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Unformatted text preview: Sec$on 5 Aircra Performance Introduc$on Image Courtesy of US Navy D.Toohey 117 Drag Polar • For subsonic aircra, the total aircra drag can be wriGen as D = DParasite + Dinduced 2 CL CD = CD ,0 + πAe CL • A straight line tangent to the drag € polar indicates where the maximum L/D is • In general, the minimum CD may be at a nonzero CL value, but usually it is small. D.Toohey CD 118 Stability Axis Xb θ α Xs Vh V γ VV = RC Horizon Zb Zs D.Toohey 119 Forces L Xb θ α Xs Vh V γ T ϕT VV = RC Horizon D Zb Zs D.Toohey W 120 Equa$ons of Mo$on for Steady Flight • For level ﬂight: γ=0 • In steady, level ﬂight, some$mes we assume angles are small and make the approxima$on that L=W, T=D. • If thrust is known and a certain ﬂight path angle is desired, you can compute necessary AoA and velocity. D.Toohey 121 Steady Un
powered Flight L D γ X b α Xs V θ Horizon W Zb RC = V sin γ RD = − RC
Zs D.Toohey 122 € Steady Un
powered Flight Photo by DG Flugzeugbau • If un
powered, T=0 • For steady condi$ons, ﬂight path angle will be nega$ve. – Flight cannot be level (γ=0) unless drag = 0 – Minimizing glide slope is achieved by maximizing L/D Glide Angle D.Toohey 123 Rate of Descent • By equa$ng aero forces with the weight, we can ﬁnd the velocity for a steady glide. • The rate of descent is the nega$ve of rate of climb: (RD =
RC) • If we assume a shallow glide slope, the aircra speed and RD can be es$mated as: V= 2W ρSCL
D.Toohey 124 € Minimum Rate of Descent and Maximum Range • Maximizing gliding range is achieved by maximizing the li to drag ra$o. This minimizes the glide angle and produces the most shallow descent. • Turns out that: • To maximize $me alo, the aircra should be ﬂying at condi$ons that produce the minimum descent rate, which is to maximize: CL3 / CD2 • These two ra$os are very important, and not just for gliding aircra. We’ll use them frequently for powered ﬂight as well. D.Toohey 125 Glider Example • Aircra data for a hypothe$cal glider is given in the table below. • What velocity should the glider ﬂy at to maximize its glide distance? If its star$ng al$tude is 1000 ., how far can it travel before reaching the ground (assume no wind)? Flying at 95 fps maximizes the L/D, which leads to the most shallow glide slope. Star$ng at 1000 , the max distance would be L/D * 1000  = 24,390 . • What velocity should the glider ﬂy at to maximize its $me alo? Flying at 75 fps maximizes the CL3/2/CD , which minimizes the descent rate. V (fps) 65.00 70.00 75.00 80.00 85.00 90.00 95.00 100.00 105.00 CL 1.59 1.37 1.20 1.05 0.93 0.83 0.74 0.67 0.61 CD 0.09 0.07 0.05 0.05 0.04 0.03 0.03 0.03 0.03 CL/CD 18.53 20.26 21.73 22.89 23.71 24.21 24.39 24.31 24.00 CL3/2/CD 23.37 23.74 23.76 23.46 22.87 22.05 21.05 19.93 18.74 17.52 ϒ (deg) 3.09 2.83 2.63 2.50 2.41 2.37 2.35 2.36 2.39 2.44 RD (fps) 3.51 3.45 3.45 3.50 3.58 3.72 3.89 4.11 4.38 4.68 126 110.00 0.56 0.02 23.50 Aircra data generated with the following ﬂight data: W= 1200 lbs, S = 150 2 , CD=0.015+0.028*CL2, ρ=0.00238 slug/3 D.Toohey Steady Powered Flight L Xb θ α Xs Vh V γ T ϕT VV = RC Horizon D Zb Zs D.Toohey W 127 Thrust Required • For steady level ﬂight, T = D and L = W 2 L = W = q∞ SCL TR = D = q∞ SCD = q∞ S CD,0 + KCL ( )
K= 1 πAe • Using L = W, we can solve for the li coeﬃcient: € 2W CL = ρV 2 S €
2 CD = (CD ,0 + KCL ) Note: • The drag polar gives us the drag coeﬃcient: 1 q∞ = ρV 2 2 €
