Numerical Study of Turbulent Non-Premixed Jet Flame Stability S. Navarro-Martinez*1, A. Kronenburg 1,2 1 2
Department of Mechanical Engineering, Imperial College London, UK Institut für Technische Verbrennung, University of Stuttgart, Germany
Abstract The stability of a non-premixed methane flame is investigated using Large Eddy Simulations with a Conditional Moment Closure combustion model (LES-CMC). The sensitivity of the blow-out limit to co-flow velocity is investigated. The model reproduces the linear dependence on the co-flow velocity as suggested by experiments in the low velocity region. The sensitivity of the model prediction to conditional sub-grid fluctuations is assessed. Analysis of the transport budgets of the scalar equation suggests that the stabilization mechanism is not unique but a combination of large scale mixing and premixed propagation. Introduction The stabilization mechanism of lifted flames has been the subject of many studies in the last decade, where numerous experimental and numerical simulations were performed (see the recent review by Lawn [1]). There are mainly two types of stabilization mechanisms: premixed controlled, where stabilization is the result of the balance of the local turbulent flame speed and the jet velocity. The fuel and the air are assumed to be premixed at the flame base. This mechanism appears to be predominant at large lift-off heights (greater than 20 jet diameters). Kalghati [2] proposed a model for the blow-out limits of free jet flames based on estimation of the turbulent flame speed from the laminar burning velocity, SL. When the lift-off height is small, the stabilization mechanism is mixing controlled, where large-scales eddies either quench the flame or ignite it by entrainment of fresh gases. Experimental correlations of blow-out limits exist based on the ratio of mixing and chemical time scales [3]. Other possible stabilization mechanisms include turbulent stretching and auto-ignition, which occur in the presence of hot co-flows [1]. It is known that the presence of a co-flow increases mean lift-off heights [4]. Small relative increase in coflow stream velocity can produce large increase in the blow-out limits. However, very little studies have been performed to address this sensitivity in attached flames [4, 5] and none of them numerically. The stability limits suggest that flames with stream co-flows greater than 3 SL, laminar flame velocity, are not sustained. In the work by Leung and Wierzba [5], several regimes were identified: At low co-flow velocities, the blow-out limit increases linearly with the velocity. If the velocity is increased, the blow-out limit decreases and at higher velocities, there is a bifurcation in the flame behaviour depending on the initial conditions of the flame Numerical studies have not fully investigated the experimental correlations and have been usually limited to few parametric studies. The partially premixed character of these flames excludes fast-chemistry approaches and more advance combustion sub-models are required that can represent accurately the turbulence-chemistry interaction. Examples include: [6, __________________________________
∗Corresponding author:
[email protected]
Proceedings of the European Combustion Meeting 2009
7]. There are very few Large Eddy Simulations of lifted flames with cold co-flow in the literature [8, 9]. Specific Objectives The present work has two objectives. The first is to capture extinction limits with a LES-CMC approach [10], and second to investigate the effect of co-flow velocity in extinction limits. The methodology has been previously used to study lifted flames stabilized by auto ignition [15]. Multidimensional RANS-CMC has been able to reproduce lifted flames successfully [6, 7]. LES allows for modelling of unsteady effects and can better represent the large scale turbulent mixing occurring at the flame base. The test case investigated here is a turbulent lifted flame based on the experiments by Leung and Wierzba [5] which consists of a jet of pure methane issuing into an air co-flow. Different simulations are performed with different co-flow conditions. During each simulation the jet velocity is progressively increased until; blow-out occurs, mimicking the experimental settings. The stability region analysed corresponds to the low co-flow velocities Uco < 0.03 Ujet. The paper is structured as follows: First the numerical method is briefly outlined, followed by a description of the experimental and numerical set-up and then a discussion of the results organized around stabilization mechanisms through analysis of transport budgets.
