Boundary Integral Equations for Potential Flow Past ...

Computational Methods and Function Theory Volume 11 (2011), No. 2, 375–394

Boundary Integral Equations for Potential Flow Past Multiple Aerofoils Mohamed M. S. Nasser (Communicated by Thomas K. DeLillo) Abstract. Two uniquely solvable boundary integral equations for calculating the incompressible potential flow past multiple aerofoils with smooth boundaries are presented. The kernels of the integral equations are the Neumann kernel and the adjoint Neumann kernel. Numerical examples reveal that the present method offers an effective solution technique for the potential flow problem. Keywords. Potential flow, multiple aerofoils, Riemann-Hilbert problem, generalized Neumann kernel. 2000 MSC. Primary 30E25; Secondary 76B10.

1. Introduction In this paper, we shall consider the two dimensional, steady-state, irrotational flow around multiple aerofoils. We assume also that the fluid is incompressible and free from viscosity and the boundaries of the aerofoils are stationary and impervious. For a single cylindrical aerofoil, the analytical solution of such potential flow problem is known [2, 10, 11]. For two cylindrical aerofoils, the analytical solution is also known [26]. Recently, Crowdy [4, 5] derived the analytical solution of the potential flow problem around any finite number of cylindrical aerofoils. Using Crowdy’s method to solve the potential flow problem around aerofoils other than cylinders requires the knowledge of the conformal mapping from the region exterior to the aerofoils to a region exterior to cylindrical aerofoils. This paper presents a boundary integral method to solve the potential flow problem around general shape aerofoils. Hence conformal mappings can be totally avoided. The boundary integral equation methods are inexpensive, flexible techniques for investigating the potential flow problem. The work of Hess and Smith [9] may be Received November 15, 2010, in revised form April 5, 2011. Published online July 25, 2011. c 2011 Heldermann Verlag ISSN 1617-9447/$ 2.50

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considered as the starting point of this method (see e.g. [3]). The method became one of the most frequently used numerical methods for calculating 2D and 3D potential flow. Various boundary integral equations for studying the external potential flow problem have been discussed in the literature. These integral equations have been derived by first formulating the potential flow problem as a boundary value problem such as a Dirichlet problem, a Neumann problem or a Riemann-Hilbert problem [2, 3, 4, 5, 9, 10, 11, 15, 16]. In this paper, we shall present two uniquely solvable boundary integral equations for determining the complex potential F and the complex velocity W = F 0 of the flow. The complex potential F is an analytic function in the flow region. The function F , which is in general a multi-valued function, can be written in terms of an auxiliary single valued analytic function f . Then, based on the boundary condition that Im F is constant on the boundary of each aerofoil [14, p. 180], the function f and its derivative f 0 can be considered as solutions of certain Riemann-Hilbert problems. We shall then use the results on the relation between the Riemann-Hilbert problem and the boundary integral equations with the generalized Neumann kernel [25, 20, 22] to obtain two uniquely solvable boundary integral equations for determining the auxiliary function f . The kernels of the integral equations obtained are the Neumann kernel and the adjoint Neumann kernel. The plan of this paper is as follows: after the presentation of some auxiliary material in Section 2, we rewrite in Section 3 the complex potential F in terms of an auxiliary function f . In Sections 4 and 5, we present two integral equations with the Neumann kernel and the adjoint Neumann kernel, respectively, for determining the function f . Section 6 presents several numerical examples and Section 7 presents short conclusions.

