Airfoils with boundary layer suction, design and off-design cases

Aerosp. Sci. Technol. 3 (1999) 403–415  1999 Éditions scientifiques et médicales Elsevier SAS. All rights reserved S1270-9638(99)00105-4/FLA

Airfoils with boundary layer suction, design and off-design cases Richard Eppler a a Institut A für Mechanik, Universität Stuttgart, Pfaffenwaldring 9, 70550 Stuttgart, Germany

Received 27 January 1999; revised and accepted 11 October 1999

Abstract

The design case of an airfoil with distributed suction specifies, together with the suction compartments and the suction pressure, the porosity of the surface. To find the porosity for the design case and to evaluate the off-design cases, it is necessary to know the relationships between the pressure difference across the surface, the porosity, and the (average) suction velocity. These relationships are derived under the assumption that the porosity consists of round holes of small diameter. The off-design cases are presented for several designs.  1999 Éditions scientifiques et médicales Elsevier SAS airfoils / boundary-layer suction / design / off-design / pipe inlet flow

Zusammenfassung

Tragflügelprofile mit Grenzschicht-Absaugung, Entwurf und Off-Design Fälle. Der Entwurfsfall eines Profils mit kontinuierlicher Absaugung bestimmt, zusammen mit den Absaugekammern und ihren Drücken, die Porosität der Oberfläche. Der Druckunterschied zwischen Kammer und Außenströmung ändert sich mit dem Flugzustand, die Porosität bleibt unverändert. Die Beziehung zwischen dem Druckunterschied, der Porosität und der (mittleren) Absaugegeschwindigkeit wird benötigt, um die Porosität aus dem Entwurfsfall festzulegen, und um die Absaugegeschwindigkeit unter anderen (off-design) Bedingungen zu berechnen. Diese Beziehung wird unter der Voraussetzung hergeleitet, daß die Porosität aus Kreislöchern kleinen Durchmessers besteht. Für verschiedene Entwürfe werden vom Entwurf abweichende Bedingungen durchgerechnet.  1999 Éditions scientifiques et médicales Elsevier SAS Tragflügelprofile / Grenzschichtabsaugung / Entwurf / Off-design / Rohr-Einlauf

1. Introduction Boundary-layer suction is a very effective method of reducing the viscous effects on wings, which cause not only the viscous drag but also limit the maximum lift coefficient. Two different means of suction have been investigated in the past: discrete suction through slots and distributed suction. Discrete suction allows an abrupt pressure increase at the location of the slot. Airfoils with this type of suction can be designed without adverse pressure gradients over a wide range of lift coefficients. Such airfoils are intended to achieve laminar boundary layer along the entire airfoil surface [4]. This approach is, however, limited to Reynolds numbers for

which transition does not occur for constant pressure. Moreover, such airfoils have extremely blunt leading edges, and a failure of the suction system would have a severe effect. The required suction was, in the few available experiments, 30% higher than that predicted by theory [11]. Distributed suction is realised by a porous surface through which the air is sucked into one or more compartments. This approach requires much more technical effort than discrete suction. Nevertheless, distributed suction has been successfully applied more often, see [5,14], for example. The present paper considers only distributed suction. The porous surface is formed by holes of small diame-

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ter. It has been shown that such surfaces are a good approximation to continuous suction if the diameter of the holes is small enough (and the number of holes is large enough). Today, such a surface is not difficult to produce, for example, using laser drilling machines. If the potential flow about an airfoil is known for a certain angle of attack, the suction velocity vo (s) can be determined along an arc length s of the airfoil by means of suction laws in such a way that, for example, boundarylayer transition is prevented, or separation of the turbulent boundary layer is prevented. For the Reynolds numbers of gliders or light aircraft, prevention of the transition always prevents laminar separation. A major practical concern is that the suction laws apply only to one design case. No aircraft flies at a single condition. Gusts, maneuvers, weight changes, takeoff, etc. change the flight conditions. The porosity of the surface and the geometry of the suction compartments can not be changed, and the compartment pressures only to a very limited degree. These features are normally determined by the design case. It is thus necessary to consider many off-design cases for which the porosity, and the suction-compartment geometry are given, and the variation of the pressure in the compartments is limited. In the present paper, variation of the pressure in the compartments is not considered. The present paper is an expanded version of reference [10].

Figure 1. Sketch of airfoil with porous surface and many suction compartments.

