Interplay of hydrodynamic instabilities and turbulence in premixed flames

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Interplay of hydrodynamic instabilities and

turbulence in premixed flames

Rachele Lamioni

Mechanical and Aerospace Engineering Department

Sapienza University of Rome

XXXII Ciclo 2016-2019

Advisor: Prof. Francesco Creta

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This thesis is dedicated to Nonno Cecco,

who can’t be here to celebrate with me, for the love he gave me

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Acknowledgements

Foremost, I would like to express my sincere gratitude to my advisor Prof. Francesco Creta. He is one of the best advisors a student could wish for, he has always been there to guide and help me, specially in times of great confusion.

I would also to thank my colleagues, Pasquale Eduardo Lapenna, Giuseppe Indelicato, Riccardo Malpica Galassi and Pietro Paolo Ciottoli who tolerated my innumerable questions and were always there to pull out ideas and keep a positive environment in the office. I am also sincerely grateful to Prof. Antonio Attili, Lukas Berger and Konstantine Kleinheinz for their valuable suggestions and discussions during the visit in RWTH university in Aachen. During my time at RWTH, I always came across a good atmosphere, whether through research collaborations or the academic resources. On other important thank goes to my friend Daniela, who has always been close to me, without ever abandoning me. Thank also to Francesca, a friend in my heart.

My work would also not have been complete without the support of my parents who despite being away never made me feel alone and unfortunately always became my punching bag during times of frustration. My family’s confidence in my abilities always pushed me to strive for excellence. I dedicate this job to grandfather Cecco, who would have said "Brava cittina, ma io non ci capisco niente in queste cose".

Lastly but not less important, a particular thank to Simone, an important person who accompanied me for most of my PhD course, often becoming my outburst for every problem.

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Abstract

This thesis is devoted to the numerical investigation of premixed Flames subject to intrinsic hydrodynamic instabilities as well as turbulence. Laminar as well as turbulent premixed combustion can be largely influenced by the onset of hydrodynamic instabilities which can cause a significant increase in flame corrugation and, as a result, in the turbulent flame speed, especially when low turbulence intensity level are present. Indeed, such instability is responsible for the formation of sharp folds and creases in the flame front and for the wrinkling observed, undergo certain conditions, over the surface of expanding flames. Hydrodynamic instability is a result of thermal expansion across the flame and its role is particularly dominant in large-scale flames, when flames are constrained by domains larger than several hundred times the flame thickness. The understanding of these phenomena and their interaction with turbulence can play a potentially significant role in practical combustion systems such as gas turbines and, more generally, in industrial and domestic burners. The aim of this thesis is to develop a numerical tool capable of simulating the propagation of turbulent premixed flames under the influence hydrodynamic instabilities and gather qualitative and quantitative data on flame properties such as morphology and global propagation.

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List of Figures

1.1 Schematic showing the premixed flame structure at successive levels of detail: (a) the hydrodynamic flame sheet; (b) the transport reaction sheet; (c) detailed structure including the reaction zone [81]. . . 5 1.2 Perturbed flame front [36]. . . 6 1.3 Bunsen flame configuration: solid line L = 0; dashed lines display the

effect of positive Markstein length. . . 8 1.4 Dispersion relation: blue line is the DL hypothesis, where all wave numbers

are unstable; purple line is the model that includes the curvature and strain effects. . . 10 1.5 Numerical simulation of perturbed 2D planar flame: Darrieus-Landau

mechanism (DL) [75]. . . 11 1.6 Illustration of hydrodynamic (Landau-Darrieus) instability [17] . . . 13 1.7 Illustration of diffusive-thermal instability, adapted from [27] . . . 13 1.8 Experimental conditions for which the flame propagation is influenced

by both turbulence and DL instability. The conditions are plotted with respect to regimes defined by Chaudhuri [26] and study by [128]. . . 17 1.9 Images taken from high speed imaging of growth of instability. Framing

rate 500 images/s, wavelength 2 cm, flame speed 11.5 cm/s. [37]. . . 18 1.10 Typical OH-PLIF images of CH4/air flames: (a) Low pressure and

turbulence, (b) high pressure and high turbulence and (c) high pressure and laminar condition [73]. . . 19 1.11 Mie scattering images of propane–air Bunsen flames at atmospheric

pressure and Re= 5000: (1) Subcritical regime, (2) Supercritical regime [121]. 19

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0.275 and 0.300 m. [9]. . . 21 1.14 Instantaneous OH-PLIF images of C3H8/air flames at different turbulence

conditions [130]. . . 22 1.15 (a) Downward propagation of a stoichiometric premixed propane air flame

in a vertically oriented Hele-Shaw cell. (b) Destabilization of a flat initial condition. (c) Cusp creation. (d) Cusps merging. (e) Lean flame close to the stability threshold [2]. . . 23 1.16 Experiments of expanding flame of three H2/02/He/Ar mixtures at

equivalence ratio φ = 0.4, performed in in a well-vetted fan-stirred, constant-pressure, dual-chambered vessel. Three levels of propagation:(i) propagation without cells (smooth propagation),(ii) propagation with initial cellular structure (transition stage) and (iii) propagation with fully developed cellular structure (saturated stage) [85]. . . 24 1.17 Flame profiles at consecutive times, in a flame-stationary frame obtained

from Michelson-Sivashinky equation [34]. . . 26 1.18 Flame surface evolution starting with randomly perturbed initial

con-ditions; the flame shape at four consecutive times is shown in (a) and the corresponding amplitude is shown (b); the symbols identify the four instances selected in(a). [98] . . . 27 1.19 Two dimensional simulations of freely propagating flames: temperature

distribution for a flame front for different domain sizes [129] . . . 28 1.20 Premixed hydrogen/air flames: (a) flame front locations at intervals of 2

time units, (b) iHRR and flame length normalized by the planar flame values for φ = 1.0 and atmospheric pressure [54]. . . 29 1.21 Three-dimensional DNS with Bunsen configuration at different

thermody-namic pressures: instantaneous isosurfaces of reaction progress variable colored by Gaussian (left) and Mean (right) curvatures, for different pressure values [71]. . . 30

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1.22 Thermal diffusive instability in a 2D DNS of freely propagating flame: images show the fields of heat release rates and temperature, respectively [55]. 31 1.23 Thermal diffusive instability in two dimensional DNS of premixed lean

hydrogen flames: normalized temperature Θ at a t = 200τF after initial

perturbation [13]. . . 31 1.24 Results obtained in the context of numerical simulations performed for

the present thesis: numerical dispersion relations using different Lewis numbers with single step chemistry. . . 32 2.1 Spectral element mapping from canonical 7th-order quadrilateral domain

ˆΩ to physical subdomain Ωe[68] . . . 39

2.2 Experimental set-up. Left: Configuration of the burner with characteristic dimensions, bluff body geometry, mixing manifold and air plus methane inlets. Right: Sketch of optical arrangement showing laser sources, optical devices, acquisition systems and related hardware [122]. . . 42 3.1 (a) Number of unstable wavelengths ncas a function of δ and Le. Stability

region corresponds to nc≤1. (b) Direct estimation of L /`Dfrom the slope

of the linear regression model (dashed line) between (non-dimensional) local normalized flame speed and stretch rate. Color map shows the normalized joint p.d.f. of flame speed and stretch obtained from a simulated flame at

