Flow and added small-scale topologies in a turbulent premixed flame L. Cifuentes*, A. Kempf* and C. Dopazo** luis.cifuentes@uni-due.de *University of Duisburg-Essen, Chair of Fluid Dynamics, Duisburg - Germany **University of Zaragoza, Fluid Mechanics Area, Zaragoza - Spain Abstract The relative motion of two points on adjacent iso-surfaces is examined in terms of local flow topologies of a turbulent premixed flame. The three invariants of the gradients of the flow velocity and of the added normal displacement speed vector are calculated. Joint probability density functions of second and third invariants show comparable magnitudes for the flow and added small-scale topologies. Flow and added enstrophies are obtained to examine the behavior of focal smallscale topologies across the iso-scalar surfaces. Introduction Flame topologies are of a fundamental interest in premixed turbulent combustion [1-3]. A number of previous studies have used Direct Numerical Simulation (DNS) data of turbulent premixed flames [4-5] to analyze statistical behaviors of the local flow topologies based on the invariants of the velocity-gradient tensor A ij [6]. However, the motion of a premixed flame can be significantly influenced by molecular diffusion and reaction rates, which create small-scale topologies and affect the geometries of the iso-scalar surfaces. The present study analyzes the flow topologies in the neighborhood of the enstrophy interface and detects nodal and focal topologies due to the flow and to the displacement speed of the iso-scalar surfaces. This study aims at exploring the general classification of small-scale flame structures based on the invariants of A ij and the added velocity-gradient tensor A a ij. The physical insights obtained from this study will help to understand how molecular diffusion and chemical reaction rates might substantially contribute to the relative motion of two points on adjacent iso-surfaces. Enstrophies ascribed to the flow and added vorticity components are also examined and yield further insight of focal small-scale topologies within a turbulent flame. Mathematical formalism In premixed flames, the total velocity v i c of a point x at time t on an iso-surface c(x, t) = Γ is expressed as [7]: v i c (x, t) = v i (x, t) + S d (x, t)n i (x, t) (1)
In Eq. (1), v i is the fluid velocity, S d is the normal propagating speed relative to the fluid, and n i = ( c/ x i )/ c is the unit vector normal to the iso-scalar surface. The time rate of change of an infinitesimal non-material vector, connecting point x on c(x, t) = Γ and point x+r on c(x, t) = Γ + ΔΓ, is: dr i = [v dt i c (x + r, t) v c i (x, t)] = r v i j = A c x ij r j (2) j In Eq. (2), the term v c c i / x j = A ij is the ij component of the total velocitygradient tensor, which can be decomposed as: c A c ij = v i c = v i + (S dn i ) a = A x j x j x ij + A ij j (3) The term A ij = v i / x j is the flow velocity-gradient tensor, and A a ij = (S d n i )/ x j is the added velocity-gradient tensor due to the spatial variations of the displacement velocity vector S d n i. Consequently, Eq. (2) can be recast as: d r i dt = S ij r j + 1 ε n 2 ijk ω j r k + S i d r x j + 1 (n S d j 2 i + S d n x j x j ) r j + 1 ε i 2 ijkω a j r k (4) In Eq. (4), the term S ij is the flow strain rate tensor, ε ijk is the Levi-Civita alternating tensor, ω i = ε ijk ( v k / x j ) is the flow vorticity vector, n i x j is the curvature tensor and ω a i = ε ijk ( S d x j )n k is the added vorticity vector [7]. The first term on the right side of Eq. (4) includes linear dilatation and angular