International Journal of Heat and Fluid Flow 32 (2011) 285–297 Contents lists available at ScienceDirect International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff Numerical investigation of the spray–mesh–turbulence interactions for high-pressure, evaporating sprays at engine conditions Tommaso Lucchini *, Gianluca D’Errico, Daniele Ettorre Internal Combustion Engine Group, Dipartimento di Energia, Politecnico di Milano, Via Lambruschini 4, 20156 Milan, Italy a r t i c l e i n f o Article history: Received 22 December 2009 Received in revised form 13 July 2010 Accepted 17 July 2010 Available online 8 August 2010 Keywords: Eulerian–Lagrangian methodology Adaptive mesh refinement Diesel spray a b s t r a c t This work presents a numerical methodology to simulate evaporating, high pressure Diesel sprays using the Eulerian–Lagrangian approach. Specific sub-models were developed to describe the liquid spray injection and breakup, and the influence of the liquid jet on the turbulence viscosity in the vicinity of the nozzle. To reduce the computational time and easily solve the problem of the grid dependency, the possibility to dynamically refine the grid where the fuel–air mixing process takes place was also included. The validity of the proposed approach was firstly verified simulating an evaporating spray in a constant-volume vessel at non-reacting conditions. The availability of a large quantity of experimental data allowed us to investigate in detail the effects of grid size, ambient diffusivity and used spray sub-models. In this way, different guidelines were derived for a successful simulation of the fuel–air mixture formation process. Finally, fuel injection and evaporation were simulated in an optical engine geometry and computed mixture fraction distributions were compared with experimental data. Ó 2010 Elsevier Inc. All rights reserved. 1. Introduction The achievement of well atomized liquid fuel spray and the effectiveness of the evaporation and fuel–air mixing process are of great importance in many industrial processes which include combustion systems, such as rocket engines, gas turbines, industrial furnaces and internal combustion engines (Schmehl et al., 1999; Park and Reitz, 2009). In particular among the latter, a detailed definition of the fuel injection strategy is a primary requisite for the control of the combustion process in Diesel engines in a wide range of operating conditions to preserve high thermal efficiencies and reduce the pollutant emissions. Within this context, accurate experimental studies are extensively performed to clarify the main physical and chemical processes governing the combustion process in Diesel engines such as fuel–air mixing, auto-ignition, flame development (Siebers and Higgins, 2002; Koss et al., 1993) and the influence of in-cylinder charge motions (Genzale et al., 2008). However, these detailed investigations are typically performed in simplified configurations, like constant-volume vessels or optical engines, and therefore they cannot be extensively used from a design perspective to optimize a real engine geometry together with the possible injection strategies. For this reason, Computational Fluid Dynamics (CFD) usually integrates the experimental activity carried out at the test-bench * Corresponding author. Tel.: +39 02 23998636; fax: +39 02 23998050. E-mail address: tommaso.lucchini@polimi.it (T. Lucchini). 0142-727X/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.ijheatfluidflow.2010.07.006 to investigate the combustion process and is now part of the industrial design and analysis. The use of detailed chemistry seems absolutely necessary to describe new combustion modes, predict the main pollutant emissions (Kong et al., 2007) and evaluate the effects of multiple injections. Furthermore, a fundamental prerequisite for a successful CFD simulation is a correct prediction of the fuel–air mixing process because it influences both the auto-ignition and mixing-controlled combustion phases. The need to keep an acceptable compromise in terms of computational time and accuracy justifies the wide use of the Eulerian–Lagrangian approach, where the spray is composed by a discrete number of computational parcels, each one formed by an arbitrary number of droplets with the same properties. Each parcel evolves in the computational mesh according to the mass, momentum and energy exchange with the continuous gas phase which is treated in an Eulerian way. Additional phenomenological sub-models are then required to