Modelling of Spray Flames with Doubly Conditional Moment Closure

In the Conditional Moment Closure combustion model, transport equations are solved for the conditional moments Q _α (Z ^c ;x, t) ≡〈Y _α (x, t)|Y ^c (x, t) = Z ^c 〉 of the reactive scalars Y _α , which are conditionally averaged on a subset of the space of all reactive scalars Y ^c ⊂Y [10]. The probability density function (pdf) of the conditioning variables is presumed and mean quantities of the reactive scalars are calculated by integrating the conditional moments with the pdf. In contrast to conventional CMC, in the present DCMC approach the reactive scalars are conditioned on two conditioning variables.

The DCMC model equation for spray flames is derived following the approach by Mortensen and Bilger [12] who used a separated flow model to incorporated the effects of spray evaporation. With a separated flow model the local instantaneous balance equations for a multi-phase flow are written by introducing a phase indicator function 𝜃 _k (x, t), which is unity in the region occupied by phase k and zero everywhere else [18, 19]; its governing equation is as follows. Here π _k is the volumetric rate of phase change per unite volume. Then the continuity equation and the species transport for the phase k take the following form [12], where \(\hat {V}_{k,\alpha } \equiv \nabla \theta _{k} \cdot \mathbf {V}_{k,\alpha } = \nabla \theta _{k} \cdot (-D_{k,\alpha }/Y_{k,\alpha })\nabla Y_{k,\alpha }\) is the diffusion velocity across the phase interface. It can be related to π _k through the jump conditions at the interface [18, 19], In this work the subscript k may only refer either to the gas or to the liquid phase and is, therefore, omitted in further derivations.

The present DCMC approach uses the mixture fraction ξ and the reaction progress variable c as conditioning variables. Whilst the mixture fraction is a passive scalar with respect to chemical reaction, it is produced through droplet evaporation. It is defined to be 0 in air and 1 in undiluted fuel vapour. For a pure liquid fuel the jump condition at the phase interface (5) for the mixture fraction gives \(\xi \hat {V}_{\xi }=(1-\xi )\varPi \) [12], which leads to the following transport equation of the mixture fraction, The progress variable is defined as a linear expression of one reactive scalar Y _ψ , normalised by its un-reacted and equilibrium values, \(Y_{\psi }^{0}(\xi )\) and \(Y_{\psi }^{\text {Eq}}(\xi )\) respectively [20, 21]. If Y _ψ is governed by an equation of the same form as Eq. 3, then the instantaneous transport equation of the reaction progress variable c = c _ψ is, where N _ξ = D _ξ ∇ξ ⋅∇ξ, N _c = D _c ∇c ⋅∇c and N _{ξ c} = D∇ξ ⋅∇c are the scalar dissipation rates. The first and second line of Eq. 8 are identical with the original result by Bray et al. [21], which was extended in the present work to include the spray source terms. Note that in the derivation of Eq. 8 it is assumed that D _c = D _ψ = D _ξ . For the sake of brevity, the entire term in the second line of Eq. 8 is written as \(\theta \rho \dot {\omega }_{c}^{*}\). The presence of an evaporation source in the c-equation was discussed by Domingo et al. [7]. Using the jump conditions for Y _ψ and ξ, the first term in the third line can be expressed as the product of the evaporation mass source ρπ and a function, which can be evaluated for any ξ and c, since Y _ψ (ξ, c) is known from Eq. 7. In this expression δ _ψ is the mass fraction of species ψ in the liquid phase, which is equals 1 if c _ψ was based on the fuel and 0 otherwise. By grouping the terms in the Eq. 8, the c-equation finally appears in a a very similar form compared to Eq. 3 with one term representing the apparent reaction rate and two separate source terms due to evaporation.

The DCMC equation is the transport equation of the doubly conditional moments Q _α ≡〈Y _α (x, t)|ξ(x, t) = η, c(x, t) = ζ〉. It is derived using the joint-pdf method assuming moderately high Reynolds number and invoking the primary closure hypothesis for the diffusive fluxes in conditional space [10]. This leads to, where 〈⋅|η, ζ〉 denotes density-weighted conditional averaging and \(\widetilde {p}\) is the density-weighted pdf, defined as \(\bar {\rho }\ \widetilde {p}(\eta )=\langle \rho |\eta ,\zeta \rangle p\). Moreover, \(\bar {\theta }\) is the volume fraction of the gaseous phase, which is ≈ 1 in a dilute spray, and δ _α signifies the species mass fraction in the liquid phase, which is 1 for the liquid fuel species and 0 otherwise.

