NASA NACA-TN-2114-1950 Theoretical lift and damping in roll of thin wings with arbitrary sweep and taper at supersonic speeds supersonic leading and trailing edges《在超音速下 带有任意掠角和锥形薄.pdf

1、!ti r_i_ and ifWhen the Mach line is behind the leading edge_ that is,4kmA 2_ +i 2_m - I J4kmIf A (l + )(mFor tapered wings:CL_ =i): %.%B_m 2 - i 2A(k 2 - i)k_m 2 - 1 cos-1 -_/(_, - 1)(kin + 1)cos_ 1 4 2) - J2)(A4)(AS)+cos-1 l_+m IF4km - A(k - 1)-12j m + i I-4A (k - l) _lk(_ + l(A6)Provided by IHSNo

2、t for ResaleNo reproduction or networking permitted without license from IHS-,-,-18 NACA TN 2114For unswept leading edges or for unswept trailing edgesFor untapered wings k = k = I:= 4 m2(_m2 2) cos-i 1CL_BA_m 2 - i_ (m 2 - i) m- I2(m- i) +m ,2+k = _ or O:(A7)(A8)Fo_ulas for CZpIf the Mach line is c

3、oincident with the leading edge, that is,B cot A = I_ there result:For tapered wings:CZp =J3k3(1 - k) 3 + 2j2k3(1 - k)2(9k - 8_(1 - k) 32Jk3(l - k)(15k 2 - 32k + 12) + 12k4(k 2 + 4)_- +(1 k)34_3(i - k)(23k 2 + lOk + 2) + 4kj2(41k 2 - 5k - i)_35+, 3_I _i-k) + 4k4_K-12k2J(29k - i) + 240kSjI_ J(i + k)

4、+35 1 3J2(I + k)3Ji(continued on next page)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN 2114 198kS(k2+ 4)j3(l _ k2)3_ _ i3(1 - k2) 3 -_I i J(l - k) + 2_k - cs-i 2k -512 X( )3j2k(41k 3 + 63k 2 +llk ii) - Jk(13k 2 + 29k - 41 48k(k 1)2+_J2(3k2

5、 + 3k + 20) + 4Jk(k2 - 21k - 2) + 4k2k(23k + 3)_x64(k + 1)2 _(A9)For unswept leading edges or_for unswept trailing edges; k = _ or O:CZp - 3_B(I + k) - I05j2 Jj3 2 + 4 -3_BJ3(I + k)J64 _ 2 - _ + _ + k cos -I j + +2 pI%312 +4 $12 I_ (_o)Provided by IHSNot for ResaleNo reproduction or networking permi

6、tted without license from IHS-,-,-20 NACA TN 2114For untapered wings_ _ = k = i:16 8CZp - _BA 3 3-_ + (A + I)2(13A 2 - 22A + 13) cos-i A - i ,+768 A + i/37A 3 41A 2 123A 3_7_ (All)2-_ 1920+_7-Y_-and ifWhen the Mach line is behind the leading edge 3 that is, B cot A i,4kmA (i + X)(m - i):For tapered

7、wings: _ ,l_-128m4k3_k 2 - m2(3k2 + i)_c_p = NI3_j3(I+ _)(m,2 _ 1)3/2(I _ ke)3128k3m4(1+ k2 2m,2k2)3_J3(I + k)(l+k2)2(m2 - 1)(k2m 2 - 1)128kP_4-_4+_2k2(k2+ 3_ -I-l3_J3(i + X)(I - k2)3(m2k 2 - 1)3/2 cos km-Y -k)4_4k% 9 12k3m, 4 k2m,3(_17 k2km2(-Sk 3 - 48k 2 + 3k + 2) - m(lOk 3 + 45k 2 + 12k + 5) -k2m

8、3(-3k 2 + 14k + 5) + km2(-3k 3 - 8k2 + k - 6) - m(6k 3 + 7k2 +8k + 3) - (3k 2 + 2k + 3)_ + 32k2m2j2(l - k)3_12k3m 4 -9k2m3(3 + k) + 3km2(k 2 - 7k - 6) + 3m(2k 2 - 5k - 1) -3(_-_)_+ 2_6k3_,3j(_ _)2_2=,3(_ k)+ _,2(2 -k -k2)+k2m ,3(_9k2 _ lOk + 35) + km2(3k 3 _ 32k 2 _ 5k + 18) + m(6k 3 -37k 2 + 4k + 3

9、) + (3k 2 - 14k +3_96J3(i + k)k(1 - k)3(m 2 1)(km+ i)3k(_+ 1)(=-(AI4Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-24 NACA TN 2114For unswept leading edges or for unswept trailing edges; k = _ or O:-Z I3 + k2_k(Z _ k)2(Sk _ 3) + 6J(1 - k)(k2 _ 4k +c

