Item 0949

OTHER: Aerodynamics - General - Disk Area [A]

Sketch of Various Disk Arraignments:

Outside Positions on Defining the 'Disk Area' of Helicopters with Twin Rotors:

The consensus of the aerodynamists appears to be that on tandem, side-by-side, interleaving and intermeshing helicopters, the total disk area is the sum of the two isolated rotors. There is no adjustment made for overlap until the consideration of power

There is disagreement as to what constitutes the disk area of a coaxial helicopter. Some consider it as the area of two separate disks, whereas others consider it as the area of a single disk with twice the disk loading.

There is little information on intermeshing helicopters.

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Helicopter Theory (1980)~ [Source ~ HT p.119]

Wayne Johnson considers the total disk area of all twin rotor helicopters to be the sum of the area of two isolated disks, irrespective of overlap. His only exception to this is coaxial, where he considers it as having just one-half the disk area of the isolated rotors and twice the disk loading.

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Stepniewski (1984) ~ [Source ~ RWA, book I, page 112+ & book II, page 184+]

Stepniewski considers the disk area of the twin rotor helicopter to be the sum of the area of two isolated disks, irrespective of overlap. His disk loading is also based on the above calculation for disk area. He then adjusts this by the overlap. His range of overlap in this book is from 0% (side-by-side configuration) to 40% (interleaving configuration). He does not go as far as the intermeshing configuration; i.e. >50%.

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Calvert Synchropter (1999)~ University of Maryland 1999 AHS Student Competition:

It considers the total disk area to be the sum of the area of two isolated rotors,

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Leishman (2000)~ [Source ~ PHA p.69+]

xx

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Stepniewski (2001)~ [Source ~ ABC Synchropter p.xx]

It appears that he considers the disk area the coaxial helicopter to be the sum of the two isolated disks, just like the above. He does note that some aerodynamists consider the coaxial disk area to be half that of two isolated disks, as Wayne Johnson does. His range of overlap is extended in this paper from 0% (side-by-side configuration) on through to 100% (coaxial configuration).

With his ABC synchropter he considers the disk area as the sum of the two separate disks.

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Leishman (2002)~ [Source ~ AHS 2002, Appendix]

Overlap Interference Factor

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Leishman (2006)

AERODYNAMIC OPTIMIZATION OF A COAXIAL PROPROTOR

My thoughts; (from a number of years ago)

1. Current methodology for determining the disk area of single rotors does not differentiate between a rotor with two blades and a rotor that is identical except that it has four blades. The distinction between the two rotors is made later when considering rotor solidity. In addition, some consider the area of a coaxial with two closely spaced 3-blade rotors as the area of the two disks, yet consider a comparably sized single 6-blades rotor as the area of the one disk. It appears that the same methodology should apply to all rotor configurations. For example; a 2-blade Interleaving configuration is part way between two side-by-side 2-blade rotors and an identical single 4-blade rotor. Therefore, its total area should be two times the area of one rotor, minus one times the area of the overlap.
2. The disk is considered as an Actuator Disk, when it's area is used for Momentum Theory calculations. Therefore, when working with dual rotor interactions, there appears to be little justification for developing and using complex algorithms, since the actuator disk considers thrust loading and velocity as being uniform over the disk. In other words, there is no reason to use overly complex calculations when working with simplistic input values?

The following is based on these two thoughts.

Disk Area Calculations:

Parameters:

• Both dual rotor diameters are equal.
• For use with Momentum Theory ~ Hover
• Area of single rotor disk. AONE = π * R2

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Single:

Total effective system area: ASYS = AONE

Side-by-Side:

Total effective system area: ASYS = AONE * 2

Interleaving:

The distance between the rotor hubs is the hypotenuse of the gap and the stagger = g2 + ds2

Note: The gap (g ) is zero for the interleaving configuration

Total effective system area: ASYS = (AONE - area of overlap segment) * 2

Tandem:

The distance between the rotor hubs is the hypotenuse of the gap and the stagger = g2 + ds2

Note: The area of the segment for the tandem configuration will be slightly less than that of a comparable Interleaving configuration with the same stager. Therefore, the total disk area of the tandem will be slightly greater. This is because the distance between the two rotor hubs is slightly greater. It accounts for the reduction of the streamtube of the upper rotor when it is level with the lower rotor's disk.

Total effective system area: ASYS = (AONE - area of overlap segment) * 2

Intermeshing:

The distance between the rotor hubs is the hypotenuse of the gap and the stagger = g2 + ds2

Note: The gap (g ) is zero for the intermeshing configuration. This is because the advantage of the actual gap will be offset by the reduced vertical component of the angled disk's thrust. [Vee / 2] [Λ / 2 ] ~ Note that this is a generalist assumption, and later the gap (g ) might be modified by a constant (κsyn) to give a more accurate value.

Total effective system area: ASYS = (AONE - area of overlap segment) * 2

Look at what Johnson's was thinking when he extended the coaxial κindh to 0.56 in [ABC Synchropter p.51]

Coaxial:

This is to be worked on.

Very Small Gap: I think that two coaxial disk with a very small gap should have the same disk area as one of its two rotors. This is because there probably is very little difference in thrust between two 3-blade coaxial disks in very close proximity and one 6-blade single disk, with similar specifications.

