A Thickened Stochastic Fields Approach for Turbulent Combustion Simulation

The Thickened Flame equation [5] is obtained by applying the transformation x′ = Fx and t′ = Ft/E [5] to Eq. 1, where the convection velocity v is given by the solution of the similarly-transformed Navier-Stokes equations [7]. In most previous applications of the Thickened Flame approach, following Refs. [5, 6], the thickened scalar transport equation Eq. 2 has been coupled with unthickened LES making the assumption that Ev is equal to the resolved velocity from the LES simulation \(\tilde {\mathbf {u}}\). The effect of thickening factor F and efficiency function E can be understood by considering the solution of a stationary freely-propagating planar premixed flame (∂/∂t′ = ∂/∂t = 0). The flame thickness given by Eq. 2 is thickened by the factor F and the propagation speed is faster by a factor E compared respectively to the flame thickness δ_L and flame speed S_L given by solution of Eq. 1.

Thickening the species transport equations by factor F with E = 1 has the attractive feature that numerical resolution requirements are reduced while laminar flame speeds are unaffected. The turbulent flame speed, however, depends on the increase in flame surface area caused by wrinkling of the flame front. The amount of wrinkling depends (at least) on the ratio of turbulent velocity fluctuations to the laminar flame speed u′/S_L, and the ratio of the turbulence length scales to the laminar flame thickness, L_T/δ_L. Thickening the flame front to Fδ_L reduces the degree to which the turbulence will wrinkle the flame. The efficiency function E can then be used as a correction factor that increases the local propagation speed in order to compensate for the loss of resolved flame surface area resulting from application of the thickening transformation.

The local propagation speed of the reaction-front resolved by the LES simulation is described as the sub-filter turbulent flame speed S_TΔ [6]. The ratio of the sub-filter turbulent flame speed S_TΔ and the laminar flame speed S_L is assumed to be equal to wrinkling factor Ξ, which is equal to the projection of the sub-filter scale flame area in the direction of propagation, Modelling for the sub-filter flame wrinkling in the context of Thickened Flame modelling has been proposed initially by Colin et al. [5] and Charlette et al. [6] as functions of the non-dimensional sub-filter velocity fluctuations u′/S_L and the non-dimensional filter size Δ_TF/δ_L, where Δ_TF = Fδ_L is the effective filter scale in the Thickened Flame model. In general, the effective filter scale Δ_TF implied by the Thickened Flame model can be different from the filter length scale Δ used in modelling of the LES momentum or Stochastic Fields equations. Thickening the flame front by factor F reduces the non-dimensional filter size to Δ/Fδ_L, resulting in a reduction in the sub-filter turbulent flame speed by factor 1/E as illustrated in Fig. 1. The efficiency function E is defined in Ref. [5] as the ratio of the wrinkling factor in the thickened and unthickened flames,

The unmodified Stochastic Fields equation is given by Valiño et al. [8] as, where ζ_(i)(x, t) is the value of the composition vector on the i^th field. The terms on the right hand side represent the evolution of the stochastic field composition due to advection by the mean (or resolved) velocity; spatial diffusion by molecular D and turbulent D_T diffusivities; turbulent advection of fields relative to one another as modelled by a Wiener process where dW_j(i) is the component in the j^th direction of a normally-distributed Markovian random increment with zero mean and variance equal to the time step dt; unresolved scalar dissipation processes modelled by interaction by exchange with the mean (IEM) [9] with dissipation time scale \(\tau _{T}^{-1}=C_{\phi {\Delta }}(D+D_{T})/{\Delta }^{2}\); and the vector of chemical reaction source terms \( {\dot {\omega }}( {\zeta }_{(i)})\).

The Thickened Stochastic Fields equation is obtained by applying to Eq. 5, the same transformation that produces the Thickened Flame model in Section 2.1: x′ = Fx, t′ = Ft/E, and, since the Wiener increment vector dW has dimension \(\sqrt {t}\), \(\mathbf {dW}^{\prime }=\sqrt {F/E}\mathbf {dW}\): In principle the convection velocity \(\tilde {\mathbf {v}}\), turbulent diffusivity \(D^{\prime }_{T}\), and dissipation time scale \(\tau ^{\prime }_{T}\) come from solution of similarly-transformed LES momentum equations. However if, following Colin et al. [5], the thickened scalar equations are coupled with unthickened LES momentum equations then the velocity \(\tilde {\mathbf {u}}\) and turbulent diffusivity D_T from the unthickened LES should be scaled as: \(\tilde {\mathbf {v}}=\tilde {\mathbf {u}}/E\) and \(D^{\prime }_{T}=D_{T}/{EF}\). The turbulence timescale to be used in Eq. 6 is then \(\tau _{T}^{\prime -1}=C_{\phi {\Delta }}(DEF+D_{T})/(F{\Delta })^{2}\).

