High-ResolutionNumericalSimulationandAnalysisof...

Hindawi Publishing CorporationJournal of CombustionVolume 2011, Article ID 738969, 18 pagesdoi:10.1155/2011/738969

Research Article

High-Resolution Numerical Simulation and Analysis ofMach Reflection Structures in Detonation Waves in Low-PressureH2–O2–Ar Mixtures: A Summary of Results Obtained with theAdaptive Mesh Refinement Framework AMROC

Ralf Deiterding

Computer Science and Mathematics, Oak Ridge National Laboratory, P.O. Box 2008 MS-6367, Oak Ridge, TN 37831, USA

Correspondence should be addressed to Ralf Deiterding, [email protected]

Received 30 October 2010; Accepted 21 February 2011

Academic Editor: Lea-Der Chen

Copyright © 2011 Ralf Deiterding. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Numerical simulation can be key to the understanding of the multidimensional nature of transient detonation waves. However, theaccurate approximation of realistic detonations is demanding as a wide range of scales needs to be resolved. This paper describesa successful solution strategy that utilizes logically rectangular dynamically adaptive meshes. The hydrodynamic transport schemeand the treatment of the nonequilibrium reaction terms are sketched. A ghost fluid approach is integrated into the method to allowfor embedded geometrically complex boundaries. Large-scale parallel simulations of unstable detonation structures of Chapman-Jouguet detonations in low-pressure hydrogen-oxygen-argon mixtures demonstrate the efficiency of the described techniques inpractice. In particular, computations of regular cellular structures in two and three space dimensions and their development undertransient conditions, that is, under diffraction and for propagation through bends are presented. Some of the observed patternsare classified by shock polar analysis, and a diagram of the transition boundaries between possible Mach reflection structures isconstructed.

1. Introduction

Reacting flows have been a topic of on-going research sincemore than one hundred years. The interaction betweenhydrodynamic flow and chemical kinetics can be extremelycomplex and even today many phenomena are not very wellunderstood. One of these phenomena is the propagation ofdetonation waves in gaseous media. While detonations prop-agate at supersonic velocities between 1000 and 2000 m/s,they inhibit nonnegligible instationary substructures in themillimeter range. Experimental observations can provideonly limited insight and it is therefore not surprisingthat the understanding of the multidimensionality has im-proved little since the first systematic investigations [1,2]. An alternative to laboratory experiments is numericalsimulations of the governing thermo- and hydrodynamicequations. But the additional source terms modeling detailednonequilibrium chemistry are often stiff and introduce new

and extremely small scales into the flow field. Their accuratenumerical representation requires finite volume meshes withextraordinarily high local resolution.

In this paper, we summarize a significant body of mul-tiyear work [3–8] devoted to simulating multidimensionaldetonations with detailed and highly stiff chemical kineticson parallel machines with distributed memory, especially onclusters of standard personal computers. The objective ofall presented computations is to resolve the transverse-wavestructures, that form the characteristic cellular structuresintrinsic to detonation propagation in two and three spacedimensions, accurately and without computational ambigu-ity. We sketch the design of AMROC (Adaptive Mesh Refine-ment in Object-oriented C++), a freely available dimension-independent mesh adaptation framework for time-explicitCartesian finite volume methods on distributed memorymachines [9, 10], and discuss briefly the locality-preservingrigorous domain decomposition technique it employs [11].

2 Journal of Combustion

0

0.1

0.2

0.3

0.4

ρ(k

g/m

3)

15 10 5 0

298

1000

2000

3000

(K)

ρ (left axis)T (right axis)

ZND solution

Figure 1: Density and temperature field of a self-sustaininghydrogen-oxygen detonation (dCJ ≈ 1626.9 m/s, lig ≈ 1.404 mm)calculated with the ZND theory (cf. Section 4.2). The abscissa showsthe distance behind the detonation front in mm.

The framework provides a generic implementation of theadaptive mesh refinement (AMR) algorithm after Berger andColella [12] designed especially for the solution of hyperbolicfluid flow problems on logically rectangular grids. The ghostfluid approach is integrated into the refinement algorithm toallow for complex moving boundaries represented implicitlyby additional scalar level set variables [13, 14]. Briefly, wedescribe the employed numerical methods and the treat-ment of the non-equilibrium reaction terms. In Section 6,several two- and three-dimensional simulations of cellulardetonation structures in low-pressure hydrogen-oxygen-argon mixtures simulated in purely Cartesian geometryand with embedded static boundaries are presented. Thepaper concludes by demonstrating that the high-resolutionadaptive computations allow a detailed quantitative analysisof the observed transverse-wave patterns. It is found thatnonreactive shock wave reflection theory [15], extended tothermally perfect gas mixtures, is applicable to classify theobserved transient triple point structures unambiguously astransitional and double Mach reflection patterns.

2. Detonation Theory

A detonation is a shock-induced combustion wave thatinternally consists of a discontinuous hydrodynamic shockwave followed by a smooth region of decaying combustion.The adiabatic compression due to the passage of the shockrises the temperature of the combustible mixture abovethe ignition limit. The reaction results in an energy releasedriving the shock wave forward. In a self-sustaining detona-tion, shock and reaction zone propagate essentially with anidentical speed dCJ that is approximated to good accuracy bythe classical Chapman-Jouguet (CJ) theory, see for example,[16]. But up to now, no theory exists that describesthe internal flow structure satisfactory. The Zel’dovich-von

λ

A

B

D

C

L

Figure 2: Regular detonation structure at three different time stepsdrawn on top of the triple point trajectories.

Neumann-During (ZND) theory is widely believed to repro-duce the one-dimensional detonation structure correctly (seee.g., Figure 1), but already early experiments [1] uncoveredthat the reduction to one space dimension is not evenjustified in long combustion devices. It was found thatdetonation waves usually exhibit nonnegligible instationarymultidimensional substructures and do not remain pla-nar. The multidimensional instability manifests itself ininstationary shock waves propagating perpendicular to thedetonation front. A complex flow pattern is formed aroundeach triple point, where the detonation front is intersected bya transverse shock. Pressure and temperature are increasedremarkably in a triple point and the chemical reaction isenhanced drastically giving rise to an enormous local energyrelease. Hence, the accurate representation of triple points isessential for safety analysis, but also in technical applications,for example, in the pulse detonation engine. Some particularmixtures, for example, low-pressure hydrogen-oxygen withhigh argon dilution, are known to produce very regulartriple point movements. The triple point trajectories formregular “fish-scale” patterns, so-called detonation cells, witha characteristic length L and width λ (compare Figure 2).

Figures 2 and 3 display the hydrodynamic flow patternof a detonation with regular cellular structure as it is knownsince the early 1970s, see for example, [2] or [17]. Figure 3shows the periodic wave configuration around a triple pointin detail. It consists of a Mach reflection, a flow patternwell known from nonreactive supersonic hydrodynamics(see e.g., [15, 18] for details). The undisturbed detonationfront is called the incident shock, while the transverse wavetakes the role of the reflected shock. The triple point is drivenforward by a strong shock wave, called Mach stem. Machstem and reflected shock enclose the slip line, the contactdiscontinuity.

