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(1)

3. An Introduction

to the Hovering Theory

2020

Prof. SangJoon Shin

(2)

 I. An intro to hovering theory

 II. Momentum Considerations

 III. Rotor Figure of Merit

 IV. Blade – Element Consideration

 V. Effect of profile drag on F.M.

 VI. Effect of Rotor Tip Speed and Solidity on F.M.

Overview

(3)

An intro to hovering theory

 No dissymmetry of velocities across the rotor disk

 Momentum theory by Rankine and Froude

- greater physical significance, can be more easily grasped.

(4)

Momentum Considerations

 1. Development of thrust

β€’ Increased velocity of the air from its initial value V to its value at the airscrew disk, which arises from the production of thrust, is called the β€œinduced” or β€œdownwash” velocity β†’ v

β€’ Thrust developed by the airscrew is then equal to the most of air passing through the disk in unit time, multiplied by the

total increase in velocity caused by the action of the airscrew.

(5)

Momentum Considerations

 1. Development of thrust (Contd.)

β€’ Assumptions

β‘  Infinite number of blades, β€œactuator disk”, no loss of thrust at the blade tips

β‘‘ Power required = axial kinetic energy imported to the air. No blade friction, profile-drag losses, rotational energy ignored.

β‘’ Infinitely thin disk, no discontinuities in velocity

β–²Gradual contraction of the slipstream

(6)

Momentum Considerations

 2. Induced velocity relationships 𝑉 ∢ π‘£π‘’π‘™π‘œπ‘π‘–π‘‘π‘¦ π‘œπ‘“ π‘‘β„Žπ‘’ π‘Žπ‘–π‘Ÿπ‘ π‘π‘Ÿπ‘’π‘€

π‘Žπ‘‰ ∢ π‘–π‘›π‘π‘Ÿπ‘’π‘Žπ‘ π‘’ 𝑖𝑛 π‘£π‘’π‘™π‘œπ‘π‘–π‘‘π‘¦ π‘Žπ‘‘ π‘‘β„Žπ‘’ π‘‘π‘–π‘ π‘˜

𝑏𝑉 ∢ π‘–π‘›π‘π‘Ÿπ‘’π‘Žπ‘ π‘’ 𝑖𝑛 π‘£π‘’π‘™π‘œπ‘π‘–π‘‘π‘¦ π‘œπ‘£π‘’π‘Ÿ π‘“π‘Ÿπ‘’π‘’ π‘Žπ‘–π‘Ÿ π‘Žπ‘‘ π‘Žπ‘› 𝑖𝑛𝑓𝑖𝑛𝑖𝑑𝑒𝑙𝑦 π‘™π‘Žπ‘Ÿπ‘”π‘’ π‘‘π‘–π‘ π‘‘π‘Žπ‘›π‘π‘’

𝑇 𝑉 + π‘Žπ‘‰ = βˆ†πΎ. 𝐸. (1)

𝑇 = 𝜌𝐴(𝑉 + π‘Žπ‘‰) 𝑏𝑉 (2)

βˆ†πΎ. 𝐸. = 1

2𝜌𝐴 𝑉 + π‘Žπ‘‰ [ 𝑉 + 𝑏𝑉 2 βˆ’ 𝑉2] (3) (1), (2) β†’ (3) : πœŒπ΄π‘‰ 1 + π‘Ž 𝑏𝑉 𝑉 1 + π‘Ž = 1

2𝜌𝐴3 1 + π‘Ž 𝑏2 + 2𝑏

∴ 𝑏 = 2π‘Ž (4)

Work done in unit time by the thrust of the

airscrew on the air

Slip stream in unit time

Change of the axial momentum of air in unit time

1

2x mass of air passing

through the disk difference in the squares of the velocities infinitely in front of and behind the disk x

(7)

Momentum Considerations

 3. Induced velocity in hovering

Hover : 𝑉 ∢ 0, 4 β†’

2 ∢ β†’ 𝑇 = πœŒπœ‹π‘…2𝜈 2𝜈 (5)

𝜈 = 𝑇

2πœŒπœ‹π‘…2 (6)

β€’ Induced power loss : π‘‡πœˆ

β€’ Disk loading 𝑇

πœ‹π‘…2 should be kept as low as possible for efficient hovering performance

β€’ Uniform inflow assumption β†’ minimum induced power loss for a given thrust

π‘Žπ‘‰ = 𝜈 𝑏𝑉 = 2𝜈

(8)

Momentum Considerations

 4. Ideal and actual losses

β€’ Momentum theory neglects:

