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First Half Objectiveirst Half Objective Turbulent flame model for carbon-burning in SNe Ia

Previous study (APS 2008) examined fully-resolved small-scale

carbon-burning thermonuclear flames at high turbulence levels

Present objective is to investigate much larger length scales

Construct a turbulent flame model for large-scale distributed burning

Theoretical treatment of burning in distributed regime

Based on Damkhler scaling (1940)

Predict scaling relations for turbulent flame speed and width

Three-dimensional simulations to test predictions

Aspden, Bell, Woosley, Distributed Flames in SNe Ia, ApJ, 710, 2010

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First Half Outlineirst Half Outline

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Turbulent intensity

and integral length

Laminar flame

speed and width

Karlovitz number

Damkhler number

Regime Diagramegime Diagram

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Distributed burning regime

Start

Larger

scales

(i.e.hig

herDa

)

Model

Transitio

n

Previous study

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Theoryheory LargeLarge Damkhleramkhler For larger DaT, turbulence cannot broaden the flame any further A limiting behaviour is reached

Burns as a turbulently broadened effective unity Lewis number flame

Local flame speed and width are constant (higher due to enhanced area)

(on scale ofl)

Refer to this kind of burning as a -flame

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Norm

alisedlength

Normalisedvelocity

TurbulentDamkhlernumber TurbulentDamkhlernumber

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Theoryheory Lambda flamesLambda flames Can we predict the -flame properties?

Depend solely on turbulence intensity and burning time scale

Dimensional analysis four quantities in two units

Two dimensionless quantities

Both are identically equal to one, which implies

What is the turbulent nuclear burning time scale?

Reference case with turbulent intensity and integral length

Measure the turbulent flame speed only

Use the relation

Then, by definition

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Modified Regime Diagramodified Regime Diagram

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Transition at -point

-flames

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Numerical Solverumerical Solver Written at Center for Computational Sciences and Engineering

Based on 3D variable-density incompressible Navier-Stokes solver

Extended for low Mach number SNe flames

Cartesian finite-volume discretisation

Predictor-corrector approach

Advection-diffusion and chemistry are operator split

Approximate projection for divergence constraint

Overall second-order accurate in space and time

Adaptive mesh refinement to focus resolution on regions of interest

Parallelised performs well up to several thousand processors

Capable of implicit LES calculations dont need a turbulence model

Aspden et. al, CAMCoS 3 (2008)

Further details can be found in Bell et. al, JCP 195 (2004)

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SchematicchematicThree dimensional box

Fuel below ash

Propagates downwards

High aspect ratio

Forced throughout

Periodic sides

Solid base

Outflow at top

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Procedurerocedure Aim is to simulate larger length scales (Da) keeping Ka fixed

Resolution requirements become relaxed for distributed flames

Mixing due to turbulence, relevant scales grow with integral length

Start with high Karlovitz number case from ApJ paper(256x256x2048)

Reduce resolution by a factor of 8 (32x32x256)

i.e. computational cell size 8 times larger

Use turbulent flame speed as diagnostic check

Use the new cell size to run in a domain 8 times larger

Adjust turbulent intensity accordingly to fix energy dissipation rate (Ka)

Repeat

Limited by DaT expect to be reasonably valid for DaT1, relevant scales are fixed

Seven cases A-G

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Slices of Densitylices of Density

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L=150cm 12m 96m 768m 6144m 49km 393km

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Turbulent Flame Speedsurbulent Flame Speeds

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Normalisedtime

A

B

C

D

E

F

G

Turbulentflamespeed

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Turbulent Flame Speedsurbulent Flame Speeds

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TurbulentDamkhlernumber(DaT)

Norm

alized

turbulentflame

speed(sT/u)

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First Half Conclusionsirst Half Conclusions Formulated and verified scaling relations at high Ka

More importantly extending to high Da

Can predict constant local flame speed and width

Perfectly suited to level set approach Suggested an approach to describe the flame speed

Tens of thousands times larger than original study in each dimension

Overall flame speed is highly fluctuating

Possible that these fluctuations may lead to run-away

Aspden, Bell, Woosley, Distributed Flames in SNe Ia, ApJ, 710, 2010

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Morton Taylor Turner Theoryorton Taylor Turner Theory Idealized thermal

Represent thermal as sphere of radius b(t) at height zb(t)

Entrainment assumption

Fluid is entrained into the thermal at a rate proportional to the rise height

Mixing is sufficiently fast that entrained fluid is mixed instantaneously

Conservation equations for volume, momentum and buoyancy

Interesting property evolves in a cone

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Initial Conditionsnitial Conditions 864 cm cube domain

Resolutions up to 40963

Base grid 5123 + 3 levels AMR

Initial bubble radius about 14cm

Perturbation to break symmetry

Fuel density 1.5e7g/cm3

Gravity 109 cm/s2

Solid base, outflow elsewhere

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Inert Thermalnert ThermalVorticityorticity3d Renderingd Rendering

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Inert Thermalnert ThermalTracer Slicesracer Slices

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Buoyant Bubbleuoyant BubbleTracer Slicesracer Slices

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Bubble Heightubble Height vss Radius (Cone)Radius (Cone)

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Bubbleradius

Normalized

bubbleh

eight

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Bubble Heightubble Height vss TimeTime

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Normalizedtime

Square

rootnormalized

height

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Second Half Conclusionsecond Half Conclusions Modified MTT theory to account for burning

Appears to provide good predictions

But requires immense simulations

Further work required for generality of entrainment coefficient

Generalization of theory for application to full stars

Ambient stratification (variations in density and pressure)

Background turbulence

Straightforward to formulate one-dimensional system of ODEs

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