The Mach stem is always much stronger than the incidentshock, which results in a considerable reduction of theinduction length lig, the distance between leading shock andmeasurable reaction. The shock front inside the detonationcell travels as two Mach stems from point A to the line BC. Inthe points B and C, the triple point configuration is inverted

Journal of Combustion 3

nearly instantaneously and the front in the cell becomes theincident shock. Along the symmetry line AD, the change issmooth and the shock strength decreases continuously. InD, the two triple points merge exactly in a single point. Theincident shock vanishes completely and the slip line, whichwas necessary for a stable triple point configuration betweenMach stem and incident shock, is torn off and remainsbehind. Two new triple points with two new slip lines developimmediately after D.

3. Governing Equations

The appropriate model for detonation propagation in pre-mixed gases with realistic chemistry are the inviscid Eulerequations for multiple thermally perfect species with reactivesource terms [16, 19]. These equations form a system ofinhomogeneous hyperbolic conservation laws that reads

∂tρi +∇ · (ρi�u) =Wiωi,

∂t(ρ�u)

+∇ · (ρ�u⊗ �u) +∇p = 0,

∂t(ρE)

+∇ · ((ρE + p)�u) = 0,

(1)

with i = 1, . . . ,K . Herein, ρi denotes the partial density ofthe ith species and ρ = ∑K

i=1 ρi is the total density. Theratios Yi = ρi/ρ are called mass fractions. We denote thevelocity vector by �u and E is the specific total energy. Weassume that all species are ideal gases in thermal equilibrium,and the hydrostatic pressure p is given as the sum ofthe partial pressures pi = RTρi/Wi with R denotingthe universal gas constant and Wi the molecular weight,respectively. The evaluation of the last equation requiresthe previous calculation of the temperature T. As detailedchemical kinetics typically require species with temperature-dependent material properties, each evaluation of T involvesthe approximate solution of an implicit equation by Newtoniteration [3].

The chemical production rate for each species is derivedfrom a reaction mechanism of J chemical reactions as

ωi =J∑

j=1

(νrji − ν

fji

)⎡

⎣kfj

K∏

l=1

(ρlWl

)νfjl

− krj

K∏

l=1

(ρlWl

)νrjl

⎤

⎦, (2)

with νf /rji denoting the forward and backward stoichiometric

coefficients of the ith species in the jth reaction. The rate

expressions kf /rj (T) are calculated by an Arrhenius law, see

for example, [16].

4. Numerical Methods

We use the time-operator splitting approach or method offractional steps [20] to decouple hydrodynamic transportand chemical reaction numerically. This technique is mostfrequently used for time-dependent reactive flow computa-tions. The homogeneous Euler equations and the usually stiffsystem of ordinary differential equations

∂tρi =Wiωi(ρ1, . . . , ρK ,T

), i = 1, . . . ,K , (3)

Head ofreaction zone

E

F

GMach stem

Triple point

Incident shock

Transversewave

Trajectory

Figure 3: Enlargement of the Mach reflection pattern around aperiodical triple point configuration. E: reflected shock, F: slip line,G: diffusive extension of slip line with flow vertex.

are integrated successively with the data from the precedingstep as initial condition. The advantage of this approach isthat a globally coupled implicit problem is avoided and atime-implicit discretization, which accounts for the stiffnessof the reaction terms, needs to be applied only local ineach finite volume cell. We use a semi-implicit Rosenbrock-Wanner method [21] to integrate (3) within each cell.Temperature-dependent material properties are derivedfrom lookup tables that are constructed during start-up ofthe computational code. The expensive reaction rate expres-sions (2) are evaluated by a mechanism-specific Fortran-77function, which is produced by a source code generator ontop of the Chemkin-II library [22] in advance. The codegenerator implements the reaction rate formulas without any

loops and inserts constants such as νf /rji directly into the code.

Since detonations involve supersonic shock waves, weuse a finite volume discretization that achieves a properupwinding in all characteristic fields (cf. [4, 6, 7]). Thescheme utilizes a quasi-one-dimensional approximate Rie-mann solver of Roe-type [23] and is extended to multiplespace dimensions via the method of fractional steps, seefor example, [24]. To circumvent the intrinsic problem ofunphysical total densities and internal energies near vacuumdue to the Roe linearization, cf. [25], the scheme has thepossibility to switch to the simple, but extremely robustHarten-Lax-Van Leer (HLL) Riemann solver. Negative massfraction values are avoided by a numerical flux modification[26]. Finally, the occurrence of the disastrous carbuncle phe-nomena, a multidimensional numerical crossflow instabilitythat destroys every simulation of strong grid-aligned shocksor detonation waves completely and has been documentedfirst by Quirk [27], is prevented by introducing a smallamount of additional numerical viscosity in a multidimen-sional way, cf. [28]. A detailed derivation of the entire Roe-HLL scheme including all necessary modifications can befound in [3]. This hybrid Riemann solver is extended toa second-order accurate method with the MUSCL-Hancock


ρi, j−1

uj−1

pj−1

ρi, ju j

p j

ρi, j2w − uj

p j

ρi, j−1

2w − uj−1

pj−1

uj

w

2w − uj

Figure 4: Moving wall boundary conditions for Euler equations inone space dimension.

variable extrapolation technique by Van Leer (see [24] for adetailed derivation).

4.1. Embedded Complex Boundaries. Higher-order shock-capturing finite volume schemes are most efficient on rect-angular Cartesian grids. In order to consider geometricallycomplex moving boundaries within the Cartesian upwindscheme outlined above, we use some of the finite volumecells as ghost cells to enforce immersed boundary conditions,cf. [13] or [29]. Their values are set immediately before theoriginal numerical update to model moving embedded wallsand can be understood as part of the numerical scheme. Theboundary geometry is mapped onto the Cartesian mesh byemploying a scalar level set function ϕ that stores the signeddistance to the boundary surface and allows the efficientevaluation of the boundary outer normal in every mesh pointas �n = ∇ϕ/|∇ϕ| [30]. A cell is considered to be a valid fluidcell within the interior if the distance in the cell midpoint ispositive and is treated as exterior otherwise. The numericalstencil by itself is not modified, which causes a slight diffu-sion of the boundary location throughout the method andresults in an overall nonconservative scheme. We alleviatesuch errors and the unavoidable staircase approximationof the boundary with this approach effectively by usingthe dynamic mesh adaptation technique described in Sec-tion 5 to also refine the Cartesian mesh appropriately alongthe boundary. Other authors have also presented cut-celltechniques that utilize the correct boundary flux [31, 32];however, the proposed numerical circumventions of thesevere time step restriction in time-explicit schemes, that canresult from small cells created by the boundary intersection,are inherently complicate and most approaches have notbeen extended successfully to three space dimensions yet.

