Aerothermodynamics of high speed flows AERO 0033 1 Lecture 4: Flow with discontinuities, oblique shocks Thierry Magin, Greg Dimitriadis, and Johan Boutet Thierry.Magin@vki.ac.be Aeronautics and Aerospace Department von Karman Institute for Fluid Dynamics Aerospace and Mechanical Engineering Department Faculty of Applied Sciences, University of Liège Room B52 +1/443 Wednesday 9am 12:00pm February May 2017 Magin (AERO 0033 1) Aerothermodynamics 2016-2017 1 / 27
Outline 1 Oblique shocks 2 Shock reflection and shock interaction 3 Prandtl-Meyer expansion Magin (AERO 0033 1) Aerothermodynamics 2016-2017 2 / 27
Oblique shocks Outline 1 Oblique shocks 2 Shock reflection and shock interaction 3 Prandtl-Meyer expansion Magin (AERO 0033 1) Aerothermodynamics 2016-2017 2 / 27
Oblique shocks Mach waves in a supersonic nozzle [Meyer, 1908] X-15 model fired into a wind Tunnel at M= 3.5 [NASA] Magin (AERO 0033 1) Aerothermodynamics 2016-2017 3 / 27
Oblique shocks Supersonic bullet passing under a perforated plate The bow shock wave on the bullet, in passing over the holes in the plate, sends out weak disturbances above the place which coalesce into a Mach wave above the plate [Stanford U] Magin (AERO 0033 1) Aerothermodynamics 2016-2017 4 / 27
Oblique shocks Supersonic bullet passing under a perforated plate The bow shock wave on the bullet, in passing over the holes in the plate, sends out weak disturbances above the place which coalesce into a Mach wave above the plate [Stanford U] Magin (AERO 0033 1) Aerothermodynamics 2016-2017 5 / 27
Oblique shocks Source of oblique waves A source constantly emits sound waves as it moves through a stationary gas at speed u Propagation of disturbances in subsonic and supersonic flow When u < a, the source always stays inside the family of circular sound waves and the waves continuously move ahead of the source When u > a, the source is constantly outside the family of circular sound waves, it is moving ahead of the wave front. The wave fronts form a disturbance envelope called Mach wave The Mach angle µ is determined as sin µ = 1 M Magin (AERO 0033 1) Aerothermodynamics 2016-2017 6 / 27
Oblique shocks Steady 2D discontinuity Rankine-Hugoniot jump relations for steady 2D discontinuity 2 u n2 = 1 u n1 2 un2 2 + p 2 = 1 un1 2 + p 1 2 u n2 u t2 = 1 u n1 u t1 2 u n2 H 2 = 1 u n1 H 1 u n1 = u n2 = 0: contact discontinuity, slip line ) p 2 = p 1 u n1, u n2 6= 0: oblique shock ) u t2 = u t1 and H 2 = H 1 2 u n2 u t2 = 1u n1 u t1 ) tan( ) = tan 2 / 1 Magin (AERO 0033 1) Aerothermodynamics 2016-2017 7 / 27
Oblique shocks Oblique shock relations The total enthalpy is H = h + 1 2 (u2 n + ut 2 )= 1 2 The jump relations 2 u n2 = 1 u n1 +1 1 a 2 2 u 2 n2 + p 2 = 1 u 2 n1 + p 1 h 2 + 1 2 u2 n2 = h 1 + 1 2 u2 n1 = H 1 2 u2 t are identical to the normal shock relations ) For a given shock angle and incoming Mach number M 1,allflow quantities behind the shock can be directly computed by the 1D relations, 2 = u n1 1 = ( +1)M2 n1 u n2 2+( 1)Mn1 2 T 2 T 1 =( p 2 p 1 )/( 2 1 )= Mn2 2 = 1+ 1 2 M 2 n1 Mn1 2 1 2 and p 2 p 1 =1+ 2 +1 (M2 n1 1) h 1+ 2 +1 (M2 n1 1)i apple 2+( 1)M 2 n1 ( +1)M 2 n1 based on normal components M n1 = M 1 sin and M n2 = M 2 sin( ) Magin (AERO 0033 1) Aerothermodynamics 2016-2017 8 / 27
