MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS Overview of Compressible Flows: Critical Mach Number and Wing Sweep April 25, 2011 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

EXAMPLES: INFLUENCE OF COMPRESSIBILITY

WHEN IS FLOW COMPRESSIBLE?

WHEN IS FLOW COMPRESSIBLE?

EXAMPLES: COMPRESSIBLE INTERNAL FLOW

EXAMPLE: H2 VARIABLE SPECIFIC HEAT, CP

COMPRESSIBILITY SENSITIVITY WITH g

PRESSURE COEFFICIENT, CP Use non-dimensional description instead of plotting actual values of pressure Pressure distribution in aerodynamic literature often given as Cp So why do we care? Distribution of Cp leads to value of cl Easy to get pressure data in wind tunnel Shows effect of M∞ on cl

EXAMPLE: CP CALCULATION See §4.10

COMPRESSIBILITY CORRECTION: EFFECT OF M∞ ON CP Cp at a point on an airfoil of fixed shape and fixed angle of attack For M∞ < 0.3, r ~ const Cp = Cp,0 = 0.5 = const Flight Mach Number, M∞

COMPRESSIBILITY CORRECTION: EFFECT OF M∞ ON CP Effect of compressibility (M∞ > 0.3) is to increase absolute magnitude of Cp as M∞ increases Called: Prandtl-Glauert Rule For M∞ < 0.3, r ~ const Cp = Cp,0 = 0.5 = const M∞ Prandtl-Glauert rule applies for 0.3 < M∞ < 0.7

EXAMPLE: SUPERSONIC WAVE DRAG F-104 Starfighter

CRITICAL MACH NUMBER, MCR As air expands around top surface near leading edge, velocity and M will increase Local M > M∞ Flow over airfoil may have sonic regions even though freestream M∞ < 1 INCREASED DRAG!

CRITICAL FLOW AND SHOCK WAVES MCR Sharp increase in cd is combined effect of shock waves and flow separation Freestream Mach number at which cd begins to increase rapidly called Drag-Divergence Mach number

CRITICAL FLOW AND SHOCK WAVES ‘bubble’ of supersonic flow

CRITICAL FLOW AND SHOCK WAVES MCR

EXAMPLE: IMPACT ON AIRFOIL / WING DRAG Only at transonic and supersonic speeds Dwave= 0 for subsonic speeds below Mdrag-divergence Profile Drag Profile Drag coefficient relatively constant with M∞ at subsonic speeds

AIRFOIL THICKNESS SUMMARY Note: thickness is relative to chord in all cases Ex. NACA 0012 → 12 % Which creates most lift? Thicker airfoil Which has higher critical Mach number? Thinner airfoil Which is better? Application dependent!

CAN WE PREDICT MCR? A Pressure coefficient defined in terms of Mach number (instead of velocity) PROVE THIS FOR CONCEPT QUIZ In an isentropic flow total pressure, p0, is constant May be related to freestream pressure, p∞, and static pressure at A, pA

CAN WE PREDICT MCR? Combined result Relates local value of CP to local Mach number Can think of this as compressible flow version of Bernoulli’s equation Set MA = 1 (onset of supersonic flow) Relates CP,CR to MCR

HOW DO WE USE THIS? 1 3 2 Plot curve of CP,CR vs. M∞ Obtain incompressible value of CP at minimum pressure point on given airfoil Use any compressibility correction (such as P-G) and plot CP vs. M∞ Intersection of these two curves represents point corresponding to sonic flow at minimum pressure location on airfoil Value of M∞ at this intersection is MCR 1 3 2

IMPLICATIONS: AIRFOIL THICKNESS Note: thickness is relative to chord in all cases Ex. NACA 0012 → 12 % Thick airfoils have a lower critical Mach number than thin airfoils Desirable to have MCR as high as possible Implication for design → high speed wings usually design with thin airfoils Supercritical airfoil is somewhat thicker

THICKNESS-TO-CHORD RATIO TRENDS Root: NACA 6716 TIP: NACA 6713 Thickness to chord ratio, % F-15 Root: NACA 64A(.055)5.9 TIP: NACA 64A203 Flight Mach Number, M∞

ROOT TO TIP AIRFOIL THICKNESS TRENDS Boeing 737 Root Mid-Span Tip http://www.nasg.com/afdb/list-airfoil-e.phtml

SWEPT WINGS All modern high-speed aircraft have swept wings: WHY?

WHY WING SWEEP? V∞ V∞ Wing sees component of flow normal to leading edge

V∞ V∞,n V∞ V∞,n < V∞ WHY WING SWEEP? W W Wing sees component of flow normal to leading edge

SWEPT WINGS: SUBSONIC FLIGHT Recall MCR If M∞ > MCR large increase in drag Wing sees component of flow normal to leading edge Can increase M∞ By sweeping wings of subsonic aircraft, drag divergence is delayed to higher Mach numbers

SWEPT WINGS: SUBSONIC FLIGHT Alternate Explanation: Airfoil has same thickness but longer effective chord Effective airfoil section is thinner Making airfoil thinner increases critical Mach number Sweeping wing usually reduces lift for subsonic flight

SWEPT WINGS: SUPERSONIC FLIGHT If leading edge of swept wing is outside Mach cone, component of Mach number normal to leading edge is supersonic → Large Wave Drag If leading edge of swept wing is inside Mach cone, component of Mach number normal to leading edge is subsonic → Reduced Wave Drag For supersonic flight, swept wings reduce wave drag