describe the various physical processes taking place in the sub-grid length scales: atomization, secondary breakup, drag, evaporation, heat transfer, collision and turbulent dispersion (Kolaitis and Founti, 2006; Beck and Watkins, 2002). Despite the Lagrangian approach is widely applied and a wide range of submodels is available, simulation results show a strong dependency on the used sub-models and on the grid size (Nordin, 2001; Abraham, 1997). In particular, there are still well known uncertainties related to:  fuel atomization at high pressure conditions;  relevant length scales close to the injector nozzle; 286 T. Lucchini et al. / International Journal of Heat and Fluid Flow 32 (2011) 285–297  turbulence–spray interaction;  interpolation of the gas phase quantities at parcel positions; and  definition of an optimum grid size. the computational time and the grid dependency is presented in detail. 2.1. Gas phase equations Different approaches were proposed to address all these problems. The use of atomization models is necessary to describe the breakup of the fuel jet, where secondary droplets are stripped from the liquid core depending on the conditions inside the nozzle (turbulence, cavitation) and instabilities due to the aerodynamic forces. However, most of the spray sub-models require different tuning constants which usually need to be adjusted depending on the operating conditions. Among the other sources of uncertainties, grid-dependency plays a major role and is strictly related to the Eulerian–Lagrangian approach. The reasons for such dependency are mainly due to the difficulty in having a spatial resolution which guarantees a cell volume bigger than the volume of the droplets they contain, but which is also able to accurately resolve the gas phase development near the nozzle (Tonini et al., 2008). Other possible sources of error are represented by the lack of statistical convergence in the Lagrangian treatment of the liquid phase (Stiesch, 2003) and the relevance of the used interpolation process, whereby the gas quantities known at Eulerian nodes are estimated at parcel locations (Golovitchev and Nordin, 2001). As a consequence, the capability of predicting the correct fuel and air mixing prior to ignition has a very critical numerical dependency. Several improvements have been proposed to overcome this known limitation by introducing new collision algorithms (Schmidt and Rutland, 2000; Schmidt and Senecal, 2002) or by combined hybrid Eulerian–Eulerian and Eulerian–Lagrangian approaches (Vallet et al., 2001) or by making use of an ‘ locally refined mesh in the spray region (Xue et al., 2008). Within this context, the authors intend to present a CFD methodology to simulate the fuel– air mixing process in Diesel engines with a reduced dependency on the grid size. To this end, specific sub-models were developed to describe the liquid jet atomization and the interaction between turbulence and spray. The possibility to dynamically refine the grid during the penetration of the liquid and vapor spray was also included in the proposed methodology in order to reduce the computational time. The 3D simulations were performed using the OpenFOAMÒ, (OpenFOAM, 2009; Weller et al., 1998) technology together with Lib-Engine, which is a set of libraries and solvers, specifically tailored for engine simulations, developed by the Internal Combustion Engine group of Politecnico di Milano (Lucchini et al., 2008; Montenegro et al., 2007). The capabilities of the proposed approach were firstly verified by simulating the air–fuel mixture formation process in the SANDIA combustion chamber (Idicheria and Pickett, 2007). The comparison of the simulation results with available set of experimental data allowed to determine the best mesh size and to identify its relationship with the relevant length scales governing the fuel–air mixing process. Finally, experimental data of mixture fraction distribution taken in an optical engine running under non-reacting conditions were used to validate the proposed model under evaporating conditions. 