4Galilean 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
5Energy eq :The changes across an oblique shock wave are governed by the normalcomponent of the free-stream velocity.
6Same 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 &
9Note :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
103.(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
114.3 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
124.4 Shock Polar –graphical explanations c.fPoint A in the hodograph plane represents the entire flowfieldof region 1 in the physical plane.
13Shock 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
15Important 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.
16ref：1. Ferri, Antonio, “Elements of Aerodynamics of Supersonic Flows” , 1949. 2. Shapiro, A.H., ”The Dynamics and Thermodynamics of CompressibleFluid Flow”, 1953.
174.5 Regular Reflection from a Solid Boundary (i.e. the reflected shock wave is not specularly reflected)Ex 4.7
184.6 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
234.7 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
24Assume and are known & are known if solutionif Assume another
254.8 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
274.9 Mach Reflection( for )( for )A straightoblique shockA regular reflection isnot possibleMuch reflectionFlow parallel to the upperwall & subsonicfor M2
284.10 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
294.11 Three – Dimensional Shock Wave Immediately behind the shock at point AInside the shock layer , non – uniform variation.
304.12 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.
31i.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.