Figure 2. Porous surface, open area shaded.

2. Porosity, suction velocity and suction drag The airfoil shape is assumed to be given. At the design angle of attack αD , the potential-flow velocity distribution UD (s) can be computed, where s is the arc length along the airfoil contour. A suction law then yields the suction velocity v0D (s) that is realised by means of a porous surface and suction compartments as shown in figure 1. The pressure p(s) outside of the airfoil can be calculated using the Bernoulli equation p(s) +

ρ 2 ρ 2 U (s) = p∞ + U∞ . 2 2

(1)

The pressure pc (s) in the compartments must be lower than p(s) if suction is intended. The porous surface is produced by drilling many small holes in the surface as shown in figure 2. The porosity P (s) is defined by open area . P (s) = total area

The average velocity u¯ in the holes will be analysed in more detail in the next section. The air in a suction compartment has the pressure pc and (nearly) zero velocity. It must be compressed to p∞ and accelerated to U∞ , see figure 3. The power required to accomplish this corresponds to a drag, called the suction drag, which is added to the viscous drag. The pump power dP for an airfoil segment of length ds is   ρ 2 . (4) dP = −vo ds p∞ − pc + U∞ 2 A velocity Uc is defined according to

(2)

The suction velocity vo depends on the pressure difference 1p = p(s) − pc (s) and the porosity. For suction, vo is negative, vo = −u¯ P .

Figure 3. Sketch of suction compartment.

(3)

pc +

ρ 2 ρ 2 ρ Uc = p∞ + U∞ = p + U 2, 2 2 2

(5)

which means that 1p = p − pc =


ρ 2 U − U2 . 2 c

(6)

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3. Boundary-layer computation and suction laws All the boundary-layer computations have been performed using the method of [7]. The same method has been used in several other papers, for example [1,2,6]. This method is based on the integral momentum and energy equations ([12,18]) vo U0 = cf + , U U

(13)

vo , U

(14)

 Z∞ u dy, 1− δ1 = U

(15)

δ20 + (2 + H12)δ2 Figure 4. Sketch of airfoil with one suction compartment.

A velocity V is introduced by V 2 = Uc2 − U 2 ,

δ30 + 3δ3 = cD + (7)

where 0 denotes d/ds and

which is the velocity that would be present in the hole without viscosity. Suction is achieved if Uc > U . The suction drag due to dP is dDS =

dP , U∞

0

cd S = − 0

Uc2 (s) vo (s) ds, 2 U∞ c U∞

Uc2 c , 2 Q U∞

δ3 =

cQ = − 0

vo ds. U∞ c

(10)

(11)

Although a single compartment is the simplest arrangement, more compartments allow lower cds . The suction power PS for an aircraft is ρ 3 PS = U∞ 2

(17)

0

(9)

where cQ is the suction coefficient ZS

  2  u u 1− dy, U U

0

Z∞

where S is the total arc length of the airfoil and c is the chord. If only one suction compartment is used, the suctiondrag coefficient is cd S =

(16)

δ2 =

(8)

and the suction-drag coefficient is ZS

  u u 1− dy, U U

Z∞

δ1 , (18) δ2 δ3 (19) H32 = . δ2 To solve equations (13) and (14), three functions cf (δ2 , δ3 , U ), cD (δ2 , δ3 , U ), and H12 (H32 ), must be known. They are different for laminar and turbulent boundary layers. In the laminar case, the boundary-layer profiles u(x, y) are approximated by a family of profiles depending on a thickness and a shape parameter. In [7], the functions are derived from special solutions of the boundarylayer equations (Hartree profiles) and precisely approximated by the functions H12 =

H12 = 4.02922 − 583.60182 − 724.55916H32
p 2 + 227.1822H32 H32 − 1.51509 [for H32 6 1.57258],

2 H12 = 79.870845 − 89.582142H32 + 25.715786H32

[for H32 > 1.57258],

Zb cdS (y)c(y) dy,

(12)

−b

where 2b is the wing span, y is the coordinate in the spanwise direction, and c = c(y) is the chord. It should be 3 , which noted that the suction power is proportional to U∞ yields high PS for high speeds, even if cQ decreases with increasing Reynolds number.

