brentner KS, Farassat F. 1998 Analytical comparison of the acoustic analogy and Kirchhoff formulation for moving surfaces. AIAA J. 36, 1379–1386. (doi:10.2514/2.558) 4. Yao HD, Eriksson LE, Davidson L, Grundestam O, Peng SH, Eliasson PE. 2012 Aeroacoustic assessment of conceptual low-noise high-lift wing configurations. In AIAA 2012-0383, Proc. 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Nashville, TN, 9–12 January 2012. Reston, VA: American Institute of Aeronautics and Astronautics. 5. Nebenführ B, Yao H, Peng SH, Davidson L. 2013 Hybrid RANS/LES simulations for aerodynamic and aeroacoustic analysis of a multi-element airfoil. In Proc. 19th AIAA/CEAS Aeroacoustics Conference, Berlin, Germany, 27–29 May 2013. Reston, VA: American Institute of Aeronautics and Astronautics. 6. di Francescantonio P. 1997 A new Kirchhoff formulation for transonic rotor noise. J. Sound Vib. 202, 491–509. (doi:10.1006/jsvi.1996.0843) 7. Brentner KS, Farassat F. 2003 Modeling aerodynamically generated sound of helicopter rotors. Prog. Aerosp. Sci. 39, 83–120. (doi:10.1016/S0376-0421(02)00068-4) 8. Strawn RC, Biswas R, Lyrintzis AS. 1996 Helicopter noise predictions using Kirchhoff methods. J. Comput. Acoust. 4, 321–339. (doi:10.1142/S0218396X96000106) 9. Chae S, Yee K, Yang C, Aoyama T, Jeong S, Obayashi S. 2010 Helicopter rotor optimization for the improvement of aeroacoustic performance in hover. J. Aircraft 47, 1770–1783. (doi:10. 2514/1.C000283) 10. Lyrintzis AS, Mankbadi R. 1996 Prediction of the far-field jet noise using Kirchhoff’s formulation. AIAA J. 34, 413–416. (doi:10.2514/3.13079) 11. Uzun A, Lyrintzis AS, Blaisdell GA. 2004 Coupling of integral acoustic methods with LES for jet noise prediction. Int. J. Aeroacoust. 3, 297–346. (doi:10.1260/1475472043499290)
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12. Pan FL, Uzun A, Lyrintzis AS. 2008 Surface integral methods in jet aeroacoustics: refraction corrections. J. Aircraft 45, 381–387. (doi:10.2514/1.23513) 13. Bodony DJ, Lele SK. 2008 Current status of jet noise predictions using large-eddy simulation. AIAA J. 46, 364–380. (doi:10.2514/1.24475) 14. Spalart PR, Shur ML. 2009 Variants of the Ffowcs Williams–Hawkings equation and their coupling with simulations of hot jets. Int. J. Aeroacoust. 8, 477–492. (doi:10.1260/14754 7209788549280) 15. Pierce AD. 1981 Acoustics: an introduction to its physical principles and applications. New York, NY: McGraw-Hill. 16. Morgans RP. 1930 The Kirchhoff formula extended to a moving surface. Lond. Edinb. Dublin Philos. Mag., J. Sci. Ser. 7 9, 141–161. (doi:10.1080/14786443008564988) 17. Morino L. 1974 A general theory of unsteady compressible potential aerodynamics. NASA Contractor Report no. CR-2464. See http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa. gov/19750004821.pdf. 18. Morino L. 1985 Mathematical foundations of integral methods. In Computational methods in potential aerodynamics. New York, NY: Springer. 19. Morino L, Freedman MI, Deutsch DJ, Sipcic SR. 1985 An integral equation method for compressible potential flows in an arbitrary frame of reference. In Computational methods in potential aerodynamics. Southampton, UK: Computational Mechanics Publications. 20. Farassat F, Myers MK. 1988 Extension of Kirchhoff’s formula to radiation from moving surfaces. J. Sound Vib. 123, 451–460. (doi:10.1016/S0022-460X(88)80162-7) 21. Farassat F, Posey J. 2007 A fast method of deriving the Kirchhoff formula for moving surfaces. J. Acoust. Soc. Am. 122, 2965. (doi:10.1121/1.2942569) 22. Lyrintzis AS. 1994 Review: the use of Kirchhoff’s method in computational aeroacoustics. J. Fluids Eng. 116, 665. (doi:10.1115/1.2911834) 23. Mani R. 1990 The radiation of sound from a propeller at angle of attack. NASA Contractor Report no. CR-4262. See http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/ 19900012286.pdf. 24. Ghorbaniasl G, Lacor C. 2012 A moving medium formulation for prediction of propeller noise at incidence. J. Sound Vib. 331, 117–137. (doi:10.1016/j.jsv.2011.08.018) 25. Lee YW, Lee DJ. 2014 Derivation and implementation of the boundary integral formula for the convective acoustic wave equation in the time domain. J. Acoust. Soc. Am. 136, 2959. (doi:10.1121/1.4898427) 26. Farassat F. 1996 The Kirchhoff formulas for moving surfaces in aeroacoustics— the subsonic and supersonic cases. NASA Technical Memorandum no. 110285. See http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19970001733.pdf. 27. Farassat F, Myers MK. 2011 Multidimensional generalized functions in aeroacoustics and fluid mechanics—part 1: basic concepts and operations. Int. J. Aeroacoust. 10, 161–200. (doi:10.1260/1475-472X.10.2-3.161) 28. Wells VL, Han AY. 1995 Acoustics of a moving source in a moving medium with application to propeller noise. J. Sound Vib. 184, 651–663. (doi:10.1006/jsvi.1995.0339) 29. Farassat F, Brentner KS, Dun MH. 1998 A study of supersonic surface sources - the Ffowcs Williams–Hawkings equation and the Kirchhoff formula. In AIAA 98-2375, Proc. 4th AIAA/CEAS Aeroacoustics Conference, Toulouse, France, 2–4 June 1998. Reston, VA: American Institute of Aeronautics and Astronautics. 30. Gelfand IM, Shilov GE. 1964 Generalized functions, volume 1: properties and operations. San Diego, CA: Academic Press. 31. Farassat F, Succi GP. 1983 The prediction of helicopter discrete frequency noise. Vertica 7, 309–320.