2. Notations and auxiliary material We consider an unbounded multiply connected region G in the extended complex plane C exterior to m (≥ 1) simply connected regions Gj , j = 1, 2, . . . , m. We assume that the region G is filled with an irrotational incompressible fluid flow and the bounded regions Gj , j = 1, 2, . . . , m, represents m aerofoils or obstacles in the flow path. We assume that the boundaries Γj := ∂Gj of the aerofoils are smooth closed Jordan curves. The orientation of the boundary Γ = ∂G is such that G is always on the left of Γ, i.e., the curves Γ1 , . . . , Γm always have clockwise orientations. The curve Γj is parametrized by a 2π-periodic twice continuously differentiable complex function ηj (t) with non-vanishing first derivative dηj (t) 6= 0, t ∈ Jj := [0, 2π], j = 1, 2, . . . , m. dt S − We define the total boundary Γ by Γ := m j=1 Γj and the bounded region G S by G− := C \ G = m j=1 Gj . We define also the total parameter domain J as

(1)

η˙ j (t) =

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the disjoint union of the intervals Jj . Hence, a parametrization of the whole boundary Γ is defined as the complex function η defined on J by   η1 (t), t ∈ J1 , (2) η(t) := ...   ηm (t), t ∈ Jm . Let H be the space of all real Hölder continuous functions on the boundary Γ. In view of the smoothness of η, a function φ ∈ H can be interpreted via ˆ := φ(η(t)), t ∈ J, as a real Hölder continuous 2π-periodic functions φ(t) ˆ φ(t) of the parameter t ∈ J, i.e.,  ˆ  φ1 (t), t ∈ J1 , ˆ := ... (3) φ(t)  ˆ φm (t), t ∈ Jm , with real Hölder continuous 2π-periodic functions φˆj defined on Jj ; and vice versa. Here and in what follows, for complex-valued or real-valued functions ψ defined on the boundary Γ and for t ∈ J, we will not distinguish between ψ(η(t)) and ψ(t). Let χ[j] be the piecewise constant function defined on J by ( 1, if t ∈ Jj , (4) χ[j] (t) := 0, if t ∈ / Jj , for j = 1, 2, . . . , m. Then, we define the space S by (5)

S = span{χ[1] , . . . , χ[m] }.

It follows from the definition of the space S that a function h ∈ S if and only if h can be written as   h1 , t ∈ J1 , (6) h(t) = ...   hm , t ∈ Jm , with real constants h1 , . . . , hm . For simplicity, the real piecewise constant function h defined on J by (6) will be denoted by (7)

h = (h1 , · · · , hm ).

Let the function A be a complex-valued function defined on Γ with A 6= 0. The generalized Neumann kernel formed with A is defined by 1 A(s) η(t) ˙ (8) N (s, t) := Im . π A(t) η(t) − η(s)

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We also define a real kernel M by 1 A(s) η(t) ˙ (9) M (s, t) := Re . π A(t) η(t) − η(s) The kernel N is continuous and the kernel M has a cotangent singularity type (see [25, 20] for more details). Hence, the operators Z (10) Nµ(s) := N (s, t)µ(t) dt, s ∈ J, J

is a Fredholm integral operator and the operator Z (11) Mµ(s) := M (s, t)µ(t) dt,

s ∈ J,

J

is a singular integral operator. The function A˜ defined by ˙ ˜ = η(t) A(t) A(t)

(12)

is known as the “adjoint function” to the function A (see [25, 20]). Then, the ˜ formed with A˜ is defined by generalized Neumann kernel N ! ˜ 1 A(s) η(t) ˙ ˜ (s, t) := Im (13) N . ˜ η(t) − η(s) π A(t) ˜ by We define also the real kernel M (14)

˜ (s, t) := 1 Re M π

Since A(t) η(s) ˙ = A(s) η(s) − η(t)

A(t) η(t) ˙ A(s) η(s) ˙

˜ A(s) η(t) ˙ ˜ A(t) η(t) − η(s)

! .

˜ η(t) ˙ A(s) η(t) ˙ =− , ˜ η(t) − η(s) η(s) − η(t) A(t)

˜ (s, t) by the adjoint kernel N ∗ (s, t) of the kernel N (s, t) is related to the kernel N ˜ (s, t). (15) N ∗ (s, t) := N (t, s) = −N ˜ (s, t) Similarly, the adjoint kernel M ∗ (s, t) of M (s, t) is related to the kernel M by ˜ (s, t). (16) M ∗ (s, t) = −M ˜ and the singular operator M ˜ be defined as in (10) Let the Fredholm operator N and (11). Then (15) and (16) imply that ˜ ˜ (17) N∗ = −N, M∗ = −M, where N∗ and M∗ are the adjoint operators to the operators N and M, respectively.

