16.885J/ESD.35J Aircraft Systems Engineering Introduction to Aircraft Performance and Static Stability Prof. Earll Murman September 18, 2003 Today’s Topics • Specific fuel consumption and Breguet range equation • Transonic aerodynamic considerations • Aircraft Performance – Aircraft turning – Energy analysis – Operating envelope – Deep dive of other performance topics for jet transport aircraft in Lectures 6 and 7 • Aircraft longitudinal static stability Thrust Specific Fuel Consumption (TSFC) lb of fuel burned • Definition: TSFC (lb of thrust delivered)(hour) • Measure of jet engine effectiveness at converting fuel to useable thrust • Includes installation effects such as – bleed air for cabin, electric generator, etc.. – Inlet effects can be included (organizational dependent) • Typical numbers are in range of 0.3 to 0.9. Can be up to 1.5 • Terminology varies with time units used, and it is not all consistent. – TSFC uses hours – “c” is often used for TSFC lb of fuel burned – Another term used is c t (lb of thrust delivered)(sec) Breguet Range Equation • Change in aircraft weight = fuel burned dW ctTdt ct TSPC/3600 T thrust • Solve for dt and multiply by Vf to get ds VfdW VfW dW VfL dW ds Vfdt ctT ctT W ctD W • Set L/D, ct, Vf constant and integrate W 3600 L TO R Vf ln TSFC DW empty Insights from Breguet Range Equation W 3600 L TO R Vf ln TSFC DW empty 3600 represents propulsion effects. Lower TSFC is better TSFC V L represents aerodynamic effect. L/D is aerodynamic efficiency fD V La M L . a is constant above 36,000 ft. M L important fD f fD f fD W ln TO represents aircraft weight/structures effect on range Wempty Optimized L/D - Transport A/C “Sweet spot” is in transonic range. 20 Losses due to 10 shock Max (L/D) Concorde waves 1 2 3 Mach No. Ref: Shevell Transonic Effects on Airfoil Cd, Cl Cd Region I. II. III. M M 1.0 M cr drag 8 divergence M<1 I. M < M 8 cr M<1 V M>1 II. M < M < M 8 cr 8 drag divergence M>1 M<1 V III. M > M 8 Separated flow 8 drag divergence Strategies for Mitigating Transonic Effects • Wing sweep – Developed by Germans. Discovered after WWII by Boeing – Incorporated in B-52 • Area Ruling, aka “coke bottling” – Developed by Dick Whitcomb at NASA Langley in 1954 • Kucheman in Germany and Hayes at North American contributors – Incorporated in F-102 • Supercritical airfoils – Developed by Dick Whitcomb at NASA Langley in 1965 • Percey at RAE had some early contributions – Incorporated in modern military and commercial aircraft Basic Sweep Concept • Consider Mach Number normal to leading edge sin P=1/ Mf P Mf Mn=Mfcos/ P = Mach angle, the direction / disturbances travel in supersonic flow • For subsonic freestreams, Mn < Mf - Lower effective “freestream” Mach number delays onset of transonic drag rise. • For supersonic freestreams –Mn < 1, / > P - Subsonic leading edge –Mn > 1, / < P - Supersonic leading edge • Extensive analysis available, but this is gist of the concept Wing Sweep Considerations Mf > 1 • Subsonic leading edge – Can have rounded subsonic type wing section • Thicker section • Upper surface suction • More lift and less drag • Supersonic leading edge – Need supersonic type wing section • Thin section • Sharp leading edge Competing Needs • Subsonic Mach number – High Aspect Ratio for low induced drag • Supersonic Mach number – Want high sweep for subsonic leading edge • Possible solutions – Variable sweep wing - B-1 – Double delta - US SST – Blended - Concorde – Optimize for supersonic - B-58 Supercritical Airfoil Supercritical airfoil shape C p keeps upper surface velocity C p, cr from getting too large. Uses aft camber x/c to generate lift. V 8 Gives nose down pitching moment. Today’s Performance Topics • Turning analysis – Critical for high performance military a/c. Applicable to all. – Horizontal, pull-up, pull-down, pull-over, vertical – Universal M-Z turn rate chart , V-n diagram • Energy analysis – Critical for high performance military a/c. Applicable to all. – Specific energy, specific excess power – M-h diagram, min time to climb • Operating envelope • Back up charts for fighter aircraft –M-Z diagram - “Doghouse” chart – Maneuver limits and some example – Extensive notes from Lockheed available. Ask me. Horizontal Turn ht pa t h W = L cosI, I = bank angle Flig Level turn, no loss of altitude R 2 2 1/2 2 1/2 F = (L -W) =W(n -1) θ r z Where n { L/W = 1/ cosI is the load factor measured in “g’s”. 