Method and system for inverse-tapered high-speed rotors
A method of designing a rotor blade for a rotary aircraft includes establishing a target thrust-weighted solidity of the rotor blade, calculating a reverse flow circle of the rotor blade at a target forward speed of the rotary aircraft, inboard of the reverse flow circle, decreasing chord of the rotor blade relative to a baseline rotor blade possessing the target thrust-weighted solidity, and outboard of the reverse flow circle, increasing the chord of the rotor blade relative to the baseline rotor blade. The rotor blade possesses the target thrust-weighted solidity of the baseline rotor blade and improved stall margin or speed margin at the target forward speed relative to the baseline rotor blade.
1. A method of manufacturing a rotor blade for a rotary aircraft, the method comprising:
establishing a target thrust-weighted solidity of the rotor blade;
calculating a reverse flow circle of the rotor blade at a target forward speed of the rotary aircraft; and
wherein, inboard of the reverse flow circle of the rotor blade, the chord of the rotor blade is less than a chord of a rectangular baseline rotor blade possessing the target thrust-weighted solidity;
wherein, outboard of the reverse flow circle, the chord of the rotor blade is greater than the chord of the rectangular baseline rotor blade;
wherein the rotor blade possesses the target thrust-weighted solidity of the baseline rotor blade and improved stall margin or speed margin at the target forward speed relative to the baseline rotor blade; and
manufacturing the rotor blade with the chord inboard of the reverse flow circle and the chord outboard of the reverse flow circle.
2. The method of claim 1 , comprising:
establishing a transition radius (“τ”) about which a transition from a decreased chord to an increased chord occurs;
wherein the transition radius is outside the reverse flow circle.
3. The method of claim 2 , wherein:
the transition radius is defined as r/R=τ;
r is an arbitrary radial position on the rotor blade from a rotor-blade root; and
R is the span of the rotor blade.
4. The method of claim 3 , wherein:
Δc1 is a linear measurement of decreased chord inboard of the transition radius;
Δc2 is a linear measurement of increased chord outboard of the transition radius; and
Δ
c
2
=
Δ
c
1
[
τ
2
1
-
τ
3
]
.
5. The method of claim 1 , wherein the steps are performed in the order listed.
6. The method of claim 2 , comprising:
modifying the rotor blade by:
increasing the decreased chord from a position inboard of the transition radius to the transition radius; and
decreasing the increased chord from a position outboard of the transition radius to the transition radius; and
verifying that thrust-weighted solidity of the modified rotor blade equals the target thrust-weighted solidity.
7. The method of claim 2 , wherein τ is greater than 0.8.
8. A rotor blade for a rotary aircraft, the rotor blade comprising:
first chord inboard of a reverse flow circle of the rotor blade, wherein the reverse flow circle of the rotor blade is calculated at a target forward speed of the rotary aircraft;
second chord outboard of the reverse flow circle;
wherein the rotor blade possesses a target thrust-weighted solidity of a rectangular baseline rotor blade;
wherein the rotor blade possesses improved stall margin at a target forward speed relative to the rectangular baseline rotor blade; and
wherein chord of the rectangular baseline rotor blade inboard of the reverse flow circle is greater than the first chord and chord of the rectangular baseline rotor blade outboard of the reverse flow circle is less than the second chord.
9. The rotor blade of claim 8 , comprising:
a transition radius (“τ”) about which a transition from the first chord to the second chord occurs;
wherein the transition radius is outside the reverse flow circle.
10. The rotor blade of claim 9 , wherein:
the transition radius is defined as r/R=τ;
r is an arbitrary radial position on the rotor blade from a rotor-blade root; and
R is the span of the rotor blade.
