Large Eddy Simulation of a turbulent premixed flame stabilized by a backward facing step.

Transcription

1 Large Eddy Simulation of a turbulent premixed flame stabilized by a backward facing step. Angelo Murrone and Dominique Scherrer ONERA, 29 Avenue de la Division Leclerc, BP. 72, Châtillon, France Large-Eddy-Simulation (LES) is becoming a standard tool for computational fluid dynamics. Here we are interested in the LES of a premixed air-methane flame stabilized by a backward-facing step. The simulation is realized with the compressible CEDRE solver using generalized unstructured meshes and developed at the ONERA. Although less numerous than those using structured grids, the example of LES on unstructured grids reported have largely increased in the last decade. 1 5 But for reactive flows, up to now very few studies on unstructured meshes have been published. 2, 6, 7 The objective of the preliminary computations presented in this paper is to show the capacity of the numerical strategy to deal with unsteady turbulent combustion. The solver uses essentially second order accuracy upwind schemes for the spatial discretization while the implicit time integration is based on a Runge-Kutta type algorithm. Then the turbulent combustion modeling is based on the coupling between an hybrid combustion model Arrhenius/EBU and the standard Smagorinski model for the subgrid closure. In the first part of the study, we show that a minimun mesh size is necessary to ensure the computation. We also determinate an optimal maximum time step for the implicit method. This last one allows to reduce the computational cost which remains a crucial point of LES. In the second part, we present a fully three-dimensional reactive computation of a turbulent premixed air-methane flame stabilized by a backward-facing step. Introduction Large-Eddy-Simulation (LES) is becoming a standard tool to study the dynamics of turbulent flames. Here we are interested in computing the LES of a turbulent flame in an experimental combustion chamber. Up to now, this type of computations have been made with the Reynolds-Avareged-Navier-Stokes (RANS) method which does not allow to clearly compute combustion instabilities or interaction between the turbulence and the flame. The simulations have been performed with the code CEDRE which solves the Compressible Multicomponent Navier-Stokes unsteady equations on generalized unstructured meshes. The numerical method uses Monotone-Upwind-Schemesfor-Conservations-Laws (MUSCL) 8 of second order accuracy. The time integration is based on a Runge-Kutta type algorithm and the linear system is solved by the GMRES strategy. Note that the combustion selected is subsonic (M 0.1) and that the implicit method has been a real advantage to decrease the computational cost of the LES. For the inert case, the Monotically-Integrated-LES (MILES) approach 9 11 has been used because the upwind schemes have intrinsic subgrid turbulence models coupled naturally to the resolved scales. Moreover the influence of a subgrid modeling would not appear if one uses second order accuracy upwind schemes. 12 On the other hand for the reactive case, the Smagorinsky model 13 has been selected because the subgrid viscosity µ t is used to compute a turbulent diffusivity λ t = (µ t C p )/P r t which is not negligible in the energy equation and particularly for reactive cases. For the combustion model, two methods exists for premixed flame in LES computation ; the first one is the G-equation method 14 which consists to put the flame thickness to zero and to propagate the front as a thin interface and the second one is the Thickened-Flame (TF) model 2, 15 which consists to thicken the flame at the LES level grid size while keeping the same speed propagating of the unthickened flame. Here we choose a more pragmatic approach coupling the hybrid combustion model Arrhenius/EBU and the standard Smagorinski model. The present strategy only needs to classical models and there is no thickness of the flame. An important point of the strategy is that computing a front flame thickness δl 0 of the order of 0.1 mm on a LES mesh size of the order of 1 mm with upwind schemes does not induce major numerical difficulties. The outline of the present paper is as follows. In the first section, we rewrite the equations of the Large-Eddy- Simulation for compressible reactive flows. Then, we also describe the modeling of the terms producing turbulent combustion. In particular, we recall the Smagorinski model and we show how to combine it with the hybrid combustion model Arrhenius/EBU. In the second section, we study the influence of the time step t and the mesh size on the accuracy of the results. The simulations have been performed in the two-dimensional inert context because of LES cost computations. Nethertheless, we assume that the trends are the same in three dimensions and for the reactive case. angelo.murrone@onera.fr domnique.scherrer@onera.fr 1 OF 10