2. Model description The mass, momentum and energy equations are solved for a compressible, multi-component gas flow using the RANS approach (Nordin, 2001; Stiesch, 2003). Conservation of mass: @q þ r  ðqUÞ ¼ q_ s @t ð1Þ Conservation of species mass fractions:   @ qY i þ r  ðqUY i Þ  r  ðl þ lt ÞrY i ¼ q_ si þ q_ chem i @t ð2Þ Conservation of momentum: h  i @ qU þ r  ðqUUÞ ¼ rp þ r  ðl þ lt Þ rU þ ðrUÞT @t    2 þ qg þ Fs  r  ðl þ lt Þ trðrUÞT 3 ð3Þ Conservation of energy: @ qh Dp þ r  ðqUhÞ  r  ½ða þ at Þrh ¼ Q_ s þ @t dt ð4Þ The source terms q_ s ; q_ si ; Fs ; Q_ s account for mass, momentum and energy exchange between the gas and the liquid phases. The turbulence viscosity lt is provided by the turbulence model and the reader can find a more detailed description of Eqs. (1)–(4) are discretized with the second-order, finite-volume method on a polyhedral mesh (Jasak, 1996; Ferziger and Peric, 2002). The discretization schemes adopted in this work will be shortly presented in Section 5.1. Discretization of Laplacian, convection and temporal derivatives terms can be performed with the different schemes originally available in the code (Jasak, 1996). The PISO algorithm is used for the pressure–velocity coupling (Issa, 1986). The turbulence viscosity lt is provided by the turbulence model. 2.2. Liquid phase equations The properties of each parcel (position, velocity, temperature, . . .) are determined for each time step by solving the mass, momentum and energy equations in a Lagrangian way. Multi-component sprays are supported, hence realistic fuel surrogates can be simulated. Droplet momentum is influenced by drag and gravity forces, while their evaporation is estimated through the D2-law and by suitable relaxation times calculated under standard and boiling conditions. The energy equation accounts for heat transfer and evaporation. Here, the equations for the liquid phase are shortly presented; the reader is referred to Nordin (2001) for further details. Droplet momentum equation: md dud pD2 qC d jud  ujðud  uÞ þ md g ¼ dt 8 ð5Þ Droplet mass equation (under standard evaporation): In the Eulerian–Lagrangian approach, the spray is described by a discrete number of computational parcels, each one representing droplets with the same properties. The spray parcels evolve into the computational domain according to the mass, momentum and energy exchange with the continuous gas phase which is threaten in an Eulerian way. The governing equations are now briefly summarized while the proposed methodology to reduce dmd md ; ¼ dt se dD D ¼ dt 3s e ð6Þ Droplet mass equation (under boiling conditions): dmd md ; ¼ dt sb dD D ¼ dt 3s b ð7Þ T. Lucchini et al. / International Journal of Heat and Fluid Flow 32 (2011) 285–297 Droplet energy equation: md dhd _ d hv ðT d Þ þ pDjNuðT  T d Þf ¼m dt ð8Þ 2.3. Spray sub-models Additional phenomenological models are required to describe the various physical processes taking place in the sub-grid length scales: atomization, secondary breakup, drag, evaporation, heat transfer, collision and turbulent dispersion. During the injection duration, large and spherical drops (called blobs or parent parcels) are continuously added to the gas phase. Their diameter is comparable to the size of the nozzle hole and their velocity depends on the injected mass flow rate profile. The number of parcels introduced per each time step is equal to:  NðtÞ ¼ max 1; Dt Ntot ðt eoi  tsoi Þ  ð9Þ where Ntot is the total number of parcels specified by the user, Dt is the time-step value, teoi and tsoi represent the end and the start time of injection. The model is able to handle multiple injections that are commonly used in modern Common-Rail Diesel engines. The injected blobs represent the liquid jet and their atomization is simulated by using a modified version of the Huh–Gosman model (Huh and Gosman, 1991; Bianchi and Pelloni, 1999; Bianchi et al., 2001). The turbulence developed in the nozzle hole induces perturbations on the jet surface that grow due to aerodynamic forces until they detach from the liquid core forming new small droplets. In Fig. 1a the atomization process of a liquid jet is illustrated, while Fig. 1b shows how spray atomization is modeled with the Lagrangian approach. Instabilities on the jet surface cause the reduction of the blob diameter (light gray), and child parcels (dark gray) are created from them when the amount of stripped mass is sufficiently high (1–10% of the primary parcel mass) (Reitz, 1987). Primary parcels are only subjected to the atomization process, they do not evaporate and no drag force acts on them (Kralj, 1995). Diameters of the blobs are reduced according to the following expression: dD LA ¼ C 5 dt sA ð10Þ 287 where LA and sA are the breakup length and time, while C5 is a model constant. LA is proportional to the turbulence length scale while sA is a linear combination of the turbulence time scale (st) and the wave growth time scale (st) due to the Kelvin–Helmoltz instability (Reitz, 