11. The rotor blade of claim 10 , wherein:
wherein the chord of the rectangular baseline rotor blade inboard of the transition radius is greater than the first chord and the chord of the rectangular baseline rotor blade outboard of the transition radius is less than the second chord;
Δc1 is a linear measurement of a difference between the chord of the rectangular baseline rotor blade inboard of the transition radius and the first chord;
Δc2 is a linear measurement of a difference between the chord of the rectangular baseline rotor blade outboard of the transition radius and the second chord; and
Δ
c
2
=
Δ
c
1
[
τ
2
1
-
τ
2
]
.
12. The rotor blade of claim 9 , comprising:
a first transition region in which chord increases from a position inboard of the transition radius to the transition radius; and
a second transition radius in which chord decreases from a position outboard of the transition radius to the transition radius; and
wherein thrust-weighted solidity of the rotor blade, including the first and second transition regions, equals the target thrust-weighted solidity.
13. The rotor blade of claim 9 , wherein τ is greater than 0.8.
14. The rotor blade of claim 8 , wherein the rotor blade is scaled about a pitch change axis of the rotor blade.
15. The rotor blade of claim 8 , wherein the rotor blade exhibits twist over a length of the rotor blade.
16. The rotor blade of claim 8 , wherein a leading edge of the rotor blade comprises a stainless-steel abrasion strip.
17. The rotor blade of claim 8 , comprising a uni-directional composite trailing edge adjacent an outboard end thereof.
18. The rotor blade of claim 8 , comprising an erosion zone formed via at least one of electroforming nickel, 3D spray coating on a formed substrate, additive manufacturing, and cold spray metallic technology.
19. The rotor blade of claim 10 , wherein:
the chord of the rectangular baseline rotor blade with a desired thrust-weighted solidity is modified to retain the baseline thrust-weighted solidity after the planform chord is modified according to the methodology:
Δ
c
2
=
Δ
c
1
[
τ
2
1
-
τ
3
]
;
inboard chord reductions Δc1 are traded for outboard chord increases Δc2; and
τ represents a radial station r/R; and
the modified planform transitions from a chord reduction Δc1 to a chord increase, Δc2.
20. The rotor blade of claim 19 , wherein:
the methodology
Δ
c
2
=
Δ
c
1
[
τ
2
1
-
τ
3
]
is used to achieve a target value of retreating blade stall margin,
C
L
2
C
L
1
,
according to the methodology:
L baseline, Ψ=270 =L modified, Ψ=270 which results in the relationship:
C
L
2
C
L
1
-
∫
μ
R
R
(
r
R
-
μ
)
2
·
c
baseline
(
r
)
·
dr
∫
μ
R
R
(
r
R
-
μ
)
2
·
c
modified
(
r
)
·
dr
stall margin, represented by
C
L
2
C
L
1
,
is achieved for the retreating blade at advance ratio μ.
21. The rotor blade of claim 20 , wherein:
the methodology
Δ
c
2
=
Δ
c
1
[
τ
2
1
-
τ
2
]
is used to achieve a target value of rotor thrust margin at high advance ratio according to the methodology:
L baseline, Ψ=270 =L modified, Ψ=270 which results in the relationship:
C
L
2
C
L
1
-
∫
μ
R
R
(
r
R
-
μ
)
2
·
c
baseline
(
r
)
·
dr
∫
μ
R
R
(
r
R
-
μ
)
2
·
c
modified
(
r
)
·
dr
stall margin
C
L
2
C
L
1
of the retreating blade is inferred to provide overall rotor thrust margin of similar magnitude at advance ratio μ.
22. The rotor blade of claim 19 , wherein:
the methodology
Δ
c
2
=
Δ
c
1
[
τ
2
1
-
τ
2
]
is used to achieve a target value of rotor flight speed margin according to the methodology defined for the blade at ψ=270°, where:
L baseline, μ1 =L modified,μ2 , results in the relationship:
∫
R
μ
1
R
(
r
R
-
μ
1
)
2
·
c
baseline
(
r
)
·
dr
=
∫
R
μ
2
R
(
r
R
-
μ
2
)
2
·
c
modified
(
r
)
·
dr
flight speed margin is inferred from the difference in advance ratio (μ 2 −μ 1 ).