The transformation of the Stochastic Fields equation has the effect that the solution for a steady-state planar freely-propagating turbulent flame modelled by Eq. 6 is thickened by factor F and the propagation speed is increased by factor E relative to the solution of Eq. 5. The thickening factor F can therefore be set in order to obtain satisfactory numerical resolution on a particular computational grid. The efficiency function E should then be set in order to account for the reduction in resolved flame surface area that results from thickening of the Stochastic Fields equation.

The specification of the efficiency function for the Thickened Stochastic Fields model relates to the wrinkling of the reaction fronts in the Stochastic Fields solution, rather than the wrinkling of physical flames considered in the conventional Thickened Flame model. The characteristic thickness \(\delta _{c^{*}}\), and propagation speed \(S_{c^{*}}\), of the reaction fronts in the Stochastic Fields solution are in general different from the thickness and speed of the corresponding laminar flame. However, the wrinkling dynamics of the reaction fronts are assumed to be governed by the same function, Ξ_Δ. This assumption is justified because in both Thickened Flame and Thickened Stochastic Fields, the wrinkling dynamics of reaction fronts in the flamelet regime are dominated by the combination of resolved convection, sub-filter turbulent transport, diffusion, and reaction processes. The efficiency function for the Thickened Stochastic Fields model (Eq. 6) is then given by, where the effective filter scale of the thickened stochastic fields is \({\Delta }_{TSF}=F\delta _{c^{*}}\). In general Δ_TSF can be different from the filter scale Δ used to evaluate the model for the turbulent diffusivity in Eq. 6.

Previous studies have developed models for the function Ξ_Δ on the basis of theory and empirical information from direct numerical simulations and laboratory measurements of flame response [5, 6]. The purpose of the TSF approach however is to provide simulation results that maintain the same flame propagation speeds as the underlying Stochastic Fields modelling when the computational grid spacing is increased. The modelling for Ξ_Δ should not seek to improve the agreement between the Stochastic Fields model and DNS or experiment, rather it should fit to predictions of the underlying unthickened Stochastic Fields model. Improving the physical accuracy of the underlying Stochastic Fields modelling is outside the scope of the present study. The functional dependence of the wrinkling factor on the filter-scale turbulence properties \({\Xi }_{{\Delta }}(u^{\prime }_{{\Delta }}/S_{L},{\Delta }/\delta _{L})\) is therefore obtained from Stochastic Fields simulations across a range of conditions. The set up of one-dimensional Stochastic Fields simulations in order to obtain data for \(\delta _{c^{*}}/\delta _{L}\) and \(S_{c^{*}}/S_{L}\) is presented in the next Section.

The dependence of the sub-filter scale turbulent flame speed and reaction-front thickness on filter-scale turbulence properties is evaluated in a one-dimensional Stochastic Fields simulation of a freely-propagating planar turbulent flame. The one-dimensional approach neglects the effects that curvature and bulk strain have on the local propagation of reaction-fronts in Stochastic Fields LES of premixed combustion, but has the advantage that the simulations are computationally inexpensive compared to three-dimensional LES and still represent the transport processes normal to the resolved reaction front that dominate the dynamics of the resolved reaction front.

The premixed combustion kinetics are modelled using a one-step reaction model for methane-air flames, The fuel reaction rate is modelled by the Arrhenius law, where T, \(Y_{CH_{4}}\), \(Y_{O_{2}}\), \(M_{CH_{4}}\), \(M_{O_{2}}\) and R denote temperature, fuel and oxygen mass fractions, corresponding molar masses and the universal gas constant, respectively. The pre-exponential factor, the activation energy and the model exponents are A = 1.1 × 10¹⁰ (cgs), E_a = 20,000 cal/mol, \(n_{CH_{4}}= 1.0\) and \(n_{O_{2}}= 0.5\). The use of such simple chemical modelling is justified by the focus of the present study on evaluation of the numerical resolution requirements of the Stochastic Fields approach, rather than assessing the physical accuracy of the Stochastic Fields approach.

