Analysis of the spatio-temporal scales of laminar premixed flames near equilibrium

Al-Khateeb, Ashraf N.; Powers, Joseph M.; Paolucci, Samuel

doi:10.1080/13647830.2012.724180

Cited by 7 publications

(5 citation statements)

References 29 publications

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“…For equilibria with positive chemical eigenvalues, λ i,0 > 0, (3.8) identifies critical length scales where the diffusion-modified eigenvalue will be zero, c = mπ Dτ i,0 ; this result is similar to that of Al-Khateeb, Powers, and Paolucci [44]. We note that our boundary conditions and basis functions allow a symmetry in the dynamics in the Galerkin projection of our system.…”

Section: Methodssupporting

confidence: 69%

One-Dimensional Slow Invariant Manifolds for Fully Coupled Reaction and Micro-scale Diffusion

Mengers¹,

Powers²

2013

SIAM J. Appl. Dyn. Syst.

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Abstract. The method of slow invariant manifolds (SIMs), applied previously to model the reduced kinetics of spatially homogeneous reactive systems, is extended to systems with diffusion. Through the use of a Galerkin projection, the governing partial differential equations are cast into a finite system of ordinary differential equations to be solved on an approximate inertial manifold. The SIM construction technique of identifying equilibria and connecting heteroclinic orbits is extended by identifying steady state solutions to the governing partial differential equations and connecting analogous orbits in the Galerkin-projected space. In parametric studies varying the domain length, the time scale spectrum is shifted, and various classes of nonlinear dynamics are identified. A critical length scale is identified below which the spatially homogeneous one-dimensional SIM models the long time dynamics of the system. At this critical length scale, a bifurcation in the slow dynamics of the system is identified; additional real nonsingular steady state solutions are found which lead to a diffusion-modified one-dimensional SIM. At these longer lengths, the spectral gap in the time scales indicates that an appropriate manifold for a reduction technique is higher than one-dimensional. This is shown for two examples: a simple chemical reaction mechanism, and the Zel'dovich reaction mechanism of NO production. These examples are evaluated in the spatially homogeneous case (a one-term projection), a two-term projection capturing the coarsest effects of diffusion, and a high-order projection that is fully resolved. 1. Introduction. Numerical simulations of the partial differential equations (PDEs) that model multiscale continuum physics are prevalent across many fields of engineering. To obtain results with fidelity to the underlying continuum model, discrete simulations must generally resolve the entire range of scales present, both spatial and temporal; a large disparity in these scales is typically referred to as stiffness. The computational costs associated with these stiff simulations grow with the disparity of scales [1].In recent decades, there have been efforts in model reduction techniques to decrease the computational costs of simulating reactive flows while maintaining as much consistency with

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Section: Methodssupporting

confidence: 69%

One-Dimensional Slow Invariant Manifolds for Fully Coupled Reaction and Micro-scale Diffusion

Mengers¹,

Powers²

2013

SIAM J. Appl. Dyn. Syst.

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“…With the finest length scale of reaction given by Powers & Paolucci (2005) for this mixture finest 10 −4 cm, and the diffusivity of the mixture D mix 10 cm 2 s −1 , one can estimate the frequency of the disturbance for which diffusion clearly dominates as ν D mix / 2 finest = 1 GHz. Useful insights on the relative importance of advection, reaction and diffusion in hydrogen-air chemistry is given by Al-Khateeb et al (2013) in the context of a laminar flame. There, it is shown that diffusion clearly influences the various reaction time scales on length scales given by a classical Maxwellian model, i √ D mix τ i , where i is the reaction length scale associated with the chemical time scale τ i .…”

Section: Discussionmentioning

confidence: 99%

“…Furthermore, it was demonstrated by Romick, Aslam & Powers (2012) that for the one-step kinetics model the long time behaviour of the detonation is affected by the addition of viscosity; these viscous effects delay the transition to instability, and in the regime where multiple frequency oscillations exist, viscosity can play a dramatic role. Moreover, Al-Khateeb, Powers & Paolucci (2013) suggested that hydrogen-air mixtures have reaction length scales present which have time scales associated with them over which both chemistry and diffusion can be important. This suggests that viscosity has a role to play in the dynamics of detonations; however, the role of viscosity in pulsating detonations of a detailed kinetics model has not been quantified in a detailed manner.…”

mentioning

confidence: 99%

Verified and validated calculation of unsteady dynamics of viscous hydrogen–air detonations

