Centre of Pressure and Aerodynamic Centre

Aerodynamic Fundamentals, Definitions and Aerofoils

3.10 Centre of Pressure and Aerodynamic Centre

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Aerodynamic Fundamentals, Definitions and Aerofoils 105 NACA Four, Five and Six series aerofoil

NACA 4412

C_I

C_I α

4412

441 2

23015

23015 230

441 15

2 stall

642–415

642–415 64

–412 5

NACA 23015

NACA 64₂–415

C_d

Figure 3.29 Comparison of three NACA aerofoils [6].

3.9.2 Comparison of Three NACA Aerofoils

The NACA 4412, NACA 23015 and NACA 642-415 are three commonly used aerofoils – there are many different types of aircraft that use one of these. Figure 3.29 compares these three popular NACA aerofoils (source NACA) to examine what is needed to make the choice.

The NACA 23015 has sharp stalling characteristics; however, it can give a higher sectional lift,C_land lower sectional moment,C_m, than others. Drag-wise, the NACA 64₂-415 has a bucket to give the lowest sectional drag. The NACA 4412 is the oldest and, for its time, was the favourite. Of these three examples, the NACA 64₂-415 is the best for gentle stall characteristics and low sectional drag, offsetting the small drag due to the relatively not so high moment coefficient.

The five-digit NACA 23015 gives the highestC_lmaxat higher angle of attack,𝛼, but has abrupt stall not suit- able for the ab initio club trainer but serves the utility category of aircraft. The benefits of the newer six-digit NACA 64₂-415 offers all round benefits with gentler stall characteristics including a lowerC_m and a drag bucket to offer lower drag at lowC_L compared to four-digit NACA 4414. However, the four-digit NACA series is tolerant to production variation, hence it is popular in small aircraft designs as it is cheaper to manufacture.

The advanced NACA six-digit series aerofoil came after NACA five-digit series aerofoil making this suc- cessful aerofoil to fall behind.

An aerofoil designer must produce a suitable aerofoil that encompasses the best of all five qualities – a diffi- cult compromise to make. Flaps are also an integral part of the design. Flap deflection effectively increases the aerofoil camber to generate more lift. Therefore, a designer also must examine all five qualities at all possible flap and slat deflections.

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Stall c.p. position

Angle of attack

LE c/4

Angle of attack reducing TE

Locus of c.p. moving aft with reduced angle of attack

hinged (point of rotation) hinged

(point of rotation)

c.p.

c.p.

c.p.

freestream velocity

toward TE

quarter chord stall

Lift

Figure 3.30 Movement of centre of pressure with change in lift.

nose-down moment may move thec.p. rearward outside of the aerofoil. Low resultant force at low angle of attack requires a large moment arm to balance, pushing thec.p. further aft. At zero lift, thec.p. approaches infinity. For a typical aerofoil, thec.p. stays within the aerofoil for the operating range. Figure 3.30 depicts the movement of centre of pressure,c.p., with a change in lift. The figure shows that, past 𝛼_max, the C_l drops rapidly – if not drastically – when stall is reached. Stalling starts at reaching𝛼_max. It will be shown in Section 3.10.1 thatc.p. is always aft of the quarter chord. This makes for a realistic aerofoil, as the aft is loaded with a nose-down (−ve) moment. The extent of aft loading depends on how far behind is the minimumc.p.

point. The second digit of the NACA six series gives the minimumc.p. point in tenth of the chord from the LE.

The aerodynamic centre, ac, is concerned with moments about a point, typically on the chord line (Figures 3.31). It does not deal with force by itself. However, it is noticed that the moment about the quarter chord of the aerofoil is invariant to the angle of attack (dC_m/d𝛼=0) until stall occurs. This point is invariant to𝛼change and is known as thea.c., which is a natural reference point through which all forces and moments are defined to act. The‘a.c.’is close to the quarter chord point. There could be minor variations of the position of thea.c. around the quarter chord among aerofoils. To standardise, the fixed point at the quarter chord from the LE is also measured. In term of coefficient, these areC_{m_ac}andC_m1/4are measured in wind tunnel tests. A symmetric aerofoil hasC_{m_ac}=0 but there is small variation ofC_m1/4with𝛼variation. The relation betweenC_{m_ac}and C_m1/4can expressed in mathematical relations as dealt with in Section 3.10.1. Aerofoil characteristics given in Appendix F show gives both theC_{m_ac}and theC_m1/4.