Oblique shocks Shock polar The M 1 polar relation derived based on trigonometry: ) tan( ) = tan 2 / 1 tan tan 1+tan tan tan = apple tan = = tan, with = apple 1 1+ tan 2 tan M 2 1 " 2+( 1)M 2 1 sin 2 ( +1)M 2 1 sin2 M1 2 sin2 1 2 cot ( + cos 2 )+2 # Magin (AERO 0033 1) Aerothermodynamics 2016-2017 9 / 27
Oblique shocks Magin (AERO 0033 1) Aerothermodynamics 2016-2017 10 / 27
Oblique shocks Maximum deflection angle for oblique shocks For any given M 1, there is a maximum deflection angle max.ifthe physical geometry is such that > max, then no solution exists for a straight oblique shockwave. Instead the shock will be curved and detached. Magin (AERO 0033 1) Aerothermodynamics 2016-2017 11 / 27
Oblique shocks Strong shock and weak shock solutions For any given < max,there are 2 values of predicted Changes across the shock are more severe for the larger value of (strong shock) The weak shock is favoured and usually occur unless, for instance, the back pressure is increased by some independent mechanism In the weak shock solution, M 2 > 1exceptforasmallregionnear max If =0,then = /2 (normalshock)or = µ (Mach wave, evanescent shock) Magin (AERO 0033 1) Aerothermodynamics 2016-2017 12 / 27
Oblique shocks Detached shock in front of a blunt body The shape of the detached shock in front of a blunt body wave can be obtained by means of computational fluid dynamics simulations e Evanescent shock (zero deflection: sin =1/M 1,Machwave) d Weak shock, M 2 > 1 c Weak shock, M 2 =1 c Maximum flow deflection point dividing weak (M 2 > 1) and strong shock solutions b Strong shock a Normal shock Magin (AERO 0033 1) Aerothermodynamics 2016-2017 13 / 27
Oblique shocks Prandtl s relation for oblique shocks A normal Bernoulli constant a 2 = a 2 1 +1 u2 t is introduced to express conservation of the normal component of the total enthalpy h 2 + 1 2 u2 n2 = h 1 + 1 2 u2 n1 = 1 2 +1 1 a 2 Prandtl s relation for oblique shocks is a 2 = u n1 u n2 A critical normal Mach number can be defined as M n = u n /a,and Prandtl s relation for oblique shocks is expressed as M n1m n2 =1 Notice that M t1 6= M t2 since a 1 6= a 2 The critical tangential Mach number defined as M t = u t /a is constant: M t1 = M t2 since a = constant Magin (AERO 0033 1) Aerothermodynamics 2016-2017 14 / 27
Shock reflection and shock interaction Outline 1 Oblique shocks 2 Shock reflection and shock interaction 3 Prandtl-Meyer expansion Magin (AERO 0033 1) Aerothermodynamics 2016-2017 14 / 27
Shock reflection and shock interaction Shock polar in hodograph plane (exact oblique shock theory) The shock polar in the hodograph plane (u 2, v 2 )=(V 2 cos, V 2 sin ), with V 2 = u2 2 + v 2 2 1/2 is derived from Prandtl s relation: u n1 u n2 = a 2 1 +1 u2 t u n1 = u 1 sin v 2 u n2 = u n1 cos u t = u 1 cos = u 1 sin v 2 cos ) u 2 1 sin2 u 1 v 2 tan = a 2 1 +1 u2 1 cos2 Defining non-dimensional velocities ū 2 = u 2 /a, v 2 = v 2 /a,and ū 1 = u 1 /a, this relation is expressed as v 2 2 =(ū 2 ū 1 ) 2 ū 1 ū 2 1 1+ 2 +1 ū2 2 ū 1 ū 2 Magin (AERO 0033 1) Aerothermodynamics 2016-2017 15 / 27
