Japan-Korea Joint Workshop on Rotorcraft
Sponsored by Japan Helicopter Society and KSAS VTOL Committee
2023.02.10 (Friday)

Aerodynamics Theories for Estimating Flight Profileof eVTOL Aircraft
Prof. Zhong Lei
Suwa University of Science, Japan
Contact: lei@rs.sus.ac.jp
Original Papers:
1. Yuto Fukumine，Zhong Lei，: “Estimation of eVTOL Performance Using Rotorcraft Theory”, the 33rd
Congress of the International Council of the Aeronautical Sciences, Stockholm, Sweden, 2022.9
2.

Haruki Takizawa, Zhong Lei: “Operational Feasibility Study of Air Taxi Service using eVTOL in Japan” the33rd Congress of the International Council of the Aeronautical Sciences, Stockholm, Sweden, 2022.9

Outline of content• Research Objectives and Approach• eVTOL Airplane of UAM• Synthesis of Propeller and Rotor Theories• Results and Discussions• Conclusion

Research ObjectiveObjectives
 to investigate flight performance of representative eVTOL aircrafts, which
including hovering, climb and descent, and forward flights,
 to provide technical information for aircraft design and operational feasibility.
The method is based on theories of rotorcraft and fixed-wing aircraft that aremodified to be used for the eVTOL aircraft.
It is expected that this tool may be used for design and operation of eVTOL aircrafts.

Fig. Turboprop propulsion

Fig. Electric propulsion

3

Research Approach• Information of eVTOL aircraft are very limited.
• A synthesis of rotary blade theory was developed to estimate flight performanceusing statistical data, empirical rules, aircraft engineering theory.
• Calculations were adjusted for overall consistency.
Aircraft data collection• Data published by aircraft maker• Data read from published graph• New, articles, web resource and so on

Performance evaluation：• fixed-wing and helicopter engineering theory• Constraints of shape, performance• Satisfaction of published specifications

15重量出力比(MTOW/Pmax)、単位kg/kW

Parameter determination：• Database of existing aircraft• Statistical and empirical formula• overall consistency with published data

20kW級 100kW級 500kW級
lightweightaircraftMotor
glider

10T-Cubic

business
prop
Large
drone

5dronehelicopter

eVTOL

tilt rotor

0Results

10

100

1,000

10,000

最大離陸重量（MTOW）、単位kg

100,000

4

Characteristics of eVTOL Aircraft
Multicopter type:• coupled rotors, no wing
• flight speed direction are controlled bychanging the angle of the rotor axis ofrotation

Table. eVTOL aircraft investigated in this study

Vectored thrust type:• coupled tilt-rotors, wing
• transit lift in vertical flight and thrust in
forward flight by changing the axis ofrotation of the rotor.
Lift & cruise type:• coupled fixed-rotors and wing
• generates lift on the fixed rotor• generates thrust in forward direction by
the propeller• generates lift on the wing during cruise5Flight Mission① Takeoff and vertical flight

Like conventional helicopter, the flight profile ofeVTOL aircraft may be divided into several phases.

② Climb and forward flight

Mission Includes altitude, flight time, range, speed.

③ Cruise at specified altitude• Take-off: Climb vertically at hover power(nohorizontal movement)④ Descent and forward flight• Climb: Climb to cruise altitude⑤ Vertical descent and landing• Cruise: Flight phase that occurs when the aircraft
levels after a climb to a cruise altitude and before
it begins to descend⑥ Hovering• Descent: Aircraft begins approach to final landing.
Has both horizontal and vertical component③④• Landing: Vertical landing at hoverpower (no horizontal movement)• Hovering: Stay in the air by rotor⑥⑤

Main mission flight profile②①6

Synthesis of Propeller and Rotor Theories: Fixed-Wing Aircraft
Induced velocity 𝑤 = 2𝑣𝑖

Advance ratio 𝐽 =According to Bernoulli’s principle𝑉Ω𝑅Ω = 2𝜋𝑛111∆𝑝 = 𝐻1 − 𝐻 = 𝑝 + 𝜌 𝑉 + 𝑤 2 − 𝑝 + 𝜌𝑉 2 = 𝜌 𝑉 + 𝑤 𝑤 = 2𝜌 𝑉 + 𝑣𝑖 𝑣𝑖
222Thrust generated by propeller is calculated by momentum equation
𝑇 = ∆𝑝𝑆 = 𝜌𝑆 𝑉 + 𝑣𝑖 𝑤 = 2𝜌𝑆 𝑉 + 𝑣𝑖 𝑣𝑖Then, induced velocity may be solved𝑣𝑖
𝑉=−+𝑣𝑠2𝑣𝑠𝑉2𝑣𝑠2+1

At static condition （𝑉 = 0）
𝑣𝑠 =𝑇2𝜌𝑆

propeller disc7

Synthesis of Propeller and Rotor Theories: Helicopter
Break velocity into normal (𝑛) and tangential (𝜏) components.

