Slides "Effect of fuel mass fraction heterogeneity on the detonation propagation speed"

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Effect of fuel mass fraction heterogeneity
on the detonation propagation speed
25th International Congress of Theoretical and Applied Mechanics
Alberto Cuadra *, César Huete & Marcos Vera
*acuadra@ing.uc3m.es
Grupo de Mecánica de Fluidos
Universidad Carlos III de Madrid
Milano |August 22-27, 2021

Effect of fuel mass
fraction heterogeneity
on the detonation
propagation speed
A. Cuadra (presenter)
Grupo de Mecánica de
Fluidos |UC3M
1Introduction
Detonation in
heterogeneous
gaseous mixture
Influences on the
propagation properties
Introduction
Reactive Rankine-Hugoniot equations
1. Fresh homogeneous mixture with
Mu≥ Mcj = (1 + Q)1/2+Q1/2.
2. Obtain the Rankine-Hugoniot jump
conditions from the conservation
equations and assuming a calorically
perfect gas, namely
Rd=ρd
ρu
=(γ+ 1) M2
u
γM2
u+ 1 −[(M2
u−1)2−4QM2
u]1/2,
Pd=pd
pu
=γM2
u+1+γ(M2
u−1)2−4QM2
u1/2
γ+ 1 ,
for density and pressure, respectively.
planar detonation

Effect of fuel mass
fraction heterogeneity
on the detonation
propagation speed
A. Cuadra (presenter)
Grupo de Mecánica de
Fluidos |UC3M
Introduction
2Detonation in
heterogeneous
gaseous mixture
Influences on the
propagation properties
Detonation in heterogeneous gaseous mixture
How will heterogeneities in the gas mixture affect the detonation
properties?
Faster detonation speed!
•The deviation of the fuel mass
fraction δYuintroduce two
source of perturbations:
1. δρ,
2. δqu.
•New contributions:
•acoustic,
•rotational,
•entropic.
•Linear analysis: weak
deviations
|δYu|=|Yu− hYui|  1.

Effect of fuel mass
fraction heterogeneity
on the detonation
propagation speed
A. Cuadra (presenter)
Grupo de Mecánica de
Fluidos |UC3M
Introduction
3Detonation in
heterogeneous
gaseous mixture
Influences on the
propagation properties
Detonation in heterogeneous gaseous mixture
The slopes Wand Hplay a pivotal role in the turbulence generation.
destructive interference constructive interference

Effect of fuel mass
fraction heterogeneity
on the detonation
propagation speed
A. Cuadra (presenter)
Grupo de Mecánica de
Fluidos |UC3M
Introduction
Detonation in
heterogeneous
gaseous mixture
4Influences on the
propagation properties
Influences on the propagation properties
•Average density and pressure are lower while average velocity and Mach
are higher.
•Do they affect the detonation propagation speed?
hρdi
hρui=Rd1 + ¯2δR,hpdi
pu
=Pd1 + ¯2δP,hudi
uu
=1
Rd
1 + ¯2δU.
−3
−2,5
−2
−1,5
−1
−0,5
0
10−210−1100101
−3
−2,5
−2
−1,5
−1
−0,5
0
10−210−1100101
0
0,5
1
1,5
2
2,5
10−210−11001010
0,5
1
1,5
2
2,5
10−210−1100101
Figure 1: Three-dimensional second-order correction to the RH jump conditions (a) δR, (b) δP, (c) δU,
and (d) δMas a function of the overdrive parameter Mu/Mcj −1. Computations correspond to
γ= 1.2,Q= 1 ,and |W||H|( ), |H||W|( ), W=H( ) and W=−H
( ).

Effect of fuel mass
fraction heterogeneity
on the detonation
propagation speed
A. Cuadra (presenter)
Grupo de Mecánica de
Fluidos |UC3M
Introduction
Detonation in
heterogeneous
gaseous mixture
5Influences on the
propagation properties
Influences on the propagation properties
2nd-order variations
δS|δhMdi=0 =−1
√RdPddMd
dMu−1δU+1
2δR − 1
2δP.(1)

Effect of fuel mass
fraction heterogeneity
on the detonation
propagation speed
A. Cuadra (presenter)
Grupo de Mecánica de
Fluidos |UC3M
Introduction
Detonation in
heterogeneous
gaseous mixture
5Influences on the
propagation properties
Influences on the propagation properties
2nd-order variations
δS|δhMdi=0 =−1
√RdPddMd
dMu−1δU+1
2δR − 1
2δP.(1)
new equilibrium
point correction
by second-order
turbulent �luctuations

Effect of fuel mass
fraction heterogeneity
on the detonation
propagation speed
A. Cuadra (presenter)
Grupo de Mecánica de
Fluidos |UC3M
Introduction
Detonation in
heterogeneous
gaseous mixture
5Influences on the
propagation properties
Influences on the propagation properties
2nd-order variations
δS|δhMdi=0 =−1
√RdPddMd
dMu−1δU+1
2δR − 1
2δP.(1)
Figure 2: Three-dimensional second-order correction of the detonation propagation velocity δS,
according to (1), as a function of the overdrive parameter Mu/Mcj −1. Computations are provided
for γ= 1.2,˜
Q= 1 (a) and ˜
Q= 10 (b) evaluated for |W||H|( ), |H||W|( ),
W=H( ) and W=−H( ).

Key takeaways
•The thin-detonation limit allows the description of the transient evolution in
analytical form.
•Turbulence generation provides second-order corrections to the averaged
Rankine–Hugoniot jump conditions.
•In almost most of the cases of interest, these (purely) hydro-averaged
perturbations translate into faster propagation speeds.