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The solvability of boundary integral equations with the generalized Neumann kernel is determined by the index (winding number) of the function A (see [25, 20]). In this paper, we shall assume that the function A is given by t ∈ J.

A(t) = 1,

For this case, the kernel N (s, t) is the well-known Neumann kernel and the kernel N ∗ (s, t) is the adjoint Neumann kernel. Hence, from [20], dim(Null(I − N)) = 0,

dim(Null(I + N)) = m.

3. The external potential flow problem Suppose that F (z) is the complex potential and W (z) = F 0 (z) is the complex velocity of the flow where z = x+i y ∈ G∪Γ. Then the velocity field is u = (u, v) where u(x, y) − i v(x, y) = W (z). The real functions φ(x, y) and ψ(x, y) defined by φ(x, y) + i ψ(x, y) = F (z) are known as the velocity potential and the stream function associated with the incompressible flow. The families of curves φ(x, y) = constant,

ψ(x, y) = constant

are known as the equi-potential curves and the stream lines, respectively, [10, p. 8]. By the Bernoulli Theorem, the pressure coefficient Cp (z) is given by (18)

Cp (z) = 1 − W (z)W (z), (j)

z ∈ G ∪ Γ. (j)

The vertical lift force Fy and horizontal force Fx on the boundary component Γj are given by Blasius’s formula (see [5], [2, p. 433] and [10, p. 112]) Z iρ (j) (j) (19) Fx − i Fy = − W 2 (η) dη, j = 1, 2, . . . , m, 2 Γj where ρ is density of the fluid. The circulation χj of the fluid along the boundary component Γj is given by (See [15, 16]) Z (20) χj = − W (η) dη, j = 1, 2, . . . , m. Γj

(Note that the boundaries Γj are assumed to be clockwise oriented). If the free stream velocity is of unit magnitude and making angle α to the positive real axis, then W (z) can be written as (21)

W (z) = e− i α + w(z),

z ∈ G ∪ Γ,

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where w(z) is the complex disturbance velocity due to the aerofoils Γ1 , . . . , Γm . The velocity disturbance w(z) in unconfined flow is required to vanish far away from the aerofoils, i.e., w(∞) = 0. Hence, the function F (z) can be written as F (z) = e− i α z + F1 (z),

z ∈ G ∪ Γ,

where the F1 (z) is the anti-derivative of the function w(z), i.e., F10 (z) = w(z). The analytic function F1 (z) is in general a multi-valued function on G. However, it can be written as m X (22) F1 (z) = f1 (z) + aj log(z − zj ) j=1

where f1 (z) is a single-valued analytic function in G, zj ∈ Gj are fixed points and the constants aj are chosen to ensure that Z f10 (η) dη = 0, (23) j = 1, 2, . . . , m. Γj

Hence, we have Z Z Z m X 0 0 ak F (η) dη = F1 (η) dη = Γj

Γj

k=1

Γk

dη = −2π i aj , η − zk

which, in view of (20), implies that the constants aj are given by Z Z 1 1 i χj 0 aj = − F (η) dη = − W (η) dη = − , j = 1, 2, . . . , m. 2π i Γj 2π i Γj 2π Let c := f1 (∞) and f (z) := i(f1 (z) − c),

(24)

z ∈ G ∪ Γ.

Hence, f (z) is single-valued analytic function in G with f (∞) = 0 and Z (25) f 0 (η) dη = 0, j = 1, 2, . . . , m. Γj

Thus the complex potential F (z) can be written as (26)

−iα

F (z) = e

z − i f (z) −

m X i χj j=1

2π

log(z − aj ) + c,

z ∈ G ∪ Γ,

and the complex velocity W (z) is given by (27)

W (z) = e

−iα

0

− i f (z) −

m X i χj j=1

1 , 2π z − aj

z ∈ G ∪ Γ.