10、b : B(z- _)3(z+ _)41+ _(-3_2+8_-6)j l .-_I (A_IujFor k = O:128k3m ,4 _ I + k2 - 2k2m 2= ik2-Z _CZP 3_BA;3(-T- k2)2 _j2 l i) i)m2(3k 2 +l) - 4k2 1(i - k2)(m 2 - l)3/2 cs-i m-+k4m2(k 2 + 3) - 4k2(1 - k2)(k2m 2 - l) 3/2(_6)For untapered wings; k = k = i:= -16 4(Pm4 - 4m 3 - 2m 2 + 9m) + 8A3m(m - 1)2(m

11、2 - i) -CZp _A,3B12A,2m,3(m , _ 1)2 + 4Am3(m - 1)2 _ ,192(m - 1)2(m 2 - 1)_m,2 _ 1m4(-3m 4 + lOre2 + 8) m4(-m6 + 4m4 - 8m2) cos -I _I_144(m 2 - 1)3 + 48(m 2 - 1)_ 2 - 1 m .j(AI7)+Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN 2114 25REFERENCE

12、Si. Lagerstrom, P. A., Wall, D., and Graham, M. E.: Formulas inThree-Dimensional Wing Theory. Rep. No. SM-II901; DouglasAircraft Co., Inc., July 8, 1946.2. Lagerstrom, P. A._ and Graham,Martha E.: SomeAerodynamic Formulasin Linearized Supersonic Theory for Dampingin Roll and Effect ofTwist for Trape

13、zoidal Wings. Rep. No. SM-13200,Douglas AircraftCo., Inc., March 12_ 1948.3. Jones, Arthur L., and Alksne, Alberta: The DampingDue to Roll ofTriangular; Trapezoidal, and Related Plan Forms in SupersonicFlow. NACATN 1548, 1948.4. Cohen; Doris: The Theoretical Lift of Flat Swept-BackWings atSupersonic

14、 Speeds. NACATN1555, 1948.5. Harmon, Sidney M.: Stability Derivatives at Supersonic Speedsof ThinRectangular Wings with Diagonals ahead of Tip MachLines. NACARep. 925, 1949.6. Malvestuto, Frank S., Jr., and Margolis, Kenneth: TheoreticalStability Derivatives of Thin SweptbackWings Tapered to a Point

15、with Sweptbackor Sweptforward Trailing Edges for a Limited Rangeof Supersonic Speeds. NACATN 1761_ 1949.7. Malvestuto, Frank S., Jr._ Margolis, Kenneth, and Ribner, Herbert S.:Theoretical Lift and Dampingin Roll of Thin SweptbackWings ofArbitrary Taper and Sweepat Supersonic Speeds. Subsonic Leading

16、Edges and Supersonic Trailing Edges. NACATN1860, 1949.8. Harmon_Sidney M.: Theoretical Relations between the StabilityDerivatives of a Wing in Direct and in Reverse Supersonic Flow.NACATN 1943, 1949.9. Brown, Clinton E.: The Reversibility Theoremfor Thin Airfoils inSubsonic and Supersonic Flow. NACA

17、TN 19443 1949.i0. Hayes, Wallace D.: Reversed Flow Theoremsin Supersonic Aerodynamics.Rep. No. AL-755, North American Aviation, Inc., Aug. 20, 1948.ii. Glauert, H.: A Non-Dimensional Form of the Stability Equations ofan Aeroplane. R. BA 4B cotA (See fig. 1.)7cot A _- i; IB cot ATE = (i + k)(l + B co

18、t A) dMach llne_Y2 _3,_ ./“4 t /cjs“ $/XFormula for _ contributed byv_(_ - _-) YaRegionseesketch)C B2m - i-i x - B2myy) cosB(mx - y) r_ + (rex+ y) cos-IYa) cos -I mXa + Ya(2Bm + i)mxa - Ya + 2-mya(x a + BYa)(Bm + i)f-Ya)-VCL mx - cos -I mXa + Ya(2Rr“ + i) COS- 1 -rexa + Bem2y a + h(B2m 2 - i_k.-7_y-

19、_ “ _m(m_-y_)cs-I_=_.B%2y_.+h(B2_+l) _ 1(mx,_+ 2h + Ya) + 2 -_-_(xa + By_)(,_,+ l)Bm(mXa + Ya + 2h) J+ Ya + 2h) cos -I mXa + ya(21_ - I) + 2hmXa + Ya + 2hProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-28 NACA TN 21147“ Ii cv xx1i-!o _-_ ,-4i%itu L_Jal+, , !1+n+u it_+ +: _+8%,+I+v c_+-%n+ii+ _8 _ vv+_“ +i+uiv+,-Ii“+i%v+c4IProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-