δ= 0.008 and using θ∗= 0.8. . . 46

3.2 Schematic of the computational domain. . . 48 3.3 Two-dimensional simulations of slot burner flames with σ = 8, Le =

1.2 and of different nondimensional thickness δ = `D/L with `D the

flame thickness and L the burner diameter. The displayed field is non-dimensional temperature θ. Flames with δ < δc where δc = 0.015 is

the critical value, exhibit DL instability. (a) δ = 0.0275 > δc, (b) δ =

0.015 ∼ δc, (c) δ = 0.008 < δc, (d) δ = 0.004 < δc. Note superadiabatic

temperatures (θ > 1) issuing from highly curved flame crests. The domain shown is smaller than the entire actual computational domain. . . 50 3.4 Curvature of iso-contour θ = θ∗ along the flame coordinate s. Upper

and lower panel display curvature profiles respectively for the stable and unstable flames displayed in Fig. 3.3. . . 51

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3.7 Turbulent burning velocity versus δ/δc for two values of turbulence

intensity. Inset: time-averaged concentration iso-contours Y = Y∗_{= 0.9}

for δ = .008, .015, .00675 and u0

0= 1.5. . . 54

3.8 Critical nondimensional flame thickness δc as a function of equivalence

ratio φ for propane/air mixtures, using different dispersion relation models (notation defined in the text) and different exponents γ for temperature dependence of transport coefficients. Dashed curves represent the nondi-mensional flame thickness δ = `D/Lfor two Bunsen diameters: L = 18 mm

(lower dashed curve) and L = 9 mm (upper dashed curve). All pertinent data for propane/air mixtures was taken from Ref. [123] while diffusivities were taken from Ref. [10]. . . 58 3.9 Mie scattering images of C3H8/Air flames at Re = 5000. Upper panels:

Bunsen diameter L = 18 mm. Lower panels: L = 9 mm. Flames a-d and I-IV correspond to φ = 0.8, 1.1, 1.4, 1.5 respectively. . . 60 3.10 Skewness of flame curvature versus δ/δcfor experimental C3H8/Air flames

at Re = 2500 − 7000 using two Bunsen diameters L = 9 and 18 mm. Range of equivalence ratio for smaller diameter is φ ∈ [0.8, 1.7] while for larger diameter φ ∈ [1.1, 1.7]. . . 61 3.11 Turbulent burning velocity versus δ/δc for experimental C3H8/Air flames

at Re = 2500 − 7000 using two Bunsen diameters L = 9 and 18 mm. . . . 62 3.12 Turbulent burning velocity normalized with laminar flame speed versus

δ/δc. Notation is identical to Fig. 3.10 and 3.11. . . 63

3.13 Enhancement (ratio) of turbulent burning velocity between Bunsen diam-eter L = 18 mm flames and L = 9 mm flames. . . 63 4.1 Upper panel: number of unstable wavelengths nc as function of the

expansion ratio σ and flame thickness δ. Lower panel: critical value of flame thickness δc as a function of σ for different Lewis number Le values. 68

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4.2 Skewness of curvature p.d.f. as a function of turbulence intensities u0

for different flame thicknesses: δ = 0.008 (◦), δ = 0.024 (_N), δ = 0.030 (×) and δ = 0.036 (•). Note that u0 = 2.5 is shown only to highlight

qualitative trends and it is not analyzed in detail in the paper. . . 70 4.3 Instantaneous temperature field. Left panel: unstable flame U (δ = 0.008).

Right panel stable flame S (δ = 0.036). . . 71 4.4 Single, laminar DL corrugation for a freely propagating flame of thickness

δ= 0.008. Left panel: normal velocity component and streamlines. Solid

line: progress variable isoline in the fresh zone (cu= 0.1). Right panel:

vorticity field. Solid line: progress variable isoline in the burnt zone (cb= 0.98). . . 72

4.5 Flame induced flow properties along a flame isoline plotted as a function of the flame curvature κ. Left panel: tangential Kτ

aT, normal K

n aT and total components of the strain rate aT on cu= 0.1 iso-line. Right panel:

vorticity ω on cb= 0.98 iso-line. . . 73

4.6 Probability density functions of displacement speed ˆSd. Unstable flames

are displayed using dashed lines (−−) and open symbols, stable flames using continuous lines (—) and filled symbols. Turbulence intensity:

u0_{= 0.5 (}

,), u0= 1.0 (◦,•), u0 _{= 1.5 (}

O,H) and u0= 2.0 (M,N). . . 75

4.7 Probability density functions of displacement speed components. Top panel: unstable flames U (δ = 0.008). Bottom panel: stable flame S (δ = 0.036). Turbulence intensity: u0 = 0.5 (

, ), u0 = 1.0 (◦, •), u0= 1.5 (

O,H) and u0= 2.0 (M,N). . . 76

4.8 Panel (a): mean progress variable field c(x, y) (only half shown for each flame due to symmetry about slot axis) for (left to right) stable flame (δ = 0.036, u0

0= 0.5), unstable flame (δ = 0.008, u00= 0.5), stable flame

(δ = 0.036, u0

0= 1.5), unstable flame (δ = 0.008, u00= 1.5). Bold lines:

c= 0.1, thin lines: instantaneous c = 0.1 at a selected instant (continuous:

stable; dashed: unstable). Panel (b): global consumption speed ST ,GC vs.

turbulence intensity u0

0 for slot flames of variable thickness δ, normalized

by reference value S0

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dashed lines (−−) and open symbols, stable flames using continuous lines (—) and filled symbols. Turbulence intensity: u0 _{= 0.5 (}

,), u0= 1.0 (◦,•), u0 _{= 1.5 (}

O,H) and u0= 2.0 (M,N). . . 80

4.10 Joint p.d.f.’s of strain rate aT and curvature for different turbulence

intensities u0 _{= 0.5 − 2.0 increasing from top to bottom. Left panels:}

unstable flames. Right panels: stable flames. Filled symbols: aT for the

reference laminar DL corrugation. . . 81 4.11 Joint p.d.f.’s of strain rate component Kn

aT and curvature for different turbulence intensities u0 _{= 0.5 − 2.0 increasing from top to bottom. Left}

panels: unstable flames. Right panels: stable flames. Filled symbols: Kn aT for the reference laminar DL corrugation. . . 82 4.12 Joint p.d.f.’s of strain rate component Kτ

aT and curvature for different turbulence intensities u0 = 0.5 − 2.0 increasing from top to bottom. Left

panels: unstable flames. Right panels: stable flames. Filled symbols: Kτ aT for the reference laminar DL corrugation. . . 83 4.13 Statistical analysis of slot flame simulations. Panel (a): p.d.f. of flame

curvature (c = 0.1 iso-line) for stable (continuous line)/unstable (dashed line) flames at varying turbulence intensity (2) u0

0 = 0.5, (◦) 1.0, (+)

