ISIS18 Rouen 2008 An Analytical Solution of Weak Mach Reflection (1.1 < M i < 1.5) by John M. Dewey University of Victoria, Canada

ISIS18 Rouen 2008 OBJECTIVE Input Mi ΘwΘw

ISIS18 Rouen 2008 OBJECTIVE Output TPT RS MS SS χ

ISIS18 Rouen 2008 OBSERVATION 1 Dewey & McMillin JFM 1985 U1U1 u1u1 Limit M i < 1.5

ISIS18 Rouen 2008 OBSERVATION 2 Dewey & McMillin JFM 1985 K Limit M 1 > 1.1 Θ w > 9 °

ISIS18 Rouen 2008 Classical Solution u2u2 u3u3 P2P2 P3P3 χ P 2 = P 3 u 2 // u 3

ISIS18 Rouen 2008 Dewey & van Netten (1994) showed that for 1.05 < M i < 1.6 there is a parabolic relationship between the triple point trajectory angle and the wedge angle viz, χ + Θ w = χ g + Θ w 2 Θ tr - χ g Θ tr 2

ISIS18 Rouen 2008 Triple Point Trajectory Angle

ISIS18 Rouen 2008 Base of Mach Stem Speed (Mg)

ISIS18 Rouen 2008 Solution overlaid on Shadowgraph M i = Θ w = 24.6 o

ISIS18 Rouen 2008 Solution Displayed as Spreadsheet

ISIS18 Rouen 2008 It has not been possible to find a model that gives a realistic solution with β 2 = β 3

ISIS18 Rouen 2008 Rikanati et al, Phys. Rev. Let., 2006 Shock-Wave Mach-Reflection Slip-Stream Instability: A Secondary Small-Scale Turbulent Mixing Phenomenon

ISIS18 Rouen 2008 Rikanati et al, 2006

ISIS18 Rouen 2008 Rikanati et al – Spread Angle

ISIS18 Rouen 2008 Θ spread compared with Θ parabolic MiMi ΘwΘw Θ spread β 3 – β

ISIS18 Rouen 2008 Compare P model, R model and Experiment

ISIS18 Rouen 2008 Compare P model, R model and experiment

ISIS18 Rouen 2008 CONCLUSIONS 1. The objective of finding an analytical solution for weak Mach reflections in terms of the shock Mach number and wedge angle only, has been achieved 2. The solution requires that the flows on the two sides of the slip stream be non-parallel 3.The spread angle calculated using the Rikanati et al (2006) analysis is, on average, approximately one fifth of that required to provide a solution that agrees with experiment

ISIS18 Rouen 2008

OBJECTIVE From an input of only M i & Θ w to provide a complete description of a weak Mach reflection i. e. positions and velocities of the reflected and Mach stem shocks; triple point trajectory angle, & slip stream angle

ISIS18 Rouen 2008 At glancing incidence, i.e. Θ w = 0 therefore, χ g = A Ben Dor (1991) gives χ + Θ w = A + B Θ w 2

ISIS18 Rouen 2008 χ + Θ w = χ g + B Θ w 2 At transition from RR to MR χ = 0 so Θ tr = χ g + B Θ tr 2 and B = (Θ tr - χ g )/ Θ tr 2 Θ tr = Θ det or Θ sonic

ISIS18 Rouen 2008 Comparison with Experiments Initial Model N.B. Model gives β 2 = β 3

ISIS18 Rouen 2008 Preliminary Comparison with Experiments

ISIS18 Rouen 2008 Ongoing Work 1. Use the sonic criterion instead of detachment to find the triple-point-trajectory angle 2. Make further comparisons with experimental and numerical simulation results 3. Continue to seek a solution in which β 2 = β 3

ISIS18 Rouen 2008

The Velocity Plane M i is the unit distance MiMi aoao U 1 /a o a 1 /a o

ISIS18 Rouen 2008 Mach Number of Reflected Shock MiMi aoao U 1 /a o a 1 /a o U 1 /a o M R = V R /a 1 = V R /a o /(a 1 /a o ) V R /a o χ ΘwΘw Assume χ is known

ISIS18 Rouen 2008 P 2 = P 3 M m MiMi P 1 /P o P2P2 P3P3 M R P 2 /P 1 P 2 /P o = P 3 /P o M m MmMm χ

ISIS18 Rouen 2008 Normal to Mach Stem at Triple Point MiMi MmMm VTVT Comp M i // TPT = Comp M m // TPT = V T /a o Gives direction of normal to M m (δ) χ

ISIS18 Rouen 2008 Centre and Base of Mach Stem K G

ISIS18 Rouen 2008 Complete Solution in terms of M i & Θ w only MiMi ΘwΘw χ MRMR MmMm MGMG

ISIS18 Rouen 2008 Mach Number of any point on Mach Stem MiMi ΘwΘw χ MRMR K ε MεMε