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Simulation of flame acceleration and DDT in H2-air mixture with a flux limiter centred method

Knut Vaagsaether, Vegeir Knudsen and Dag Bjerketvedt

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• Outline– Introduction

– Models and numerics

– Physical experiments

– Numerical experiments

– Conclusion

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• The goal of this work is to simulate the explosion process from a weak ignition source through flame acceleration and DDT to a detonation

• The simulation tool is based on large eddy simulation (LES) of the filtered conservation equations with a 2. order centred TVD method

• Numerical results are compared to experimental results with pressure records

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• Filtered conservation equations of mass, momentum and energy

0~

ii

uxt

jijiijij

iji

j uuuuxx

puu

xt

u ~~~~~

EuEu

x

T

xx

puuE

xt

Eii

iij

ii

i

~~~

~~~

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• Turbulence model, by Menon et.al.

DPx

k

xku

xt

k

it

t

ii

i

Pr

~

j

iij x

uP

~

2

3

kCD

ijijkkijtij kSS 3

2~

3

1~2

2

1

kCst

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• In addition to the mass, momentum, energy and k, two other variables are conserved

– Two reaction variables, α and z

– α is a variable for the production of radicals where no energy is released

– z is a variable for the consumption of radicals (exothermal reactions)

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– α is only solved for the unreacted gas

– α keeps track of the induction time

– If α is below 1, no exothermal reaction is taking place

– If α reaches 1 an exothermal reaction occurs

– The production term of α is an Arrhenius function and can be assumed to be 1/τ

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• The exothermal reactions are handeled in two ways

– If the flame is a deflagration wave, a Riemann solver is used to calculate the states at each side of the flame

– The Riemann solver use the burning velocity as the reaction rate

– If the flame is a detonation wave or α reaches 1, another reaction model is used, presented by Korobeinikov (1972)

RT

QEzpk

RT

Ezpk

dt

dz 2223

2222 exp1exp

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• Burning velocity as a function of velocity fluctuations, presented by Flohr and Pitsch (2000)

• This model is developed for lean premixed combustion in gas turbine combustors

4

1

2

1

DaPrReA1Lt SS

u

Recu

Da 52.0A

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• Flame tracking with the G-equation

• Where vf is the local particle velocity in front of the flame

• G is negative in the unburned gas

• The G0 surface is set to be immediately in front of the flame

GSGvt

GT

f

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• Solvers– A flux limiter centered method (FLIC) to solve the

hyperbolic part of the equations, an explicit 2nd order TVD method

– Central differencing for the diffusion terms

– Godunov splitting for dimensions, diffusion terms and sub-models

– 4. order RK for ODEs

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• Experimental setup– 30% hydrogen in air

– 1 atm, 20°C

– Closed tube

– 10.7 cm ID

– Spark plug ignition at p0

– 0.5 m between sensors

– 1.5 between p0 and p1

– 3 cm orifice in obstacle

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• Experimental results, pressure records

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• Numerical setup– Same conditions as physical experiments

– Assume cylindrical coordinates• 2D

• Axis-symmetric

– Carthesian, homogeneous grid

– CV length 2 mm (~50 000 CV)

– CFL number 0.9

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• Comparison of pressure history at sensor p0

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• Comparison of pressure history at p2

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• Density in a 240 mm X 107 mm area • Time difference is 0.025 ms • DDT occurs between image 1 and 2

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• Mach number at center line behind the obstacle as the flame reaches the opening

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• Discussion and conclusion– The pressure in the ignition end of the tube is simulated

with some accuracy, even with these assumtions

– The detonation wave is simulated very accurate compared to the experiments which means that the Korobeinikov model is good enough for this work

– A DDT is simulated

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• Discussion and conclusion– Some discrepancies between numerical and physical

results in the ignition part (deflagration)• 2D

• Boundary conditions for the G-equation

• Burning velocity model

– The DDT is simulated too late• 2D

• Induction time

• Errors in pressure from the ignition part

• Is it possible with LES?

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• Further work

– 3D simulation should be performed

– Boundary conditions for the G-equation?

– Burning velocity model

– Adaptive mesh refinement

– A new model for the induction time