Inflow ratio

Tip speed ratio𝜇𝜏 =𝑉𝜏
𝑉∞ cos 𝛼=Ω𝑅
Ω𝑅

𝜆=𝑉𝑛𝑣𝑖+= 𝜇𝑛 + 𝜆𝑖
Ω𝑅 Ω𝑅

rotating disk

Induced velocity at hovering𝑣ℎ =𝑇2𝜌𝜋𝑅 2and𝜆ℎ =𝐶𝑇2

Induced velocity at forward flight𝑣𝑖 =𝑣ℎ2𝑣ℎ2=𝑈𝑉𝜏 2 + 𝑉𝑛 + 𝑣𝑖 2𝑉∞

Substitute into the equation for the inflow ratio 𝜆 is𝜆 = 𝜇𝑛 +𝐶𝑇2 𝜇𝜏2 + 𝜆2

Solved by iteration𝑉𝜏𝑉𝑐𝑉𝑛𝛼𝑣𝑖8Flight Performance Estimation Using the Synthesis Theory
Flight Mission

Force balance

𝐿 = 𝑊 cos 𝜃𝐹𝑃 ,Specification: range, weight, motor power
Dimension: fuselage, wing, rotor, propeller

Momentum equation

Aerodynamic forces of the fixed wing:
Lift, drag, pitching moment

Equation of inflow ratio

Angle of attack
Induced velocity, rotating speed of blade
Thrust of rotor or propeller

Solve a set of equations including• Force balance equation• Momentum equation• Blade element theory
Specification and constraints
satisfied？
Results：• Induced velocity, rotating speed,、• Blade lift coeff., pitching angle,• Motor torque, power• Range, endurance

𝑇 = 𝐷 + 𝑊 sin 𝜃𝐹𝑃

𝑇 = 2𝜌𝐴𝑣𝑖 𝑉∞2 + 2𝑉∞ 𝑣𝑖 sin 𝛼 + 𝑣𝑖2
𝜆 = 𝜇𝑛 +𝐶𝑇2 𝜇𝜏2 + 𝜆2

Relation of thrust coefficient and lift coeff.
Modifyparameters

Total power required① induced power

1𝐶𝑇 = 𝜎 𝐶𝐿
6

𝑃 = 𝑃𝑖 + 𝑃0 + 𝑃𝑝 + 𝑃𝑐
𝑃𝑖 = 𝑇 × 𝑣𝑖

② Profile power
113𝑃0 = 𝜌𝑁𝑏 𝑐𝑅 Ω𝑅 3 1 + 𝐾𝜇2 𝐶𝑑0 = 𝜌𝐴𝑉𝑡𝑖𝑝
𝜎 1 + 𝐾𝜇2 𝐶𝑑088③ Airframe parasite power
11𝑃𝑝 = 𝐷 × 𝑉∞ = 𝜌𝑉∞2 𝑆𝐶𝐷𝑓 × 𝑉∞ = 𝜌𝑉∞3 𝑆𝐶𝐷𝑓
22④ climb power
𝑃𝑐 = 𝑊 × 𝑉𝑐

9Validation : Performance Estimation of UH-60A (Black Hwak)• The method is validated by the flighttest data of UH-60A helicopter.

 Flight test
ー Estimation

 Flight test
ー Estimation

main rotor

tail rotor• The predicted power coefficients ofthe main and tail rotors arecompared and agreed well with theflight data.
Main rotor
SC1095/
airfoil
SC1094
radius [m]8.18
chord length [m]0.527
number of blades 4rotating speed at258
hovering [rpm]Tail rotor

SC1095
1.68
0.250
41190
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Investigation of Flight Performance of eVTOL Airplanes
11 eVTOL airplanes were investigated, 3 of them were typically picked up,
based on the data available in the middle of 2022. Red data are estimated.

11

Disc Loading and Weight-to-Power Ratio
𝑇~1𝑃

Multicopter

10

Lift & cruise

𝑇2𝜌𝐴

2

Characteristics
Multicopter type：• Large disc area
• Low disc loading

111

95648
3

Vectored thrust7① EHang 216② Volocity③ AirCitybus④ S4⑤ Maker⑥ VX4⑦ 7-Seater⑧ Cora⑨ Allia-250⑩ DreamMaker⑪ VF1500

Vectored thrust type：• Allaircrafts
hassimilarperformance, except Lilium Jet
Lift & cruise type：• High efficiency• Rotor for hovering• Propeller for wing-borne flight

Generally, it is recommended that theweight-to-power ratio• be larger than 5 kg/kW at hovering• be larger than 20 kg/kW at cruising12Overall Performance(based on the data available in the middle of 2022)• Lift & Cruise type has the best performance at cruising.
• #7, Lilium Jet has bad performance, especially at hovering.
profile power
Very large
disc loading

0.6

0.4

4
induced power

Small discloading

0.2

0.2
0.15

profile power
airframe power

0.1

2.5 Multicopter

21.50
1 2 3 4 5 6 7 8 9 1011

Fig. Power at hovering flight

Fig. Power at cruising flight𝑟 = 𝑅total /𝐸total

30.5

1 2 3 4 5 6 7 8 9 10 11

Cruising rate

Vectored Thrust

10.05003.5

R/E [km/kWh]0.80.25

induced power
Power/Weight [kW/kg]

Power/Weight [kW/kg]

1

Lift & Cruise

Multicopter < Vectored Thrust < Lift & Cruise

Where, 𝑅total : range, 𝐸total : consumed energy

1 2 3 4 5 6 7 8 9 1011

Fig. Cruising rate13

Results and Discussions: Hovering• Power is increased as the weight at hovering.
• Power-to-weight ratio is increased with diskloading.