For the inviscid Euler equations (1), the boundary con-dition at a rigid wall moving with velocity �w is �u · �n = �w · �n.Enforcing the latter with ghost cells, in which the discretevalues are located in the cell centers, requires the mirroring ofthe primitive values ρi, �u, p across the embedded boundary.The normal velocity in the ghost cells is set to (2�w·�n−�u·�n)�n,while the mirrored tangential velocity remains unmodified.The construction of the velocity vector within the ghost cellstherefore reads

�u′ = (2�w ·�n− �u ·�n)�n +(�u ·�t

)�t

= 2((�w − �u) · �n)�n + �u

(4)

Figure 5: Interpolation from interior cells to construct mirroredvalues to be used within internal ghost cells (gray).

with �t denoting the boundary tangential. This constructionof discrete values in ghost cells (indicated by gray) for (1) isdepicted in one space dimension in Figure 4.

The utilization of mirrored ghost cell values in a ghostcell center �x requires the calculation of spatially interpolatedvalues in the point

�x = �x + 2ϕ�n (5)

from neighboring interior cells. For instance in two spacedimensions, we employ a bilinear interpolation betweenusually four adjacent cell values, but directly near theboundary the number of interpolants needs to be decreased,cf. Figure 5. It has to be underlined that an extrapolation insuch situations is inappropriate for hyperbolic problems withdiscontinuities like detonation waves that necessarily requirethe monotonicity preservation of the numerical solution.Figure 5 highlights the reduction of the interpolation stencilfor some exemplary cases close to the embedded boundary.The interpolation locations according to (5) are indicated bythe origins of the red arrows. See [7] for further details.

4.2. Meshes for Detonation Simulation. Numerical simula-tions of detonation waves require computational meshesthat are able to represent the strong local flow changesdue to the reaction correctly. In particular, the shock of adetonation wave with detailed kinetics can be very sensitiveto changes of the reaction behind, and if the mesh is toocoarse to resolve all reaction details correctly, the RiemannProblem at the detonation front is changed leading to awrong speed of propagation. We make a simple discretizationtest in order to illustrate how fine computational meshesfor accurate detonation simulations in fact have to be.Figure 1 displays the density and temperature profile for thefrequently studied H2 : O2 : Ar Chapman-Jouguet detonationwith molar ratios of 2 : 1 : 7 in the unburned mixture with


0

0.03

0.06

0.09

YH

2O

5 0

1 Pts/lig

(a)

0

2e−6

4e−6

6e−6

YH

2O

2

5 0

4 Pts/lig

(b)

Figure 6: Representation of two mass fraction distributions on grids with different mesh widths for the detonation shown in Figure 1. Thedots represent the values in the center of a finite volume. The abscissas show the distance behind the detonation front in mm.

initial temperature T0 = 298 K and pressure p0 = 6.67 kPa.Figure 6 shows the distributions of YH2O and YH2O2 accordingto the ZND detonation model for this detonation discretizedwith different grids. Apparently, a resolution of 4 finitevolumes per induction length (4 Pts/lig with lig ≈ 1.404 mm)would not be sufficient to capture the maximum of theintermediate product H2O2 correctly. This requires at least5 to 6 Pts/lig; however, in triple points significantly finerresolutions can be expected. As discretizations of realisticcombustors with such fine uniform meshes typically wouldrequire up to 109 points in the two- and up to 1012 points inthe three-dimensional case, the application of a dynamicallyadaptive mesh refinement technique is imperative.

5. Adaptive Mesh Refinement

In order to supply the required temporal and spatial reso-lution efficiently, we employ the block-structured adaptivemesh refinement (AMR) method after Berger and Colella[12], which is tailored especially for hyperbolic conservationlaws on logically rectangular finite volume grids. We haveimplemented the AMR method in a generic, dimension-independent object-oriented framework in C++. It is calledAMROC (Adaptive Mesh Refinement in Object-orientedC++) and is free of charge for scientific use [9] (Notethat the most recent version V2.0 of AMROC is part ofthe Virtual Test Facility software [10], available at http://www.cacr.caltech.edu/asc.). An efficient parallelization strat-egy for distributed memory machines has been found, andthe codes can be executed on all systems that provide the MPIlibrary.

Figure 7: Structured AMR employs a hierarchy of successivelyembedded rectangular subgrids.

5.1. Structured AMR Method. Instead of replacing single cellsby finer ones, as it is done in cell-oriented refinement tech-niques, structured AMR methods follow a patch-orientedapproach. Cells being flagged by various error indicators(shaded in Figure 7) are clustered with a special algorithm[33] into nonoverlapping rectangular grids. Refinementgrids are derived recursively from coarser ones, and a hier-archy of successively embedded levels is thereby constructed,


see for example, Figure 7. All mesh widths on level l are rl-times finer than on level l − 1, that is, Δtl := Δtl−1/rl andΔxn,l := Δxn,l−1/rl with rl ≥ 2 for l > 0 and r0 = 1, anda time-explicit finite volume scheme (in principle) remainsstable on all levels of the hierarchy. The recursive integrationorder visualized in Figure 8 is an important difference tousual unstructured adaptation strategies and one of the mainreasons for the high efficiency of the approach.

The numerical scheme is applied on level l by callinga single-grid routine in a loop over all subgrids. The subgridsbecome computationally decoupled by employing additionalghost cells around each computational grid. Three types ofdifferent ghost cells have to be considered in the sequentialcase, see Figure 9. Cells outside of the root domain are usedto implement physical boundary conditions. Ghost cellsoverlaid by a grid on level l have a unique interior cell an-alogue and are set by copying the data value from the grid,where the interior cell is contained (synchronization). Onthe root level, no further boundary conditions need to beconsidered, but for l > 0 also internal boundaries can occur.They are set by a conservative time-space interpolation fromtwo previously calculated time steps of level l − 1.

Beside a general data tree that stores the topology of thehierarchy, the AMR method requires at most two regulararrays assigned to each subgrid. They contain the discretevector of state for the actual and updated time step. Theregularity of the data allows high performance on vectorand superscalar processors and cache optimizations. Smalldata arrays are effectively avoided by leaving coarse level datastructures untouched when higher-level grids are created.Values of cells covered by finer subgrids are overwritten byaveraged fine grid values subsequently. This operation leadsto a modification of the numerical stencil on the coarse meshand requires a special flux correction in cells abutting a finegrid. The correction replaces the coarse grid flux along thefine grid boundary by a sum of fine fluxes and ensures thediscrete conservation property of the hierarchical methodat least for purely Cartesian problems without embeddedboundaries. See [12] or [3] for details.