(20)

cf =

cf =

(21)

ν 2 2.512589 − 1.686095H12 + 0.3915406H12 U δ2
3 − 0.031729H12 (22) [for H32 < 1.57258],
ν 2 1.372391 − 4.226253H32 + 2.221687H32 U δ2 (23) [for H32 > 1.57258],

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cD =

ν 15.707852 − 20.521103H32 U δ2
2 + 6.8377961H32 ,

(24)

where ν is the kinematic viscosity of air. In the turbulent case, empirical functions from [13,17,19], have been modified in [7] to satisfy some asymptotic properties that become important only if suction is applied. This yields 11H32 + 15 , (25) 48H32 − 59   U δ2 −0.232 −1.26 H12 e , (26) cf = 0.045716 (H12 − 1) ν   U δ2 −1/6 . (27) cD = 0.0100 (H12 − 1) ν

H12 =

Transition requires only switching from equations (20)– (24) to equations (25)–(27). In [7], a local transition criterion was used that does not consider the instability history of the boundary layer. An improved empirical criterion is presented in [9] where a ‘contribution’ B to transition is defined. Once the stability limit H32N (U δ2 /ν) is exceeded, the contribution   U δ2 B H32 , ν   U δ2 1.7 0.612 r e (28) = 0.9225(H32N − H32)2 ν is integrated over s and transition is assumed to occur if Zs B ds > 15.

(29)

sN

A roughness factor r allows the effect of distributed surface roughness or wind-tunnel turbulence to be simulated. With this method, suction laws can also be simply formulated. A ‘desired’ H32S is introduced in [8] by H32S

  U δ2 U δ2 =9 . = a + b ln ν ν

(30)

The parameters a and b allow different suction laws to be specified. For example, a = 1.44 and b = 0.2924 prevents the laminar boundary layer from becoming unstable and thus prevents transition. If, for a turbulent boundary layer, b = 0, and a > 1.6 is specified, separation is prevented. The parameters a and b can be varied in order to minimize the suction drag. Usually, it is better to begin the suction earlier because the suction is more effective for thin boundary layers. The suction laws can be satisfied by introducing (30) into (13) and (14). This is, however, only possible if (30) is satisfied from the very beginning of the suction area.

Otherwise, problems with the initial conditions arise. These can be prevented by using (13) and (14) as usual and by (numerically) substituting into them vo cDS − (b + ψ)cfS = U b−1+9 0

δ2 U [b − 9 + H12 (b + 9) + F (9 − H32 )] . + U b−1+9 (31) Here, the factor F is set F = 5 for U 0 < 0 and F = 0 for U 0 > 0, and the coefficients cDS and cfS are computed using in (24) and (22) or (23) H32S from (30). By doing so, (30) is not satisfied from the beginning of the computation, but the solution converges to the desired suction law. In the examples presented later, no difference can be seen between the desired and the achieved H32(x). 3.1. Example Several comparisons between the present, approximative method and exact solutions are given in [7]. It has been demonstrated that the method works very well if no abrupt changes of the pressure gradient or the suction velocity are present. Recently, numerical solutions of the Navier–Stokes equations have become available that allow the method to be tested for more extreme cases, for example [3]. The following example shows results for a flat plate with a section having constant suction velocity vo . The ramps at the beginning and the end of the suction sections are very steep. Three different suction velocities have been evaluated. The results from a numerical solution of the Navier–Stokes equation [3,15,20] are compared with those from the present approximative solution in figure 5. The highest suction velocity yields very abrupt changes in the shape parameter H12 , where the approximative method can not be very precise. Even in this case, however, the velocity gradients ∂u/∂y at the wall agree well. The velocity gradient is, with respect to laminar separation, the most significant parameter. It can, therefore, be concluded that the approximative method allows the design and off-design cases to be studied with adequate precision. 4. Pipe inlet flow The flow in the holes that form the porosity is dominated by viscosity. In pipes, the Hagen-Poiseulle flow (HPF) occurs except at the very beginning. The holes of the porous surface are relatively short compared to their diameter, and the flow can not be described as HPF. Accordingly, the ‘pipe inlet flow’ that gradually forms the HPF must be analysed. An approximate solution which corresponds to the approximate solution of the boundarylayer equations is used. The assumed velocity distribution in the pipe is shown in figure 6.

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Figure 5. Boundary-layer development on a flat plate with a suction section. Solid line: present approximative method; dashed line: numerical solution of the Navier–Stokes equations.

