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Blade-Element Theory for Propeller in Forward Flight.

In order to derive a Blade-Element/Momentum procedure valid for helicopter hover and vertical

climb as well as propeller static conditions and propeller forward ight, it is necessary to make a

few generalizations to the simple helicopter theory. To begin, consider the blade-element diagram

for the propeller case, and allow (the forward velocity) to be large relative to (or comparable to)

the rotational speed, . We will also allow the induced velocity ( ) to have a swirl component

as well as an axial one, and to a rst approximation is induced perpendicular to the apparent

oncoming ow. We will de ne

= arctan

(1)

= arctan

p 2 + 2 2

p 2 + 2 2

(2)

Note that , the induced angle of attack, is nearly always a small angle (the exception being

under very highly loaded static conditions) whereas is generally not small. Also note that the

angle designated as in our earlier analysis (which we had assumed to be small for the helicopter

case) is approximately + .

The axial component of the induced velocity is cos . (If were small, as in the helicopter

development, 1 .) Incorporating this into the expression for the thrust obtained using the

momentum theory for an annulus of thickness , we arrive at

= 2 cos ( + cos ) 2 (3)

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Now it is necessary to formulate an expression for the elemental thrust generated by a blade

element. First, recall that

= cos( + ) sin( + ), and (4)

= [ cos( + ) + sin( + )] (5)

Using the fact that is small and, for now, neglecting the drag component,

cos (6)

where

= 12

2 + 2 2 ( ) (7)We justify the neglect of the drag at this juncture since we are comparing an inviscid expression

to another inviscid expression. In e ff ect, we are stating that the viscous drag does not a ff ect

the magnitude of the induced velocity . This is not entirely true since the drag force causes a

negative component of thrust, thus requiring that a higher lift be produced than would be otherwise

predicted. The drag, therefore, indirectly a ff ects the value of the induced velocity.

Equating the elemental thrust from the momentum analysis to that obtained using the blade-

element method gives

2 + 2 2 ( ) = 8 ( + cos ) (8)

Rearranging and letting

p 2 + 2 2 and making all values dimensionless leads to the

quadratic equation for the induced angle

2 + + 8 2 8 2 ( ) = 0 (9)All dimensionless variables are de ned as before, with = , = , and = . The

new dimensionless value, = p 2 + 2 2 = q 2 + 2 . Note that if , reduces to

and equation 9 becomes that for the problem of the helicopter in vertical climb. Solving for

results in

= 12 +

8 2 + s

14 +

8 2

2

+

8 2 ( ) (10)

Equation 10 is general and can be used for any helicopter or propeller in axial ight (including

static loading or hover). Notice that, in this case, the induced in ow ratio (Leishmens ) is not

well-de ned and it is useful to solve directly for the induced angle of attack.

Equations 10 and 7, along with an equivalent expression for (= 12 ( 2 + 2 2 ) ), can

be used in equations 4 and 5 and integrated over the blades to obtain the total thrust and power

required for a given propeller under given operating conditions.

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Prandtls Tip Loss Factor for Propellers.

To add the tip-loss factor to the propeller blade-element analysis, begin with the equivalent to

Prandtls assertion, i.e. , that = 2 cos ( + cos ) 2 . Following the above steps

leads to the equation involving :

8 ( + ) = ( ), (11)

which can be rearranged to yield a "quadratic" for . However, since is a function of , the

equation is transcendental and must be solved through iteration or another root- nding method

(such as MATLABs fsolve fuction). Equation 11 is likely more easily solved than the equivalent

quadratic form as it avoids having a zero denominator when = 0 at the blade tip. The tip-loss

function remains given as = 2 arccos(exp( )) , where is given more generally as

= 2 1

where now + .

Compressibility Corrections

The correction to the lift-curve slope to account for Mach-number e ff ects is, ideally

= 0

p 1 2

where 0 is the lift-curve slope for low speed ( 0.) For a propeller or rotor in vertical ight,

= q 2 + 2 where is the forward Mach number and is the rotational Mach number.Using the expressions for and gives =

0

q 1 2tip 2(12)

Correction to the pro le drag term requires knowledge of the critical Mach number for the

rotor airfoil (freestream Mach number at which sonic velocity rst occurs on the airfoil surface).A typical correction factor can be determined by tting the vs Mach number curve to an

appropriate polynomial. Critical Mach number depends on the airfoil angle of attack (or ) and

thickness. De ne = crit . Then a sixth-order curve t appropriate for 0 75 1 07 is

given by

= 14 7487 111 95 + 350 502 2 580 051 3 + 535 651 4 261 909 5 + 53 0095 6 (13)

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