5.2. Parallelization. Up to now, various reliable implemen-tations of the AMR method for single processor computershave been developed, see for instance [34] or [35]. Even theusage of parallel computers with shared memory is straight-forward, because a time-explicit scheme allows the parallelcalculation of the gridwise numerical update, cf. [33].However, the question for an efficient parallelization strategybecomes more delicate for distributed memory architecturessince on such machines the costs for communication cannot be neglected. Due to the technical difficulties in imple-menting dynamical adaptive methods in distributed memoryenvironments, only few parallelization strategies have beenconsidered in practice yet, cf. [36] or [37].

In the AMROC framework, we follow a rigorous domaindecomposition approach and partition the AMR hierarchyfrom the root level on. The key idea is that all higher leveldomains are required to follow this “floor-plan”. A carefulanalysis of the AMR algorithm uncovers that the only parallel

Level 2r2 = 2

Level 1r1 = 4

Root levelr0 = 1

3 4 6 7 9 10 12 13

1

2 5 8 11

Time

Regridding of finer levels

Base level stays fixed

Figure 8: Structured AMR allows for recursively refined temporalintegration.

InterpolationSynchronizationPhysical boundary

Figure 9: Sources of ghost cell values at the boundaries of an AMRsubgrid.

operations under this paradigm are ghost cell synchro-nization, redistribution of the AMR hierarchy, and theapplication of the previously mentioned flux correctionterms. Interpolation and averaging, but in particular thecalculation of the flux corrections, remain strictly local, cf.[3, 11]. Currently, we employ a generalization of Hilbert’sspace-filling curve [38] to derive load-balanced root leveldistributions at runtime. The entire AMR hierarchy is con-sidered by projecting the accumulated work from higherlevels onto the root level cells. Although rigorous domaindecomposition does not lead to a perfect balance of workloadon single levels, good scale-up is usually achieved formoderate CPU counts. Figure 10 shows a representative scal-ability test for a three-dimensional spherical shock waveproblem for the numerically inexpensive Euler equationsfor a single polytropic gas without chemical reaction. Roe’sapproximate Riemann solver within the multidimensionalWave Propagation Method [39] is used as efficient single-grid scheme. The test was run on a Pentium-4-2.4 GHz dualprocessor cluster connected with Quadrics Interconnect.The base grid has 323 cells and two additional levels with


10

100

1000

Seco

nds

/tim

est

ep

4 8 16 32 64 128 256

CPUs

Figure 10: Representative AMROC scale-up test for fixed problemsize. The abscissa shows the number of processors.

refinement factors 2 and 4. The adaptive calculation uses∼7.0 M cells in each time step instead of 16.8 M cells in theuniform case. The calculation on 256 CPUs employs between∼1,500 and ∼1,700 subgrids on each level. Displayed arethe average costs for each root level time step, whichinvolves two time steps on the middle level and eight onthe highest. All components of the dynamically adaptivealgorithm, especially regridding and parallel redistributionare activated to obtain realistic results. Although we utilize asingle-grid update routine in Fortran 77 in a C++ frameworkwith full compiler optimization, the fraction of the timespent in this Fortran routine is 90.5% on four and still 74.9%on 16 CPUs. Hence, Figure 10 shows a satisfying scale-up forat least up to 64 CPUs.

AMROC’s hierarchical data structures are derived fromthe DAGH (Distributive Adaptive Grid Hierarchies) package[37] and are implemented completely in C++. A redesignof major parts of DAGH was necessary to allow the AMRalgorithm as it was described in the previous section.Currently, AMROC consists of approximately 46,000 lines ofcode in C++ and approximately 6,000 lines for visualizationand data conversion.

6. Numerical Results

Self-sustaining CJ detonations in low-pressure hydrogen-oxygen mixtures with high-argon dilution are ideal can-didates for fundamental detonation structure simulations,because they are known to produce very regular detonationcell patterns [2]. In the following, we will simulate suchmixtures with the objective of resolving the Mach reflectionstructures around triple points with high accuracy. Althoughthe detailed hydrodynamic shock wave structure of periodi-cally oscillating detonations in rectangular two-dimensionalchannels has been fairly well established by now [3, 40,41], numerous open questions remain for non-rectangulargeometries and three space dimensions.

6.1. Two-Dimensional Cellular Structure. We start by repro-ducing the detonation structure result of Hu et al. [41], utiliz-ing the dynamically adaptive computing approach presented

(a)

(b)

Figure 11: Color plot of temperature and schlieren plot of densityon refinement in 1st half of a detonation cell.

(a)

(b)

Figure 12: Color plots of temperature and schlieren plots of densityon refinement in 2nd half of a detonation cell.

in the previous sections. As initial conditions, the analyticalsolution according to the one-dimensional ZND theory forthe CJ detonation of Figure 1 is extended to multiple spacedimensions, and transverse disturbances are initiated byplacing a small rectangular unreacted pocket behind thedetonation front, cf. [40] or [3]. Throughout this paper, onlythe hydrogen-oxygen reaction mechanism extracted fromthe larger hydrocarbon mechanism assembled by Westbrook[42] is used. The mechanism consists of 34 elementaryreactions and considers the 9 species H, O, OH, H2, O2,H2O, HO2, H2O2, and Ar. According to the ZND theory,the induction length, the distance between leading shockand head of reaction zone in one space dimension, is


Table 1: Breakdown of computational time for the two-dimensional cellular structure simulation.

Task %

Fluid dynamics 33.5

Chemical kinetics 36.8

Boundary setting 26.8

Recomposition 2.4

Misc. 0.5

Total [h CPU] 2150

lig ≈ 1.404 mm for this mechanism in the above configura-tion. The detonation velocity is ∼1626.9 m/s.

Our shown dynamically adaptive computation has effec-tively four times higher resolution (44.8 Pts/lig) than earliercalculations [40] and is similarly resolved as the result by Huet al. [41] on a uniform mesh. Unfortunately, no technicaldetails are reported for this simulation. In our case, thecalculation was run on a small Beowulf-cluster of 7 Pentium3-850 MHz-CPUs connected with a 1 Gb-Myrinet networkand required 2150 h CPU time (12.8 days wall time).

We simulate a physical time of 800 μs. After an initialperiod of ∼200 μs, the simulation starts to exhibit veryregular detonation cells with oscillation period ∼32 μs.Exploiting the spatial periodicity, we concentrate on thehigh-resolution simulation of a single cell. The calculationis done in a frame of reference attached to the detonationand requires just the computational domain 10 cm × 3 cm.The adaptive run uses a root level grid of 200 × 40cells and two refinement levels with r1,2 = 4, where∼3550 root level time steps (automatically chosen based onCCFL ≈ 0.95) with average Δt0 ≈ 0.225μs are taken (Allcomputations herein used time steps automatically chosenbased on the largest Courant-Friedrichs-Lewy (CFL) numberCCFL encountered throughout the last root level time step.In evaluating CCFL, our practical implementation considersall hierarchical time steps, yet only adjustment of Δt0 ispermitted. This choice avoids having to deal with additionalsteps during the recursive integration procedure, cf. Figure 8.If the maximal CFL number exceeds a given threshold, thesoftware has the capability of repeating the entire root leveltime step.) A physically motivated combination of scaledgradients and heuristically estimated relative errors is appliedas adaptation criteria. These criteria are used similarly togenerate the refinement of all computations throughoutthis paper (see [3, 7] for details). Note that in order tosimulate the shock wave patterns around the triple points,it suffices to concentrate the computational expense on thedetonation front. The flow behind a detonation is basicallysupersonic, and resolution can be dropped downstreamwithout affecting the approximation of the front structures.Two typical snapshots with the corresponding refinementclose to the end of the simulation, when the oscillation isperfectly regular, are displayed in the Figures 11 and 12.

