This article was downloaded by: [Uppsala universitetsbibliotek]On: 19 November 2014, At: 04:56Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Combustion Theory and ModellingPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/tctm20

Acoustic timescale characterisationof symmetric and asymmetricmultidimensional hot spotsMichael Drew Kurtza & Jonathan David Regelea

a Department of Aerospace Engineering, Iowa State University,Ames, IA 50011, USAPublished online: 07 Nov 2014.

To cite this article: Michael Drew Kurtz & Jonathan David Regele (2014): Acoustic timescalecharacterisation of symmetric and asymmetric multidimensional hot spots, Combustion Theory andModelling, DOI: 10.1080/13647830.2014.971058

To link to this article: http://dx.doi.org/10.1080/13647830.2014.971058

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to orarising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Combustion Theory and Modelling, 2014

http://dx.doi.org/10.1080/13647830.2014.971058

Acoustic timescale characterisation of symmetric and asymmetricmultidimensional hot spots

Michael Drew Kurtz and Jonathan David Regele∗

Department of Aerospace Engineering, Iowa State University, Ames, IA 50011, USA

(Received 17 June 2014; accepted 17 September 2014)

In this work, two-dimensional hot spots are modelled by combining a linear temperaturegradient with a constant temperature plateau. This approach retains the simplicity of alinear temperature gradient, but captures the effects of a local temperature maximumof finite size. Symmetric and asymmetric plateau regions are modelled using bothrectangular and elliptical geometries. A one-step Arrhenius reaction for H2–air is usedto model the reactive mixture. Plateaus with different ratios of excitation to acoustictimescales, spanning two orders of magnitude, are simulated. Even with clear differencesin behaviour between one and two dimensions, the a priori prescribed hot spot timescaleratio is shown to characterise the 2-D gasdynamic response. The relationship betweenone and two dimensions is explored using asymmetric plateau regions. It is shown that1-D behaviour is recovered over a finite time. Furthermore, the duration of this 1-Dbehaviour is directly related to the asymmetry of the plateau.

Keywords: multidimensional hot spot; detonation initiation; acoustic timescale; tem-perature gradient; thermal stratification

1. Introduction

Hot spots can be regarded as locations of autoignition within a reactive mixture. Althoughthe term hot spot implies a region of warmer temperature, localised regions of more reactivefuel–air mixtures have a similar effect [1,2]. Hot spots are commonly observed in internalcombustion (IC) and homogeneous charge compression (HCCI) engines [3,4] as well as indetonation initiation studies [1,5,6].

Hot spots have been studied for many years, especially their role in causing detonationswithin non-uniform reactive mixtures. Thermal explosion theory [7–9] provides criticalconditions for which a hot spot will form a detonation. Using the induction-domain equationalong with one-step kinetics, theoretical descriptions of hot spot evolution to detonationshave been constructed [10,11]. Jackson, Kapila, and Stewart [12] investigate the evolutionof two different types of localised hot spots, depending on where they are in a system, i.e.at a wall or internal.

Zel’dovich showed that a spatial gradient in ignition delay time and can result in theformation of a detonation [13]. Shock wave amplification due to coherent energy release(SWACER) expands upon the Zel’dovich mechanism showing that a shock wave will beamplified and form a detonation when the chemical reaction energy is released immediatelyfollowing the shock wave [1,14,15].

∗Corresponding author. Email: jregele@iastate.edu

C© 2014 Taylor & Francis

Dow

nloa

ded

by [

Upp

sala

uni

vers

itets

bibl

iote

k] a

t 04:

56 1

9 N

ovem

ber

2014

2 M.D. Kurtz and J.D. Regele

Kassoy [16–18] used asymptotic analysis of the Navier–Stokes equations to describethe thermomechanical response of fluid to a localised heat addition and the ramificationsupon the surrounding fluid for both inert and reactive systems. He defined the ratio of theheating time to the acoustic timescale. The fluid response depends on a wide spectrum ofpower additions as well as the ratio of heating time to acoustic time. These range fromsmall acoustic waves, generated when the heat addition is much smaller than the internalenergy of the gas, to shock waves or even blast waves with a large heat addition.

Kassoy and colleagues [19–21] studied detonation initiation by using a spatially resolvedthermal power deposition on the acoustic timescale of the deposition volume in a reactivemixture with high activation energies. It was found that, depending on the ratio of the heatrelease time to the acoustic timescale of the hot spot, the thermomechanical response canvary from creating minor acoustic disturbances to the formation of spontaneous waves thatultimately form a detonation.

It is common to analyse hot spots as linear temperature gradients with one-step anddetailed kinetics [13,22–25]. Sharpe and Short [24] and Liberman et al. [25] showed thatthe critical gradient length required to form a detonation found via detailed kinetic modelsis two to three orders of magnitude larger than that found using simplified one-step reactionmodels, meaning detailed kinetics are required to predict quantitatively the critical gradientlength.

The spatial structure of an autoignition hot spot can also have a significant influenceon its reaction [26]. Sankaran and colleagues [27–29] used a thermally stratified initialtemperature field to show that ignition at local temperature maxima (hot spots) can reacteither isobarically or isochorically. Kurtz and Regele [26] modelled 1-D hot spots witha linear temperature gradient by joining the surrounding fluid with a higher temperatureplateau region. This simplified model captures the effect of a local maximum of finitesize and allows for the characterisation of hot spot spatial structure using the ratio of theexcitation and acoustic times. The acoustic time is equal to the time it takes for the headof an expansion wave created at the edge of the plateau to reach the centre of the hot spotand the time associated with chemical heat release is called the excitation time [22,30–32].If the excitation time is long in comparison to the acoustic time, expansion waves arecontinuously emitted from the edge of the plateau during the reaction, which reduces thereaction rate inside the plateau such that a nearly isobaric reaction occurs. On the otherhand, if the reaction time is short relative to the acoustic time, the reaction completesbefore the expansion wave has time move into the plateau region, reduce the temperature,and slow the reaction. In this case, a nearly isochoric reaction occurs and a shock wave isproduced near the plateau edge. The shock preheats the gradient region, and reduces theignition delay time. If shock preheating occurs in phase with the reaction wave a detonationis formed [20]. This behaviour is typical of deflagration-to-detonation transition (DDT)processes [33–41].

Although the one-dimensional hot spot model provides fundamental insight, hot spotsare inherently multidimensional and expansion waves propagate in different directions. Theprimary objective of this work is to extend the hot spot model to multiple dimensions anddemonstrate how multidimensional effects modify the behaviour of the hot spot model.