A similar transport equation can be derived for the conditional enthalpy Q _h . This equation is presented in the Appendix and for completeness, the terms of molecular transport in the Q _h -equation are also shown. The analogue term is neglected in Eq. 10 due to the high Reynolds number assumption.

In Eq. 10, \(\dot {\omega }_{c}^{*}\) represents the apparent reaction progress variable source term in a flow with non-uniform mixture fraction. Since Y _ψ = Y _ψ (ξ, c), according to the definition in Eq. 7, \(Y_{\psi }^{\prime \prime }= 0\). Thus, the conditional apparent reaction rate is given as,

Lewis number effects have been argued to be important for CMC of premixed flames [22]. Replacing the usual unity Lewis number assumption with the assumption of constant Le _α for each species, adds a factor to the scalar dissipation rate terms in Eq. 10 and also adds a differential diffusion term, whose influence on major species might be small however [22]. We recall that in the derivation of the c-equation (8), it was assumed that Le _c =Le _ξ =Le _ψ . This constraint can only be approximately valid in the general case of non-unity Lewis number and limits the choice of progress variable to major species with a Lewis number similar to Le _ξ . If this limitation is respected and Le _ψ is sufficiently close to Le _ξ , the error due this assumption will be small compared to the effect of non-unity Le _α which for minor species can be substantially different from Le _ξ .

Considering the effects of liquid fuel evaporation adds several terms to the DCMC equation (10), including doubly conditional terms equivalent to the terms first derived by Mortensen and Bilger [12]. Additional new terms appear to account for the effects of evaporation on the reaction progress variable. These terms contain the function \(\mathscr {C}(\eta ,\zeta )\), which is given by Eq. 9.

In the present derivation, the primary closure hypothesis is specifically applied to fluxes of the reactive scalar in conditional space only. This is in line with the original derivation [23], where it is clearly specified that closure is achieved by assuming a “first-order diffusion relation” for these fluxes. Source terms from evaporation or the reaction progress variable are not treated through this approximation, which leads to the appearance of terms containing the conditional correlations 〈Y α″π^″|η, ζ〉 and \(\langle Y_{\alpha }^{\prime \prime }\dot {\omega }_{c}^{*\prime \prime }|\eta ,\zeta \rangle \). In this way, Mortensen and Bilger [12] derived the conditional correlation term of mass fraction and evaporation rate using the joint-pdf method, whilst it seems that this term does not appear when the decomposition method is used. Indeed, the conditional correlation term of mass fraction and reaction progress variable source does not occur in the CMC equation for premixed flames derived by Mantel and Bilger [24] using the decomposition method and it has not been addressed in more recent work [22, 25] either. The effect of the conditional correlation term of the evaporation rate has not been studied yet and it has been neglected so far [13, 26].

Equation 10 represents the unclosed DCMC equation in its general form, whose derivation only requires very light modelling assumptions, viz. the primary closure hypothesis, moderate to high Reynolds number and constant Lewis numbers. Whilst the accuracy of first order closure for the doubly conditional reaction source term has been demonstrated [16], there is, in general, very little experience with the sub-models for DCMC. In some cases, however, it might be possible to generalise models used in conventional CMC or to adapt them from other combustion models that use a similar parametrisation, such as for instance mixture fraction-progress variable flamelet models.

In order to apply the DCMC model to a preliminary test case, in the present work the modelling choices detailed below are made. The selection of sub-models was partly driven by their extremely limited availability. In this sense, the set of models proposed is a first suggestion and more work will be necessary in the future to improve and validate sub-models for DCMC.