A breakdown of the computational time spent in thecomputationally most expensive parts for this simulationis given in Table 1. The numerical integration is split intothe two main steps of the fractional step method: the fluid

Table 2: Breakdown of the computational time for the first three-dimensional rectangular structure simulation.

Task %

Fluid dynamics 36.7

Chemical kinetics 20.3

Boundary setting 38.4

Recomposition 1.9

Misc. 2.7

Total [h CPU] 3800

Figure 13: Isolines of density on refinement levels.

dynamic update and the integration of the reactive sourceterm. A ratio between these portions of 1.1 is in good agree-ment with timings reported by Oran and coworkers [40]and underlines that a reasonable load-balancing is achievedwithout considering the number of subcycling time stepsin the source term integration in the workload estimationexplicitly. The third important portion in Table 1 is the timefor setting the ghost cell values at subgrid boundaries, whichis dominated (∼90%) by parallel synchronization.

6.2. Three-Dimensional Cellular Structure. On 24 Athlon-1.4 GHz double-processor nodes (2 Gb-Myrinet intercon-nect), our approach allowed a sufficiently resolved com-putation of the three-dimensional cellular structure of ahydrogen-oxygen detonation. The maximal effective resolu-tion of this calculation is 16.8 Pts/lig and the run required3800 h CPU time (3.3 days wall time). Our adaptive resultsare in perfect agreement with the calculations by Tsuboi et al.[43] for the same configuration obtained on a uniform meshon a superscalar vector machine. A snapshot of the regulartwo-dimensional solution of the preceding section is used toinitialize a three-dimensional oscillation in the x2-directionand disturbed with an unreacted pocket in the orthogonaldirection. We use a computational domain of the size 7 cm×1.5 cm × 3 cm that exploits the symmetry of the initial data,but allows the development of a full detonation cell in thex3-direction. Again, we simulate a physical time of 800 μs.The AMROC computation uses a two-level refinement with


(a) (b)

Figure 14: Schlieren plots of density, mirrored at x2 = 0 cm for visualization, 5.0 cm < x1 < 7.0 cm.

L1′ L2′ L1′

L2 L1 L2

L1′ L2′ L1′

L2 L1 L2

L1′ L2′ L1′

MI II MI II

MM MI MM MI

MI II MIII

MIMMMIMM

IIMIIIMI

Figure 15: Schematic front view of the periodic triple point linestructure. The gray square shows the situation of (a) of Figure 14;the black dotted square corresponds to (b) at the same time.

r1 = 2 and r2 = 3 on a base grid of 140 × 12 × 24 cells (cf.Figure 13) and utilizes between 1.3 M and 1.5 M cells, insteadof 8.7 M cells like a uniformly refined grid. 3431 root leveltime steps with average size Δt0 ≈ 0.233μs are taken. Notehow Table 2 clearly reflects the increased expense in solvingthe hydrodynamic equations in three space dimensions.

After a settling time of about 20 periods (t ≈ 600μs), aregular cellular oscillation with identical strength in x2- andx3-direction can be observed. In both transverse directions,the strong two-dimensional oscillations is present and forcesthe creation of rectangular detonation cells of 3 cm width.Like in the two-dimensional case, the oscillation periodis ∼32 μs. The transverse waves form triple point lines inthree space dimensions. During a complete detonation cell,the four lines remain mostly parallel to the boundary and

hardly disturb each other, cf. Figure 14. The characteristictriple point pattern can therefore be observed in all planesperpendicular to a triple point line. Each triple point line isdriven forward by a Mach stem line into an incident shockregion. A region that is a Mach stem in one direction can beeither a Mach stem or an incident shock in the orthogonaldirection. The same is true for the incident shock, andwe consequently have three different types of rectangularshock regions at the detonation front: Mach stem-Mach stem(MM), Incident-Incident (II), and mixed type regions (MI)[44]. An MM region is bounded only by Mach stems andexpands in the x2- and x3-direction. An II rectangle is formedonly by incident shocks and shrinks in both directions. TheMI region is of mixed type and expands in one direction,while it shrinks in the other. Figure 15 displays the differentrectangular shock regions for a specific snapshot. It dependson the initial flow field whether the upper or the lowersituation of Figure 14 appears in the simulation. UnlikeWilliams et al. [44], who presented a similar calculation foran overdriven detonation with simplified one-step reactionmodel, we notice no phase-shift between both transversedirections. In all our computations, only this regular three-dimensional mode or a purely two-dimensional mode withtriple point lines just in x2- or x3-direction did occur. Thethree-dimensional mode of propagation, called “rectangular-mode-in-phase”, has also been found in experiments withhydrogen-oxygen CJ detonations [45].

In order to verify the independence of the numericalsolution from the computational grid, we carry out asimilar three-dimensional simulation on a grid rotated by45 degree and utilize periodic boundary conditions inthe x2- and x3-directions. The domain has the size 7 cm× 3

√2 cm × 3

√2 cm and the base grid the resolution

140 × 34 × 34. With a two-level refinement with factorsr1 = 2 and r2 = 3 and time step parameters as before,the simulation has the same effective resolution as theprevious one and is consequently about four times more


(a)

(b)

Figure 16: Schlieren plots of YOH in the first half of a detonationcell for a rectangular (a) and a diagonal (b) structure simulation,5.0 cm < x1 < 7.0 cm.

expensive. The lower graph of Figure 16 shows a typicalsnapshot that corresponds well to the situation shown inthe upper graph, which illustrates that the “diagonal-mode-in-phase”, sometimes observed in experiments, is only adifferent, boundary-condition-dependent, appearance of thesame three-dimensional mode of oscillation. Diagonal andrectangular oscillation are simultaneously present in theperiodic situation depicted in Figure 15, and it depends onlyon the size and orientation of the observation window whichoscillation type is actually seen. The graphs of Figure 16 showschlieren plots of the mass fraction for YOH overlaid by a blueisosurface of the density, which visualizes the inductionlength lig. The induction length shows significantly largervariations than in the two-dimensional case. The oscillationperiod is identical in two- and three space dimensionswhich underlines that the basic two-dimensional instabilityis exactly preserved in three space dimensions, but that itsmanifestation in the hydrodynamic flow field is different.