To this end, two 2-D models will be used to investigate the effect of a constant temper-ature plateau on a surrounding gradient. First, a rectangular, constant temperature plateauregion surrounded by a linear temperature gradient will be implemented as a direct exten-sion of the 1-D model [26]. For these cases, the expansion waves generated by the plateaureaction will be orthogonal to each other and nearly planar along the axes at early times.Eventually, the expansion waves merge, forming one wave which reaches the centre in

Dow

nloa

ded

by [

Upp

sala

uni

vers

itets

bibl

iote

k] a

t 04:

56 1

9 N

ovem

ber

2014

Combustion Theory and Modelling 3

roughly the acoustic time of the reaction products. It is anticipated that until this time thebehaviour along the x- and y-axes will be very similar to the 1-D model results. The secondmodel, accounting for curvature effects seen in real systems, will feature an elliptic plateauregion surrounded by a linear temperature gradient. Unlike the rectangular cases, expansionwaves will reduce the pressure and temperature on the axes from the moment the reactionbegins. While the current two-dimensional results may suggest similar behaviour in threedimensions, no explicit assessment of three dimensions is contained here, and this awaitsfurther attention.

The mathematical model used to form the solution is presented in Section 2, followedby a discussion of the important length and timescales in Section 3. These are used toformulate the model problems in Section 4, which are solved by the numerical methodspresented in Section 5. Finally, results are presented in Section 6, followed by conclusions.

2. Mathematical model

The reactive fluid mixture being simulated is best described using the reactive Navier–Stokes equations. The current work focuses on the autoignition of hot spot reactions,the spontaneous waves they create, and the interaction of these phenomena. Transporteffects can be considered negligible, reducing the governing equations to the reactive Eulerequations. This assumption is justified explicitly in the next two sections. In addition, asimple one-step Arrhenius reaction model is used to describe the chemistry since predictingthe critical gradient length is not the focus of this study. Thus, the 2-D reactive Eulerequations, in conservative form, are

∂U

∂t+ ∂Fx

∂x+ ∂Fy

∂y= S, (1)

where

U =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

ρ

ρu

ρv

ρeT

ρY

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

, Fx =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

ρu

ρu2 + p

ρuv

(ρeT + p)uρYu

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

, Fy =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

ρu

ρuv

ρv2 + p

(ρeT + p)vρYv

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

, S =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

0000

−W

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

. (2)

Here ρ is the density, ρu is the x-momentum, ρv is the y-momentum, eT is the total chemicalenergy, and Y is the fuel mass fraction. The total energy is defined as the sum of internal,kinetic and chemical energy, which is expressed as

eT = p

ρ(γ − 1)+ 1

2u2 + Yq. (3)

The reaction rate is modelled using a simple one-step Arrhenius law and is given by

W = BρYe−Ea/RT , (4)

where B is the pre-exponential factor, q is the heat of reaction, and Ea is the activationenergy. Temperature is found with the ideal gas law, p = ρRT.

Dow

nloa

ded

by [

Upp

sala

uni

vers

itets

bibl

iote

k] a

t 04:

56 1

9 N

ovem

ber

2014

4 M.D. Kurtz and J.D. Regele

3. Length and timescales

The amount of time it takes a reactive mixture to be consumed (excitation time) and thetime it takes for fluid motion to be induced (acoustic time) have been successfully used tocharacterise a number of reactive thermomechanical problems [18,21,26]. The relationshipbetween these two timescales determines the process by which the mixture reacts and theeffect it will have upon its surroundings.

Consider a volume of premixed, reactive mixture at some initial temperature and pres-sure. The excitation time, τ e, is defined roughly as the time it takes for the fuel to beconsumed after the ignition delay time, τ i, has elapsed [22,30–32]. Here τ is used to differ-entiate chemical timescales from others. The consumption of the fuel does not have exactstart and end times, so the excitation time needs to be approximated. Thus, the slope of thefuel mass fraction when half the fuel has been consumed, dY(Y=0.5)/dt , is used to extrapolatelinearly the start and end times, the difference of which is defined to be the excitation time.This gives an excitation time of

τe = −(

dY(Y=0.5)

dt

)−1

. (5)

The amount of time it takes for mechanical disturbances, such as expansion waves, topropagate some characteristic length l through a mixture with local sound speed a is termedthe acoustic timescale, given by

ta = l

a. (6)

As in previous work [16,20,21,26], the ratio of the heat release time to the acoustictime th/ta can be used to characterise the thermomechanical response of a fluid to spatiallyresolved heat addition. In the present case the excitation time is equal to the heat releasetime. The excitation-to-acoustic-timescale ratio can then be written as a ratio of lengthscales

τe

ta= τea

l= L

l, (7)

where L, defined as L = τ e · a [26], is the distance travelled by an acoustic wave during theexcitation period and τ e is chosen as the isochoric excitation time, τ v

e . Ratios greater thanone will result in weak acoustic wave generation, ratios less than one can generate strongcompression waves, and those roughly equal to one generate compression between these twodelineating behaviours. The characteristic length l, used to define the plateau sizes in thiswork, can be obtained from Equation (7) by prescribing the excitation-to-acoustic-timescaleratio τ e/ta for a given plateau temperature.

The viscous and conductive timescales, defined by Kassoy [16] and rewritten [26] interms of the acoustic Reynolds number, Rea = lao/νo, and Prandtl number, Pr = νo/κo, areexpressed as

τv = l2

νo

= Reata (8)

and

τc = l2

κo

= ReaPrta. (9)

Dow

nloa

ded

by [

Upp

sala

uni

vers

itets

bibl

iote

k] a

t 04:

56 1

9 N

ovem

ber

2014

Combustion Theory and Modelling 5

Table 1. Arrhenius reaction rate parameters [42] presented interms of the given reference state (subscript o).

Parameter Value

Po 1 atmTo 300 Kγ 1.4R 398 J/(kg K)ao 347 m/sB 2.26E+9 s−1

Ea 33 γ RTo

q 12 γ RTo

Here, νo and κo as the characteristic dynamic viscosity and thermal diffusivity. Theserelations show that the timescale ratios associated with molecular transport to acoustictimescales are proportional to Rea.

Common combustion systems typically have Rea = O(104) and Prandtl numbers of0.7. Equations (8) and (9) demonstrate the acoustic timescale is four orders of magnitudesmaller than τv and τ c. This suggests that transport effects are negligible and that the flowcan be modelled using the reactive Euler equations.

4. Problem statement

A stoichiometric mixture of hydrogen–air is modelled using a one-step Arrhenius reactionrate with parameters given in Table 1 with respect to a reference thermodynamic state. Here,R is the universal gas constant, calculated for the weighted average of a stoichiometric H2–air reactive mixture.

Two 2-D hot spot models, rectangular and elliptical, are used to extend the analysisperformed in the previous 1-D study [26]. Both models are used to explore symmetricand asymmetric hot spots. Hot spots with symmetric plateaus will be considered to showhow multiple dimensions affect the hot spot characterisation. Asymmetric plateaus will beexamined to explore the effects of asymmetry and non-unity hot spot aspect ratio.

The two hot spot models are composed of a constant temperature plateau region sur-rounded by a linear temperature gradient. This approach models the effect of a rounded peakin temperature on the surrounding region with a more substantial temperature gradient. The1-D temperature distribution used in the previous study [26] is given by

T =⎧⎨⎩

Tp, if 0 ≤ x < l

Tp + (Ta − Tp)(x − l)/lg, if l ≤ x < l + lgTa, if x ≥ l + lg,

(10)

where Tp is the plateau temperature, Ta is the ambient temperature, l is the plateau size, andlg is the gradient length.