1.5, (4) 2.0. Panel (b): Induced flow field in unstable freely propagating laminar flame. . . 85 4.14 Joint p.d.f.’s of normal velocity component vn and curvature for different

turbulence intensities u0 _{= 0.5 − 2.0 increasing from top to bottom. Left}

panels: unstable flames. Right panels: stable flames. . . 86 4.15 Instantaneous vorticity fields with superimposed reference progress variable

isolines cb= 0.98 (dashed line) and cu= 0.1 (continuos line). Left panel:

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4.16 Probability density functions of vorticity conditioned to: cu= 0.1 iso-line

(top panel) and cb = 0.98 iso-line (bottom panel). Unstable flames are

displayed using dashed lines (−−) and open symbols, stable flames using continuous lines (—) and filled symbols. Turbulence intensity: u0 = 0.5

(,), u0 = 1.0 (◦,•), u0_{= 1.5 (}

O,H) and u0= 2.0 (M,N). . . 88

4.17 Instantaneous vorticity magnitude and curvature along the flame coordi-nate s for different turbulence intensities u0= 0.5 − 2.0 increasing from

top to bottom. Unstable flames are displayed in the left panels while the stable flames in right panels. . . 90 4.18 Joint p.d.f.’s of vorticity and curvature for different turbulence intensities

u0= 0.5−2.0 increasing from top to bottom. Unstable flames are displayed

in the left panels while the stable flames in right panels. The black dots represent a single, laminar DL corrugation. . . 91 4.19 Generalized flame surface density, scaled with flame thickness, δΣgen,

conditioned to mean progress variable. Continuous lines: stable flame

δ = 0.36 > δc. Dashed lines: unstable flame δ = 0.08 < δc. Symbols

indicate different turbulence intensities: (2) u0

0 = 0.5, (◦) 1.0, (+) 1.5,

(4) 2.0. Bold continuous line, δ|∇c| conditioned to instantaneous progress variable c (laminar flamelet structure). . . 93 4.20 hρSdi_s/ρuSL0 term (stretch factor I0) conditioned to mean progress

vari-able. Continuous lines: stable flame δ = 0.36 > δc. Dashed lines: unstable

flame δ = 0.08 < δc. Symbols indicate different turbulence intensities: (2)

u0

0= 0.5, (◦) 1.0, (+) 1.5, (4) 2.0. . . 94

4.21 Statistical analysis for an experimental propane-air Bunsen flame at Reynolds number Re = 1500, diameter L = 14 mm and two different equivalence ratios, φ = 1.5 corresponding to a hydrodynamically unstable flame (left panels) and φ = 0.8 corresponding to a stable flame (right panels). Panel (a): joint-pdf of flame strain KS and flame front curvature

κ (left: unstable flame; right: stable). Inset: p.d.f. of flame curvature

for unstable flame (dashed line) and stable flame (bold continuous line). Panel (b): joint-pdf of normal-to-front velocity component un and flame

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5.1 Characteristic cutoff wavelength λc as a function of equivalence ratio φ

scaled in terms of `D for various fuels. . . 101

5.2 Representative length scales involved in a (A) high-Ka and (B) large scale turbulent premixed flame simulation. . . 102 5.3 Comparison between analytical (solid line) and numerical (circles

simula-tions, dash-dotted line parabolic fit) dispersion relation. . . 104 5.4 Overview of flames morphology and computational domains, from left to

right: small/large scale slot Bunsens flame (SB/LB) and small/large scale statistically planar flame (SP/LP). . . 107 5.5 (a) P.d.f. of the mean curvature KM and (b) p.d.f. of the shape factor

SF, both variable are conditioned to c∗. . . 108

5.6 Joint p.d.f.’s of mean KM and Gaussian KG curvatures of SB flame (left

panel) and LB flame (right panel). . . 109 5.7 Probability density functions of displacement speed ˆSd scaled by the

laminar unstretched flame speed S0

L. . . 110

5.8 Probability density functions of displacement speed components scaled by the laminar unstretched flame speed S0

L. . . 110

5.9 Probability density functions of tangential strain rate (left panel) and total stretch (right panel). . . 112 5.10 Joint probability density functions of tangential strain rate and the mean

curvature for the SB flame (left panel) and LB flame (right panel). . . 112 5.11 Top panel: c fields for SB (left) and LB (right) cases, solid black lines

indicate the c∗ _{isoline used for S}

T ,GC evaluation. Bottom panel: time

evolution of turbulent flame speed ST/SL0: LP ( ), SP ( ), and a 2D

version of LP (–), the corresponding dashed lines represent hST ,GCi. The

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5.12 Conditional averages of FSD (upper panel) scaled with the thermal thickness and wrinkling factor (lower panel): LB (), SB (), LP ( ) and

SP ( ). . . 115

5.13 Spatial distribution of Σ for SB (left panel) and LB (right panel) with superimposed four slices (z = 0.5L) of instantaneous realizations of the flame front identified by c = c∗. For clarity only half of the domain is

shown in the x direction. . . 116 6.1 Top panel: temporal evolution of the iso-contour of temperature

corre-sponding to the peak reaction rate for an unstable flame at unburned temperature Tu = 300K, equivalence ratioφ = 0.7, and pressure p = 8

atm, initially perturbed with a wavelength λ equal to 12 thermal flame thickness. The axis are scaled with thermal flame thickness lT . Bottom

panel: time evolution of the amplitude for a stable (Tu= 300K, φ = 0.7,

and p = 1 atm, blue) and an unstable flame (Tu= 300K, φ = 0.7, and

p= 8 atm, blue), both initially perturbed with a wavelength λ = 12lT .

All the results are obtained with the multi-step chemistry. . . 123 6.2 Growth rate of the perturbation ω(k) for different wavelength λ = 1/k

(dispersion relation) for two flames at different pressure and equivalence ratio φ = 0.7 obtained with multi-step chemistry DNS and unity Lewis numbers: p = 1 atm (red circles) and p = 8 atm (blue triangles). The lines are parabolic fits to the DNS data. The results are shown in dimensional form (top panel) and non- dimensionalized with the thermal flame thickness and flame time (bottom panel). . . 125 6.3 Comparison of the pressure dependency of the thermal (triangles), diffusive

(diamonds), and reaction-layer (circle) flame thickness with the cut-off length scale of the hydrodynamics instability in linear (a) and logarithmic (b) scale. In (b) the results are normalized with the values at p = 1 atm; the dashed lines indicate different power-law pressure scaling. (c) Pressure dependency of the Zeldovich number (pentagons), expansion ratio σ (open circles), and laminar flame speed SL (open triangles). All results are

shown for the methane-air flame at Tu = 300K, φ = 0.7, unity Lewis

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corresponding to a methane air flame at Tu = 300K, φ = 0.7 (black

pentagons). The results are scaled with the thermal (a) and diffusive thickness (b). (c) Ratio of the thermal and diffusive thicknesses for the same cases. . . 128 6.5 Top panel: normalized heat release rate HRR/HRRmax as function of the

normalized temperature θ = (T − Tu)/(Tb− Tu) for the ,multi- step model

and in bottom panel normalized reaction rate for the one-step model as function of the temperature and deficient reactant mass fraction. The condition are those of a methane air flame with Tu = 300K, φ = 0.7,