Momentum balance

dI dp = −πR 2 − 2πR|τ |, dx dx

(34)

where 2uo µ δ is the shear stress at the wall, µ is the viscosity, ρ is the density of the air, and the momentum of the flow at a station x τ =−

Figure 6. Assumed velocity distributions in the pipe inlet.

ZR

It is specified by  u(r) = uo − uo

r − (R − δ) δ

u2 2πr dr = ρπR 2 u¯ 2

I (λ) = ρ

2 ,

(32)

1−

14 4 2 15 λ + 15 λ N2

(35)

0

depends only on δ and u. ¯

where ( {f } =

f

if f > 0,

0

if f < 0,

Navier–Stokes equation at the wall

and R = D/2 is the radius of the pipe. The following conditions are applicable.

 2  ∂ u 1 ∂u ∂p +µ + = 0. − ∂x r ∂r r=R ∂r 2 Using the nondimensional variables

Mass balance ξ= uo =

u¯ 1 − 23 λ + 16 λ2

=

u¯ , N

(33)

where u¯ is the average velocity in the pipe and λ = δ/R.

x , R

5=

p

ρ 2, ¯ 2u

it follows that 4(λ + 1) d5 =− , dξ Re λ2 N

(36)

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where N comes from (33) and Re =

ρR u¯ . µ

(37)

4.1. Bernoulli equation in the center line of the pipe (r = 0) duo dp + ρuo = 0, dx dx or, using the above nondimensional variables, 2 duo dλ d5 =− . dξ N u¯ dλ dξ

(38)

As an initial condition, δ = 0 is assumed along with constant velocity u over the entire cross section of the pipe, which yields infinite shear stress for x = 0 and r = R. This singularity is not critical, and it is not present in the real flow. It slightly overpredicts the pressure difference along the pipe. More precise solutions of the pipe inlet problem are presently investigated. In [16], the assumption (32) for the velocity distributions is used in (33), (36), and (38) to obtain an approximate solution. In the present paper, the momentum balance (34) was used instead of (38). This solution probably is not new, but no reference was found. Using (34) dI dλ dI = dξ dλ dξ

1 dλ = . dξ Re λ2 (18 − 34λ + 21λ2 − 4λ3 )

(40)

The numerical integration of (40) yields ξ(λ). From (36) follows 5(λ) and also 5(ξ ). Due to the factor 1 − λ in (40), the HPF λ = 1 is approached asymptotically, whereas the solution in [16] yields the HPF after a finite length. The nondimensional pressure difference in the pipe 15P (ξ ) = 5(0) − 5(ξ ) is shown in figure 7 for both approaches. The present solution yields lower 1p which means probably a lower overprediction than [16] (Schiller). In both cases, 15P (ξ ) is a unique function of ξ¯ = ξ/Re only where Re comes from (37). The approximation (41) 15P (ξ ) = 1 + 16ξ¯ − f (ξ¯ ) is used, where 1−

0.13021+ξ¯

1 + 25ξ¯

1p ρ 2 ¯ 2u

90(1 − λ)N 2

f (ξ¯ ) =

The normal problem of the suction is not yet solved by this result. The geometry of the hole, defined by the length L and the radius R, and the pressure difference 1p = p − pc are given, and u¯ must be evaluated. The Reynolds number in Re in (37) is then not known a priori. Moreover, the development of u¯ at the beginning of the pipe requires an additional pressure difference. Thus,

(39)

and from (33), (34), and (36), it follows that

ξ¯

Figure 7. The pressure function of the pipe inlet flow.


1/3 .

(42)

= 15P + 1

(43)

holds for 1p. It is again simpler to use, instead of 1p, the velocity V from (7) which is the velocity in the pipe without viscosity. This velocity is known along with 1p. Therefore, using the equations L , R ξL ζ= , ReV it follows from (41) that

ρR V , µ u¯ Re = ReV , V

ξL =

ReV =

2 + 16

  V V2 V ζ −f ζ = 2 u¯ u¯ u¯

or 

u¯ V

2

  2   1 u¯ u¯ Vζ = 0. (44) + 8ζ − 1 + f V 2 V u¯

This is a transcendental equation for u/V ¯ that can be easily solved by iteration because the term f from (42) with is small. The solution is shown in figure 8. The solution can be approximated by the function

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409

Figure 9. Airfoil 41 with velocity distributions. Figure 8. Pressure functions of the pipe inlet flow, from (44), approximation (45) and error.