Figure 1 illustrates the initial conditions for the 1-D model [26], which will also beused in this work. It consists of a higher temperature plateau, a gradient region, and alow temperature ambient fluid region. The pressure is initially constant at p = 1 atm, thetemperature is given by Equation (10), and the density is determined using the idealgas equation of state p = ρRT. The initial conditions within the plateau region are

Dow

nloa

ded

by [

Upp

sala

uni

vers

itets

bibl

iote

k] a

t 04:

56 1

9 N

ovem

ber

2014

6 M.D. Kurtz and J.D. Regele

Figure 1. The initial 1-D temperature profile and thermodynamic state that is used to construct theinitial conditions in two dimensions. (Colour online.)

Tp = 1260 K, pp = 1 atm, and ρp = 0.24 kg/m3. The isochoric excitation time is cal-culated as τe = τ v

e = 0.46 μs, giving L = τ e · ap = 0.46 μs · 712 (m/s) = 0.34 mm forthe initial hot spot conditions. Here ap is the plateau sound speed. The gradient length isdefined as lg = (Ta − Tp)/S = 30L, where Ta is the temperature of the surroundings andthe gradient slope is S = −119 K/mm. A parametric study was performed to determine thegradient length lg that is slightly too short to initiate a detonation wave.

The 1-D temperature gradient is extended to two dimensions in the rectangular andelliptical models shown in Figure 2. Both the rectangular and elliptical models will haveinitial temperature profiles on the x-axis that are identical to the 1-D condition listed inEquation (10). The rectangular case serves as an idealised model for direct comparisonwith the 1-D results because there are no curvature effects on the axes. The second model

IV

V

Figure 2. Initial temperature distribution diagrams for rectangular and elliptical hot spot models.

Dow

nloa

ded

by [

Upp

sala

uni

vers

itets

bibl

iote

k] a

t 04:

56 1

9 N

ovem

ber

2014

Combustion Theory and Modelling 7

features an elliptical plateau as a more realistic hot spot that accounts for curvature effectspresent in real systems. Each of these two different models can be made symmetric bymaking the height and width of the hot spot plateau regions equal lx = ly = l.

Figure 2(a) illustrates the five different regions required to describe the initial conditionfor the rectangular model. Region I is the temperature plateau. Regions II and III are regionswhere linear temperature gradients exist in only the y- and x-directions, respectively. Thetemperature in region IV is defined by a plane that intersects the right edge of region IIand the top edge of region III. Finally, region V is the ambient temperature Ta. The initialtemperature distribution is

T (x, y) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

Tp, if 0 ≤ x < lx and 0 ≤ y < lyTp + S(y − ly), if 0 ≤ x < lx and ly ≤ y < ly + lgTp + S(x − lx), if lx ≤ x < lx + lg and 0 ≤ y < lyTp + S

(x − lx + y − ly

), if x ≥ lx and ly ≤ y < lx + ly + lg − x

Ta, otherwise,(11)

where the slope S, gradient length lg, and temperatures Tp and Ta are identical to thosedescribed above for the 1-D case.

In Figure 2(b), the initial condition is composed of three different regions. The plateauis in region I, the gradient is in region II, and the ambient fluid is in region III. The initialtemperature for the elliptical model is

T (x, y) =⎧⎨⎩

Tp, if d ≤ 0Tp + (Ta − Tp) d / lg, if 0 ≤ d < lgTa, if d ≥ lg,

(12)

where d = d(x, y) is the normal distance from the edge of the plateau inside the gradientregion. As illustrated in Figure 2(b), the distance d is related to any position (x, y) throughthe relation

1 = x2

(lx + d)2+ y2

(ly + d)2, (13)

where the nonlinear algebraic equation is solved numerically. Each unique value of drepresents a different ellipse of constant temperature inside the temperature gradient region.

To establish a baseline behaviour for the temperature gradient in 2-D, a case withouta plateau region, lx = ly = 0, will be run. Multidimensional effects are explored usingsymmetric plateau regions with equal side lengths, lx = ly = l. Three different cases areused with each model, letting l = L/10, L and 10L, which correspond to timescale ratiosτ e/ta = 10, 1 and 0.1, respectively. The plateau reaction generates a compression wavedirected away from the plateau and an expansion wave directed towards the plateau centre.

In general, hot spots are not usually symmetric. Sankaran and colleagues [27–29]focused on initial temperature fields with asymmetric hot spots. To study the effects ofthis aspect ratio, additional cases will be run with both models letting ly > lx. The lengthsly and lx correspond, respectively, to the height and width of the rectangular plateau andthe semi-major and semi-minor axes of the elliptical plateau. This results in two differentacoustic timescales for an asymmetric plateau region. On the horizontal axis the acoustictimescale is defined as txa = lx/a. Similarly, the vertical direction has an acoustic timescaletya = ly/a.

Dow

nloa

ded

by [

Upp

sala

uni

vers

itets

bibl

iote

k] a

t 04:

56 1

9 N

ovem

ber

2014

8 M.D. Kurtz and J.D. Regele

The three aspect ratios examined are tya /txa = ly/ lx = 2, 5 and 10. To study this effect,

a partially confined plateau (τe = txa ) has been selected as the fixed excitation-to-acoustic-timescale ratio along the x-axis. An unconfined plateau along the x-axis (τe � txa ) wouldallow rapid fluid expansion in the x-direction to occur during the reaction timescale, makingadditional expansion due to multidimensional effects difficult to isolate. On the other hand,having a confined plateau (τe � txa ) in the x-direction readily reproduces limiting 1-Dbehaviour. Therefore, the chosen excitation-to-acoustic-timescale ratio along the x-axis,τe/txa = 1, is the most pertinent to examine. As the aspect ratio t

ya /txa increases, behaviour

along the x-axis is expected to approach the 1-D model behaviour.

5. Numerical method

The dynamically adaptive wavelet-collocation method (AWCM) is used to perform allcalculations in this study [43,44]. With this method, every point is associated with a wavelet[44]. This allows the grid to be adapted such that wavelets with coefficients greater than acertain threshold parameter ε will be retained for computation while others are discarded.This allows a higher level of resolution for less computational power. A revised hyperbolicsolver [45], developed for the AWCM, is used to solve the reactive Euler equations. Thescheme is second-order accurate in smooth regions and between first- and second-orderaccurate near shocks and contact discontinuities. Van Leer flux limiting is used to minimisenumerical diffusion while maintaining nonlinear stability [46,47].

A base grid of 40 × 40 points is used with a maximum of nine refinement levels. Thenumber of required refinement levels was determined by increasing the resolution untilthe half-reaction location changed by only 1% after one acoustic time for the confinedsquare case (τ e � ta). This results in an effective grid with N = 10,240 × 10,240 effectivegrid points, corresponding to a grid resolution of x = 1.3 μm at the finest level. Thegrid resolution is kept constant for all other cases. Further refinement of the grid does notprovide any noticeable improvement in reaction front trajectory, which suggests that thesolution is grid independent.

