1 Introduction

In the CFD simulation of turbulent premixed combustion using RANS and LES, the size of computational cells is usually too big to fully resolve the laminar flame structure embedded in the turbulent flow field. The ratio of flame thickness to cell size decreases further at elevated pressures due to the drop of diffusivities and heat conductivity with pressure. Thus combustion (subgrid) models are required for the simulation of premixed combustion processes in most technical applications.

For better understanding of basic phenomena and for model development, it is often useful in premixed combustion to study the case of a single reaction progress variable \(c=(T-T_u)/(T_b-T_u)\) assuming single-step irreversible chemistry and adiabatic combustion. \(T_u,T_b\) are the unburnt and fully burnt temperatures, respectively. The c transport equation is given by

$$\rho \frac{\partial c}{\partial t}+\rho u \frac{\partial c}{\partial x}=\frac{\partial }{\partial x}\left( \frac{\lambda }{c_p} \frac{\partial c}{\partial x}\right) -\frac{{{\dot{\omega }}}_F}{Y^{1}_{F}}$$
(1)

where \(\rho\), u, c are density, velocity and progress variable, and \(\lambda\), \(c_p\) are the heat conductivity and specific heat at constant pressure. For Arrhenius chemistry and Lewis number \(Le=1\), the chemical source term can be written as Poinsot and Veynante (2005):

$$\frac{{\dot{\omega }}_F}{Y^{1}_{F}}=B_1 T^{\beta _1} e^{-\frac{\beta }{\alpha }}\rho (1-c)exp\left( -\frac{\beta (1-c)}{1-\alpha (1-c)}\right)$$
(2)

where \(\alpha =\frac{T_b-T_u}{T_b}\) represents the normalized temperature raise and \(\beta =\alpha T_{a1}/T_b\) is a measure of the activation temperature \(T_{a1}\). The temperature exponent \(\beta _1\) in Eq. (2) is usually taken as \(\beta _1=0\) or \(\beta _1=1\). The continuity equation requires \(\rho u = const. = \rho _u s_L\) at steady state and we have \(\rho \sim 1/T \sim 1/\left( 1-\alpha (1-c)\right)\) for constant pressure combustion. One can rescale the spatial coordinate (Poinsot and Veynante 2005) according to \(\xi = \int _0^x \rho _u s_L c_p / \lambda dx\), yielding a simpler differential equation for c:

$$\frac{\partial c}{\partial \xi }=\frac{\partial ^2 c}{\partial \xi ^2}+\omega (c)$$
(3)

with

$$\omega (c)=\Lambda \left( 1-\alpha (1-c)\right) ^{\beta _1-1} (1-c)exp\left( -\frac{\beta (1-c)}{1-\alpha (1-c)}\right)$$
(4)

The prefactor \(\Lambda\) in Eq. (4) represents the eigenvalue of the transport equation which needs to be chosen such that the boundary conditions \(c = 0\) for \(\xi \rightarrow -\infty\) and \(c = 1\) for \(\xi \rightarrow +\infty\) are fulfilled. Asymptotic analysis of Eqs. (3, 4) in the limit of large \(\beta\) by Zeldovic, Frank Kamenetski and von Karman yielded \(\Lambda \sim \beta ^2/2\). An improved result was derived by Williams (2018) for \(\beta _1=1\):

$$\Lambda _{Williams}=\frac{\beta ^2}{2}+\beta (3\alpha -1.344)$$
(5)

The turbulent flow field will wrinkle and stretch the flame front, modifying the volume specific fuel consumption rate. For high activation energies, the reaction front will remain thin even for quite high Karlowitz numbers (Nilsson et al. 2019). A variety of subgrid models have been developed to represent the effect of subgrid flame folding/stretching on the fuel consumption rate. The artificially thickened flame model (Colin et al. 2000) makes the flame front resolvable on the LES grid by increasing the diffusion and heat conduction coefficients and reducing the strength of the reaction term, leaving the flame speed unchanged. The effect of non resolved subgrid flame wrinkling needs to be taken into account by an efficiency function. Other approaches are flame surface density (transport) models, where the RHS of Eq. (1) is replaced by \(\rho _u \left\langle {s_c}\right\rangle \Sigma_f\) with flame surface density \(\Sigma_f\) and \(\left\langle {s_c} \right\rangle\) being a surface averaged flame speed. \(\Sigma_f\) is either determined by a transport equation or approximated as \(\Sigma_f = \Xi \mid \nabla c\mid\) and evoking algebraic models for the wrinkling factor \(\Xi\) (Poinsot and Veynante 2005; Ma et al. 2013). Other researchers have proposed to pretabulate the filtered source term (Fiorina et al. 2010) for given LES cell size \(\Delta\) or extract more accurate expressions for \(\overline{\omega (c)}\) by 1-D approximate deconvolution (Domingo and Vervisch 2015).

Probability density functions p(c) with normalization condition \(\int _0^1 p(c) dc=1\) allow to evaluate cell averages of any quantity z(c) as \(\overline{z(c)}=\int _0^1 z(c) p(c) dc\). In classic BML theory the pdf is assumed to take the form

$$p_{BML}(c)=A \delta (c) + B \delta (1-c) + \gamma (c)$$
(6)

with \(\gamma (c) \ll 1\). For \(\gamma (c)\rightarrow 0\) one obtains \(B \sim {\overline{c}}\) and \(A \sim (1-{\overline{c}})\). Unfortunately, in the limit \(\gamma (c)=0\) accurate mean values of quantities vanishing at both ends \(c=0,1\) (like the chemical source term \(\omega (c)\)) cannot be evaluated by \(p_{BML}(c)\). Consequently, the calculation of the mean fuel consumption rate requires knowledge of \(\gamma (c)\). A 1-D flamelet pdf can be defined through (Bray et al. 2006):

$$p_f(c)=\frac{1}{N}\frac{1}{\partial c/\partial \xi }$$
(7)

with normalization factor \(N=\int _{c^-}^{c^+}\frac{dc}{\partial c/\partial \xi }\) where \(c^-,c^+\) represent the c values on the edges of the computational cell. All ingredients of \(p_f(c)\) can be evaluated only numerically for Arrhenius chemistry. Bray et al. (2006), Salehi and Bushe (2010) use ad hoc lower and upper bounds \(c^-=\epsilon >0\), \(c^+=1-\epsilon\) to determine N, claiming that N depends only weakly on \(\epsilon\). Another popular presumed pdf is the beta pdf

$$p_\beta (c)=\frac{c^{a-1}(1-c)^{b-1}\Gamma (a+b)}{\Gamma (a)\Gamma (b)}$$
(8)

where the coefficients \(a,b>0\) can be chosen to reproduce given values of mean and variance of c. In case of a beta pdf, \(c^-=0\) and \(c^+=1\). The beta pdf is mostly used to model diffusion processes and does a good job in representing typical pdf shapes occurring in this case. As a representation of the flamelet pdf, it has been shown (Bray et al. 2006) to deliver inaccurate results for the mean reaction term particularly for large c variance.

Knudsen et al. (2010) find a good correlation between the DNS source term filtered to a LES grid and the beta pdf value (with mean and variance in the beta pdf evaluated from the DNS) for small ratios of \({\Delta _{LES} / \Delta _{DNS}} < 5\), while large errors result for larger ratios \({\Delta _{LES} / \Delta _{DNS}}\). Domingo et al. (2005) report that their flamelet pdf with ad hoc choice of the integration limits \(c^-,c^+\) delivered unphysical parameters A, B for small values of c variance. They switched to a beta pdf in those cases. In the current contribution we hope to clarify some of these issues.

The paper is structured as follows: We first present the simplified reaction source term introduced by Ferziger and Echekki (1993), which yields analytical results for many quantities of interest. We then introduce a new source term \(\omega _m(c)\) with similar analytic capability, which however approximates the Arrhenius source term and the flamelet profile much closer. As first applications, we derive expressions for the \(\Lambda\) eigenvalue of the Arrhenius source term and for the dependence of the laminar flame speed \(s_L\) on Lewis number and Arrhenius parameters, which are more accurate than classic expressions from the literature. We compare flamelet and beta pdf’s for different filter widths \(\Delta /\delta _f\). We discuss the relation between pdf, wrinkling factor and correction factor for multidimensionally wrinkled flames and we demonstrate the modification to the pdf by flame wrinkling using a simple model of a sinusoidally wrinkled flame. Finally, we give some conclusions and an outlook to further work.