Unity Lewis number is assumed for simplicity in this first application of the model. For the transport in physical space the well-tested sub-models for conventional CMC can be easily adopted. The diffusion approximation is used for 〈u ^″ Y α″|η, ζ〉 [10] and the conditional velocity is modelled as \(\langle \mathbf {u}|\eta ,\zeta \rangle =\widetilde {\mathbf {u}}\), since gradients of \(\widetilde {\xi }\) are small. For the Favre pdfs of the conditioning variables, β-pdfs are presumed and, assuming statistical independence, the joint-pdf is computed as \(\widetilde {p}(\eta ,\zeta )=p_{\beta }(\eta ; \widetilde {\xi }, \widetilde {\xi ^{\prime 2}}) p_{\beta }(\zeta ; \widetilde {c}, \widetilde {c^{\prime 2}})\). This assumption is made for simplicity since accurate modelling of the joint-pdf in the present three-stream mixing problem plus droplet evaporation would be very complex and is not attempted here. The choice of sub-models for the doubly conditional scalar dissipation rates is very limited. In the present work, we follow the suggestions by Nguyen et al. [27]. For the scalar dissipation rate of the mixture fraction they assumed that N _ξ is primarily imposed by the mixing with little dependence on chemistry and, thus, 〈N _ξ |η, ζ〉≈〈N _ξ |η〉, which is modelled as a bell curve given by the Amplitude Mapping Closure (AMC) model [28]. Again following Nguyen et al. [27], 〈N _c |η, ζ〉 is modelled as the product of two bell curves, b(η) centred on the stoichiometric mixture fraction η = ξ _st and G(ζ), centred on ζ = 0.5; for η ≥ 2ξ _st it is zero. This model is a simple approximation of N _c -values from premixed flamelet tabulation. The conditional cross-scalar dissipation rate is not considered, in line with the assumption of statistically independent conditioning variables in the modelling of the pdf. A dilute spray is assumed, i.e. \(\bar {\theta }\approx 1\) but the doubly conditional evaporation source terms are not included in the DCMC equation. Finally radiation and wall heat losses are neglect. Together with the unity Lewis number assumption and in the absence spray source terms, the transport equation of the conditionally averaged enthalpy Q _h becomes trivial and does not need to be calculated. The conditional temperature Q _T is then calculated from Q _h and the conditional species mass fractions Q _α .

The DCMC combustion model is coupled to the flow field solver. An Euler-Lagrangian approach is followed, where the gaseous phase is computed through an unsteady Reynolds-averaged Navier-Stokes (RANS) simulation with the standard k-ε turbulence model and the parcels of liquid droplets are tracked as Lagrangian particles. The flow field solver integrates the momentum equation as well as the Favre-mean and variance equations of the conditioning variables, ξ and c. Note that (⋅)^′ denotes the fluctuation around the Favre mean, i.e. \(Y=\widetilde {Y}+Y^{\prime }\), to distinguish it from the conditional fluctuation in the DCMC equation. Here, the reaction progress variable is based on the mass fraction of carbon dioxide, \(Y_{\psi }=Y_{\text {CO}_{2}}\). The eddy diffusivity is calculated as \(D_{T}= \mu _{T}/(\bar {\rho }\text {Sc}_{T})\), with a constant turbulent Schmidt number Sc _T = 0.7; for the molecular diffusivity \(D= \mu /(\bar {\rho }\text {Sc})\), with Sc = 0.7. In the case of the mixture fraction, the scalar dissipation rate is modelled as for a passive scalar, \((\widetilde {\varepsilon }/\widetilde {k})\ \widetilde {\xi ^{\prime 2}}\) [29]. The scalar dissipation rate of c is closed using the model by Kolla et al. [30], where the model coefficients are set as described by Kolla and Swaminathan [31], notably β ^′ = 6.7 and \(K_{c}^{*}= 0.85\tau \). To account for non-uniform mixture τ, \({S_{L}^{0}}\) and \({\delta _{L}^{0}}\) are evaluated at the local mean mixture fraction \(\widetilde {\xi }\) [32]. For this purpose, the laminar flame speed and the thermal laminar flame thickness are pre-computed using the commercial software Cosilab and tabulated as functions of mixture fraction, \({S_{L}^{0}}(\xi )\) and \({\delta _{L}^{0}}(\xi )\) respectively; \(\tau (\widetilde {\xi })\) is calculated based on the conditional temperature Q _T (η, ζ) with \(\eta =\widetilde {\xi }\) and the Karlovitz number is \(\text {Ka}=[{\delta _{L}^{0}}(\widetilde {\xi })/{S_{L}^{0}}(\widetilde {\xi })]/\sqrt {\nu /\widetilde {\varepsilon }}\).