(a)

(b)

Figure 17: Schlieren plots of the density for a detonation diffractingout of the two different tubes. (a) Detonation failure for the width8 λ, (b) re-ignition for 10 λ.

6.3. Structure of Diffracting Detonations. Experiments haveshown that the behavior of planar CJ detonations propagat-ing out of tubes into unconfinement is determined mainlyby the width of the tube. For square tubes, the critical tubewidth has been found to be of the order of 10-times thecell width, that is, 10 λ [17]. For widths significantly below10 λ, the process of shock wave diffraction causes a pressuredecrease at the head of the detonation wave below the limitof detonability across the entire tube width. Hydrodynamicshock and reaction front decouple and the detonation decaysto a shock-induced flame. This observation is independentof a particular mixture. While the successful transmission ofthe detonation is hardly disturbed for tubes widths 10 λ,a backward-facing wave reignites the detonation in thepartially decoupled region for widths of ∼10 λ and createsconsiderable vortices.

We are interested in the decoupling of shock and reactionand also in the reignition phenomenon. Therefore, we sim-ulate the two-dimensional diffraction of the H2 : O2 : Ar CJdetonation for the tube widths 8 λ and 10 λ. A periodicallyreproduced snapshot of the regular oscillating detonation ofSection 6.1 propagating into unreacted gas at rest is usedas initial condition. This is a reasonable idealization forthe flow situation in real detonation tubes directly beforethe experimental setup. The symmetry of the problem isexploited by simulating just one half.

The adaptive simulations utilize a base grid of 508× 288cells and use four levels of refinement with r1,2,3 = 2, r4 = 4.


(a)

(b)

Figure 18: Density distribution on refinement levels after thedetonation has left the tube (a) and distribution of computationaldomain to 48 CPUs (b) for 10 λ.

(a)

(b)

Figure 19: Schlieren plots of the density (left) show the partialdecoupling of shock and reaction zone in the bend at t = 200μs (a)and the reflected transverse re-ignition wave at t = 250μs (b). Thecorresponding dynamic refinement is shown in the right graphs.

Figure 20: Triple point trajectories for the regular oscillatinghydrogen-oxygen detonation propagating through a 60-degree pipebend of 5 λ. The upper graph is an enlargement of the triple pointtrajectories in the bend.

The calculations correspond to a uniform computation with∼150 M cells and have an effective resolution of 25.5 Pts/lig inthe x1-direction (with respect to the initial detonation). Bothruns are stopped at tend = 240μs after 730 root level timesteps, with average step size Δt0 ≈ 0.329μs, when the flowsituations of interest are clearly established. The completedecoupling of shock wave and flame front for 8 λ is visible inthe upper graph of Figure 17, while the lower graph clearlyexhibits the re-ignition wave. It is interesting to note that there-ignition wave itself is a detonation. The triple point tracksfor 10 λ (not shown) uncover that it has developed out of thetransverse wave of an initial triple point.

The enormous efficiency of the refinement is visualizedin the upper graph of Figure 18 for the setup with tubewidth 10 λ. At tend, the calculation shown in Figure 18 uses∼3.0 M cells on all levels, where ∼2.4 M cells are insideone of the 2,479 grids of the highest level. The bottomgraph of Figure 18 displays the complexity of the domaindecomposition (indicated by color) to the 48 processors.These simulations were also run on the same parallelcomputer system as the three-dimensional computations andrequired ∼4500 h CPU (3.9 days wall time).

6.4. Propagation through a Pipe Bend. In order to demon-strate the enormous potential of the entire approach evenfor non-Cartesian problems, we finally discuss a case studythat combines highly efficient dynamic mesh adaptation withthe embedded boundary method sketched in Section 4.1.A two-dimensional regularly oscillating detonation similarto Section 6.3 but with initial pressure p0 = 10.0 kPa andλ = 1.6 cm is placed into a pipe of width 5 λ. The pipebends at different angles and has the inner radius 9.375 λ.First, we consider the bending angle ϕ = 60◦. When thedetonation propagates through the bend, it gets compressedand consequently overdriven near the outer wall, but acontinuous shock wave diffraction occurs near the inner wall.Similar to the simulation of Section 6.3, the diffraction causes


(a) (b)

(c) (d)

Figure 21: Color plots of the temperature for the regular oscillating hydrogen-oxygen detonation propagating upwards a pipe bend of width5 λ. Snapshots are taken in 50 μs intervals and start at t = 100μs (a).

a pressure decrease below the limit of detonability that leadsto a continuous decoupling of shock and reaction front. Thiseffect is visible particularly in the second graph of Figure 21at t = 150μs. The detonation exits the bend before the decayto a flame occurs across the entire tube width. A reignitionwave arises from the successfully transmitted region andreinitiates the detonation in the decoupled area, see thirdgraph of Figure 21 and first schlieren graph of Figure 19for t = 200μs. It propagates in the direction normal to thepipe middle axis and causes a strong shock wave reflectionas it hits the inner wall, compare last graph of Figure 21 andsecond schlieren graphs of Figure 19 for t = 250μs. Afterthe partial reignition event, the front structure convergesback to the initial oscillation with five regular cells acrossthe tube width. Figure 20 shows the simulated triple pointtrajectories through the course of the entire simulated timeof tend = 400μs and the convergence back to the equilibriumstate can be inferred.

This simulation uses a base grid of 1200 × 992 cells,four levels of refinement with r1,2,3 = 2, r4 = 4, andhas an effective resolution of 67.6 Pts/lig. Approximately7.1 M to 3.4 M cells on all and 4.8 M to 1.8 M cells on thehighest level are used instead of ∼1,219 M in the uniform

case. The number of subgrids on the highest level variesbetween ∼2,600 and ∼800. The calculation was run on 64dual-processor nodes of the ASC Linux cluster at LawrenceLivermore National Laboratory and required nevertheless∼70,000 h CPU (∼23 days wall time), due to the necessaryhigh local resolution in space and time. Approximately4000 time steps were taken on the root level, yielding anaverage root level time step of Δt0 ≈ 0.1μs, but note thatdue to automatic time step steering based on CCFL ≈ 0.6the used root level time steps vary between ∼0.085 μs and∼0.120 μs. The extraordinary high efficiency of our meshrefinement approach in capturing only the essential featuresnear the detonation front is illustrated by the right graphsof Figure 19. The effective resolution at the detonation frontis so high that the simulation succeeds in capturing evensecondary triple points throughout the entire run. Figure 22shows a zoom into a triple point as it enters the bend and allfeatures are fully resolved.