Numerical modelling A spatial density weighted filter is applied to the Navier-Stokes equations giving the conservation equations in standard notation: Continuity ~ ∂ρ ∂ρ u j + =0 ∂t ∂x j
(1)
Momentum ∗ ~~ ~ ∂ρ u~i ∂ρ ui u j ∂P ∂τ ij ∂τ ij − =− + + ∂xi ∂x j ∂x j ∂x j ∂t
(2)
Mixture fraction ~ ~~ ∂ρ ξ ∂ρ u jξ ∂ + = ∂t ∂x j ∂x j
with Cχ=2. The unconditionally filtered reactive species are obtained by integration over mixture fraction space 1 ~ ~ ~ "2 (10) Yk = ∫ Qk P ξ , ξ sgs ;η dη ,
~ ⎡ ∂ξ ⎤ ⎢ ρ (D + Dsgs ) ⎥ ∂x j ⎥⎦ ⎢⎣ (3) The sub-grid stress tensor, τ*, is determined using the standard Smagorinksy type closures, with a sub-grid viscosity proportional to the filtered strain rate, viz: ~ ν sgs = (CΔ )2 S , (4) where the constant is taken C=0.09 and Δ is the filter width. The scalar sub-grid transport is modelled similarly using the gradient approach introducing a subgrid diffusivity Dsgs ,which is obtained from νsgs using a constant turbulent Schmidt number of 0.4 [11].
(
where the sub-grid PDF is reconstructed with the calculated mean from Eq. (3) and the sub-grid variance (8) assuming a β-PDF distribution in η space. The conservation equations (2) and (3) are solved using a finite volume code in a staggered storage arrangement. For the momentum equations, an energy conserving discretization scheme is used and all other spatial derivative are approximated by second order central differences. The scalar equations (3) and (5) are solving using a second order TVD scheme with a Van Leer limiter to avoid over and undershoot of scalars.
The LES-CMC equations are written as [10] χ~ ∂ 2Qk ~ ∂Qk ~ + vη ⋅ ∇Qk = η + Wη + eY + eD , (5) ∂t 2 ∂η 2 where Qk represents the conditionally filtered mass fraction of the k-species [12]. The variable η represent the phase space of the mixture fraction, and as a subscript indicates a conditionally filtered variable. The terms in the left hand side of Eq. (5) correspond to the evolution-advection of the conditional moments. On the right hand side, the moments change due to diffusion in mixture fraction space, due to the conditional scalar dissipation χ, and chemical reaction term W, which is closed using first order closure approximations [12]. The last two terms represent conditional transport due to velocity-scalar fluctuations and molecular diffusion, which is modelled using a gradient approach eY = −
(
)
Experimental and Numerical Set-up The test case under consideration is based on the burner by Leung and Wierzba [5]. A confined methane jet flame is initiated by a spark with a fixed co-flow velocity and fuel discharge velocity. Several fuel dilutions were tested, however in the present study only pure methane is considered. The jet diameter, d0, is 2 mm, with an initial bulk jet velocity of 25 m/s. The Reynolds number is 6120 based on jet characteristics. Uncertainties in the flow measurements were within 5%. Several air co-flow streams were tested in the experiments although in the present work only the linear region is observed with co-flow velocities, Uco, ranging from 0.01 to 0.1 m/s. In the linear region (or Region I after [5]), the attached flame is lifted and blow-out limits depend linearly on the co-flow velocity. The linear dependency is preserved independently of the nozzle diameter.
~ D sgs ∇ ⋅ ρu"Y " η P (η ) ~ ≈ ∇ ⋅ D sgsη∇Q k + ~ η ∇Q k ⋅ ∇ ρP (η ) ~ ρ P (η ) ρP (η )
(
)
0
)
(6) Here, P(η) is the instantaneous sub-grid mixture fraction probability density function (PDF). The last term in Eq. (6) is traditionally neglected as PDF gradients are expected to be smaller than conditional moment gradients [6]. A similar approach is used for the closure of the diffusive term. e D ≈ ∇ ⋅ (Dη∇Q k ) . To obtain a
The calculations were performed on a cylindrical grid with 128 × 80 × 48 cells in axial, radial and azimuthal direction respectively. The computational domain is 100 × 20 d0 in axial and radial direction respectively. Although the actual experimental test section was wider (~30 d0) wall effects are neglected in the present study and the resolved domain is assumed to be wide enough to avoid interference of the boundary condition with the region of interest. The mixture fraction space is discretized with 100 bins, refined around stochiometric mixture fraction (ξst~ 0.054). Two CMC meshes were considered, one with 32 × 20 cells in axial and radial direction and other with 64 CMC cells in axial direction only. No angular dependence of the conditional moments is considered. Previous investigations [7, 13] highlight the importance of radial moments to accurately capture the lift-off height. The ratio ΔCMC/ΔLES is 4 and 2 for the two meshes respectively. The conditional moments are initialized in mixture fraction space with a flamelet solution obtained with a low strain rate. The inflow conditions Q(y=0, r, t; η) are taken as a flamelet solution in a jet diameter region around the nozzle rim and pure mixing everywhere else.