See Required Power Comparison for an example. This example shows an increase in power of 33% when going from two blades to 4 blades and this is close to the gap factor of 1.28.

This method should result in compatibility with the other configurations above and with Solidity Ratios and disk loading for all of them.

Very Large Gap: At a very large gap ( 1 or 2 times the rotor diameter) the upper rotor's streamtube diameter will be 50% of the area that it is at its actuator disk. Therefore, it will also be 50% of the area of the lower actuator disk that it is passing through. The total effective system area (ASYS) will be larger in this situation. As a ludicrous thought, if the gap was 2 miles then the two disks could be treated as two side-by-side disks. Perhaps some calculation of the gap could be used to set the total effective system area from between that of a single disk (AONE) to that of side-by-side disks (2 * AONE). In other words the (AONE) multiplier goes from 1 at zero gap to 2 at infinity gap.

Review Leishman's 2002 AHS report.

This will give a gap factor range of 1.41 to 1.28, with adjustment for improvement. Look at the middle of HT p.119 for final calculations. Note that Leishman and other thing that these values are now incorrect.

Total effective system area: ASYS = (2 * AONE) - Area of streamtube???

Revise the Access database functions and reference them here.

Also make the appropriate changes to the Helicopter form and coding.

Public Function disk_overlap_area(ds, D As Single) As Single 'AOV

'This is to calculate the area of overlap for tandem rotors.

'From Principles of Helicopter Aerodynamics, page 71

'There is no reason why it will not work for Side-by-side as well.

'This should produce the same result, or double the result, as FUNCTION: area_of_segment()

'ds is the Stagger (horizontal distance between the 2 masts)

'D is the diameter of each disk.

Dim mm As Single 'Twin rotor overlap fraction

Dim Theta As Single

Dim A As Single

Dim AOV As Single 'Disk overlap area.

On Error GoTo disk_overlap_area_Err

Theta = Cos(ds / D) 'theta = cos -1 (ds/D)##NOTE THAT THE NEGATIVE MUST STILL be put IN

mm = (2 / PI) * (Theta - ((ds / D) * Sin(Theta)))

A = PI * ((D / 2) ^ 2)

AOV = mm * A

disk_overlap_area = AOV

disk_overlap_area_Exit:

Exit Function

disk_overlap_area_Err:

MsgBox Err.Description

Resume disk_overlap_area_Exit

End Function

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Public Function area_of_segment(R, CtrToCtr As Single) As Single

'Segment - Part of a circular area bounded by an arc and its chord.

Dim AoS, AoT As Single

Dim Angle As Single 'In radians

On Error GoTo area_of_segment_Err

If (CtrToCtr > (R * 2)) Then 'If circles do not intersect

area_of_segment = 0

Else

Angle = Arccos((CtrToCtr / 2) / R) 'cos(Angle) = (CtrToCtr/2)/R

Angle = 2 * Angle

AoS = (Angle / 2) * R ^ 2 'Area of sector.

AoT = (Sin(Angle) / 2) * (R ^ 2) 'Area of triangle.

area_of_segment = AoS - AoT

End If

area_of_segment_Exit:

Exit Function

area_of_segment_Err:

MsgBox Err.Description

Resume area_of_segment_Exit

End Function

The following must be reviewed and possibly removed ~ at some point.

Function: total_disk_area(RotorConfig, R, Gap, Stagger) in General module.

Coaxial; ASYS = AONE Coaxial is considered same size as single rotor. This may be valid and required for subsequent coding, or perhaps the Gap factor K = 1.41 - ((1.41 - 1.28) * (gg / (2 * R))) should be included here; when using more advanced calculation then momentum theory..

Prouty uses this disk area and method for calculating Momentum Theory - Hover. [Source ~ RWP1 p.5] The above is used in the Access database coding, but I suspect that the results will be a slightly conservative for multiple rotors since;

• it does not consider the elimination of the tail rotor.
• the contraction of the stream tube of the upper rotor at the lower rotor.
• the opposed swirl.

Offsetting some of the above is the decreased coefficient of lift in the overlapping area.

Disk Area - Effective: [AE]

The effective disk area of the helicopter.

Database:

Public Function effective_disk_area(RotorConfig As Integer, R As Single, gg, ds As Single) As Single

Public Function twin_rotor_factor(Config, gg, ds, R, B As Single) As Single

Configurations:

Single Rotor: AE = AS

Side by side: AE = AS * 2

Tandem: AE =

Coaxial: AE = (AS * 2) / kOV

Intermeshing: AE = (AS * 2) / kOV

Overlap Factor for Twin Rotor: [kOV]

kOV = twin_rotor_factor

The following is the twin rotor factor for intermeshing only

SR = ds / R

kOV = 1.46 - 0.253 * SR

Note:

See Stepniewski's Rotor-Wing Aerodynamics, and Leishman's loose pages on Dual Rotor Momentum Theory, and other material in white 3-ring binder.

For a tandem helicopter, the Disk Loading is based on the sum of the two Disk Areas. I.e. there is no consideration for overlap (gap and stagger). [Source ~ RWA Book 1, p.195]