2 Ferziger/Echekki Source Term

Ferziger and Echekki (1993) proposed to set the source term to zero in the pure diffusion region \(0 \le c < 1-1/\beta\) and to use a linear term \(\omega (c)=\Lambda _E (1-c)\) in the reaction region \(1-1/\beta \le c \le 1\). Due to the simple form of the source term, the solution of the differential equation

$$\frac{\partial c}{\partial \xi }=\frac{\partial ^2 c}{\partial \xi ^2}+\Lambda _E (1-c) H(c-(1-1/\beta ))$$
(9)

where H(x) represents the Heaviside jump function, are pure exponentials in \(\xi\). Choosing the location \(\xi =0\) for \(c=1-1/\beta\) puts the pure diffusion region at negative \(\xi\) and the reacting region at positive \(\xi\). Matching function values and derivatives of \(c(\xi )\) at \(\xi =0\), the \(c(\xi )\) profile becomes

$$c_E(\xi ) = \left\{ \begin{array}{lr} \left( 1-\frac{1}{\beta }\right) exp(\xi )&{} : \xi <0 \\ 1-\frac{1}{\beta } exp(-\xi (\beta -1))&{} : \xi > 0 \end{array} \right.$$
(10)

The eigenvalue of Eq. (9) is given by \(\Lambda _E=\beta (\beta -1)\). Figure 1 shows a comparison of the Echekki \(c_E(\xi )\) profile and its source term with the Arrhenius source term and the numerically integrated \(c_A(\xi )\) profile for \(\beta _1=0\) and a relatively low \(\beta =6\), \(\alpha =9/11\) corresponding to a density jump \(T_b/T_u=4.5\). The \(c(\xi )\) profiles are in reasonable agreement.

Fig. 1
figure 1

Left: Arrhenius (full) and Echekki (dashed) source terms versus c. Right: Numerical Arrhenius solution \(c_A(\xi )\) and Echekki \(c(\xi )\) with difference (gray)

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3 New Analytical Flamelet Profile \(c_m(\xi )\) and Source Term \(\omega _m(c)\)

Although the Echekki solution admits analytical evaluations of a number of quantities, which are available only numerically with the Arrhenius source term, it has some disadvantages:

  • \(c_E(\xi )\) is only a reasonable approximation to \(c_A(\xi )\) for small to medium \(\beta\)

  • the pdf is not a good approximation to the Arrhenius pdf in the reactive c region

  • the piecewise definitions of \(c_E(\xi )\) and its inversion \(\xi _E(c)\) makes calculations quite cumbersome

It is therefore our goal to find an improved approximation to the Arrhenius \(\omega _A(c)\), which guarantees the correct boundary conditions of \(c(\xi )\) and still admits analytical integration of Eq. (3). This can be achieved by postulating functional forms of \(c(\xi )\) with analytical inverse \(\xi (c)\) and calculating \(\omega (c)\) according to:

$$z(\xi )=\frac{\partial c}{\partial \xi }-\frac{\partial ^2 c}{\partial \xi ^2}$$
(11)

and \(\omega (c)=z(\xi (c))\). Obviously, the resulting \(\omega (c)\) should be positive over the whole interval \(0< c < 1\). The Echekki \(c_E(\xi )\) fulfills this requirement but has the above disadvantages. A more elegant and admissible \(c(\xi )\) is given by:

$$c_m(\xi )=[1+exp(-m*\xi )]^{-1/m}$$
(12)

Similar to the Echekki \(c_E(\xi )\) with parameter \(\beta\), \(c_m(\xi )\) has a free parameter m which can be used to mimic \(c_A(\xi )\) for different Arrhenius parameters \(\alpha ,\beta ,\beta _1\). For large m, \(c_m(\xi )\) approaches a step function \(H(\xi )\) centered at \(\xi =0\), which is the correct thin flame limit. The profile \(c_m(\xi )\) can be inverted easily:

$$\xi _m(c)=\frac{1}{m} ln\left( \frac{c^m}{1-c^m}\right)$$
(13)

The profile yields a thermal flame thickness of:

$$\delta _{f,m}=\frac{1}{\left( dc_m(\xi )/d\xi \right) _{max}}=\frac{(m+1)^{\frac{m+1}{m}}}{m}$$
(14)

and the reaction source term is given by:

$$\omega _m(c)=(m+1)(1-c^m)c^{m+1}$$
(15)

Note that this source term is identical to that of the KPP-Fisher equation for \(m=1\), generalizations exist which correspond to the case \(m\ne 1\). Equation (11) represents the travelling wave equation of these equations with suitably rescaled spatial coordinate \(\xi\). Solutions of this equation and their behaviour have been studied in Kyrychko and Blyuss (2009), Fan (2002).

4 Determination of Parameter m

To determine values of m which “optimally” represent a given Arrhenius source term \(\omega _A(c)\) with parameters \(\alpha ,\beta ,\beta _1\), we generated numerical solutions of Eq. (3) for \(\beta _1=0,1\) for a range of parameters (\(\alpha ,\beta\)) and determined m by numerical minimization of \(\left[ c_m(\xi )-c_A(\xi -\xi _0)\right] ^2\). Note that the numerical shift \(\xi _0\), which is necessary to align numerically generated \(c_A(\xi\)) with \(c_m(\xi )\), is irrelevant since the position of the flame front on the \(\xi\) axis is arbitrary. The fitted m are found to be well represented by the following expressions valid in the physically relevant ranges of \(4< \beta < 30\) and \(1/2<\alpha <1\):

$$\beta _1=0: m_0=\frac{1}{5} (4 \alpha +4 \beta -5)$$
(16)
$$\beta _1=1: m_1=\frac{1}{12} \left( 5 \sqrt{\beta (4 \alpha +\beta )}+10 \alpha +5 \beta -18\right)$$
(17)

Figure 2 shows comparisons of \(\omega _A(c)\) with \(\omega _m(c)\) and of \(c_A(\xi )\) with \(c_m(c)\) for \(\beta _1=0,\alpha =9/11,\beta =6,18\) using these m correlations. Results for \(\beta _1=1\) are too similar to be shown here. The new source term is always shifted slightly towards \(c=1\) compared to the Arrhenius one. Note however the excellent agreement between \(c_A(\xi )\) and \(c_m(\xi )\).

Fig. 2
figure 2

Left: Arrhenius (full) and new (dashed) source terms. Right: Numerical Arrhenius solution \(c_A(\xi )\) and new \(c(\xi )\) with difference (gray)

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Table 1 Typical m values for small and large activation energy flames
Full size table

Table 1 shows typical m values for small and large activation energy flames as analysed in Poinsot and Veynante (2005).

5 Application: Derivation of Improved \(\Lambda\) Eigenvalue

As first applications of this model we present a new approach to derive improved approximations for the laminar flame eigenvalues \(\Lambda _{0,1}\). In contrast to results from classic asymptotic analyses (which are derived in the limit of large activation energy, i.e. large \(\beta\) values), the new expressions are also accurate for moderate values of \(\beta\).