6. Results

The symmetric cases are presented first to study the multidimensional effects independentof asymmetry effects. Results for the square cases are expected to behave in a relativelyplanar fashion on the x- and y-axes, prior to the acoustic time. Additional expansion in thecircular cases is expected to lessen this effect. In Section 6.2, the effect of asymmetry isexplored by altering the aspect ratio between t

ya and txa for both rectangular and elliptical

models. As the aspect ratio becomes larger, 1-D behaviour along the shorter axis is expectedto occur for a longer time period.

In each section, a series of temperature contours from select cases are presented toshow the hot spot reaction and illustrate its evolution. Due to the qualitative nature ofthe information conveyed with these contours, only one case from each model will bepresented. After this, it is most constructive to view results from a 1-D slice along theshorter axis (the x-axis), where a direct comparison with 1-D results from the previousstudy [26] can be made. Each section will investigate the 2-D rectangular model first soas to compare directly with the 1-D results. The results from the more realistic ellipticalmodel are presented second.

Dow

nloa

ded

by [

Upp

sala

uni

vers

itets

bibl

iote

k] a

t 04:

56 1

9 N

ovem

ber

2014

Combustion Theory and Modelling 9

Figure 3. Selected time steps of the temperature contours for (a) temperature (K) and (b) pressure(p/po) for the square case with τ e/ta = 0.1. The time shown in the upper-right corner of each frameis the normalised time, (t − τ i)/τ e. (Colour online.)

6.1. Symmetric effects

To study the effects of multiple dimensions, four symmetric (txa = tya = ta) cases are con-

sidered with each model (square and circular). First, a case is run with no plateau region(lx = ly = 0) to establish the baseline behaviour of the temperature gradient in 2-D. This willbe referred to as the ‘gradient-only’ case. Three other cases are examined with excitation-to-acoustic-timescale ratios of τ e/ta = 10, 1, and 0.1. The τ e = 10ta case will be termed the‘unconfined’ case, τ e = ta the ‘partially confined’ case, and τ e = 0.1ta the ‘confined’ case.

6.1.1. Square model

Figure 3(a) contains a sequence of temperature contours displaying the hot spot evolutionof the confined case (τ e/ta = 0.1) for the square model. A blue line is superimposed on thelocation of the half-reaction to show the reaction progress. The simulation time has beenshifted by the ignition delay period τ i, normalised by the excitation time τ e, and is labelled atthe top right of each frame. Unreacted fuel mixture appears red, while combustion productsare yellow. A shock wave can be seen as the temperature discontinuity it creates.

For 0.9 < (t − τ i)/τ e < 2.75 in Figure 3(a), the half-reaction location (blue line)coincides with the burnt–unburnt gas interface. This suggests that the reaction is coupledwith the shock that is formed from the consumption of the reactants in the plateau region.At (t − τ i)/τ e = 4.0 the reaction front is no longer coupled with the shock everywhere in thedomain and by (t − τ i)/τ e = 6.5 there is clear separation of the fluid that is preheated fromthe shock and the burnt combustion products as indicated in the figure. For 0 < (t − τ i)/τ e ≤ 6.5 the shock and reaction are coincident along the x- and y-axes, which indicates thewave is propagating as a detonation in those regions. At (t − τ i)/τ e = 9.0, the shock andreaction fronts decouple along the x- and y-axes too. Unlike the 1-D analysis [26], whichshows detonation formation for both the confined (τ e/ta = 0.1) and partially confined(τ e/ta = 1) cases, only the confined case results in detonation formation in two dimensions.

Dow

nloa

ded

by [

Upp

sala

uni

vers

itets

bibl

iote

k] a

t 04:

56 1

9 N

ovem

ber

2014

10 M.D. Kurtz and J.D. Regele

0 1 2 3 4 5 60

2

4

6

8

10

12

x− l (mm)

(t−

τ i)/

τ e

(a) Half-reaction location.

0 2 4 6 8 10 12

1

2

3

4

5

6

(t −τi)/τe

u/a

o

τrv τ

rp

(b) Propagation velocity.

2D: Gradient only

2D: τe=10t

a

2D: τe=t

a

2D: τe=0.1t

a

1D: Gradient only

1D: τe=10t

a

1D: τe=t

a

1D: τe=0.1t

a

Figure 4. Square model: comparison of the half-reaction location and the propagation velocity ofthe reaction wave along the x-axis of all 2-D cases. The corresponding 1-D cases are included withthe same line style sans markers. Two vertical dotted lines are included to represent the reaction timesfor an isochoric (τ v

r ) and isobaric (τpr ) homogeneous reactor. (Colour online.)

Figure 3(b) shows the corresponding confined case pressure contours. Upon reaction ofthe plateau region at (t − τ i)/τ e = 1, compression and expansion waves, observed first inframe (t − τ i)/τ e = 2.13, travel outwards and inwards from the edge of the plateau region.As expected, these waves are mostly planar initially. At (t − τ i)/τ e = 2.13, Figure 3(b)shows this planar behaviour. The head of the inward travelling expansion wave, definedas the location where the pressure has decreased to 90% of the isochoric plateau reactiontemperature, is marked by a green dashed curve. If the expansion wave is assumed to travelat the speed of sound through combustion products that are at the isochoric post-combustiontemperature, the expansion wave will reach the centre of the hot spot at (t − τ i)/τ e = 4.8.Figure 3(b) shows that the expansion wave does reach the centre in that time, as indicatedby the green line disappearing between frames 4.0 and 5.25.

The behaviour of the partially confined, unconfined and gradient-only cases is bestillustrated by plotting the half-reaction location on the axes and the speed of the spontaneouswave relative to the 1-D cases. Figure 4 shows the half-reaction location shifted by theplateau size, x1/2 − l = x(Y = 0.5) − l, and reaction propagation velocity, us = dx1/2/dt ,normalised by the reference sound speed, ao, along the x-axis for each square case. 1-Dcases with corresponding plateau acoustic timescales are included with the same line style,without line markers. The time axis is shifted by the ignition delay time and normalisedby excitation time so that zero corresponds to the end of the ignition delay time and onecorresponds to the end of the reaction.

Two vertical dotted lines are included in the velocity plot representing the reactiontimes, defined τ r = τ i + τ e/2, for isochoric (τ v

r ) and isobaric (τpr ) homogeneous reactors.

These times align well with the initial peak in velocity that occurs due to the reaction ofthe confined and unconfined plateaus, respectively. As expected, this suggests the confinedplateau reacts in a constant volume process and the unconfined plateau in a nearly constantpressure process. The addition of a larger plateau causes a larger volume of fluid to reactmore quickly, resulting in a more confined reaction, and a larger initial transient propagationvelocity. After the plateau reacts the reaction wave either accelerates until it reaches amaximum velocity after which it decelerates due to low initial temperatures, or it deceleratesimmediately.

Dow

nloa

ded

by [

Upp

sala

uni

vers

itets

bibl

iote

k] a

t 04:

56 1

9 N

ovem

ber

2014

Combustion Theory and Modelling 11

As illustrated in Figure 4(a), the gradient-only case has the slowest travelling reactionwave and penetrates the smallest distance into the domain. Alternatively, the confinedcase has the largest velocity and travels farthest into the domain. Overall, the additionaldimension allows the fluid to expand in a direction orthogonal to the direction of reactionwave propagation. This extra expansion reduces the inertial confinement of the system andresults in a reaction process that takes longer. As can be expected, this suggests that inmultiple dimensions a shallower gradient is required to form a detonation.

