High Speed Flow
High Speed Flow AE 6020
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This 0 page Class Notes was uploaded by Demond Hoppe on Monday November 2, 2015. The Class Notes belongs to AE 6020 at Georgia Institute of Technology - Main Campus taught by Staff in Fall. Since its upload, it has received 8 views. For similar materials see /class/234310/ae-6020-georgia-institute-of-technology-main-campus in Aerospace Engineering at Georgia Institute of Technology - Main Campus.
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Date Created: 11/02/15
Hypersonic flow introduction Van Dyke Hypersonic ow is flow past a body at high Mach number where nonlinearity is an essential feature of the flow Also understood for thin bodies that if 17 is the thicknesstochord ratio of the body M r is of order 1 Special Features Thin shock layer shock is very close to the body The thin region between the shock and the body is called the Shock Layer Entropy Layer Shock curvature implies that shock strength is different for different streamlines stagnation pressure and velocity gradients rotational flow The Hypersonic Tunnelquot For Airhreathing Propulsion 0 Mlmmum Snead mo ma pi 0mm Pressure Best Sustained cram 120 lt x psi Sank Boom a Allllude 0 H L Maxlmum Spam 750 ps39 Dynnmrc Pleasure 4 Mach Numhey 1 Altitude VelocityAltitude Map For ReEntry Typical reentry case Very little deceleration until Vehicle reaches denser air Deliberater so to avoid large fluctuations in aerodynamic loads and landing point Velocity Atmosphere Troposphere 0 lt z lt 10km Stratosphere 10 lt z lt 50km Mesosphere 50 lt z lt 80km Thermosphere z gt 80km lonosphere 65 lt 365 km Contains ions and free electrons 60 ltz lt 85 km NO 85 ltz lt 140 km NO 02 140 ltz lt 200 km NO 02 0 Zgt 200 km N O A Simple Model for Variation of density with altitude dp pgdz pR A M Neglect dissociation and ionization Molecular weight is constant Assume isothermal T constant poor assumption g quz p RT g e lo A 2 p po 99 RT Nonlifting body moving at velocity V which is inclined at angle 6 to the xaxis D 2 md7 DCost9 dt 8 L22 DSinB mg U dt2 mdizz 1 UZC SsinO m dtz 2P D g m 39 39 39 CBS IS the BaIIIstIc Parameter Assuming that the drag force is gtgt weight and that 6 is constant because gravitational force is too weak to change the flight path much U 1 poCDS gMZ L g ue 2msinOeXp RT High Angle of Attack Hypersonic Aerodynamics mmgalleryof uidmechanicscomlshockslsmnhlm u I mm mm Vlmlnunewllum 7 z m Waxmmm 7 mm Nnmmllb nm Mu mm 417 Ly httpMMMscienli ccagecom magesphotoshpersonic owjpg Crocco s Theorem I V S Vho a X 1 5 Implies vorticity in the shock layer Viscous Layer Thick boundary layer merges with shock wave to produce a merged shockviscous layer Coupled analysis needed High Temperature Effects Very large range of properties temperature density pressure in the flowfield so that speci c heats and mean molecular weight may not be constant Low Density Flow Most hypersonic ight except of hypervelocity projectiles occurs at very high altitudes A L Above 120 km continuum assumption is poor Below 60 km mean free path is less than 1mm Knudsen N0 ratio of Mean Free Path to characteristic length A a an 7Jpg Summary of Theoretical Approaches Newtonian Flow Flow hits surface layer and abruptly turns parallel to surface Normal force decomposed into lift and drag Modified Newtonian Flow Account for stagnation pressure drop across shock Local Surface Inclination Method Cp at a point is calculated from static pressure behind an oblique shock caused by local surface slope at freestream Mach number Tangent Cone approach similar to local surface slope arguments Mach number independence Shockexpansion relations and Cp become independent of Mach number at very high Mach number Blast wave theory Energy of Disturbance caused by hypersonic vehicle is like a detonation wave Hypersonic similarity Allows developing equivalent shock tube experiments for hypersonic aerodynamics Local Surface Inclination Methods Approximate methods over arbitrary configurations in particular where Cp is a function of local surface slope Newtonian Aerodynamics Newton 1687 concept was that particles travel along straight lines without Interaction with other particles let pellets from a shotgun On striking a surface they would lose all