The reaction source term in the mean and variance equation of the progress variable, \(\widetilde {\dot {\omega }_{c}^{*}}\) and \(\widetilde {c^{\prime }\dot {\omega }_{c}^{*\prime }}\), are calculated by integrating \(\langle \dot {\omega }_{c}^{*}|\eta ,\zeta \rangle \) (11) with the pdf in the conditional space.

In this work an emphasis is put on the modelling of the spray combustion and we aim to provide closure for the complete set of evaporation terms that appear in the Favre-averaged transport equations of the conditioning variables. The mean evaporation source is computed by summing over the evaporated mass of all droplets in a CFD cell, where V represents the cell volume. The Favre-averaged transport equations contain several other evaporation source terms, which require modelling. In particular, for the mixture fraction variance equation, Giusti and Mastorakos [33] pointed out that both spray source terms could have a significant effect in the inner flame region. In single-conditional CMC, the terms \(\widetilde {\xi \varPi }\) and \(\widetilde {\xi ^{2}\varPi }\) are modelled either by summing \(\xi _{s,i}^{k} \dot {m}_{d,i}\) (k = 1 or 2) for all droplets in one cell [34] or, alternatively, by assuming that 〈π|η〉 has the shape of a δ-function at the average surface mixture fraction 〈ξ _s 〉 [35]. In both cases, it is assumed that ξ _s ≈ Y _F,sat, calculated for the droplet temperature T _d . This assumption is, however, only correct if the droplets evaporate upstream of the flame [13]. In general, \(Y_{F,\text {sat}} \leq \xi _{s} \leq Y_{F,\text {Eq}}^{-1}(Y_{F,\text {sat}})\), where \(Y_{F,\text {Eq}}^{-1}\) is the inverse function of the equilibrium fuel mass fraction Y _F,Eq(ξ). This reflects the case of a droplet evaporating in burned gases where some reaction products are present besides the fuel vapour at the droplet surface. In principle, this effect can be accounted for by integrating the spray source term in doubly conditional space. For a generic spray source term of the type \(\widetilde {\mathcal {F}\varPi }\), Eq. 21 is exact if \(\langle \mathcal {F}^{\prime \prime }\varPi ^{\prime \prime }|\eta ,\zeta \rangle = 0\), notably for \(\mathcal {F}=\xi \) or \(\mathcal {F}=\mathscr {C}(\xi ,c)\) etc. Fig. 1 shows a contour plot of the spray source term of the \(\widetilde {c}\)-equation in conditional space, \((\mathscr {C}(\eta ,\zeta )+\zeta )\), which can be directly evaluated using the conditional moments from CMC. Considering the conditional evaporation term shows that two necessary conditions for a source term of the progress variable are fulfilled: first, droplet evaporation in unburned mixture does not impact the progress variable since \((\mathscr {C}(\eta ,\zeta )+\zeta )= 0\) for ζ = 0 and, second, \((\mathscr {C}(\eta ,\zeta )+\zeta )\leq 1\) for ζ = 1, which signifies c is automatically bounded at 1 with respect to this term.

In order to devise a model for 〈π|η, ζ〉 it is recognised that the droplet temperature computed by the evaporation model fixes the fuel mass fraction – not the mixture fraction – on the droplet surface. This means that the droplet surface corresponds to the isoline of Q _F (η, ζ) = Y _{F s} in conditional space (represented by black lines in Fig. 1). Since evaporation only happens on the droplet surface, 〈π|η, ζ〉 must be zero everywhere except on this isoline and \(\widetilde {\mathcal {F}\varPi }\) can be calculated as a line integral. In Eq. 22, it was further assumed that π(η, ζ) is constant along this isoline. This is consistent with the evaporation model, where \(\dot {m}_{d}\) is computed based on the droplet temperature and the temperature far from the droplet surface, thus ignoring the exact temperature distribution in the near field. Note that the model of 〈π|η, ζ〉 presented above is only used to close the Favre-averaged evaporation source terms, but it is not used as a sub-model in the DCMC equation (see Section 2.2).

The evaporation of the Lagrangian droplets was computed according to the model by Abramzon and Sirignano [36] with Stefan flow correction and non-unity Lewis number in the film; for the liquid droplets, the approximation of infinite conductivity is made. where B _M and B _T are the Spalding mass and heat transfer numbers respectively. Sh and Nu are calculated using the Frössling correlation. The quantities ρ _G , C _{p G} , C _{p V} , μ _G , λ _G , D _G are evaluated at reference conditions for the gaseous boundary layer, computed according to the 1/3 rule for temperature and species; ρ _L , C _{p L} , latent heat L _V and the fuel saturation vapour pressure \(p_{F}^{\text {sat}}\) are computed at the droplet temperature T _d . The droplet experiences sphere drag and the impact of turbulence on the droplet motion is mimicked through stochastic dispersion for isotropic turbulence. No model for secondary break-up of droplets was used.

