This theoretical study determines the aerodynamic loads on an aerofoil with a prescribed porosity distribution in a steady incompressible flow. A Darcy porosity condition on the aerofoil surface furnishes a Fredholm integral equation for the pressure distribution, which is solved exactly and generally as a Riemann–Hilbert problem provided that the porosity distribution is Hölder-continuous. The Hölder condition includes as a subset any continuously differentiable porosity distributions that may be of practical interest. This formal restriction on the analysis is examined by a class of differentiable porosity distributions that approach a piecewise, discontinuous function in a certain parametric limit. The Hölder-continuous solution is verified in this limit against analytical results for partially porous aerofoils in the literature. Finally, a comparison made between the new theoretical predictions and experimental measurements of SD7003 aerofoils presented in the literature. Results from this analysis may be integrated into a theoretical framework to optimize turbulence noise suppression with minimal impact to aerodynamic performance.

1. Introduction

The study of aerodynamic loads on permeable aerofoils can be motivated by the need for passive design structures to reduce the aerodynamic self-noise of fluid-loaded bodies. The trailing edge is an unavoidable source of this self-noise for aerodynamic structures and is the subject of a large body of research developed to model, measure and mitigate the noise due to the edge interaction with turbulent eddies [1–6]. Turbulence noise can be reduced by changing the acoustical impedance near the edge [2]. A number of theoretical studies have sought to predict the impact of the edge boundary condition on the trailing-edge scattering mechanism. The seminal work of Ffowcs Williams & Hall [3] showed that the far-field acoustic intensity of a turbulent source in the presence of an impermeable rigid half plane is M⁻³ louder than a turbulent eddy in free space with no solid boundaries, where M is the eddy Mach number. Crighton & Leppington [7] confirmed this result using a different approach based on the Wiener–Hopf method and showed that a sufficiently limp edge scatters a weaker field of intensity M⁻² louder than a free-space turbulent eddy; this Mach-number dependence is identical to the turbulence noise scaling of an edgeless perforated screen in the ‘acoustically transparent’ low-porosity limit identified by Ffowcs Williams [8]. Howe [4] also used the Wiener–Hopf method to predict the scattered field from an elastic edge, including its critical dependence on the coincidence frequency. Using the poroelastic plate model of Howe [5], Jaworski & Peake [6] examined the scattering of turbulent noise sources from a poroelastic half-plane. Motivated by the unique wing attributes of silent owl species, these authors showed that trailing-edge porosity and elasticity can be tuned to effectively eliminate the predominant scattering mechanism of trailing edge noise. Recent analytical work by Ayton [9] extended these results to examine the effects of finite chord for partially porous aerofoils. Also, Cavalieri et al. [10] developed a boundary element framework to investigate the elasticity and porosity of finite-chord aerofoils on the scattered acoustic field, noting the complementary noise reduction in high- and low-frequency ranges due to elasticity and porosity effects, respectively. The aforementioned works consider only stationary bodies and represent porosity with the Rayleigh conductivity of a thin perforated surface, which neglects any viscous effects within the pores. Weidenfeld & Manela [11] predicted that porous noise reductions can indeed persist when a viscous Darcy porosity condition is applied to a pitching aerofoil. However, to be useful in the design of any practical application, these aeroacoustic works need a complementary assessment of porosity on the aerodynamics.