Profile power
Induced power
𝑃𝑊~𝑊2𝜌𝐴

A large rotating area results in alower disk loading, thus a benefitof power-to-weight ratio.

Example 1 Example 2 Example 3

Fig. Power breakdown at hovering
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Results and Discussions: Climb and Descent

Suitable rate of climb and descentshould be selected to reduce power
consumption.

Generally, the rate is recommended
to be larger than 4 m/s.

Altitude: 0~150 ft
Energy consumption [kWh]At vertical climb and descent, total
power consumption is decreased
with increase of climb and descentrate, respectively.

descent

climb
Example 3
Example 2Example 1

Suitable rate

Suitable rate

Rate of climb [m/s]Fig. Estimation results of total power consumption
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Results and Discussions: Performance at Cruising• Multicopter Volocity is suitable for shortrange.

Example 1〇 cruisePower requied [kW]• The wing has significant effects on thecruise speed and necessary power atcruising.

300
250

Example 3200150
Example 210050Volocity

S4

Alia 250

00100
200
300
flight speed [km/h]

400Fig. Power required in level flight16

Results and Discussions: Flight Profile of Motor Output• Different characteristics for
fixed-pitch / variable-pitch
blade, fixed / tilt rotor, rotor /propeller• Large changes in output ofpower, torque, and rotating
speed at different phases

It suggests that different typeshould match different motor.

PowerTorqueRotating speed

Example 1

Example 2

Example 3⑥Flight phases
17Results and Discussions: Range and Endurance
Factors which most influence range and endurance, include• flight profile : altitude, hovering time, climb rate, cruise speed• aerodynamics : rotor, propeller and airframe• propulsion : efficiency, energy density, battery capacity (especially for eVTOL)
Both lift & cruise type (b) and vectored thrust type (c) may fly more than 1 hour and 150 miles.

It is suggested that both of them may be used as UAM.
Cruise range
Cruise time40
2000100
200300
Cruising speed[km/h](multicopter)(a) Example
Volocity 1(multicopter)

100

Cruise range

80

80

Cruise time

60

100

500

100

Cruise range
Cruise time

500

400

80

400

60

300

60

300

40

40200

40

200

2020100

20100

000000 100 200 300 400
Cruising speed[km/h]

02 (vectoredthrust)(b) Example
S4 (vectoredthrust)(lift && cruise)cruise)(c) Example
Alia 2503(liftCruise range [km]Cruise time [min]60

100 200 300 400
Cruising speed[km/h]
18

Results and Discussions: Power Density of Motor
With consideration of cost and performance, power density of motor 3~4kW/kg
is suitable for eVTOL airplane.

Endurance [min]range [km]ExampleExamplepower density of motor

range [km]range [km]power density of motor

Endurance [min]ExampleEndurance [min]range [km]Endurance [min]Endurance [min]power density of motor

Endurance [min]power density of motor

Examplepower density of motor

range [km]power density of motor

range [km]power density of motor

Endurance [min]rangerange [km]Endurance [min]endurance

ExampleExamplerange [km]ExampleExamplepower density of motor

19Results and Discussions: Energy Density of Battery
Energy density of battery may be determined to match specified range and endurancewith consideration of cost and performance.

Energy Density of Battery

Endurance [min]range [km]Endurance [min]Energy Density of Battery

ExampleExampleExampleEnergy Density of Batteryrange [km]range [km]range [km]Energy Density of Battery

Endurance [min]Energy Density of Battery

Endurance [min]range [km]Endurance [min]ExampleEnergy Density of Battery

Endurance [min]Energy Density of Battery

range [km]Endurance [min]range [km]Endurance [min]range

ExampleExamplerange [km]ExampleExampleendurance

Energy Density of Battery

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Conclusion
The theory of rotorcraft aerodynamics was extended to calculate rotor, propeller
and tiltrotor, and a computational tool was developed to estimate various typesof eVTOL aircrafts.
Some suggestions for eVTOL design and operation.
• Generally, it is recommended that the ratio of weight-to-power should
be larger than 5 kg/kW at hovering, be larger than 20 kg/kW at cruising.
• Suitable rates of climb and descent should be selected to reduce power
consumption. Generally, the rate is recommended to be larger than 4 m/s.
• Energy density of motor 3~4 kW/kg is suitable for eVTOL airplane.
• Energy density of battery should be larger than 200Wh/kg, which is dependent
on the eVTOL aircraft.
• It suggests that different type of eVTOL aircraft should match different motor.
• Multicopter eVTOL aircraft is suitable for short range with low speed.
• The vectored thrust eVTOL aircraft and lift & cruise eVTOL aircraft are more
21
efficient for longer range with high speed.

Hope this computation tool may be used to help designand operation of eVTOL aircraft.

Thanks for your attention

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