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**Helicopter Aerodynamics and Performance**

Preliminary Remarks © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

The problems are many.. © L. Sankar Helicopter Aerodynamics

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**A systematic Approach is necessary**

A variety of tools are needed to understand, and predict these phenomena. Tools needed include Simple back-of-the envelop tools for sizing helicopters, selecting engines, laying out configuration, and predicting performance Spreadsheets and MATLAB scripts for mapping out the blade loads over the entire rotor disk High end CFD tools for modeling Airfoil and rotor aerodynamics and design Rotor-airframe interactions Aeroacoustic analyses Elastic and multi-body dynamics modeling tools Trim analyses, Flight Simulation software In this work, we will cover most of the tools that we need, except for elastic analyses, multi-body dynamics analyses, and flight simulation software. We will cover both the basics, and the applications. We will assume familiarity with classical low speed and high speed aerodynamics, but nothing more. © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Plan for the Course PowerPoint presentations, interspersed with numerical calculations and spreadsheet applications. Part 1: Hover Prediction Methods Part 2: Forward Flight Prediction Methods Part 3: Helicopter Performance Prediction Methods Part 4: Introduction to Comprehensive Codes and CFD tools Part 5: Completion of CFD tools, Discussion of Advanced Concepts © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Text Books Wayne Johnson: Helicopter Theory, Dover Publications,ISBN References: Gordon Leishman: Principles of Helicopter Aerodynamics, Cambridge Aerospace Series, ISBN Prouty: Helicopter Performance, Stability, and Control, Prindle, Weber & Schmidt, ISBN Gessow and Myers Stepniewski & Keys © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Grading 5 Homework Assignments (each worth 5%). Two quizzes (each worth 25%) One final examination (worth 25%) All quizzes and exams will be take-home type. They will require use of an Excel spreadsheet program, or optionally short computer programs you will write. All the material may be submitted electronically. © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Instructor Info. Lakshmi N. Sankar School of Aerospace Engineering, Georgia Tech, Atlanta, GA , USA. Web site: Address: © L. Sankar Helicopter Aerodynamics

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**Earliest Helicopter.. Chinese Top**

© L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Leonardo da Vinci (1480? 1493?) © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Human Powered Flight? © L. Sankar Helicopter Aerodynamics

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**D’AmeCourt (1863) Steam-Propelled Helicopter**

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**Paul Cornu (1907) First man to fly in helicopter mode..**

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**De La Cierva invented Autogyros (1923)**

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**© L. Sankar Helicopter Aerodynamics**

Cierva introduced hinges at the root that allowed blades to freely flap Hinges Only the lifts were transferred to the fuselage, not unwanted moments. In later models, lead-lag hinges were also used to Alleviate root stresses from Coriolis forces © L. Sankar Helicopter Aerodynamics

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**Igor Sikorsky Started work in 1907, Patent in 1935**

Used tail rotor to counter-act the reactive torque exerted by the rotor on the vehicle. © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Sikorsky’s R-4 © L. Sankar Helicopter Aerodynamics

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**Ways of countering the Reactive Torque**

Other possibilities: Tip jets, tip mounted engines © L. Sankar Helicopter Aerodynamics

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**Single Rotor Helicopter**

© L. Sankar Helicopter Aerodynamics

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**Tandem Rotors (Chinook)**

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**Coaxial rotors Kamov KA-52**

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**© L. Sankar Helicopter Aerodynamics**

NOTAR Helicopter © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

NOTAR Concept © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Tilt Rotor Vehicles © L. Sankar Helicopter Aerodynamics

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**Helicopters tend to grow in size..**

AH-64A AH-64D Length 58.17 ft (17.73 m) Height 15.24 ft (4.64 m) 13.30 ft (4.05 m) Wing Span 17.15 ft (5.227 m) Primary Mission Gross Weight 15,075 lb (6838 kg) 11,800 pounds Empty 16,027 lb (7270 kg) Lot 1 Weight © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