Lei= 1 and the results are shown for different pressures. . . 129

6.6 Visualization of the temperature field in a stable and an unstable slot-burner flame. . . 131

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caution”

— Wernher Von Braun

1

Introduction

The conversion of chemical energy into heat and ultimately into mechanical or propulsive energy is achieved in a multitude of devices through combustion of fuels in air. Combustion may be described as the complex interplay of chemical as well as diffusive and convective transport processes. Being the convective fields turbulent in nature in most devices of interest (other than in controlled experimental setups), such processes are referred to as turbulent combustion. The premixedness of reactants or their initial

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conditions used in practical devices ranging from power generation burners to aeronautical or rocket combustion chambers. Computational Fluid Dynamics (CFD) is a fundamental tool in research due to its extreme flexibility and the availability of a wide range of existing models and the ever increasing computational resources with respect to the experiments which are generally more expensive. The numerical simulation of a compressible reactive flow, occurring in such devices at typical operating conditions, is still an open problem in the scientific community. The main reason lies in the fact that diffusion, advection, acoustics and chemical reactions introduce a broad range of spatial and temporal scales. Direct numerical simulations (DNS) of these multi-scale problems are presently feasible only for a range of prototypical flame configurations. This approach is a useful tool for fundamental research since it can be considered a numerical experiment [63, 96] from which it is possible to extract information difficult or impossible to obtain experimentally. Thus DNS simulations are useful in the development of turbulent combustion models for practical applications, such as sub-grid scale (SGS) models for large eddy simulation (LES) and simplified models for methods that solve Reynolds averaged Navier-Stokes

equations (RANS) useful at system design level.

In the context of multi-scale phenomena a relatively unexplored subject is the interplay of the, so called, intrinsic premixed flame instability with turbulence in large scale turbulent premixed combustion. Combustion processes are characterized by considerable variations in temperature and density due to the large amount of heat released by chemical reactions. This significant heat release inevitably leads to gas expansion, inducing velocities that affect the flame propagation itself. Thermo-diffusive and hydrodynamic effects all participate in combustor and flame instability. Hydrodynamic effects are due to the Darrieus-Landau (DL) instability, where the large jump in density induces velocity changes and deviation of streamlines which ultimately amplify perturbations, thereby wrinkling the flame. The DL instability is itself coupled with thermal-diffusive effects,

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which can be stabilizing or destabilizing. For Lewis numbers smaller than unity, thermal-diffusive effects can cause flame destabilization, causing a small scale cellular structure with localized quenching. The subject of instability in premixed combustion has generated an extensive and diversified literature, which will be discussed in the following sections.

Additional hydrodynamic instability mechanisms exist, such as the Rayleigh-Taylor (R-T) [46, 66] instability and Saffman-Taylor (S-T) [64, 65]. The former (R-T) is caused by a preferential acceleration - under the action of the acoustic pressure gradient - of the lower density burnt gas into the higher density unburnt gas. The second one (S-T) is well-known like a classical viscous fingering problem, caused by the viscous effect in the fluid between two parallel plates. The R-T instability often develops into a thermo-acoustic instability, where the perturbations to the flame due to R-T effects create acoustic waves which resonate in the acoustic duct, and in turn affect the flame.

More generally, the broad subject of instability in fluid flow, which encompasses flame instability and comprises Darrieus-Landau, thermal-diffusive as well as Rayleigh-Taylor, Richtmyer-Meshkov, Kelvin-Helmholtz instabilities, has captured significant scientific interest and is of practical relevance in diverse areas of science [131].

The objective of this work, however, is focalized exclusively on the interplay of the intrinsic hydrodynamic flame instability with an incident turbulent field, aiming to characterize all of the main features of turbulent premixed flame propagation and the operative conditions which would favor the onset of instabilities. In particular, a clear identification of the regime where the turbulence dominates over instability effects and vice versa, still needs to be assessed, and to this end, systematic DNS data will be gathered.

1.1 Structure of laminar premixed flames

The simplest description of a flame front is a model where one considers two fluid domains separated by a discontinuity surface. All transport processes inside the flame front are taken into account by means of jump conditions. In a flame stationary frame (Fig. 1.1) the fresh mixture approaches the flame with velocity uu= s0uand temperature Tu, and

leaves the flame with velocity u0

b and temperature T 0

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can be used, with Yu being the concentration of the fresh mixture and Yb0 = 0

indicating its complete consumption upon crossing the flame. Furthermore, since s0 u is

much smaller than the speed of sound, compressibility effects are negligible, and the combustion process can be assumed to be isobaric.

The flame structure can be considered at three levels of detail. First of all, concerning the flame-sheet or hydrodynamic level, fresh and burnt states are related by the Rankine–Hugoniot relations (Fig. 1.1 (a)), where the flame is simply an interface separating two fluid dynamical states. At this flame sheet level, the temperature and reactant concentration change discontinuously from Tu to Tb0, and from Yuto Yb0= 0,

respectively. At the next, more detailed, transport dominated level, the flame sheet of Fig. 1.1 (a) is expanded to reveal a preheat zone, having characteristic thickness lD and

governed by heat and mass diffusion processes, as shown in Fig. 1.1 (b). Here, as the mixture approaches the flame, it is gradually heated up by the heat released by chemical reactions, which is advected upstream. Temperature profile increases continuously until

T0 is reached and the profile is nonlinear due to the presence of convective transport,

acting concurrently to diffusion. The third and final level of detail includes a thin reaction zone (defined as Ti ≤ T ≤ T ≤ Tb0) in which the reaction rate is non-negligible.

Other multidimensional configurations can be conceived such as that depicted in Fig. 1.3, which is applicable to a steady Bunsen burner flame. The laminar burning velocity, SL,u, defined as the normal flame front velocity with respect to the unburned

mixture, can be therefore measured as

SLu= Vn= V sinα (1.2)

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1. Introduction26 CHAPTER 2. PREMIXED FLAMES 5

0521870526c07 CUFX045/Law Printer: cupusbw 0 521 87052 6 July 19, 2006 17:12

242 Laminar Premixed Flames

T_u Y_u T_b Y = 0_b Reaction rate Inner, reaction region (in) Outer, downstream, equilibrium region (+) Outer, upstream, transport region (-) D R (c) (b) T_u Y_u T_b Y = 0_b Reaction sheet D Y_u T_u T_b Y = 0_b Flame sheet (a) = u_u s_u = u_u s_u = u_u s_u x x x l l l o o o o o o o o o o o o o o o

Figure 7.2.1. Schematic showing the premixed flame structure at successive levels of detail: (a) The hydrodynamic, flame-sheet level; (b) the transport, reaction-sheet level; and (c) detailed structure including the reaction zone.

deficient reactant, then a one-reactant reaction with

Reactant → Products (7.2.1) can be used, with Yubeing the concentration of the fresh mixture and Ybo≡ 0

indi-cating its complete consumption upon crossing the flame. Furthermore, since so uis

much smaller than the speed of sound, the combustion process can be assumed to be isobaric in accordance with the previous discussion in Section 5.2.4 on the properties of low-speed, subsonic flows, and in the previous section on the small pressure change across a weak deflagration wave.

