shock instability in gases characterizedby inelastic collisions

7/30/2019 Shock Instability in Gases Characterizedby Inelastic Collisions

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Shock Instability in Gases Characterized

by Inelastic Collisions

Nick Sirmas

Supervisor: Matei Radulescu

December 7, 2012


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Instability in Strong Shock Waves

Stability of shock waves important to understand in combustion andaerospace industry Combustion: Non-uniform state behind shock may result in hot-spots and autoignition

of fuel

Hypersonic vehicles need to account for fluctuation of properties

Nick Sirmas


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Nick Sirmas

Instability has been studied extensively in detonations(exothermic

reactions) and is understood

Experiments have also shown instability in strong shock waves

(hypersonic) undergoing thermal relaxation which is not well understood

These instabilities have been seen with endothermic reactions: Ionization

Dissociation

Vibrational relaxation

Instability has also been shown where there is no dissociation or

ionization, with the presence of heavy molecules(CO2, C3H8, CCL2F2)


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Nick Sirmas

Flow over cylinder in propane, stable M=5.5(left) and unstable M=10(right)

(Hornung and Lemieux, 1999)


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Structure of Strong Shockwaves in Molecular

Gases

Nick Sirmas


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Structure of Strong Shockwaves in Molecular

Gases

Nick Sirmas

Ex. Apollo re-entry capsule, M=36 Assuming ideal jump Ts250 To

Ideal jump, Ts60000K=10x suns surface

With thermal relaxation, Ts11000K

Dissociation:

O22O 2000K 4000K

N22N 4000K 9000K

Ionization:

NN++e- T>9000K

OO++e-


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Dissipative Collisions in Granular Media

Nick Sirmas

Endothermic collisions between molecules in gases similar to dissipativecollisions between particle in granular media

Granular particles collide inelastically, controlled by the coefficient of

restitution,


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Shocks in Granular Media

Rapid granular flows leading to shock waves:

Industrial processing and packaging

Explosives

Astrophysical phenomena

Nick Sirmas

Frost et al.(2011)Swinney & Rericha(2004)


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Shocks in Granular Media

Unique structure of shock waves in granular mediaKamenetsky et al.(2000)

Goldshtein, Shapiro & Gutfinger(1995)

Nick Sirmas


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Shock Instability in Molecular

and Granular Gases

Both molecular and granular gases show instability in shock

waves with the presence of dissipative collisions

Similar mechanism?

Nick Sirmas

Unstable flow over cylinder in propane

(Hornung and Lemieux, 1999)

Unstable jet formation in granular media

(Frost et al., 2011)


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Present Study

Investigate the structure of shock waves in dissipative granular

gases where kinetic energy preserved in equilibrium zone

Develop a simple model to see the effect energy dissipation has on the

stability of shock waves(2D)

See effect that a thermalized equilibrium zone has on structure ofshock wave

See if any instabilities we see can be predicted using general shock

theory for instability

Investigate other possible underlying mechanisms

Nick Sirmas


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Hard Particle Molecular Dynamics

Nick Sirmas

Colliding hard disks(2D) and

hard spheres(3D) serve as a

good approximation for

molecular models of gases,

liquids and granular media Kinetic theory well

established(Chapman &

Cowling)


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Molecular Dynamics Model

Nick Sirmas

Use a 2D, dissipative, hard disk system

Relaxing shock waves initiated by piston, travelling at velocity up


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Nick Sirmas

Event Driven Molecular Dynamics algorithm (Poschel and Schwager):

1. Compute time until next collision with particle/wall or particle/particle

2. Evolve particles and boundaries(piston) to calculated time

3. Calculate and implement post-collision velocities

4. Step 1


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Data collection at each defined time step Velocities and positions of hard disks recorded to calculate macroscopic

properties of temperature, pressure, density and velocity flow fields

Ensemble averaging between simulations

Nick Sirmas


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EDMD model for piston driven shockwaves

Sirmas et al.(2012) Shock Waves in

Dense Hard Disk Fluids Shock Waves

All collisions elastic

Shock jump conditions determined using hard

disk EOS

Nick Sirmas


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Model Description for Inelastic Collisions

Nick Sirmas

inelastic

elastic

max)()(

max)()(

uuu

uuu

NBNA

NBNA

1

'' )()()()(

TATANBNA

uuuu

)()()()( '' TATANBNA uuuu


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Results

Nick Sirmas

Evolution of the shock structure demonstrating instability. Simulation with

30,000 hard disks, and up=20, umax=10, =0.95


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Results: Structure of Shock

Nick Sirmas

Thermalized region in equilibrium

state leads to convective rolls seen

by the velocity streamlines for the

unstable flow


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Parameters for instability

Nick Sirmas

What controls the structure and instability:

What effect does umax, up, and have on relaxation length, lR?

What effect does umax, up, and have on the spacing between bumps,


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Results: Relaxation Length

Nick SirmasTemperature profiles for similar values ofup/umax=2, for different

=0.95

lr


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Results: Relaxation Length

Nick Sirmas

Computed relationship between up/umaxand relaxation length, lR, for different


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Results: Bump Spacing

Nick Sirmas

=0.50

Computed relationship between bump spacing and relaxation length

lR for up/umax=2.25

=0.70

=0.80

=0.90

=0.95


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Cause of Instability?

The Usual Suspects

Nick Sirmas

Current predictions for shock

instability associated with the

shock Hugoniot

The Hugoniot represents all

possible end states

Intersection of Rayleigh line

represents the post shock

state(B) from initial(A)

Slope of Rayleigh depends on

the strength of shock


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The Usual Suspects

Nick Sirmas

Unstable Hugoniot forms for:

1. Shocked media undergoing

phase transition

2. Shocked gases undergoing

dissociation

-Positive slope corresponding to DYakov

Kontorovich Instability


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Shock Hugoniot Analysis

Nick Sirmas


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Possible Mechanism:

Clustering Instability

Nick Sirmas

Cooling of granular gas shown to become unstable through clusteringinstability, (Goldhirsch and Zanetti, 1993)

Homogenous cooling of granular gas undergoing clustering.

60000 hard disks with =0.5


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Comparison with Clustering Instability

Nick Sirmas

Shock theory does not predict instability, maybe clustering instability in charge

What is the time to clustering across shock?

How does it compare with clustering instability by Goldhirsch and Zanetti?

Time to clustering of shocked conditions in case of homogenous cooling(5% dev. Haffs Law)

tclustering


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Nick Sirmas

Relationship between temperature and time across the shockwave, normalized by local mean free time, for =0.9

()

=

+ Solve for R in:


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Nick Sirmas

Fitting exponential decay across the shock wave:

Exponential time constant for the decay of energy across theshock wave, for different and u

p/u

max. Plotted with range of

time to clustering for same


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Conclusion

The present study has identified the presence of shock instability in

dissipative gases (high density non-uniformities, convective rolls)

Characteristic spacing of clusters proportional to the relaxation length

scale (dependent on up/umaxand )

The instability is not predicted by other instability criteria, suggesting thatthe instability seen here is the clustering instability seen by Goldhirsch and

Zanetti, with similar time scales present

Nick Sirmas


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Future Work

Adjust simulations for hard-spheres(3D) to rule out the instability being an

artifact in only 2D

Adjust the molecular model to look and compare with realistic gases

undergoing thermal relaxation

Derive continuum solution

Ni k Si

shock instability in gases characterizedby inelastic collisions

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