3.1.1 RANS closure
The planar turbulent premixed flame is statistically homogeneous parallel to the flame and it is investigated initially using a one-dimensional Reynolds Averaged Stochastic Fields simulation. The turbulence is assumed to be uniform throughout the domain and across the flame, and specified by setting the ratio of the rms velocity fluctuation to the laminar flame speed
u′/
SL, and the ratio of the integral length scale of the turbulence normalised by the thermal thickness of the laminar flame,
LT/
δL. The turbulent diffusivity and mixing time scale required in Eq.
1 are then modelled by the following relations:
$$ D_{T}=C_{\mu}u^{\prime}L_{T}, $$
(2)
and
$$ \tau_{T}=\frac{C_{\phi}L_{T}}{2u^{\prime}}. $$
(3)
in which
Cμ = 0.09 [
18] and
Cϕ = 2.0 (
μL/
μT + 1), where the factor (
μL/
μT + 1) models behaviour at low Reynolds numbers [
19].
3.1.2 LES closure
Large Eddy Simulation is inherently three-dimensional. However the resolved flame front in LES of premixed turbulent combustion is typically thin with respect to the radius of curvature of the resolved flame front, across much of the flame surface area. Therefore, over much of the flame area, it is valid to approximate the molecular and sub-filter scale turbulent transport within the resolved flame front as one-dimensional in the direction perpendicular to the resolved flame front.
In order to investigate resolution requirements in the LES context, one-dimensional Stochastic Fields simulations are performed using sub-models for the sub-filter scale turbulent diffusivity and dissipation time scale that depend on a notional LES filter length scale Δ and the corresponding sub-filter scale velocity fluctuation
\(u_{{\Delta }}^{\prime }\). The one-dimensional simulations may be interpreted loosely as representing the transport along a line passing perpendicularly through a LES-resolved flame front, assuming that the propagation of the LES-resolved flame front is quasi-steady and unaffected by other flame fronts, by curvature, or by resolved strain (except to the extent that the resolved strain results in generation of sub-filter scale velocity fluctuations characterised by
\(u_{{\Delta }}^{\prime }\)). The sub-filter scale diffusivity and dissipation time scales in Eq.
1 are then modelled by Eqs.
2 and
3, replacing the turbulence length scale
LT with the filter scale Δ, and the turbulent velocity
u′ with the sub-filter scale velocity
\(u^{\prime }_{{\Delta }}\), and setting coefficients
Cμ = 0.09 and
Cϕ = 2.0 (
μL/
μT + 1) as in the RANS case.
In both the RANS and LES context, the turbulent premixed combustion regime is characterised by the Karlovitz number, Ka. In terms of integral length scale, the Karlovitz number can be approximated by
$$ \text{Ka}=\left[\left( \frac{u^{\prime}}{S_{L}}\right)^{3}\frac{\delta_{L}}{L_{T}}\right]^{1/2} $$
(4)
which can be obtained though conventional scaling analysis. In the LES context, for filter length scales in the inertial sub-range of the turbulence spectrum, the filter scale Karlovitz number is given by,
$$ \text{Ka}_{{\Delta}}=\left[\left( \frac{u_{{\Delta}}^{\prime}}{S_{L}}\right)^{3}\frac{\delta_{L}}{{\Delta}}\right]^{1/2}. $$
(5)
With the assumption of constant dissipation in the inertial range, the Karlovitz number is scale invariant [
16] and equal to the integral scale Karlovitz number, Ka
Δ = Ka. Since LES relies on selection of a filter length scale in the inertial sub-range, the effect of choosing different ratios of the filter length scale to laminar flame thickness (Δ/
δL) for simulation of a particular turbulent flame regime can be investigated by fixing Karlovitz number and evaluating the corresponding sub-filter scale velocity fluctuation as,
$$ u_{{\Delta}}^{\prime}=S_{L}\text{Ka}_{{\Delta}}^{2/3}\left( \frac{{\Delta}}{\delta_{L}}\right)^{1/3}. $$
(6)
The RANS model described previously is recovered in the limit that Δ →
LT, giving
\(u_{{\Delta }}^{\prime }\to u^{\prime }\).
In the following one-dimensional study, combinations of Karlovitz numbers, Ka ∈ [0.5, 1, 5, 10, 20, 30, 40, 50], and three characteristic, normalised length scales LT/δL of [1, 2.5, 5.0] are used for the one-dimensional simulations. These parameters span a range of premixed combustion regimes that would be typical of currently advanced LES of spark ignition and industrial gas turbines using between 10 and 100 million cells, as well as the conditions in the three-dimensional LES test case studied here.