In addition, the 1-D behaviour observed from adding a plateau region to a gradient isalso observed in these 2-D cases. Namely, more confined plateaus will react more quickly,emit stronger compression waves, and cause the reaction to propagate further into thedomain at a larger velocity. This effect is seen in Figure 4 as increasingly confined plateaus(τ e/ta decreasing) are added to the gradient. Since the 1-D gradient-only case is muchcloser to forming a detonation alone, the addition of a partially confined plateau causes itto detonate, whereas a confined plateau is required to form a detonation with this slope in2-D.

When the excitation time is much smaller than the acoustic time, as in the confinedcase (τ e = 0.1ta), the inward travelling expansion will reach the centre of the plateauwell after the reaction is complete. This occurs at (t − τ i)/τ e = 4.8, which is consistentwith what is observed in Figure 3(b) where the expansion head disappears between frames4.0 and 5.25. Figure 4(b) shows that, after this time, two-dimensional expansion effectscome into play, causing the reaction to decelerate and separate from the 1-D result. Thismeans the symmetry axes for 2-D hot spots behave similarly to the 1-D hot spots until theexpansion wave has reached the centre. Once the expansion wave has reached the centre,multidimensional expansion causes the reaction to occur more slowly than in the 1-D model.

6.1.2. Circular model

Here, a more realistic circular plateau is considered, which uses the same four plateauexcitation-to-acoustic-timescale ratios as the square case. Figure 5(a) contains temperaturecontours for the circular confined case. The blue line, which indicates the reaction front, iscollocated with the temperature discontinuity until (t − τ i)/τ e = 4.5. Since the temperaturediscontinuity is created by the shock wave, this indicates that the reaction and shockwave are coupled and a detonation wave exists during this time. Figure 5(b) provides thecorresponding pressure contours. As in the square case, a shock wave propagates outwardsand an expansion (marked in frame 2.13) propagates inwards. Here, these two waves havecurvature throughout the domain, which is different than square cases where there was aperiod of time when the expansion waves were planar.

Figure 6 shows the half-reaction location and reaction propagation velocity for eachcircular case. The overall behaviour of the circular cases is similar to the square cases.Figure 6(b) shows there is a large initial acceleration indicated by the first velocity max-imum, after which all 2-D cases decelerate, except for the confined case. This initialmaximum is associated with the reaction of the plateau region. The amount of fluid to beconsumed in the excitation time, τ e, for the confined case is much larger than that of theunconfined and partially confined cases. This causes the first maximum in the confined caseto be significantly larger than the others.

Figure 6(b) shows that once the confined plateau reaction is complete there is a localminimum in propagation velocity (near the vertical line τ r

p). Following this minimum thereis a slight increase in propagation speed, which indicates the propagation of a detonation.The propagation speed reaches a maximum at a time (t − τ i)/τ e = 4.25. For (t − τ i)/τ e = 5,

Dow

nloa

ded

by [

Upp

sala

uni

vers

itets

bibl

iote

k] a

t 04:

56 1

9 N

ovem

ber

2014

12 M.D. Kurtz and J.D. Regele

Figure 5. Selected time steps of the temperature contours for (a) temperature (K) and (b) pressure(p/po) for the circular case with τ e/ta = 0.1. The time shown in the upper-right corner of each frameis the normalised time, (t − τ i)/τ e. (Colour online.)

the propagation speed begins to decrease, which indicates that the detonation propagates foronly a short period of time before the reaction ceases to be coupled with the shock. The 1-Dbehaviour along the x-axis only lasts until (t − τ i)/τ e ≈ 2.75 compared to (t − τ i)/τ e ≈ 5.25seen in the square confined case, which shows that the additional expansion in the circularcase that is not present initially in the square case causes the reaction to proceed moreslowly.

It is illustrative to compare the globally integrated instantaneous heat release per unitarea, defined as Q(t)/A = ∫

lp+lgW (t)q dx, as a function of time to contrast the different

cases. Figure 7 shows the global heat release along the x-axis for the circular and square

0 1 2 3 4 5 60

2

4

6

8

10

12

x− l (mm)

(t−

τ i)/

τ e

(a) Half-reaction location.

0 2 4 6 8 10 12

1

2

3

4

5

6

(t −τi)/τe

u/a

o

τrv

τrp

(b) Propagation velocity.

2D: Gradient only

2D: τe=10t

a

2D: τe=t

a

2D: τe=0.1t

a

1D: Gradient only

1D: τe=10t

a

1D: τe=t

a

1D: τe=0.1t

a

Figure 6. Circular model: comparison of the half-reaction location, the propagation velocity ofthe reaction wave, and the global heat release along the x-axis of all 2-D cases. The correspond-ing 1-D cases are included with the same line style sans markers. Two vertical dotted lines areincluded to represent the reaction times for an isochoric (τ v

r ) and isobaric (τpr ) homogeneous reactor.

(Colour online.)

Dow

nloa

ded

by [

Upp

sala

uni

vers

itets

bibl

iote

k] a

t 04:

56 1

9 N

ovem

ber

2014

Combustion Theory and Modelling 13

τ τ

τ

τ

τ

τ

τ

τ

Figure 7. Comparison of global heat release along the x-axis of square and circular 2-D cases. Twovertical dotted lines are included to represent the reaction times for an isochoric (τ v

r ) and isobaric(τp

r ) homogeneous reactor. (Colour online.)

cases. The initial maximum in all cases is due to the plateau reaction, after which the heatrelease decreases. Both the square and circular confined cases develop into detonations,which cause the heat release to increase again after the local minimum that occurs atapproximately (t − τ i)/τ e = 1.5. However, additional expansion in the circular confinedcase causes heat to be released at a slower rate compared to the square confined case,beginning around (t − τ i)/τ e = 1.5.

6.2. Asymmetric effects

The height of the plateau region is increased in both the rectangular and elliptical modelsto analyse the effect of asymmetry on hot spot reaction. As discussed earlier, the primaryfocus will be on the partially confined case with the pocket width lx constructed such thattxa = τe. The plateau height ly is increased to raise the aspect ratio between t

ya and txa . Three

cases with tya /txa = ly/ lx = 2, 5 and 10 are considered with the rectangular and elliptical

hot spot models to demonstrate the effects of asymmetry. As this ratio is increased, 1-Dbehaviour should be recovered along the short axis until multidimensional expansion slowsthe reaction. As in the last section, the rectangular model will be investigated first as a directextension of 1-D, followed by the more realistic elliptical model.

6.2.1. Rectangular model

Figure 8 contains a sequence of temperature and pressure contours which show the reactionof the hot spot with aspect ratio t

ya = 10txa for the rectangular model. The time, labelled

at the top right of each frame, has been shifted by the ignition delay time, τ i, so zerocorresponds to the end of the ignition delay period. The time has also been normalised bytxa , since all cases have the same acoustic time along the x-axis.