- Profile-drag losses

- Nonuniformity of induced flow (tip losses) - Slipstream rotational losses

β€’ Listing of actual losses:

- Profile-drag loss Β· Β· Β· Β·30%

- Nonuniform inflow Β· Β· Β· 6%

- Slipstream rotation Β· Β· Β· 0.2%

- Tip loss Β· Β· Β· Β·3%

(9)

Rotor Figure of Merit

 Efficiency of a lifting rotor

β€’ Actual power required to produce a given thrust

β€’ Minimum possible power required to produce thrust ← ideal rotor

𝑀 = π‘šπ‘–π‘›π‘–π‘šπ‘’π‘š π‘π‘œπ‘ π‘ π‘–π‘π‘™π‘’ π‘π‘œπ‘€π‘’π‘Ÿ π‘Ÿπ‘’π‘žπ‘’π‘–π‘Ÿπ‘’π‘‘ π‘‘π‘œ β„Žπ‘œπ‘£π‘’π‘Ÿ

π‘Žπ‘π‘‘π‘’π‘Žπ‘™ π‘π‘œπ‘€π‘’π‘Ÿ π‘Ÿπ‘’π‘žπ‘’π‘–π‘Ÿπ‘’π‘‘ π‘‘π‘œ β„Žπ‘œπ‘£π‘’π‘Ÿ = π‘‡πœˆ

𝑃 (7)

(6) β†’ (7) 𝑀 = 1

2 𝑇 𝑃

𝑇

πœŒπœ‹π‘…2 (8)

β€œFigure of merit”

(10)

Rotor Figure of Merit

 1. Ideal F.M.

β€’ Ideal rotor : Producing thrust with the minimum amount of power

β†’ 𝑀𝑖 = 1

β€’ Zero profile drag but non-uniform inflow distribution β†’ non-unique F.M.

β€’ Max F.M. of untwisted blades ≃ 0.94 - relation between D.L. and P.L.

𝑀 = 1

2 𝑇 𝑃

𝑇

πœŒπœ‹π‘…2 = 1

2𝑃. 𝐿. 𝐷.𝐿.

𝜌 (9)

β€’ Assuming sea-level, 𝑃. 𝐿. = 38𝑀 1

𝐷.𝐿. (10) β–²M=1 : upper limit for any rotor

M=0.75 : typical of good rotor M=0.5 : poor rotors

(11)

 2. Non-dimensional F.M.

β€’ Non-dimensional quantities :

𝑇 = πΆπ‘‡πœ‹π‘…2𝜌(Ω𝑅)2

𝑄 = πΆπ‘„πœ‹π‘…2𝜌(Ω𝑅)2𝑅 (11)

𝑃 = πΆπ‘ƒπœ‹π‘…2𝜌(Ω𝑅)2

𝐢𝑄 = 𝐢𝑃

𝑀 = 1

2

𝐢𝑇3ΰ΅—2

𝐢𝑄 = 0.707𝐢𝑇

3ΰ΅— 2

𝐢𝑄 (12)

Rotor Figure of Merit

(12)

Blade – Element Consideration

 Limitation of the momentum theory - No info about the blade design

- Profile-drag losses ignored

β€’ Effective A.o.A. 𝛼𝑒 = 𝛼 βˆ’ πœ– (πœ– = ΀𝑀 𝑉)

β€’ πœ™: inflow A.o.A. = 𝜈 + π‘‰πœˆ

β€’ πœƒ: blade pitch angle

β€’ πœƒ = 𝛼 + πœ™

↑Climb velocity

(13)

Effect of Profile Drag on F.M.

 1. Expression for thrust - Differential lift

𝑑𝐿 = 𝐢𝑙(𝑏𝑐 π‘‘π‘Ÿ)(1

2πœŒπ‘‰π‘…2) (13)

- Simplification : Flow through the disk is small compared with the tangential velocity

sin πœ™ = πœ™, cos πœ™ = 1, 𝑉𝑅 = Ξ©π‘Ÿ (14)

- Blade-element lift

𝐢𝑙 = π‘Žπ›Όπ‘Ÿ = π‘Ž πœƒ βˆ’ πœ™ (15)

(14), (15) β†’ (13) : 𝑑𝑇 = 𝑑𝐿 = 𝑏1

2𝜌 Ξ©π‘Ÿ 2π‘Ž πœƒ βˆ’ πœ™ 𝑐 π‘‘π‘Ÿ (16)

(14)

Effect of Profile Drag on F.M.