The constant c has no effect on the velocity field. Hence, to determine the potential function F (z), it is only required to determine the auxiliary function

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f (z). Then, the stream function is given by −iα

ψ(x, y) = Im[e

m X χj ln |z − zj | + constant z] − Re[f (z)] − 2π j=1

and the velocity potential is given by −iα

φ(x, y) = Re[e

m X χj z] + Im[f (z)] + arg(z − zj ) + constant. 2π j=1

In the next two sections, we shall present two methods for calculating the function f (z). The first method is based on a boundary integral equation with the Neumann kernel and the second method is based on a boundary integral equation with the adjoint Neumann kernel.

4. An integral equation with the Neumann kernel The complex potential F (z) must satisfy the condition that Im F (z) is constant on the boundary of each aerofoil so that the boundaries of the aerofoils are streamlines [14, p. 180] (see also [4, 5]). Thus ˆ Im[F (η)] = h,

η∈Γ

ˆ ∈ S . Then, it follows from (26) that f (z) with a piecewise constant function h is a solution of the Riemann-Hilbert problem Re[A(t)f (η(t))] = Im[e

−iα

m X χj ˆ ln |η(t) − zj | + Im[c] − h(t) η(t)] − 2π j=1

where A = 1. Hence, the boundary values of the function f (z) are given by (28)

A(t)f (η(t)) = γ(t) + h(t) + i µ(t)

where m X χj ln |η − zj |, 2π j=1

(29)

γ := Im[e− i α η] −

(30)

ˆ ∈ S, h := Im[c] − h

and µ is unknown function. Then, it follows from [20] (see also [25, Eq. (38)]) that the function µ is the unique solution of the integral equation µ − Nµ = −Mγ

(31) and the function h is given by (32)

1 h = [Mµ − (I − N)γ]. 2

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By obtaining h and µ, we obtain the boundary values of f (z) from (28). Then, the values of the function f (z) can be calculated for z ∈ G by the Cauchy integral formula. Z 1 γ + h + iµ dη. (33) f (z) = 2π i Γ η − z Since h is a piecewise constant function, then we have for z ∈ G, Z 1 h dη = 0. 2π i Γ η − z Thus, to calculate the values of the function f (z) for z ∈ G, it is not required to find the function h since Z 1 γ + iµ (34) f (z) = dη. 2π i Γ η − z

5. An integral equation with the adjoint Neumann kernel An integral equation with the adjoint Neumann kernel can be derived for determining the auxiliary function f (z). By differentiating both sides of (28) with respect to the parameter t, we obtain 0 η(t)f ˙ (η(t)) = γ 0 (t) + i µ0 (t)

(35) where (36)

0

−iα

γ (t) = Im e

m X χj η(t) ˙ − Re 2π j=1

η(t) ˙ η(t) − zj

,

and µ0 is an unknown function. We define a complex-valued function f2 (z) for z ∈ G by m X χj 1 0 (37) f2 (z) = f (z) + . 2π z − z j j=1 Then f2 (z) is analytic in G with f2 (∞) = 0 and has the boundary values (38)

η(t)f ˙ 2 (η(t)) = φ(t) + i ψ(t),

where φ(t) := Im e− i α η(t) ˙

is a known function and m X χj η(t) ˙ ψ(t) := µ (t) + Im 2π η(t) − zj j=1 0

is an unknown function, i.e., f2 (z) is a solution of the Riemann-Hilbert problem ˜ ˜ Re[A(t)f ˙ = η(t). ˙ Then, in view of (38), it 2 (η(t))] = φ(t) where A(t) = η(t)/A(t) follows from [20] (see also [25, Eq. (42)]) that ψ satisfies the integral equation ˜ ˜ (39) (I − N)ψ = −Mφ.

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˜ will not The integral equation (39) can be modified so that the operator M appear in the right-hand side. Let the complex-valued function g(z) be defined for z ∈ G− by g(z) = − i e− i α .