AH-64A AH-64D Length 58.17 ft (17.73 m) Height 15.24 ft (4.64 m) 13.30 ft (4.05 m) Wing Span 17.15 ft (5.227 m) Primary Mission Gross Weight 15,075 lb (6838 kg) 11,800 pounds Empty 16,027 lb (7270 kg) Lot 1 Weight Hover In-Ground Effect (MRP) 15,895 ft (4845 m) [Standard Day] 14,845 ft (4525 m) [Hot Day ISA + 15C] 14,650 ft (4465 m) [Standard Day] 13,350 ft (4068 m) [Hot Day ISA + 15 C] Hover Out-of-Ground Effect (MRP) 12,685 ft (3866 m) [Sea Level Standard Day] 11,215 ft (3418 m) [Hot Day 2000 ft 70 F (21 C)] 10,520 ft (3206 m) [Standard Day] 9,050 ft (2759 m) [Hot Day ISA + 15 C] Vertical Rate of Climb (MRP) 2,175 fpm (663 mpm) [Sea Level Standard Day] 2,050 fpm (625 mpm) [Hot Day 2000 ft 70 F (21 C)] 1,775 fpm (541 mpm) [Sea Level Standard Day] 1,595 fpm (486 mpm) [Hot Day 2000 ft 70 F (21 C)] Maximum Rate of Climb (IRP) 2,915 fpm (889 mpm) [Sea Level Standard Day] 2,890 fpm (881 mpm) [Hot Day 2000 ft 70 F (21 C)] 2,635 fpm (803 mpm) [Sea Level Standard Day] 2,600 fpm (793 mpm) [Hot Day 2000 ft 70 F (21 C)] Maximum Level Flight Speed 150 kt (279 kph) [Sea Level Standard Day] 153 kt (284 kph) [Hot Day 2000 ft 70 F (21 C)] 147 kt (273 kph) [Sea Level Standard Day] 149 kt (276 kph) [Hot Day 2000 ft 70 F (21 C)] Cruise Speed (MCP) © L. Sankar Helicopter Aerodynamics

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**Power Plant Limitations**

Helicopters use turbo shaft engines. Power available is the principal factor. An adequate power plant is important for carrying out the missions. We will look at ways of estimating power requirements for a variety of operating conditions. © L. Sankar Helicopter Aerodynamics

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**High Speed Forward Flight Limitations**

As the forward speed increases, advancing side experiences shock effects, retreating side stalls. This limits thrust available. Vibrations go up, because of the increased dynamic pressure, and increased harmonic content. Shock Noise goes up. Fuselage drag increases, and parasite power consumption goes up as V3. We need to understand and accurately predict the air loads in high speed forward flight. © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Concluding Remarks Helicopter aerodynamics is an interesting area. There are a lot of problems, but there are also opportunities for innovation. This course is intended to be a starting point for engineers and researchers to explore efficient (low power), safer, comfortable (low vibration), environmentally friendly (low noise) helicopters. © L. Sankar Helicopter Aerodynamics

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**Hover Performance Prediction Methods**

I. Momentum Theory © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Background Developed for marine propellers by Rankine (1865), Froude (1885). Extended to include swirl in the slipstream by Betz (1920) This theory can predict performance in hover, and climb. We will look at the general case of climb, and extract hover as a special situation with zero climb velocity. © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Assumptions Momentum theory concerns itself with the global balance of mass, momentum, and energy. It does not concern itself with details of the flow around the blades. It gives a good representation of what is happening far away from the rotor. This theory makes a number of simplifying assumptions. © L. Sankar Helicopter Aerodynamics

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**Assumptions (Continued)**

Rotor is modeled as an actuator disk which adds momentum and energy to the flow. Flow is incompressible. Flow is steady, inviscid, irrotational. Flow is one-dimensional, and uniform through the rotor disk, and in the far wake. There is no swirl in the wake. © L. Sankar Helicopter Aerodynamics

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**Control Volume is a Cylinder**

Station1 Disk area A 2 V+v2 3 V+v3 4 V+v4 Total area S © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Conservation of Mass © L. Sankar Helicopter Aerodynamics

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**Conservation of Mass through the Rotor Disk**