Shock reflection and shock interaction Proof Considering the relation tan = u 1 u 2,obtainedbasedonthegreytriangle,and v 2 the trigonometric relations cos 2 1 = 1+tan 2 and sin 2 = tan2 1+tan 2,onegets u1 2 (u 1 u 2 ) 2 v2 2 +(u 1 u 2 ) 2 u 1 (u 1 u 2 )=a 2 1 u1 2v 2 2 +1 v2 2 +(u 1 u 2 ) 2 Dividing by a 2 ū 2 1 (ū 1 ū 2 2 ) ū 1(ū 1 ū 2 )[ v 2 2 +(ū 1 ū 2 ) 2 ]= v 2 2 +(ū 1 ū 2 ) 2 1 +1 ū2 1 v 2 2 After some algebra v 2 2 (1 + 2 +1 ū2 2 ū 1 ū 2 )=(ū 2 ū 1 ) 2 (ū 1 ū 2 1) No deflection ( = 0, v 2 = 0) u 2 = u 1 : evanescent shock (Mach line) ū 2 =1/ū 1 : normal shock (Prandtl relation) Deflection angle 0 < < max :2valuesforv 2 Strong shock solution (always supersonic flow) Weak shock solution (either subsonic or supersonic flow) Maximum deflection angle = max Deflection angle > max : no solution (detached shock) Magin (AERO 0033 1) Aerothermodynamics 2016-2017 16 / 27
Shock reflection and shock interaction For M 1 =2 =0 Point A : evanescent shock, Point F: normal shock 0 < = 10 < max, Point B: weak shock, Point E: strong shock = max, Point D: maximum deflection angle Point C: separation between supersonic and subsonic regimes (critical Mach number M behaves as the local Mach number M) Magin (AERO 0033 1) Aerothermodynamics 2016-2017 17 / 27
Shock reflection and shock interaction Shock polar in pressure-deflection plane Pressure-deflection diagram: locus of all possible static pressure behind an oblique shock wave as a function of the deflection angle It is a parametric curve depending on the free stream Mach number p M 1 : 2 p 1 =1+ 2 +1 (M2 1 sin2 1), where = (M 1, ) Points B and B : weak shock solutions depending on the sign of Points E and E : strong shock solutions Points G and G : expansion shock (not physical) Magin (AERO 0033 1) Aerothermodynamics 2016-2017 18 / 27
Shock reflection and shock interaction Shock reflection from a solid boundary Consider an oblique shock wave deviating the flow from region 1 to 2 through an angle incident on a solid wall at point I The streamline is deflected from region 2 to 3 through a reflected shock to leave the flow parallel to the wall M 1 =2and = 10 ) M 2 =1.641 and p 2/p 1 =1.707 M 3 =1.641 and =+10 ) M 2 =1.285 and p 3/p 1 =2.805 In the pressure-deflection plane M 2 < M 1:thereflectedshockis weaker than the oblique shock p 3 p 2 > p 2 p 1 The shock is not specularly reflected Magin (AERO 0033 1) Aerothermodynamics 2016-2017 19 / 27
Shock reflection and shock interaction Approximate solution based on the characteristic theory On s + characteristic between regions 1 and 2 Isentropic relation (M 2 )= (M 1 )+ =26.379 10 =16.379 ) M 2 =1.651 p 2 = 1+! 1 2 M2 1 1 p 1 1+ 1 =1.705 2 M2 2 On s characteristic between regions 2 and 3 (M 3 )= (M 2 ) =16.379 10 =6.379 ) M 3 =1.308 Isentropic relation p 3 = 1+! 1 2 M2 1 1 p 1 1+ 1 =2.795 2 M2 3 Magin (AERO 0033 1) Aerothermodynamics 2016-2017 20 / 27
Shock reflection and shock interaction Singular shock reflection from a solid boundary When the deflection is larger than the maximum deflection angle max for the M 2 value, no regular shock reflection is allowed M 1 =2and = 16 ) M 2 =1.403 and p 2/p 1 =2.308 M 2 =1.403 and =+16 ) no oblique shock solution Anormalshockisformedatthewalltoallowthestreamlinesto continue parallel to the wall (Mach stem) Away from the wall, this normal shock transits into a curved shock which intersects the incident shock Singular reflexion: the reflected shock is curved in reality Slip line betweenregions3and4 results from di erent entropy jump Intersection between two polars: p 3 = p 4 (< p 5) ) <16 Subsonic region: no analytical result Triple point T: incident & reflected shocks + Mach stem (+slip line) Magin (AERO 0033 1) Aerothermodynamics 2016-2017 21 / 27
