Advances in the nonlinear dynamics of gaseous detonations

HAL Id: hal-02557909

https://hal-amu.archives-ouvertes.fr/hal-02557909

Submitted on 29 Apr 2020

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Advances in the nonlinear dynamics of gaseous detonations

Paul Clavin

To cite this version:

Paul Clavin. Advances in the nonlinear dynamics of gaseous detonations. Autumn Sciences Days at University of Almeria in honor of Manuel Garcia VELARDE September 24-27 2019, Sep 2019, Almeria, Spain. �hal-02557909�

Paul Clavin

Institut de Recherche sur les Phénomènes Hors Équilibre

Aix-Marseille Université, CNRS & Centrale Marseille

, France

Advances in the nonlinear dynamics of

gaseous detonations

Autumn Sciences Days at University of Almeria in honor of Manuel Garcia VELARDE

September 24-27 2019

Understanding complex phenomena ?

A physicist approach: reduce the complexity ending with simplified equations

to be solved analytically and compare with experiments Direct numerical simulations ?

Examples: combustion phenomena

Hopefully 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 ?

Safety of nuclear power plant

Explosions…..supernovae….

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

Background in compressible fluids Planar shock waves and detonations

1860-1940

Back to school

thickness: few mean free paths

distance ( D v

)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

thickness few mms Arrhenius law _e

E

k

T 10 reaction rate e

coll

elastic collisions

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

Planar detonation

(inner structure in steady state)

sonic condition

D

D _CJ

no solution subsonic condition

in the burnt gas

self sustained supersonic wave supersonic overdriven wave

propagation velocity

q_m cpTu

1 D_CJ

au

2 + 1 2

q_m cpTu

>1

D > a

Zoom detonation!

front

shock detonation!

velocity piston!

velocity end of heat!

release zone N

detonation thickness

shock reaction!

rate Neumann!

state

U_p

U_N

U_b T_N

T_u

T_b T

l = U

⇥ t

(T

)

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reaction time

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t

⇡ t

⇥ e

t

E/kbT ⇡10

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flow velocity/shock

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U

⇡ a

sound speed

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⌫

⇡ a

⇥ t

Reynolds nb

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U

l

⌫ ⇡ a

⇥ t

⌫

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⇡ t

t

⇡ e

1

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Reactive Euler Equations

Negligible heat conduction and viscosity

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) ³ Wrong ! Under estimated by a factor 10 ⁶

Experiments (1975)-(1980)

10 ⁸

Back to our problem !

Korobeinikov (1971) Detonation = discontinuity

(zero detonation thickness) No critical energy !

Numerics in spherical geometry

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

reactive mixture: numerics D

reactive mixture: numerics Korobeinikov (1971) in ert

blast w av e

radius propagation velocity

D

r

D E

u

1/2

1 r

Taylor (1941)

velocity

radius

density Sedov 1946-Taylor 1950

Finite radius spheric = planar

Zeldovich 1942 -Taylor 1950

radius >> detonation thickness

Turning point

critical radius detonation thickness

8

1 e

10

Arrhenius factor

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

no solution subsonic

Turning point

sonic condition

sonic condition subsonic condition

subsonic condition

Curvature effect (steady state approximation)

mi sfire

attractor ini tiat

ion E

Arrhenius factor

critical radius

detonation thickness 10

critical energy

Zeldovich value 10

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

Experiments and DNS show a sensitivity to details

unsteadiness promotes failure trajectory 2

unsteadiness promotes re-ignition trajectory 3

The analysis is in progress

1D dynamics in planar geometry

PC & F.A. Williams (2002-2009)