Figure 2.1: Schematic showing the premixed flame structure at successive levels of detail: (a) The hydrodynamic, flame-sheet level; (b) the transport, reaction-sheet level; (c) detailed structure including the reaction zone

Figure 1.1: Schematic showing the premixed flame structure at successive levels of detail: (a)

the hydrodynamic flame sheet; (b) the transport reaction sheet; (c) detailed structure including the reaction zone [81].

SLu= v∗·n − Vf (1.3)

where v∗ is the local gas velocity just ahead of the front, n is the flame normal and

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Figure 1.2: Perturbed flame front [36].

Consider a single-valued perturbed flame front (i.e with no folds or pockets), Fig. 1.2. A F iso-surface (iso-contour in two dimensions) is selected to represent the front. It can be expressed as the zero-level set of the scalar function:

F(x, t) = F (x, y, t) = 0 (1.4)

in the Fig.1.2 the zone F < 0 represents the unburned gas and F > 0 the burnt gas re-gion. Thus: dF dt = ∂F ∂t + ∇F · dx dt = 0 (1.5) where dx

dt = Vf is the local absolute flame propagation speed. Defining the normal

absolute propagation speed Vf = n · Vf we write the flame speed as:

Sf = v∗·n − Vf (1.6)

where the front normal is defined as n = ∇F/|∇F | thus

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yielding the following level set equation for F (x, t) and thus the flame front:

∂F ∂t + v

∗_{· ∇F} _{= S}

f|∇F | (1.8)

this equation describes a surface propagating in space with a velocity Sf (normal to

itself) relative to the local unburned flow velocity v∗_{. The velocity S}

f considers the

fact that the laminar speed SL is not always equivalent to the unstretched laminar

flame speed, in this expression the effect of the local curvature and strain should be considered [87]. The combined effects are often referred as flame stretch, K , defined by the fractional rate of change of a flame surface element A [125]:

K = 1 A

dA

dt (1.9)

Example: Bunsen configuration

Now we consider the Bunsen configuration (curved flame) like an example for the level set equation eq. 1.8.

Equation 1.8 at steady state yields:          vu· ∇F = Sf|∇F | ∇F = (1, −fy)T vu(y) = SLq1 + fy2 (1.10) As a first approximation, we suppose that Sf = SL, where SL is a constant flame

speed and that the flow is uniform vu(y) = V → V2 = SL2(1 + fy)2. The condition

for the Bunsen configuration develops the following form:        fy= ±  V2_{− S}2 L S2 L 1/2 f = 0 at y = ±a 2 (1.11) f = (V2_{− S}2 L) S2 L 1/2_a 2− |y| (1.12)

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y

a/2 a/2

V

L

Figure 1.3: Bunsen flame configuration: solid line L = 0; dashed lines display the effect of

positive Markstein length.

Where a is the Bunsen diameter, V is the unburnt speed, SL is the laminar flame

speed and H is the Bunsen height.

When y = 0 the solution fails because Sf = SL is unrealistic as curvature effects

cannot be neglected. We can introduce the Markstein correction [87], who incorporated a dependence of the flame speed on curvature:

Sf = SL− Lκ (1.13)

where κ = −∇ · n is the flame curvature and L is the Markstein length, a coefficient on the order of flame thickness, which mimics the effects of diffusion and chemical reactions occurring inside the flame zone.

n = ∇F |∇F | = (1, −fy) T 1 q1 + f2 y (1.14) κ= − ∂n ∂x + ∂n ∂y = fyy (1 + f2 y)3/2 (1.15) The normal is defined positive towards unburnt mixture. We now consider a new form for flame velocity Sf introducing a corrective term: the Markstein length (Eq. 1.13). This

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               vu(y) = V = SL− L fyy (1 + f2 y)3/2 ! (1 + f2 y) 1/2 V(1 + f2 y) = SL(1 + fy2) 3/2_{− Lf} yy f = 0 at y= ±a 2 f0(0) = 0 (1.16)

Figure 1.3 displays the flame profiles under the effect of positive Markstein lengths. The level-set analysis described in this section views the flame as a gasdynamic interface, similarly to the hydrodynamic flame models [91, 94], used to study the linear stability of premixed flame. The hydrodynamic asymptotic model is based on a multi-scale analysis that exploits the disparity between the hydrodynamic length scale L, which is a measure of the characteristic size of the flame, such as the domain, and the diffusion length scale

lD = Dth/SL, which represents the diffusive flame thickness.

1.2 Linear stability analysis: hydrodynamic theory

The stability of the planar flame under small perturbation has been studied, among others, by Pelce & Clavin [100], Frankel & Sivashinsky [53] and Matalon and Matkowsky [91]. The latter study yields the hydrodynamic flame model, where the flame is a gasdynamic discontinuity separating burnt and unburnt gases (see Fig. 1.2), a general form of the dispersion relation is written in Eq. 1.17 [36]. The dispersion relation describes the the growth rate of a disturbance of wavenumber k.

(σ + 1)ω2_{+ [(1 + kL)2kσ]ω − k}2_σ_{(σ − 1 − 2kLσ) = 0} _(1.17)

where σ = ρu/ρb is the expansion ratio, and L the Markstein length. Expanding

the dispersion relation expression is series of k and considered L = 0, we assume that Sf = SL = 1 in accordance with the original Darrieus-Landau hypothesis [40,

77], ω assume the following form:    ω= ω0k ω0= −σ+√σ3+ σ2_{− σ} σ+ 1 (1.18)

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ω(k) = ω0SLk − ASL(L − Lc)k + o(k ) (1.19)

where the first term describes the hydrodynamic o DL mechanism while the second term depicts the diffusive effects, A is function of σ. The Lc separates stabilizing (L > Lc)

from destabilizing (L < Lc) thermal diffusive effects.

The growth rate of the disturbance with respect to the wave number is shown in Fig. 1.4. −0.5 0 0.5 1 1.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 ω (k) [1/s] k [1/m]

Figure 1.4: Dispersion relation: blue line is the DL hypothesis, where all wave numbers are

unstable; purple line is the model that includes the curvature and strain effects.

The dispersion relation in eq. 1.19 provides an expression for the critical wavenumber

kc or, equivalently, the critical wavelength λc = 2π/kc obtained for ω = 0.

The critical wave number separates the unstable from the stable wave numbers regions, so that any flame unconstrained by a lateral domain smaller than λc will effectively

inhibit instabilities. Figure 1.4 shows a stabilizing effect of the Markstein correction (purple line) with respect to the Darrieus-Landau model (blue line).

An example of the nonlinear evolution of hydrodynamic instability is shown in Fig. 1.5. This shows a steadily propagating solution of a direct numerical simulation of an initially perturbed planar flame using a single step chemistry [75] model.