Flow through the rotor disk = Thus v2=v3=v There is no velocity jump across the rotor disk The quantity v is called induced velocity at the rotor disk © L. Sankar Helicopter Aerodynamics

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**Global Conservation of Momentum**

Mass flow rate through the rotor disk times Excess velocity between stations 1 and 4 © L. Sankar Helicopter Aerodynamics

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**Conservation of Momentum at the Rotor Disk**

V+v Due to conservation of mass across the Rotor disk, there is no velocity jump. Momentum inflow rate = Momentum outflow rate Thus, Thrust T = A(p3-p2) p2 p3 V+v © L. Sankar Helicopter Aerodynamics

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**Conservation of Energy**

Consider a particle that traverses from Station 1 to station 4 We can apply Bernoulli equation between Stations 1 and 2, and between stations 3 and 4. Recall assumptions that the flow is steady, irrotational, inviscid. 1 V+v 2 3 4 V+v4 © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

From an earlier slide # 36, Thrust equals mass flow rate through the rotor disk times excess velocity between stations 1 and 4 Thus, v = v4/2 © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Induced Velocities V The excess velocity in the Far wake is twice the induced Velocity at the rotor disk. To accommodate this excess Velocity, the stream tube has to contract. V+v V+2v © L. Sankar Helicopter Aerodynamics

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**Induced Velocity at the Rotor Disk**

Now we can compute the induced velocity at the rotor disk in terms of thrust T. T = Mass flow rate through the rotor disk * (Excess velocity between 1 and 4). T = 2 r A (V+v) v There are two solutions. The – sign Corresponds to a wind turbine, where energy Is removed from the flow. v is negative. The + sign corresponds to a rotor or Propeller where energy is added to the flow. In this case, v is positive. © L. Sankar Helicopter Aerodynamics

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**Induced velocity at the rotor disk**

© L. Sankar Helicopter Aerodynamics

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**Ideal Power Consumed by the Rotor**

In hover, ideal power © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Summary According to momentum theory, the downwash in the far wake is twice the induced velocity at the rotor disk. Momentum theory gives an expression for induced velocity at the rotor disk. It also gives an expression for ideal power consumed by a rotor of specified dimensions. Actual power will be higher, because momentum theory neglected many sources of losses- viscous effects, compressibility (shocks), tip losses, swirl, non-uniform flows, etc. © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Figure of Merit Figure of merit is defined as the ratio of ideal power for a rotor in hover obtained from momentum theory and the actual power consumed by the rotor. For most rotors, it is between 0.7 and 0.8. © L. Sankar Helicopter Aerodynamics

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**Some Observations on Figure of Merit**

Because a helicopter spends considerable portions of time in hover, designers attempt to optimize the rotor for hover (FM~0.8). We will discuss how to do this later. A rotor with a lower figure of merit (FM~0.6) is not necessarily a bad rotor. It has simply been optimized for other conditions (e.g. high speed forward flight). © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Example #1 A tilt-rotor aircraft has a gross weight of 60,500 lb. (27500 kg). The rotor diameter is 38 feet (11.58 m). Assume FM=0.75, Transmission losses=5% Compute power needed to hover at sea level on a hot day. © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Example #1 (Continued) © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Alternate scenarios What happens on a hot day, and/or high altitude? Induced velocity is higher. Power consumption is higher What happens if the rotor disk area A is smaller? Induced velocity and power are higher. There are practical limits to how large A can be. © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Disk Loading The ratio T/A is called disk loading. The higher the disk loading, the higher the induced velocity, and the higher the power. For helicopters, disk loading is between 5 and 10 lb/ft2 (24 to 48 kg/m2). Tilt-rotor vehicles tend to have a disk loading of 20 to 40 lbf/ft2. They are less efficient in hover. VTOL aircraft have very small fans, and have very high disk loading (500 lb/ft2). © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Power Loading The ratio of thrust to power T/P is called the Power Loading. Pure helicopters have a power loading between 6 to 10 lb/HP. Tilt-rotors have lower power loading – 2 to 6 lb/HP. VTOL vehicles have the lowest power loading – less than 2 lb/HP. © L. Sankar Helicopter Aerodynamics

