Numerically constrained one-dimensional interaction of a propagating planar shock wave

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Numerically constrained one-dimensional
interaction of a propagating planar shock wave
A. Chatterjee
Department of Aerospace Engineering, Indian
Institute of Technology, Bombay Mumbai 400076,
INDIA
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1D problem numerically constrained interaction
of a propagating planar shock wave

• Rightward planar propagating shock wave
• uc(x, t) arbitrary imposed flowfield
  downstream of shock wave
• constrains development of
  flowfield (u(x,t)) behind
• propagating shock wave
• xsw current position of shock wave

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Algorithm
Proposed algorithm Unsteady 1D Euler equations
in x1 , x2
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Validation
Test Case Constrained Interaction of
Planar (rightward) Propagating Shock wave

• Unsteady interaction of Mach 3 shock wave with
  Sine entropy wave
• (Shu Osher)

Initial Conditions
Vl (rl,ul,pl)(3.8571143, 2.629369, 10.333333)
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Solution Methodologies
Without Constrain (regular solution)
in -5 5
With Constrain
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Numerical Validation
Shock/entropy wave interaction( time1.8)
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Numerical Validation
Shock/entropy wave interaction( time1.8)
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Application 2D Shock-vortex interaction problem
An initially planar shock wave interacts with a
2D compressible vortex superposed on ambient
resulting in creation of acoustic waves and
secondary shock structures.
( Compressible vortex model )
Experimental Condition (Dosanjh Weeks,
1965) Ms 1.29 Umax 177 m/s
(Mv0.52) r1 0.277 cm r2
1.75 cm
Strong interaction with secondary shock formation
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Application a possible constrained numerical
experiment

• Solving numerically a reduced model of complex
  unsteady shock wave
• phenomenon with appropriate constrains
• Demonstrate role of purely translational motion
  of an initially planar shock
• wave in secondary shock structure formation
  when interacting with 2D
• compressible vortex
• Planar shock wave interact with 1D flow field
  (uc)
• uc represents initial flowfield along vortex
  model normal to shock wave

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Application ..

• uc controls development of the flowfield
  behind shock wave (example of an
• arbitrary constraining flowfield)
• Ignores shock wave (and vortex) deformation

Computational Domain
0, 20
Initial position of shock 8. 25 cm Properties
behind the shock R-H condition No. of cells
900 equally spaced uc constraining flowfield
ahead of shock centered at 10.0 cm
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uc downstream of normal shock
Case 1 2 y ? 0.45
Case 3 4 y ? 1.25
Vortex center y0.0
Velocity distribution along horizontal lines
(cases 1 to 4)
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Results
Case 1 2 ? 1.25 (farther from vortex
center)
T1 Start of the simulation
T6 Shock wave almost out of domain
Pressure Profiles (Case 1)
generation of acoustic waves
Pressure Profiles (Case 2)
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Results
Case 3 4 ? 0.45 (near vortex center)
Pressure Profiles (Case 3)
generation of upstream moving shocklet
Pressure Profiles (Case 4)
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Conclusions

• An algorithm proposed for constrained one
  dimensional interaction of a planar propagating
  shock wave.
• Validated for 1D shock-entropy wave interaction.
• Constraining flowfield can be arbitrary.
• Allows setting up a constrained numerical
  experiment otherwise not possible.