4 Galilean Invariance :The tangential component of the flow velocity is preserved.Superposition of uniform velocity does not change static variables.Continuity eq :Momentum eq :parallel to the shockThe tangential component of the flow velocity ispreserved across an oblique shock waveNormal to the shock
5 Energy eq :The changes across an oblique shock wave are governed by the normalcomponent of the free-stream velocity.
6 Same algebra as applied to the normal shock equction For a calorically perfect gasandSpecial casenormal shockNote：changes across a normal shock wave the functions of M1 onlychanges across an oblique shock wave the functions of M1 &
9 Note :1. For any given M1 ，there is a maximum deflection angleIfno solution exists for a straight oblique shock waveshock is curved & detached,2. Ifstrong shock solution (large )M2 is subsonicweak shock solution (small )M2 is supersonic except for a small region near, there are two values of β for a given M1
10 3.(weak shock solution)4. For a fixed→Finally, there is a M1 below which no solutions are possible→shock detached5. For a fixed M1andShock detachedEx 4.1
11 4.3 Supersonic Flow over Wedges and Cones Straight oblique shocks3-D flow, Ps P2.Streamlines are curved.3-D relieving effect.Weaker shock wave thana wedge of the same ,P2, , T2 are lowerIntegration (Taylor &Maccoll’s solution, ch 10)The flow streamlines behind the shock arestraight and parallel to the wedge surface.The pressure on the surface of the wedgeis constant = P2Ex 4.4 Ex 4.5 Ex4.6
12 4.4 Shock Polar –graphical explanations c.fPoint A in the hodograph plane represents the entire flowfieldof region 1 in the physical plane.
13 Shock polarIncreases to(stronger shock)Locus of all possible velocities behind the oblique shockNondimensionalize Vx and Vy by a*(Sec 3.4, a*1=a* adiabatic )Shock polar of all possible for a given
15 Important properties of the shock polar For a given deflection angle , there are 2 intersection points D&B(strong shock solution) (weak shock solution)tangent to the shock polarthe maximum lefleation angle for a givenFor no oblique shock solution3. Point E & A represent flow with no deflectionMach linenormal shock solutionShock wave angle5. The shock polars for different mach numbers.
16 ref：1. Ferri, Antonio, “Elements of Aerodynamics of Supersonic Flows” , 1949. 2. Shapiro, A.H., ”The Dynamics and Thermodynamics of CompressibleFluid Flow”, 1953.
17 4.5 Regular Reflection from a Solid Boundary (i.e. the reflected shock wave is not specularly reflected)Ex 4.7
18 4.6 Pressure – Deflection Diagrams -locus of all possible static pressure behind an oblique shock wave as afunction deflection angle for givenupstream conditions.Shock wave – a solid boundaryShock – shockShock – expansionShock – free boundariesExpansion – expansionWave interaction
23 4.7 Intersection of Shocks of Opposite Families C&D:refracted shocks(maybe expansion waves)Assumeshock A is strongerthan shock Ba streamline going throughthe shock system A&Cexperience or a differententropy change than astreamline going through theshock system B&D1.and have(the same direction.In general they differ in magnitude. )Dividing streamline EF(slip line)Ifcoupletely sysmuetricno slip line
24 Assume and are known & are known if solutionif Assume another
25 4.8 Intersection of Shocks of the same family Will Mach wave emanate from A & Cintersect the shock ?Point A supersonicintersectionPoint CSubsonic
26 (or expansion wave)A left running shock intersectsanother left running shock
27 4.9 Mach Reflection( for )( for )A straightoblique shockA regular reflection isnot possibleMuch reflectionFlow parallel to the upperwall & subsonicfor M2
28 4.10 Detached Shock Wave in Front of a Blunt Body From a to e , the curved shock goesthrough all possible oblique shockconditions for M1.CFD is needed
29 4.11 Three – Dimensional Shock Wave Immediately behind the shock at point AInside the shock layer , non – uniform variation.
30 4.12 Prandtl – Meyer Expansion Waves Expansion waves are theantithesis of shock wavesCentered expansion fanSome qualitative aspects :M2>M12.3. The expansion fan is a continuous expansion region. Composed of an infinitenumber of Mach waves.Forward Mach line :Rearward Mach line :4. Streamlines through an expansion wave are smooth curved lines.
31 i.e. The expansion is isentropic. ( Mach wave) Consider the infinitesimal changes across a very weak wave.(essentially a Mach wave)An infinitesimally small flow deflection.
32 …tangential component is preserved.as…governing differential equation for prandtl-Meyer flowgeneral relation holds for perfect, chemically reacting gasesreal gases.