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**Non-Dimensional Forms**

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**Non-dimensional forms..**

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**© L. Sankar Helicopter Aerodynamics**

Tip Losses A portion of the rotor near the Tip does not produce much lift Due to leakage of air from The bottom of the disk to the top. One can crudely account for it by Using a smaller, modified radius BR, where R BR B = Number of blades. © L. Sankar Helicopter Aerodynamics

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**Power Consumption in Hover Including Tip Losses..**

© L. Sankar Helicopter Aerodynamics

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**Hover Performance Prediction Methods**

II. Blade Element Theory © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Preliminary Remarks Momentum theory gives rapid, back-of-the-envelope estimates of Power. This approach is sufficient to size a rotor (i.e. select the disk area) for a given power plant (engine), and a given gross weight. This approach is not adequate for designing the rotor. © L. Sankar Helicopter Aerodynamics

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**Drawbacks of Momentum Theory**

It does not take into account Number of blades Airfoil characteristics (lift, drag, angle of zero lift) Blade planform (taper, sweep, root cut-out) Blade twist distribution Compressibility effects © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Blade Element Theory Blade Element Theory rectifies many of these drawbacks. First proposed by Drzwiecki in 1892. It is a “strip” theory. The blade is divided into a number of strips, of width Dr. The lift generated by that strip, and the power consumed by that strip, are computed using 2-D airfoil aerodynamics. The contributions from all the strips from all the blades are summed up to get total thrust, and total power. © L. Sankar Helicopter Aerodynamics

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**Typical Blade Section (Strip)**

dT r dr R Root Cut-out © L. Sankar Helicopter Aerodynamics

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**Typical Airfoil Section**

Line of Zero Lift q V+v f Wr aeffective = q - f © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Sectional Forces Once the effective angle of attack is known, we can look-up the lift and drag coefficients for the airfoil section at that strip. We can subsequently compute sectional lift and drag forces per foot (or meter) of span. UT=wr UP=V+v These forces will be normal to and along the total velocity vector. © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Rotation of Forces DT DL DFx V+v Wr DD © L. Sankar Helicopter Aerodynamics

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**Approximate Expressions**

The integration (or summation of forces) can only be done numerically. A spreadsheet may be designed. A sample spreadsheet is being provided as part of the course notes. In some simple cases, analytical expressions may be obtained. © L. Sankar Helicopter Aerodynamics

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**Closed Form Integrations**

The chord c is constant. Simple linear twist. The inflow velocity v and climb velocity V are small. Thus, f << 1. We can approximate cos(f ) by unity, and approximate sin(f ) by ( f ). The lift coefficient is a linear function of the effective angle of attack, that is, Cl=a(q-f) where a is the lift curve slope. For low speeds, a may be set equal to 5.7 per radian. Cd is small. So, Cd sin(f) may be neglected. The in-plane velocity Wr is much larger than the normal component V+v over most of the rotor. © L. Sankar Helicopter Aerodynamics

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**Closed Form Expressions**

© L. Sankar Helicopter Aerodynamics

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**Linearly Twisted Rotor: Thrust**

Here, we assume that the pitch angle varies as © L. Sankar Helicopter Aerodynamics

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**Linearly Twisted Rotor**

Notice that the thrust coefficient is linearly proportional to the pitch angle q at the 75% Radius. This is why the pitch angle is usually defined at 75% R in industry. The expression for power may be integrated in a similar manner, if the drag coefficient Cd is assumed to be a constant, equal to Cd0. Induced Power Profile Power © L. Sankar Helicopter Aerodynamics

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**Closed Form Expressions for Ideally Twisted Rotor**

Same as linearly Twisted rotor! © L. Sankar Helicopter Aerodynamics

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**Figure of Merit according to Blade Element Theory**

High solidity (lot of blades, wide-chord, large blade area) leads to higher Power consumption, and lower figure of merit. Figure of merit can be improved with the use of low drag airfoils. © L. Sankar Helicopter Aerodynamics