We use the fact that by construction \(\omega _m\) already guarantees the correct limiting behaviour \(c(\xi \rightarrow -\infty )=0\) and \(c(\xi \rightarrow +\infty )=1\). The optimal m values are found to be closely correlated to \(c_{max}\), the location of the maximum of \(\omega _A(c)\), which is independent of prefactor \(\Lambda\). Therefore a choice of the Arrhenius \(\Lambda\) which minimizes the difference between \(\omega _A(c)\) and \(\omega _m(c)\) for given m should provide a good approximation to the correct Arrhenius flame eigenvalue \(\Lambda\). Normally one would look for a minimimum of the integral of the squared difference between \(\omega _A(c)\) and \(\omega _m(c)\). Since this does not admit an analytic solution we solve the following equation for \(\Lambda\):

$$\int _0^1\left( \omega _A(c)-\omega _m(c)\right) =0$$
(18)

This equation is acceptable in this case due to the undulary difference of \(\omega _A(c)\) and \(\omega _m(c)\). For \(\beta _1=0\) we get with

$$\begin{aligned} L1&=(\alpha +\beta ) e^{\frac{\beta }{\alpha }} \left( {Ei}\left( -\frac{\beta }{\alpha }\right) -{Ei}\left( \frac{\beta }{(\alpha -1) \alpha }\right) \right) \\ L2&=(\alpha -1) \alpha e^{\frac{\beta }{\alpha -1}}+\alpha \\ \Lambda _0&=\frac{-\alpha ^3 m_0}{2 (m_0+2)\left( L1+L2\right) } \end{aligned}$$
(19)

while \(\beta _1=1\) yields with

$$\begin{aligned} K1&=\alpha \left( (\alpha -1) \left( \alpha ^2+\alpha +\beta \right) e^{\frac{\beta }{\alpha -1}}+\alpha +\beta \right) \\ K2&=\beta (2 \alpha +\beta ) e^{\frac{\beta }{\alpha }} \left( {Ei}\left( -\frac{\beta }{\alpha }\right) -{Ei}\left( \frac{\beta }{(\alpha -1) \alpha }\right) \right) \\ \Lambda _1&= \frac{\alpha ^4 m_1}{(m_1+2) \left( K1+K2\right) } \end{aligned}$$
(20)

Ei(z) represents the exponential integral function. To simplify the expressions in Eqs. (19, 20), we note that \(exp[\beta /(\alpha -1)]\rightarrow 0\) for large \(\beta\) and \(\alpha <1\) and that the function \(f(z)=1/(e^z Ei(-z)(z+2)z+z+1)\) with \(z=\beta /\alpha\) can be approximated accurately by a second order polynomial in z in the interesting range \(4<z<40\). The following expressions for \(\Lambda _0,\Lambda _1\) approximate Eqs. (19, 20) very closely in this z range:

$$\Lambda _0= -\frac{(4 \alpha +4 \beta -5) \left( \alpha ^2-4 \alpha \beta -\beta ^2\right) }{8 \alpha +8 \beta +10}$$
(21)
$$\Lambda _1= -\frac{\left( 2 \alpha ^2+6 \alpha \beta +\beta ^2\right) \left( 9 \sqrt{\beta (4 \alpha +\beta )}+2 \alpha (5 \alpha -9)-9 \beta \right) }{6 \sqrt{\beta (4 \alpha +\beta )}-4 \alpha (5 \alpha +3)-6 \beta }$$
(22)

For comparison to classical results, we perform a Taylor series in \(1/\beta\), yielding the following leading order terms:

$$\Lambda _0\sim \frac{\beta ^2}{2}+\left( 2 \alpha -\frac{5}{4}\right) \beta +\frac{1}{16} \left( -8 \alpha ^2-60 \alpha +25\right)$$
(23)

and

$$\Lambda _1\sim \frac{\beta ^2}{2}+\left( 3 \alpha -\frac{6}{5}\right) \beta + \alpha ^2 -\frac{24 \alpha }{5}+\frac{18}{25}$$
(24)

Note the similarity of the first two terms in Eq. (24) to the Williams result Eq. (5), which was derived by asymptotic expansions in \(1/\beta\) and assuming \(\beta _1=1\). The Williams factor \((3\alpha -1.344)\) is replaced by \((3\alpha -6/5)\) with additional terms in \(\alpha\) only. For \(\beta _1=0\), the term multiplying \(\beta\) is replaced by \(2\alpha -5/4\) with different \(\alpha\) terms. While the Williams \(\Lambda\) works well for \(\beta > 8\) and \(\beta _1=1\), the new expressions are also valid for smaller \(\beta >4\) and provide a much better approximation in the case of \(\beta _1=0\). Figure 3 shows the numerically integrated \(c_A(\xi )'s\) using the Arrhenius source term \(\omega _A(c)\) for \(\alpha =9/11, \beta =6, \beta _1=0\) with Eigenvalues \(\Lambda _0\) and \(\Lambda _{Williams}\). The violation of the boundary condition \(c(\xi )=0\) for \(c\rightarrow -\infty\) when using \(\Lambda _{Williams}\) is apparent.

Fig. 3
figure 3

Numerical \(c_A(\xi )\) solution using \(\Lambda _0\) [black, Eq. (19)] and Williams \(\Lambda\) [gray, Eq. (5)]

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6 Application: Lewis Number Dependence of Laminar Flame Speed

In the case of \(Le \ne 1\), an additional transport equation for the normalized fuel concentration Y(x, t) needs to be solved:

$$\rho \frac{\partial Y}{\partial t}+\rho u \frac{\partial Y}{\partial x}=\frac{\partial }{\partial x}\left( \rho D \frac{\partial Y}{\partial x}\right) +\frac{{{\dot{\omega }}}_F}{Y^{1}_{F}}$$
(25)

Defining \(\phi =1-Y\) and \(Le=\lambda /(\rho c_p D)\), Eq. (25) can be cast in the following form:

$$\rho \frac{\partial \phi }{\partial t}+\rho u \frac{\partial \phi }{\partial x}=\frac{1}{Le}\frac{\partial }{\partial x}\left( \frac{\lambda }{c_p} \frac{\partial \phi }{\partial x}\right) -\frac{{{\dot{\omega }}}_F}{Y^{1}_{F}}$$
(26)

emphasising the similarity to Eq. (1) except for the denominator 1/Le. In the chemical source term \(\omega (c)\) the factor \((1-c)\) in front of the exponential in Eq. (4) needs to be replaced by \((1-\phi )\) (Poinsot and Veynante 2005; Ferziger and Echekki 1993):

$$\frac{{\dot{\omega }}_F}{Y^{1}_{F}}=B_1 \left( 1-\alpha (1-c)\right) ^{\beta _1-1} e^{-\frac{\beta }{\alpha }}\rho (1-\phi )exp\left( -\frac{\beta (1-c)}{1-\alpha (1-c)}\right)$$
(27)

Again rescaling the spatial coordinate according to \(\xi = \rho _u s_L c_p / \lambda\), we obtain:

$$\frac{\partial c}{\partial \xi }=\frac{\partial ^2 c}{\partial \xi ^2}+\omega (c,\phi )$$
(28)

and

$$\frac{\partial \phi }{\partial \xi }=\frac{1}{Le}\frac{\partial ^2 \phi }{\partial \xi ^2}+\omega (c,\phi )$$
(29)

where now the chemical source term depends not only on c but also on \(\phi\):

$$\omega (c,\phi )=\Lambda _{Le} \left( 1-\alpha (1-c)\right) ^{\beta _1-1} (1-\phi )exp\left( -\frac{\beta (1-c)}{1-\alpha (1-c)}\right)$$
(30)

The prefactor \(\Lambda _{Le}=\lambda /(c_p \rho _u^2 s_L^2)\) in the source term represents the Lewis-dependent eigenvalue of the transport equations, which needs to chosen such that the boundary conditions \(c,\phi = 0\) for \(\xi \rightarrow -\infty\) and \(c,\phi = 1\) for \(\xi \rightarrow +\infty\) are fulfilled simultaneously. For \(Le=1\), it is obvious that \(\phi (\xi )\equiv c(\xi )\) solves both Eqs. (28, 29) with the given boundary conditions and \(\Lambda\) chosen as described above.

From numerical integration of Eqs. (28, 29), varying \(\Lambda _{Le}\) to fulfill the boundary conditions we find (maybe not surprisingly) that \(c(\xi )'s\) for \(Le \ne 1\) are almost identical to \(c(\xi )\) for Le=1, while the \(\phi (\xi )\) are more diffuse for \(Le<1\) and less diffuse for \(Le>1\). We find also, that \((1-\phi )\) is closely proportional to \((1-c)\) in the region where the source term \(\omega (c,\phi )\) is notably different from zero.