The generalized aerofoil aerodynamic theory of Woods [12] considered the aerodynamics of porous foils in inviscid, steady and subsonic flow, where the pressure jump across the wall of a hollowed aerofoil was linearly related to the local normal flow velocity. However, in contrast to this study and passive porous aerofoil experiments considered herein, Woods assumed a prescribed pressure distribution along the interior surface of the aerofoil, whereas here the upper and lower surfaces of the aerofoil communicate through the Darcy boundary condition. The Darcy boundary condition holds for small Reynolds numbers based upon the pore permeability and seepage velocity [11] and is tacitly assumed to be valid in the analysis herein. From intuition and according to measurements by Geyer et al. [13,14], the aerodynamic performance of a porous aerofoil is expected to be worse than for a non-porous aerofoil, where an increase in the extent of the porous material decreases the lift and increases the drag. Numerical computations by Bae & Moon [15] corroborate these trends, demonstrate the ability of porous trailing edges to suppress tonal peaks in the acoustic signature, and suggest that the optimization of the porosity distribution could enable greater noise reductions (e.g. [16]). Hence, there is a potential trade-off between the acoustical benefits of porosity and its negative impact on aerodynamic performance. Recent experimental work by Geyer & Sarradj [14] investigated the aerodynamic noise from aerofoils with a finite-length porous trailing edge in an effort to incorporate the acoustic advantages of porosity. They showed that, depending on the porous material, aerofoils with porosity at the trailing-edge section only can still lead to a notable noise reduction, while maintaining a certain level of aerodynamic performance over a fully-porous aerofoil. The impact of a finite region of uniform porosity along the aft portion of aerofoil has been examined theoretically by Iosilevskii [17,18], resulting in closed-form expressions for pressure distribution, lift and pitching-moment coefficients, and seepage drag of the aerofoil. However, it is unknown what impact a variation in porosity distribution would have on the aerofoil performance, which may be optimized for noise suppression in conjunction with an external aeroacoustic analysis.

The aerodynamic impact of a functional porosity gradient is addressed in this work. The essential singular integral equation is derived and solved exactly for the broad class of Hölder-continuous porosity distributions. The resulting general expression for the pressure distribution may be evaluated numerically and is evaluated here in closed form for the special case of uniform porosity. Furthermore, analytical and numerical evaluations of this general result in the limit of a discontinuous porosity jump are demonstrated to match the analytical work of Iosilevskii [18]. Lastly, the pressure distribution for a porous SD7003 aerofoil is integrated to furnish a lift prediction, which is compared and contrasted against the experimental data of Geyer et al. [13].

2. Mathematical model

Consider a thin aerofoil under the assumption of small disturbances in a two-dimensional, steady incompressible flow. The solution to the flow field may be written as the linear combination of two velocity potential functions [19],

ϕ = ϕ^{t} + ϕ^{l},

2.1

where ϕ^t denotes the flow field correction due to aerofoil thickness, and ϕ^l is the lifting flow field due to aerofoil camber and angle of attack. The symmetry of the thickness problem requires the same pressure distribution above and below the aerofoil, and thus no pressure jump exists across the aerofoil. This fact holds regardless of whether or not the aerofoil is porous. Therefore, porosity does not affect the solution of the thickness problem presented in the classical literature [19,20]. However, the thickness of a generalized porous aerofoil can be absorbed into the porosity distribution function and is represented in the lifting problem. The lifting problem is now formulated and solved.

Suppose a mean background flow velocity oriented in the x-direction such that $U = U \hat{i}$ , and the local flow rate, w_s, directed along the unit normal to the aerofoil surface, $\hat{n} = (- \partial z_{a} / \partial x, 1)$ , is given by

w_{s} = (\nabla ϕ + U) \cdot \hat{n} .

2.2

The function z_a(x) defines the camber line of the wing, e.g. for a flat aerofoil at angle of attack α, dz_a/dx=−α. Classical linear aerodynamic theory requires the ratios of the flow perturbation velocities relative to U and the local slope of the aerofoil to be small [21], say, O(λ), such that O(λ²) terms are neglected. (Note that this ordering scheme permits the present analysis to hold for weakly compressible flows provided that the Mach number is also O(λ). The interested reader may wish to consult Van Dyke [22] and references therein for consideration of Mach number expansions, which are not pursued in detail here.) After neglecting higher-order terms, (2.2) becomes

w (x, z) = w_{s} + U \frac{d z_{a}}{d x},

2.3

where w(x,z)=∂ϕ/∂z. The perturbed flow velocity in the field is also related to the bound vorticity distribution on the aerofoil, γ(x), by [21]

w (x, z) = - \frac{1}{2 π} \int_{- 1}^{1} \frac{(x - ξ) γ (ξ)}{{(x - ξ)}^{2} + z^{2}} d ξ,

2.4

where x and z have been non-dimensionalized by the aerofoil semi-chord. For an aerofoil with a non-dimensional Darcy-type porosity distribution R(x), the local flow rate is linearly proportional to the porosity and vorticity distribution [23]:

w_{s} = ρ U C R (x) γ (x) .