2 But Fr = (W/g)(V /R) So radius of turn is φ L 2 2 1/2 R = V /g(n -1) z φ Fr And turn rate Z = V/R is 2 1/2 R Z= g(n -1) / V W Want high load factor, low velocity Pull Up Pull Over Vertical θ W R R L L θ L R θ W W Fr = (L-W) =W(n-1) Fr = (L +W) =W(n+1) Fr = L =Wn 2 2 2 = (W/g)(V /R) = (W/g)(V /R) = (W/g)(V /R) 2 2 2 R = V /g(n-1) R = V /g(n+1) R = V /gn = = Z g(n-1)/ V Z g(n+1)/ V Z = gn/ V Vertical Plan Turn Rates 2.5 Let TÝ Z Vertical plane turn rate 2.0 Kθ = Horizontal plane turn rate Pull Over 2 1/2 KZ = (n+1)/(n -1) θ 1.5 Pullover fro m inverted attitude Vertical maneuver Vertical Maneuver 1.0 tude Turn rate ratio, K rate ratio, Turn Pull-up from level atti 2 1/2 KZ = n/(n -1) 0.5 Pull Up 2 1/2 0 KZ = (n-1)/(n -1) 1 2 3 4 5 6 7 8 9 Load factor, in 'g's For large n, KZ » 1 and for all maneuvers Z#gn/ V 2 Similarly for turn radius, for large n, R # V /gn. For large Z and small R, want large n and small V Z#gn/ V = gn/aM so Z ~ 1/ M at const h (altitude) & n 2 Using R # V /gn, Z#V/R = aM/R. So Z ~ M at const h & R For high Mach numbers, the turn radius gets large Rmin and Zmax 1/2 1/2 Using V = (2L/USCL) = (2nW/USCL) 2 R # V /gn becomes R = 2(W/S)/ gUCL W/S = wing loading, an important performance parameter 2 And using n = L/W = UV SCL/2W Z # gn/ V= g UV CL/2(W/S) For each airplane, W/S set by range, payload, Vmax. Then, for a given airplane Rmin = 2(W/S)/ gUCL,max Zmax = g UV CL,max /2(W/S) Higher CL,max gives superior turning performance. 2 But does n CL,max = UV CL,max/2(W/S) exceed structural limits? V-n diagram CL < CLmax 2n W V* max UfCL, max S Structural damage Load factor n Load Highest Positive limit load factor possible Z Stall area Lowest max q > q possible R 0 V V* 8 Negative limit load factor Stall area Structural damage CL < CLmax Each airplane has a V-n diagram. Source: Anderson Summary on Turning • Want large structural load factor n • Want large CL,MAX • Want small V • Shortest turn radius, maximum turn rate is “Corner Velocity” • Question, does the aircraft have the power to execute these maneuvers? Specific Energy and Excess Power Total aircraft energy = PE + KE 2 Etot = mgh + mV /2 Specific energy = (PE + KE)/W 2 He = h + V /2g “energy height” Excess Power = (T-D)V Specific excess power* = (TV-DV)/W = dHe/dt Ps = dh/dt + V/g dV/dt Ps may be used to change altitude, or accelerate, or both * Called specific power in Lockheed Martin notes. Excess Power Power Required 2 PR = DV = qS(CD,0 + C L/SARe)V Power PA Excess power 2 PR = qSCD,0V + qSVC L/SARe 3 2 2 = USCD,0V /2 + 2n W /USVSARe Parasite power Induced power required required V 8 Power Available Excess power PA = TV and Thrust is approximately depends upon constant with velocity, but varies velocity, altitude linearly with density. and load factor Altitude Effects on Excess Power PR = DV = (nW/L) DV = nWV CD/CL 2 From L= USV CL/2 = nW, get 1/2 V = (2nW/ USCL) Substitute in PR to get 3 3 2 3 1/2 PR = (2n W C D/ USC L) So can scale between sea level “0” and altitude “alt” assuming CD,CL const. 1/2 1/2 Valt = V0(U0/Ualt) , PR,alt = PR,0(U0/Ualt) Thrust scales with density, so PA,alt = PA,0(Ualt/U0) Summary of Power Characteristics • He = specific energy represents “state” of aircraft. Units are in feet. – Curves are universal • Ps = (T/W-D/W)V= specific excess power – Represents ability of aircraft to change energy state. – Curves depend upon aircraft (thrust and drag) – Maybe used to climb and/or accelerate – Function of altitude – Function of load factor • “Military pilots fly with Ps diagrams in the cockpit”, Anderson A/C Performance Summary Factor Commercial Military Fighter General Aviation Transport Transport Take-off Liebeck hobs = 35’ hobs = 50’ hobs = 50’ hobs = 50’ Landing Liebeck Vapp = 1.3 Vstall Vapp = 1.2 Vstall Vapp = 1.2 Vstall Vapp = 1.3 Vstall Climb Liebeck Level Flight Liebeck Range Breguet Range Radius of action*. Breguet for prop Uses refueling Endurance, E (hrs) = R (miles)/V(mph), where R = Breguet Range Loiter Turning, Emergency handling Major Emergency Maneuver performance handling factor Supersonic N/A N/A Important N/A Dash Service 100 fpm climb Ceiling Lectures 6 and 7 for commercial and military transport * Radius of action comprised of outbound leg, on target leg, and return.