Ignition in a supersonic hydrogen–air mixing layer interacting with an oblique shock wave is investigated analytically under conditions such that the postshock flow is supersonic and the peak postshock temperature before ignition remains below the crossover temperature. The study requires consideration of the flow structure in the postshock ignition kernel found around the point of maximum temperature, which is assumed in this study to lie at an intermediate location across the mixing layer, as occurs in mixing layers subject to significant viscous dissipation. The ignition kernel displays a balance between the rates of chemical reaction and postshock flow expansion, including the acoustic interactions of the chemical heat release with the shock wave leading to increased front curvature. The problem is formulated with account taken of the strong temperature dependence of the chemical heat-release rate characterizing the ignition chemistry in the low-temperature regime analyzed here. It is shown how consideration of a two-step reduced chemical–kinetic mechanism derived in previous work leads to a boundary-value problem that can be solved analytically to determine ignition as a fold bifurcation, with the turning point in the diagram of peak perturbation induced by the chemical reaction as a function of the Damköhler number providing the critical conditions for ignition.

1. Shock-Induced Ignition of a Hydrogen-Air
Supersonic Mixing Layer
11th European Fluid Mechanics Conference, Seville, Spain.
C. Huete, A.L. Sánchez & F.A Williams
Grupo de Mecánica de Fluidos, UC3M
Mechanical and Aerospace Engineering. UCSD.
September 12th, 2016
C. Huete, A.L. Sánchez, & F.A Williams Shock-Induced Ignition of a H2-Air Supersonic ML

2. Motivation
The problem is motivated by SHCRAMJETS engines
reflected waves
hot air stream
H2 fuel stream
incident oblique shock
temperature-rise and
radical-production
ignition?
wedge
mixing layer
speed-up
Because of the very high speed of the gas stream, the
residence time of the reactants in the combustor is short
ignition cannot be achieved by relying on diffusion and heat
conduction alone
Shock waves may help to heat the mixture and speed-up the mixing
process! (Marble et al. 1987, Menon 1989; Lu & Wu 1991; Marble 1994; Nuding
1996; Brummund & Nuding 1997; GÂťenin & Menon 2010; Zhang et al. 2015))
C. Huete, A.L. Sánchez, & F.A Williams Shock-Induced Ignition of a H2-Air Supersonic ML

3. Problem configuration
Complete problem: interaction of a mixing layer with an oblique shock
supersonic hot air
supersonic H2 stream
mixing layer
M (z)
σ∞
z
x
∼ Re−1/2
L
T (z)
L
Outer problem: interaction of a tangential discontinuity with an
oblique shock (Landau & Lifshitz)
σ∞
M∞
M−∞
σ−∞
M
∞
M
−
∞
incident shock
reflected rarefaction
transmitted shock
σ∞
M∞
M−∞
σ−∞
M
∞
M
−
∞
incident shock
reflected shock
transmitted shock
C. Huete, A.L. Sánchez, & F.A Williams Shock-Induced Ignition of a H2-Air Supersonic ML

4. Problem configuration
Complete problem: interaction of a mixing layer with an oblique shock
supersonic hot air
supersonic H2 stream
mixing layer
M (z)
σ∞
z
x
∼ Re−1/2
L
T (z)
L
Outer problem: interaction of a tangential discontinuity with an
oblique shock (Landau & Lifshitz): ZOOM OUT
σ∞
M∞
M−∞
σ−∞
M
∞
M
−
∞
incident shock
reflected rarefaction
transmitted shock
σ∞
M∞
M−∞
M (z)
z
µ
µ
ν
φ
σ(z)
x
n
s
streamlines
C. Huete, A.L. Sánchez, & F.A Williams Shock-Induced Ignition of a H2-Air Supersonic ML

5. Characterization of the non-reactive mixing layer
The continuity, momentum, and species conservation equations are
rewritten, for the pre-shock mixing layer, in terms of the selfsimilar
variable η = z/[(µ∞/ρ∞)x/U∞]1/2
, namely
−
η
2
d
dη
(RU) +
d
dη
(RV ) = 0
R V −
η
2
U
dU
dη
=
d
dη
µ
dU
dη
R V −
η
2
U
dY
dη
= −
dJ
dη
η
15
10
5
−5
−10
0
1.510.5
temp. peak
Y
R
T
RCp V −
η
2
U
dT
dη
=
d
dη
µCp
P r
dT
dη
−
1 − W2
W2
J
dT
dη
−
α(γ − 1)
W2γ
d (W JT )
dη
Soret + Dufour effects
+ (γ − 1)M
2
1 µ
dU
dη
2
Shear effect
with J = − RD
P rLe
dY
dη
+ α
Y (1 − Y )
T
dT
dη
Soret effect
being the scaled diffusion flux.
When shear-induced temperature rise overcomes outer temperature
boundaries the ignition kernel is placed in rich-inner regions.
C. Huete, A.L. Sánchez, & F.A Williams Shock-Induced Ignition of a H2-Air Supersonic ML

