shock-induced ignition of a hydrogen-air supersonic mixing layer

Shock-Induced Ignition of a Hydrogen-AirSupersonic Mixing Layer

11th European Fluid Mechanics Conference, Seville, Spain.

C. Huete, A.L. Sánchez & F.A Williams

Grupo de Mecánica de Fluidos, UC3MMechanical and Aerospace Engineering. UCSD.

September 12th, 2016

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

Motivation

The problem is motivated by SHCRAMJETS engines

reflected waves

hot air stream

H2 fuel stream

incident oblique shocktemperature-rise andradical-production

ignition?

wedge

mixing layer speed-up

Because of the very high speed of the gas stream, theresidence time of the reactants in the combustor is shortignition cannot be achieved by relying on diffusion and heatconduction alone

Shock waves may help to heat the mixture and speed-up the mixingprocess! (Marble et al. 1987, Menon 1989; Lu & Wu 1991; Marble 1994; Nuding1996; Brummund & Nuding 1997; GÂťenin & Menon 2010; Zhang et al. 2015))


Problem configuration

Complete problem: interaction of a mixing layer with an oblique shock

supersonic hot air

supersonic H2 streammixing layer

M ′(z)

σ∞

zx

∼ Re−1/2L

T ′(z)

L

Outer problem: interaction of a tangential discontinuity with anoblique shock (Landau & Lifshitz)

σ∞M ′∞M ′−∞

σ−∞

M∞M

−∞

incident shockreflected rarefaction

transmitted shock

σ∞M ′∞M ′−∞

σ−∞

M∞M

−∞

incident shockreflected shock

transmitted shock



Complete problem: interaction of a mixing layer with an oblique shock

supersonic hot air

supersonic H2 streammixing layer

M ′(z)

σ∞

zx

∼ Re−1/2L

T ′(z)

L

Outer problem: interaction of a tangential discontinuity with anoblique shock (Landau & Lifshitz): ZOOM OUT

σ∞M ′∞M ′−∞

σ−∞

M∞M

−∞


transmitted shock

σ∞

M ′∞

M ′−∞

M ′(z)

z

µµ

νφ

σ(z)

x

ns

streamlines


Characterization of the non-reactive mixing layerThe continuity, momentum, and species conservation equations arerewritten, for the pre-shock mixing layer, in terms of the selfsimilarvariable η = z/[(µ′∞/ρ′∞)x/U ′∞]1/2, namely

−η

2

d

dη(RU) +

d

dη(RV ) = 0

R

(V −

η

2U

) dU

dη=

d

dη

(µ

dU

dη

)R

(V −

η

2U

) dY

dη= −

dJ

dη

η

15

10

5

−5

−10

0

1.510.5

temp. peak

Y

R

T

RCp

(V −

η

2U

) dT

dη=

d

dη

(µCp

Pr

dT

dη

)−

1 −W2W2

JdT

dη−

α(γ − 1)

W2γ

d (WJT )

dη︸︷︷︸Soret + Dufour effects

+ (γ − 1)M21µ

(dU

dη

)2︸︷︷︸Shear effect

with J = − RDPrLe

(dYdη + α

Y (1 − Y )

T

dT

dη︸︷︷︸Soret effect

) being the scaled diffusion flux.

When shear-induced temperature rise overcomes outer temperatureboundaries the ignition kernel is placed in rich-inner regions.



σ∞M ′∞M ′−∞

σ−∞

M∞M

−∞


transmitted shock

σ∞

M ′∞

M ′−∞

M ′(z)

z

µµ

νφ

σ(z)

x

ns

streamlines

Assessment of critical ignition conditions:Computation of chemically frozen base flow, including post-shocktemperature distributionInvestigation of existence of slowly reacting solutions withδTT0

∼ T0RgEa

= β−1 � 1

Ignition kernel defined by the competition of chemical heat releasewith the cooling associated with the post-shock expansion resultingfrom the mean shock-front curvature


Problem formulation

For Re = ρ′∞δmU′∞/µ

′∞ � 1, the compressible flow is inviscid in the

nonslender interaction region.

γM2o − 1

γM2o

∂p

∂s− ∂T

∂s+ ∂V

∂n= 0

γM2o∂V

∂s+ ∂p

∂n= 0,

∂T

∂s− γ − 1

γ

∂p

∂s= −

N∑i=1

(hoi

ρocpTo

)(CiUo/δ

)∂Ci∂s

= CiUo/δ

,

Complemented withboundary conditions (shock front + upper incoming perturbationsalong Mach lines C−)simplified chemistry model for the H2-air mixture below the crossovertemperature.


Problem formulation

If M > 1 in the postshock region, the flow can be described in termsof characteristic equations, with three different characteristic linescrossing any given point.