u¯ = −4ζ + V + 1−

q √

Compute:

16ζ 2 + 0.5
1/3 ζ
1 − 0.06+ζ 0.5 , 1 + 90ζ + 500ζ 2

(45)

which is also shown in figure 8. The difference between the iterated and the approximate solutions is very small. It can hardly be realized from the two lines. The difference, multiplied by 10, is therefore also shown in figure 8. It is less than 1% and, thus, smaller than the uncertainty due to the approximate solution and the initial conditions at the beginning of the pipe. 5. Applications 5.1. Variation of lift coefficient Many off-design cases occur due to changes in angle of attack without changes of the velocity or Reynolds number. Examples are wind-tunnel experiments with constant tunnel speed, gusts in flight, and aircraft weight changes. The evaluation of these off-design cases is performed in several steps. The following parameters are given and remain constant: Airfoil, Chord Reynolds number Rec , Suction-compartment geometries and associated pressures, Geometry R, L of holes. The design case must first be evaluated. Specify: αD , U (αD , s), and suction law (30), Uc (s) > U (αD , s).

Boundary-layer development according to (13) to (29) including vo (s) according to (31), V 2 (s) = Uc2 (s) − U 2 (s), R V Rec , ReV (s) = c U∞ L , ζ= R ReV u(s) ¯ from (45), V (s) vo (s) . P (s) = u(s) ¯ The off-design cases are then evaluated. Specify: α 6= αD . Compute: U (α, s), V 2 (s) = Uc2 (s) − U 2 (α, s), ¯ (s), as above, ReV , ζ , and u(s)/V ¯ vo (s) = P (s)u(s), boundary-layers developments. 5.1.1. Example This example, although not appropriate for practical applications, shows typical effects. The example airfoil 41 is shown in figure 9. Suction is applied only on the upper surface. The velocity is constant from the leading edge to 0.35 c at αD = 6◦ (c` = 0.66), which is the design case. Only one suction compartment is specified, from 0.2 c to 0.96 c. The maximum velocity for the design case is Umax = 1.463. The pressure pc in the suction compartment should be as high as possible (and Uc according to (5) as low as possible) to prevent unnecessarily high suction drag according to (9). Therefore, Uc = 1.52 was selected.

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Figure 10. Boundary-layer development for airfoil 41 at α = 6◦ , Rec = 106 , and constant vo /U∞ = 0.0021.

The off-design cases are computed for α = 7◦ and α = 8◦ . The velocity distributions for these α’s are also shown in figure 9. Two different approaches have been used to maintain a laminar boundary layer. First, constant vo /U∞ = 0.0021 was specified. The value of vo (s) was varied and determined as low as possible to keep the boundary laminar. The results are shown in figure 10. The upper diagram shows the boundary-layer development. This diagram is helpful for judging the stability of the boundary layer. The stability limit is included in this diagram. It occurs at higher Reδ2 for higher H32 . The laminar-separation limit, which corresponds to H32 = 1.51509, and the old transition criterion are shown as broken lines. The boundarylayer development curve begins at the leading-edge stagnation point, which corresponds to H32 = 1.62, and proceeds upward (increasing Reδ2 ). The lower diagram shows H32 (x) and vo (x) in the suction region. Only near the trailing edge does transition occur, but no separation of the turbulent boundary layer. The suction coefficient is cQ = 0.00148. The second approach uses the suction law (30), with a = 1.4, b = 0.02924 which keeps the boundary layer laminar. This can be seen in the boundary-layer development shown in figure 11. No suction is applied where the boundary layer is to the right of the suction law. The potential flow velocity is constant in this region (see figure 9), and, accordingly, the Blasius boundary layer de-

Figure 11. Boundary-layer development for airfoil 41 at α = 6◦ , and Rec = 106 , and variable vo (s)/U∞ to prevent transition according to suction law.

