Estimation of three-dimensional flame surface densities from planar images in turbulent premixed combustion

Abstract

Turbulence motions are, by nature, three-dimensional while planar imaging techniques, widely used in turbulent combustion, give only access to two-dimensional information. For example, to extract flame surface densities, a key ingredient of some turbulent combustion models, from planar images implicitly assumes an instantaneously two-dimensional flow, neglecting the unresolved flame front wrinkling. The objective here is to estimate flame surface densities from two-dimensional measurements assuming that (1) the flow is statistically two dimensional; (2) the measuring plane is a plane of symmetry of the mean flow, either by translation (homogeneous third direction as in slot burners for example) or by rotation (axi-symmetrical flows such as jets) and (3) flame movements in transverse directions are similar. The unknown flame front wrinkling is then modelled from known quantities. An excellent agreement is achieved against direct numerical simulation (DNS) data where all three-dimensional quantities are known, but validations in other conditions (larger DNS, experiments) are required.

Appendices

Appendix A: isotropic distribution of the unit vector normal to the flame front

The objective of this appendix is to investigate some consequences of the assumption of an isotropic distribution of the unit vector normal to the flame front (i.e. all directions are equally probable). According to our notations (Sect. 3.1; Fig. 2), this normal vector is determined from the two angles θ and ϕ such as −π ≤ θ ≤ +π and −π/2 ≤ ϕ ≤ +π/2. The surface S of the sphere of radius unity defined by the extremity of the vector n is given by:

$$ S = \int\limits_{-\pi}^{\pi}\int\limits_{-{\pi}/{2}}^{\pi/2} \cos\phi d\phi d\theta = 4 \pi $$

(36)

while the probability density functions p(θ) and p(ϕ), defining the distribution of both angles verifies:

$$ \int\limits_{-\pi}^{\pi}\int\limits_{-{\pi}/{2}}^{\pi/2} p\left(\phi\right) p\left(\theta\right) d\phi d\theta = 1 \quad \hbox{with} \quad \int\limits_{-\pi}^{\pi}p\left(\theta\right) d\theta =1 \quad \hbox{and} \quad \int\limits_{-{\pi}/{2}}^{\pi/2} p\left(\phi\right) d\phi = 1 $$

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Assuming an isotropic distribution of the vector n means that the probability that this vector points on an elementary surface $\cos\phi d\phi d\theta$ is $p(\theta)p(\phi) d\phi d\theta = \cos\phi d\phi d\theta / S.$ Then:

$$ p\left(\theta\right) = \frac{1}{2\pi} ; \qquad p\left(\phi\right) = \frac{\cos\phi}{2} $$

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The mean value of $\cos{\phi}$ becomes:

$$ \langle \cos\phi \rangle_s = \int\limits_{-{\pi}/{2}}^{\pi/2} \cos\phi p\left(\phi\right) d\phi = \int\limits_{-{\pi}/{2}}^{\pi/2} \frac{1 + \cos\left(2\phi\right)} {4} d\phi = \frac{\pi} {4} $$

(39)

According to Eq. 11, the flame surface density Σ is directly related to its two-dimensional estimate Σ_(x,y) as:

$$ \Sigma = \frac{4}{\pi}\Sigma_{(x,y)} $$

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showing that the flame surface Σ is about 30% higher than its two-dimensional estimate.

The mean value of cos²ϕ is given by:

$$ \langle \cos^2\phi \rangle_s = \int\limits_{-{\pi}/{2}}^{\pi/2} \cos^2\phi p\left(\phi\right) d\phi = \frac{1} {2}\int\limits_{-{\pi}/{2}}^{\pi/2}{\left(1 - \sin^2\phi\right) \cos\phi} d\phi = \frac{2}{3} $$

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Replacing $\langle \cos^2\phi \rangle_s$ by $\langle \cos\phi \rangle_s^2$ in this situation introduces an error of about 7.5% on $\langle \cos^2\phi \rangle_s.$ An isotropic distribution of the unit vector normal to the flame front corresponds to:

$$ {\langle{n_{x_{(x,y)}}}\rangle}_s^{2D} = {\langle{n_{y_{(x,y)}}}\rangle}_s^{2D} =0 $$