 1. Expression for thrust (Contd.) - Assumption

πœƒ = πœƒπ‘‘ 𝑅

π‘Ÿ β†’ (18)

(17), (18) β†’ (16) : 𝑑𝑇 = 𝑏1

2𝜌 Ξ©π‘Ÿ 2π‘Žπ‘…

π‘Ÿ πœƒπ‘‘ βˆ’ πœ™π‘‘ 𝑐 π‘‘π‘Ÿ (19)

Integrating over blade radius, assuming constant c

𝑇 = 𝑏

2 𝜌Ω2π‘Žπ‘…

2 πœƒπ‘‘ βˆ’ πœ™π‘‘ 𝑐 (20)

𝑇 = πΆπ‘‡πœ‹π‘…2𝜌 Ω𝑅 2

Uniform inflow distribution

β€œideal twist”

(15)

Effect of Profile Drag on F.M.

 1. Expression for thrust (Contd.)

𝐢𝑇 = π‘Ž

4 𝑏𝑐

πœ‹π‘… πœƒπ‘‘ βˆ’ πœ™π‘‘ (21)

- β€œSolidity” : ratio of the total blade area to the rotor disk area 𝜎 = 𝑏𝑐𝑅

πœ‹π‘…2 = 𝑏𝑐

πœ‹π‘… (22)

(22) β†’ (21) 𝐢𝑇 = 𝜎

4 π‘Ž πœƒπ‘‘ βˆ’ πœ™π‘‘ (23)

β†’ Thrust of an ideally twisted, constant-chord blade

(16)

Effect of Profile Drag on F.M.

 2. Expression for torque

β€’ Drag of the blade element

β€’ Torque

𝑑𝑄 = 𝑑𝐷 βˆ™ π‘Ÿ = 𝑏1

2𝜌 Ξ©π‘Ÿ 2𝑐(π·π‘‘π‘œ + πœ™π‘π‘™)π‘Ÿπ‘‘π‘Ÿ (24)

β€’ Blade-section AoA for an ideally twisted blade π›Όπ‘Ÿ = 𝑅

π‘Ÿ πœƒπ‘Ÿ βˆ’ πœ™π‘‘ = 1

π‘₯ πœƒπ‘Ÿ βˆ’ πœ™π‘‘ (25)

Profile drag Induced drag

component of lift in the plane of rotation due to the tilt of the lift vector caused by the inflow velocity

(17)

Effect of Profile Drag on F.M.

 2. Expression for torque (Contd.)

β€’ Constant drag coefficient assumption

β†’ Highly optimistic rotor performance then blade stall is present

β†’ assume πΆπ‘‘π‘œ = 𝛿 : average blade profile-drag coefficient 𝐢𝑙 = π‘Žπ‘…

𝑙 πœƒπ‘‘ βˆ’ πœ™π‘‘ , πœ™ = πœ™π‘‘ 𝑅

π‘Ÿ , πΆπ‘‘π‘œ = 𝛿 (26)

(26) β†’ (24) : 𝑑𝑄 = 𝑏1

2𝜌Ω2π‘Ÿ3𝑐 𝛿 + πœ™π‘‘ 𝑅2

π‘Ÿ2 πœƒπ‘‘ βˆ’ πœ™π‘‘ π‘Ž π‘‘π‘Ÿ (27)

Integrating : Q = 𝑏

4 𝜌Ω2π‘Ÿ4𝑐 𝛿

2 + π‘Žπœ™π‘‘ πœƒπ‘‘ βˆ’ πœ™π‘‘ (28)

(28) = (11) : 𝐢𝑄 = 𝜎

4 𝛿

2 + π‘Žπœ™π‘‘ πœƒπ‘‘ βˆ’ πœ™π‘‘ (29)

(23) β†’ (29) : 𝐢𝑄 = πœŽπ›Ώ

8 + πœ™π‘‘πΆπ‘‡ (30)

(18)

Effect of Profile Drag on F.M.

 2. Expression for torque (Contd.)

It is necessary to replace πœ™π‘‘ by parameters that are known or easily

(6) : 𝜈 = 𝑇

2πœŒπœ‹π‘…2 = πΆπ‘‡πœ‹π‘…2𝜌(Ω𝑅)2

2πœŒπœ‹π‘…2 = Ω𝑅 𝐢𝑇

2 (31)

πœ™π‘‘ = 𝜐

Ω𝑅 (32)

(31), (32) : πœ™π‘‘ = 𝐢𝑇

2 (33)

(33) β†’ (30) : 𝐢𝑄 = 𝐢𝑇

3ΰ΅— 2

2 + πœŽπ›Ώ

8 (34)

Induced↑ loss

↑ Profile-drag

loss

(19)

Effect of Profile Drag on F.M.