A detailed chemical mechanism for ethanol combustion [37] with 57 species and 383 reversible reactions is used. The mechanism performs well in predicting ignition delays and laminar flame speeds at ambient pressure when compared to experimental data. It has been successfully used in a CMC simulation by Giusti and Mastorakos [33].

The computational fluid dynamics toolbox OpenFOAM-2.3.0 is used to solve for the Favre-averaged flow field variables, \(\widetilde {\mathbf {u}}\), \(\widetilde {\xi }\), \(\widetilde {\xi ^{\prime 2}}\), \(\widetilde {c}\) and \(\widetilde {c^{\prime 2}}\) and to track the Lagrangian droplets. The coupling of the flow field solver and the DCMC model is effectuated as shown in a schematic in Ref. [38]. In addition to \(\bar {\rho }\) and \(\widetilde {T}\), the DCMC model also returns \(\widetilde {Y_{F}}\), \(\widetilde {\dot {\omega }_{c}^{*}}\), \(\widetilde {c^{\prime }\dot {\omega }_{c}^{* \prime }}\) and the Favre-averaged evaporation terms (22) to the flow field solver. The motion and evaporation of the Lagrangian droplets are solved at the at beginning of each time step.

Applying the closure models and simplifying assumptions presented in Section 2.2, the DCMC equation (10) reduces to, Eq. 25 is integrated with a finite volume method whilst employing an operator splitting procedure. The fractional step approach is well established in CMC modelling and the significance of operator splitting errors in spray flames was investigated by Wright et al. [39]. The operator splitting approach allows for higher computational efficiency, since the stiff chemistry term is decoupled from the non-stiff convective terms. First, the transport in physical space by advection and turbulent diffusion, as well as the dilatation term (terms 2, 3 and 4 in the first line of Eq. 25 respectively) are computed. An upwind scheme is used for the advection term. Second, transport in conditional space (lines 2 and 3 in Eq. 25) is integrated for every DCMC cell. Finally, the chemical reaction source is solved independently for every (η, ζ)-node in every DCMC cell. For the second and the third sub-step the solver VODPK is used. This operator splitting procedure automatically assures that Q _ψ conserves its linear dependence on ζ according to the definition of the progress variable.

A 1D DCMC grid was used to discretise the physical domain with 35 DCMC cells along the burner axis for 0 < z/d < 5. The transfer of data from the fine CFD mesh to the coarse DCMC grid is achieved by integrating the conditional scalar dissipation rates over the volume of the DCMC cell [40].

The conditional space, \(D=\{(\eta ,\zeta )\in \mathbb {R}^{2}: 0\le \eta \le 1, 0\le \zeta \le 1\}\), is discretised with 51 η-nodes, clustered around the stoichiometric mixture fraction η = ξ _st (for the reaction of ethanol with air ξ _st ≈ 0.1006) and 41 ζ-nodes, which are more closely spaced at ζ = 1. The derivatives in conditional space are discretised with second-order finite differences apart from ∂/∂ ζ, which is computed with an upwind scheme. Dirichlet boundary conditions are set on all four sides of the conditional domain; mixing line and equilibrium condition for ζ = 0 and 1 respectively, air at η = 0 and fuel vapour at the boiling point at η = 1. The equilibrium condition was approximated by solving single-conditioned, non-premixed “0D-CMC” equation, similar to the non-premixed flamelet equation, with a very low scalar dissipation rate N _ξ,max = 1 1/s, compared to the critical scalar dissipation rate of approximately 367 1/s [41]. The initial condition for the doubly conditional moments in the DCMC cells is computed by solving the 0D-DCMC equation, that is to say the DCMC equation (25) with prescribed, fixed scalar dissipation rates and for a spatially homogeneous case, i.e. without the terms that represent transport in physical space, until steady state is reached. The solutions of this equation are similar to the results presented by Nguyen et al. [27]. As DCMC initial condition a very weakly strained solution of the 0D-DCMC equation was used; notably, N _ξ,max = 1 1/s and N _c,max = 200 1/s. The set of conditional moments used as inlet boundary condition is identical with the initial condition. Hence the conditional moments in the DCMC cell located at z/d = 0, shown later in Figs. 4 and 5, are representative of the DCMC initial condition.