velops with constant H32 = 1.57258. The beginning of the suction corresponds to the point where this vertical segment bends to the right into the inclined straight line from the suction law. A short section of the line is there in the unstable region, but this does not cause transition because the instability is very small near the stability limit. The lower diagram in figure 11 shows that suction begins at x/c = 0.2, and vo (s) differs much from the constant value of the first approach. The cQ = 0.00128 is 18% lower. This approach is thus considerably better. In the upper diagram of figure 11, the inclined straight section from the suction corresponds to the section 0.2 < x/c < 0.96. This large section of the airfoil is thus mapped into a short section of the boundary-layer development diagram, whereas the short section in front of the suction yields a much longer line. This is a typical property of this diagram. The off-design cases, α = 7◦ and α = 8◦ , are shown in figure 12 without the diagrams Reδ2 (H32 ). Increasing α means increasing U and decreasing pressure outside of the airfoil. The Uc in the suction compartment was not much larger than the maximum U for α = 6◦ . This reduces vo , for α = 8◦ even vo > 0 results between x/c = 0.2 and x/c = 0.3. This has drastic effects. For α = 7◦ the transition occurs already at x/c = 0.52, for α = 8◦ at x/c = 0.46. In both cases, vo is changed very little behind x/c = 0.55. This prevents the separation

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because U/U∞ is independent of U∞ . In the specification of the design case, U∞D and the chord Reynolds number RecD are therefore also important. From (5), it follows that 2 2 = Uc2D (s) − U∞ Uc2 (s) − U∞ D

and Uc2 (s) =1+ 2 U∞



Uc2D (s) 2 U∞ D

 −1

Re2cD Re2c

.

(47)

The design and off-design cases must, therefore, be treated in different ways. The following parameters are given and remain constant: Airfoil Suction-compartment geometries and associated pressures (and, accordingly, UcD (s)/U∞D ), Geometry of the porosity holes. The design case is treated as before, only RecD and U∞D must be specially regarded.

Figure 12. H32 (x), and vo (x) (broken line) for airfoil 41 at α = 7◦ and α = 8◦ .

Specify:

αD , RecD , suction law.

Compute:

U (αD , s)/U∞D , Boundary-layer development according to (13) to (29), including vo (s) according to (31),

of the turbulent boundary layer. But this airfoil would also without suction show the turbulent separation very close to the trailing edge. The suction causes in this case even additional drag. This can be prevented by increasing Uc which, however, would increase the suction drag according to (10).

CLD , V 2 (s) = Uc2D (s) − U 2 (αD , s), ReV (s) =

5.2. Variation of lift coefficient and Reynolds number Many off-design conditions occur when an aircraft must fly at different speeds, for example, under climb and cruise conditions. A typical example is a glider which circles in thermals at high CL and penetrates at high speeds and low CL . For gliders, the application of boundary-layer suction to increase performance has been discussed recently. See [4], for example. For an aircraft ρ 2 U CL S = W 2 ∞ is valid for steady flight where, CL is the lift coefficient, S is the wing area and W is the aircraft weight. Thus, U∞ and s CLD (46) Re = ReD CL vary with CL . Moreover, the pressure in a suction compartment is normally independent of U∞ , whereas U depends on U∞

ζ=

R V RecD , c U∞D

L , R ReV

u(s) ¯ V from (44), ¯ P (s) = vo (s)/u(s). The off-design cases are then evaluated. Select:

α 6= αD .

Compute:

U (α, s), CL , p Rec = RecD CLD /CL , p U∞ = U∞D CLD /CL , Uc (s) according to (47), V 2 (s) = Uc2 (s) − U 2 (α, s), ¯ , as above, ReV , ζ , and u(s)/V ¯ vo (s) = P (s)u(s), boundary-layer development.