Inspection of Eq. (30) shows that if \(I_{Le}(1-\phi )=(1-c)\) with constant \(I_{Le}\), the source term in Eq. (28) would be identical to Eq. (4) and the solution for \(c(\xi )\) would indeed be the same as in the case \(Le=1\) provided that \(\Lambda _{Le}/I_{Le}=\Lambda _{Le=1}(\alpha ,\beta )\). Equation (29) can be cast in a formally similar form to Eq. (28) by rescaling the spatial coordinate in Eq. (29) as \(\eta = Le \xi\):

$$\frac{\partial \phi }{\partial \eta }=\frac{\partial ^2 \phi }{\partial \eta ^2}+\frac{1}{Le}\omega (c,\phi )$$
(31)

Inserting the Ansatz \((1-c)=I_{Le}(1-\phi )\) into Eq. (30) yields

$$\frac{\partial \phi }{\partial \eta }=\frac{\partial ^2 \phi }{\partial \eta ^2} +\frac{\Lambda _{Le}}{Le} \left( \frac{1}{1-\alpha I_{Le}(1-\phi )}\right) ^{\beta _1-1} (1-\phi )exp\left( -\frac{\beta I_{Le}(1-\phi )}{1-\alpha I_{Le}(1-\phi )}\right)$$
(32)

which is formally identical to Eq. (28) if we replace \(\phi\) by c, \(\alpha\) by \(\alpha '=I_{Le} \alpha\), and \(\beta\) by \(\beta '=I_{Le} \beta\). The Eigenvalue of the equation to fulfill the boundary condition must therefore be equal to \(\Lambda _{Le=1}(\alpha ',\beta ')\). This yields the following equations for \(\Lambda _{Le}\) and \(I_{Le}\):

$$\frac{\Lambda _{Le}(\alpha ,\beta )}{I_{Le}(\alpha ,\beta )}=\Lambda _{Le=1}(\alpha ,\beta )$$
(33)

and

$$\frac{\Lambda _{Le}(\alpha ,\beta )}{Le}=\Lambda _{Le=1}(\alpha I_{Le},\beta I_{Le})$$
(34)

To derive an equation for the dependence of the laminar flame speed \(s_L\) on \(Le,\alpha ,\beta\), we now replace \(\Lambda _{Le=1}\) in Eqs. (33, 34) for \(\beta _1=0\) by Eq.  (21). This yields a quadratic equation for \(I_{Le,0}\):

$$I_{{Le},0} \left( \alpha ^2-4 \alpha \beta -\beta ^2\right) \left( \frac{4 \alpha +4 \beta -5}{Le (8 \alpha +8 \beta +10)}-\frac{I_{{Le},0} (4 \alpha I_{{Le},0}+4 \beta I_{{Le},0}-5)}{8 \alpha I_{{Le},0}+8 \beta I_{{Le},0}+10}\right) =0$$
(35)

Solving for \(I_{{Le},0}\) and remembering that the Eigenvalue of the transport equation is related to the laminar flame speed through \(\Lambda _{Le}=\lambda /(c_p \rho _u^2 s_L^2)\), we obtain a relationship for the variation of \(s_L\) with \(Le,\alpha ,\beta\) through \(s_L(Le)=s_L(Le=1)/\sqrt{I_{Le}}\):

$$\begin{aligned} g&=4(\alpha +\beta )-5 \\ h&=\frac{4(\alpha +\beta )+5}{4(\alpha +\beta )-5} \\ \frac{s_L(Le)}{s_L(Le=1)}&= \frac{\sqrt{\sqrt{g^2 \left( g^2 (h+1)^2+60 g h (h+1) Le+100 h^2 Le^2\right) }-g ^2(h+1)-10g h Le}}{2 \sqrt{5 g}} \end{aligned}$$
(36)

A similar slightly more complex relation can be obtained for \(s_L\) in the case of \(\beta _1=1\). The dependence of \(s_L(Le)/s_L(Le=1)\) on Le is much stronger than on \(\alpha ,\beta\). Figure 4 shows \(s_L(Le)/s_L(Le=1)\) versus Le for \(\alpha =0.818\), typical values of \(\beta\) and \(\beta _1=0,1\). We can see that the differences between \(s_L\) for \(\beta _1=0\) and \(\beta _1=1\) in this range of Le numbers are minimal (of the order of 1% at \(Le=2\), but increasing at higher Le). A Taylor series of \(s_L^2\) in inverse powers \(\beta\) shows that to leading order \(s_L(Le)/s_L(Le=1) \rightarrow \sqrt{Le}\) for large \(\beta\) in agreement with classical results, see Williams (2018), Ferziger and Echekki (1993).

Fig. 4
figure 4

\(s_L(Le)/(s_L(Le=1))\) versus Le for \(\beta =6\) (black), \(\beta =12\) (dashed) and \(\beta =20\) (gray); left:\(\beta _1=0\), right:\(\beta _1=1\)

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7 Analytic Flamelet pdf’s

In the following, we derive the subgrid flamelet pdf of a freely propagating flat premixed flame within a CFD cell of width \(\Delta\). Figure 5 shows the \(c(\xi )\) profile of a 1-D-flame front in a cell of width \(\Delta \sim 2*\delta _f\). Note that for consistency all spatial quantities like the cell width \(\Delta\) and the thermal flame thickness \(\delta _f\) have to be calculated in units of \(\xi\).

Due to the monotonicity of \(c(\xi )\), the cell average \({\overline{c}}\) is a unique function of \(\Delta\) and of the position of the flame front within the cell. We denote the position of the left cell boundary as \(\xi ^-\) and the corresponding c value as \(c^-=c(\xi ^-)\). The position of the right cell boundary is \(\xi ^+=\xi ^-+\Delta\) with \(c^+=c(\xi ^+)\).

Fig. 5
figure 5

LES cell width for integration of premixed reaction source term

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The flamelet pdf p(c) is given by

$$p(c) = \left\{ \begin{array}{lr} 0&{} : c<c^-\\ \frac{1}{N}\frac{1}{dc/d\xi }&{} : c^-< c < c^+\\ 0&{} : c>c^+\\ \end{array} \right.$$
(37)

with normalisation factor N. The normalisation condition for p(c) yields

$$1=\int _0^1p(c)dc=\frac{1}{N}\int _{c^-}^{c^+}\frac{1}{dc/d\xi }dc=\frac{1}{N}\int _{\xi ^-}^{\xi ^+} d\xi =\frac{\xi ^+-\xi ^-}{N}=\frac{\Delta }{N}$$
(38)

for any flamelet pdf, yielding \(N=\Delta\). This is also evident when calculating the mean of a variable z(c) from p(c):

$$\overline{z(c)}=\int _0^1z(c)p(c)dc=\frac{1}{\Delta } \int _{c^-}^{c^+}\frac{z(c)}{dc/d\xi } dc=\frac{1}{\Delta } \int _{\xi ^-}^{\xi ^+}z(\xi ) d\xi$$
(39)

The last term is identical to a simple spatial mean of z within the cell.

Many of the analytic results presented below are only possible if \(c^-\), \(c^+\) and \(1/(dc/d\xi )\) are available as analytical functions of c. This requirement is fulfilled both by the Echekki \(c_E(\xi )\) and by the new profile \(c_m(\xi )\). For the Echekki \(c_E(\xi )\), we get:

$$p_E(c)=\frac{1}{\Delta }\left( \frac{1}{dc/d\xi }\right) _E = \left\{ \begin{array}{lr} \frac{1}{\Delta }\frac{1}{c}&{} : c<1-\frac{1}{\beta }\\ \frac{1}{\Delta }\frac{1}{(\beta -1)(1-c)}&{} : c > 1-\frac{1}{\beta } \end{array} \right.$$
(40)

The new profile \(c_m(\xi )\) yields:

$$\frac{dc_m(\xi )}{d\xi }= c(1-c^m)$$
(41)

and

$$p_m(c) = \left\{ \begin{array}{lr} 0&{} : c<c^-\\ \frac{1}{\Delta }\frac{1}{c(1-c^m)}&{} : c^-\le c \le c^+\\ 0&{} : c>c^+\\ \end{array} \right.$$
(42)

The functional form of the new flamelet pdf \(p_m(c)\) may look similar to a beta pdf \(p_\beta (c)\) but there are notable differences. The first factor 1/c could be realized in \(p_\beta (c)\) by setting \(a=0\), which is however not admissible in this context since the pdf would not be integrable at \(c=0\). Note that the factor 1/c arises as a consequence of the pure diffusion in the preheat region \(c \rightarrow 0\) and is independent of the Arrhenius parameters \(\alpha ,\beta ,\beta _1\). The second factor \(1/(1-c^m)\) is responsible for the characteristic asymmetry of the flamelet pdf, its integral also diverges logarithmically \(c=1\). In contrast to the beta pdf, the power m is on c, not on \((1-c)\).

