Linear-Eddy Model Formulated Probability Density Function and Scalar Dissipation Rate Models for Premixed Combustion

Tsui, Hong Pak; Bushe, W. Kendal

doi:10.1007/s10494-014-9554-4

“…These surrogate pdf's appear to be a good approximation of the model pdf in the reactive c region for not too large wrinkling amplitudes and large Δ∕ f . Shapes of premixed flame pdf's derived from DNS data (Salehi and Bushe 2010;Salehi et al 2013;Jin et al 2008) and from application of the (stochastic) linear eddy model to numerically derived 1D flamelet pdf's (Tsui and Bushe 2014) look similar to the pdf's derived from this simple analytical model. It appears that the latter can reproduce the effects of turbulent flame wrinkling on p(c) at least qualitatively.…”

Section: Illustration Of Wrinkled Flame Pdfmentioning

confidence: 73%

A New Analytic pdf for Simulations of Premixed Turbulent Combustion

Pfitzner

¹

2020

Flow Turbulence Combust

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A new reaction rate source term $$\omega _m(c)$$ ω m ( c ) for modelling of premixed combustion with a single progress variable c is proposed. $$\omega _m(c)$$ ω m ( c ) mimics closely the Arrhenius source term $$\omega _A(c)$$ ω A ( c ) for a large range of activation energies and density ratios while admitting analytic evaluation of many quantities of interest. The analytic flame profile $$c_m(\xi )$$ c m ( ξ ) very closely approximates the numerically integrated Arrhenius flame profiles $$c_A(\xi )$$ c A ( ξ ) . An important feature of $$c_m(\xi )$$ c m ( ξ ) is that it is analytically invertible into a $$\xi _m(c)$$ ξ m ( c ) . Analytic estimates of the laminar flame Eigenvalue $$\Lambda$$ Λ and of the Le dependence of the laminar flame speed $$s_L$$ s L are derived, which are more accurate than classic results based on asymptotic analyses. The flamelet pdf $$p(c)=1/(\Delta *c*(1-c^m))$$ p ( c ) = 1 / ( Δ ∗ c ∗ ( 1 - c m ) ) for a flat laminar flame front in a LES cell of width $$\Delta$$ Δ is derived. The exact mean of the reaction rate $$\overline{\omega (c)}$$ ω ( c ) ¯ is compared to a beta pdf result, which is shown to be inaccurate for large ratios of filter width to flame thickness $$\Delta /\delta _f$$ Δ / δ f and particularly for high activation energy flames. For multidimensional flame wrinkling we derive the exact relationship $$p(c)=p_{1D}(c)I(c)\Xi (c)$$ p ( c ) = p 1 D ( c ) I ( c ) Ξ ( c ) between the 3D pdf p(c), the 1D flat flame pdf $$p_{1D}(c)$$ p 1 D ( c ) , a correction factor I(c) for change of inner flame structure and a geometrical wrinkling factor $$\Xi (c)$$ Ξ ( c ) . We show that the c dependence of these quantities cannot be neglected for small $$\Delta /\delta _f$$ Δ / δ f . A simple model of a sinusoidally wrinkled flame front qualitatively demonstrates the effect of flame wrinkling on p(c). We show that for large $$\Delta /\delta _f$$ Δ / δ f , a wrinkling of the reaction zone almost constantly increases p(c) in the reaction zone by a wrinkling factor $$\Xi ^*$$ Ξ ∗ (defined for the surface of the isosurface of maximum heat release) while reducing it near $$c=0,1$$ c = 0 , 1 as required for normalisation of p(c). The 1D p(c) evaluated using a reduced filter width $$\Delta '=\Delta /\Xi ^*$$ Δ ′ = Δ / Ξ ∗ may be a good approximation of the wrinkled flame pdf for evaluation of $$\overline{\omega (c)}$$ ω ( c ) ¯ for such cases.

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“…Such a formulation is compatible with several current models for the turbulence and chemistry interactions for turbulent premixed combustion [1][2][3][4]. A detailed description of the LEM PDF formulation can be found in [11], while a brief overview of the construction of the LEM PDF is provided in Section 4. Furthermore, we define a normalised variance,c 1.…”

Section: Lem Simulation Methodsmentioning

confidence: 99%

“…The mapping introduces variations in the temperature gradients at a given temperature (Figure 3(b)); as a result, the probability of the flame existing at any given state varies with different temperature profiles. When averaged, the PDF decreases to zero at each boundary more gradually [11].…”

Section: Lem Pdf Constructionmentioning

confidence: 98%

“…Equation 7 is particularly interesting to our analysis because the LEM temperature profiles can be used to construct both χ c |c * and P(c * ). A detailed description of the methodology used in constructing the PDF and SDR models can be found in [11]. There are two observable behaviors from the conditionally averaged SDR models from the LEM.…”

Section: Probability Density Functionmentioning

confidence: 99%

“…To address this issue, a one-dimensional turbulent method was proposed to take the place of the typical laminar flame calculation to tabulate pseudo-turbulent PDF models for RANS and LES closures. The Linear-Eddy Model, an inexpensive one dimensional stochastic mixing model, has demonstrated the ability to capture important effects from the interaction between chemistry and turbulence on the PDF distributions sufficiently well [9][10][11]. This model provides us with a mechanism to investigate the general flame characteristics at very high turbulence intensities, much beyond the capability of current DNS strategies.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Direct comparison of PDF and scalar dissipation rates between LEM simulations and experiments for turbulent, premixed methane air flames

Tsui

¹

,

Kamal

²

,

Hochgreb

³

et al. 2016

Combustion and Flame

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We present a direct comparison between the predicted and measured probability density functions (PDF) of the reaction progress variable and conditioned values of the scalar dissipation rates (SDR) in premixed turbulent flames. The predictions are based on simulations of premixed flames using the linear-eddy model (LEM), parameterised by a wide range of integral length scales and turbulent Reynolds numbers. The experimental results are highly spatially resolved temperature and species data from the Cambridge-Sandia swirl burner. The LEM simulations display remarkable accuracy in capturing the features observed experimentally. Further, the results reveal that the LEM calculated PDF and SDR for premixed flames remain relatively steady under a variety of turbulent conditions, including variations in the integral length and turbulent Reynolds number. In general, it appears to be practical to use representative pseudo-turbulent PDF and SDR models for a range of turbulence intensities and length scales.

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