1 D

Dt + u

x = 0, D

Dt u = 1 p x

D

Dt ln T ( 1)

ln p = q

c

T

˙w

t

, D

Dt Y = ˙w t

D

Dt t + u x

reaction rate heat released/mass

progress variable

Constitutive equations

1 D

Dt ln p ± 1 a

D

Dt u = q

c

T

˙w(Y, T ) t

D

Dt t + (u ± a) x

Hyperbolic equations: Eqs. of the characteristics

1-D dynamics of the inner structure of planar detonations

Nonlinear acoustic waves generated by an exothermal reaction

p/ = (c

c

)T

Entropy wave: ^{˙w(Y, T )}

energy

3 nonlinear modes: Entropy wave + 2 acoustic waves 1-D compressible, inviscid and reactive gas mixture

t + u

r Y = w(T, ˙ Y ) t

chemical kinetics entropy wave

+ energy eq. for T heat release per unit mass progress variable

t + u

r ln T ( 1)

ln p = q

c

T

˙

w(T, Y ) t

˙

w : non-dimensional reaction rate

acoustic waves 1

p t + (u ± a)

x p ± 1

a t + (u ± a)

x u = q

c

T

˙ w t

a ( 1)c

T

sound speed

heat release rate

too complicated system for a general analytical solution 4 nonlinear equations & 4 unknowns: p, T, u, Y

Newtonian approximation

small heat release: a cst.

(t) x

= 0

D induction heat release oscillations

oscillations induction

Reaction rate (Realtive units)

Relative distance

D = 3100 m/s, TN = 2027 K D = 3000 m/s, TN = 1925 K D = 2900 m/s, TN = 1825 K

feedback loop

shock wave heat release direct

influence

entropy wave acoustic waves entropy wave

downstream running acoustic wave

upstream running acoustic wave

1-D dynamics of the inner structure of planar detonations Set of feedback loops

Threshold of the planar instability of the CJ regime:

Small heat release Transonic flow

Two-time scales: The upstream running acoustic wave controls the loops 3rd simplification:

The two other modes involves much shorter time lags

quasi-instantaneous

distribution of the rate of heat release for different velocity of the leading shock mean flow

me

Integral equation for the propagation velocity ^˙a

(t)

˙a

: distribution of the rate of heat release for (constant)

G(z, ˙a

)

(t) x

= 0

D induction heat release oscillations

oscillations induction

Reaction rate (Realtive units)

Relative distance

D = 3100 m/s, TN = 2027 K D = 3000 m/s, TN = 1925 K D = 2900 m/s, TN = 1825 K

distributions in steady state for different ˙at

2˙a

(t) =

G(z , ˙a

(t + z ))dz

Unperturbed flow velocity of burnt gas (intrinsic dynamics):

velocity of the lead shock time lag / upstream running acoustic wave

2˙a

(t) =

G(z , ˙a

(t + z ))dz

Typical solution for the dynamics when increasing the activation energy

period doubling

intermittency

Theoretical analyses of cellular structures 1996-2017

one-dimensional instability 1996-2002

Propagation

Leading shock front

Heat release

zone

Triple point Trajec

tory of triple point

Secondary shock

1-D oscillatory instability + transverse propagation and Mach stem formation

B.Maxwell, R.Bhattacharjee, S. Lau-Chapelaine, S.Falle, G. Sharpe and M. Radulescu Journal of Fluid Mechanics 2017

Schlieren images

Successive frames 11µs apart

CH

+ 2O

p = 3.5kP

High activation energy: Irregular structure and transient phenomena 2D geometry experiment

Intermittent formation

burning pockets of fresh mixture of

engulfed in the burned gas

The role of unsteadiness in the decay to the CJ regime when the piston is brought to instantaneous halt

PC & B. Denet (2018)

0

t₁

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t₁

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t<latexit sha1_base64="iXVXUtrMH2CbP1JXikf+Io21XPY=">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</latexit><latexit sha1_base64="Ax4Cm9kGORb34Vh0x7sukfUMpek=">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</latexit><latexit sha1_base64="Ax4Cm9kGORb34Vh0x7sukfUMpek=">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</latexit><latexit sha1_base64="aaSeuniZyqDSTPK/KZQFSb1cY6s=">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</latexit> ₂

t₂

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x

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x

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u_o

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0

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t

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1-D (planar) centered rarefaction wave when the piston is suddenly arrested

gas at r est

rarefaction wave

gas in motion at constant velocity

pistoninmotion pistonstopped

t= 0