1987). The initial value of st for each parent parcel depends on the nozzle flow conditions (Huh and Gosman, 1991), then the turbulence decay is described by a simplified version of the k–e model. For each parent parcel, the amount of mass stripped is increased at each time step by the quantity: Dmstripped ¼ ql   1 pNd D3new  D3old 6 ð11Þ where Nd is the number of droplets in the parent parcel, ql is the liquid density, Dold and Dnew are the diameters of the parent parcel before and after the Eq. (10). When a new child parcel is created, the amount of stripped mass for the blob is set to zero. The child parcel diameter is taken from a Rosin–Rammler distribution whose mean value is a function of the nozzle flow Reynolds number (Bianchi et al., 2001): Renoz ¼ ql U inj Dinj lliq ð12Þ where Uinj is the injection velocity, Dinj is the parcel diameter and lliq is the liquid density. Since parent droplets continuously reduce their diameter because of atomization, they will considerably reduce their Weber number. When the Weber number falls below the threshold of 1000, these parent droplets are taken in charge by the secondary breakup model (Reitz, 1987). An additional condition was also introduced to check if a droplet is suitable to be treated by the atomization model or not: when the droplet diameters become less than the 20% of the injector diameter, the droplets are treated by the secondary breakup model. The proposed approach considers separated models for the primary and secondary breakup of the spray and this is substantially different to the cases where only one model is used for both the purposes. In particular, the atomization model takes into account the effects of turbulent and aerodynamic instabilities on the liquid jet surface and produces secondary droplets whose size is rather small (610 lm) compared to the nozzle diameter. The secondary breakup model operates only on such droplets and this results in a more realistic way to handle this process according to the Kelvin–Helmoltz and Rayleigth Taylor instabilities. The drag force acting on the child droplets is modeled by using the correlation proposed in Kralj (1995). To correctly describe the mass and energy exchange between the liquid and the surrounding gas, Eq. (6) requires expressions for the Sherwood and Nusselt numbers which are modeled according to the approach described in Crowe et al. (1998). To reduce the results sensitivity to the turbulence model (Stiesch, 2003), effects of turbulent dispersion were not considered. Collision models were also not used, because of their limited effects on the Sauter mean radius (SMR) of an evaporating spray, as it is illustrated in Baumgarten (2006). 3. Eulerian–Lagrangian coupling Fig. 1. Application of the Lagrangian approach to model the atomization process. The interaction between the Lagrangian and the Eulerian phases determines the fuel/air mixture formation process and modifies the flow-field in the computational domain. The phase coupling can be distinguished into ‘gas-to-liquid’ and ‘liquid-to-gas’. The first one includes the interpolation process, whereby gas quantities known at the Eulerian nodes are estimated at the parcel locations. The ‘liquid-to-gas’ refers to the summation of the particle source terms in the Eulerian conservation laws. 288 T. Lucchini et al. / International Journal of Heat and Fluid Flow 32 (2011) 285–297 tion, while excessively refined meshes lead to an unphysical fast diffusion of momentum from the liquid to the gas phase, resulting in high gas velocities and spray penetration. Hence, the best mesh size always represents a compromise between the two aforementioned requirements and convergence of the computed results cannot be achieved. The problem of grid dependency was extensively investigated and the most popular proposed solutions are listed below: vertex i α i X βi 1. Use of a specific collision mesh to deal with droplet-to-droplet interactions (Schmidt and Senecal, 2002). 2. Distribution of the momentum and evaporation Lagrangian source terms within a sphere of influence for each droplet (Beard et al., 2000). 3. Modeling the relative gas–liquid velocity using the gas-jet theory (Abani et al., 2009). base i Fig. 2. Interpolation of the velocity field at the parcel locations. 