(40)

Then g(z) is analytic in the bounded exterior region G− and has the boundary values ˆ (41) η(t)g(η(t)) ˙ = φ(t) + i φ(t), where ˆ := − Re e− i α η(t) φ(t) ˙ .

(42)

Then, it follows from [20] that φˆ satisfies the integral equation ˜ φˆ = Mφ. ˜ (43) (I + N) By adding (43) to (39) we conclude that the function ˆ (44) ϕ(t) := ψ(t) − φ(t) satisfies the integral equation ˜ = −2φˆ (I − N)ϕ

(45)

which in view of (17) can be written as ˆ (I + N∗ )ϕ = −2φ.

(46)

Since dim(Null(I + N)) = m, the integral equation (46) is not uniquely solvable. To make it uniquely solvable, we shall impose additional conditions. Assume that the circulations χ1 , χ2 , . . ., χm are given. Then, by (25) and (37), the function f2 (z) must satisfy Z (47) f2 (η) dη = − i χj , j = 1, 2, . . . , m. Γj

We define an integral operator J by Z X m (48) Jµ(s) := χ[j] (s)χ[j] (t)µ(t) dt J j=1

where the functions χ[j] are defined by (4). Hence, by (38) and (47), the unknown function ψ satisfies Jψ = −χ := −(χ1 , . . . , χm ). Similarly, it follows from (42) that Jφˆ = 0. Hence, the function ϕ satisfies the additional condition (49)

Jϕ = −χ.

By adding (49) to (46), we obtain the integral equation (50) (I + N∗ + J)ϕ = −2φˆ − χ

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which is uniquely solvable (see [22]). By solving the integral equation (50) for ϕ, we conclude from (38) and from the definitions of the functions φ, ψ, φˆ and ϕ that the boundary values of the analytic function f2 (z) are given by −iα η(t)f ˙ η(t) ˙ + i ϕ(t), 2 (η(t)) = − i e

η(t) ∈ Γ.

By (27) and (37), we have W (z) = e− i α − i f2 (z),

(51)

z ∈ G ∪ Γ.

Thus, the boundary values of the analytic function W (z) are given by (52)

η(t)W ˙ (η(t)) = ϕ(t),

η(t) ∈ Γ.

Then, for z ∈ G, the values of the function W (z) can be calculated by the Cauchy integral formula Z 1 ϕ dη −iα (53) W (z) = e − , z ∈ G. 2π i Γ η˙ η − z Then the pressure coefficient Cp (z) is given for z ∈ G ∪ Γ by (18). For η ∈ Γ, we have ϕ(t)2 (54) Cp (η) = 1 − , η = η(t) ∈ Γ. |η(t)| ˙ 2 (j)

(j)

The vertical lift force Fy (η) and horizontal force Fx (η) are given for η ∈ Γj by Z ϕ2 (t) iρ (j) (j) (55) Fx (η) − i Fy (η) = − dt, j = 1, 2, . . . , m, 2 Jj η(t) ˙ where ρ is the density of the fluid. This method can be also used to calculate the auxiliary function f (z) and hence the potential function F (z). By solving the integral equation (50) for ϕ, the function µ0 is given by m X η(t) ˙ χ j 0 ˆ − Im (56) µ (t) = ϕ(t) + φ(t) . 2π η(t) − z j j=1 Hence, by (35) and (36), the boundary values of the function f 0 (z) are given (57)

0

−iα

η(t)f ˙ (η(t)) = − i e

m X χj η(t) ˙ η(t) ˙ − + i ϕ(t). 2π η(t) − zj j=1

In view of (25), the function f 0 (z) has a single-valued anti-derivative function f (z) in G (see [12, p. 88]). Since f (∞) = 0, we have Z 1 η (58) f (z) = − f 0 (η) log 1 − dη, z ∈ G, 2π i Γ z where the branch of logarithm is chosen which is equal to zero for z = ∞.