6.5. Triple Point Analysis. The high resolution of the lastsimulation admits a considerable refinement of the triplepoint pattern introduced in Section 2. We start by analyzing


(a) (b)

(c) (d)

Figure 22: Successively enlarged schlieren plots of the density on refinement regions at t = 150μs for the detonation propagating throughthe pipe bend. The simulation resolves the detailed structure around each triple point, cf. Figure 23.

the shock reflection pattern around a triple point for theperfectly regularly oscillating detonation before entering thebend, which obviously also corresponds to the flow situationof Section 6.1. In order to carry out a quantitative Machreflection pattern analysis, it is necessary to map the velocityfield of the simulation into a frame of reference attachedto the triple point. However, the reliable estimation of thetriple point speed v from a single output time step isnonapparent. When evaluating v, we take advantage of thefact that the triple point is formed at the tip, where the Machstem intersects the incident region, and that the obliqueshock relations [15] between two points in regions A and B(cf. Figure 24) close to the triple point must hold true. Werequire only the two relations

ρAuA sin(φB) = ρBuB sin

(φB − θB

), (6)

pA + ρAu2A sin2(φB

) = pB + ρBu2B sin2(φB − θB

). (7)

Inserting (6) into (7) allows the elimination of uB sin(φB −θB), which yields

uA = 1sinφB

√√√√ρB

(pB − pA

)

ρA(ρB − ρA

) . (8)

As the gas is initially at rest, the triple point velocity is v =−ua, and φB , the angle of inflow, is given as the angle betweenMach stem front and the triple point trajectory, which can bemeasured from visualizations such as those shown in Figures25 and 26.

Figure 23 zooms into the situation shortly before thecollision of two triple points. The enlargement shows clearlythat the shock wave pattern around each triple point isof double Mach reflection (DMR) (aka “strong”) type. Asdepicted in Figure 24, the essential flow features around triplepoint T are inflow (A), region B behind the Mach stem M,transverse wave region (C), and region D downstream of theincident shock I. Regions B and C are separated by the slipline S, a contact discontinuity. Regions C and D are separatedby the reflected shock R. The characteristic for the DMRpattern is a high supersonic velocity in region C that leadsto the formation of a further shock R′ creating a secondarytriple point T′ on the transverse wave. The very weak slip lineS′, emanating from T′ between regions E and F, can hardly beinferred. Note that the reflected shock R is the incident shockI′ for the secondary triple point T′.

Since the triple point is far ahead of the reaction region,changes in mixture are neglected in evaluating the Machnumber M in the triple point pattern. As it can be expected in


p/p0 ρ/ρ0 T [K] u [m/s] MA 1.00 1.00 298 1775 5.078B 31.45 4.17 2248 447 0.477C 31.69 5.32 1775 965 1.153D 19.17 3.84 1487 1178 1.533E 35.61 5.72 1856 901 1.053F 40.61 6.09 1987 777 0.880

(a)

2.3

2.4

2.5

2.6

2.7

2.8

85.2 85.3 85.4 85.5 85.6 85.7

FE

C BAD

(b)

Figure 23: States of a double Mach reflection structure shortlybefore the next collision in a regularly oscillating CJ detonation in aH2 : O2 : Ar/2 : 1 : 7 mixture at initially 298 K and p0 = 10 kPa.

M′

D

F

E

CR′

S′ S

B

M

T′ I′ RT

φB

I

A

Figure 24: Sketch of a triple point with double Mach reflectionstructure.

a DMR [15],MC is clearly greater than 1. A further importantquantity for triple point structures is the strength S of thetransverse wave [19] that is defined as

S := pC − pDpD

. (9)

20 25 30 35 40

(a)

37 38 39 40

(b)

Figure 25: (a) Cellular structure after the bend (shown rotated)for ϕ = 15◦, (b) triple point re-initiation with change from TMRto DMR. The physical time steps 200 μs, 210 μs, and 220 μs aredisplayed.

For the present computation, S is rather constantthroughout one regular detonation cell and varies marginallyaround 0.65.

6.6. Triple Point Structures under Transient Conditions. Thediffraction and compression of the detonation wave at thebend leads to changes of the transverse wave strength S.While the state C behind the reflected shock R remainsinitially unchanged, the change in geometry alters theincident state D. Near the inner bend wall (upper boundaryin Figure 25), the incident pressure pD drops, leading toa considerable increase in S. The resulting triple pointstructure is of strengthened DMR type with larger deto-nation cells (cf. upper graph of Figure 25) and increasedMach number MC. In the compression region near theouter wall (lower boundary in Figure 25), however, pDincreases drastically leading to a considerable decrease of S.As a consequence, the flow in region C decelerates and thesecondary shock R′ between C and E is no longer necessaryfor a stable configuration. The triple point now exhibitsa transitional Mach reflection (TMR) or “weak” pattern.Characteristic for the TMR structure is that the flow in regionC is just barely supersonic with respect to T [15]. TMRstructures are also found in the diffraction region when asufficiently large angle ϕ causes partial detonation failure.

The lower graph of Figure 25 depicts the interesting caseof triple point reinitiation behind the bend for ϕ = 15◦. Thenew triple point in the center of the graph is formed as a weakstructure (TMR) but strengthens into a DMR after its first


20 25 30 35 40

(a)

26 27 28 29 30

(b)

Figure 26: (a) Cellular structure immediately after the bend (shownrotated) for ϕ = 30◦, (b) triple point quenching and failure as SMRat 140 μs and 150 μs simulated physical time.

0

0.5

1

1.5

2

S

1 2 3 4 5 6 7 8

M

SMR

TMR

DMR

RR

Figure 27: Transition boundaries of shock wave reflection phenom-ena for the nonreactive H2 : O2 : Ar mixture at initially 298 K and10 kPa.

collision. Figure 26 visualizes the quenching of triple pointsat the outer bend wall that occurs for all larger values of ϕ.As can be inferred from the lower graph of Figure 26, failingtriple points with vanishing trajectories seem to exhibit aweak structure.

6.7. Transition Criteria of Mach Reflection Phenomena. Thetwo-dimensional oblique jump conditions [15] are satisfied

across each discontinuity depicted in the sketch of Fig-ure 24. We solve the resulting nonlinear system of equationsnumerically under the assumption that the compositionof the H2 : O2 : Ar mixture is invariant, but note that thetemperature dependency of the material properties is fullyconsidered. It is well known (cf. [15]) that changes in Machreflection type occur when the flow in certain regions of thetriple point structure becomes supersonic in the frame ofreference of the triple point T or T′. Any Mach reflectionrequires a supersonic flow in region B measured in the frameof reference of the primary triple point, that is, MT

B > 1, whilefor MT

B < 1 a regular reflection (RR) occurs. A single Machreflection (SMR) pattern encompassing just the four statesA to D is found for MT

C < 1; for MTC > 1 a transitional or a

double Mach reflection pattern occurs. In a DMR, the flow inC is also supersonic in the frame of reference of the secondarytriple point T′ (MT′

C > 1), however, it is subsonic with MT′C <

1, in a TMR. Consequently, the secondary triple point T′

is not fully developed in a TMR and the discontinuities R′

and S′ are absent. For given inflow velocity v, the transitionpoints MT

B/C = 1 are unique and can be determined rathereasily by solving the corresponding set of oblique shockrelations. The evaluation of the TMR/DMR transition pointMT′

C = 1 is more involved as knowledge of the velocity vector�a of T′ relative to T is required. The oblique shock relationsbetween C and D readily yield an ≡ 0; however, the tangentialvelocity at of T′along R is a free parameter that needs to bespecified separately.