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1. Introduction 11 Flow Turbulence Combust

Fig. 4 Single, laminar DL corrugation for a freely propagating flame of thickness δ = 0.008. Left panel:

normal velocity component and streamlines. Solid line: progress variable isoline in the fresh zone (cu= 0.1). Right panel: vorticity field. Solid line: progress variable isoline in the burnt zone (cb= 0.98)

The right panel of Fig.5shows the vorticity, conditioned to an iso-c chosen toward the burnt gases (cb= 0.98), as a function of flame curvature. Note that curvature values are less

negative for such a choice of isoline on the burnt side of the flame. The diagram highlights that for negative curvature (cusp-tip region), vorticity spans a wide range of values, from the largely negative/positive of the left/right side of the cusp-tip to the ω_{∼ 0 values of the} horizontal cusp-tip. A narrower range of ω values are experienced in the positive curvature region which represents the wide troughs of the laminar DL corrugation.

Fig. 5 Flame induced flow properties along a flame isoline plotted as a function of the flame curvature κ.

Left panel: tangential Kτ

aT, normal K

n

aT and total components of the strain rate aT on cu= 0.1 iso-line. Right

panel: vorticity ω on cb= 0.98 iso-line

Figure 1.5: Numerical simulation of perturbed 2D planar flame: Darrieus-Landau mechanism

(DL) [75].

The most complete formulation of the dispersion relation was obtained by [90]:

ωk= SLkω0− SLlD[B1+ β(Leef f−1)B2+ P rB3]k2 (1.20)

Here P r is the Prandtl number and the coefficients B1, B2and B3 depend only on

the expansion ratio σ. The three terms in the square bracket correspond to thermal, molecular, and viscous diffusion, respectively. Thermal diffusion term has always a stabilizing influence. The viscous term (P rB3) has also a stabilizing effect. On the

other hand, the effect of molecular diffusion depends on the mixture composition, or the effective Lewis number of the mixture. Therefore the second term can induce stabilizing or destabilizing effects. In order to stabilize the short wavelength disturbances Leef f

must be grater than a critical value Le∗_{, which depends on the particular mixture. For}

Leef f < Le∗, according to this formulation, the flame appears unconditionally unstable

and in order to stabilize short wavelength disturbances the linear stability analysis should be carried to higher order. A formulation of this kind, with higher order stabilizing diffusive effects, however, has not been derived. This was only explored for weak thermal expansion by Sivashinsky [114] giving rise to a dispersion relation of the form:

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1.2.1 Hydrodynamic instability

Historically, Darrieus [40] in 1938 first recognized that the gas expansion produced by heat release in a wrinkled premixed flame, considered as a thin interface propagating towards the fresh gas with the constant laminar speed SL, will deviate flow lines across

the front away from the normal to the flame. This would always accelerate (decelerate) the flow ahead of concave (convex) flame portions, augmenting any flame perturbation. This instability mechanism leading to cellular flame development is termed hydrodynamic instability or Darrieus-Landau instability [40, 77] and is illustrated in Fig. 1.6. Taking a closer look, if a flame is locally perturbed (red solid line), Fig. 1.6, forming convex and adjacent concave sections, flow behind the flame is deflected due to expansion across the flame. Ahead of the concave section streamlines converge, whereas they diverge ahead the convex section. This accelerates and decelerates the flame locally in the concave and convex sections, respectively, and thus amplifies flame wrinkling (red dashed line). This unconditional hydrodynamic instability was predicted independently in 1944 by Landau [77]. This result was published 65 years ago against the fact that stable planar laminar flames had been observed in the laboratory since the work of Mallard [37]. The Darrieus-Landau (DL) theory assumes that the flame is infinitesimally thin and propagates normal to itself at a constant speed SL relative to the unburnt gas,

unaffected by hydrodynamic disturbances. The study of Markstein [87], later assumed a finite thickness and a dependence of the flame speed on the local curvature of the front through a phenomenological constant known as the Markstein length. The more rigorous asymptotic treatment of Clavin e Williams [31] and Matalon e Matkowsky [91] exploits the multi-scale nature of the problem characterized by two different length scales: the diffusion length lD representing the flame thickness, where lD= Dth/SL with Dth

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example, with the average size of the wrinkles on the flame front or with the geometrical

dimensions of the combustion vessel. Physics and Chemical Kinetics of H2–Air Explosions in Tubes

Reactants Products

S_L S σ_L

Figure 2.8: Illustration of hydromechanic (Landau-Darrieus) instability.

ignition, an initial decrease in cellular lengthscale, often termed wavelength, is typical. It is followed by a state of quasi-stationary topology, where cells grow and refine dynamically as described by Bradley et al. [10, 11]. This distortion of the flame front is known to enhance the overall reaction rate and thus support FA [20].

In lean mixtures (15 and 20 vol. %) separated flame islands with quenching in intermediate cracks are observed in Fig. 2.7. At 25 vol. %, no local quench-ing is observed anymore. With risquench-ing H2concentration the wavelength of the

cellularity increases and flame fronts becomed more stable.7

An instability mechanism leading to cellular flame development is hydrody-namic instability, also termed Landau-Darrieus instability [26, 88, 114]. It is illustrated in Fig. 2.8. If a flame is locally perturbed (red solid line), left part of Fig. 2.8, forming convex and adjacent concave sections, flow behind the flame is deflected due to expansion across the flame as depicted in the right part of Fig. 2.8. Behind the convex section streamlines converge, whereas they di-verge behind the concave section. This accelerates and decelerates the flame locally in the convex and concave sections, respectively, and thus amplifies flame wrinkling (red dashed line).

Additionally, diffusive instability [96] needs to be taken into account. It interacts with the hydrodynamic instability, either supporting or damping flame wrinkling. If the diffusivities of the limiting component of a mixture

7_{In addition, these images show the influence of buoyancy at low H}

2concentration. Lean flames are oriented

towards the top of the channel, while stoichiometric and rich mixtures cause rather symmetric flames with re-spect to the channel centerline.

26

Figure 1.6: Illustration of hydrodynamic (Landau-Darrieus) instability [17]

1.2.2 Thermal diffusive instability

Thermal diffusive instabilities are the consequence of the disparity between thermal conductivity of mixture and molecular diffusivity of the controlling reactant [81] . This can cause local changes in mixture composition and reaction rate and subsequently in flame speed, ultimately causing flame front wrinkling. Since the instability is caused by the active modification of the diffusional structure of the flame, the cell size is expected to be order of the flame thickness [21, 30, 92, 115].Physics and Chemical Kinetics of H2–Air Explosions in Tubes

heat species Reactants Products Reactants Products Le < 1 Ma < 0 (unstable) Le > 1 Ma > 0 (stable) heat species

Figure 2.10: Illustration of diffusive-thermal instability, adapted from [20].

of thermal diffusivity a and diffusion coefficient D of the limiting species in

the mixture forms the Lewis number Le

9

_:

Le =

a

D

.

(2.30)

Lewis numbers smaller than about unity enhance flame wrinkling whereas

Lewis numbers larger than unity damp it. Experimental values for the Lewis

number in H

2

–air are given in Fig. 2.11. It can be seen that transition from

stabilizing to destabilizing occurs close to stoichiometry.

Since most other flammable gas mixtures have a Lewis number close to or

larger than unity, H

2

takes a special position with the highest propensity for

cellular flame development due to the high diffusivity of H

2

.