6. Problem configuration
σ∞
M∞
M−∞
σ−∞
M
∞
M
−
∞
incident shock
reflected rarefaction
transmitted shock
σ∞
M∞
M−∞
M (z)
z
µ
µ
ν
φ
σ(z)
x
n
s
streamlines
Assessment of critical ignition conditions:
Computation of chemically frozen base flow, including post-shock
temperature distribution
Investigation of existence of slowly reacting solutions with
δT
T0
∼ T0
RgEa = β−1
1
Ignition kernel defined by the competition of chemical heat release
with the cooling associated with the post-shock expansion resulting
from the mean shock-front curvature
C. Huete, A.L. Sánchez, & F.A Williams Shock-Induced Ignition of a H2-Air Supersonic ML

7. Problem formulation
For Re = ρ∞δmU∞/µ∞ 1, the compressible flow is inviscid in the
nonslender interaction region.
γM2
o − 1
γM2
o
∂ˆp
∂s
−
∂ ˆT
∂s
+
∂ ˆV
∂n
= 0
γM2
o
∂ ˆV
∂s
+
∂ˆp
∂n
= 0,
∂ ˆT
∂s
−
γ − 1
γ
∂ˆp
∂s
= −
N
i=1
ho
i
ρocpTo
˙Ci
Uo/δ
∂Ci
∂s
=
˙Ci
Uo/δ
,
Complemented with
boundary conditions (shock front + upper incoming perturbations
along Mach lines C−
)
simplified chemistry model for the H2-air mixture below the crossover
temperature.
C. Huete, A.L. Sánchez, & F.A Williams Shock-Induced Ignition of a H2-Air Supersonic ML

8. Problem formulation
If M > 1 in the postshock region, the flow can be described in terms
of characteristic equations, with three different characteristic lines
crossing any given point.
∂I±
∂s
±
1
√
M2
o − 1
∂I±
∂n
= −
γM2
o
M2
o − 1
N
i=1
ho
i
ρocpTo
˙Ci
Uo/δ
∂ ˆT
∂s
−
γ − 1
2γ
∂
∂s
(I+
+ I−
) = −
N
i=1
ho
i
ρocpTo
˙Ci
Uo/δ
.
with I±
= ˆp ±
γM2
o√
M2
o −1
ˆV and C±
: s + n
tan φo
= 1
tan φo
± 1
tan µo
(n − ns)
provided with boundary conditions at the shock front:
Y = Yf (n)
ˆT = ˆT + AT
−
BT
B−
A− ˆM +
BT
B−
I−
I+
= A+
−
B+
B−
A− ˆM +
B+
B−
I−
C. Huete, A.L. Sánchez, & F.A Williams Shock-Induced Ignition of a H2-Air Supersonic ML

9. The chemistry description
Ignition occurring at post-shock temperatures below the crossover
temperature in fuel-rich mixtures. Some kinetics simplifications lead
us to a simple two-step reduced description involving the reactions
(Boivin et al. Comb. & Flame, 2012)
2H2 + O2 → 2H2O, H2 + O2 → H2O2
with rates given by
ωI = k1CM1 CH2O2 = k1(To)CM1 CH2O2 eβ ˆT
ωII = k3
k1
k2
1/2
C
1/2
M1
CH2 C
1/2
H2O2
= k3(To) k1(To)
k2(To)
1/2
C
1/2
M1
CH2 C
1/2
H2O2
eβ ˆT
in the Frank-Kamenetskii linearization of the rates about To.
Reaction B n Ta
1 H2O2+M → OH+OH+Ma k0 7.60 1030 -4.20 25703
k∞ 2.63 1019 -1.27 25703
2 HO2+HO2 → H2O2+O2 1.03 1014 0.0 5556
1.94 1011 0.0 -709
3 HO2+H2 → H2O2+H 7.80 1010 0.61 12045
C. Huete, A.L. Sánchez, & F.A Williams Shock-Induced Ignition of a H2-Air Supersonic ML

10. Ignition kernel
σ∞
M∞
M−∞
M (z)
z
µ
µ
ν
φ
σ(z)
x
n
s
streamlines
δT
T0
∼ β−1
T = T0
IGNITION KERNEL
δ = sin φ0
sin σ0
dM
dz
−1
η
ξ
Frozen-flow temperature profile (Huete et al. JFM, 2015)
ˆTF = −ΓT n2
−
γ − 1
γ
Λ s +
n
tan φo
,
shows that for ˆT ∼ β−1
, extends over streamwise distances of order
β−1
and much larger transverse distances of order β−1/2
, suggesting
ξ =
γ − 1
γ
Λβ s +
n
tan φo
and η = Γ
1/2
T β1/2
n
as the relevant stretched coordinates, indicating that that the ignition
kernel is thin in the streamwise direction when Γ
1/2
T Λβ1/2
.
C. Huete, A.L. Sánchez, & F.A Williams Shock-Induced Ignition of a H2-Air Supersonic ML

