EVALUATION OF EULER-EULER AND EULER-LAGRANGE STRATEGIES FOR LARGE-EDDY SIMULATIONS OF TURBULENT REACTING FLOWS

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1 EVALUATION OF EULER-EULER AND EULER-LAGRANGE STRATEGIES FOR LARGE-EDDY SIMULATIONS OF TURBULENT REACTING FLOWS M. García 1, Y. Sommerer 1, T. Schönfeld 1 and T. Poinsot 1 CERFACS - Toulouse, France IMFT - Toulouse, France Abstract Keywords This paper describes a study of Euler-Lagrange two-phase flow solver for Large- Eddy Simulations (LES). In modern LES for reacting two-phase flows, the coupling between the dispersed and the gaseous phase is not only a physical problem where all submodels for drop evaporation, combustion, wall interaction must be specified. It is also a numerical problem that, especially for strong coupling between the two phases, must be performed at each iteration of an unsteady computation and run efficiently on massively parallel machines. These issues are investigated here by implementing a droplet solver into a LES compressible code and assessing the results in terms of physics but also of numerical efficiency. The tests are performed for dispersed particles in homogeneous turbulence: results show that the code scales very well when the particles are homogeneously distributed but leads to large numerical losses if particles are located in one part only of the computational domain (as it is often the case for reacting flows). Results demonstrate the need for dynamic load balancing in such cases. Parallel computing, unstructured grids, two-phase flows, Lagrangian method, solid particles. INTRODUCTION Today, the RANS (Reynolds Averaged Navier Stokes) equations are routinely solved to design combustion chambers, for both gaseous and liquid fuels. Recently, in order to provide better accuracy for the prediction of mean flows but also to give access to unsteady phenomena occurring in combustion devices (such as instabilities, flashback or quenching), Large-Eddy Simulation (LES) has been extended to reacting flows. The success of these approaches for gaseous flames in the last years [1,, 3, 4, 5, 6, 7, 8, 9, 10, 11] is a clear illustration of their potential. LES gives access to the large scale structures of the flow, which reduces the importance of modelling and naturally captures a significant part of the physics controlling these flames. Even though LES has already demonstrated its potential for gaseous flames, its extension to two-phase flames is still largely to be done. There are many reasons which slow down the application of LES to two-phase reacting flows:

2 the physical submodels required to describe the atomization of a liquid fuel jet, the dispersion of droplets, their interaction with walls, evaporation and combustion are as difficult to build in LES as in RANS because they are essentially subgrid scale phenomena for which even the basic mechanisms are often not well understood. the numerical implementation of two-phase flow LES remains a challenge. The equations for both the gaseous and the dispersed phases must be solved together at each time step in a strongly coupled manner. This differs from classical RANS where the resolution of the two phases can be done in a weak procedure, bringing first the gas flow to convergence, then the droplets and finally iterating until convergence of both phases. This paper mainly focuses on the second issue: the efficiency of the coupled solvers for the two phases, especially in the context of parallel computers. Recent tests performed at CERFACS show that a modern LES code can run gaseous reacting flows with speedups of the order of 4900 on 5000 processors (see Events section). Maintaining a similar parallel efficiency for a two-phase flow solver raises questions which have never been addressed before. The first question is related, for such architectures, to the paradigm used to describe the two-phase flow: most RANS codes use Euler- Lagrange (EL) methods in which the flow is solved using an Eulerian method and the particles are tracked using a Lagrangian approach. An alternative technique is to use two-fluid models in which both the gas and the dispersed phases are solved using an Eulerian method (Euler-Euler or EE). The drawbacks and advantages of each method are discussed below: The EL approach is more intuitive because it highlights trajectories of particles (or droplets) which appeal to the CFD user. However, these particles are often not real particles but parcels which can contain thousands of droplets because present computers can not handle the millions of droplets which are created by standard fuel injectors every second. Therefore models are required for these parcels which sometimes makes modelling more difficult [1]. The EE approach requires an initial basic modelling effort which is larger than for the EL method [13]. When a two-fluid model is used, the EE model faces difficulties in handling droplet clouds with extended size distributions because it basically assumes that all droplets in a given cell move at the same speed.