3.1. Gas to liquid coupling Two different interpolation techniques were chosen to compute the gas quantities at the parcel positions. The cell-point-face approach is used to estimate the gas velocities at the parcel locations (Nordin, 2001). Firstly, the velocity field, which is stored at the mesh cell centers, is interpolated both at the mesh faces and mesh points locations. Then, a tetrahedron is built with the closest points around each parcel (cell centers, face centers, mesh points), as it is illustrated in Fig. 2. The distance between the parcel and one of the tetrahedron faces is ai, while ai + bi represents the distance between the same face and its opposite vertex. The interpolated velocity at the parcel position x is: uðxÞ ¼ 4 X i¼1 bi a1 þ bi ui ð13Þ When this interpolation technique is used, each parcel experiences a continuous velocity field within the computational domain, such that the grid dependency of the spray simulations is reduced. For stability reasons, the same technique cannot be employed to estimate the gas phase temperature, pressure and density at the parcel positions: these quantities are assumed to have the same values of the cell center where the parcel is found (Nordin, 2001). 3.2. Liquid to gas coupling Each parcel is tracked along its path by using the face-to-face algorithm. This makes possible to identify all the cells crossed by each parcel during one time step and to split the Lagrangian source terms of the Eulerian equations accordingly. This technique was proved to increase the accuracy and stability of Diesel spray simulations (Nordin, 2001). 4. Techniques to improve the accuracy and reduce the computational time of high-pressure spray simulations It is widely accepted (Lippert et al., 2005; Aneja and Abraham, 1998) that the accuracy of the Eulerian–Lagrangian spray simulations is related to the various sub-models involved and to the inadequate spatial resolution hindering the coupling between the gas and liquid phases. This second aspect is the most important since it often requires an unphysical adjustment of the spray sub-models coefficients. Grid dependency plays also a very important role in spray simulation, since coarse meshes are not able to correctly describe the interaction between the liquid and gas phase causing an underestimation of the spray penetra- None of these techniques was used in this work. The first one was not considered since collision has no relevant effects on the evolution of an evaporating spray. The radius of the sphere of influence is still a function of the grid size and for this reason this technique does not completely solve the grid-dependency problem even if collision models are used. The third approach appears rather promising and will be evaluated in future works as well as the possible advantages of such techniques when evaporating sprays have to be simulated with multiple injections. Another problem of spray simulation was illustrated in Abraham (1997) and is related to the dependency of the computed results on the initial ambient diffusivity: leff ¼ l þ lt ; lt ¼ qC l k2 e ð14Þ This is because the grid size generally adopted is much bigger than the nozzle diameter (2–5 times), which represents the relevant length scale that needs to be discretized to achieve a satisfactory agreement with the experimental data. Within this context, the following solutions were identified to address the problems of grid dependency and dependency on the ambient viscosity:  Use of spray-adapted grids when possible, such that the spray penetrates almost perpendicular into the cells.  Limitation of the turbulence length scale in the liquid jet region, since the liquid jet governs the relevant length scales.  Introduction of an adaptive local mesh refinement technique to keep an acceptable grid size and, at the same time, to model the relevant length scales which are typical of the fuel–air mixture formation process. 4.1. Use of spray-adapted grids The orientation of the grid relative to the spray axis may also influence the results since the number of nearest nodes (and thus the total gas volume) that are included in the direct exchange of mass, momentum and energy is dependent on the angle between spray axis and grid. If the momentum transferred to the gas is distributed to a larger number of nodes, the increase of gas velocity in the spray region is slower. If there are less nodes, the gas velocity will increase faster, and the relative velocity between droplets and gas will decrease more quickly. These effects may cause different penetration lengths with identical sprays, which differ only in orientation with respect to the mesh. For this reason, one possibility of reducing the effect of grid orientation is the use of sprayadapted grids, where the grid arrangement is adjusted