Temperature dependent properties are modelled with NASA polynomials and, due to the inherent unity Lewis number assumption in the Stochastic Fields formulation, the Schmidt and Prandtl numbers are both set equal to 0.7, while the mixture kinematic viscosity is modelled with Wilkes law. These assumptions lead to a laminar flame speed S_L = 0.38 ms^− 1, a thermal thickness of 0.408 mm and a burnt gas adiabatic temperature T_b = 2328 K in atmospheric stoichiometric conditions.

The Stochastic Fields equation is implemented within the block-structured BOFFIN computational fluid dynamics code [3, 17]. The code is a second order accurate finite volume method based on fully implicit low-Mach-number formulation using a staggered storage arrangement. For the momentum equation convection term, an energy conserving discritization scheme is used and all other spatial derivatives are approximated by standard second order central differences. A TVD scheme is used for the convection terms in the scalar conservation equations. The stochastic field equations are solved using a weak first order temporal approximation with accuracy \(\mathcal {O}\left (\sqrt {{\Delta }{t}}\right )\) based on the Euler–Maruyama scheme [18]. The Wiener process is approximated by time-step increments \(dt^{1/2}{\eta ^{n}_{i}}\) where \({\eta ^{n}_{i}}\) is a {− 1, 1} dichotomic random vector [19]. The chemical source terms are solved using a Newton method-based stiff solver.

The turbulent Bunsen flame is simulated using two different computational grids: a fine grid characterised by 0.5 mm grid spacing at the inlet, and a coarse grid characterised by 1.0 mm grid spacing at the inlet. Both grids are Cartesian, with a region of uniform transverse grid spacing around the inlet with transverse extent equal to twice the nozzle diameter. The axial grid spacing increases linearly in the axial direction. Sixteen stochastic fields are used in the three-dimensional simulations. The computational time step for the respective cases are 4.6μs and 2.3μs. The turbulent inflow is modelled with the digital filter based method of Klein et al. [20] using the mean and rms velocity profiles from [11].

Artificial thickening of the scalar equations is typically necessary only in regions of the flow containing thin reaction fronts. Thickening in regions where it is not required leads to an unnecessary loss of simulation fidelity, for example by over-predicting the rate of fuel-air premixing in partially-premixed combustion systems. To overcome this, Durand et al. [21] introduced dynamic thickening by a flame sensor Ω to remove the effects of thickening away from reaction fronts. This flame sensor is given by where c is a relevant progress variable equal to zero in the reactants and unity in the products. Ω > 0 indicates the presence of a flame front and Ω = 0 its absence. The thickening factor in the scalar transport equations are then rewritten as where F₀ is the thickening factor determined as a function of the grid spacing and the desired number of grid points within the reaction front.

A flame sensor can also be applied in the Thickened Stochastic Fields approach. However, evaluating Eq. 17 using the filtered progress variable obtained by ensemble averaging the stochastic fields causes the individual stochastic fields’ reaction fronts to be thickened to different extents, depending on their position in the resolved flame front. This is illustrated in Fig. 5, which presents the variation of the progress variable of selected individual stochastic fields ζ_(i) with respect to the ensemble averaged progress variable \(\tilde {c}\) for a one-dimensional flame with low sub-filter variance (Ka = 50 and Δ/δ_L = 1) and a flame with high sub-filter variance (Ka = 0.5 and Δ/δ_L = 5). The flame sensor value given by Eq. 17 is also presented. Thickening of the stochastic fields is most necessary in the range 0.5 < ζ_(i) < 0.8 where the reaction rate is greatest. Figure 5 shows that stochastic fields with their reaction fronts near to the leading or trailing edge of the resolved flame front might not be thickened to the extent intended.

In order to ensure sufficient thickening of every field, the flame sensor can be evaluated for every field and the maximum value then applied to all fields This approach yields the desirable behaviour that Eq. 17 is recovered in the limit of negligible sub-filter progress variable variance.

To ensure a smooth variation of Ω through the resolved flame front in simulations with a limited number of stochastic fields, Eq. 19 is modified to broaden the region of influence of each field according to where β is a parameter greater than unity that broadens the thickening zone for each field and determines how quickly the sensor approaches unity for c > 0 and c < 1. This broadening of the thickening zone of the individual fields increases the overlap between regions of large Ω_(i) and together with Eq. 20, setting β = 5 provided a smooth variation of Ω through the resolved flame front in the simulations presented below.