2 Finally, we present a fully three-dimensional computation of the turbulent premixed flame stabilized by a backward facing step. Governing equations Filtered equations and large eddy simulation The Reynolds Avareged Navier-Stokes (RANS) equations, closed with appropriate models, allow only for the determination of mean quantities, that may differ from instantaneous one. Strong unsteady mixing effects, resulting from the rolling up of shear layers, are observed in turbulent flames, and the knowledge of steady statistical means is indeed not always sufficient to describe turbulent combustion. An alternative is to use the Large-Eddy-Simulation (LES). Its objective is to explicitely compute the largest structures of the flow (typically the structures larger than the mesh size) while the effects of the smaller one are modeled. In LES, the revelant quantities Q are filtered in the spectral space (components greater than a given cut off frequency are supressed) or in the physical space (weighted averaging in a given volume). The filtered operation is defined by : ρ Q(x) = ρq(x )F (x (x )dx (1) where F is the LES filter. So filtering the instantaneous balance equations leads to : ρ t + ρũ j = 0 (2.1) ρũ i t + ρũ jũ i = [ρ(ũ i u j ũ i ũ j )] p x i + τ ij + F i (2.2) ρỹk t ρ H t + ρũ jỹk + ρũ j H = [ ρ( x ] u j Y k ũ j Ỹ k ) F j k + ω k (2.3) j = [ ] ρ(ũjh ũ j H) p t + ( ) Fj h x + u iτ ij + u j F j (2.4) j Modeling of the turbulent combustion The unknown quantities are the unresolved Reynolds stresses (ũ i u j ũ i ũ j ), species fluxes ( u j Y k ũ j Ỹ k ), enthalpy fluxes (ũjh ũ j H) and also the chemical reaction rate ωk. The closure of all these SGS terms is described below. The Smagorinski model In the momentum equation (2.2), we have to close the non linear term ρ(ũ i u j ũ i ũ j ) which is the SGS stress tensor. Similar expressions appear in equations (2.3)-(2.4) and stand for the SGS species and heat flux tensors. These terms have to be closed and the most common and simplest explicit SGS model is that of Smagorinski (1963). The isotropic part of ρ(ũ i u j ũ i ũ j ) can be neglected under the assumption of low compressibility effects in the SGS fluctuations. 13 The deviatoric part T ij is expressed by an eddy viscosity hypothesis : T ij = 2µ t ( S ij 1 3 S kk ) (3) where S ij is the resolved strain rate tensor of the filtered velocity : S ij = 1 2 ( ũi + ũ j x i ) (4) An algebraic model for the eddy viscosity µ t can be derived from adimensional arguments to be : µ t = ρ(c s ) 2 S (5) where is the filter width, C s a constant and S = 2 S ij Sij. To complete the definition of the SGS viscosity, we have to specify the grid filter. For unstructured grids, there is no reliable criteria to define the width of the filter and the mesh size has been selected. 2 OF 10