3 In very dense zones where the topology of the flow might differ from a cloud of droplets, the EE approach is often preferred. For RANS computations, EE techniques are commonly used for fluidized beds [14, 15] or for chemical reactors [16, 17, 18]. In terms of computer implementation the EL approach is not well-suited to parallel computers: since two different solvers must be coupled, the complexity of the implementation on a parallel computer increases drastically compared to a single-phase code. Two methods may be used for LES: (1) task parallelization in which certain processors compute the gas flow and others compute the droplets flow and () domain partitioning in which droplets are computed together with the gas flow on geometrical subdomains mapped on parallel processors. Droplets must then be exchanged between processors when leaving a subdomain to enter an adjacent domain. For LES, it is easy to show that only domain partitioning is efficient on large grids because task parallelization would require the communication of very large three-dimensional data sets at each iteration between all processors. However, codes based on domain partitioning are difficult to optimize on massively parallel architectures when droplets are clustered in one part of the domain (typically, near the fuel injectors). Moreover, the distribution of droplets may change during the computation: for a gas turbine reignition sequence, for example, the chamber is filled with droplets when the ignition begins thus ensuring an almost uniform droplet distribution; these droplets then evaporate rapidly during the computation, leaving droplets only in the near injector regions. This leads to a poor speedup on a parallel machine if the domain is decomposed in the same way for the entire computation. As a result, dynamic load balancing strategies are required to redecompose the domain during the computation itself to preserve a high parallel efficiency [19]. On the other hand, EE techniques are naturally parallel because the flow and the droplets are solved using the same solver [0]. The history of RANS development has shown that both EE and EL are useful and either is found today in most commercial codes. Moreover, coupling strategies between EE and EL methods within the same application are considered for certain cases. In the present work, we focus on the EL technique for LES and investigate the key issue of parallel efficiency. This question has already been identified by other groups as a critical issue to run LES for reacting two-phase flows on massively parallel computers [19].

4 In the present paper, a Lagrangian solver is first coupled with a compressible LES code for reacting flows on hybrid grids (the AVBP code of CER- FACS: see Parallelization is performed using domain partitioning. The partitioning algorithm is the same as the one used for gaseous flows and does not take into account the additional computing cost induced by the droplets. The implementation is first verified by computing the dispersion of non evaporating droplets distributed homogeneously in homogeneous turbulence which is a classical test case [0, 1, ]. For this homogeneous case, the parallel efficiency is tested. More realistic cases are then tested where the droplets are distributed non-uniformly in the domain: this situation is more representative of real combustion chambers where droplets will be present mainly near the fuel injector. For this type of loading, adequate speedups are more difficult to obtain, which confirms the need for dynamic load balancing. NUMERICAL AND PHYSICAL MODELLING This section describes briefly the governing equations used in the AVBP solver for the gaseous and dispersed phases for the two-phase flow Euler-Euler [3] and Euler-Lagrangian model for dilute flows with one-way coupling between the two phases. The particle-tracking scheme and the interpolation algorithm are also included at the end of the section. Two-phase flow EE model Gaseous phase The continuity, momentum and energy equations for the gaseous phase are: (ρ g E g ) t (ρ g u g,i ) t ρ g t + (ρ gu g,j ) = 0 (1) + (ρ gu g,i u g,j ) = p g x i + τ g,ij () + (u g,j(ρ g E g + p g )) = (τ g,iju g,i ) q g,j. (3) Pressure is obtained from the equation of state p g = ρ g rt g where r is computed from the mass fractions r = R/W with W the air molecular weight. The Newtonian viscous tensor is defined as: ( ug,i τ g,ij = ρ g ν g + u g,j ) x i 3 δ u g,i ij (4) x i