The DCMC model is applied to a piloted ethanol spray flame recently studied experimentally by Kariuki and Mastorakos [17]. The presence of droplet evaporation and, thus, the existence of mixture inhomogeneities, combined with substantial pre-vaporisation and premixing makes this flame a suitable candidate to test the present DCMC model.

In the experimental set-up, the ethanol spray is injected into the main flow of air ≈ 38 cm upstream of the nozzle with a diameter d = 42 mm. The flame is stabilised at the end of the nozzle through an annular pilot of diameter ≈ 51 mm and width ≈ 6 mm. The pilot is a open premixed methane-air flame with an approximately stoichiometric fuel-air ratio and a cold volume flow rate of 33.5 L/min. In the present work, we consider two reacting cases with overall equivalence ratios (liquid fuel to main air flow) ϕ = 0.62 and 0.82 (corresponds to ξ ≈ 0.065 and 0.084 respectively) and a non-reacting case. In all three cases the air-flow rate is 235 L/min and the liquid fuel mass injected in the cold case is the same as in the richer flame. Hence, the flames have the character of a pilot-stabilised turbulent premixed jet flames with a bulk velocity U _b ≈ 3.04 m/s, calculated using the area of the nozzle.

The three-dimensional computational domain stretches − 1 < z/d < 19 along the burner axis, where z = 0 is the position of the nozzle outlet and the pilot and the main flow inlet is retracted by one nozzle diameter; the diameter of the entire domain is 33d. Turbulence levels are set according to Laser Doppler Anemometry (LDA) measurements to u ^′/U _b ≈ 0.18 and estimating L _T ≈ d/3. Since the degree of pre-vaporisation of the ethanol spray was not measured in the experiment, the mixture fraction at the inlet had to be estimated. According to a separate RANS simulation of the upstream part of the burner corresponding to the richer flame and the cold case, the inlet boundary condition for the mixture fraction was set to \(\widetilde {\xi }= 0.04\) and \(\widetilde {\xi ^{\prime 2}}= 0\). Even though the injected amount of liquid fuel is smaller in the leaner case, the same boundary condition is also used for the simulation of the flame with ϕ = 0.62. This will allow us to explore the sensitivity of the simulation results to the level of pre-vaporisation and premixing in the upstream region of a spray flame; in the rich flame approximately half the fuel is pre-vaporised compared to two thirds in the leaner flame. For the main flow, the inlet boundary condition for the reaction progress variable upstream of the nozzle is set to \(\widetilde {c}= 0\). For the purpose of the present paper whose aim is to discuss the model and its application, the uncertainty related to the inlet boundary condition is considered satisfactory.

The pilot is an open flame, which is not retracted relative to the main flow. Hence, dilatation in the pilot flame does not lead to a significant increase of the mean axial velocity but to an increase in width of the hot pilot stream as it is expected for a usual triangular flame. Following a single mixture fraction based approach in the simulations, the pilot is modelled as a premixed stoichiometric ethanol-air flame. The annular pilot flame itself is not resolved but instead modelled as a uniform inlet boundary condition with a laminar inflow of burnt gases, i.e. \(\widetilde {\xi }=\xi _{\text {st}}\), \(\widetilde {\xi ^{\prime 2}}= 0\) and \(\widetilde {c}= 1\), with a mass flow rate corresponding to the experimental configuration. In the experimental rig the annulus that stabilises the pilot is very narrow and the pilot flow is broadened by the dilatation across the flame. If the width of the original annulus was used for the uniform inlet of hot gas, whilst the pilot mass flow rate was kept equal to its value in the experiments, the axial velocity of the pilot flow would be greatly over-estimated. Instead, the surface area of the pilot inlet is increased to assure a realistic axial velocity for the pilot flow. For this purpose, the surface of the pilot inlet is three times the surface area of the annulus in the rig.

A small, laminar co-flow of 0.1 m/s is set around the burner nozzle, where \(\widetilde {\xi }= 0\) and \(\widetilde {c}= 1\). Walls are considered adiabatic.