2.5

The combination of equations (2.3)–(2.5) evaluated at the aerofoil surface (z=0) furnishes a Fredholm integral equation for the vorticity distribution,

2 ρ U C R (x) γ (x) - \frac{1}{π} ⨍_{- 1}^{1} \frac{γ (t)}{t - x} d t = - 2 U \frac{d z_{a}}{d x},

2.6

where constants ρ, U and C define the air density, mean flow velocity, and the porosity coefficient, respectively.

The dimensionless pressure jump is linearly related to the vorticity distribution by [21]

p (x) = \frac{p_{u} (x) - p_{l} (x)}{(1 / 2) ρ U^{2}} = - 2 \frac{γ (x)}{U},

2.7

where p_u and p_l denote the dimensional pressure distributions above and below the wing.

According to the Darcy boundary condition, the local flow velocity directed along the unit normal to the aerofoil surface is

w_{s} = - \frac{ζ}{μ n d} (p_{u} - p_{l}),

2.8

where ζ, μ, n and d denote the permeability of the solid porous medium, the dynamic viscosity of the fluid, the open area fraction of the porous material and the thickness of the aerofoil, respectively. For real aerofoils in ordinary scenarios, the values of κ, μ and n are constant, but the thickness d=d(x) varies along the chord.

Also, equations (2.5) and (2.7) together yield

w_{s} = - C R (x) (p_{u} - p_{l}) .

2.9

Therefore, we can define the multiplication of the porosity coefficient, C, and porosity distribution, R(x), based on the physical properties of the aerofoil and surrounding fluid as follows:

C R (x) = \frac{ζ}{μ n d} .

2.10

By substitution of (2.7) into (2.6), the following integral equation is obtained:

ρ U C R (x) p (x) - \frac{1}{2 π} ⨍_{- 1}^{1} \frac{p (t)}{t - x} d t = 2 \frac{d z_{a}}{d x} .

2.11

The above equation has been non-dimensionalized using U and

\frac{1}{2} ρ U^{2}

as the velocity and pressure scales, respectively. This integral equation is identical to that examined by Iosilevskii, eqn (9) in [17] with a change of variable ϵ=ρUC and R(x)=H(x−a), where H(x) is the Heaviside function and a is the chordwise location where the non-porous and uniformly porous segments meet. The class of singular integral equations with Cauchy kernels in the form of (2.11) can be formulated and solved as a Riemann–Hilbert problem [24,25].

3. Solution of the aerofoil pressure distribution

The integral equation (2.11) is now solved as a Riemann–Hilbert problem. Comparing (2.11) with the canonical singular integral eqn (47.1) in [24],

A (x) p (x) + \frac{B (x)}{π i} \int_{L} \frac{p (t)}{t - x} d t = f (x),

3.1

we have

A (x) = δ R (x) = \frac{ψ (x)}{2}, B (x) = - \frac{i}{2}, f (x) = 2 \frac{d z_{a}}{d x},

3.2

where the dimensionless parameter δ=ρUC embodies the interaction between fluid and aerofoil porosity, while ψ(x)=2δR(x) also contains the aerofoil geometry effects. Note that L is a smooth contour that contains points t and x, and A(x), B(x) are functions given on L. To make progress, the following assumptions must be satisfied [24]:

(i) The line L consists of a finite number of (smooth) non-intersecting contours, which is here a single open contour from −1 to 1 on the real axis.
(ii) The functions A(x) and B(x) must be Hölder-continuous. A function h is Hölder-continuous when there are non-negative real constants α and β such that the relation
$| h (x) - h (y) | \leq α | x - y |^{β}$ 3.3
holds everywhere on L.
(iii) The sum and difference functions S(x)=A(x)+B(x) and D(x)=A(x)−B(x) do not vanish anywhere on L.

The index κ of the Fredholm integral equation (2.11) is identically zero, κ≡0 [24,26].

