5. Factors Affecting
Hovering/Vertical Flight
2020
Prof. SangJoon Shin
I. Effects of Blade Twist and Taper
II. Optimum Hovering Rotor
III. Effect of Climb on Induced-power Losses
IV. Ground Effect
V. Effect of Engine Supercharging (excluded)
Overview
1
2
Effects of Blade Twist and Taper
- Induced / profile-drag losses can be minimized in hover
→ by operating with the largest diameter and slowest turning rotor compatible with structural criteria and operating considerations
- Additional improvement --- choice of the proper blade geometry (blade planform, pitch distribution)
→ max-thrust for a given power input
Nonuniformity of induced flow minimized Blade twist --- (-), washout
Taper --- greater root chord
3
Effects of Blade Twist and Taper
1. Twist effect
▲Effect of twist on inflow distribution and section A.o.A. against (𝑟𝑎𝑑𝑖𝑢𝑠)2
▲beneficial in reducing rotor losses, although not change in 𝐶𝑄𝑜 is insignificant
4
Effects of Blade Twist and Taper
1. Twist effect (Contd.)
However, detrimental effect when thrust coefficient ≃ 0 since (-) thrust
Linear -12º twist : max possible induced power loss reduction, while yielding identical profile-drag losses
Highly twisted blades : detrimental in other flight conditions
▲Percent increase in thrust for an untapered blade →Table
▲Table
5
Effects of Blade Twist and Taper
2. Taper effect
- Larger chord at the inner portion → more uniform inflow distribution
▲Effect on over-all effects
▲Table Percent increase in thrust
Large helicopters: small efficiency gains are significant, tapering highly desirable
Effects of Blade Twist and Taper
3. combined twist and taper
▲Table Percent increase in thrust by both twist and taper
6
▲Table Percent in thrust → max. possible increase : 2%
7
Effects of Blade Twist and Taper
4. Solidity effects
𝜎 = 0.042, 0.060, similar amounts of benefits for thrust increase
5. Partial taper
▲Inflow distribution between partially and full tapered rotors → similar
8
Effects of Blade Twist and Taper
6. Twist in forward flight
(-) twist →
improve the forward flight efficiency as well as delay blade stalling at high forward speeds
Minimizing compressibility effect
Little effect on autorotative performance
9
Optimum Hovering Rotor
- Induced losses → ideally twisted
Profile-drag losses minimum when each blade element is at its most efficient AoA
(𝐶𝑙ൗ𝐶𝑑 is max. if 𝑉𝑡𝑖𝑝 is fixed; 𝐶𝑙3ൗ2ൗ𝐶
𝑑 is max. if 𝐷. 𝐿. is fixed)
10
Optimum Hovering Rotor
1. Thrust
𝜃 = 𝛼𝑟 + 𝜙 = 𝛼𝑟 + 𝑣
Ω𝑟 (1)
Differential thrust 𝑑𝑇 = 1
2𝜌(Ω𝑟)2𝑎𝛼𝑟𝑐 𝑑𝑟 (2)
𝛼𝑟 independent of r → chord must be adjusted so that uniform downwash can be obtained.
𝑑𝑇 varies linearly with r
𝑐 = 𝑐𝑡 𝑅
𝑟 (3)
(3) → (2) : dT = 1
2𝜌Ω2𝑟𝑎𝛼𝑟𝑐𝑡𝑅 𝑑𝑟 (4)
Integrating : T = 𝑏
2𝜌Ω2 𝑅3
2 𝑎𝛼𝑟𝑐𝑡 (5)
Non-dimensional form : 𝐶𝑇 = 𝜎𝑡
4 𝜎𝛼𝑟 = 𝜎𝑡
4 𝑐𝑙 (6)
𝜎𝑡= 𝑏𝑐𝑡
𝜋𝑅 (7)
↑
Const. ↑
Variable
11
Optimum Hovering Rotor
2. Induced torque
𝑄𝑖 = 0𝑅𝑏1
2𝜌Ω2𝑟3𝑐𝑙𝜙𝑐 𝑑𝑟 (8)
= 𝑏
2 𝜌(Ω𝑅)𝑅2
2 𝑐𝑙𝑐𝑡𝑣 (9)
Non-dimensional form : 𝐶𝑄𝑖 = 𝑄𝑡
4 𝑐𝑙 𝑣
Ω𝑅 (10)
From momentum considerations:
𝑣 = 𝑇
2𝜌𝜋𝑅2 = Ω𝑅 𝐶𝑇
2 , 𝑣
Ω𝑅 = 𝜙 = 𝐶𝑇
2 (11)
(6), (11) → (10) 𝐶𝑄𝑖 = 𝐶𝑇 ൗ
32
2 (12)
12
Optimum Hovering Rotor
3. Profile-drag torque
𝑄𝑜 = 0𝑅 1
2𝑏𝜌(Ω𝑟)2𝑐𝑑𝑜𝑐𝑟 𝑑𝑟 = 1
6𝑏𝜌(Ω𝑅)2𝑅2𝑐𝑡𝑐𝑑𝑜 (13)
𝐶𝑄𝑜 = 1
6𝜎𝑡𝑐𝑑𝑜 (14)
13
Optimum Hovering Rotor
4. Performance equation
- Optimum rotor : with twist and taper to give const. inflow and const.