11. Ignition kernel
σ∞
M∞
M−∞
M (z)
z
µ
µ
ν
φ
σ(z)
x
n
s
streamlines
δT
T0
∼ β−1
T = T0
IGNITION KERNEL
δ = sin φ0
sin σ0
dM
dz
−1
η
ξ
∂Y
∂ξ
= DY 1/2
e−η2
−ξ
eθ
∂θ
∂ξ
−
γ − 1
2γ
∂
∂ξ
(J+
+ J−
) = D(Y + ¯λY 1/2
)e−η2
−ξ
eθ
1 ±
tan µo
tan φo
∂J±
∂ξ
= γ(tan2
µo + 1)D(Y + ¯λY 1/2
)e−η2
−ξ
eθ
,
D = γβq
(γ−1)Λ
k1CM1
Cc
CH2
Uo/δ
is the relevant Damköhler number
¯λ =
ho
H2O2
2ho
H2O
k2
3
k1k2
1/2
CH2
/CM1
Cc/CH2
1/2
measures the heat released by H2O2
formation.
C. Huete, A.L. Sánchez, & F.A Williams Shock-Induced Ignition of a H2-Air Supersonic ML

12. The eigenvalue problem
Integration with the boundary condition determines D as an
eigenvalue. Defining the parameters
κ =
B−
γBT
tan µo/ tan φo − 1
tan2 µo + 1
1 −
(γ − 1)(tan2
µo + 1)
(tan µo/ tan φo)2 − 1
that measures the competition between the cooling rate associated
with the flow expansion induced by the chemical reaction and the
direct heat release of the chemical reaction, and
ϕ =
γBT
B−
tan2
µo + 1
tan µo/ tan φo − 1
2/3
Y
as the rescaled H2O2 concentration, we get
∂ϕ
∂ξ
= ∆ϕ1/2
e−η2
−ξ
eθ
∂θ
∂ξ
= κ∆(ϕ + λϕ1/2
)e−η2
−ξ
eθ
,
subject to ϕ = θ − θs = 0 at ξ = 0 and θ = (1 + κ)θs at ξ = ∞,
C. Huete, A.L. Sánchez, & F.A Williams Shock-Induced Ignition of a H2-Air Supersonic ML

13. Results
The eigenvalue problem determines whether weak-reaction solution is
not feasible, then indicating that thermal explosion occurs.
λ=0
κ=-1
κ=2
-0.5
0
1
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Δ
φ∞
λ=0.5
κ=-1
κ=2
-0.5
0
1
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Δ
φ∞
∆ =
γBT
B−
tan2
µo + 1
tan µo/ tan φo − 1
1/3
D and λ =
γBT
B−
tan2
µo + 1
tan µo/ tan φo − 1
1/3
¯λ
C. Huete, A.L. Sánchez, & F.A Williams Shock-Induced Ignition of a H2-Air Supersonic ML

14. Results
And the value of ∆ and κ are functions of the ignition-kernel base
conditions M0 and σ0.
κ = 2
κ = 2.3
κ = -1
κ = 1
Δ = 0.8
Δ = 0.77
Δ = 0.8
Δ = 0.9
Δ = 1.2
Δ = 0.7
subsonic flow
weak-shock limit
Δ = 0.6
Δ = 0.77
κ = 0
1 2 3 4 5 6 7 8 9 10
0
15
30
45
60
75
90
Mo
′
σo
C. Huete, A.L. Sánchez, & F.A Williams Shock-Induced Ignition of a H2-Air Supersonic ML

15. Aftermath
C. Huete, A.L. Sánchez, & F.A Williams Shock-Induced Ignition of a H2-Air Supersonic ML

16. Concluding Remarks
The ignition analyses that have been developed here on the
basis of a laminar flow configuration help to clarify the
manner in which shock waves may promote ignition in
supersonic mixing layers to lead to establishment of diffusion
flames in supersonic flows.
As in classical Frank-Kamenetskii theory of thermal
explosions, ignition is found to occur as a fold bifurcation, in
which the cooling processes that compete with the chemical
heat release involve inviscid gasdynamic acoustic-wave
propagation (instead of the familiar diffusive heat conduction)
Future research should address other ignition conditions
(including post-shock temperatures above crossover) as well
as influences of turbulence flow on the ignition dynamics
C. Huete, A.L. Sánchez, & F.A Williams Shock-Induced Ignition of a H2-Air Supersonic ML