Dow

nloa

ded

by [

Upp

sala

uni

vers

itets

bibl

iote

k] a

t 04:

56 1

9 N

ovem

ber

2014

14 M.D. Kurtz and J.D. Regele

Figure 8. Selected time steps of (a) temperature (K) and (b) pressure (p/po) contours for therectangular cases with ty

a /txa = 10. The time shown in the upper-right corner of each frame is the

normalised time, (t − τi)/txa . (Colour online.)

Upon reaction of the plateau region, relatively planar compression and expansion wavesare generated in the positive and negative x-directions, respectively. Frame 1.7 in Figure 8(b)shows the compression wave and the expansion wave that is reflected off the left symmetryaxis. The large aspect ratio causes a separation of timescales for fluid expansion along thelarge axis (ly) and small axis (lx), resulting in dominant behaviour along the short axis.As the reaction progresses, a detonation forms in the x-direction, evident from the shock-coupled reaction seen in frames 4.2 ≤ (t − τi)/txa ≤ 6.1 of Figure 8(a). As the reactiondecouples from the shock front around (t − τi)/txa ≈ 6.7, hot reaction products are allowedto expand in the y-direction.

To gain a better understanding of these results, a 1-D slice along the x-axis of eachcase is now compared with the 1-D results from the previous study [26]. Figure 9 showsthe half-reaction location and reaction propagation velocity for the three 2-D rectangularmodel cases with t

ya /txa = 1, 2, 5 and 10, in addition to the 1-D partially confined case.

The case with tya = 2txa looks similar to the square partially confined case with txa =

tya = ta seen in Figure 9. Figure 9(a) shows that the reaction begins slightly later than the

other cases with larger aspect ratios. The half-reaction location quickly deviates from 1-Dand other higher aspect ratio cases, but it travels further into the domain than the 2-Dsymmetric case. The case with t

ya = 5txa behaves like the 1-D case longer than the t

ya = 2txa

case, resulting in the reaction propagating further into the domain (Figure 9(a)) at a largerwave speed, seen in Figure 9(b). Figure 9(a) shows that the half-reaction location begins todeviate significantly from the 1-D location when (t − τi)/txa ≈ 3.

Figure 9(a) shows the tya = 10txa case matches the 1-D results the longest, with notable

deviations beginning around (t − τi)/txa = 7. This causes the reaction wave to propagate

Dow

nloa

ded

by [

Upp

sala

uni

vers

itets

bibl

iote

k] a

t 04:

56 1

9 N

ovem

ber

2014

Combustion Theory and Modelling 15

τ

Figure 9. Rectangular model: comparison of the half-reaction location and propagation velocity ofreaction wave along 1-D slice on x-axis for ty

a /txa = 2, 5 and 10 with 1-D results. (Colour online.)

the farthest into the domain and have a much larger velocity than the 2txa or 5txa cases. The10txa case behaves like the 1-D case long enough to form a detonation in the x-direction,just as in the 1-D case. As the aspect ratio is increased, the behaviour of the reaction inthe direction normal to the larger hot spot axis resembles 1-D behaviour, which is expectedsince the limit of infinite aspect ratio ly/lx → ∞ corresponds exactly to the 1-D case. Somedeviation prior to t

ya is seen in Figure 9. This is due to other multidimensional effects,

such as rounding of the expansion waves and expansion wave interaction with reflectedexpansion waves.

6.2.2. Elliptical model

The results of the previous section can be expanded upon by considering a more realisticelliptical asymmetric hot spot. The sequence of temperature and pressure contours inFigures 10(a) and 10(b) display the evolution of an elliptical hot spot with t

ya = 10txa . This

case results in a shock coupled reaction for times 1.7 ≤ (t − τi)/txa ≤ 6.7. This couplingdoes not last as long as the rectangular case (Figure 8). As the height of the plateau (ly)increases, the hot spot is elongated. This causes a larger portion of the reaction in theelliptical cases to be orthogonal to the x-axis, which results in behaviour similar to therectangular cases when t

ya /txa becomes large.

Figure 11 shows the half-reaction location and reaction propagation velocity for theelliptical cases with aspect ratio t

ya /txa = 1, 2, 5 and 10 as well as the 1-D partially confined

case. Similar to the asymmetric rectangular cases, increasing the aspect ratio causes 1-Dbehaviour along the x-axis for a longer period of time. This results in the reaction wavetravelling further into the domain (Figure 11(a)) at a larger velocity (Figure 11(b)). Asin the circular cases, there is an extra degree of expansion due to the initial curvature ofthe plateau region. The difference between the elliptical and rectangular plateau regions isless pronounced than the symmetric cases. The asymmetry causes the elliptical plateau tohave a larger planar region than the circular plateau. This causes the curvature effect seenin the circular case to have a smaller effect on the elliptical case, allowing its behaviourto be more similar to the asymmetric rectangular cases as the hot spot becomes moreelongated.

Dow

nloa

ded

by [

Upp

sala

uni

vers

itets

bibl

iote

k] a

t 04:

56 1

9 N

ovem

ber

2014

16 M.D. Kurtz and J.D. Regele

Figure 10. Selected time steps of (a) temperature (K) and (b) pressure (p/po) contours for theelliptical cases with ty

a /txa = 10. The time shown in the upper-right corner of each frame is the

normalised time, (t − τi)/txa . (Colour online.)

τ

Figure 11. Elliptical model: comparison with 1-D results of the half-reaction location and thepropagation velocity of the reaction wave along a 1-D slice on the x-axis for ty

a /txa = 2, 5 and 10.

(Colour online.)

7. Conclusions

Hot spots are modelled in two dimensions using a constant temperature plateau combinedwith a linear temperature gradient. Symmetric and asymmetric plateau regions are mod-elled using both rectangular and elliptical geometries. For symmetric cases, the excitation-to-acoustic-timescale ratio is used to characterise the thermomechanical response of thesurrounding fluid. Hot spots with more confined plateaus react more quickly, emit strongercompression waves, and cause the reaction to propagate further in the domain at a fastervelocity.

Dow

nloa

ded

by [

Upp

sala

uni

vers

itets

bibl

iote

k] a

t 04:

56 1

9 N

ovem

ber

2014

Combustion Theory and Modelling 17

In general, the inertial confinement in multiple dimensions is less than that observedin one dimension. It is shown that the effects of additional expansion is most severe incases with larger plateau excitation-to-acoustic-timescale ratios, τ e � ta, or smaller plateausizes. However, for cases with significantly small excitation-to-acoustic-timescale ratios,τ e � ta, or large plateau regions, the behaviour is very similar to the 1-D model results up tothe acoustic time. After the acoustic time, the effects of multidimensional expansion causethe behaviour to deviate from the 1-D model. While the effects of multiple dimensions isonly quantified in two dimensions, the results suggest that the same qualitative behaviourcan be expected in three dimensions. A full three-dimensional study is required to quantifythe differences.