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5.2.1. Example The design case is now αD = 2◦ , and ReD = 8.856 × 106 . A laminar boundary layer on the upper surface is to be maintained by suction for the design case. One suction compartment extending from 0.10c to 0.96c is assumed. The pressure in this compartment will be discussed later. As α increases, U/U∞ increases on the upper surface as in the preceding example. The effect of the increasing U/U∞ on vo (s) may be compensated by the decreasing U∞ according to (46). On the lower surface, a long extent of natural laminar flow is intended. Accordingly, favorable pressure gradient was specified to near the trailing edge for αD = 2◦ . The design case corresponds to the cruise condition for a light aircraft. The airfoil design must consider that, on the upper surface, high suction peaks for the off-design cases may cause blowing instead of suction. Toward the leading edge, decreasing velocity was therefore specified for αD = 2◦ . Airfoil 44, which satisfies all these conditions is shown along with two velocity distributions in figure 13. At α = 8◦ , U/U∞ < 1.7. For the design case, the suction law (30), with a = 1.4, and b = 0.02924 was used to prevent transition on the upper surface. The boundary-layer development Reδ2 (H32 ) and the diagram with H32 (x) and vo (x) are shown in figure 14. A distinct corner can again be seen in the boundary-layer development where the suction begins. In the lower diagram, it can be seen that the suction begins at the beginning of the suction compartment at 0.1c. This means that, for this Reynolds number, suction is already necessary to prevent transition in the region having constant pressure. Of course, much more suction is necessary to prevent transition in the region having adverse pressure gradient (i.e. x/c > 0.5). For the design case, the porosity was evaluated according to (45) and (3) with Uc = 1.4, according to (6). This value is higher than Umax on the upper surface, and suction is achieved where the surface is porous. Then the analysis was performed. The porosity and Uc were assumed to be given and the boundary layer developments were computed for many α’s. The Reynolds number was varied with α according to (46). For α = αD = 2◦ the results should be the same as those from the evaluation using the suction law. The results for α = αD are shown in figure 15. Several small differences from figure 14 can be seen, mainly in H32 near the trailing edge. The reason for these differences were investigated in detail. The boundary layer is very sensitive to slight changes in the distribution vo (s). The analysis was performed in such a way that cQ was not changed, but inside of an integration step, vo (s) was assumed to be constant in the analysis case but not in the design case. This difference in the numerical procedures was tolerated because the porosity can anyway not be realised too precisely.

Figure 13. Airfoil 44 with two velocity distributions.

Figure 14. The boundary-layer development of airfoil 44 for αD = 2◦ , RecD = 8.856 × 106 , and suction law for prevention of transition.

The diagrams containing vo (x) and H32 (x) for several other α’s are shown in figures 16 to 18. The scale for vo is adapted to the maximum value. Due to (47), the suction increases with increasing α. The boundary layer remains laminar up to α = 9◦ . At α = 10◦ H32 jumps at the beginning of the suction to high values near H32 = 1.8, which indicates turbulent boundary layer. For this α, the suction is too low to prevent transition, although it is still strong enough to prevent separation of the turbulent boundary layer. This situation continues until α = 16◦ .

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Figure 17. H32 (x) and vo (s)(x) for airfoil 44 at α = 9◦ and α = 10◦ . Figure 15. Re-analysis of the boundary-layer development on airfoil 44 for α = 2◦ , Re = 8.856 × 106 .

Figure 16. H32 (x) and vo (s)(x) for airfoil 44 at α = 3◦ and α = 7◦ .

Figure 18. H32 (x) and vo (s)(x) for airfoil 44 at α = 16◦ and α = 18◦ .

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Figure 19. Drag polars for airfoil 44 with suction. The chords c = 2 m and c = 1 m correspond to the design Reynolds numbers ReD = 8.856 × 106 and ReD = 4.428 × 106 .

Even at α = 18◦ , turbulent separation occurs at 0.9c which still causes no breakdowm of the lift. The drag polar for airfoil 44 is presented in figure 19 for RecD = 8.856 × 106 and RecD = 4.428 × 106 . The cd -values include the suction drag. This diagram demonstrates that boundary-layer suction can improve the minimum drag and the maximum lift of an airfoil. Between α = 2◦ and α = 9◦ the boundary layer remains laminar along the entire upper surface and 90% of the lower surface. The drag still increases with CL because of decreasing Re and increasing cQ . The minimum drag for RecD = 8.856 × 106 is about half that which can be achieved without suction. The improvement percentage is slightly lower for RecD = 4.428 × 106, which coincides with previous results. 6. Future aspects The present program system was developed for design and analysis of airfoils with boundary-layer suction. The suction-analysis portion is based on given porosity of the surface and given suction-compartment geometries and the associated pressures. The suction-design portion requires also the specification of the suction-compartment geometries and the associated pressures, and, additionally, a suction law so that the porosity can be evaluated. The variation of Reynolds number with lift coefficient is included for the case of an aircraft having constant weight. The aircraft-oriented boundary-layer developments for the off-design cases can thus be evaluated. Moreover, the computing times are so low that the airfoil program can be included in an iteration or optimization scheme. The definition of appropriate objective functions for such optimizations requires further research, however.

Acknowlegements The present work was supported by Deutsche Forschungsgemeinschaft in the framework of Memorandum of Understanding between DFG and Russian Fondation for Basic Research. I thank Dan M. Somers for his careful editing of the text and Peter Wassermann for providing the solutions of the Navier–Stokes equations for figure 5.

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