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**Average Lift Coefficient**

Let us assume that every section of the entire rotor is operating at an optimum lift coefficient. Let us assume the rotor is untapered. Rotor will stall if average lift coefficient exceeds 1.2, or so. Thus, in practice, CT/s is limited to 0.2 or so. © L. Sankar Helicopter Aerodynamics

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**Optimum Lift Coefficient in Hover**

© L. Sankar Helicopter Aerodynamics

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**Drawbacks of Blade Element Theory**

It does not handle tip losses. Solution: Numerically integrate thrust from the cutout to BR, where B is the tip loss factor. Integrate torque from cut-out all the way to the tip. It assumes that the induced velocity v is uniform. It does not account for swirl losses. The Predicted power is sometimes empirically corrected for these losses. © L. Sankar Helicopter Aerodynamics

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**Example (From Leishman)**

Gross Weight = 16,000lb Main rotor radius = 27 ft Tail rotor radius 5.5 ft Chord=1.7 ft (main), Tail rotor chord=0.8 ft No. of blades =4 (Main rotor), 4 (tail rotor) Tip speed= 725 ft/s (main), 685 ft/s (tail) K=1.15, Cd0=0.008 Available HP =3000Transmission losses=10% Estimate hover ceiling (as density altitude) © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Step I Multiply 3000 HP by 550 ft.lb/sec. Divide this by 1.10 to account for available power to the two rotors (10% transmission loss). We will use non-dimensional form of power into dimensional forms, as shown below: P=k Tv+ r(WR)3A [sCd0/8] Find an empirical fit for variation of r with altitude: © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Step 2 Assume an altitude, h. Compute density, r. Do the following for main rotor: Find main rotor area A Find v as [T/(2rA)]1/2 Note T= Vehicle weight in lbf. Insert supplied values of k, Cd0, W to find main rotor P. Divide this power by angular velocity W to get main rotor torque. Divide this by the distance between the two rotor shafts to get tail rotor thrust. Now that the tail rotor thrust is known, find tail rotor power in the same way as the main rotor. Add main rotor and tail rotor powers. Compare with available power from step 1. Increase altitude, until required power = available power. Answer = 10,500 ft © L. Sankar Helicopter Aerodynamics

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**Hover Performance Prediction Methods**

III. Combined Blade Element-Momentum (BEM) Theory © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Background Blade Element Theory has a number of assumptions. The biggest (and worst) assumption is that the inflow is uniform. In reality, the inflow is non-uniform. It may be shown from variational calculus that uniform inflow yields the lowest induced power consumption. © L. Sankar Helicopter Aerodynamics

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**Consider an Annulus of the rotor Disk**

Area = 2prdr Mass flow rate =2prr(V+v)dr dT = (Mass flow rate) * (twice the induced velocity at the annulus) = 4prr(V+v)vdr dr r © L. Sankar Helicopter Aerodynamics

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**Blade Elements Captured by the Annulus**

Thrust generated by these blade elements: dr r © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Equate the Thrust for the Elements from the Momentum and Blade Element Approaches Total Inflow Velocity from Combined Blade Element-Momentum Theory © L. Sankar Helicopter Aerodynamics

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**Numerical Implementation of Combined BEM Theory**

The numerical implementation is identical to classical blade element theory. The only difference is the inflow is no longer uniform. It is computed using the formula given earlier, reproduced below: Note that inflow is uniform if q= CR/r . This twist is therefore called the ideal twist. © L. Sankar Helicopter Aerodynamics

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**Effect of Inflow on Power in Hover**

constraint Variation of a functional © L. Sankar Helicopter Aerodynamics

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**Ideal Rotor vs. Optimum Rotor**