The stochastic field reaction front thicknesses reported in Fig. 3 and approximated by Eq. 13 may be used to estimate the resolution requirements for Stochastic Field LES. In the case of the Bunsen flame simulated in this study [11], the thermal thickness of a laminar stoichiometric methane-air flame is 0.41mm (using the one-step chemistry model described above), the filter length scale selected is 1mm, and the Karlovitz number is reported to range between 1 and 10 [11]. These Karlovitz numbers (corresponding to \(u^{\prime }_{{\Delta }}/S_{L}= 1.3\) and 6.2 respectively) lead to the prediction that \(\delta _{c^{*}}/\delta _{L} = 1.3 - 1.7\), given Δ = 1mm. Imposing a minimum requirement of 5 grid spacings within the stochastic field reaction front thickness \(\delta _{c^{*}}\) then dictates a maximum grid spacing of 0.11 - 0.14 mm for fully-resolved Stochastic Fields simulation. Compared to the usual practice of setting the grid spacing equal to the LES filter scale, this estimate suggests a numerically-resolved stochastic field solution would require a grid spacing of 11-14% of the filter scale, implying 360-750 times more grid points in a three-dimensional simulation. The resolution requirements of the Stochastic Fields approach may be even more daunting in simulation of industrial combustion systems for which the ratio of filter length scale to laminar flame thickness might need to be an order of magnitude greater than in the laboratory Bunsen flame example. The need for sub-filter scale resolution around the stochastic field reaction fronts suggest that the Thickened Stochastic Fields approach or an Adaptive Mesh Refinement (AMR) approach may be needed in order to obtain numerically-accurate Stochastic Fields predictions for practical systems.

A known disadvantage of thickened flame approaches is that application of the thickening factor and the efficiency function affects the chemical time scales in the fluid, and consequently changes the Damköhler and Karlovitz numbers describing the turbulence-chemistry interactions. The Damköhler number (Da= L_Ts_L/u′δ_L) and Karlovitz number (Ka\(=(u^{\prime 3}\delta _{L}/{S_{L}^{3}} L_{t})^{1/2}\)) change by E/F and (F/E³)^1/2 respectively. Since the turbulent diffusivity and micromixing terms in Eq. 6 lead to a stochastic field reaction front thickness that is greater than the corresponding laminar flame thickness by the factor \(f_{\delta }=\delta _{c^{*}}/\delta _{L}\) described by Eq. 13, the Thickened Stochastic Fields requires a thickening factor that is smaller than for the conventional Thickened Flame approach by the same factor f_δ. In the case of the Bunsen flame LES in this study, the thickening factor is expected to be 1.3-1.7 times less than in a conventional thickened flame LES with the same filter scale. The undesirable effect of thickening on the Damköhler and Karlovitz numbers is therefore generally smaller for the Thickened Stochastic Fields approach compared to the conventional Thickened Flame approach, and the correct chemical time scale and the conventional Stochastic Fields model is recovered in the limit of adequate resolution, for which no thickening is required.

Results of three simulations of the F3 Bunsen flame are presented in Figs. 6, 7 and 8. The LES filter length scale Δ is set equal to 1mm in all three cases. The first simulation solves the Stochastic Fields equation (Eq. 5) and adopts the conventional practice of setting the grid spacing equal to the filter scale, Δ_x = Δ = 1mm. The second simulation solves Eq. 5 with improved numerical resolution, using a grid characterised by Δ_x = Δ/2 = 0.5mm. The third simulation solves the Thickened Stochastic Fields equation (Eq. 6) with Δ_x = Δ = 1mm.

Figure 6 presents the instantaneous filtered temperature fields for an individual stochastic field in each of the three simulations. The images indicate that the flame thickness increases and the flame wrinkling decreases due to application of the Thickened Stochastic Fields model, and that the flame height increases. Use of “improved resolution” leads to a reduction in flame thickness and increased flame wrinkling. The different flame thicknesses in the three simulations are quantified by evaluating an indicative stochastic fields reaction front thickness given by the inverse average progress variable gradient magnitude on the stochastic fields conditioned on the progress variable giving maximum heat release, 〈|∇ζ_c,(i)|∣ζ_c,(i) = 0.68〉^− 1. Application of the Thickened Stochastic Fields model leads to the average thickness of the stochastic field reaction fronts increasing from 2.68 mm in the conventional Stochastic Fields simulation to 4.9 mm in the Thickened solution – confirming that the Thickened Stochastic Fields approach delivers the target five grid spacings within the stochastic field reaction fronts. Use of the “improved resolution” reduces the thickness to 1.8 mm, indicating that the numerical solution of the Stochastic Fields equation is not grid-independent with Δ_x = Δ [4].