to the spray direction, such that the main spray direction is as perpendicular as possible to the cells (Baumgarten, 2006). T. Lucchini et al. / International Journal of Heat and Fluid Flow 32 (2011) 285–297 4.2. Turbulent length-scale limiter Karrholm and Nordin (2005) introduced a turbulent length scale limiter to reduce the dependency of spray simulations on the initial ambient diffusivity. Following (Abraham, 1997), the turbulent length scale was limited to the orifice diameter in the cells where the spray parcels are present. A similar approach was followed by the authors in this work, in which the length scale was limited only in the cells where non-atomized droplets exist since, in other parts of the computational domain, the relevant length scales are mainly related to the existing in-cylinder flows. The turbulence integral length is defined as: lt ¼ C l k 3=2 e 6 Lsgs ð15Þ where k is the turbulent kinetic energy and e represents its dissipation rate. Lsgs is the limiter length which is set to the nozzle diameter. Since k governs the scale of the fluctuating turbulent velocity and e controls the size of the turbulent eddies, the limit was imposed as constraint on e, after solving the equations for k and e: 3=2 e P Cl k Lsgs ð16Þ In Karrholm and Nordin (2005) it was shown that this technique reduces both the dependency on the initial ambient diffusivity and on the grid size. However, the length-scale limiter mainly governs the turbulent viscosity distribution close to the nozzle, hence it has a less relevant influence on the grid independency than the used interpolation scheme. In particular, when the droplets do not experience a continuous gas velocity field, the amount of momentum exchange between the liquid and the gas phase will be strongly grid dependent. For this reason, the use of distanceweighted interpolation techniques is strongly advised. 4.3. Adaptive local mesh refinement (ALMR) When a computational meshes is generated for Diesel spray calculations, a compromise has to be found among these three contradictory requirements: 289 at the mesh face centers (Jasak, 1996; Lippert et al., 2005). An arbitrary level of refinement can be chosen by the user, and a maximum number of cells should be specified in order to control the mesh size. Refined cells are marked with a group number, in this way the mesh can be easily unrefined when the values of the error estimator are outside the specified interval (OpenFOAM, 2009). 4.3.2. Refinement criterion The geometric field used as a refinement criterion is represented by the total fuel mass fraction (liquid and gas) in each cell: Y lþg ¼ mf ;l þ qY tf V cell qV cell ð17Þ where mf,l is the liquid mass of all the parcels belonging to the cell, Ytf is the fuel mass fraction in the continuous phase, q is the gas phase density and Vcell is the cell volume. The lower threshold value was set to 102 while the higher was 1. This allows an adequate refinement of the mesh close to the nozzle in the first time steps. 4.3.3. Solution procedure During each time step, the solution of the governing equations and the mesh management proceeds as follows: 1. The refinement criterion is evaluated in all the cells of the computational mesh. Cells whose values belong to the specified refinement interval are marked. 2. Refined cells, which do not satisfy any more the refinement criterion, are identified and the corresponding mesh is unrefined. 3. The mesh topology is modified by refining and unrefining the corresponding regions. Only the cells whose refinement level is under the maximum one are refined. 4. The computed flow-field is interpolated from the old mesh to the new refined mesh with an inverse, distance-based technique (Lucchini et al., 2008). 5. The governing equations are discretized and solved on the new mesh. 6. The time step is advanced and all the steps are repeated until the end of the simulation. 5. Results and discussion  reproduction of the main geometry details (e.g. piston bowl, cylinder head,. . .);  refinement to the best size where fuel–air mixture formation takes place;  limitation of the minimum mesh size to reduce the computational time. Within this context, adaptive local mesh refinement (ALMR) becomes very interesting since it enables a high mesh resolution where fuel–air mixing takes place while the overall grid size is weakly increased (Lippert et al., 2005; Jasak, 1996). In this work, such technique was applied by the authors, defining opportune refinement criteria according to which the topology of the polyhedral mesh is changed. Previous examples of use of such techniques for other fields of applications are illustrated in Jasak (1996). 