Thea.c. offers much useful information. At thea.c., although the dC_m/d𝛼=0, it has some moment (except symmetrical aerofoil) and is not thec.p., which is always aft of thea.c. Thea.c. is an useful parameter in stability analyses when aircraft CG has to be taken into account. The higher the positive camber, the more lift is generated for a given angle of attack; however, this leads to a greater nose-down moment. To counter this nose-down moment, conventional aircraft have a horizontal tail with the negative camber supported by an elevator. For tailless aircraft (e.g. delta wing designs in which the horizontal tail merges with wing), the TE is given a negative camber as a ‘reflex’. This balancing is known astrimmingand it is associated with the type of drag known astrim drag. Aerofoil selection is then a compromise between having good lift characteristics and a low moment.

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Aerodynamic Fundamentals, Definitions and Aerofoils 107

0 (LE)

Angle of attack, α chord

0 (LE) 1/4 Angle of attack 1/2

–ve camber+ve camber

0

Pitching moment

3/4 1 (TE)

1/4 1/2 3/4 Chord fraction

Lift (l) l

a.c.

c.p.D chord

x C_{m_c/4}

N

x_c.p.

x_a.c.

x_c/4 airflow

Figure 3.31 Aerodynamic centre invariant near the quarter chord (fractional chord position about which the moment is taken).

When series of aerofoil are stacked up side by side to form a wing that is to be integrated with the aircraft, then contribution of isolated wing weight has to be considered its moment characteristics. Typically, wing Mean Aerodynamic Chord (MAC) (Section 4.8) represents its reference geometry to compute wing aerody- namic characteristics. The point of CG location is about which the wing moment is considered. The CG can be ahead or aft of the quarter chord point of the wing MAC. Chapter 18 explains that static stability requires to simultaneously satisfy (i) that aerofoilC_mhas to be positive at𝛼=0 and (ii) dC_m/d𝛼has to be negative as well. A typical cambered aerofoil has a negativeC_m at𝛼=0. Therefore, the wing will require an additional surface behind it (a small wing like horizontal tail at aft fuselage or a reflex wing TE) to keep nose up to retain static stability. The wing and empennage are dealt with in Chapters 4 and 6, respectively.

3.10.1 Relation Between Centre of Pressure and Aerodynamic Centre

Typically, the a.c. is within 22–27% of the aerofoil chord from theLE. Since the point varies from aerofoil to aerofoil, therefore, for standardisation, test results are given about a fixed point of quarter chord,c/4, from theLE. From the test result of a aerofoil moment around the quarter chord, the aerodynamic centre can be accurately determined and also given in a graph (Appendix F).

Figure 3.31 shows a typical aerofoil (NACA six series) with quarter chordc/4and the aerodynamic centre, a.c., shown at a distancex_c/4andx_a.c.from theLE, respectively. At thex_c/4, the sectional lift,L, and moment M_c/4are shown in the diagram (drag is small and its moment contribution is negligibly small).

3.10.1.1 Estimating the Position of the Aerodynamic Centre,a.c.

As mentioned before, the quarter chordx_c/4is at a fixed position and test results ofC_{m_c/4}are available [6]. Also, C_{m_ac}is invariant with respect to angle of attack,𝛼(i.e.C_l). Taking the moment about thea.c. the moment equilibrium gives (moment contribution by drag is small as the moment arm is negligible). The aerofoil is

k k considered weightless and the definitions of the aerodynamic coefficients are given in Equation (1.2).