The general solution for the pressure distribution on an aerofoil with a Hölder-continuous porosity distribution R(x) is now pursued. Following the procedure of [24], define the set of auxiliary functions

G (x) = \frac{A (x) - B (x)}{A (x) + B (x)} = \frac{ψ (x) + i}{ψ (x) - i}

3.4

and

Γ (x) = \frac{1}{2 π i} ⨍_{- 1}^{1} \frac{\log G (t)}{t - x} d t = ⨍_{- 1}^{1} \frac{k (ψ (t))}{t - x} d t,

3.5

to obtain the fundamental function Z(x),

\begin{aligned} Z (x) & = \sqrt{A^{2} (x) - B^{2} (x)} x^{- κ / 2} e^{Γ (x)} \end{aligned}

3.6

\begin{aligned} = \frac{\sqrt{1 + ψ^{2} (x)}}{2} \exp (⨍_{- 1}^{1} \frac{k (ψ (t))}{t - x} d t), \end{aligned}

3.7

where

k (ψ (x)) = (1 / π) \cot^{- 1} ψ (x)

for real ψ(x).

Substitution of equations (3.4)–(3.6) into the general solution given by (47.13) in [24] yields

p (x) = A^{*} (x) f (x) - \frac{B^{*} (x) Z (x)}{π i} ⨍_{- 1}^{1} \frac{f (t) d t}{Z (t) (t - x)} + B^{*} (x) Z (x) P_{κ - 1} (x),

3.8

where P_κ−1(x) is an arbitrary polynomial of degree not greater than κ−1 (P_κ−1(x)≡0 for κ=0), and

A^{*} (x) = \frac{A (x)}{A^{2} (x) - B^{2} (x)} = \frac{2 ψ (x)}{1 + ψ^{2} (x)}

3.9

and

B^{*} (x) = \frac{B (x)}{A^{2} (x) - B^{2} (x)} = \frac{- 2 i}{1 + ψ^{2} (x)} .

3.10

Finally, the substitution of equations (3.9), (3.10) into (3.8) gives the following pressure distribution for an aerofoil with the prescribed porosity distribution R(x):

\begin{aligned} p (x) & = \frac{4 ψ (x)}{1 + ψ^{2} (x)} \frac{d z_{a}}{d x} \\ - \frac{4}{π \sqrt{1 + ψ^{2} (x)}} \exp (⨍_{- 1}^{1} \frac{k (ψ (t))}{t - x} d t) \\ \times ⨍_{- 1}^{1} \frac{d z_{a} / d t}{\sqrt{1 + ψ^{2} (t)} \exp (⨍_{- 1}^{1} (k (ψ (ξ)) / (ξ - t)) d ξ) (x - t)} d t . \end{aligned}

3.11

Recall that ψ(x)=2ρUCR(x) and

k (ψ (x)) = (1 / π) \cot^{- 1} ψ (x)

for real ψ(x). Equation (3.11) supplies the pressure jump across a thin aerofoil with any Darcy-type porosity distribution, provided that this distribution is Hölder-continuous. We note that the Hölder condition includes any continuously differentiable porosity distributions that may be of practical interest.

In general, (3.11) must be evaluated numerically, but analytical progress can be made for a uniformly porous aerofoil. In the next section, the general solution (3.11) furnishes closed-form expressions for the pressure distribution over a uniformly porous aerofoil. The theoretical result (3.11) is also shown to hold in the discontinuous limit of a partially porous aerofoil, where the Hölder continuity condition formally breaks down.

4. Special cases

In this section, the general solution (3.11) is demonstrated for aerofoils with uniform porosity, and for partially porous aerofoils composed of a non-porous leading-edge section attached to a trailing-edge section of uniform porosity.