Figure 27 visualizes the transition boundaries of thedifferent shock wave reflection phenomena evaluated numer-ically for varying inflow Mach number. Since the estimationof the exact minimally possible value of at in a DMR isa considerable problem in itself, the TMR/DMR transitionline is evaluated for the fixed value at = 100 m/s, whichputs a reasonable upper transition boundary on the TMRregion (dotted in Figure 27). Specially geared to detonations,results are displayed for the relative transverse wave strengthS. We use the simulation results discussed in Sections 6.5and 6.6 as a first verification for the found quantitative tran-sition boundaries. Throughout an unperturbed detonationcell only the DMR structure occurs, which is confirmedby plotting (with stars) the point at the end of a cell (cf.Figure 23) and an additional one immediately after triplepoint collision. The open circles mark the triple pointreinitiation visualized in the lower graph of Figure 25, andthe transition boundaries clearly confirm the transition fromTMR to DMR. The failing triple point, shown in the lowergraph of Figure 26, is marked with closed triangles indicatingthat the SMR pattern seems unstable for a detonationsubstructure and occurs only immediately before the triplepoint disappears.

7. Conclusions

We have described an efficient solution strategy for thenumerical simulation of gaseous detonations with detailedchemical reaction. Beside the application of the time-operator splitting technique and the construction of a robust


high-resolution shock capturing scheme, the key to thehigh efficiency of the presented simulations is the appli-cation of a block-structured adaptive mesh refinementmethod and its parallelized implementation in our softwareframework AMROC. AMROC provides the required highlocal resolution dynamically and follows a parallelizationstrategy tailored especially for the emerging generation ofdistributed memory architectures. An embedded boundarymethod utilizing internal ghost cells extends the frameworkeffectively to non-Cartesian problems.

Two- and three-dimensional detonation structure simu-lations for hydrogen-oxygen-argon mixtures have been dis-cussed in detail, where all temporal and spatial scales relevantfor the complex process of detonation propagation andcellular structure formation from subscale Mach reflectionpatterns were successfully resolved. All presented simulationresults have been obtained on Linux-Beowulf-clusters ofmoderate size. The reliability of the mesh adaptation methodhas been verified for periodically oscillating detonationstructure simulations in a frame of reference attached tothe detonation front. While the savings from dynamicmesh adaptation are necessarily modest for these types ofsimulations, they have been demonstrated to be enormousfor detonation structure simulations in more complexgeometries that have to be carried out in an Eulerian frameof reference. Depending on the required maximal resolution,increasingly larger savings in mesh size can be achieved.The large-scale two-dimensional structure simulations inpipe bends exhibit reductions of the finest mesh size of atleast a factor of 250 and up to 680. These computationshave an effective resolution that captures even secondarytriple points reliably throughout the whole simulation andallows the unambiguous classification of the Mach reflectionstructures around triple points. Under two-dimensionaltransient conditions, transitional (aka “weak”) and doubleMach reflection patterns (aka “strong” structures) have beenobserved.

Nonreactive but thermally perfect shock wave reflectiontheory was used successfully to predict the transition bound-aries between Mach reflection patterns depending on theinflow velocity. Exemplary transient triple points structuresfrom the simulations are compared with the theoreticalresults and good agreement between observed and predictedMach reflection type is found. Future work in understandingthe occurrence of Mach reflection patterns in self-sustaineddetonation waves will concentrate on deriving a physicalestimate for the smallest possible relative tangential velocityat between primary and secondary triple point in a doubleMach reflection in real gases to complete the theoreticalmodel and on comparing the substructures of the regularlyoscillating situations in two and three space dimensions.First detailed investigations [8] indicate that the fully three-dimensional case shows lower values for S and therefore atransitional rather than a double Mach reflection as basicpropagation pattern. This result also suggests that fullypredictive detonation simulations in general will have tobe carried out in three space dimensions. Since it wasdemonstrated that our computational approach permits theaccurate numerical simulation of detonation structures in

technically relevant two-dimensional devices and the studyof idealized three-dimensional configurations on currentcapacity computing systems, this ultimate goal might alreadybe attainable on available petaFLOPS supercomputers.

As an outlook on computational detonation research ingeneral, we finally want to mention that the quantification ofthe influence of diffusive processes on the detailed structureof the primary slip line and its extension (denoted F and G inFigure 3) in Mach reflection patterns in detonation waves hasrecently received some attention. This topic is of particularinterest for mixtures that lead to very irregularly oscillatingcellular structures, for example, hydrocarbon fuels, cf. [46,47]. The slip line is formed by a shear layer flow that isunstable to perturbations (which is commonly known asa Kelvin-Helmholtz instability). A fully converged solution,that predicts the physically accurate behavior of this shearlayer at all subscales, cannot be obtained with an inviscidmodel. For the resolutions and second-order scheme usedin this paper, stability and structure of the triple point sliplines are still clearly determined by the numerical diffusionintroduced by the approximation; yet under increasinglyfiner grid resolution a point is invariably reached from whichon the correct physical diffusion needs to be considered.Instead of the Euler equations (1), the fully reactive Navier-Stokes equations (cf. [16]) need to be employed, whichcan be accomplished technically rather easily for detonationwave simulation by supplementing the presented numericalmethod with a conservative approximation of the physicaldiffusion fluxes and incorporation of an additional stabil-ity condition. Utilizing such an approach in AMROC incombination with an upwind scheme of very high order,we have recently succeeded in performing a large-scale two-dimensional adaptive computation that resolves all physicallength scales along the slip line of an isolated transitionalMach reflection pattern that is created by the reflection ofa planar detonation wave at an inclined wedge [48]. Notethat this pseudo-DNS (A true direct numerical simulation(DNS) would require accurate resolution of all physicallength scales involved. This is presently not feasible for thethickness of the shock waves, which is of the order of threeto five mean free path lengths.) result has been obtained forthe simplified case of a single exothermic reaction betweentwo calorically perfect gases and not for a realistic chemicalreaction mechanism, as used throughout this paper.

Acknowledgments

This work has been sponsored by the Office of AdvancedScientific Computing Research, U.S. Department of Energy(DOE) and was performed at the Oak Ridge NationalLaboratory, which is managed by UT-Battelle, LLC underContract no. DE-AC05-00OR22725, and by the DFG highpriority research program “Analysis and Numerics of Con-servation Laws”, grant Ba 840/3-3. The large-scale computa-tions of Sections 6.4 to 6.7 have been carried out while theauthor was at the California Institute of Technology and wassupported by the ASC program of the Department of Energyunder subcontract no. B341492 of DOE contract W-7405-ENG-48.


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