A contour map of the thickening factor used in the Thickened Stochastic Fields model is presented in Fig. 7. The thickening factor is unity outside of the flame since the flame sensor tends to zero. Inside the flame, the value of the thickening factor varies due to the dependence on the local value of \(u^{\prime }_{{\Delta }}/S_{L}\) in Eq. 13. The peak value of the thickening factor within the flame increases with downstream distance: in the near-field the peak thickening factor is typically in the range 2-4, and it is mostly in the range 2-6 downstream of x/D = 5. The thickening factor tends to increase downstream because the Karlovitz number and \(u^{\prime }_{{\Delta }}/S_{L}\) decay away from the jet inlet, leading to the combustion becoming more flamelet-like and thereby increasing the resolution requirement. These thickening factor values may be compared with the fixed thickening factor F = 8 used in Ref. [22] for Thickened Flame simulations of the F3 flame with a similar grid spacing, Δ = Δ_x = 0.8mm. Figure 7 shows that the thickening factor required by the Thickened Stochastic Fields approach can be significantly less than the value required in the Thickened Flame model, reducing unwanted effects of the thickening on the chemical time scale, the Damköhler number and the Karlovitz number.

The radial profiles of the predicted mean methane mass fraction and mean axial velocity are shown for the three simulations in Fig. 8. The mean axial velocity shows reasonable agreement with the experimental data in each case, with predictions of the Thickened Stochastic Fields approach showing the closest agreement. Due to thermal expansion, the mean velocity field is affected by the flame location. The Thickened Stochastic Field predicts a longer flame that is closer to the experimental observations and this leads to a correspondingly better prediction of the mean velocity field.

The methane mass fraction profiles in Fig. 8 show that the two Stochastic Field simulations over-predict the rate of flame propagation, leading to the shorter flame length. The improved resolution case with Δ_x = 0.5mm displays faster flame propagation than the less well-resolved case with Δ_x = 1.0mm. Under-resolution affects the flame speed in three main ways [4]: first, numerical diffusion increases the local propagation speed of the individual stochastic field reaction fronts; second, thickening of the stochastic field reaction fronts caused by the numerical diffusion reduces the amount of flame wrinkling produced by a given velocity field; third, the numerical viscosity reduces the strength of small-scale eddies that wrinkle the reaction fronts. Since the present Bunsen flame exhibits combustion in the flamelet regime, the scales of the reaction fronts are generally smaller than the scales of the velocity field that wrinkles the flame. This suggests that a small degree of under-resolution primarily will affect the reaction front propagation speed, without affecting the flame wrinkling significantly – leading to an over-prediction of flame speed. More significant under-resolution increasingly will tend to affect the amount of flame wrinkling – reducing the predicted flame speed. The present Stochastic Fields LES of the F3 Bunsen flame with a 1 mm filter scale was predicted to require a maximum 0.11-0.14 mm grid spacing in order to resolve the stochastic field reaction fronts fully. This suggests that, with either the 1 mm grid spacing or the “improved resolution” 0.5 mm grid spacing, the present Stochastic Fields simulations remain substantially under-resolved. The observation that the predicted flame speed increases when refining the grid spacing to 0.5 mm indicates that this change in resolution affects the flame speed primarily through an increase in flame wrinkling, thereby taking the predicted flame propagation speed further away from the experimental observations. Further grid refinement beyond Δ_x = Δ/2 is not expected to have a significant further effect on the strength of eddies in the velocity field [23]. However, since the reaction fronts are expected to be around five times thinner in a fully-resolved Stochastic Fields simulation compared to in the “improved resolution case”, we anticipate that the flame propagation speed may start to reduce towards the speed seen in the experiment as the grid is refined further.

Full-resolution Stochastic Fields simulations however are not currently available. Since the computational effort for a fully-resolved simulation would be (0.5mm/0.11mm)³ = 93 times greater than for the “improved resolution” case, the computational resource for such a computation is prohibitive, even for this laboratory flame configuration. Such fully-resolved Stochastic Fields simulations of premixed turbulent combustion will be facilitated by application of AMR approaches, however introduction of AMR-Stochastic Fields LES is beyond the scope of this study. The Thickened Stochastic Fields approach however accounts for the loss of flame surface area due to the thickening applied in the model, and does not suffer from numerical diffusion effects since the thickened reaction fronts are well-resolved. Overall the Thickened Stochastic Fields prediction is in good agreement with the experiment. The simple addition of thickening terms in the Thickened Stochastic Fields equation thereby yields a computationally-efficient and accurate alternative to full-resolution Stochastic Fields simulations, which remain computationally prohibitive without adoption of AMR techniques.