3 Then in the total energy equation, the effect of the subgrid fluctuations (ũjh ũ j H) are given by a Fourier law (ψ t λ t T ). The SGS conductivity coefficient λ t (added to the laminar one) is evaluated as : λ t = C pµ t P r t (6) where P r t is a SGS Prandtl number. A similar treatment is realized for the terms ( u j Y k ũ j Ỹ k ) in the species equations using a Fick law (ψ t Dt k Y k ) where the molecular diffusivity is given by Dt k = µt Sc t and Sc t is the Schmidt number. The hybrid combustion model Arrhenius/EBU Now, the only term to be closed is the filtered rate production ω k due to the combustion. So here, we present the closure of this term by the hybrid combustion model Arrhenius/EBU. We consider a premixed methane-air combustor at lean regime with φ = 0.8. φch 4 + 2O N 2 φco 2 + 2φH 2 O + 2(1 φ)o N 2 (7) To reduce the computational cost of the LES, we define the specy F = φch 4 + 2O N 2 for the fresh gases while B = φco 2 + 2φH 2 O + 2(1 φ)o N 2 designate the burnt gases, and the rewrite the reaction under the form : F B (8) The starting point of the model is an Arrhenius law for air-methane combustion but the idea consists to consider that this cinetic is not applied to the fresh gases F but to an additional specy F referred as the mixed fresh gases in the sense of fresh gases mixed with burnt gases B. So, the modeling consists to split the reaction into two steps (Arrhenius/EBU) as : F + B F + B EBU (9.1) F B Arrhenius (9.2) The mixing is due to the turbulence and the production rate of step (9.1) is based on the classical Eddy-Break-Up modeling for premixed flame : ω F ρ 1 τ t Ỹ F (1 ỸF ) (10) To evaluate the frequency 1/τ t, we do not dispose of the terms k and ε used in RANS modeling and we propose the following expression 1/τ t = C s Tij 2 with T ij = S ij 1 S 3 kk and S ij the resolved strain rate tensor. Then the second step (9.2) is an Arrhenius cinetic based on a single step reaction proposed by Westbrook and Dryer 16 and applied to the mixed fresh gases F : ω F A[Y F ] a exp( E a RT ) (11) The EBU model introduces a characteristic time τ t drived by the turbulence intensity mixing between fresh and burnt gases. This time τ t is larger than the time characteristic τ c 1/A of the Arrhenius law. This first step acts as a limitation of the combustion and traduces the fact that the mixture at the subgrid scale level is not perfect. Finally the laminar diffusivity λ is increased in a natural way by adding a turbulent diffusivity λ t to get a turbulent flame. This last quantity λ t is computed thanks to the eddy viscosity µ t given by the Smagorinski model 13 presented before. So, in the LES Thickened-Flame (TF) model for premixed flame, 2 the diffusivity λ is multiplied by a factor F while the pre-exponential factor of the chemical Arrhenius law is decreased by the same previous factor. As a consequence, the laminar flame speed remains the same (s 0 l λa) while the thickness (δl 0 λ/a) becomes F δl 0. This method allows to have a flame thickeness of the same order of the mesh size LES Filter. Here we propose to keep the original thickness of the flame because the upwind schemes do not lead to major numerical problems. Netherheless, the front flame thickness is much δ 0 smaller than the LES mesh size. So, we propose to examine the behaviour of the numerical method when computing the front flame. This is the purpose of the next section. 3 OF 10

4 Influence of the mesh on the one-dimensional laminar flame speed So here we propose to study the numerical discretization of the one-dimensional laminar flame given by the Arrhenius law. Figure 1 displays the laminar flame velocity s 0 l with respect to the mesh size. As the mesh size tends to zero, the velocity converges to a value given by the Arrhenius law the order of 45 cm.s 1. On the other hand, as the mesh size increases, the results become worse and the flame speed tends to an asymptotic value the order of 15 cm.s 1. This last value, much smaller than the real one, is due to an important error of the numerical scheme. In Figure 2, we show the structure of the front flame for different computation mesh sizes. The profils of mass fractions, temperature, rate production and velocity are displayed. We now try to understand the behaviour described before. When is increased, there are less and less points of discretization in the front flame. As a consequence, the front cannot be stabilized on the mesh and the maximum of the exponential term source is not constant. This produces undesirable fluctuations of pressure and velocity which make the laminar flame velocity decrease. Now we try to evauate the influence of this underesolved front flame on the LES of the turbulent flame. For the three dimensional mesh size = 1 mm taken in our simulations, the error on the laminar velocity is the order of 30%. Moreover, in the LES computation, the flame is turbulent (λ t >> λ) and as a consequence the thickness is superior to the laminar one (s 0 t λ t /A >> s 0 l λ/a). So in conclusion the error on the flame velocity is much minus than 30% and could be compensate by increasing C s the constant of the Smagorinski model. Fig. 1 Influence of the mesh size on the one-dimensional laminar flame velocity s 0 l. Fig. 2 Flame structures on different mesh size = 10, 50, 100, 150 µm. 4 OF 10