The effects of asymmetry were studied as well. As the aspect ratio is increased, thebehaviour along the shorter axis retains 1-D behaviour for longer durations. This is due tothe separation of timescales between the two axes. As the aspect ratio is increased, fluidexpansion along the long axis takes more time than that along the short axis, which resultsin dominant, 1-D behaviour along the short axis for times prior to the acoustic time of thelong axis. Therefore, when the aspect ratio t

ya /txa is large, results from the 1-D study [26]

become increasingly accurate.

FundingThe research reported in this paper is partially supported by the HPC equipment purchased through:NSF MRI [grant number CNS 1229081]; NSF CRI [grant number 1205413].

References[1] J.H.S. Lee, Initiation of gaseous detonation, Annu. Rev. Phys. Chem. 28 (1977), pp. 75–104.[2] T. Echekki and J.H. Chen, Direct numerical simulation of autoignition in non-

homogeneous hydrogen–air mixtures, Combust. Flame 134 (2003), pp. 169–191. Availableat http://linkinghub.elsevier.com/retrieve/pii/S0010218003000889.

[3] M. Yao, Z. Zheng, and H. Liu, Progress and recent trends in homogeneous charge compressionignition (HCCI) engines, Prog. Energy Combust. Sci. 35 (2009), pp. 398–437. Available athttp://linkinghub.elsevier.com/retrieve/pii/S0360128509000197.

[4] J.E. Dec, Advanced compression-ignition engines – undersdanding the in-cylinder pro-cesses, Proc. Combust. Inst. 32 (2009), pp. 2727–2742. Available at http://linkinghub.elsevier.com/retrieve/pii/S1540748908001739.

[5] A.K. Oppenheim and R.I. Soloukhin, Experiments in gasdynamics of explosions, Annu. Rev.Fluid Mech. 5 (1973), pp. 31–58.

[6] E.S. Oran and V.N. Gamezo, Origins of the deflagration-to-detonation transition in gas-phase combustion, Combust. Flame 148 (2007), pp. 4–47. Available at http://linkinghub.elsevier.com/retrieve/pii/S0010218006001817.

[7] A.G. Merzhanov, On critical conditions for thermal explosion of a hot spot, Combust. Flame10 (1966), pp. 341–348. Available at http://dx.doi.org/10.1016/0010-2180(66)90041-1.

[8] P.H. Thomas, An approximate theory of ‘hot spot’ criticality, Combust. Flame 21 (1973), pp.99–109.

[9] M.B. Zaturska, Thermal explosion of interacting hot spots, Combust. Flame 25 (1975), pp.25–30. Available at http://linkinghub.elsevier.com/retrieve/pii/0010218075900656.

[10] J. Clarke, Fast flames, waves and detonation, Prog. Energy Combust. Sci. 15 (1989), pp.241–271. Available at http://linkinghub.elsevier.com/retrieve/pii/0360128589900105.

[11] N. Nikiforakis and J.F. Clarke, Quasi-steady structures in the two-dimensional initia-tion of detonations, Proc. R. Soc. Lond. A 452 (1996), pp. 2023–2042. Available athttp://dx.doi.org/10.1098/rspa.1996.0107.

[12] T.L. Jackson, A.K. Kapila, and D.S. Stewart, Evolution of a reaction center in an explosivematerial, SIAM J. Appl. Math. 49 (1989), pp. 432–458.

Dow

nloa

ded

by [

Upp

sala

uni

vers

itets

bibl

iote

k] a

t 04:

56 1

9 N

ovem

ber

2014

18 M.D. Kurtz and J.D. Regele

[13] Ya.B. Zel’dovich, V.B. Librovich, G.M. Makhviladze, and G.I. Sivashinskiy, On the onset ofdetonation in a nonuniformly heated gas, J. Appl. Mech. & Tech. Phys. 11 (1970), pp. 264–270.Available at http://link.springer.com/article/10.1007/BF00908106.

[14] J.H.S. Lee and I.O. Moen, The mechanism of transition from deflagration to detonation in vaporcloud explosions, Prog. Energy Combust. Sci. 6 (1980), pp. 359–389.

[15] J.H.S. Lee, R. Knystautas, and N. Yoshikawa, Photochemical initiation of gaseous detonations,Acta Astronaut. 5 (1978), pp. 971–982.

[16] D.R. Kassoy, The response of a compressible gas to extremely rapid transient, spatially resolvedenergy addition: An asymptotic formulation, J. Engrg. Math. 68 (2010), pp. 249–262. Availableat http://www.springerlink.com/index/10.1007/s10665-010-9402-z.

[17] D.R. Kassoy, Mechanical disturbances arising from thermal power deposition in a gas withapplication to supercritical liquid rocket engine combustion instability, in Proceedings of the51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and AerospaceExposition, 7–10 January 2013, Grapevine, TX, Paper Number AIAA-2013-0567. CurranAssociates, Red Hook, NY, 2013, pp. 1–13.

[18] D.R. Kassoy, Non-diffusive ignition of a gaseous reactive mixture following time-resolved,spatially distributed energy deposition, Combust. Theory Model. 18 (2014), pp. 1–16. Availableat http://www.tandfonline.com/doi/abs/10.1080/13647830.2013.863385.

[19] A.A. Sileem, D.R. Kassoy, and A.K. Hayashi, Thermally initiated detonation throughdeflagration to detonation transition, Proc. Roy. Soc. London A: Math. Phys. & En-gng Sci. 435 (1991), pp. 459–482. Available at http://rspa.royalsocietypublishing.org/cgi/doi/10.1098/rspa.1991.0156.

[20] D.R. Kassoy, J.A. Kuehn, M.W. Nabity, and J.F. Clarke, Detonation initiation on the microsec-ond time scale: DDTs, Combustion Theory and Modelling 12 (2008), pp. 1009–1047. Availableat http://www.tandfonline.com/doi/abs/10.1080/13647830802045080.

[21] J.D. Regele, D.R. Kassoy, and O.V. Vasilyev, Effects of high activation energies on acoustictimescale detonation initiation, Combust. Theory Model. 16 (2012), pp. 650–678. Available athttp://www.tandfonline.com/doi/abs/10.1080/13647830.2011.647090.

[22] X. Gu, D. Emerson, and D. Bradley, Modes of reaction front propagation from hotspots, Combust. Flame 133 (2003), pp. 63–74. Available at http://linkinghub.elsevier.com/retrieve/pii/S0010218002005412.

[23] A.K. Kapila, D.W. Schwendeman, J.J. Quirk, and T. Hawa, Mechanisms of detonation formationdue to a temperature gradient, Combust. Theory Model. 6 (2002), pp. 553–594. Available athttp://www.tandfonline.com/doi/abs/10.1088/1364-7830/6/4/302.

[24] G.J. Sharpe and M. Short, Detonation ignition from a temperature gradient for a two-step chain-branching kinetics model, J. Fluid Mech. 476 (2003), pp. 267–292. Available athttp://www.journals.cambridge.org/abstract_S0022112002002963.

[25] M.A. Liberman, A.D. Kiverin, and M.F. Ivanov, On detonation initiation by a temperaturegradient for a detailed chemical reaction models, Phys. Lett. A 375 (2011), pp. 1803–1808.Available at http://linkinghub.elsevier.com/retrieve/pii/S0375960111003410.