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6. Numerical results In this section, the methods described above will be applied to five test regions. For comparison, we have chosen the first three regions which have been considered in Crowdy [5]. Since the functions Ak and ηk are 2π-periodic, a reliable procedure for solving the integral equations (31) and (50) numerically is by using the Nyström method with the trapezoidal rule [1]. If the integrands in the integral equations (31) and (50) are k-times continuously differentiable, then the rate of convergence of the trapezoidal rule is O(1/nk ) where n is the number of collocation points used in the discretization of each boundary component Γj . For analytic integrands, the rate of convergence is better than 1/nk for any positive integer k (see e.g., [13, p. 83]). Since the smoothness of the integrands in (31) and (50) depends on the smoothness of the function η(t) in (2), results of high accuracy can be obtained for very smooth boundaries. Solving the integral equations is then reduced to solving a linear system (59)

Ax = y.

Since the integral equations (31) and (50) are uniquely solvable, then for sufficiently large values of n, the linear system (59) is also uniquely solvable [1]. The linear system (59) is solved using the Gauss elimination method. The computational details are similar to previous works [18, 19] in connection with numerical conformal mapping of multiply connected regions. In Section 4, we have described how the method based on the integral equation (31) can be used to calculate the potential function F (z). The complex velocity W (z) can then be computed by numerically differentiating the complex potential F (z). For the method based on the integral equation (50), we have described in Section 5 how the method can be used to calculate both the potential function F (z) and the velocity function W (z). Numerical tests show that the first method yields more accurate results than the second method for computing the function F (z) especially for points near to the boundary. The accuracy of the first method for points near to the boundary is expected since the density function in the Cauchy integral (33) is a boundary value of an analytic function in G (see [7, § 3] for details). In the numerical calculations below, the streamlines were computed using the method based on the integral equation with the Neumann kernel (31) and the forces were computed using the method based on the integral equation with the adjoint Neumann kernel (50). Example 1. The region G is the unbounded region exterior to the unit circle (see Figure 1) Γ1 : η1 (t) = e− i t , where 0 ≤ t ≤ 2π.

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This example has been considered in [5, Fig. 2]. The streamlines for α = 0 and for several values of the circulation χ1 are shown in Figure 1.

(a)

(b)

(c)

Figure 1. Uniform flow past the circle in Example 1 with α = 0 and: (a) χ1 = 0, (b) χ1 = −10, (c) χ1 = −20. Example 2. The region G is the unbounded region exterior to the three circles (see Figure 2) Γ1 : η1 (t) = e− i t , Γ2 : η1 (t) = −4 i +e− i t , Γ3 : η1 (t) = 4 i +e− i t ,

where 0 ≤ t ≤ 2π. This example has been considered in [5, Figures 7–9]. The streamlines for α = 0 and for several values of the circulations χj , j = 1, 2, 3, are shown in Figure 2. (j) Figure 3 shows the vertical lift forces Fy on the boundaries Γj as a function of the separation of the centers of Γj for ρ = 1 and for several values of circulations χj . (j) For the horizontal force components, we obtain Fx = 0. Example 3. The region G is the unbounded region exterior to the three circles (see Figure 4) Γ1 : η1 (t) = e− i t , Γ2 : η1 (t) = 4 + e− i t , Γ3 : η1 (t) = −4 + e− i t ,

where 0 ≤ t ≤ 2π.

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(a)

(b)

387

(c)

Figure 2. Uniform flow past the three circles in Example 2 with α = 0 and (a) χ1 = χ2 = χ3 = 0, (b) χ1 = χ2 = χ3 = −15, (c) χ1 = χ2 = χ3 = −25. 3 F(2) y

6

2

F(3) y

F(1) y

F(1) y

1 5 0

F(2) y

F(3) y −1 2

4 4

6

(a)

8

10

5

10

15

20

25

30

(b)

Figure 3. Forces on the three circles in Example 2 given as a function of the separation distance between centers with α = 0, ρ = 1 and: (a) χ1 = χ2 = χ3 = −1, (b) χ1 = χ2 = χ3 = −5.