 3. M as a function of 𝐢𝑇 and 𝛿 (34) β†’ F.M.

𝑀 = 0.707𝐢𝑇

3ΰ΅— 2

𝐢𝑄 = 0.707 𝐢𝑇

3ΰ΅— 2 𝐢𝑇3ΰ΅—

2 2 +πœŽπ›Ώ8

(35)

β€’ If 𝛿 = 0(zero profile drag)

β†’ M=1, M is independent of rotor operating conditions, such as disk loading or tip speed

(20)

Effect of Profile Drag on F.M.

 3. M as a function of 𝐢𝑇 and 𝛿 (Contd.)

β€’ At a small 𝐢𝑇 operation, M would be small – profile drag is fixed and large compared to the numerator.(a)

β€’ At 𝐢𝑇 increases, relative importance of the 𝛿 term decreases – M increases until at a large enough 𝐢𝑇.(b)

β€’ little thrust, or very high tip speed

β€’ (b) rate of increase of thrust > that in power required.

β–²Variation of M w.r.t.𝐢𝑇 for typical values of 𝜎, 𝛿

β€’ (c) at very high thrust values, rate of change of F.M. is not as rapid

β€’ ← induced losses become a larger percentage of the total loss.

β€’ However, In practice, large increases in G β†’ stall of the blade section

β€’ β†’ large increase in profile-drag, falling off of thrust β†’ large decrease in F.M.

(21)

Effect of Rotor Tip Speed and Solidity on F.M.

 Most efficient hovering rotor:

β€’ infinite diameter, zero rotational speed

β†’ Profile-drag losses ≃ 0, induced losses ≃ 0

β€’ Impossible due to practical considerations

β†’ structural, blade weight, size limitations

(22)

 1. Optimum combinations of 𝑉𝑑𝑖𝑝 and 𝜎

Combination of 𝑉𝑑𝑖𝑝 and 𝜎 for minimum profile-drag losses

i) rotor should operate at mean lift coefficient closest to the stall AoA.

ii) rotor should operate at lowest feasible 𝑉𝑑𝑖𝑝

𝐢𝑇 ~ (𝑉𝑑𝑖𝑝)2 and 𝐢𝑃~ (𝑉𝑑𝑖𝑝)3 β†’ for a given thrust, smallest profile-drag power loss is obtained at the smallest 𝑉𝑑𝑖𝑝

Thrust also ∝ 𝐢𝐿 β†’ thrust should be produced at high 𝐢𝐿 and low 𝑉𝑑𝑖𝑝

β†’ high 𝜎

Effect of Rotor Tip Speed and Solidity on F.M.

(23)

 2. Effect of 𝑉𝑑𝑖𝑝

𝑉𝑑𝑖𝑝 = 600𝑓𝑝𝑠 β†’ 400𝑓𝑝𝑠

β†’ power required decrease 50hp (26%), 800lb (25%) increase of

thrust for a constant power and 𝜎 for minimum profile-drag losses.

β€’ Also vertical rate of climb 200fpm

β†’ 1150fpm

Effect of Rotor Tip Speed and Solidity on F.M.

(24)

 3. Relationship between 𝐢𝐿, 𝐢𝑇 and 𝜎

W = 𝐢𝐿 Χ¬0𝑅𝑏1

2𝜌(Ξ©π‘Ÿ)2𝑐 π‘‘π‘Ÿ = 𝐢𝑇 πœ‹π‘…2𝜌(Ω𝑅)2 1

6𝐢𝐿𝜌Ω2𝑅3𝑏𝑐 = 𝐢𝑇 πœ‹π‘…2𝜌(Ω𝑅)2 𝐢𝐿 = 6𝐢𝑇

𝜎 (36)

Lower πΆπ‘‡Ξ€πœŽ : increase in πΆπ‘‡Ξ€πœŽ by reduction in 𝜎 β†’ gain in F.M.

Higher πΆπ‘‡Ξ€πœŽ : increase in 𝜎 β†’ gain in F.M.

β–²F.M. vs πΆπ‘‡Ξ€πœŽ

Effect of Rotor Tip Speed and Solidity on F.M.

(25)

 3. Relationship between 𝐢𝐿, 𝐢𝑇 and 𝜎 (Contd.)

β€’ Practical consideration on 𝑉𝑑𝑖𝑝

- Sudden power failure, more K.E. in the blades

- Larger coning angles and poor psychological effects

β†’ efficient and smooth operation at high speeds ↔ hover design requirements

Design compromise

Effect of Rotor Tip Speed and Solidity on F.M.

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