5 and the heat diffusion is calculated from the classical Fourier law: q g,j = λ g T g. (5) Dispersed phase The continuity, momentum and energy equations for the dispersed phase are: (α p ρ p ) t (α p ρ p u p,i ) t (α p ρ p E QB ) t + (α pρ p u p,j ) = 0 (6) + (α pρ p u p,i u p,j ) = p QB x i + (α pτ QB,ij ) + I p,i (7) + (α pρ p u p,j E QB ) = α pρ p τ p E QB ( p QB δ ij + α p τ QB,ij ) u p,i + ( ) α p ρ p κ QB E QB (8) where α p = nπd 3 /6 with n the number of particles by unit volume, τ p is the particles relaxation time (to be defined in the following section) and κ QB = 5τ p E QB /3 where E QB δqp is the quasi-brownian energy which accounts for the uncorrelated particles velocity as described in [13]. The quasi-brownian pressure and stress tensor p QB and τ QB,ij are defined by analogy with the kinetic theory of gases. p QB is calculated through an equivalent equation of state: p QB = 3 α pρ p E QB (9) and τ QB,ij has a definition similar to the gaseous viscous tensor: ( up,i τ QB,ij = ρ p ν QB + u p,j ) x i 3 δ u p,i ij x i (10) where ν QB = τ p E QB /3 is the quasi-brownian viscosity and I p,i is the standard drag force (see [3] for details).

6 Two-phase flow EL model Gaseous phase The compressible Navier-Stokes equations are solved in their conservative form for the gaseous phase. The continuity, momentum and energy equations are the same as for the gaseous phase of the EE model (see Eqs. 1, and 3). Dispersed phase The dispersed phase consists of particles which are assumed to be rigid spheres. If the density of particles is much larger than the fluid density, the forces acting on particles reduce to drag and gravity [1] (though the latter is not considered in this study). Particle size is assumed small compared to the smallest turbulence length scale, the Kolmogorov scale, while collisions among particles are neglected. The analysis presented in this work is limited to one-way coupling from gas-to-particle phases with the assumption of small particle loadings. The particle equations of motion can then be written for a single particle as: x p,i t u p,i t = u p,i (11) = u p,i ũ g,i τ p (1) where ũ g,i is the fluid velocity at the position of the particle assuming that the flow field is locally undisturbed by the presence of this particle [4, 5]. Due to the very small droplet Reynolds number measured in the simulation, the particle relaxation time τ p is defined as the characteristic time for Stokes drag: τ p = d pρ p 18µ g. (13) Locating particles in elements of arbitrary shape In order to interpolate the discreted flow fields from the grid point onto the particle position (inside a cell) it is necessary to know the cell the particle is located in as well as the particle exact location within the cell. This is straightforward for regular, uniform Cartesian grids where the physical coordinates can be transformed into a uniform computational space. However,

7 this is no longer valid for unstructured grids. In order to decide if the particle is located inside the element or not, the scalar product is taken between the vector starting from the vertex of the element to the particle position and the inward normal vector of the corresponding edge. Then, the particle is inside the element if all the scalar products of each edge are positive. If any of them is negative, the particle falls outside the element in that direction. Search algorithm for particles on unstructured grids During each time step of the simulation, the particles change their position. But before they can contribute information to the grid or sample field information from it, their new host element (or cell) must be identified. Therefore, they must be traced to the grid and the idea is to exploit as an initial, most likely, guess that host element before the particle was moved. Under the reasonable condition that a particle can only travel to a neighboring cell during one single time step, it is sufficient to limit the search to the cell the particle was located in at the end of the previous timestep and to its adjacent cells. More precisely, the closest point of the mesh to the particle location is evaluated and only the elements surrounding to that point are considered. In case of search failure, the elements surrounding all the close points are considered. Such an approach is usually referred to as modified brute-force algorithm [6] and has been retained for this study. Interpolation of gaseous-phase properties For this particular case, hexahedral elements are used, so that, flow variables such as the velocity vector or the fluid viscosity, are evaluated at particle position via trilinear Lagrange interpolation from the values at the cell vertices. The AVBP solver The AVBP solver is a finite volume code based on a cell-vertex formulation. It solves the laminar and turbulent compressible Navier-Stokes equations in two and three space dimensions for hybrid and unstructured grids. Steady state or unsteady flows may be simulated, and the variations of molecular weights and heat capacities with temperature and mixture composition are accounted for. In this work, a Lax-Wendroff scheme was used for the numerical discretization in space and time.