35

40

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This example has been considered in [5, Fig. 10–12]. The streamlines for α = 0 and for several values of the circulations χj , j = 1, 2, 3, are shown in Figure 4. (j) Figure 5 shows the vertical lift forces Fy and the horizontal force components (j) Fx on the boundaries Γj as a function of the separation of the centers of Γj for ρ = 1 and for several values of circulations χj .

(a)

(b)

(c) Figure 4. Uniform flow past the three circles in Example 3 with α = 0 and (a) χ1 = χ2 = χ3 = 0, (b) χ1 = χ2 = χ3 = −5, (c) χ1 = χ2 = χ3 = −10.

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6

F(2) =F(3) y y F(1) y

4

F(2)

3

x

2 F(1) 0

F(2) =F(3) y y

5

1

0.5

389

1

x

0

F(1) y (2)

Fx

F(1) x

−1 −0.5 2

F(3) x

−2 4

6

(a)

8

10

−3 2

F(3) x 4

6

8

10

(b)

Figure 5. Forces on the three circles in Example 3 given as a function of the separation distance between centers with α = 0, ρ = 1 and: (a) χ1 = χ2 = χ3 = −1, (b) χ1 = χ2 = χ3 = −5. Example 4. The region G is the unbounded region exterior to the three curves (see Figure 6) Γ1 : η1 (t) = (1 − 0.2 cos(2t) − 0.2 sin(5t))e− i t , Γ2 : η1 (t) = −4 i +(1 − 0.2 cos(2t) − 0.2 sin(5t))e− i t , Γ3 : η1 (t) = 4 i +(1 − 0.2 cos(2t) − 0.2 sin(5t))e− i t ,

where 0 ≤ t ≤ 2π. The streamlines for α = 0 and for several values of the circulations χj , j = 1, 2, 3, (j) are shown in Figure 6. Figure 7 shows the vertical lift forces Fy on the boundaries Γj as a function of the separation of the centers of Γj for ρ = 1 and for several values of circulations χj . For the horizontal force components, we obtain (j) Fx = 0. Example 5. The region G is the unbounded region exterior to the three curves (see Figure 8) Γ1 : η1 (t) = (1 − 0.2 cos(2t) − 0.2 sin(5t))e− i t , Γ2 : η1 (t) = 4 + (1 − 0.2 cos(2t) − 0.2 sin(5t))e− i t , Γ3 : η1 (t) = −4 + (1 − 0.2 cos(2t) − 0.2 sin(5t))e− i t ,

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(a)

CMFT

(b)

(c)

Figure 6. Uniform flow past the three curves in Example 4 with α = 0 and (a) χ1 = χ2 = χ3 = 0, (b) χ1 = χ2 = χ3 = −15, (c) χ1 = χ2 = χ3 = −25.

8 6

6 4

F(1) y

2 0 −2

F(1) y

5

F(3) y

−4 −6 2

F(3) y

F(2) y

4 4

6

(a)

8

10

F(2) y 5

10

15

20

25

30

(b)

Figure 7. Forces on the three curves in Example 4 given as a function of the separation distance between centers with α = 0, ρ = 1 and: (a) χ1 = χ2 = χ3 = −1, (b) χ1 = χ2 = χ3 = −5.

where 0 ≤ t ≤ 2π.

35

40

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The streamlines for α = 0 and for several values of the circulations χj , j = 1, 2, 3, (j) are shown in Figure 8. Figure 9 shows the vertical lift forces Fy and the hori(j) zontal force components Fx on the boundaries Γj as a function of the separation of the centers of Γj for ρ = 1 and for several values of circulations χj .

(a)

(b)

(c) Figure 8. Uniform flow past the three curves in Example 5 with α = 0 and (a) χ1 = χ2 = χ3 = 0, (b) χ1 = χ2 = χ3 = −5, (c) χ1 = χ2 = χ3 = −10.

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6

F(2) =F(3) y y F(1) y

4 (2)

F(1) y

3

Fx

2 0

F(2) =F(3) y y

5

1

0.5

CMFT

(1) Fx

1 0

F(2) x

F(1) x

−1 −0.5

2

F(3) x

−2 4

6

(a)

8

10

−3 2

F(3) x 4

6

8

10

(b)

Figure 9. Forces on the three curves in Example 5 given as a function of the separation distance between centers with α = 0, ρ = 1 and: (a) χ1 = χ2 = χ3 = −1, (b) χ1 = χ2 = χ3 = −5.