(a) Uniformly porous aerofoils

For the uniformly porous aerofoil, R(x)=1 and ψ=2δ is a constant. Therefore, (3.5) becomes

Γ (x) = k (2 δ) ⨍_{- 1}^{1} \frac{d t}{t - x} = \ln {(\frac{1 - x}{1 + x})}^{k (2 δ)},

4.1

and the pressure distribution obtained by (3.11) can be written in the following form:

p (x) = \frac{8 δ}{1 + 4 δ^{2}} \frac{d z_{a}}{d x} - \frac{4}{π (1 + 4 δ^{2})} {(\frac{1 - x}{1 + x})}^{k (2 δ)} ⨍_{- 1}^{1} \frac{d z_{a} / d t}{x - t} {(\frac{1 + t}{1 - t})}^{k (2 δ)} d t .

4.2

This result holds generally for any camber line z_a(x). However, if we restrict ourselves to a flat aerofoil at angle of attack α, such that dz_a/dx=−α, then the pressure distribution is

p (x) = \frac{- 4 α}{\sqrt{1 + 4 δ^{2}}} {(\frac{1 - x}{1 + x})}^{k (2 δ)} .

4.3

The obtained pressure distribution (4.3) for the uniformly porous aerofoil is the same as the result of Iosilevskii [18] for an aerofoil with constant porosity that was determined using an independent asymptotic approach. According to (4.3), increasing the porosity parameter decreases the pressure distribution over the uniform-porosity aerofoil, as illustrated in figure 1a. For δ≫1, the pressure distribution becomes increasingly flat with the value

p (x) \sim \frac{- 2 α}{δ},

4.4

and all of the substantial variations in pressure jump are shifted closer to the leading and trailing edges. This trend can be seen in the pressure distributions normalized by the high porosity limit (4.4) shown in figure 1b. Note that the singular behaviour of the normalized pressure jump near the leading edge (

x \to - 1

) in this case is

p (x) \sim \frac{- 2^{k (2 δ) + 2} α}{\sqrt{1 + 4 δ^{2}}} {(1 + x)}^{- k (2 δ)},

4.5

while the regular behaviour near the trailing edge (

x \to 1

) is approximated by

p (x) \sim \frac{- 2^{- k (2 δ) + 2} α}{\sqrt{1 + 4 δ^{2}}} {(1 - x)}^{k (2 δ)} .

4.6

Figure 1. Normalized pressure distribution of a uniformly porous flat aerofoil for different porosity parameters δ: (a) pressure jump normalized by angle of attack, −p(x)/α; (b) pressure jump normalized by the high porosity limit, −p(x)/(2α/δ). (Online version in colour.)

The limiting case of a non-porous aerofoil, where the porosity coefficient C=0 and $k (0) = \frac{1}{2}$ , recovers the well-known pressure distribution for a non-porous aerofoil [21]:

p (x) = - 4 α \sqrt{\frac{1 - x}{1 + x}} .

4.7

Note that all integrated loads such as lift, pitching moment and seepage drag can be determined for the uniformly porous aerofoil from the pressure distribution provided by (4.2).

(b) Partially porous aerofoils

The general result (3.11) for a generic Hölder-continuous porosity distribution is now demonstrated to also hold in the discontinuous limit of a partially porous thin aerofoil. The aerodynamic impact of a finite, uniform porosity distribution along the aft portion of an aerofoil has been examined theoretically by Iosilevskii [17,18], resulting in closed-form expressions for pressure distribution, lift and pitching-moment coefficients, and seepage drag of the aerofoil, which can be reproduced numerically in the discontinuous limit of a continuous porosity distribution. Attention is again given below only to the pressure distribution, from which all of the aerodynamic coefficients can be derived.

Suppose a thin aerofoil with the following prescribed differentiable porosity distribution (cf. [11,27]):

\frac{ψ (x)}{2 ρ U} = C R (x) = \frac{1}{2} (1 + \tanh [r (x - a)]) .

