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An Analytical Solution of Weak Mach Reflection (1.1 < M i < 1.5) by John M. Dewey University of Victoria, Canada. OBJECTIVE. Input. Mi. Θ w. OBJECTIVE. Output. RS. MS. SS. TPT. χ. OBSERVATION 1 Dewey & McMillin JFM 1985. u 1. U 1. Limit M i < 1.5. OBSERVATION 2
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An Analytical Solution of Weak Mach Reflection (1.1 < Mi < 1.5) by John M. Dewey University of Victoria, Canada ISIS18 Rouen 2008
OBJECTIVE Input Mi Θw ISIS18 Rouen 2008
OBJECTIVE Output RS MS SS TPT χ ISIS18 Rouen 2008
OBSERVATION 1 Dewey & McMillin JFM 1985 u1 U1 Limit Mi < 1.5 ISIS18 Rouen 2008
OBSERVATION 2 Dewey & McMillin JFM 1985 K Limit M1 > 1.1 Θw > 9° ISIS18 Rouen 2008
Classical Solution P2 = P3 u2 // u3 P2 P3 u2 u3 χ ISIS18 Rouen 2008
Dewey & van Netten (1994) showed that for 1.05 < Mi < 1.6 there is a parabolic relationship between the triple point trajectory angle and the wedge angle viz, χ + Θw = χg + Θw2 Θtr - χg Θtr2 ISIS18 Rouen 2008
Triple Point Trajectory Angle ISIS18 Rouen 2008
Base of Mach Stem Speed (Mg) ISIS18 Rouen 2008
Solution overlaid on Shadowgraph Mi = 1.402 Θw = 24.6o 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 β2 = β3 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
Θspread compared with Θparabolic 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 • 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 Mi & Θw 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
χ + Θw = A + B Θw2 At glancing incidence, i.e. Θw = 0 therefore, χg = A Ben Dor (1991) gives ISIS18 Rouen 2008
χ + Θw = χg + B Θw2 At transition from RR to MR χ = 0 so Θtr = χg + B Θtr2 and B = (Θtr - χg)/ Θtr2 Θtr = Θdet or Θsonic ISIS18 Rouen 2008
Comparison with Experiments Initial Model N.B. Model gives β2 = β3 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 β2 = β3 ISIS18 Rouen 2008
The Velocity Plane Mi is the unit distance Mi U1/ao a1/ao ao ISIS18 Rouen 2008
Mach Number of Reflected Shock Assume χ is known Mi U1/ao a1/ao ao MR = VR/a1 = VR/ao/(a1/ao) VR/ao χ Θw U1/ao ISIS18 Rouen 2008
P2 = P3 Mm Mi P1/Po MR P2/P1 P2/Po = P3/Po Mm Mm P2 P3 χ ISIS18 Rouen 2008
Normal to Mach Stem at Triple Point Mi VT Comp Mi // TPT = Comp Mm // TPT = VT/ao Mm Gives direction of normal to Mm (δ) χ ISIS18 Rouen 2008
Centre and Base of Mach Stem K G ISIS18 Rouen 2008
Complete Solution in terms of Mi & Θw only Mi MR Mm χ MG Θw ISIS18 Rouen 2008
Mach Number of any point on Mach Stem Mi MR Mε K ε χ Θw ISIS18 Rouen 2008