8 APPLICATION TO HOMOGENEOUS ISOTROPIC TURBU- LENCE WITH PARTICLES The bulk of the simulations for this study was performed on a 64 3 uniform grid, the length of the computational domain is π 10 3 m in the three directions and has a cell size of x = m. Parallel simulations were performed on a Compaq AlphaServer SC with, 4, 8, and 1 processors and.48 million particles trajectories were computed at each time step corresponding approx. to 10 particles per cell. This number is a minimum value if proper sampling (specially for combustion) must be achieved in each cell. Initialization of the gaseous phase The gaseous phase is initialized with a divergence free velocity obeying a Passot-Pouquet spectrum [7] for the kinetic energy: ( ) k 4 E(k) = C e (k/ke). (14) ke Here k is the wave length and k e corresponds to the most energetic wave length. The gaseous Navier-Stokes equations are run up to t 0 = s which corresponds to half an eddy turnover time. This is necessary to ensure that the generated homogeneous isotropic turbulence (HIT) flow is indeed a solution of the Navier-Stokes equations before inserting the dispersed phase. Table 1. Flow field parameters at time t 0. N x/η u g [m/s] qg [m /s ] ε g [m /s 3 ] τ ε [s] Table 1 summarizes the main characteristics of the gaseous field obtained at time t 0. In this table, η is the Kolmogorov scale, u g is the fluctuating turbulent velocity, and qg and ε g are the gaseous kinetic energy and dissipation rate defined as: q g = 1 u g,iu g,i, ε g = ν g ug,i u g,i (15) respectively, and ν g is the kinematic viscosity defined by ν g = m /s. The last two parameters are used to form the Eulerian time macro-scale or turnover time, τ ε = q g/ε g. Averages of turbulence quantities obtained over the computational domain are denoted.

9 Figure 1. Snapshot (top) and D slice (bottom) of the initial particle distribution for the homogeneous (left) and non-homogeneous (right) cases. Initialization of the dispersed phase The dynamics of the particles in HIT depends on the Stokes number which is the ratio between the particle relaxation time and a characteristic time scale of turbulence. In the present study we define a Stokes number based on the turnover time, St ε = τp τ. For all simulations the particles diameter, d ε p, was set to m which gives τ p = s and St ε = This small value for St ε indicates that the particles initial velocity is close to the gaseous phase velocity so that it is a reasonable assumption to initialize the dispersed phase velocity field with the gaseous one. For EE simulations, this implies that the quasi-brownian energy is initialized with a value close to zero.

10 In both approaches, particles are randomly placed within the computational domain and periodic boundary conditions are used in all directions. Figure 1 shows the initial particle distribution for the two cases studied in this work: the homogeneous distribution, where all processors share the same number of particles, the non-homogeneous distribution, where only one processor contains all particles. These two cases correspond to limit situations: for the homogeneous loading case, high parallel efficiency should be obtained for the EL approach since all processors will have the same number of particles. On the other hand, the non-homogeneous case mimics a real combustor in which a few processors only located in the fuel injectors will contain most particles while others will have no particles at all. Comparison of dispersed two-phase flows Figure reports the temporal evolution of the kinetic energy of Lagrangian particles: q p Lagrangian = 1 N p u p,i u p,i (16) N p i=1 where N p is the total number of particles. This is compared with the particle kinetic energy obtained in the Eulerian simulation (note that prime in Eq. 15 is omitted here as in HIT u p,i = 0): q p Eulerian = 1 u p,i u p,i. (17) Note that qp Eulerian is obtained by adding the correlated part (mesoscopic energy) q p, and the uncorrelated part (or quasi-brownian energy) δqp [13]. The results show an overall good agreement between the two approaches and confirm that both methods provide the same result for this academic test case. As expected for this range of Stokes numbers [0], the particle kinetic energy qp obtained with both methods is larger than the gaseous phase kinetic energy qg.