4.8

The porosity distribution given in (4.8) is continuously differentiable and therefore automatically Hölder-continuous, and the general solution (3.11) for the pressure distribution is valid. We note that

\tanh [r (x - a)] \to \pm 1

r \to \infty

for

x ≷ a

, enabling the pressure distribution (3.11) to be written in the following form in the case of a thin aerofoil with parabolic camber line, in which dz_a/dx=−α−βx, as

r \to \infty

\begin{aligned} p (x) & = \frac{- 4 (α + β x) ψ (x)}{1 + ψ^{2} (x)} + \frac{4}{π \sqrt{1 + ψ^{2} (x)}} {| \frac{a - x}{1 + x} |}^{1 / 2} {| \frac{1 - x}{a - x} |}^{\cot^{- 1} C / π} \\ \times ⨍_{- 1}^{1} \frac{α + β t}{\sqrt{1 + ψ^{2} (t)} (x - t)} {| \frac{1 + t}{a - t} |}^{1 / 2} {| \frac{a - t}{1 - t} |}^{\cot^{- 1} C / π} d t, \end{aligned}

4.9

where

\frac{ψ (x)}{2 ρ U} \to {\begin{cases} 0 & for x < a, \\ C & for x > a . \end{cases}

4.10

From (4.10), note that as $r \to \infty$ the aerofoil with given porosity distribution (4.8) represents a partially porous thin aerofoil, composed of an impermeable forward segment connected to an aft permeable section with a constant porosity distribution. Figure 2 illustrates the porosity distribution (4.8) for the illustrated case of a=−0.5 with $r = 10, \infty$ and the resulting pressure distribution for a flat aerofoil (β=0). In the limit $r \to \infty$ , the present model is validated by the independent asymptotic analysis of Iosilevskii [18] for partially porous aerofoils.

Figure 2. Porosity and pressure distributions of a thin aerofoil with a prescribed differentiable porosity distribution given by (4.8) with a=−0.5: (a) porosity distributions for r=10 and $r \to \infty$ ; (b) pressure distributions for r=10 and the singular limit as $r \to \infty$ for the flat aerofoil. The dashed line indicates Iosilevskii’s result, eqn (13) in [18]. (Online version in colour.)

Note that the porosity distribution (4.8) is an example of a Hölder-continuous function that behaves as a discontinuous piecewise function in the limit $r \to \infty$ to represent a partially porous aerofoil. Other types of functions, e.g. piecewise continuous functions among others, may be used to attain the same result.

5. Comparison with experimental data for porous SD7003 aerofoils

This section compares and contrasts the obtained theoretical result for lift coefficient using the pressure distribution (3.11) against experimental measurements by Geyer et al. [13] of aerofoils constructed of uniform porous material at various flow speeds U. The chord-based Reynolds number varies between approximately 4×10⁵ and 8×10⁵, and the Mach number lies in the range 0.07–0.14. Their experimental study cut slabs of porous textiles into a modified semi-symmetrical SD7003 aerofoil shape. This process was repeated to create a set of aerofoils, each of which was constructed using a single textile. Each textile has an intrinsic air flow resistivity, r, which can be measured from a static pressure drop test of a uniform slab of material using [13]

r = \frac{Δ p}{w_{s} d},

5.1

where Δp and d denote the pressure drop and the thickness of the porous sample, respectively. According to the theoretical model and equation (5.1), the porosity coefficient, C, and porosity distribution, R(x), can be written in terms of the flow resistivity of the textile and the thickness distribution of the SD7003 aerofoil section:

C = \frac{1}{r} and R (x) = \frac{1}{d (x)} .

5.2

The slope of the camber line, dz_a/dx, and thickness distribution, d(x), of the SD7003 aerofoil are represented in the theoretical model by curve fits to aerofoil coordinate data in [28]. These curves are based upon standard formulae describing NACA aerofoils and are presented in appendix A.

In the experimental study, aerofoils are placed in an open jet wind tunnel such that its spanwise extent is greater than the nozzle diameter, which is circular and of Witoszynski type. In an attempt to make a comparison with the present theoretical model, the measured lift force, F_L, on the wing is converted into a lift coefficient,

c_{L} = \frac{F_{L}}{(1 / 2) ρ U^{2} l S},

5.3

where l and S denote the chord length and estimated wetted wing span, respectively.

