Scaling and renormalization for the Kolmogorov-Petrovskii-Piskunov equationwith turbulent convection (original) (raw)
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A statistical theory is developed for the structure and propagation velocity of premixed flames in turbulent flows with scales large compared with the laminar flame thickness. The analysis, free of usual closure assumptions, involves a regular perturbation for small values of the ratio of laminar flame thickness to turbulence scale, termed the scale ratio ε, and a singular perturbation for large values of the non-dimensional activation temperature β. Any effects of the flame on the flow are considered to be given. In this initial study, molecular coefficients for diffusion of heat and reactants are set equal. The results identify convective-diffusive and reactive-diffusive zones in the flame and predict thickening of the flame by turbulence through streamwise displacement of the reactive-diffusive zone. Profiles for intensities of temperature fluctuations and for streamwise turbulent transport are obtained. A fundamental quantity occurring in the analysis is the longitudinal displac...
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Physics of Fluids, 2011
In this paper, we analyze two effects caused by the Lagrangian nature of turbulent transfer which are usually ignored in the theory of turbulent premixed combustion. These effects are ͑i͒ the nonequilibrium behavior of the turbulent diffusion coefficient, which is important for modeling the initial stage of combustion ͑for example, in the spark ignition engine͒, and ͑ii͒ the existence of a traveling front of turbulent diffusion with finite speed, which controls the velocity of the steady state flame in strong turbulence. However, in order to derive simple and exact results, the hydrodynamical and the combustion subproblems are stated to be independent by assuming a constant density so that a passive chemical reaction is actually considered. First, we derive a parabolic diffusion equation with both diffusion and chemical source terms expressed by Lagrangian characteristics of turbulence. We show that, in general, the diffusivity of product particles is not zero in the moment of their generation by chemical transformation and this result is important for combustion theories that relate the formation of the initial flame with the development of the diffusion coefficient. Afterward, a hyperbolic diffusion equation based on hydrodynamics is derived with turbulent diffusion front velocity U f = ͗uЈ 2 ͘ 1/2 , where ͗uЈ 2 ͘ 1/2 is the root mean square of turbulent velocity fluctuations, and we analyze the relationships between U f and the speed of the steady state premixed flame U t ss. In particular, for the flamelet combustion mechanism, we obtain U t ss = ͑U f 2 + S L 2 ͒ 1/2 , where S L is the normal laminar flame speed. This result shows that, in moderate turbulence ͑͗uЈ 2 ͘ 1/2 ϳ S L ͒, the usually assumed relation U t ss = U f + S L is not consistent with an accurate statistical analysis and more when U f Ӎ S L gives a percent error around 40%, which cannot be neglected in applications. In strong turbulence case ͑͗uЈ 2 ͘ 1/2 ӷ S L ͒, the value of the flame speed is very close to that of the diffusion front velocity.
Analysis and Comparison of Large Time Front Speeds in Turbulent Combustion Models
Predicting turbulent flame speed (the large time front speed) is a fundamental problem in turbulent combustion theory. Several models have been proposed to study the turbulent flame speed, such as the G-equations, the F-equations (Majda-Souganidis model) and reaction-diffusion-advection (RDA) equations. In the first part of this paper, we show that flow induced strain reduces front speeds of G-equations in periodic compressible and shear flows. The F-equations arise in asymptotic analysis of reaction-diffusion-advection equations and are quadratically nonlinear analogues of the G-equations. In the second part of the paper, we compare asymptotic growth rates of the turbulent flame speeds from the G-equations, the F-equations and the RDA equations in the large amplitude ($A$) regime of spatially periodic flows. The F and G equations share the same asymptotic front speed growth rate; in particular, the same sublinear growth law Aoverlog(A)A\over \log(A)Aoverlog(A) holds in cellular flows. Moreover, in two...