5 Numerical Results Description of the geometry, meshes, and boundary conditions Geometry and meshes The computational domain consists of a streamwise length L x = 1.4 m including an inlet section L i = 0.1 m prior to the sudden expansion, vertical height L z = 0.1 m and spanwise width L y = 0.1m. The high step is equal to h = m conducing to an expansion ratio ER = L z /(L z h) equal to The Reynolds number Re h = U 0 h/ν based on the step height h and the inlet velocity U 0 = 50 m.s 1 is the order of The Mach number M = U 0 /a at the entry is the order 0.1 and decreases after the step because of the diminution of magnitude velocities in the recirculation and elevation of temperature due to the combustion. The three dimensional unstructured mesh represented in Figure 4 is generated with PROAM. The two dimensional mesh represented in Figure 3 is a fully structured mesh built with the ICEMCFD mesher. For the 3D mesh, we have = 1mm in the refined zone near the step and = 2mm overall. A local raffinement has been possible thanks to the formulation of the schemes for general unstructured mesh. For the 2D structured mesh, is constant in the streamwise direction the mesh except in the neighborhood of step nose where = 0.25 mm. Boundary conditions There are four types of boundary conditions in the computation. The first one is the subsonic entry where we impose the temperature T = 520 K and the following profil velocity for the shear layer : ( ) 6 z L z+h 2 u(z) = 1.168U 0 1 L z h (12) 2 The second one is the outflow P = Pa. These two boundary conditions are imposed with a relaxation time τ R in order to limit acoustic waves reflexions like as in the NSCBC method. 17 Then the upper and lower boundary conditions are wall and the lateral sides are slipping in the three dimensional case. Fig. 3 ICEMCFD structured mesh of the chamber for the inert two-dimensional case. Fig. 4 PROAM general unstructured mesh of the chamber for the reactive three-dimensional case. 5 OF 10

6 Influence of the time step and the mesh size on the two-dimensional non reactive flow results Mean flow variables In this section we propose to study the influence of the mesh size and the time step t on the accuracy of the results. The experiments are made for the two dimensional backward facing step inert case. So let us start with the influence of the time step t. In particular, we choose to use an implicit time integration and we want to determinate the maximum time step which can be used without loss of accuracy. So we have made several runs at CF L = 2,..., 25. The first point investigated is the influence of the time step on the mean reattachment location X r (see Figure 5). To determine the mean reattachment location X r, several methods are possible and here X r is defined by the location at which the mean velocity u = 0 at the first grid point away from the wall. Let us remark that the reattachment length X r is not constant in time. Here, we present the mean values of X r obtained with the different computations respectively. As the time step decreases, X r becomes independant of t and tends to an asymptotic value X r = 3.9 h. This last value is obtained for the two computations at CF L = 2 and CF L = 5. The experimental value obtained is X r = 4.3 h and the numerical converged value X r = 3.9 h leads to less than 9% of error. On the other hand, as the time step increases, the results become worse and do not converge to a physical value. In conclusion, the results do not suffer from a lack of accuracy until CF L = 8 which corresponds to a time step of the order of the convective time characteristic. So the optimal maximum time step seems to be obtained for CF L = 5. To end this paragraph, we propose to show the influence of the mesh size on the mean flows results. The conclusions of this study is that for this case the mesh size in the refined zone near the step, the mesh size does not have to be superior to = 0.5 mm. If the mesh is larger, the vortex shedding phenomena is not identify by the computation and one reason could be that the mesh is too large to damp the contrarotation vortex. Note that this minimun size is larger in three dimensions and that the computation with = 1 mm has been possible. Figure 6 compares the mean pressure and horizontal velocity fields obtained with the meshes (h = 0.25 mm, h = 0.5 mm). It is clear that the results obtained with the mesh (h = 0.5 mm) are not available and conduce to a large surestimation of the mean reattachment lenght due to the fact that the vortex shedding is not correctly damped. This mechanism is described in the following section where we study the instantaneous flow. 6 OF 10

7 Fig. 5 Influence of the computational time step t on the mean reattachment Length Xr. Fig. 6 Mean Pressure (left) and X-Velocity (Right) field for different mesh sizes = 0.25 mm, 0.5 mm. Instantaneous flow The principal phenomen observed in this type of flow separation and reattachment is the vortex shedding from the reattachment region. The vortical structures from the separated shear layer tend to contact the surface approximatively at half the time mean reattachment length, where they grow and coalesce to be shedded periodically from the reattachment zone. Here a spectral analysis of the vertical velocity signal exhibit a frequency of the order of f 120Hz. This value compare well with the experimental results and also with the theoretical values for two dimensional study of this type of flow. In effect in, 18 it is mentioned that the averaged frequency f of vortex shedding was measured to be 0.6 to 0.7U 0 /X r. Here we found f 0.4X r = 120Hz for CF L < 8. (see Figure 7). Finally we present the instantaneous modulus of vorticity (see Figure 8) for the computation at CF L = 2 and during a cycle of the global phenomena. The sequence of events are clearly identified : vortex formation, merging and shedding. 7 OF 10