8 Mean Value of Chemical Source Term

For the new \(c(\xi )\) flame profile and pdf, the mean of the reaction source term \(\omega _m(c)\) evaluates analytically as:

$$\overline{\omega _m(c)}=\int _0^1 \omega _m(c) p_m(c) dc=\frac{1}{\Delta } \int _{c^-}^{c^+}\frac{\omega _m(c)}{c(1-c^m)}dc=\frac{(c^+)^{m+1}-(c^-)^{m+1}}{\Delta }$$
(43)

An analytical result can also be derived for the mean of the sum of laminar diffusion and reaction source terms, which is often modelled together e.g. in flame surface density models:

$$\overline{\frac{\partial ^2 c}{\partial \xi ^2}+\omega (c)}=\overline{\frac{\partial c}{\partial \xi }}=\frac{1}{\Delta } \int _{c^-}^{c^+}\frac{\frac{\partial c}{\partial \xi }}{dc/d\xi }dc=\frac{1}{\Delta }\int _{c^-}^{c^+}dc=\frac{c^+-c^-}{\Delta }$$
(44)

This result again is true for all flamelet pdf’s. For small \(\Delta\), this expression obviously approaches the correct limit \(dc/d\xi\). In flame surface density models the mean of the sum of diffusion and source terms is modelled as \(\rho _u s_L \mid \nabla c \mid\), which in our context is equivalent to \(dc/d\xi\) because \(\rho _u=1=s_L\) due to the scaling of \(\xi\).

9 Evaluation of Means and Variances of c

For evaluation of mean and variance of c from \(p_m(c)\) the following integrals are defined:

$$I_n(c,m)=\int \ c^n\left( \frac{1}{dc/d\xi }\right) _m dc=\int \frac{c^n}{c(1-c^m)}dc$$
(45)

Specifically, we have

$$I_0(c,m)= log (c)-\log \left( 1-c^m\right) /m$$
(46)
$$I_1(c,m)= c \ _2F_1\left( 1,\frac{1}{m},1+\frac{1}{m};c^m\right)$$
(47)
$$I_2(c,m)= \frac{c^2}{2}\, _2F_1\left( 1,\frac{2}{m},1 +\frac{2}{m};c^m\right) =\frac{1}{2} I_1 \left( c^2,\frac{m}{2}\right)$$
(48)

where \(_2F_1(a,b,c;z)\) is the hypergeometric function, which can be evaluated numerically from its power series definition for \(z<1\). Note that its last argument \(c^m \in [0,1]\). The last equality in Eq. (48) results from addition theorems of \(_2F_1(a,b,c;z)\). For integer m, \(I_1(c,m)\) reduces to expressions containing only powers of c, logarithms and trigonometric functions, e.g.:

$$I_1(c,4)=\frac{\tan ^{-1}(c)+\tanh ^{-1}(c)}{2}$$
(49)

The means of c and \(c^2\) in a cell of width \(\Delta\) evaluate as:

$$\overline{c_m},_\Delta =\int _0^1 c p_m(c) dc=\frac{1}{\Delta } \int _{c^-}^{c^+} \frac{1}{(1-c^m)} dc=\frac{ I_1(c^+,m)-I_1(c^-,m)}{\Delta }$$
(50)

and

$$\overline{c_m^2},_\Delta =\frac{I_2(c^+,m)-I_2(c^-,m)}{\Delta }=\frac{1}{2 \Delta }\left( I_1 \left( {c^+}^2,\frac{m}{2}\right) -I_1 \left( {c^-}^2,\frac{m}{2}\right) \right)$$
(51)

from which the variance \(\overline{c_m^2},_\Delta -(\overline{c_m},_\Delta )^2\) can be calculated.

10 Numerical Evaluation of \({\overline{c}}\)

The numerical evaluation of \(I_n(c,m)\) becomes numerically ill-defined for c values approaching 1. The reason is that the series expansion of the hypergeometric function \(_2F_1(a,b,c;z)\) does not converge well for \(z\rightarrow 1\). \({\overline{c}}\) can be evaluated in c or \(\xi\) space:

$$\overline{c_m},_\Delta =\int _0^1c p_m(c)dc=\frac{1}{\Delta }\int _{c^-}^{c^+}\frac{c}{dc/d\xi }dc=\frac{1}{\Delta }\int _{\xi ^-}^{\xi ^+} c(\xi ) d\xi$$
(52)

where \(\xi ^+=\xi ^-+\Delta\), \(c^-=c_m(\xi ^-),c^+=c_m(\xi ^+)\) and \(\xi ^-=\xi _m(c^-)\), \(\xi ^+=\xi _m(c^+)\). The indefinite integral of \(c_m(\xi )\) is given by:

$$\int c_m(\xi ) d\xi =e^{x} \left( 1+e^{m x}\right) ^{\left( 1-\frac{1}{m}\right) } \, _2F_1\left( 1,1;1+\frac{1}{m};-e^{m x}\right)$$
(53)

We now choose a small \(\epsilon\) such that \(1-c_m(\xi _\epsilon )<\epsilon\). By replacing \(c_m(\xi )\) in the integrand of Eq. (53) by 1 for \(\xi >\xi _\epsilon\), we can assure that the hypergeometric function is only evaluated for values of its last argument far enough from 1 so that the series still converges. In the results presented here, we have used \(\epsilon =10^{-6}\). Explicitly, \(\xi _\epsilon =\log \left( 1-(1-\epsilon )^{-m}\right) /m\).

11 Evaluation of Favre Averages

At constant pressure, the ratio of density \(\rho\) to the unburnt density \(\rho _u\) is proportional to 1/T, yielding

$$\overline{\left( \frac{\rho }{\rho _u}\right) }=\int _0^1 \frac{1-\alpha }{1-c(1-\alpha )}p(c)dc$$
(54)

and for the Favre average of c

$${\tilde{c}}=\frac{\overline{c\rho }}{{\overline{\rho }}}= \frac{\int _0^1 \frac{c(1-\alpha )}{1-c(1-\alpha )}p(c)dc}{\int _0^1 \frac{1-\alpha }{1-c(1-\alpha )}p(c)dc}$$
(55)

Those two quantities cannot be evaluated analytically for arbitrary values of m, but integer (and some half-integer) values of m yield analytic results in terms of powers of c, logarithmic and trigonimetric functions. Results are too unwieldy to be shown here. Figure 6 shows \({\tilde{c}}\) as function of \({\overline{c}}\) for various values of \(\Delta\) together with the BML limit \({\tilde{c}}=\alpha {\overline{c}}/(1-\alpha {\overline{c}})\) for \(m=4, \alpha =3/4\). We can see that the BML limit is approached only for \(\Delta \gg \delta _f\) (\(\delta _f=1.87\) for \(m=4\)).