4.3.1. Refinement strategy To preserve the quality of the mesh, only hexahedral and degenerated hexahedral cells (wedges) can be refined. An initial computational mesh has to be provided by the user and the size should be fine enough to correctly reproduce the geometrical domain to be simulated and the main details of the initial flow-field (Jasak, 1996). A geometric field is chosen as an error estimator and when its values lie in a user-specified interval the parent cell is split into eight child cells by introducing new nodes at the cell centroid and The proposed approach was validated with two different sets of experimental data at non-reacting conditions. Experiments conducted in an optical, constant-volume vessel with a single-component fuel (C7H16) were used to tune the spray model constants and to identify the optimum refinement levels. Then the model was verified simulating the injection and the fuel–air mixture formation process in an optical engine using a two-component fuel, including a comparison of the equivalence ratio distributions. 5.1. SANDIA combustion chamber A preliminary investigation was performed at constant-volume conditions to determine the best mesh size. The computational domain reproduces a quarter of the combustion chamber (50.46  50.46  108 mm) and the thermodynamic conditions in the vessel at SOI are summarized in Table 1. Liquid fuel was provided according to the experimental mean injection profile through a 0.1 mm orifice. The initial mesh size was 4 mm, resulting in 3600 cells, and simulations were run with different refinement levels (1–4) with the mean cell size ranging from 0.25 to 2 mm. The chosen time-step value is 5  104 ms ensuring a maximum Courant number value of 0.15. When ALMR was used, a maximum number of 60,000 cells was allowed and all the simulations end at 1.5 ms after SOI. This value 290 T. Lucchini et al. / International Journal of Heat and Fluid Flow 32 (2011) 285–297 Table 1 Initial ambient conditions for the non-reacting case. Case qamb (kg/m3) T (K) [O2] 1 14.8 1000 0% was chosen because it is greater than the typical auto-ignition delay times that are encountered in Diesel engines (Idicheria and Pickett, 2007). If the vapor penetration is correctly predicted for such a long time, the combustion model will probably succeed for a wide range of operating conditions (in-cylinder pressure, temperature and EGR). First-order numerical schemes were used for these simulations to preserve the results stability and to be consistent with the typical case setup which is used for the simulations of Diesel engines. The chosen refinement interval ranges from 103 to 1 to describe the spray and fuel vapor evolution during the first part of the simulation. Table 2 illustrates the minimum size and the overall cell number at 1.5 ms. When four levels were used, refinement was performed only in the first part of the spray region because the maximum number of cells was reached. The number of cells of the corresponding uniform meshes used is shown in Table 3. The use of ALMR drastically reduces the number of required cells, and, as a consequence, also a reduction of the computational time can be achieved. The consistency of the adaptive local mesh refinement technique was firstly verified. For each level of refinement, the com- Table 2 Minimum mesh size and total number of cells of the grids tested using ALMR. Refinement levels Minimum size Cells @1.5 ms 1 2 3 4 2.0 1.0 0.5 0.25 4076 6211 20,470 61,182 Table 3 Cell size of the fixed meshes tested. Mesh Minimum size (mm) Cell size Uniform mesh, cell size 1 2 3 4 2.0 1.0 0.5 0.25 7975 25,125 95,125 410,125 33,750 270,000 2,203,416 17,627,328 puted liquid and vapor penetrations were compared with the ones obtained by a simulation performed on a fixed mesh with the same minimum size where the fuel–air mixing process takes place. To save computational time and avoid fine meshes where nothing happens, all the tested fixed meshes increase progressively their resolution close to the injector axis as shown in Fig. 3. For this reason, their size is much lower than a uniform mesh with the same minimum size, which is displayed in the fourth column of Table 3. Figs. 4 and 5 compare the computed liquid and vapor penetrations using ALMR and fixed meshes with the same minimum size. Despite the grid dependency exhibited by the computed liquid and vapor penetrations, it is possible