Asymptotic analysis of direct initiation of gaseous detonations

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Submitted on 29 Apr 2020

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Asymptotic analysis of direct initiation of gaseous detonations

Paul Clavin

To cite this version:

Paul Clavin. Asymptotic analysis of direct initiation of gaseous detonations. In honor to Amable Liñan and Enrique Alarcon STAMS2019 meeting, Sep 2019, Madrid, Spain. �hal-02557916�

Paul Clavin

Institut de Recherche sur les Phénomènes Hors Équilibre

Aix-Marseille Université, CNRS & Centrale Marseille, France

Asymptotic analysis of direct initiation of gaseous detonations

In honor to Amable Liñan and Enrique Alarcon STAMS2019 meeting, Madrid 28-29 March 2019

Gaseous detonations are not easy to initiate

What minimum energy should be deposited by an external source

to initiate a gaseous detonation in an open space ?

Background in compressible fluids Planar shock waves and detonations

1860-1940

inert shock followed by an exothermal reaction zone

in gas at ordinary conditions: 1800 3400 m/s, 15 30 bar, 2500 3700 K

Deflagration = combustion subsonic waves

reaction-diffusion waves

Detonations = combustion supersonic waves

Old scientific topic (end of 19th - extensively studied since the mid 20th)

Comprehensible explanation of the multidimensional geometry and complex dynamics of the wave fronts has been elusive. Understanding is recent (nonlinear analyses)

Reactive Euler equations

thickness: few mean free paths

distance (D vp)t

FRESH MIXTURE!

!!!!!!! AT REST UNIFORM FLOW!

OF BURNED GAS

MOVING!

PISTON

D > a _u

SUPERSONIC!

FRONT

D > a _u

v _p

!!!!!!!!!OVERDRIVEN DETONATION!

!!!!!!!!!!!!!!!!!!!!REACTING GAS!

!

PISTON SUPORTED SUPERSONIC WAVE

FIRE

u _b = D v _p u _b < a _b

reaction zone lead shock

Abel 1870, Berthelot et Vielle 1881, Mallard et Le Chatelier 1881, Mikhel’son 1893, Chapman 1899, Jouguet 1904, Vielle 1900, Zel’dovich 1940, von Neumann1942, Döring 1943,

Zoom

Detonation Front

Piston velocity End of heat

release zone Shock

Shock

Detonation velocity

N

Neumann state

Reaction rate

Detonation thickness

Overdriven regime / Self sustained wave

Planar detonation

Sonic condition

U

= a

Michelson (1893) Rayleigh (1910)

0

Michelson- Rayleigh line Hugoniot after

heat release

Hugoniot of shock

CJ Michelson- Rayleigh line

Marginal solution

the so called Chapman-Jouguet wave

Mikhel’son

(1893)

supersonic without shock non physical

subsonic

under ordinary conditions

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Arrhenius law _e

reaction rate 1/t_r e ^E/kBT/t_coll

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Re = a²_ut_r/ t_r/t_coll 1 reactive Euler eqs.

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Selection mechanism of the self sustained propagation (CJ wave) by the rarefaction wave in the burnt gas

(planar geometry and inner structure of the detonation in steady state)

pistoninmotion pistonstopped

t= 0 t= 0

Do> DCJ

D^ot+x^o

t_i

x_o x

(u^bo+a^bo)t

0

D^CJ t

x u

t < t_i t > t_i

t

0

D^ot+x^o

x^f(t)

x^f(t) xf

x_{f i} (DCJ apo)t

ub_CJ =DCJ ab_CJ

ubCJ

ubo

ub

(u^bo+a^bo)t

(u^b+a^b)t

uo(x, t) u(x, t)

C_o^trail₊

C_o^lead₊

Fresh mixture at rest in the burnt gas

Burnt at restgas

C⁺ C⁺

a_pot a^po

t

xf

xf

D^CJ t+

x^f Rarefaction wave

space time

space flow velocity

D(t) D_CJ D_CJ

1 t² Levin & Chernyi (1967), Li˜nan, Kurdyumov & Sanchez (2012), P.C. Denet (2018)

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What minimum energy should be released by an external source

to initiate a gaseous detonation in open space (3D) ?

Korobeinikov (1971) Detonation = discontinuity

(zero detonation thickness) No critical energy !

Numerics in spherical geometry

D_CJ

reactive mixture: numerics D_CJ

reactive mixture: numerics Korobeinikov (1971)

Conclusion: the critical energy is due to modifications of the inner structure

blast wave

radius propagation velocity

D

r

D E

u

1/2 1 r^3/2

Taylor (1941) ^density

velocity

radius

Zeldovich-Taylor spherical CJ solution

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What minimum energy should be released by an external source

to initiate a gaseous detonation in open space (3D) ?

Zeldovich criterion (1956) critical energy chemical energy

unit volume (detonation thickness)

Wrong ! Under estimated by a factor 10

Experiments (1975)-(1980)

10

Turning point

sonic condition

sonic condition subsonic condition

subsonic condition

subsonic condition

Curvature effect (steady state approximation)

Marginal CJ condition in the parameter space «radius-velocity  »

L. He & P.C (1994)

sonic condition

D

D_CJ

no solution subsonic

Arrhenius factor

critical radius detonation thickness

16 e ( ² 1)

E

k_BT 10²

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Turning point

sonic condition

sonic condition subsonic condition

subsonic condition

Curvature effect (steady state approximation)

misfire

attractor initiat

ion

Arrhenius factor

critical radius

detonation thickness 10² critical energy

Zeldovich value 10⁶ E

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Role of unsteadiness of the inner structure ? Dynamics close to the critical radius ?

DNS show sensitivity to details

unsteadiness promotes failure trajectory 2

unsteadiness promotes re-ignition trajectory 3

L. He (1995-1996), Lee & Higgins (1999)

Eckert, Quirk & Shepherd (2000)

« the primary failure mechanism is found to be unsteadiness »

Problem under investigation not yet fully resolved:

THEORETICAL STUDY OF THE UNSTEADINESS OF THE INNER DETONATION STRUCTURE

DURING THE DIRECT INITIATION

GENERAL FORMULATION OF THE PROBLEM