Figure 3 compares the predicted and measured lift coefficients of a porous SD7003 aerofoil at zero angle of attack with various physical porosity properties. The lift coefficient measured experimentally for the non-porous aerofoil (δ=0), c_L≃0.07, is less than the expected value based on the theory, c_L=0.0974. One would not expect these numbers to match exactly, as Geyer et al. [13] themselves indicated that angle-of-attack corrections to their raw lift data were abandoned due to their experimental configuration. However, both theory and experiment show qualitatively that the lift coefficient decreases with increasing porosity parameter δ as expected. For small δ, the experimental measurements agree well with the theoretical model, and changes to the lift coefficient become less sensitive to the porosity parameter as it increases. For porosity parameter values above the approximate value δ≈0.01, the theoretical predictions and the experimental data diverge: the experiments yield a positive lift for all δ considered, yet the model predicts negative lift at large δ. This latter trend suggests that there may be a predominant physical flow feature of porous aerofoils with high porosity values that is not considered by the present model. High porosity values may invalidate the small pore-based Reynolds number restriction required by the Darcy boundary condition and the merit the investigation of more general porosity laws, such as the Ergun model [15,29]. The sensitivity of the aerodynamic loads to the choice of porosity boundary condition at large δ is beyond the scope of this work and is the subject of ongoing research.

In the theoretical model, we observe the change to negative lift coefficient and reverse pressure distribution after some porosity parametric value δ₀, which depends on the mean camber line of the aerofoil. As is shown in figure 4, the singular pressure distribution at the leading edge starts from positive infinity, dips to negative values away from the leading edge, and then changes sign at a point ahead of the trailing edge. The location of the sign change moves towards the trailing edge as the porosity parameter δ increases. Note that the lift coefficient remains negative for large porosity constants. Therefore, for any cambered aerofoil there is a porosity constant δ₀ beyond which the aerofoil produces a negative lift coefficient for δ>δ₀. The change of sign in the pressure distribution occurs due to the aerofoil camber, as discussed in appendix B for the uniformly porous special case. Porous symmetric aerofoils at positive angle of attack maintain a positive pressure distribution and integrated lift for all porosity parameters.

Figure 4. Pressure distribution of a porous SD7003 aerofoil at zero angle of attack for various porosity constants δ, based on the theoretical model. (Online version in colour.)

6. Conclusion

This paper presents the exact solution for the pressure distribution over an aerofoil in a steady incompressible flow with a prescribed Hölder-continuous porosity distribution. Aerodynamic loads coefficients, lift, moment coefficients and seepage drag can be obtained in closed form for the special case of a uniformly porous aerofoil. Previous analytical results for partially porous aerofoils are recovered by the new general solution for certain limiting cases of piecewise-continuous and differentiable porosity distributions, which further verifies the present results. A comparison of the lift prediction for a porous SD7003 aerofoil against available experimental data indicates good agreement for sufficiently small values of the non-dimensional porosity parameter that depends on the flow and porosity of the aerofoil material. For large values of the porosity parameter, the model predicts negative lift, a phenomenon due to the camber of porous aerofoils and not the angle of attack. Experimental data at large porosity parameter values are positive for all available data and suggest a missing physical feature in the present model at these high porosity cases that is the subject of future investigation. Further extensions of the current work could also include unsteady aerofoil motions, which would rely on the general solution of (2.6), where the theoretical frameworks of Theodorsen [30] and Jaworski [31] could be appropriate.

Data accessibility

This work does not contain any original experimental data, and all of the theoretical results can be generated from the equations provided in the paper.

Authors' contributions

R.H. derived the mathematical model and its solution and generated the numerical results. J.W.J. revised the mathematical model. Both authors contributed to the writing of the manuscript and to the analysis of the results. All authors gave final approval for publication.

Competing interests

We declare we have no competing interests.

Funding

This research was supported in part by a Faculty Innovation Grant (FIG) to J.W.J. from Lehigh University.

Acknowledgements

The authors gratefully acknowledge the inspirational discussions with Prof. Philip Blythe and Prof. Gil Iosilevskii at the early stages of this work. The authors thank and acknowledge Dr Thomas Geyer for sharing his experimental data to furnish a comparison with the present theory. The insightful comments of the anonymous reviewers concerning details at the aerofoil leading edge and the applicability of the present results to weakly compressible flows are gratefully acknowledged.