7. Conclusions In this paper, two boundary integral equations (31) and (50) have been presented for determining the complex potential F and the complex velocity W . The derivation of the integral equations is based on expressing the potential function F in terms of an auxiliary function f . The function f and its derivative f 0 satisfy certain Riemann-Hilbert problems which can be solved by boundary integral equation with the generalized Neumann kernel that have been derived and studied in [25, 20]. The first integral equation (31) is a Fredholm integral equation of the second kind with the Neumann kernel. Once the solution of the integral equation is computed, the auxiliary function f is then given by the Cauchy integral formula. Then, the complex potential F can be computed from f . The complex velocity W can be computed by numerically differentiating the function F . The second integral equation (50) is a Fredholm integral equation of the second kind with the adjoint Neumann kernel. This equation has been derived in [21] for a single aerofoil and was extended in this paper to multiple aerofoils (see also [17]). Once the solution of the integral equation is computed, the complex velocity W is then given by the Cauchy integral formula. The complex potential F can be computed as an anti-derivative of the function W . Several numerical examples have been solved using the methods presented. The first three examples have been considered by Crowdy [5]. In the last two examples, we consider aerofoils whose boundaries have complicated geometry. The numerical examples illustrate that the proposed methods yield approximations of high accuracy.

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In the numerical examples presented, the linear system (59) was solved using the Gauss elimination method. However, it is difficult or impossible to use the Gauss elimination method to solve the linear system (59) if the number of aerofoils is large or if the aerofoils lie closed to each other where more discretization points are needed. In these cases, it is recommended to use an alternative efficient method for solving the linear system (59) such as the GMRES iterative method powered by the Fast Multiple Method (FMM) (see e.g. [6, 8, 23, 24]. Acknowledgement. The author acknowledges useful discussions with Professor Ali H. M. Murid. Thanks are also due to anonymous referees for suggesting several improvements.

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18. M. M. S. Nasser, A boundary integral equation for conformal mapping of bounded multiply connected regions, Comput. Methods Funct. Theory 9 (2009), 127–143. , Numerical conformal mapping via a boundary integral equation with the general19. ized Neumann kernel, SIAM J. Sci. Comput. 31 (2009), 1695–1715. , The Riemann-Hilbert problem and the generalized Neumann kernel on unbounded 20. multiply connected regions, The University Researcher (IBB University Journal) 20 (2009), 47–60. 21. M. M. S. Nasser, A. Murid and N. Amin, A boundary integral equation for the 2D external potential flow, Int. J. Appl. Mech. Eng. 11 (1) (2006), 61–75. 22. M. M. S. Nasser, A. H. M. Murid, M. Ismail and E. M. A. Alejaily, A boundary integral equation with the generalized Neumann kernel for Laplace’s equation in multiply connected regions, Appl. Math. Comput. 217 (2011), 4710–4727. 23. V. Rokhlin, Rapid solution of integral equations of classical potential theory, J. Comput. Phys. 60 (2) (1985), 187–207. 24. Y. Saad and M. H. Schultz, GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comp. 7 (3) (1986), 856–869. 25. R. Wegmann and M. M. S. Nasser, The Riemann-Hilbert problem and the generalized Neumann kernel on multiply connected regions, J. Comp. Appl. Math. 214 (2008), 36–57. 26. B. R. Williams, An exact test case for the plane potential flow about two adjacent lifting airfoils, Aeronautical research council reports and memoranda, No. 3717, 1971. Mohamed M. S. Nasser E-mail: mms [email protected] Address: King Khalid University, Faculty of Science, Department of Mathematics, P. O. Box 9004, Abha, Saudi Arabia. Ibb University, Faculty of Science, Department of Mathematics, P. O. Box 70270, Ibb, Yemen.