WING SWEEP COMPARISON F-100D English Lightning

SWEPT WINGS: SUPERSONIC FLIGHT M∞ < 1 SU-27 q M∞ > 1 ~ 26º m(M=1.2) ~ 56º m(M=2.2) ~ 27º

WING SWEEP DISADVANTAGE At M ~ 0.6, severely reduced L/D Benefit of this design is at M > 1, to sweep wings inside Mach cone Wing sweep beneficial in that it increases drag-divergences Mach number Increasing wing sweep reduces the lift coefficient

TRANSONIC AREA RULE Drag created related to change in cross-sectional area of vehicle from nose to tail Shape itself is not as critical in creation of drag, but rate of change in shape Wave drag related to 2nd derivative of volume distribution of vehicle

EXAMPLE: YF-102A vs. F-102A

EXAMPLE: YF-102A vs. F-102A

CURRENT EXAMPLES No longer as relevant today – more powerful engines F-5 Fighter Partial upper deck on 747 tapers off cross-sectional area of fuselage, smoothing transition in total cross-sectional area as wing starts adding in Not as effective as true ‘waisting’ but does yield some benefit. Full double-decker does not glean this wave drag benefit (no different than any single-deck airliner with a truly constant cross-section through entire cabin area)

EXAMPLE OF SUPERSONIC AIRFOILS http://odin.prohosting.com/~evgenik1/wing.htm

SUPERSONIC AIRFOIL MODELS Supersonic airfoil modeled as a flat plate Combination of oblique shock waves and expansion fans acting at leading and trailing edges R’=(p3-p2)c L’=(p3-p2)c(cosa) D’=(p3-p2)c(sina) Supersonic airfoil modeled as double diamond Combination of oblique shock waves and expansion fans acting at leading and trailing edge, and at turning corner D’=(p2-p3)t

APPROXIMATE RELATIONS FOR LIFT AND DRAG COEFFICIENTS

http://www. hasdeu. bz. edu http://www.hasdeu.bz.edu.ro/softuri/fizica/mariana/Mecanica/Supersonic/home.htm CASE 1: a=0° Shock waves Expansion

CASE 1: a=0°

CASE 2: a=4° Aerodynamic Force Vector Note large L/D=5.57 at a=4°

CASE 3: a=8°

CASE 5: a=20° At around a=30°, a detached shock begins to form before bottom leading edge

CASE 6: a=30°

DESIGN OF ASYMMETRIC AIRFOILS

QUESTION 9.14 Compare with your solution Consider a diamond-wedge airfoil as shown in Figure 9.36, with half angle e=10° Airfoil is at an angle of attack a=15° in a Mach 3 flow. Calculate the lift and wave-drag coefficients for the airfoil. Compare with your solution

EXAMPLE: MEASUREMENT OF AIRSPEED Pitot tubes are used on aircraft as speedometers (point measurement) Subsonic M < 0.3 Subsonic M > 0.3 Supersonic M > 1 M < 0.3 and M > 0.3: Flows are qualitatively similar but quantitatively different M < 1 and M > 1: Flows are qualitatively and quantitatively different

MEASUREMENT OF AIRSPEED: INCOMPRESSIBLE FLOW (M < 0.3) May apply Bernoulli Equation with relatively small error since compressibility effects may be neglected To find velocity all that is needed is pressure sensed by Pitot tube (total or stagnation pressure) and static pressure Comment: What is value of r? If r is measured in actual air around airplane (difficult to do) V is called true airspeed, Vtrue Practically easier to use value at standard seal-level conditions, rs V is called equivalent airspeed, Ve Static pressure Dynamic pressure Total pressure Incompressible Flow

MEASUREMENT OF AIRSPEED: SUBSONIC COMRESSIBLE FLOW (0. 3 < M < 1 If M > 0.3, flow is compressible (density changes are important) Need to introduce energy equation and isentropic relations

MEASUREMENT OF AIRSPEED: SUBSONIC COMRESSIBLE FLOW (0. 3 < M < 1 How do we use these results to measure airspeed? p0 and p1 give flight Mach number Instrument called Mach meter M1 = V1/a1 V1 is actual flight speed Actual flight speed using pressure difference What are T1 and a1? Again use sea-level conditions Ts, as, ps (a1 = (gRT)½ = 340.3 m/s) V is called Calibrated Velocity, Vcal

MEASUREMENT OF AIRSPEED: SUPERSONIC FLOW (M > 1) Rayleigh Pitot Tube Formula

EXAMPLE: SUBSONIC AND SUPERSONIC FLIGHT Flight at four different speeds, pitot measures p0 = 1.05, 1.2, 3 and 10 atm What is flight speed if flying in 1 atm static pressure and Tambient = 288 K (a = 340 m/s)? Determine which measurements are in subsonic or supersonic flow p0/p = 1.893 is boundary between subsonic and sonic flows 1.05 atm → p0/p = 1.05 → subsonic Use compressible flow form, M = 0.265, V ~ 90 m/s ~ 200 MPH Could use Bernoulli which will provide small error (~ 1%) and give V directly Compressible form requires knowledge of speed of sound (temperature) Apply Bernoulli safely? p0/p < 1.065 1.2 atm → p0/p = 1.2 → subsonic M = 0.52, V ~ 177 m/s ~ 396 MPH Use of compressible subsonic form justified (Bernoulli ~ 3% error) 3 atm → p02/p1 = 3 → supersonic M1 = 1.39, V ~ 473 m/s ~ 1057 MPH (Bernoulli ~ 22% error) 10 atm → p02/p1 = 10 → supersonic M1 = 2.73, V ~ 928 m/s ~ 2076 MPH (MCO → LAX in 1 hour 30 minutes)