Shock reflection and shock interaction Shock interaction Consider two oblique shocks of the same intensity deviating the flow from region 1 to 2 through an angle and interacting at point I The streamlines are deflected from region 2 to 3 through refracted shocks to leave the flow parallel to the symmetry plane M 1 =1.95 and = 10 ) M 2 =1.594 and p 2/p 1 =1.694 M 2 =1.594 and =+10 ) M 3 =1.233 and p 3/p 1 =2.783 In the pressure-deflection plane M 2 < M 1: the refracted shock is weaker than the oblique shock p 3 p 2 > p 2 p 1 No slip line due to the symmetry of the problem Magin (AERO 0033 1) Aerothermodynamics 2016-2017 22 / 27
Shock reflection and shock interaction Approximate solution based on the characteristic theory On s + characteristic between regions 1 and 2 (M 2 )= (M 1 )+ = 24.992 10 = 14.992 ) M 2 =1.604 On s characteristic between regions 2 and 3 (M 3 )= (M 2 ) = 14.992 10 =4.992 ) M 3 =1.256 Magin (AERO 0033 1) Aerothermodynamics 2016-2017 23 / 27
Shock reflection and shock interaction Singular shock interaction ( type II interference, Edney, 1968) When the deflection is larger than the maximum deflection angle max for the M 2 value, no regular shock refraction is allowed M 1 =1.95 and = 14 ) M 2 =1.440 and p 2/p 1 =2.069 M 2 =1.440 and =+14 ) no oblique shock solution Anormalshockisformedbetweenthetwoincidentsshockstoallow the streamlines to continue parallel to the symmetry plane (Mach disk for revolution flow) Away from the wall, this normal shock transits into a curved shock which intersects the incident shock Two triple points T 1 and T 2 Two slip lines 1 and 2 between regions 3 and 4 result from di erent entropy jump Intersection between two polars: p 3 = p 4 ) <14 Magin (AERO 0033 1) Aerothermodynamics 2016-2017 24 / 27
Prandtl-Meyer expansion Outline 1 Oblique shocks 2 Shock reflection and shock interaction 3 Prandtl-Meyer expansion Magin (AERO 0033 1) Aerothermodynamics 2016-2017 24 / 27
Prandtl-Meyer expansion When a supersonic flow is turned away from itself, an expansion wave is formed M 2 > M 1 an expansion corner is a means to increase the Mach number p 2 < p 1, 2 < 1, T 2 < T 1 The expansion fan is a continuous expansion region composed of an infinite number of Mach waves and bounded by two Mach waves The expansion is isentropic Magin (AERO 0033 1) Aerothermodynamics 2016-2017 25 / 27 Prandtl-Meyer expansion
Prandtl-Meyer expansion The Mach number of a Mach wave is given by (M) = (M 1 ) = (M 1 )+ where is the flow angle (see characteristics theory in lecture 2) The Prandtl-Meyer q function qis an increasing function of the Mach number: (M) = +1 1 arctan 1 +1 (M2 1) arctan p M 2 1 In particular, the flow in region 2 is given by (M 2 )= (M 1 )+ 2 Magin (AERO 0033 1) Aerothermodynamics 2016-2017 26 / 27
Prandtl-Meyer expansion The maximum turning angle is obtained when the end Mach number approaches infinity. In this case, Prandtl-Meyer function becomes (M!1)= 2 ( s +1 1 The maximum turning angle = (M!1) (M 1 ) When the deflection angle exceeds the maximum angle, the flow in this case behaves as if there is almost a maximum angle and in that region beyond the flow will become a vortex street 1) Magin (AERO 0033 1) Aerothermodynamics 2016-2017 27 / 27