AoA along the span
𝐶𝑄 = 𝐶𝑇 ൗ
3 2
2 + 1
6𝜎𝑡𝑐𝑑𝑜 (15)
- Conventional weighted solidity 𝑐𝑒 = 3
2𝑐𝑡, 𝜎 = 3 2𝜎𝑡
(15) → 𝐶𝑄 = 𝐶𝑇 ൗ
3 2
2 + 1
9𝜎𝑐𝑑𝑜 (16)
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Effects of Blade Twist and Taper
5. design of optimum rotor
1. Determine the chosen airfoil section AoA so that the profile drag is a min 2. Calculate 𝜎𝑡 for the design 𝐶𝑇, 𝜎𝑡 = 4𝐶𝑇
𝐶𝑙
3. Twist 𝜃 = 𝛼𝑟 + 𝑅
𝑟
𝐶𝑇
2 , taper 𝑐 = 𝑐𝑡 𝑅
𝑟
(𝑐𝑡 = 𝜎𝑡 𝜋𝑅
𝑏 )
15
- Climb : less induced power since needs to accelerate the mass of air less to produce the same thrust
- Descent: greater induced loss
Effect of Climb on Induced-power Losses
16
1. Equation for induced velocity in climb 𝑉ℎ: induced velocity in hover
𝑉𝑣: induced velocity in climb (31), chap.3 → 𝑉ℎ = Ω𝑅 𝐶𝑇
2 (17)
- Total thrust is the same in both hover and climb
𝑑𝑇 = 2𝜋𝑟𝑑𝑟𝜌2𝑉ℎ2 = 2𝜋𝑟𝑑𝑟𝜌 𝑣𝑣 + 𝑉𝑣 2𝑣𝑣 𝑣ℎ2 = 𝑉𝑣𝑣𝑣 + 𝑣𝑣2
𝑣ℎ = −𝑉𝑣+ 𝑉𝑣
2+4𝑣ℎ2
2 (18)
(17) → 𝑣𝑣 = −𝑉𝑣+ 𝑉𝑣
2+2𝐶𝑇(Ω𝑅)2
2 (19)
Effect of Climb on Induced-power Losses
17
2. Experimental check of induced velocity in climb
Total power required = induced + profile + power required to lift the aircraft of 𝑉𝑣
𝑃𝑣 = 𝑃𝑖 + 𝑃𝑜 + 𝑇𝑉𝑣
𝑃𝑣
𝑃ℎ = 𝑇𝑣𝑣+𝑇𝑉𝑣+𝑃𝑜
𝑇𝑉ℎ+𝑃𝑜 → Fig. 5-8
Effect of Climb on Induced-power Losses
18
3. Effect of climb on power required
𝑃𝑣+𝑊𝑉𝑣
𝑃ℎ : climb power that would be expected if there were no changes in the induced or profile-drag power losses
1.26−1.14
1.26−1.00 = 46% saving
- Increase in power required actually
≃ 1
2 × 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝐸 𝑉𝑣 = 2(𝑒𝑥𝑐𝑒𝑠𝑠 𝑝𝑜𝑤𝑒𝑟
𝑔𝑟𝑜𝑠𝑠 𝑤𝑒𝑖𝑔ℎ𝑡)
Effect of Climb on Induced-power Losses
19
Ground Effect
- Beneficial for overloaded helicopters to take- off and hover
1. Theoretical treatment
a) Near ground, induced velocity is reduced → decrease in induced power or increase in
thrust
- cylindrical vortex and image vortex cylinder B.C. of zero vertical velocity, opposite induced Velocities of the same magnitude
Potential theory → influence of 2 vortex systems
▲Cylindrical vortex and image vortex cylinder
20
Ground Effect
1. Theoretical treatment (Contd.) (36), chap.4 → part of the profile-drag torque that varies with 𝐶𝑇
2 3
𝛿1 𝑎
𝐶𝑇 𝐵 + 4
𝜎 𝛿2 𝑎2
𝐶𝑇 𝐵2
2
Induced torque (𝐶𝑇3ൗ2)൘
(𝐵 2)
Δ𝐶𝑄∞ = 𝐶𝑇 ൗ
3 2
𝐵 2 + 2
3 𝛿1
𝑎 𝐶𝑇 𝐵2 + 4
𝜎 𝛿2 𝑎2
𝐶𝑇 𝐵2
2 (22)
- Torque coefficient that varies with 𝐶𝑇 Λ = Δ𝐶𝑄
Δ𝐶𝑄∞ ∶ 𝑟𝑎𝑡𝑖𝑜 → Fig. 5-11
↑ Free air
21
Ground Effect
2. Experimental check
R=5ft, untwisted model
- Comparison between analysis and experiments → Fig. 5-12
→ quite good
- Smaller distance above the ground, worse agreement due to stall at the
larger pitch angles ← sharp reduction in induced velocity
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Ground Effect
2. Experimental check (Contd.)
▲Tൗ𝑇∞ vs rotor height in terms of R, 𝐶𝑇∞ൗ𝜎
→increase in thrust obtained at const. power Ex) 25% increase in load when 𝑍Τ𝐷= 0.25,𝐶𝑇∞ൗ𝜎=0.1
▲prediction vs flight test → “ground cushion”