8 Fig. 7 Influence of the time step t on the vortex shedding frequency f identification. Left-top (CF L = 2), righttop(cf L = 5), left-bottom (CF L = 8) and right-bottom (CF L = 12). Fig. 8 Instantaneous Modulus of the vorticity at different times T = 0 ms, 2 ms,..., 6 ms for the inert 2D computation at t = 10 6 i.e CF L = 2. 8 OF 10

9 Three dimensional preliminary reactive flow results Here we present a fully three dimensional computation of the reactive flow. Figure 9 displays the 1500 K-iso-surface of temperature which gives an approximation of the position of the front flame. It is clearly observed that the flame is turbulent and that the vortices produce important three dimensional effects on the structure of flame. Netherless, the angle of the flame seems to be inferior of the experimental value and it is not surprising because of the approximations of the model which need to be improved. This computation is a preliminar one and could be the starting of a more detailed study about reactive Large-Eddy-Simulation using upwind schemes. Fig. 9 Instantaneous 1500 K isosurface for the 3D reactive computation at t = i.e CF L = 5. Conclusion In conclusion, we have presented in this paper model for computing unsteady turbulent combustion adapted to upwind schemes. It is based on the coupling between the hybrid combustion model Arrhenius/EBU and the standard Smagorinski model for the subgrid closure. Even if the first results seems to be available, a lot of works remain to be done to improve the method. References 1 F. Chalot, B. Marquez, M. Ravachol, F. Ducros, F. Nicoud, and T. Poinsot. A consistent finite element approach to large eddy simulation. AIAA, :, O. Colin, F. Ducros, D. Veynante, and T. Poinsot. A thickened flame model for large eddy simulations of turbulent premixed combustion. Phys Fluids, 12(7): , D.C. Haworth and K.E. Jansen. Large eddy simulation on unstructured deforming meshes : Towards reciprocating ic engines. Computers and fluids, 29(5): , N. Okong o, D. Knight, and G. Zhou. Large eddy simulations using an unstructured grid compressible navier stokes algorithm. International Journal of Computational fluid Dynamics, 13:303 26, S. Camarri, M.V. Salvetti, B. Koobus, and A. Dervieux. A low-diffusion muscl scheme for les on unstructured grids. Computers and fluids, 33: , D. Caraeni, C. Bergstrom, and L. Fuchs. Modeling of liquid fuel injection, evaporation and mixing in a gas turbine burner using large eddy simulations. Flow, Turbulence and Combustion, 65: , R. Courtois. Simulation aux grandes échelles des écoulements réactifs dans la chambre A3C. PhD thesis, Ecole Centrale of Paris, B. van Leer. Towards the Ultimate Conservative Difference Scheme v. A Second-Order Sequel to Godunov s Method. Journal of Computational Physics, 32: , J.P. Boris, F.F. Grinstein, E.S. Oran, and R.J. Kolbe. New insights into large eddy simulation. Fluid Dynam Res, 10(4-6): , C. Fureby and F.F. Grinstein. Monotonically integrated large eddy simulation of free shear layer flows. AIAA, 37(5): , F.F. Grinstein and C. Fureby. Recent progress on miles for high reynolds number flows. Journal of fluids Engineering, 124(4): , E. Garnier, M. Mossi, P. Sagaut, P. Comte, and M. Deville. On the use of shock capturing schemes for large eddy simulation. Journal of Computational Physics, 311: , G. Erlebacher, M.Y. Hussaini, C.G. Speziale, and T.A. Tang. Toward the large eddy simulation of compressible flows. Journal of Fluid Mechanics, 238:155 85, L. Duchamp De Lageneste and H. Pitsch. Large-eddy simulation of premixed turbulent combustion using a level-set approach. Proceedings of the Combustion Institute, 29(2): , P.J. O Rourke and F.V. Bracco. Two scaling transformations for the numerical computation of multidimensional unsteady laminar flames. Journal of Computational Physics, 133: , C.K. Westbrook and F.L. Dryer. Simplified Reaction Mechanisms for the Oxidation of Hydrocarbon Fuels in Flames. Combustion Science and Technology, 27:31 43, M. Baum, T. Poinsot, and D. Thévenin. Accurate Boundary Conditions for Multicomponent Reactive Flows. Journal of Computational Physics, 116: , OF 10

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