Ideal rotor has a non-linear twist: q= CR/r This rotor will, according to the BEM theory, have a uniform inflow, and the lowest induced power possible. The rotor blade will have very high local pitch angles q near the root, which may cause the rotor to stall. Ideally Twisted rotor is also hard to manufacture. For these reasons, helicopter designers strive for optimum rotors that minimize total power, and maximize figure of merit. This is done by a combination of twist, and taper, and the use of low drag airfoil sections. © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Optimum Rotor We try to minimize total power (Induced power + Profile Power) for a given T. In other words, an optimum rotor has the maximum figure of merit. From earlier work (see slide 72), figure of merit is maximized if is maximized. All the sections of the rotor will operate at the angle of attack where this value of Cl and Cd are produced. We will call this Cl the optimum lift coefficient Cl,optimum . © L. Sankar Helicopter Aerodynamics

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**Optimum rotor (continued..)**

© L. Sankar Helicopter Aerodynamics

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**Variation of Chord for the Optimum Rotor**

dT = (Mass flow rate) * (twice the induced velocity at the annulus) = 4prr(v)vdr Compare these two. Note that Cl is a constant (the optimum value). It follows that Local solidity © L. Sankar Helicopter Aerodynamics

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**Planform of Optimum Rotor**

Root Cut out Chord is proportional to 1/r Tip r r=R Such planforms and twist distributions are hard to manufacture, and are optimum only at one thrust setting. Manufacturers therefore use a combination of linear twist, and linear variation in chord (constant taper ratio) to achieve optimum performance. © L. Sankar Helicopter Aerodynamics

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**Accounting for Tip Losses**

We have already accounted for two sources of performance loss-non-uniform inflow, and blade viscous drag. We can account for compressibility wave drag effects and associated losses, during the table look-up of drag coefficient. Two more sources of loss in performance are tip losses, and swirl. An elegant theory is available for tip losses from Prandtl. © L. Sankar Helicopter Aerodynamics

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**Prandtl’s Tip Loss Model**

Prandtl suggests that we multiply the sectional inflow by a function F, which goes to zero at the tip, and unity in the interior. When there are infinite number of blades, F approaches unity, there is no tip loss. © L. Sankar Helicopter Aerodynamics

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**Incorporation of Tip Loss Model in BEM**

All we need to do is multiply the lift due to inflow by F. Thrust generated by the annulus: dr dT = = 4prrF(V+v)vdr r © L. Sankar Helicopter Aerodynamics

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**Resulting Inflow (Hover)**

© L. Sankar Helicopter Aerodynamics

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**Hover Performance Prediction Methods**

IV. Vortex Theory © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

BACKGROUND Extension of Prandtl’s Lifting Line Theory Uses a combination of Kutta-Joukowski Theorem Biot-Savart Law Empirical Prescribed Wake or Free Wake Representation of Tip Vortices and Inner Wake Robin Gray proposed the prescribed wake model in 1952. Landgrebe generalzied Gray’s model with extensive experimental data. Vortex theory was the extensively used in the 1970s and 1980s for rotor performance calculations, and is slowly giving way to CFD methods. © L. Sankar Helicopter Aerodynamics

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**Background (Continued)**

Vortex theory addresses some of the drawbacks of combined blade element-momentum theory methods, at high thrust settings (high CT/s). At these settings, the inflow velocity is affected by the contraction of the wake. Near the tip, there can be an upward directed inflow (rather than downward directed) due to this contraction, which increases the tip loading, and alters the tip power consumption. © L. Sankar Helicopter Aerodynamics

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**Kutta-Joukowsky Theorem**

G : Bound Circulation surrounding the airfoil section. This circulation is physically stored As vorticity in the boundary Layer over the airfoil DT Wr DFx V+v DT = r (Wr) G DFx= r (V+v) G © L. Sankar Helicopter Aerodynamics

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**Representation of Bound and Trailing Vorticies**

Since vorticity can not abruptly increase in space, trailing vortices develop. Some have clockwise rotation, others have counterclockwise rotation. © L. Sankar Helicopter Aerodynamics

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**Robin Gray’s Conceptual Model**

Tip Vortex has a Contraction that can be fitted with an exponential curve fit. Inner wake descends at a near constant velocity. It descends faster near the tip than at the root. © L. Sankar Helicopter Aerodynamics

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**Landgrebe’s Curve Fit for the Tip Vortex Contraction**