Fig. 6
figure 6

\({\tilde{c}}\) versus \({\overline{c}}\) for \(m=4, \alpha =3/4\) and \(\Delta =0\) (gray), \(\Delta =3\) (thin), \(\Delta =10\) (thick) and \(\Delta \longrightarrow \infty\) (dashed), i.e. the BML limit

Full size image

12 Analytic Results for \(\beta\)-pdf

The mean of the new chemical source term \(\omega _m(c)\) can also be analytically evaluated if the beta pdf \(p_\beta (c)\) is used instead of the flamelet pdf \(p_m(c)\):

$$\overline{\left( \omega _m\right) _\beta }=\int _0^1 p_\beta (c) \omega _m(c) dc =\frac{(m+1) \Gamma (a+b) \left( \frac{\Gamma (a+m+1)}{\Gamma (a+b+m+1)}-\frac{\Gamma (a+2 m+1)}{\Gamma (a+b+2 m+1)}\right) }{\Gamma (a)}$$
(56)

and

$$\begin{aligned} \overline{\left( \frac{\partial ^2 c_m}{\partial \xi ^2}+\omega _m(c_m)\right) }_\beta&=\overline{\left( \frac{\partial c_m}{\partial \xi }\right) }_\beta \\&=\int _0^1 p_\beta (c) \frac{\partial c}{\partial \xi } dc \\&=\frac{a}{a+b}-\frac{\Gamma (a+b) \Gamma (a+m+1)}{\Gamma (a) \Gamma (a+b+m+1)} \end{aligned}$$
(57)

where the beta pdf parameters a, b can be evaluated as usual from \({\overline{c}}=a/(a+b)\) and \(\overline{c^2}- {\overline{c}}^2=ab/[(a+b)^2(1+a+b)]\).

13 Approximation for Lower and Upper Bounds \(c^-\), \(c^+\) and \(\xi ^-\), \(\xi ^+\)

The evaluation of quantities from \(p_m(c)\) requires knowledge of the lower and upper bounds \(c^-\),\(c^+\) as function of the quantities provided by the CFD solver. In a CFD simulation, a transport equation provides \({\tilde{c}}\) and \(\Delta\) is calculated from the computational grid. \({\overline{c}}\) can be estimated from \({\tilde{c}}\) through Eq. (55). Thus it is desirable to obtain \(c^-,c^+\) as functions of \({\overline{c}}\) and \(\Delta\).

Because \(c^-=c_m(\xi ^-)\), \(\xi ^+=\xi ^-+\Delta\) and \(c^+=c_m(\xi ^+)=c_m(\xi _m(c^-)+\Delta )\), it suffices to derive a relation \(c^-({\overline{c}},\Delta )\). For \(m=1\), the function \(c^-({\overline{c}},\Delta )\) is given analytically by:

$$c_1^-({\overline{c}},\Delta )=\frac{e^{{\overline{c}} \Delta }-1}{e^{\Delta }-1}$$
(58)

For larger m, an analytic inversion of \({\overline{c}}(c^-,\Delta )\) is not possible.

Since an explicit expression is desirable for fast evaluation in the CFD code and for the study of the behaviour of \(c^-({\overline{c}},\Delta )\), we provide the following approximation. With

$$\delta _0(c)=\frac{1-c^m}{1-c}$$
(59)

and

$$s({\overline{c}},\Delta ,m)=e^{-\Delta /7} \left[ \left( e^{\Delta /7}-1\right) e^{2 ({\overline{c}}-1) m}+{\overline{c}}\right]$$
(60)

we define a first approximation of \(c^-_{m,a}({\overline{c}},\Delta )\):

$$c^-_{m,a}({\overline{c}},\Delta ) = c_1^-({\overline{c}},\Delta *\delta _0\left[ s({\overline{c}},\Delta ,m)\right] )$$
(61)

This approximation has the correct limiting behaviour \(c^- \rightarrow {\overline{c}}\) for \(\Delta \rightarrow 0\) and is also accurate for \(\Delta \gg \delta _f\). It is only accurate to 5% in the region \(\Delta /\delta _f \approx 20\). Figure 7 shows \(c^-_{m,a}\) and \(c^+_{m,a}=c_{m}(\xi _m(c^-_{m,a})+\Delta )\) as functions of \({\overline{c}}\) for \(m=6\) (for which \(\delta _f=1.61\)) and different values of \(\Delta\). We can see that as expected \(c^-\) and \(c^+\) are near \({\overline{c}}\) for small \(\Delta /\delta _f\), while for large \(\Delta /\delta _f\), \(c^+\) raises fast towards 1, while \(c^-\) stays near 0 and raises to 1 only near \({\overline{c}}\rightarrow 1\).

Fig. 7
figure 7

\(c^+\) and \(c^-\) as function of \({\overline{c}}\) for \(m=6\) and \(\Delta =0.1,1,5,10\) (black full, black dashed, gray full, gray dashed). Curves below the diagonal line correspond to \(c^-\), curves above are \(c^+\)

Full size image

\({\overline{c}}\) calculated with \(c^-_{m,a}({\overline{c}},\Delta )\) through Eq. (50) deviates slightly from the originally specified \({\overline{c}}\). An even better approximation \(c^-_{m,b}({\overline{c}},\Delta )\) results by applying a second fix-point iteration:

$$\begin{aligned} {\overline{c}}_{corr.}&=2{\overline{c}}-\int _{c^-_{m,a}({\overline{c}}, \Delta )}^{c^+_{m,a}({\overline{c}},\Delta )} cp_m(c)dc \\ c^-_{m,b}({\overline{c}},\Delta )&=c^-_{m,a}({\overline{c}}_{corr.},\Delta ) \end{aligned}$$
(62)

This approximation has an error below 1% and represents \(c^-,c^+\) considerably better at large \(\Delta /\delta _f\). Obviously, since \({\overline{c}}\) is available analytically as function of \(c^-\) and \(\Delta\) through Eq. (50), the relation betweem \({\overline{c}}\) and \(c^-\) can also be inverted numerically e.g. by 1-D-re-interpolation. Since \({\overline{c}}\) is a strictly monotonic function of \(c^-\) and \(\Delta\), this procedure is always well defined.

14 Comparison of Different pdf’s

A comparison of \(p_A(c)\) (evaluated numerically from \(c_A(\xi )\)) with the Echekki \(p_E(c)\) and the new pdf \(p_m(c)\) is shown in Fig. 8 together with the respective source terms \(\omega _A(c)\), \(\omega _E(c)\) and \(\omega _m(c)\). It is evident that \(p_E(c)\), featuring a slope discontinuity at \(c=1/\beta\) is not a particularly good approximation to the real \(p_A(c)\) in contrast to the new \(p_m(c)\).

Fig. 8
figure 8

Comparison of \(p_m(c)\) (full), \(p_E(c)\) (dashed) and \(p_A(c)\) (dotted) (black); source term \(\omega (c)\) in gray for comparison

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Figure 9 compares the new pdf \(p_m(c)\) and the beta pdf \(p_\beta (c)\) for two values of \(\Delta\). The exact mean and variance of the c distribution within the cell was used to calculate the parameters a, b in \(p_\beta (c)\). Similar to Bray et al. (2006), we find that \(p_\beta (c)\) does not fit the flamelet pdf well in the region where \(\omega (c)>0\), causing an over prediction of \(\overline{\omega (c)}\) for large c variance, i.e. large \(\Delta /\delta _f\). Note that in contrast to \(p_{\beta }(c)\), the shape of the flamelet pdf \(p_m(c)\) is actually independent of \({\overline{c}}\) and \(\Delta\); those parameters only influence the limits \(c^-\) and \(c^+\) and the normalisation factor \(N=\Delta\).

Fig. 9
figure 9

Flamelet (black) and beta (dashed) pdf’s with source term (gray); left: \(\Delta =2*\delta _f\), right: \(\Delta =0.5*\delta _f\)

Full size image

Both pdf’s approach the correct limit of \(p(c) \rightarrow \delta (c-{\overline{c}})\) in the DNS limit (\(\Delta<< \delta _f\)). For small ratios \(\Delta /\delta _f\), only a small part of \(p_m(c)\) is cut out near \({\overline{c}}\) while for larger \(\Delta /\delta _f\), \(c^-\) and \(c^+\) move nearer towards \(c=0,1\) and reveal a larger portion of the complete \(1/(dc_m(\xi )/d\xi )\) shape. In contrast, the beta pdf resembles a Gaussian near \({\overline{c}}\) for small variance (i.e. small \(\Delta /\delta _f\)) and switches to a double-delta type behaviour for larger \(\Delta /\delta _f\).