momentum perpendicular to the surface but retain all tangential momentum ie slide off the surface Net rate of change of momentum pwUOOZSinZHAA Cp 28in20 gt In 3D flows we replace UmSinB with Uoo 39 I7 luw lz W Shadow region Cp 0 Cp2 Shadow region is where 000 f gt 0 Remarks on Newtonian Theory Poor in low speed ow Predicts C cc 02 1 Works well as Mach number gets large and specific heat ratio y tends towards 10 Why Because shock is close to surface and velocity across the shock is very large most of the normal momentum is lost 2 Tends to overpredict GP and cd CD see figure 311 3 Works better in 3D than in 2D 4 In 3D works best for blunt bodies not good for wedges cones wings etc Modified Newtonian Was proposed by Lester Lees in 1955 as a wa of improving Newtonian theory and bringing in Mach Number and p dependence on He proposed replacing 2 with Cpm oo ax 2 0 Clamx sm 9 Here CPmax is the CP coefficient behind a Normal shock wave at the stagnation point That is p02poo CPmax 1 U 2 2pm 00 From RankineHugoniot relations L amp y12Moo2 Y1 1 y2yMoo poo 4me2 2y 1 y 1 317 Then 1 cp 390 Y M 2 oo 2 In the limit as Moo gt 00 We get 2 y 1 1 4 cp 7 1 4M4 r As 1 gt Cpmax E 2 Proposed by Newton AS 7 e1 chax Exercise Compute oIo values for configurations shown on Figures 38 36 311 and 312 using Newtonian and Modified Newtonian theories Bioonvex Airfoil yC 005 o2 xc2 Mach Number Independence Where does freestream Mach number appear in the above Only in the dependence of downstream pressure density temperature As freestream Mach number becomes large Q 1 L 131 r 1 L2ampp 2 szsinz 12 mem poo pooUm Y 1 YMOO 27Ysin2 y1 Why nondimensionalize by mem2 Because p2 0poouw2 And it allows cancellation of Mach number Examine other relations for properties downstream of the shock freestream Mach number does not appear anywhere The blast wave theory argues that the sudden addition of energy to the fluid by the body is equivalent to a high explosive of energy E being exploded at time tO A shock wave associated with the explosion spreads away from the origin with time X t U00 ln 2D problem the shock wave is a plane wave Shock wave moves outward with t Blast wave origin Hypersonic Shock amp Expansion Relations Why 1 Simpler than exact expressions for analysis 2 Key parameter is seen to be M6 where 6 is the flow turning angle for Mgtgt1 and 6ltlt1 Oblique Shock Relations 2 2 1 tan62cotf5 24 8111 f M1 cos 5 2 M 2sinzfa tan6 z2cotfa 1 gtgt M126 1 M1 1 small 3 M1 gtgt1 small 3 6 a 23 y 1 P 39 i ressureJump amp 1 277 M12 sin2 3 p1 y 1 M1 gtgt1 amp 277 M1zsin2 P1 l 1 amp1 2YM1zsin23 1 P1 Y1 1l2YM12132 1 y1 2 amp1YY1K2K2 y1 i P1 4 4 K2 Defining pressure coefficient m4 0 5 m p M12 2 4 amp P1 Y1 Y1 L 02 YKz 4 4 K2 Next u721gn1zsin2 1 v1 y 1M12 In the hypersonic limit u72125in2 V1 Y1 Also Q2 1zsin23 1 ot V1 Y 1W12 V72 sinZ V1 Y 1 Density Jump Across Shock amp y 1Il12 sin2 3 p1 y 1M12 sin2 13 2 In the hypersonic limit for large M1gtgt1 finite 3 ampi 1 p1 r1 Then Q P2 P2 2i 1W12 Sinz T1 P1 P1 y 12 amp1 P1 C a p M12 2 2 CPM M1gtgt1 y1 Hypersonic Shock Relations in the Limit of Large but Finite Mach number and small turning angle We define a similarity parameter K M10 which can be used to collapse a variety of data For large but finite M small 9 and 3 2 2 tan02cot sml231 M1 cos 3 2 becomes 9 4 16 M1202 LLH 1 J Works for finite values of M16 K Hypersonic Expansion Wave Relations From PrandtlMeyer theory 9 V2 V1 v L tanquot1 LNG24 tan391M2 1 71 r1 For M1 gtgt 1 M12 1 5 M12 Also tan 1x tan 11 From Taylor series tanquot11 1 71 71 XX 3X3 5X5 quot Va Y1 l Y1l V y1 2 2 y 1 7 1 2 Then 0 2 1 1 2 2r amp Mquot M Y 2 5 71 P1 1Y1M2 2 27 271 p1y1M10y 11y 1KY 1 P1 2 2 amp1 27 1 p1 2h1Y1KY1 2 2 6 sz yK Notethat C p ing 1 62 y Mach Number Independence Consider ow over a blunt body Where does freestream Mach number appear in the above Only in the dependence of downstream pressure density temperature As freestream Mach number becomes large Q 1 L 131 r 1 L2ampp 2 Moozsinz 12 mem poo pooUm Y 1 YMOO 27Ysin2 y1 Why nondimensionalize by mem2 Because p2 0poouw2 And it allows cancellation of Mach number Examine other relations for properties downstream of the shock freestream Mach number does not appear anywhere This Mach number independence is also observed in experiments Sphere drag coefficient for example