complete closure of Eq. (5), models are required for the conditional scalar dissipation, diffusion and velocity. Local homogeneity of the conditional moments is assumed within one CMC cell. The conditional filtered values are then obtained by employing the assumption that the conditionally filtered values can be approximate by the conditional average of the unconditional dissipation over one CMC cell, viz. (7) χ~η ≈ χ~ η . The sub-grid scalar variance is modelled assuming local equilibrium ~ ~2 (8) ξ sgs2 = Cξ Δ2 ∇ξ Where following [14], Cξ=0.1. The sub-grid scalar dissipation is then assumed proportional to (8) D ~2 (9) χ sgs = C χ sgs ξ sgs Δ2
2
In Figure 3, a detailed view of the base of the lifted flame is shown. The results are qualitatively similar to experimental images obtained at the flame base by PLIF (see reviews of Lawn [1]). The coarse CMC mesh produces an unrealistically flat flame base. It can be argued that CMC does not allow the existence of triple flames at the base and therefore prevents an accurate description of this region. However, as seen in Figure 4, the results show a characteristic lifted flame, with the diffusion region convoluted by large scale vortices. The reaction and diffusion regions are limited by the isocontours of axial velocity 3 and 13 SL, which was the most probable velocity found experimentally at the flame base by Muñiz and Mungal [4]. The flame base is located where the iso-lines are separated by approximately 1 LES cell.
The solution evolves until it reaches at stationary state, hereafter t0. Then, the mass flow rate in progressively increased until flame blow-out is observed. The jet velocity is either increased in steps of 3 m/s (and then constant for 4 ms) or constantly accelerated (see Figure 1). Between 30 and 40 ms were simulated after the stationary state is perturbed which corresponds approximately to 5 flow-through times at the nominal discharge velocity. A constant time step of 2 × 10-6 s is employed throughout the calculations. The CFL at the maximum fuel discharge velocity (80 m/s), is 0.7 while it is 0.25 at the nominal initial bulk velocity. In the present simulation the flame is considered extinguished when less than 5 % LES points have a temperature greater than 1500 K. The lift-off height was taken as the minimum axial distance with OH mass fraction greater than 0.001. Turbulent inflow conditions are generated by super-imposing angular modes to the mean flow corresponding with 1% turbulence intensity. The velocity profiles obtained are correlated in circumferential direction and time [10] and are strong enough to promote the growing of large scale KelvinHelmholtz instabilities. The chemistry mechanism employed is a reduced mechanism obtained from GRI3.0 with 15 steps and 19 species [14] including H2, H, O2, OH, H2O, HO2, H2O2, CH3, CH4, CO, CO2, CH2O, C2H2, C2H4, C2H6, NH3, NO, HCN and N2. The simulations were performed on an Intel Xeon Cluster using 32 nodes. A whole simulation from t0 takes roughly one week.
Figure 2: Evolution of the lift-off height for different co-flows velocities. The x-axis shows time after t0.
Figure 1: Boundary condition for the bulk-velocity versus time after t0. All the different set-up cases are shown.
Figure 3: Detail of instantaneous filtered temperature contours at the flame base. Two iso-contours of velocity are super-imposed corresponding to 3SL and13SL.
Results and Discussion Figure 2 shows the evolution of the lift-off height for three different co-flows using 1D-CMC. The low coflow velocity case has a larger lift-off height at the same discharge velocity, which is consistent with experiments. The 0.01 and 0.1 cases have the same behaviour up to approximately 5 ms after t0 where they bifurcate.
When 2D CMC is used, the flame quickly stabilizes at 12 jet diameters and no blow off was observed for Uco=0.1 m/s at the jet velocities considered in the experiments (