Appendix A

The slope of the mean surface of the wing, dz_a/dx, and thickness distribution, d(x), are based on the SD7003 aerofoil coordinates given in [28] and are approximated by the following expressions:

\begin{aligned} \frac{d z_{a}}{d x} & = 0.0456479 + 0.00359184 {(1 + x)}^{- 1 / 2} - 0.179623 (1 + x) + 0.287101 {(1 + x)}^{2} \\ - 0.270092 {(1 + x)}^{3} + 0.134608 {(1 + x)}^{4} - 0.0270882 {(1 + x)}^{5} \end{aligned}

A 1

and

\begin{aligned} d (x) & = - 0.00256 + 0.21524 {(1 + x)}^{1 / 2} - 0.09707 (1 + x) \\ + 0.057069 {(1 + x)}^{2} - 0.059067 {(1 + x)}^{3} \\ + 0.131062 {(1 + x)}^{4} - 0.0790595 {(1 + x)}^{5} \\ + 0.0160312 {(1 + x)}^{6} . \end{aligned}

A 2

Appendix B

The general solution (3.11) for Hölder-continuous porosity distributions is here shown to recover the theoretical results presented by Iosilevskii [18] for parabolic, uniform-porosity aerofoils. A closed-form expression is obtained for the pressure distribution over a uniformly porous aerofoil with parabolic camber line, for which the camber-to-chord ratio is β/4, and

\frac{d z_{a}}{d x} = - α - β x .

B 1

Substitution of (B1) into (4.2) leads to the closed-form expression for the pressure distribution over a uniformly porous aerofoil with a parabolic camber line. Following the evaluation of the Cauchy principal value integral,

⨍_{- 1}^{1} \frac{(- α - β x)}{x - t} {(\frac{1 + t}{1 - t})}^{k (2 δ)} d t = π \sqrt{1 + 4 δ^{2}} (α + β x + 2 k (2 δ) β) - π ψ (α + β x) {(\frac{1 + x}{1 - x})}^{k (2 δ)},

the pressure distribution is

p (x) = \frac{- 4}{\sqrt{1 + 4 δ^{2}}} (α + β (x + 2 k (2 δ))) {(\frac{1 - x}{1 + x})}^{k (2 δ)} .

B 2

This pressure distribution for the uniformly porous aerofoil is the same as the result of Iosilevskii, eqn (40) in [18], for an aerofoil with piecewise-constant porosity using an independent asymptotic analysis approach. According to (B.4), increasing the porosity parameter decreases the pressure distribution over the uniform-porosity aerofoil, as illustrated in figure 5. For δ≫1, the pressure distribution becomes linearly proportional to x with slope −2β/δ,

p (x) \sim \frac{- 2 (α + β x)}{δ},

B 3

when sufficiently far from the leading and trailing edges. For a uniformly porous aerofoil with a parabolic camber line, the lift, pitching moment and seepage drag can be predicted from the pressure distribution provided by (B.6). The sectional lift coefficient is directly calculated by

c_{L} = - \frac{1}{2} \int_{- 1}^{1} p (x) d x = 4 k (2 δ) π α + 4 {k (2 δ)}^{2} π β .

B 4

Note that the obtained lift coefficient for a uniformly porous aerofoil with a parabolic camber line in (B.6) recovers eqn (42) in [18] as

a \to - 1

. The agreement of aerodynamic loads in this limit suggests that the universal constant for the suction force acting on an impermeable leading edge (see appendix E of [18]) is unaffected by the imposition of porosity at the leading edge.

Figure 5. Normalized pressure distribution, −p(x)/β, of a uniformly porous cambered aerofoil at zero angle of attack (α=0) for different porosity constants δ. (Online version in colour.)

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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences cover image

September 2017

Volume 473Issue 2205

Article Information

DOI:https://doi.org/10.1098/rspa.2017.0266
PubMed:28989307
Published by:Royal Society
Online ISSN:1471-2946

History:

Manuscript received13/04/2017
Manuscript accepted24/08/2017
Published online27/09/2017
Published in print30/09/2017

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The steady aerodynamics of aerofoils with porosity gradients

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