∂I±

∂s± 1√

M2o − 1

∂I±

∂n= − γM2

o

M2o − 1

N∑i=1

(hoi

ρocpTo

)(CiUo/δ

)∂T

∂s− γ − 1

2γ∂

∂s(I+ + I−) = −

N∑i=1

(hoi

ρocpTo

)(CiUo/δ

).

with I± = p± γM2o√

M2o−1

V and C± : s+ ntanφo =

(1

tanφo ±1

tanµo

)(n− ns)

provided with boundary conditions at the shock front:

Y = Yf (n)

T = T ′ +(AT − BT

B−A−)M ′ + BT

B−I−

I+ =(A+ − B+

B−A−)M ′ + B+

B−I−


The chemistry description

Ignition occurring at post-shock temperatures below the crossovertemperature in fuel-rich mixtures. Some kinetics simplifications leadus 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 = k1CM1CH2O2 = k1(To)CM1CH2O2eβT

ωII = k3(k1k2

)1/2C

1/2M1

CH2C1/2H2O2

= k3(To)[k1(To)k2(To)

]1/2C

1/2M1

CH2C1/2H2O2

eβT

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 25703k∞ 2.63 1019 -1.27 25703

2 HO2+HO2 → H2O2+O2 1.03 1014 0.0 55561.94 1011 0.0 -709

3 HO2+H2 → H2O2+H 7.80 1010 0.61 12045


Ignition kernel

σ∞

M ′∞

M ′−∞

M ′(z)

z

µµ

νφ

σ(z)

x

ns

streamlines

δTT0∼ β−1

T = T0

IGNITION KERNEL

δ = sinφ0sinσ0

(dM′

dz

)−1

ηξ

Frozen-flow temperature profile (Huete et al. JFM, 2015)

TF = −ΓTn2 − γ − 1γ

Λ(s+ n

tanφo

),

shows that for T ∼ β−1, extends over streamwise distances of orderβ−1 and much larger transverse distances of order β−1/2, suggesting

ξ = γ − 1γ

Λβ(s+ n

tanφo

)and η = Γ1/2

T β1/2n

as the relevant stretched coordinates, indicating that that the ignitionkernel is thin in the streamwise direction when Γ1/2

T � Λβ1/2.C. Huete, A.L. Sánchez, & F.A Williams Shock-Induced Ignition of a H2-Air Supersonic ML

Ignition kernel

σ∞

M ′∞

M ′−∞

M ′(z)

z

µµ

νφ

σ(z)

x

ns

streamlines

δTT0∼ β−1

T = T0

IGNITION KERNEL

δ = sinφ0sinσ0

(dM′

dz

)−1

ηξ

∂Y

∂ξ= DY 1/2e−η

2−ξeθ

∂θ

∂ξ− γ − 1

2γ∂

∂ξ(J+ + J−) = D(Y + λY 1/2)e−η

2−ξeθ(1± tanµo

tanφo

)∂J±

∂ξ= γ(tan2 µo + 1)D(Y + λY 1/2)e−η

2−ξeθ,

D = γβq(γ−1)Λ

k1CM1CcCH2Uo/δ

is the relevant Damköhler number

λ =hoH2O22hoH2O

(k2

3k1k2

)1/2 (CH2/CM1Cc/CH2

)1/2measures the heat released by H2O2

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

The eigenvalue problem

Integration with the boundary condition determines D as aneigenvalue. Defining the parameters

κ = B−

γBT

tanµo/ tanφo − 1tan2 µo + 1

[1− (γ − 1)(tan2 µo + 1)

(tanµo/ tanφo)2 − 1

]that measures the competition between the cooling rate associatedwith the flow expansion induced by the chemical reaction and thedirect heat release of the chemical reaction, and

ϕ =(γBTB−

tan2 µo + 1tanµo/ tanφo − 1

)2/3

Y

as the rescaled H2O2 concentration, we get

∂ϕ

∂ξ= ∆ϕ1/2e−η

2−ξeθ

∂θ

∂ξ= κ∆(ϕ+ λϕ1/2)e−η

2−ξeθ,

subject to ϕ = θ − θs = 0 at ξ = 0 and θ = (1 + κ)θs at ξ =∞,


Results

The eigenvalue problem determines whether weak-reaction solution isnot feasible, then indicating that thermal explosion occurs.

λ=0

κ=-1

κ=2

-0.50

1

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0

0.5

1.0

1.5

2.0

2.5

3.0

Δ

φ∞

λ=0.5

κ=-1

κ=2

-0.5

01

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0

0.5

1.0

1.5

2.0

2.5

3.0

Δ

φ∞

∆ =(γBTB−


)1/3

D and λ =(γBTB−


)1/3

λ


Results

And the value of ∆ and κ are functions of the ignition-kernel baseconditions M ′0 and σ′0.

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 = 0.6Δ = 0.77

κ = 0

1 2 3 4 5 6 7 8 9 100

15

30

45

60

75

90

Mo′

σo


Aftermath


Concluding Remarks

The ignition analyses that have been developed here on thebasis of a laminar flow configuration help to clarify themanner in which shock waves may promote ignition insupersonic mixing layers to lead to establishment of diffusionflames in supersonic flows.As in classical Frank-Kamenetskii theory of thermalexplosions, ignition is found to occur as a fold bifurcation, inwhich the cooling processes that compete with the chemicalheat release involve inviscid gasdynamic acoustic-wavepropagation (instead of the familiar diffusive heat conduction)

Future research should address other ignition conditions(including post-shock temperatures above crossover) as wellas influences of turbulence flow on the ignition dynamics