deformation rates due to the flow, whereas the second one represents solid body rotation of r. The third term contains the stretch rate of an area element on the isosurface, the fourth one accounts for rates of separation or approach of two adjacent iso-surfaces, and the fifth term is the solid body rotation with ω a i. The total enstrophy corresponding to the total vorticity vector ω c i = ω i + ω a i is: E c = 1 ω 2 i c ω c i = 1 ω 2 iω i + 1 ω 2 i a ω a i + ω i ω a i = E + E a a + ω i ω i (5) The total enstrophy aggregates the flow and added enstrophies plus the scalar product of flow and added vorticity vectors. Perry and Chong [6] characterized the small-scale flow motions of point x + r about point r in terms of the three invariants (P, Q, R) of A ij. Nodal and focal topologies of a turbulent flow emerge depending on the dominance of strain or rotation. If r is not a material vector, the previous methodology can be extended by further examining the invariants (P a, Q a, R a ) of A a ij. The contributions of A ij and
A a ij are additive (A c ij = A ij + A a ij ). Hence, apart from the small-scale flow topologies, point x + r will undergo an additional motion with respect to point r due to the gradient of the normal displacement speed (S d n i )/ x j. A general analysis can be performed examining the invariants (P c Q c, R c ) of A c ij, and those of A ij and A a ij. Direct numerical simulation The three-dimensional DNS analyzed was performed by Proch with the in-house code PsiPhi [8-9]. The Schmidt number is 0.7 and the Lewis number is unity for all species. The chemistry is tabulated with the Premixed Flamelet Generated Manifolds (PFGM) approach. Viscosity, thermal conductivity, and mass diffusion coefficients are given functions of the temperature. The geometry is described by an immersed boundary technique. The DNS computes the Cambridge stratified burner [9], which consists of a central bluff-body surrounded by two co-annular premixed methane-air streams at ambient conditions and a co-flow of air (see Fig. 1a). The computational domain consisted of 1120 1200 1200 equidistant cells and a grid of 1.6 billion cells with a resolution of 100μm. Numerical values of the variables and dimensionless parameters for this simulation are presented in Table 1. Table 1. DNS dimensional variables and dimensionless parameters. Variables and dimensionless parameters Inner stream Outer stream Integral length scale, l (mm) 0.5 0.5 Velocity fluctuations rms, u (m/s) 0.9 1.8 Kolmogorov length micro-scale, η (μm) 40.9 24.3 Non-stretched laminar flame speed, S L (m/s) 0.212 0.212 Laminar flame thickness, δ L (mm) 0.565 0.565 Turbulent Reynolds number, Re 28.1 56.3 Damköhler number, Da 0.2 0.1 Schmidt number, Sc 0.7 0.7 Lewis number, Le 1.0 1.0 Results Figure 1 shows the interaction between the instantaneous enstrophy field, the enstrophy interfaces and the iso-lines c = 0.1 and c = 0.9 for the premixed turbulent annular jet flame. The normalized progress variable based on temperature c = (T T min )/(T max T min ) is used to characterize the scalar interface of interest. The progress variable is zero in the fresh reactants and unity in the fully burnt products. Enstrophy structures are less convoluted near the nozzle and they curve and radially grow between the inner and outer interfaces downstream, as a result of engulfment and entrainment. This widening of the vortical region could be explained by the behavior of the large flow structures, which grow downstream and engulf large fluid packages from the irrotational region into the rotational zone.