Cellular flame propagation has been investigated by Markstein [102]. He

de-fines the Markstein length L

M

, describing the effect of flame stretch rate on

local burning velocity. The unstretched burning velocity is further on termed

S

L

, whereas the local burning velocity of the stretched flame is S

L,S

. Employing

the Karlovitz stretch factor K [71],

K =

_A

1

F

dA

F

dt

,

(2.31)

describing the normalized rate of flame surface area change, yields the

rela-tion of flame stretch rate and burning velocity of the stretched flame

depend-9_{The following discussion neglects multicomponent diffusion, which would lead to the formulation of a}

sep-arate Lewis number for each species. The Lewis number introduced here is an effective Lewis number for the entire mixture.

28

Figure 1.7: Illustration of diffusive-thermal instability, adapted from [27]

The ratio of thermal diffusivity a and diffusion coefficient D of the limiting species in the mixture forms the Lewis number Le. For Lewis number grater than one the flame

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that a flame is diffusively unstable if it is deficient in the more mobile reactant [81]. If the diffusivities of the limiting and excess reactant differ (such as in lean hydrogen-air mixtures), then such reactants will diffuse differently into a curved flame. In particular, in the presence of preferential diffusion of the limiting reactant, this will tend to diffuse more effectively into convex flame portions, thus increasing its concentration there. In concave regions, on the other hand, one could even experience local extinction due to starvation of limiting reactant.

What occurs to local flame speed further depends on thermal diffusivity. If this is low compared to the diffusivity of the limiting species (Le < 1), then in convex regions, where the concentration of the limiting reactant is higher, the heat flux lost outwards towards the fresh gases is low and therefore the temperature in the preheat region tends to increase, thus increasing reaction rate and flame speed. This situation is illustrated in the left panel of Fig. 1.7 where convex flame portions are faster and flame wrinkling is promoted. If on the other hand Le > 1, the increased outward heat flux reduces local temperature and compensates the higher limiting reactant concentration and the flame speed is reproduced in convex regions, this decreasing the flame speed. This situation is illustrated in the right panel of Fig. 1.7 where convex flame portions are slower and flame wrinkling is inhibited.

1.3 Turbulent premixed flames and their interaction

with instability

We have seen how intrinsic instability mechanisms are responsible for the onset of cellular flames in a laminar flow. In turbulent flames eddy/flame interaction could suppress these instabilities; however, there is no consensus regarding the conditions at which eddy/flame interaction dictates, as a whole, the propagation of turbulent premixed flames. Considering a laminar flame where the propagation speed depends on the

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thermal and chemical properties of the mixture, a turbulent flame has a propagation velocity that depends in addition on the character of the flow (it depends on the turbulence characteristics).

Damkohler [39] introduced the concept of large (wrinkled flamelet) and small scale (corrugated flamelets) turbulence regimes for premixed combustion. For large scale turbulence, Damkohler assumed that the interaction between a wrinkled flame front and the turbulent flow field is purely kinematic, so that by continuity arguments [101]:

˙m = ρuSLAT = ρuSTA (1.22)

where ˙m is the mass flux of a mixture of density ρu, SL is the local laminar flame

speed of the instantaneous (wrinkled) flame area AT and ST is a global turbulent flame

speed of a front of cross sectional area A normal to the mass flux [101]. It follows that:

ST

SL

=AT

A . (1.23)

Damkohler [39] related the area increase of the wrinkled flame surface area to the velocity fluctuation divided by the laminar burning velocity, AT/A ∼ u0o/SL which leads

to ST ∼ u0o referred to as kinematic scaling.

When the turbulence scale is smaller than the flame thickness, Damköhler argued that turbulence only modifies the transport between the reaction zone and the unburnt gas. The turbulent eddies simply modify the transport process between the reaction sheet and the unburned gas. In analogy to the scaling relation for the laminar burning velocity, substituting the laminar diffusivity with the turbulent diffusivity [101]:

ST

SL

=r DT

D (1.24)

Where in Eq.1.24 SL is the laminar burning velocity and DT and D are the turbulent

and molecular diffusivities. Since the turbulent diffusivity is proportional to the product

u0

ol0 , where u00 is the turbulence intensity, l0is the characteristic length of turbulence

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The question arises as to whether the intrinsic instabilities of premixed flames, briefly introduced earlier, may interfere with this classical description of turbulent premixed propagation. While this will constitute the basic question this thesis work will try to address, we can briefly address some recent attempts. In a recent work by Yang [128] the role of Darrieus–Landau instability in the propagation of expanding turbulent flames is studied. They show the important role of hydrodynamic instability on the propagation of expanding turbulent flames for various regimes of turbulent propagation. They find three distinct regimes based on the relative role of DL instability cells and turbulence on flame propagation: an instability dominated regime, an instability–turbulence interaction regime and a turbulence dominated regime. Such regimes were shown to coincide with specific regions of the Borghi diagram, as shown in Fig. 1.8. A less recent theoretical work by Chaudhuri [26] also conjectured an instability dominated regime on the Borghi diagram identified by the red boundary in Fig. 1.8 which demarcates strong turbulence dominated regime and DL instability affected regime. In this study a parameter Γ is introduced representing the ratio of turbulent time scales to Darrieus-Landau growth rates so that when Γ < 1 the instability mechanism will dominate over the turbulence effects.

1.4 State of the art

The role of intrinsic instabilities was investigated experimentally using various config-urations namely: Bunsen burners [72, 73, 120], spherical expanding flames [113, 128] and planar configurations [29, 112] . From the numerical standpoint, Darrieus-Landau instability has been studied by means of different methods: weakly non-linear models [88, 109, 116] and hybrid-level set [50, 51] . More recently, DNS have also been employed using a one-step deficient reactant chemistry model, in both two-dimensional [129] and three-dimensional settings [80]. A one-step chemistry model greatly simplifies

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800 S. Yang, A. Saha, Z. Liu and C. K. Law

1 10 100 1000 10 000 Corrugated flamelets (instability−turbulence interaction) Thickened flamelets (turbulence dominated) Broken/distributed reaction zone Da = 1 Ka = 1 ˝ = 1 Ka = 100 Laminar

flames _{(instability dominated)}Wrinkled flamelets

LI/∂L0

Urms /SL0,u

1 10 100 1000 10000

FIGURE 11. (Colour online) Experimental conditions for which the flame propagation is

influenced by both turbulence and DL instability. The conditions are plotted with respect to regimes defined by Chaudhuri et al. (2011) and present study.

REFERENCES

ABDEL-GAYED, R. G., BRADLEY, D. & LAWES, M. 1987 Turbulent burning velocities: a general correlation in terms of straining rates. Proc. R. Soc. Lond. A 414 (1847), 389–413.

ADDABBO, R., BECHTOLD, J. K. & MATALON, M. 2002 Wrinkling of spherically expanding flames.

Proc. Combust. Inst. 29 (2), 1527–1535.

AKKERMAN, V. & BYCHKOV, V. 2005 Velocity of weakly turbulent flames of finite thickness. Combust.

Theor. Model. 9 (2), 323–351.

AKKERMAN, V., BYCHKOV, V. & ERIKSSON, L.-E. 2007 Numerical study of turbulent flame velocity.