[26] M.D. Kurtz and J.D. Regele, Acoustic timescale characterisation of a one-dimensionalmodel hot spot, Combust. Theory Model. (2014), pp. 1–20. Available at http://www.tandfonline.com/doi/abs/10.1080/13647830.2014.934922.

[27] J.H. Chen, E.R. Hawkes, R. Sankaran, S.D. Mason, and H.G. Im, Direct numerical simula-tion of ignition front propagation in a constant volume with temperature inhomogeneities I.Fundamental analysis and diagnostics, Combust. Flame 145 (2006), pp. 128–144. Availableat http://linkinghub.elsevier.com/retrieve/pii/S0010218005003342.

[28] R. Sankaran, H.G. Im, E.R. Hawkes, and J.H. Chen, The effects of non-uniform tem-perature distribution on the ignition of a lean homogeneous hydrogen–air mixture,Proc. Combust. Inst. 30 (2005), pp. 875–882. Available at http://linkinghub.elsevier.com/retrieve/pii/S0082078404002280.

[29] R. Sankaran and H.G. Im, Characteristics of auto-ignition in a stratified iso-octane mix-ture with exhaust gases under homogeneous charge compression ignition conditions, Com-bust. Theory Model. 9 (2005), pp. 417–432. Available at http://www.tandfonline.com/doi/abs/10.1080/13647830500184108.

[30] A.E. Lutz, R.J. Kee, J.A. Miller, H.A. Dwyer, and A.K. Oppenheim, Dynamic effects ofautoignition centers for hydrogen and C1, 2-hydrocarbon fuels, 22nd Symp. (Int.) Combust.(1988), pp. 1683–1693.

Dow

nloa

ded

by [

Upp

sala

uni

vers

itets

bibl

iote

k] a

t 04:

56 1

9 N

ovem

ber

2014

Combustion Theory and Modelling 19

[31] D. Bradley and G. Kalghatgi, Influence of autoignition delay time characteristics of differentfuels on pressure waves and knock in reciprocating engines, Combust. Flame 156 (2009), pp.2307–2318. Available at http://linkinghub.elsevier.com/retrieve/pii/S0010218009002272.

[32] D. Bradley, Autoignitions and detonations in engines and ducts, Philos. Trans. A Math.Phys. Eng. Sci. 370 (2012), pp. 689–714. Available at http://www.ncbi.nlm.nih.gov/pubmed/22213665.

[33] V.N. Gamezo, D. Desbordes, and E.S. Oran, Formation and evolution of two-dimensional cellular detonations, Combust. Flame 116 (1999), pp. 154–165. Available athttp://linkinghub.elsevier.com/retrieve/pii/S0010218098000315.

[34] L. Bauwens and Z. Liang, Shock formation ahead of hot spots, Proc. Combust. Inst. 29 (2002),pp. 2795–2802. Available at http://linkinghub.elsevier.com/retrieve/pii/S1540748902803418.

[35] J.M. Austin, The role of instability in gaseous detonation, Ph.D. diss., California Institute ofTechnology, Pasadena, 2003. Available at http://thesis.library.caltech.edu/2234/1/thesis.pdf.

[36] J.D. Regele, D.R. Kassoy, and O.V. Vasilyev, Numerical modeling of acoustic timescale det-onation initiation, in Proceedings of the 46th AIAA Aerospace Sciences Meeting includingthe New Horizons Forum and Aerospace Exposition, 7–10 January 2008, Reno, NV, PaperNumber AIAA-2013-1037. Curran Associates, Red Hook, NY, 2008, pp. 1–19. Available athttp://scales.colorado.edu/vasilyev/Publications/AIAA-2008-1037.pdf.

[37] J.D. Regele, D.R. Kassoy, A. Vezolainen, and O.V. Vasilyev, Purely gasdynamic multidi-mensional indirect detonation initiation using localized acoustic timescale power deposi-tion, in Proceedings of the 51st AIAA Aerospace Sciences Meeting including the New Hori-zons Forum and Aerospace Exposition, 7–10 January 2013, Grapevine, TX, Paper Num-ber AIAA-2013-1172. Curran Associates, Red Hook, NY, 2013, pp. 1–15. Available athttp://scales.colorado.edu/vasilyev/Publications/AIAA-2013-1172.pdf.

[38] J.D. Regele, D.R. Kassoy, A. Vezolainen, and O.V. Vasilyev, Indirect detonation initiation usingacoustic timescale thermal power deposition, Phys. Fluids 25 (2013), pp. 1–2.

[39] G. Thomas, Some observations on the initiation and onset of detonation, Philos. Trans.A Math. Phys. Eng. Sci. 370 (2012), pp. 715–739. Available at http://www.ncbi.nlm.nih.gov/pubmed/22213666.

[40] M. Radulescu, G. Sharpe, J. Lee, C. Kiyanda, A.J. Higgins, and R. Hanson, The ignitionmechanism in irregular structure gaseous detonations, Proc. Combust. Inst. 30 (2005), pp.1859–1867. Available at http://linkinghub.elsevier.com/retrieve/pii/S0082078404001006.

[41] J.H.S. Lee and I.O. Moen, The mechanism of transition from deflagration to detonation in vaporcloud explosions, Prog. Energy Combust. Sci. 6 (1980), pp. 359–389.

[42] S.P.M. Bane, Spark ignition: Experimental and numerical investigation with application toaviation safety, Ph.D. diss., California Institute of Technology, Pasadena, 2010. Available athttp://thesis.library.caltech.edu/5868/1/thesis_SBane.pdf.

[43] N.K.R. Kevlahan and O.V. Vasilyev, An adaptive wavelet collocation method for fluid–structureinteraction at high Reynolds numbers, SIAM J. Sci. Comput. 26 (2005), pp. 1894–1915.

[44] O.V. Vasilyev and C. Bowman, Second-generation wavelet collocation method for the solutionof partial differential equations, J. Comput. Phys. 165 (2000), pp. 660–693. Available athttp://linkinghub.elsevier.com/retrieve/pii/S0021999100966385.

[45] J.D. Regele and O.V. Vasilyev, An adaptive wavelet-collocation method for shock computations,Int. J. Comput. Fluid Dyn. 23 (2009), pp. 503–518.

[46] B. van Leer, Flux-vector splitting for the Euler equations, in Proceedings of the 8th In-ternational Conference on Numerical Methods for Engineering, Aachen, Germany, 1982,Lecture Notes in Physics Vol. 170. Springer, 1982, pp. 507–512. Available at http://link.springer.com/chapter/10.1007/3-540-11948-5_66.

[47] W.K. Anderson, J.L. Thomas, and B. Van Leer, Comparison of finite volume flux vector split-tings for the Euler equations, AIAA J. 24 (1986), pp. 1453–1460. Available at http://arc.aiaa.org/doi/abs/10.2514/3.9465.

Dow

nloa

ded

by [

Upp

sala

uni

vers

itets

bibl

iote

k] a

t 04:

56 1

9 N

ovem

ber

2014