Figure 1. Instantaneous enstrophy field, progress variable iso-lines c = 0.1 and c = 0.9, and iso-lines defining the enstrophy inner E/E max = 7.0 10 5 and outer E/E max = 1.0 10 7 interfaces. Figure 2. JPDFs of the invariants: Q and R of A ij ; Q a and R a of A a ij ; and Q c and R c of A c ij. Figure 2 shows the joint probability density functions (JPDFs) of the second and third invariants of the velocity-gradient tensor A ij, of the added velocity-gradient
tensor A a ij and of the total velocity-gradient tensor A C ij for the layers at the inner and outer enstrophy interfaces. The statistics has been performed at different axial locations above the burner, namely, Zone 1 (0-20mm), Zone 2 (20-45mm) and Zone 3 (45-70mm), as illustrated in Fig. 1. Data were analyzed for ten cross-stream planes x z in the burner midsection, considering grid points in a region ±5 mm in the radial direction from the enstrophy inner and outer interfaces. It is interesting to note in Fig. 2 that the universal teardrop shape typical of constant-density turbulent flows disappears at both interfaces. In the case of the inner enstrophy interface, the statistical distributions are displaced towards the left side, which is consistent with positive local dilatation rate close to the flame region. Figure 3. Mean values of the flow E and added E a enstrophies, scalar product ω i ω i a and total enstrophy E c across the progress variable c. Figure 3 depicts the average of the flow E and added E a enstrophies, scalar product ω i ω i a and the total enstrophy E c across the progress variable c. For c > 0.80, namely, in the burning and hot product regions, the added enstrophy grows significantly, whereas the flow enstrophy remains approximately constant and becomes much smaller as the reaction progress variable increases. This behavior implies that flow focal topologies govern in the fresh gases and tend to disappear, in favor of added focal topologies, towards the burning region and hot products. It is worth stressing that the magnitudes of flow and added enstrophies are comparable in the present case, which confirms that the rotation of r, as a solid body with an angular velocity ω i a, creates enstrophy by flame. Conclusions Contributions to the relative motion of two points on adjacent iso-surfaces can be examined in terms of small-scale flow and added topologies. The later are
classified using the gradient of the displacement speed normal vector, related to molecular diffusion and reaction rates. The magnitudes of the added invariants, Q a and R a, are comparable to those of the turbulent flow, Q and R. Alternative decomposition of A a ij can explicitly include area stretch rates and contributions to flame thickening or thinning. The added enstrophy is as important as the flow enstrophy, which indicates that a share of the total enstrophy is generated by flame. Acknowledgments The authors are grateful to Jülich Supercomputing Centre (JSC) for computational support. This project has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement No 706672 - ITPF. References [1] Günther, R., Lenze, B., Exchange coefficients and mathematical modelsof jetdiffusion flames, In Symposium (International) on Combustion. Vol. 14, No. 1, 1973, pp. 675-687. [2] Driscoll, J.F., Gulati, A., Measurement of various terms in the turbulent kinetic energy balance within a flame and comparison with theory, Combust. Flame, 72(2): 131-152 (1988). [3] Steinberg, A.M., Driscoll, J.F., Straining and wrinkling processes during turbulence-premixed flame interaction measured using temporally-resolved diagnostics, Combust. Flame, 156(12): 2285-2306 (2009). [4] Cifuentes, L., Dopazo, C., Martin, J., Jimenez, C., Local flow topologies and scalar structures in a turbulent premixed flame, Phys. Fluids, 26: 065108, (2014). [5] Wacks, D.H., Chakraborty, N., Klein, M., Arias, P. G., Im, H.G., Flow topologies in different regimes of premixed turbulent combustion: A direct numerical simulation analysis, Physical Review Fluids, 1(8): 083401, (2016). [6] Chong, M.S., Perry, A.E., Cantwell, B.J., A general classification of threedimensional flow fields, Physics of Fluids A: Fluid Dynamics, 2(5): 765-777 (1990). [7] Dopazo, C., Cifuentes, L., Martin, J., Jimenez, C., Strain rates normal to approaching iso-scalar surfaces in a turbulent premixed flame, Combust. Flame, 162(5): 1729-1736 (2015). [8] Kempf, A.M., Geurts, B.J., Oefelein, J.C., Error analysis of large-eddy simulation of the turbulent non-premixed sydney bluff-body flame, Combust. Flame, 158(12): 2408-2419 (2011). [9] Proch, F., Domingo, P., Vervisch, L., Kempf, A.M., Flame resolved simulation of a turbulent premixed bluff-body burner experiment. Part I: Analysis of the reaction zone dynamics with tabulated chemistry, Combust. Flame, 180: 321-339 (2017).