€ € • Plugging in the CL value into the drag polar, the thrust required to € maintain steady level ﬂight is: 12 2KW 2 2 TR = D = q∞ SCD = q∞ S(CD,0 + KCL ) = ρV SCD ,0 + 2 ρV 2 S
D.Toohey Parasite Drag Induced Drag 128 € Parasite Drag Total Drag Induced Drag 1 2KS ȹ W ȹ 2 T R = D = ρV SCD ,0 + ȹ 2 ȹ 2 ρV ȹ S Ⱥ 2 € Aircra data generated with the following ﬂight data: W= 3200 lbs, S = 300 2 , CD=0.02+0.05*CL2, ρ=0.00238 slug/3 D.Toohey 129 Thrust Required Cont. • Solving for the Velocity for a given required thrust: • We see that the equa$on for V and from the Drag vs. V plot that there are two veloci$es that correspond to a given required thrust, except for the minimum TR, where there is only one value. – This occurs when – Which means that… D.Toohey 130 Thrust Required Cont. • We also can ﬁnd minimum drag rela$onships by diﬀeren$a$ng TR with respect to the dynamic pressure. • The minimum drag condi$on occurs when parasite drag and induced drag are equal. D.Toohey 131 Thrust Available • • • • Thrust available represents the thrust capability of the aircra. Func$on of power plant size and type Also depends on the aircra velocity and al$tude For piston
propeller engines, thrust decreases at high speeds due to Mach eﬀects on the propeller $p For turbojet engines, thrust increases slightly with speed due to increased inlet performance and increased mass ﬂow rate with Mach Number. In this course, we oen assume that thrust from turbojets is constant with velocity. Other engine types like turboprops and turbofans have thrust varia$ons somewhere in between turbojets and piston
propeller engines. 132 • • D.Toohey Thrust Available Cont. • • • For a given aircra, the range of possible steady ﬂight veloci$es depends upon the rela$ve values of thrust required, TR, and thrust available, TA. When TA>=TR, the diﬀerence between the two is the excess thrust. Excess thrust is the thrust capability beyond what is required to maintain steady level ﬂight, and it can be used for accelera$ng or climbing. Level, un
accelerated ﬂight is only possible when TA >= TR Thrust Available Thrust Excess Parasite Drag Thrust Required Vmin Vmax Induced Drag D.Toohey Velocity 133 Power Available and Power Required • • • • For propeller driven aircra, power available is roughly constant with speed. When dealing with propeller aircra, it is usually more useful to look at available vs. required power instead of thrust. Power required is the thrust required mul$plied by the velocity. When we solve for PR, we see that it is inversely propor$onal to CL3/2 / CD ( PR is minimized when the ra$o CL3/2 / CD is maximized). W W PR= V= L /D CL / CD 2 ȹ W ȹ ȹ ȹ ρCL ȹ S Ⱥ ȹ W ȹ 3 2S C 1 PR = ȹ ∝ € 3 2 3 ȹ ρCL ȹ S Ⱥ CL CD
2 2 D
D.Toohey 134 Power Available and Power Required When CL3/2 / CD is maximized, the power required is at its minimum D.Toohey 135 Power Required Cont. • Turns out that for airplanes with parabolic drag polars, D.Toohey 136 References 1. 2. 3. C.E.Lan, J. Roskam, Airplane Aerodynamics and Performance, Design Analysis Research Corpora$on, 1997 McCormick, B.W., Aerodynamics, Aeronau4cs and Flight Mechanics, 2nd edi$on, Wiley & Sons, 1995 E. Field, MAE 154S, Mechanical and Aerospace Engineering Department, UCLA, 2001 D.Toohey 137 ...
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This note was uploaded on 02/02/2011 for the course MAE 154s taught by Professor Tooney during the Spring '09 term at UCLA.
 Spring '09
 Tooney

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