Fig. 10
figure 10

Exact \(\overline{\omega _m}\) (full) compared to mean evaluated with beta function (dashed) as function of \(\Delta\); black: m=4, gray: m=12; left:\({\overline{\omega }}\) versus \(\overline{\omega _\beta }\), right: \(\overline{\omega _\beta }/{\overline{\omega }}\)

Full size image

Figure 10 (left) shows a comparison of \(\overline{\omega _\beta (c)}\) evaluated using the beta pdf (with exact mean and variance of c to evaluate a, b) with the exact \(\overline{\omega (c)}\) (evaluated using \(p_m(c)\)) vs. \(\Delta\) for two different values of m. On the right we plot the ratio \(\overline{\omega _\beta (c)}/\overline{\omega (c)}\). \(\overline{\omega _\beta (c)}\) over predicts the exact value of \(\overline{\omega (c)}\) for \(\Delta > 0.75\). In general \(\overline{\omega _\beta (c)}/\overline{\omega (c)}\) increases with \(\Delta\) and saturates for \(\Delta \rightarrow \infty\) to a value of \(m/3+0.4\). The error introduced when using the beta pdf thus increases with increasing activation energy (i.e. \(\beta\), i.e. m).

15 Effect of 3D Flame Wrinkling

In the thin reaction zone regime, large turbulent eddies of size \(l_e \gg \delta _f\) will wrinkle the reaction zone (Fig. 11) while leaving its internal structure largely intact. This will increase the mean reaction rate in the cell approximately by the wrinkling factor \(\Xi =\Sigma /\Delta ^2\), where \(\Sigma\) is the area of the wrinkled reaction zone within the LES cell and \(\Delta ^2\) is the area of the flat flame propagating in \(\xi\) direction.

Fig. 11
figure 11

Wrinkled flame front

Full size image

For a given \(c(\mathbf {\xi })\) field within an LES cell volume \(\Omega\) of size \(\Delta ^3\), the area of an isosurface \(\Sigma\) of a certain \((c^*)\) value is defined by (Osher and Fedkiw 2005):

$$\Sigma (c^*) = \int _{\Omega } \delta (c (\mathbf {\xi }) - c^*) \mid \nabla c (\mathbf {\xi }) \mid d \mathbf {\xi }$$
(63)

Note that (Boger et al. 1998)

$$\overline{\mid \nabla c\mid }=\frac{1}{\Delta^3}\int _0^1 \Sigma (c)dc$$
(64)

We can define a generalized wrinkling factor \(\Xi (c^*)\) through:

$$\Xi (c^*) = \frac{\Sigma (c^*)}{\Delta ^2}$$
(65)

Note that \(\Xi (c^*)\) can be smaller than 1 for \(c^*\) isosurfaces which are not or only partially contained within the LES cell. The fine grained pdf of a given \(c(\mathbf {\xi })\) field is given by Poinsot and Veynante (2005), Gao and OBrien (1993)

$$p (c^*) = \frac{1}{\Delta ^3} \int _{\Omega } \delta (c (\mathbf {\xi }) - c^*) d \mathbf {\xi }$$
(66)

16 Evaluation of \(\overline{\omega (c)}\) in the Thin Reaction Zone Regime

Using the definition of the fine-grained pdf, we can evaluate \(\overline{\omega (c)}\) as

$$\overline{\omega (c)} = \int _0^1 \omega (c^*) p (c^*) dc^*= \int _0^1 \omega (c^*) \frac{1}{\Delta ^3} \int _{\Omega } \delta (c (\mathbf {\xi }) - c^*) d \mathbf {\xi } dc^*$$
(67)

We can multiply the delta function by \(1=\frac{\mid \nabla c\mid }{\mid \nabla c\mid }\):

$$\overline{\omega (c)} = \int _0^1 \omega (c^*) \frac{1}{\Delta ^3} \int _{\Omega } \frac{1}{\mid \nabla c (\mathbf {\xi }) \mid } \delta (c (\mathbf {\xi }) - c^*) \mid \nabla c (\mathbf {\xi }) \mid d \mathbf {\xi } dc^*$$

In the thin reaction zone limit, c isosurfaces within the reaction zone are largely parallel and \(\mid \nabla c (\mathbf {\xi })\mid _{c^*} \approx \mid dc / d\xi \mid _{1D,c^*}\). Replacing the c gradient in the denominator by the 1D gradient and moving it outside the \(\Omega\) integral (note that \(\mid dc / d\xi \mid _{1D,c^*}\) is constant during the \(\Omega\) integration since it depends on \(c^*\) only) yields:

$$\overline{\omega (c)} = \int _0^1 \omega (c^*) \frac{1}{\Delta } \frac{1}{\mid dc / d\xi \mid _{1D,c^*}} \frac{1}{\Delta ^2} \int _{\Omega } \delta (c (\mathbf {\xi }) - c^*) \mid \nabla c (\mathbf {\xi }) \mid d \mathbf {\xi }dc^*$$
(68)

The second integral is just \(\Sigma (c^*)\), see Eq. (63) and the first term is equal to the flat flame pdf:

$$p_{1D} (c^*) = \frac{1}{\Delta } \frac{1}{\mid dc / d\xi \mid _{1D,c^*}}$$
(69)

We obtain the following approximation in the thin reaction zone limit:

$$\overline{\omega (c)} = \int _0^1 \omega (c^*) p_{1D} (c^*) \frac{\Sigma (c^*)}{\Delta ^2} dc^*= \int _0^1 \omega (c^*) p_{1D} (c^*) \Xi (c^*) dc^*$$
(70)

using Eq. (65).

For large LES cells (\(\Delta > \delta _f\)), \(\Sigma (c^*)\) and \(\Xi (c^*)\) might be approximately constant in the c region where \(\omega (c)>0\) as we show later using a simple flame wrinkling model. Equation (70) can then be further simplified as proposed in Boger et al. (1998):

$$\overline{\omega (c)} = \Xi ^*\int _0^1 \omega (c^*) p_{1D} (c^*) dc^*= \Xi ^**\overline{\omega (c)}_{1D}$$
(71)

The assumption of a constant wrinkling factor (independent of c) is the basis of many algebraic flame surface density models.

Obviously, \(\Xi ^*\) in Eq. (71) should represent the wrinkling of the \(c^*\) isosurface representing the maximum heat release. Note that the wrinkling of c isosurfaces outside the reaction region (\(\omega (c)>0\)), e.g. within the preheat zone, will not affect \(\overline{\omega (c)}\).

17 Relation Between pdf, Wrinkling Factor and Correction Factor

We can also multiply the delta function in Eq. (67) by \(1=\frac{\mid dc/d \xi \mid _{1D,c^*}}{\mid dc/d \xi \mid _{1D,c^*}}\)

$$\overline{\omega (c)} = \int _0^1 \omega (c^*) \frac{1}{\Delta } \frac{1}{\mid dc / d\xi \mid _{1D,c^*}} \frac{1}{\Delta ^2} \int _{\Omega } \delta (c (\mathbf {\xi }) - c^*) \mid dc/d \xi \mid _{1D,c^*} d \mathbf {\xi }$$
(72)

Defining a surface-like quantity \({\hat{\Sigma }}(c^*)\)

$${\hat{\Sigma }} (c^*) = \int _{\Omega } \delta (c (\mathbf {\xi }) - c^*) \mid dc / dx \mid _{1D,c^*} d \mathbf {\xi }$$
(73)

and \({\hat{\Xi }}(c^*)={\hat{\Sigma }} (c^*)/\Delta ^2\) we obtain the following exact relationship:

$$\overline{\omega (c)} = \int _0^1 \omega (c^*) p_{1D} (c^*) {\hat{\Xi }} (c^*) d c^*$$
(74)