© L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Radial Contraction © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Vertical Descent Rate Descent is faster After the first blade Passes over the vortex Zv Initial descent is slow v © L. Sankar Helicopter Aerodynamics

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**Landgrebe’s Curve Fit for Tip Vortex Descent Rate**

qtwist,degrees: Blade twist=Tip Pitch angle – Root Pitch Angle This quantity is usually negative. © L. Sankar Helicopter Aerodynamics

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**Circulation Coupled Wake Model**

Landgrebe’s earlier curve fits (1972) were based on the thrust coefficient, blade twist (change in the pitch angle between tip and root, usually negative). He subsequently found (1977) that better curve fits are obtained if the tip vortex trajectory is fitted on the basis of peak bound circulation, rather than CT/s. © L. Sankar Helicopter Aerodynamics

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**Tip Vortex Representation in Computational Analyses**

The tip vortex is a continuous helical structure. This continuous structure is broken into piecewise straight line segments, each representing 15 degrees to 30 degrees of vortex age. The tip vortex strength is assumed to be the maximum bound circulation. Some calculations assume it to be 80% of the peak circulation. The vortex is assumed to have a small core of an empirically prescribed radius, to keep induced velocities finite. © L. Sankar Helicopter Aerodynamics

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**Tip Vortex Representation**

Control Points on the Lifting Line where induced flow is calculated Lifting Line 15 degrees Inner Wake (Optional) The x,y,z positions of the End points of each segment Are computed using Landgrebe’s Prescribed Wake Model © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Biot-Savart Law Control Point Segment © L. Sankar Helicopter Aerodynamics

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**Biot-Savart Law (Continued)**

Core radius used to keep Denominator from going to zero. © L. Sankar Helicopter Aerodynamics

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**Overview of Vortex Theory Based Computations (Code supplied)**

Compute inflow using BEM first, using Biot-Savart law during subsequent iterations. Compute radial distribution of Loads. Convert these loads into circulation strengths. Compute the peak circulation strength. This is the strength of the tip vortex. Assume a prescribed vortex trajectory. Discard the induced velocities from BEM, use induced velocities from Biot-Savart law. Repeat until everything converges. During each iteration, adjust the blade pitch angle (trim it) if CT computed is too small or too large, compared to the supplied value. © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Free Wake Models These models remove the need for empirical prescription of the tip vortex structure. We march in time, starting with an initial guess for the wake. The end points of the segments are allowed to freely move in space, convected the self-induced velocity at these end points. Their positions are updated at the end of each time step. © L. Sankar Helicopter Aerodynamics

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**Free Wake Trajectories (Calculations by Leishman)**

© L. Sankar Helicopter Aerodynamics

111
**Vertical Descent of Rotors**

© L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Background We now discuss vertical descent operations, with and without power. Accurate prediction of performance is not done. (The engine selection is done for hover or climb considerations. Descent requires less power than these more demanding conditions). Discussions are qualitative. We may use momentum theory to guide the analysis. © L. Sankar Helicopter Aerodynamics

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**Phase I: Power Needed in Climb and Hover**

Descent Climb Velocity, V © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Non-Dimensional Form It is convenient to non-dimensionalize these graphs, so that universal behavior of a variety of rotors can be studied. © L. Sankar Helicopter Aerodynamics

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**Momentum Theory gives incorrect Estimates of Power in Descent**

(V+v)/vh Descent Climb V/vh No matter how fast we descend, positive power is still required if we use the above formula. This is incorrect! © L. Sankar Helicopter Aerodynamics

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**The reason.. Climb or hover Descent: Everything inside**

V is down V is up V+v is down V+v is down V is down V is down V+2v is down V is up V is up V+2v is down Climb or hover Physically acceptable Flow Descent: Everything inside Slipstream is down Outside flow is up © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

In reality.. The rotor in descent operates in a number of stages, depending on how fast the vertical descent is in comparison to hover induced velocity. Vortex Ring State Turbulent Wake State Windmill Brake State © L. Sankar Helicopter Aerodynamics

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**Vortex Ring State (V is up, V+v is down, V+2v is down)**

