ISIS18 Rouen 2008

An Analytical Solution of

Weak Mach Reflection (1.1 < Mi < 1.5)

by

John M. Dewey

University of Victoria, Canada

ISIS18 Rouen 2008

OBJECTIVEInput

Mi

Θw

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OBJECTIVEOutput

TPT

RS

MSSS

χ

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OBSERVATION 1Dewey & McMillin JFM 1985

U1

u1

Limit Mi < 1.5

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OBSERVATION 2Dewey & McMillin JFM 1985

K

Limit M1 > 1.1Θw > 9 °

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Classical Solution

u2

u3

P2

P3

χ

P2 = P3

u2 // u3

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Dewey & van Netten (1994) showed thatfor 1.05 < Mi < 1.6

there is a parabolic relationship betweenthe triple point trajectory angle and the wedge angle

viz,

χ + Θw = χg + Θw2

Θtr - χg

Θtr2

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Triple Point Trajectory Angle

TP Trajectory Angle

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

1 1.1 1.2 1.3 1.4 1.5

Inc. Shock Mach Number

TP

An

gle

/Wed

ge

An

gle

P Model

Experiment

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Base of Mach Stem Speed (Mg)Foot Mach Stem Speed

1.00

1.10

1.20

1.30

1.40

1.50

1.60

1.70

1.0 6.0 11.0 16.0 21.0 26.0 31.0 36.0

Wedge Angle

Mg

/Mi

P Model

Experiment

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Solution overlaid on Shadowgraph

Mi = 1.402 Θw = 24.6o

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Solution Displayed as SpreadsheetMi Theta w u1/a0 c1/c0 Ki g

1.415 10 0.590239 1.124455 28.37558

Theta det Ki + Th w Mr P2,P3(/P0) Mm

48.14095 29.22843 1.016739 2.254696 1.440643

u3/c0 u2/c0 delta Beta 2 Beta 3

0.622091 0.613061 1.913175 14.60125 14.95532

Mg Mk Th w - del Epsilon Mme

1.49324 5.289248 8.086825 6 1.464264

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It has not been possible to find a modelthat gives a realistic solution with

β2 = β3

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Rikanati et al, Phys. Rev. Let., 2006

Shock-Wave Mach-Reflection

Slip-Stream Instability:

A Secondary Small-Scale

Turbulent Mixing Phenomenon

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Rikanati et al, 2006

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Rikanati et al – Spread Angle

x

xhspread 2

arctan

a

vxf

vvSf

vvSxh HiMach

2121

21

,,21

,38.0

21

2121,

vv

vvvvS

12

1221

1

1

2

1,

f

2.12tanh15.0a

v

a

vfHiMach

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Θspread compared with Θparabolic

Mi Θw Θspread β3 – β2

1.105 10.0 0.05 0.19

1.108 24.7 0.23 1.35

1.135 33.4 0.37 2.49

1.135 14.4 0.11 0.47

1.183 34.0 0.45 2.46

1.186 22.0 0.24 1.27

1.190 35.0 0.47 2.41

1.240 10.0 0.07 0.29

1.288 32.7 0.54 1.77

1.288 17.3 0.19 0.92

1.290 35.0 0.59 1.08

1.402 24.6 0.41 1.52

1.415 10.0 0.08 0.35

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Compare P model, R model and Experiment

TP Trajectory Angle

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

1 1.1 1.2 1.3 1.4 1.5

Inc. Shock Mach #

TP

An

gle

/Th

w

P Model

R Model

Experiment

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Compare P model, R model and experiment

Base Mach Stem Speed

1.00

1.05

1.10

1.15

1.20

1.25

1.30

0.0 10.0 20.0 30.0 40.0

Wedge Angle

Mg

/Mi P Model

R Model

Experiment

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CONCLUSIONS

1. The objective of finding an analytical solutionfor 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

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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

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At glancing incidence, i.e. Θw = 0therefore, χg = A

Ben Dor (1991) gives

χ + Θw = A + B Θw2

i

ig M

auMaa 201

2011tan

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χ + Θw = χg + B Θw2

At transition from RR to MRχ = 0

so

Θtr = χg + B Θtr2

and

B = (Θtr - χg)/ Θtr2

Θtr = Θdet or Θsonic

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Model Expt. 1 Model Expt. 2 Model Expt. 3Mi 1.105 1.105 1.240 1.240 1.415 1.415

χ + Θw 21.95 21.95 27.00 27.00 28.98 28.98MR 1.007 1.006 1.009 1.013 1.012 1.012β2 3.31 2.94 8.48 7.50 14.39 13.46β3 3.45 2.84 8.68 7.40 14.65 13.45

u2/a0 0.176 0.179 0.375 0.372 0.609 0.615u3/a0 0.176 0.178 0.378 0.380 0.615 0.638P2/P0 1.269 1.273 1.643 1.685 2.211 2.304P3/P0 1.269 1.278 1.643 1.663 2.211 2.304Mm 1.111 1.112 1.252 1.254 1.465 1.453MG 1.147 1.144 1.301 1.293 1.491 1.487

Comparison with ExperimentsInitial Model

N.B. Model gives β2 = β3

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Mi Theta wModel Expt Model Expt Model Expt Model Expt

+/- 0.005 +/- 0.3 +/- 0.5 +/- .01 +/- 5.0 +/- 5.01.105 10.0 22.6 22.0 1.15 1.14 26 1781.108 24.7 28.5 27.3 1.25 1.24 33 33 170 1711.135 14.4 25.3 27.0 1.21 1.21 30 36 177 1741.135 33.4 34.9 36.8 1.38 1.39 41 48 161 1561.186 22.0 29.8 28.6 1.33 1.32 37 36 172 1691.183 34.0 36.2 35.4 1.46 1.44 44 45 160 1531.240 10.0 27.4 27.0 1.30 1.30 36 1781.288 17.3 29.9 29.6 1.42 1.39 41 31 175 1731.288 32.7 36.8 36.2 1.59 1.59 48 43 159 1501.402 24.6 33.5 32.9 1.63 1.62 48 49 168 1661.415 10.0 29.2 29.0 1.49 1.49 44 178

Ki+Thw Vg/a0 Omega ir Omega im

Preliminary Comparison with Experiments

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Ongoing Work

1. Use the sonic criterion instead of detachmentto 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

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The Velocity PlaneMi is the unit distance

Mi

ao

U1/ao

a1/ao

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Mach Numberof Reflected Shock

Mi

ao

U1/ao

a1/ao

U1/ao

MR = VR/a1 = VR/ao/(a1/ao)

VR/ao

χΘw

Assume χ is known

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P2 = P3 Mm

MiP1/Po

P2

P3

MR P2/P1 P2/Po = P3/Po MmMm

χ

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Normal to Mach Stem at Triple Point

Mi

Mm

VT

Comp Mi // TPT = Comp Mm // TPT = VT/ao

Gives direction of normal to Mm (δ)

χ

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Centre and Base of Mach Stem

K

G

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Complete Solution in terms of Mi & Θw only

Mi

Θw

χ

MR

Mm

MG

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Mach Number of any point on Mach Stem

Mi

Θw

χ

MR

K

ε

Mε