2015

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The dynamics of one-dimensional, piston-driven hydrogen-air detonations are predicted in the presence of physical mass, momentum and energy diffusion. The calculations are automatically verified by the use of an adaptive wavelet-based computational method which correlates a user-specified error tolerance to the error in the calculations. The predicted frequency of 0.97 MHz for an overdriven pulsating detonation agrees well with the 1.04 MHz frequency observed by Lehr in a shock-induced combustion experiment around a spherical projectile, thus giving a limited validation for the model. A study is performed in which the supporting piston velocity is varied, and the long time behaviour is examined for an initially stoichiometric mixture at 293.15 K and 1 atm. Several distinct propagation behaviours are predicted: a stable detonation, a high-frequency pulsating detonation, a pulsating detonation with two competing modes, a low-frequency pulsating detonation and a propagating detonation with many active frequencies. In the low-frequency pulsating mode, the long time behaviour undergoes a phenomenon similar to period-doubling. Harmonic analysis is used to examine how the frequency of the pulsations evolves as the supporting piston velocity is varied. It is found that the addition of viscosity shifts the neutral stability boundary by about 2 % with respect to the supporting piston velocity. As the supporting piston velocity is lowered, the intrinsic instability grows in strength, and the effect of viscosity is weakened such that the results are indistinguishable from the inviscid predictions.

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“…One can calculate a rough estimate of the length scales for these oscillations in a manner similar to that given by Al-Khateeb et al, 20 who estimate the length scale as ℓ = √ D ik τ . For example, the fine diffusion scale diffusion mode is estimated by using the smallest diffusion coefficient, D ik ≈ O(10 1 cm 2 /s), and higher harmonics' time scale and thus yields ℓ ≈ O(10 −4 cm).…”

Section: Hydrogen-air Dynamicsmentioning

confidence: 99%

“…Mazaheri et al 19 demonstrated that diffusion can play an important role in regions of high-resolutions. Furthermore recent calculations done by Al-Khateeb et al, 20 indicate that the fine reaction length scales present in hydrogen-air mixtures can have time scales associated with them at which both chemistry and diffusion become important. Therefore, the introduction of grid-independent physical diffusion should be considered.…”

Section: Introductionmentioning

confidence: 99%

On the resolution necessary to capture dynamics of unsteady detonation

Romick

Aslam

Powers

2013

51st AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition

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The dynamics of one-dimensional, overdriven, hydrogen-air detonations predicted in the inviscid limit as well as with the inclusion of mass, momentum, and energy diffusion were investigated. The effect of resolution was studied for both shock-capturing and shockfitting in the inviscid limit, and it was found that shock-capturing required four times the amount of resolution of shock-fitting to predict the essential dynamics. Harmonic analysis was used to examine how the long time dynamics changed as the overdrive was varied in both the inviscid limit and the viscous analog. As the overdrive is lowered, the pulsation's fundamental frequency shifts smoothly from 1.03 M Hz at an overdrive of f = 1.12 to 0.71 M Hz at an overdrive of f = 1.035 before making a sudden jump to 0.11 M Hz at an overdrive of f = 1.029, where overdrive is f = D 2 o /D 2 CJ , Do the initial detonation velocity, and DCJ the Chapman-Jouguet velocity. It was found that viscous effects reduce the magnitude of the pulsations. As the strength of the inherent instability grows, due to interaction of the hydrodynamics and chemical reactions, the effect of viscosity is reduced.

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Analysis of the spatio-temporal scales of laminar premixed flames near equilibrium

Cited by 7 publications

References 29 publications

One-Dimensional Slow Invariant Manifolds for Fully Coupled Reaction and Micro-scale Diffusion

One-Dimensional Slow Invariant Manifolds for Fully Coupled Reaction and Micro-scale Diffusion

Verified and validated calculation of unsteady dynamics of viscous hydrogen–air detonations

On the resolution necessary to capture dynamics of unsteady detonation

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