The rotor pushes tip vortices down. Oncoming air at the bottom pushes them up Vortices get trapped in a donut-shaped ring. The ring periodically grows and bursts. Flow is highly unsteady. Can only be empirically analyzed. V+v is down V is up V is up © L. Sankar Helicopter Aerodynamics

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**Performance in Vortex Ring State**

Descent Climb Power/TVh Vortex Ring State Experimental data Has scatter Momentum Theory V/vh Cross-over At V=-1.71vh © L. Sankar Helicopter Aerodynamics

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**Turbulent Wake State (V is up, V+v is up, V+2v is down)**

Rotor looks and behaves like a bluff Body (or disk). The vortices look Like wake behind the bluff body. Again, the flow is unsteady, Can not analyze using momentum theory Need empirical data. V+2v is down V is up V is up © L. Sankar Helicopter Aerodynamics

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**Performance in Turbulent Wake State**

Descent Power/TVh Climb Vortex Ring State Momentum Theory Turbulent Wake State Notice power is –ve Engine need not supply power V/vh Cross-over At V=-1.71vh © L. Sankar Helicopter Aerodynamics

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**Wind Mill Brake State (V is up, V+v is up, V+2v is up)**

Flow is well behaved. No trapped vortices, no wake. Momentum theory can be used. V is up T = - 2rAv(V+v) V+2v up V is up Notice the minus sign. This is because v (down) and V+v (up) have opposite signs. The product must be positive.. © L. Sankar Helicopter Aerodynamics

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**Power is Extracted in Wind Mill Brake State**

© L. Sankar Helicopter Aerodynamics

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**Physical Mechanism for Wind Mill Power Extraction**

Lift Total Velocity Vector V+v Wr The airfoil experiences an induced thrust, rather than induced drag! This causes the rotor to rotate without any need for supplying power or torque. This is called autorotation. Pilots can take advantage of this if power is lost. © L. Sankar Helicopter Aerodynamics

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**Complete Performance Map**

Descent Power/TVh Climb Vortex Ring State Momentum Theory Turbulent Wake State V/vh Cross-over At V=-1.71vh Wind Mill Brake State © L. Sankar Helicopter Aerodynamics

126
**Consider the cross-over Point**

© L. Sankar Helicopter Aerodynamics

127
**Coning Angle Calculations**

Hover Performance Coning Angle Calculations © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Background Blades are usually hinged near the root, to alleviate high bending moments at the root. This allows the blades t flap up and down. Aerodynamic forces cause the blades to flap up. Centrifugal forces causes the blades to flap down. In hover, an equilibrium position is achieved, where the net moments at the hinge due to the opposing forces (aerodynamic and centrifugal) cancel out and go to zero. © L. Sankar Helicopter Aerodynamics

129
**Schematic of Forces and Moments**

We assume that the rotor is hinged at the root, for simplicity. This assumption is adequate for most aerodynamic calculations. Effects of hinge offset are discussed in many classical texts. © L. Sankar Helicopter Aerodynamics

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**Moment at the Hinge due to Aerodynamic Forces**

From blade element theory, the lift force dL = Moment arm = r cosb0 ~ r Counterclockwise moment due to lift = Integrating over all such strips, Total counterclockwise moment = © L. Sankar Helicopter Aerodynamics

131
**Moment due to Centrifugal Forces**

The centrifugal force acting on this strip = Where “dm” is the mass of this strip. This force acts horizontally. The moment arm = r sinb0 ~ r b0 Clockwise moment due to centrifugal forces = Integrating over all such strips, total clockwise moment = © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

At equilibrium.. Lock Number, g © L. Sankar Helicopter Aerodynamics

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**© L. Sankar Helicopter Aerodynamics**

Lock Number, g The quantity g=racR4/I is called the Lock number. It is a measure of the balance between the aerodynamic forces and inertial forces on the rotor. In general g has a value between 8 and 10 for articulated rotors (i.e. rotors with flapping and lead-lag hinges). It has a value between 5 and 7 for hingeless rotors. We will later discuss optimum values of Lock number. © L. Sankar Helicopter Aerodynamics

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