∑M=0 that is,M_ac=l× (x_a.c.−x_c∕4) +M_c∕4

In coefficient form,C_{m ac}= (C_l∕c) × (x_a.c.−x_c∕4) +C_{m c∕4} (3.16) Differentiating with respect to angle of attack𝛼, it becomes (note thatC_{m_ac}is constant),

0= (dC_l∕d𝛼) × (x_a.c.∕c−x_c∕4∕c) +dC_{m c∕4}∕d𝛼

Transposing(x_a.c.∕c−x_c∕4∕c) = −(dC_{M c∕4}∕d𝛼)∕(dC_l∕d𝛼) = −(m₀∕a₀) where,m₀=slope of theC_{M_c/4}curve anda₀=is the lift curve slope.

substituting in Equation(3.16),C_{m c∕4}=C_{m ac}+(dC_{m c∕4}∕d𝛼)

(dC_l∕d𝛼) (C_l) =C_{m ac}−(m₀)

(a₀)(C_l) (3.17) m₀anda₀can be evaluated from the test results (Appendix F).

orx_a.c.∕c=x_c∕4∕c−m₀∕a₀=0.25−m₀∕a₀(in terms of percentage of chord) (3.18) 3.10.1.2 Estimating the Position of the Centre of Pressure,c.p.

Estimatingc.p. is lot more difficult as it moves (Figure 3.30), with the operating range from mid-chord,c/2, to a.c. at aircraft stall.

Let the aerofoil section liftlact at the aerofoilc.p. Taking moment about the quarter chord,c/4.

∑M=0,that is,0=l× (x_c.p.−x_c∕4) +M_c∕4.

In coefficient form,C_l×(x_c.p.–x_c/4)_.+C_{m_c/4}=0

Substituting the value ofC_{m_c/4}from Equation (3.18) into this equation C_l× (x_c.p∕c−x_c∕4) +C_{m ac}−(m₀)

(a₀)(C_l) =0 or(x_c.p∕c−x_c∕4∕c) = −(C_{m ac})∕C_l+ (m₀)

(a₀) (3.19)

C_{m_ac}is negative for conventional aerofoil, that makesx_c.p/c>x_c/4, always.

Replacingm₀/a₀=(x_a.c./c–x_c/4/c) from the relation given below Equation (3.16) in this equation it becomes:

(x_c.p∕c−x_c∕4∕c) = −(C_{m ac})∕C_l+ (x_a.c.∕c−x_c∕4∕c)

orx_c.p∕c= (x_a.c.∕c) − (C_{m ac})∕C_l (3.20)

It can be seen that thec.p. is aft of thea.c. At thea.c., although thedC_m/d𝛼=0, it has some moment (except symmetrical aerofoil).

Example 3.1 Find the aerodynamic centre,a.c. and centre of pressure,c.p., at 6∘of angle of incidence,𝛼, for the NACA 65-410 aerofoil from the test results given in Appendix F.

Solution

From the test result graph NACA 65-410 givesdC_l/d𝛼=a₀=(1.05–0.25)/8=0.1/degree andm₀=(−0.0755−(−0.062)/[8−(−4)]= −0.0135/12= −0.00125/degree

Equation (3.17) gives,x_a.c./c=x_c/4/c+m₀/a₀=0.25−(−0.00125/0.1)≈0.2625 of the chord from the LE.

Page 186 of [6] gives≈0.262. (The idealC_lfor NACA 65-410 is 0.3.)

Then at 6∘angle,C_l=0.9 andC_{m_c/4}= −0.075 from the graph in Appendix F.

Then at angle of zero liftC_{m_ac}=C_{m_c/4}+C_l×(x_a.c./c−x_c/4/c)= −0.075−0.9×(0.012)= −0.0846.

Equation (3.20) gives,x_c.p/c=(x_a.c./c)−(C_{m_ac})/C_l=0.25+(0.0846/0.9)=0.25+0.094=0.344.

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Aerodynamic Fundamentals, Definitions and Aerofoils 109

The centre of pressure and aerodynamic centre can be analytically determined if the aerofoil properties can be analytically expressed. Within in the operating range, the aerofoil lift characteristics can be expressed as a straight line and the moment characteristic is not exactly a straight line but can be tolerated as straight line;

however the drag characteristics may not be amenable to fit as an equation of parabola resulting in inaccura- cies. To obtain industry standard results, it is recommended that test data may be used [6].

It is stressed that the readers must understand the role ofa.c. andc.p. This gives a good insight into how an aerofoil behaves.