Combust. Flame 151 (3), 452–471.

BAUWENS, C. R., BERGTHORSON, J. M. & DOROFEEV, S. B. 2017 On the interaction of the Darrieus–Landau instability with weak initial turbulence. Proc. Combust. Inst. 36 (2), 2815–2822.

BELL, J. B., DAY, M. S., SHEPHERD, I. G., JOHNSON, M. R., CHENG, R. K., GRCAR, J. F., BECKNER, V. E. & LIJEWSKI, M. J. 2005 Numerical simulation of a laboratory-scale turbulent v-flame. Proc. Natl Acad. Sci. USA 102 (29), 10006–10011.

BOUGHANEM, H. & TROUVÉ, A. 1998 The domain of influence of flame instabilities in turbulent

premixed combustion. Symp. (Int) Combust. 27 (1), 971–978.

BRADLEY, D. 1992 How fast can we burn? Proc. Combust. Inst. 24 (1), 247–262.

BRADLEY, D., LAWES, M., LIU, K. & MANSOUR, M. S. 2013 Measurements and correlations of

turbulent burning velocities over wide ranges of fuels and elevated pressures. Proc. Combust. Inst. 34 (1), 1519–1526.

BRAY, K. N. C. 1990 Studies of the turbulent burning velocity. Proc. R. Soc. Lond. A 431 (1882), 315–335.

BURKE, M. P., CHEN, Z., JU, Y. & DRYER, F. L. 2009 Effect of cylindrical confinement on the determination of laminar flame speeds using outwardly propagating flames. Combust. Flame 156 (4), 771–779.

BYCHKOV, V. 2003 Importance of the darrieus-landau instability for strongly corrugated turbulent

flames. Phys. Rev. E 68 (6), 066304.

CHAUDHURI, S., AKKERMAN, V. & LAW, C. K. 2011 Spectral formulation of turbulent flame speed

with consideration of hydrodynamic instability. Phys. Rev. E 84 (2), 026322.

CHAUDHURI, S., SAHA, A. & LAW, C. K. 2015 On flame–turbulence interaction in constant-pressure

expanding flames. Proc. Combust. Inst. 35 (2), 1331–1339.

9CC5:87' /25657 9CC423:586846 /:14:66560644:26C 16,2C D364CCC96.23:586.6C67D6 22:2362C 9CC423:586846C6

Figure 1.8: Experimental conditions for which the flame propagation is influenced by both

turbulence and DL instability. The conditions are plotted with respect to regimes defined by Chaudhuri [26] and study by [128].

the reacting part of the governing equations by considering a highly diluted reactant, completely consumed across the flame. Using such a model allows relying on results from asymptotic theory. Others numerical simulations using single-step chemistry and detailed transport are used to study premixed hydrogen/air flames in two-dimensional channel-like domains by Altantzis et al. [4]. In this study both unity Lewis number, where only hydrodynamic instability (DL) appears, and sub-unity Lewis number, where the flame propagation is strongly affected by the combined effect of hydrodynamic and thermo-diffusive instabilities are analyzed. Frouzakis et al. in [54] examine the nonlinear development of hydrodynamically unstable flames using detailed (hydrogen air) numerical simulations with different equivalence ratio.

1.4.1 Flame instabilities in experiments

A considerable number of experiments revealed DL instabilities in a variety of premixed flame configurations in the form of flame surface corrugation [37, 58, 102]. In particular, the laminar propagation of spherical flames revealed self-acceleration due to the onset of a fractal conformation, brought about by DL instability as the flame expands [57, 126]. Clanet and Seaby performed experiments [37] showing a planar downward propagating

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Kobayashy experiments, first reveled the potential impact of DL instability in turbulent premixed flame propagation, where the pressure was used to induce or suppress the DL instabilities. Figure 1.11 shows experiments of propane air Bunsen flames using particle image velocimetry diagnostics by Troiani et al. [121]. In the latter experiments hydrodynamic instability (DL) are suppressed or induced modulating the equivalence ratio of the mixture. Wu et al. [127] presents an experimental investigation on self-acceleration of cellular spherical flames at high pressure and different equivalence ratio, as shown in Figure 1.12. In this case, the equivalence ratio of the mixture is also used to induce or suppress the hydrodynamic instabilities.

the two regions of instability overlapped and it was not possible to obtain a flat laminar flame by this method of stabilization. In order to control the wavelength and the spatial phase of the cellular structures that developed when the acoustic stabilization was removed, the upstream gas flow was perturbed by placing an array of parallel wires, 2 mm in diameter, on the downstream face of the honey-comb. The object of this scheme was to excite purely 2D cells at chosen wavelengths. The spacing between the wires was chosen to be an integral divisor of the tube di-ameter (i.e., 10yn cm). However, it was found that, if the spacing of the wires was not sufficiently closes¯630%d to the naturally most unstable wavelength, cells with this latter wavelength appeared, either compounded with the forced wavelength and/or in the direction perpendicular to the forcing. Because of these two limitations, it was not possible to measure the growth rate at a fixed flame speed over a significant range of wavelengths. However, it was

FIG. 5. Decay of acoustic level in tube: (a) Natural decay; ( b) forced decay. Upper traces: Input to loudspeaker; lower traces: Acoustic pressure in tube.

growth of the 2D Darrieus-Landau instability. The apparent thickening of the flame, particularly at high cell aspect ratios, indicates the presence of slight three di-mensionality of the wrinkling. The peak-to-peak ampli-tude of the wrinkling was measured on digitized images and plotted in semilog coordinates as shown in Fig. 7.

The large scatter of the points in the early stages of the growth arise from the small amplitude of the cells, of the order of the apparent flame thickness. The nonlinearity at long times indicates the onset of saturation of the instability. The nonlinearity of the shape of the cells is clearly visible in Fig. 6 after 140 ms. These points were systematically eliminated before data reduction to obtain the growth rate. The points were fitted to an exponential function of the form

ystd ≠1₂∑µy01Dy_s ∂ est₁µ_y 02Dy_s ∂ e2st∏_{, (2)}

which is the general solution of ≠2_yy≠t2_{≠ s}2_{y with the}

initial conditions ys0d ≠ y0and ≠ys0dy≠t ≠ Dy. Here,

Dy is the rate of increase of the wrinkling at time t≠ 0, supposed equal to the measured peak-to-peak velocity modulation produced by the wires in the flow. The

FIG. 6. Images taken from high speed film of growth of instability. Framing rate 500 imagesys, wavelength 2 cm, flame speed 11.5 cmys.

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Figure 1.9: Images taken from high speed imaging of growth of instability. Framing rate 500

images/s, wavelength 2 cm, flame speed 11.5 cm/s. [37].

More recently, Bauwens et al. [9] study the effect of the DL instability in methane air flame at atmospheric pressure in a spherical configuration (Fig. 1.13). This work [9] used the expanding flame configuration to induced the DL instability. Zangh et al. [130] study the propane air premixed flames in Bunsen burner configuration with different turbulent conditions. Figure 1.13 shows the instantaneous OH-PLIF images of the flames

Interplay of hydrodynamic instabilities and turbulence in premixed flames