Combination with Eq. (70) yields

$$\overline{\omega (c)} = \int _0^1 \omega (c^*) p_{1D} (c^*) I(c^*) \Xi (c^*) dc^*$$
(75)

with the definition of a correction factor (Driscoll 2008)

$$I(c^*)={\hat{\Sigma }}(c^*)/\Sigma (c^*)={\hat{\Xi }}(c^*)/\Xi (c^*)$$
(76)

I(c) thus represents the effect of a change of the inner structure of the reaction zone from that of the corresponding 1-D flat flame. Combination of these equation yields

$$p(c^*) = p_{1D} (c^*) {\hat{\Xi }}(c^*) =p_{1D} (c^*) I(c^*) \Xi (c^*)$$
(77)

This decomposition nicely separates the contributions of the freely propagating flat flame pdf \(p_{1D} (c^*)\), correction factor \(I(c^*)\) (providing the modification of the flame inner structure due to e.g. strain) and wrinkling factor \(\Xi (c^*)\) (contributing the geometrical effect of 3D wrinkling of \(c^*\) isosurfaces). This exact equation is valid independent of the number of spatial dimensions and of the form and size of the LES cells. Such relationships make pdf formulations attractive and conceptually more general than e.g. 1-D flamelet filtered tabulations or 1-D approximate deconvolution methods. The general applicability also makes pdf methods suitable candidates for LES simulations of high pressure premixed flames, where a large amount of subgrid flame wrinkling is to be expected.

18 Illustration of Wrinkled Flame pdf

In the appendix, we present a simple model for the 3-D \(c(\mathbf {\xi })\) field of a sinusoidally wrinkled flame, which allows analytical calculations of isocontours. Implict is the assumption that the c gradient on all \(c^*\) isosurfaces is equal to that of the flat flame at \(c=c^*\).

Fig. 12
figure 12

Isocontours at distances \(d=0.25,0.5,0.75,1,1.25,1.5,1.75,2\) (gray) from single cosine wave of amplitude A (black); left: \(A=0.5\), right: \(A=1\); dashed curves are not fully contained in cell

Full size image

A wrinkling of the reaction zone \(c^*\) isosurface can push other isosurfaces partly or totally out of the LES cell as illustrated in Fig. 12. This reduces the corresponding \(\Sigma (c)\)’s and translates into a reduction of p(c) as shown in Fig. 13. Parameters used to produce these plots are given in the “Appendix”. We see that for small enough wrinkling amplitude \(A/\Delta\), p(c) is enhanced almost uniformly by the wrinkling factor \(\Xi ^*\) in the reactive c region, where \(\omega (c)>0\), while p(c) is decreased below \(p_{1D}(c)\) already for \(c>c^-\) and \(c<c^+\). Also shown in Fig. 13 are 1D pdf’s where we replaced \(\Delta\) by a smaller filter size \(\Delta /\Xi ^*\) (which automatically moves cut-off values \(c^-,c^+\) closer towards \({\overline{c}}\)). 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 \(\Delta /\delta _f\).

Fig. 13
figure 13

Subgrid pdf for for \(\Delta =4,A=1\) (left) and \(\Delta =8,A=2\) (right), black: wrinkled flame pdf; gray: flat flame pdf, black dashed: flat flame pdf for filter \(\Delta /\Xi\); reaction rate (gray dashed) for comparison

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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.

19 Effect of Flame Stretch and Curvature on p(c)

In a locally stretched flame, the reaction layer will become thinner, increasing \(\mid \nabla c \mid\) over \((dc/d\xi )_{1D}\) and thus decreasing I(c) and p(c). In contrast, a locally thickened reaction zone (e.g through turbulent mixing by very small vortices or through differential diffusion) should lead to an increase of I(c) and p(c). Such an increase of I(c) with increasing Karlowitz number has been observed in highly turbulent flame DNS’s (Luca et al. 2019).

Stronger modifications of local p(c)’s are expected in large LES cells containing regions of large flame curvature (e.g. cusps, tips of flame fingers) and regions of subgrid flame-flame interaction. Here the filtered flame front might not be represented well by the 1-D flat flame structure.

20 Summary and Conclusions

In this paper we present a chemical source term \(\omega _m(c)\) to be used in the transport equation of a single reaction progress variable c, which closely mimics the Arrhenius \(\omega _A(c)\) but yields an analytical solution \(c_m(\xi )\) with a simple analytical inverse \(\xi _m(c)\). The parameter m can be adapted so that \(c_m(\xi )\) closely mimics the Arrhenius \(c_A(\xi )\) for the relevant range of its parameters \(\alpha , \beta , \beta _1\).

We derive the flat flame premixed flamelet pdf \(p_m(c)=\frac{1}{\Delta } \frac{1}{c(1-c^m)}\) and provide an analytic correlation to evaluate the limits of the c integration \(c^-,c^+\) as function of \({\overline{c}}\) and \(\Delta\) for efficient implementation of \(p_m(c)\) into CFD codes. The integral of \(p_m(c)\) diverges logarithmically at \(c=0,1\), but these limits are never accessed. Due to the closed form of the expressions no piecewise definition of the pdf as in Salehi and Bushe (2010), Domingo et al. (2005) or \(\delta\) functions at \(c=0,1\) like in BML theory are required. The explicit form of the flamelet p(c)’s show that the 1/c-behaviour near \(c=0\) is universal, independent of Arrhenius parameters, since it is caused by the pure heat diffusion within the preheat zone.

Many quantities of interest can be evaluated analytically with this approach. We derive estimates of the laminar flame Eigenvalue \(\Lambda\) and of the Le dependence of the laminar flame speed \(s_L\) which are more accurate than classical results from the literature based on asymptotic analysis.

The cell averages of source term \(\omega _m(c)\) and of the sum of source and diffusion terms (equivalent to \(dc/d\xi\)) are evaluated analytically with the new flamelet pdf and compared to beta pdf results, which are also available in closed form when using \(\omega _m(c)\) as source term. We show that for \(\Delta /\delta _f<1\) (i.e. small c variance) the beta pdf results are reasonable, while for large \(\Delta /\delta _f\) the beta pdf strongly over predicts \(\overline{\omega (c)}\), particularly for flames of large activation energy. Due to the simplicity of the new pdf, we can expect that \(p_m(c)\) could replace the beta pdf in presumed pdf modelling of the progress variable in premixed combustion.

The effect of flame wrinkling on the pdf is investigated using the fine-grained pdf formulation. We derive the exact relationship \(p(c)=p_{1D}(c)*I(c)*\Xi (c)\), separating the effects of the 1-D flamelet pdf from those of flame thickening and geometrical flame wrinkling. The c dependence of I(c) and \(\Xi (c)\) needs to be taken into account for large wrinkling factors and if isosurfaces for different values of c are wrinkled differently, e.g. in the case of large Karlowitz number flames.

For illustration of the effect of flame wrinkling on the 3D pdf we derive a simple model of sinusoidally wrinkled thin reaction fronts where for a given \(c^*\) value \(\mid \nabla c\mid\) is identical the flat flame gradient. We show that for large \(\Delta /\delta _f\) and moderate flame wrinkling the main effect on the pdf is an increase of p(c) in the reactive region by a wrinkling factor \(\Xi ^*\), representing the wrinkling of the isosurface at maximum heat release. p(c) decreases near \(c=0,1\) as required by its normalisation condition. In this limit the wrinkled flame pdf’s resemble flat flame pdf’s evaluated at a smaller filter \(\Delta '=\Delta /\Xi ^*\) in the reaction region \(c \approx c^*\).

As the wrinkling factor \(\Xi (c)\) is mostly determined by the turbulent flow field, it may be simpler to model Ξ than other quantities like the c variance or the c scalar dissipation rate which depend on the turbulent field and the inner flame structure. Many models for (constant) \(\Xi ^*\) are available in the framework of flame surface density theory (for an overview, see e.g. Ma et al. 2013). Ultimately, the choice of modelling style may however be a matter of taste or experience.

In the future we plan to compare the derived pdf’s with data extracted from DNS’s of premixed flames at ambient and elevated pressures (e.g. Klein et al. 2018). We will investigate modelling strategies for \(\Xi (c)\) and I(c) and we will attempt to extend the theory to the case of stratified / partially premixed flames.