A transport equation for reaction rate in turbulent flows
Physics of Fluids, 2016
New transport equations for chemical reaction rate and its mean value in turbulent flows have been derived and analyzed. Local perturbations of the reaction zone by turbulent eddies are shown to play a pivotal role even for weakly turbulent flows. The mean-reaction-rate transport equation is shown to involve two unclosed dominant terms and a joint closure relation for the sum of these two terms is developed. Obtained analytical results and, in particular, the closure relation are supported by processing two widely recognized sets of data obtained from earlier Direct Numerical Simulations of statistically planar 1D premixed flames associated with both weak large-scale and intense small-scale turbulence.
On velocity and reactive scalar spectra in turbulent premixed flames
Kinetic energy and reactive scalar spectra in turbulent premixed flames are studied from compressible three-dimensional direct numerical simulations (DNS) of a temporally evolving rectangular slot-jet premixed flame, a statistically one-dimensional configuration. The flames correspond to a lean premixed hydrogen-air mixture at an equivalence ratio of 0.7, preheated to 700 K and at 1 atm, and three DNS are considered with a fixed jet Reynolds number of 10 000 and a jet Damköhler number varying between 0.13 and 0.54. For the study of spectra, motivated by the need to account for density change, which can be locally strong in premixed flames, a new density-weighted definition for two-point velocity/scalar correlations is proposed. The density-weighted two-point correlation tensor retains the essential properties of its constant-density (incompressible) counterpart and recovers the density-weighted Reynolds stress tensor in the limit of zero separation. The density weighting also allows the derivation of balance equations for velocity and scalar spectrum functions in the wavenumber space that illuminate physics unique to combusting flows. Pressure-dilatation correlation is a source of kinetic energy at high wavenumbers and, analogously, reaction rate-scalar fluctuation correlation is a high-wavenumber source of scalar energy. These results are verified by the spectra constructed from the DNS data. The kinetic energy spectra show a distinct inertial range with a −5/3 scaling followed by a 'diffusive-reactive' range at higher wavenumbers. The exponential drop-off in this range shows a distinct inflection in the vicinity of the wavenumber corresponding to a laminar flame thickness, δ L , and this is attributed to the contribution from the pressure-dilatation term in the energy balance in wavenumber space. Likewise, a clear spike in spectra of major reactant species (hydrogen) arising from the reaction-rate term is observed at wavenumbers close to δ L. It appears that in the inertial range classical scaling laws for the spectra involving the Kolmogorov scale are applicable, but in the high-wavenumber range where chemical reactions have a strong signature the laminar flame thickness produces a better collapse. It is suggested that a full scaling should perhaps involve the Kolmogorov scale, laminar flame thickness, Damköhler number and Karlovitz number.
Lagrangian properties of diffusion in the theory of turbulent combustion
Proceedings of the Sixth International Symposium On Turbulence, Heat and Mass Transfer, 2009
In this paper we analyze two effects of the Lagrangian nature of turbulent transfer that are usually ignored in the theory of turbulent combustion. They are (i) nonequilibrium behavior of the turbulent diffusion coefficient, which is important for modeling of initial stage of combustion in the SI engine, and (ii) the existence of a traveling front of turbulent diffusion with finite speed, which controls the velocity of the steady state flame in strong turbulence. First we derive the parabolic diffusion equation with both the diffusion and chemical source terms expressed by Lagrangian characteristics of turbulence. We show that chemical transformation does not influence the turbulent diffusion coefficient and this result is important for combustion theories that relate the formation of the initial flame with the development of the diffusion coefficient. Second, an hyperbolic diffusion equation based on hydrodynamics is derived and we analyze relationship between velocities of the turbulent diffusion front and the speed of the steady state premixed flame. In particular, in the case of the flamelet combustion mechanism and strong turbulence, we state that the flame speed is very closed to the theoretical value of the diffusion front velocity, which is equal to the root mean square of turbulent velocity fluctuations.
Journal of Statistical Physics, 2007
We present a systematic way to compute the scaling exponents of the structure functions of the Kraichnan model of turbulent advection in a series of powers of ξ, adimensional coupling constant measuring the degree of roughness of the advecting velocity field. We also investigate the relation between standard and renormalization group improved perturbation theory. The aim is to shed light on the relation between renormalization group methods and the statistical conservation laws of the Kraichnan model, also known as zero modes.