11 Kinetic energy [m / s ] q p EE q p EL q g EE or EL δq p 0 3 t [s] 4 5 6x10-5 Figure. Time evolution of the dispersed phase kinetic energy q p, computed with the Euler- Euler model (squares), and with the Euler-Lagrange model (solid line). Gaseous phase kinetic energy q g (circles). Quasi-Brownian kinetic energy δq p (crosses). Analysis of code scalability In this section, the scalability of the Euler-Lagrange model is analyzed by means of two basic parameters used to measure the efficiency of parallel implementation: the speedup, S run (N procs ), and the reference single-phase CPU time ratio, T run (N procs ). The former is defined as the ratio between the CPU time of a simulation with processors and the CPU time of a simulation with a given number of processors N procs : S run (N procs ) = T run() T run (N procs ). (18) The latter is defined as the ratio between the CPU time of a simulation with a given number of procs and the CPU time of the reference single-phase simulation with processors: T run (N procs ) = T run(n procs ) T single phase (). (19)

12 The values of these two parameters are summarized in Tables and 3 for the homogeneous and non-homogeneous cases (note that in the EE model, the CPU time is the same for both cases since the computational cost of this approach does not depend on the number of particles). Table. Summary of the speedup of EE and EL models. N procs Ideal scaling Single-phase Two-phase EE Two-phase EL Homog Two-phase EL Non-Homog Table 3. Summary of the CPU time ratios of EE and EL models. N procs Single-phase Two-phase EE Two-phase EL Homog Two-phase EL Non-Homog Homogeneous particle loading For the homogeneous case Figure 3 exhibits an excelent speedup for both single and two-phase models. More precisely, the good parallel implementation of the two-phase EL model can be observed from the super-linear values obtained with more than 4 processors. Figure 3 (bottom) indicates that taking into account the dispersed phase results in about twice and a half the CPU time the single-phase simulation for the EE model and about twice the CPU time for the EL approach. The differences between both models increase with the number of processors, thus, the higher the number of processors, the faster the EL model, leading to differences from 18% for a two-processors simulation up to 55% for the simulation performed with 1 processors. This is due to a

13 lower ratio of particles by processor which implies less computational operations, less memory requirements and, therefore, less cache memory defaults. These effects are stronger than the increase of the communication cost related to a higher number of particles crossing processors domain. Although the current tests have been performed over only a few processors, the results reported in Figure 3 show encouraging scaling of the EL homogeneous case. The super-linear speedup obtained for the EL homogeneous case confirms the improved behaviour of the EL approach when increasing the number of partitions. Non-homogeneous particle loading In the non-homogeneous case, the dispersed phase in the EL model is only computed by a single processor (the master node in the present study). This leads to a significant increase in CPU time compared to the homogeneous case where all parallel processors execute the same number of instructions for both gaseous and dispersed phases. The main reasons contributing to this increase are: 1) a higher number of numerical operations per processor related to the solution of Eqs. 11 and 1, such as search and interpolation procedures; ) higher memory requirements and 3) communication cost related to the flux of particles between processors. On the other hand, the EE approach does not suffer from this load balancing problem. This is confirmed in Figure 4 which shows that the speedup of the non-homogeneous two-phase EL approach drifts rapidly far from the ideal, linear curve. The drop of performance in this case is not related to large communications costs between processors but merely to the parallel load imbalance. Thus, an efficient parallel implementation of an EL model can only be achieved by applying the same principles used in parallel computing so as to reduce the computational time required to complete a task: namely, decomposition of a data set into smaller subsets executing the same instructions on separate processors. This points out the need of dynamic load balancing for real two-phase flow LES in complex geometry (see also [19]), e.g., by using multi-constraint partitioning algorithms which take into account particles loading on each processor.

14 Ideal scaling Single phase Two-phase EE Two-phase EL Speedup, S run (N procs ) N procs 3 CPU time ratio, T run (N procs ) Single-phase Two-phase EE Two-phase EL N procs Figure 3. Homogeneous particle distribution. Top: Speedup of the single and two-phase flow models, Eq. 18. Bottom: CPU time ratio with respect to the single-phase simulation, Eq. 19 (all scalings are reported relative to the corresponding two-processors simulation).

15 Ideal scaling Single phase Two-phase EE Two-phase EL Speedup, S run (N procs ) N procs 3 CPU time ratio, T run (N procs ) Single-phase Two-phase EE Two-phase EL N procs Figure 4. Non-homogeneous particle distribution. Top: Speedup of the single and twophase flow models, Eq. 18. Bottom: CPU time ratio with respect to the single-phase simulation, Eq. 19 (all scalings are reported relative to the corresponding two-processors simulation).

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