)
√( )
√(
)
From the above expressions, we can say that inflow decreases as the helicopter rise up or climb up. Mass flow rate increases as the velocity changes.
Power Needed in Climb and Hover
It is convenient to non-dimensionalize these graphs, so that universal behavior of a variety of rotors can be studied.
( )
< √( )
=
Descent condition
Thrust is always up and weight is down. The rotor to support the weight it has to push the air down. In this case is downward and V is upward. As the rotor pushes the air down, the farfield downstream also moves downward.
The mass flux is
̇ ( ) Momentum conservation gives
Thrust = Change in Momentum
Climb Velocity, V Power
Descent
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̇( ) ̇ ̇
Thrust acting on the rotor disk is upward, but on the fluid is opposite in direction.
From energy conservation
( ) ( ) ̇( ) ̇
̇( ) ( ) ̇ ( ) ̇ ( ) ̇( )
( )
Momentum Theory gives incorrect Estimates of Power in Descent.
( )
< √( )
=
In hover
In hover, climb or descent, rotor supports the same weight. So T will be the same ( )
V/v
h(V+v)/v
hClimb Descent
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(
) ( )
( ) ( )
This is a simple quadratic relationship of non-dimensional inflow w.r.t inflow at hover condition.
The solution of the above quadratic equation is
: √( ) ; ⁄
√(
) The roots are positive in both the cases. When V is less than
√( )
becomes imaginary value. This is valid when V is more than . is positive in both the roots When V is much larger than the term
√( )
Gives positive value, otherwise this becomes imaginary value.
At rotor disk,
√(
) Climb (V<0)
√(
)
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If we consider V<0 for descending condition Helicopter Up (Positive) and Down (Negative)
Assume that these expressions are valid everywhere.
Inflow diagram for vertical flight is expressed in non-dimensional form. Continuous lines are valid. In dotted line region, flow is highly mixed so it is difficult to calculate inflow value.
Dotted lines are extrapolated values. When , there is no flow through the disk.
The lines divide the plane into four regions, where the rotor operating condition is named the normal working state (climb and hover), vortex ring state, turbulent wake state and windmill brake state. For climb it was assumed that the air is moving downward throughout the flow field ( ). The flow through the disk and in the wake is downward while the flow outside the slipstream is upward. The climb solution may be expected to be valid for small rate of descent, however, where at least near the rotor the flow is all downward.
For the rotor in descent, it was assumed that the air is moving upward throughout the flow
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field ( ). For the upper branch of the descent solution, however, , so the flow is downward in the far wake while it is upward everywhere else, including outside the wake slip-stream.
In the vortex ring and turbulent wake states the flow outside the slipstream is upward while the flow inside the far wake is nominally downward. Because such a flow state is not possible, there is no valid momentum theory solution for the moderate rates of descent between V = 0 and .
The line corresponds to ideal autorotation, P=0, and is in the center of the range where momentum theory is not valid.
At hover V=0, becomes 1. So the curve starts from 1 in climb. is a constant quantity for a given rotor. It is a fixed value in climb condition. As the velocity increases the below term decreases and reaches zero asymptotically at very high velocity value.
√(
)
Expression given below is valid both in descent and climb condition
√(
)
In descent, velocity is negative so the result from the above expression keeps on increases.
The climb curve will reach technically.
Asymptotically the curve will reach the line at 45 degree. If we extrapolate to descent, the curve will be like as shown in graph.
If
Or
Then
√( )
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becomes imaginary value. We cannot have the root less than -2.
In descent
√(
) In this case both roots are valid, because
√( )
Is smaller than
When V is less than or
< 2
√( )
becomes imaginary value.
For each value of
,
we will have two roots one above the and one below it.The line is the ideal autorotation case here; for points the above ideal autorotation the rotor is absorbing power and for points below it is producing power for the helicopter.
The universal inflow curve crosses the ideal autorotation line at about ⁄ (the scatter extends over roughly ⁄ to -1.8. Real autorotation occurs at a high rate of descent, in the turbulent wake state. In the turbulent wake state the induced velocity curve can be approximated fairly well by a straight line on the ( ) ⁄ vs ( ⁄ ) plane. Joining the ideal autorotation intercept ( ⁄ ) and the windmill brake state boundary (( ) ⁄ ) gives
( )
(-2,-1) is the point where the root for the descent condition starts. The above equation represents the straight line equation.
If we substitute X as -1.6 to -1.8
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( )
We know that
( )
Autorotation is the state of rotor operation with no net power requirement. The power to produce the thrust and turn the rotor is supplied either by auxillary propulsion or by descent of the helicopter. A component of the aircraft forward velocity directed upward through the
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rotor disk supplies the power to the rotor, in the autorotative descent of the helicopter, the source of power is the decrease of the gravitational potential energy. More directly, the descent velocity upward through the disk supplies the power to the rotor. The net rotor power is zero for vertical descent in autorotation. The decrease in potential energy (TV) balances the induced (T ) and profile losses of the rotor. Neglecting the profile losses gives ideal
Typically the descent velocity is in the range of 15 to 20 m/s in autorotation.
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identified with descending flight.
(a) Normal thrusting state , ( ) ( ) ( )- (b) Vortex ring state , ( ) ( ) ( )- (c) Turbulent Wake state , ( ) ( )-
(d) Windmill brake state , ( ) ( ) ( )-
Assume that the induced velocity is constant and uniform over the entire disk. Velocity of the flow inside the slipstream is same in the direction everywhere (downward). The value of induced velocity vary depend on flight condition. This condition is defined as Normal Working state (Flow everywhere inside and outside the slipstream is same in the direction).