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$$ {\langle{m_{x_{(x,y)}}}{m_{x_{(x,y)}}}\rangle}_s^{2D} = {\langle{m_{y_{(x,y)}}}{m_{y_{(x,y)}}}\rangle}_s^{2D} = \frac{1} {2} $$

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Then, Eq. 27 gives:

$$ \Sigma = \sqrt{\frac{3}{2}} \Sigma_{(x,y)} $$

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underestimating Eq. 18 with an error lower than 4%, expected to be below experimental uncertainties. The proposed model (Eq.  27) recovers satisfactorily well the isotropic case.

Appendix B: small ϕ angles

For small ϕ angles, Taylor series expansions of cosϕ and cos²ϕ give:

$$ \cos\phi = 1 - \frac{\phi^2}{2} + {\mathcal O}\left(\phi^4\right) ; \quad \cos^2 \phi = 1 - \phi^2 + {\mathcal O}\left(\phi^4\right) $$

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leading to, when surface averaged:

$$ \langle{\cos\phi}\rangle_s^2 = 1 -{\langle\phi^2\rangle_s} + {\mathcal O}\left(\phi^4\right) ; \quad \langle{\cos^2 \phi\rangle}_s = 1 - \langle{\phi^2\rangle}_s + {\mathcal O}\left(\phi^4\right) $$

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Accordingly, the assumption ${\langle \cos^2\phi \rangle}_s \approx {\langle \cos\phi \rangle}^2_s$ holds at the fourth order.

Appendix C: usual spherical coordinates

As pointed out in Sect. 3.1, the components of the unit vector n normal to the flame surface are expressed from angles θ and ϕ which are not the usual spherical coordinates (see Fig. 2). This choice is motivated for convenience: ϕ is the off-measuring plane angle and its mean value is expected to be zero for two-dimensional mean flows, while the mean value of θ is also expected to be zero, because of the cylindrical mean shape of the flame investigated here, making easier comparisons of angle distributions as displayed in Fig. 3. Of course, all the derivation in Sect. 3 may be recast in terms of usual spherical coordinates (τ, ψ) as defined in Fig. 11. Comparing Figs. 2 and 11 shows that angles (θ, ϕ) and (τ, ψ) are related through:

$$ \tau = \frac{\pi}{2} + \theta (+ 2 k \pi) ; \quad \psi = \frac{\pi}{2} - \phi $$

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where k is an integer chosen to ensure a correct range of variations of τ, defined here as 0 ≤ τ ≤ 2π. Note that the angle θ is defined as positive when lying in the half-space x ≥0. For instantaneous two-dimensional flame fronts in the (x, y) plane, ϕ = 0 and ψ = π/2.

Fig. 11

Notations. x denotes the downstream direction while instantaneous flame front visualisations are performed in the (x, y) plane. The unit vector n normal to the instantaneous flame front is characterised from using spherical coordinates (τ, ψ), where 0 ≤ τ ≤ 2π and 0 ≤ ψ ≤ π

According to these notations, the components of vectors ${\bf n}_{(x,y)}$ and n are given by:

$$ {\bf n}_{(x,y)} = (\cos\tau,\sin\tau,0) \quad {\hbox{and}} \quad {\bf n} = (\cos\tau \sin\psi, \sin\tau \sin\psi,\cos\psi) $$

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while ${\bf n}_{(x,y)}$ and ${\bf n}^p_{(x,y)}$ are related through:

$$ {\bf n}^p_{(x,y)} = \sin\psi {\bf n}_{(x,y)} $$

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Two- and three-dimensional gradients of the progress variable c are linked by:

$$ \left| \nabla c \right|_{(x,y)} = \sin\psi \left| \nabla c \right| $$

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where $\sin\psi{\ge}0.$ Then:

$$ \Sigma_{(x,y)} = \overline{\left(\sin\psi \left| \nabla c \right| \delta\left(c-c^*\right)\right)} = {\langle \sin\psi \rangle}_s \Sigma $$

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