to see that ALMR provides almost the same results of the fixed mesh. This comparison verifies the consistency of the ALMR approach and allows to use it for the simulation of high-pressure Diesel sprays with the Lagrangian approach. The evolution of the CPU time and the number of mesh cells as a function of the simulated time is illustrated in Figs. 6 and 7 for the mesh with three levels of refinement. Fig. 6 shows that the number of mesh cells used by ALMR grows a bit faster than linearly because of the shape of the fuel vapor region, but its size remains always lower than the one of the reference fixed mesh. The effect on the CPU time can be appreciated in Fig. 7, where the data were normalized with respect to the CPU time required by the reference fixed mesh to reach 2 ms of simulated time. The achieved speed-up factor varies between 12 and 5, with its value progressively decreasing as the number of cells using ALMR grows. These results can be considered satisfactory, especially because the higher interest for Diesel engine application is in the lower time intervals after start of injection. The effects of the grid size on the computed spray penetration are summarized in Fig. 8 and compared with experimental data. Because of the evaporating conditions, the liquid length reaches a stable value and this aspect is correctly predicted by all the tested meshes. However, its value shows a moderate dependency on the grid size until 0.5 mm are used. Two possible reasons can be identified for this behavior: the use of a liquid-core based atomization model and the high evaporation rate which is typical of n-heptane. Injecting dormant parcels reduces the grid dependency because the gas phase interacts only with the secondary stripped droplets in the vicinity of the injector. Since their diameter is rather small, the secondary droplets evaporate very fastly and this explains why the computed liquid length has a limited grid dependency for this experiment. However, when a minimum size of 0.25 mm is used, Fig. 3. Structure of the fixed meshes used to verify the consistency of the ALMR approach. The methodology used to refine the fixed grids is accurately described in Jasak (1996). 291 T. Lucchini et al. / International Journal of Heat and Fluid Flow 32 (2011) 285–297 40 25 20 Speed-up 0.9 15 10 10 0.6 5 Speed-up factor 30 ALMR Mesh 3 (0.5 mm) Normalized CPU time 35 Spray penetration [mm] 15 1.2 Unif., 2 mm Unif., 1 mm Unif., 0.5 mm Unif., 0.25 mm ALMR, 1 lev. ALMR, 2 lev. ALMR, 3 lev. ALMR, 4 lev. 0.3 5 0 0 0.5 1 1.5 0 Time ASOI [ms] 0 0.5 1 1.5 2 0 Time ASOI [ms] Fig. 4. Computed spray liquid penetrations by ALMR and the corresponding uniform meshes with the same minimum size. Fig. 7. Comparison between ALMR and the reference fixed mesh in the case of 0.5 mm minimum size: CPU time required as a function of the simulated time. 90 Unif., 2 mm Unif., 1 mm Unif., 0.5 mm Unif., 0.25 mm ALMR, 1 lev. ALMR, 2 lev. ALMR, 3 lev. ALMR, 4 lev. 60 45 30 30 15 0 15 10 0 0 0.5 1 1.5 150 ALMR Mesh 4 (0.5 mm) 125 100 75 50 25 0 0 0.5 1 0 0.5 1 1.5 Time ASOI [ms] Fig. 5. Computed spray vapor penetrations by ALMR and the corresponding uniform meshes with the same minimum size. 3 20 5 Time ASOI [ms] Mesh cells*10 4 lev. (0.25 mm) 3 lev. (0.50 mm) 2 lev. (1.00 mm) 1 lev. (2.00 mm) Experimental 25 Liquid length [mm] Vapour penetration [mm] 75 1.5 2 Time ASOI [ms] Fig. 6. Comparison between ALMR and the reference fixed mesh in the case of 0.5 mm minimum size: evolution of the number of mesh cells as a function of the simulated time. Fig. 8. Comparison between experimental and computed spray liquid penetrations for the four tested grid resolutions. the main assumptions of the Lagrangian approach are not valid anymore since the void fraction becomes too small. This leads to a fast diffusion of the momentum from the liquid to the gas phase and the small droplets are convected at a higher distance from the injection before their complete evaporation. The carried out investigations confirm that it is not possible to achieve a convergent solution in terms of spray penetration when the grid is refined too much if the Lagrangian approach is used. This aspect is further clarified in Fig. 9, where the computed spray shapes are illustrated for the different refinement levels tested at 0.4 ms. That picture also shows that the high liquid penetration of the 2 mm mesh is due to a restricted number of droplets having a high breakup time. This behavior is also obtained by the fixed 2 mm mesh and for this reason it is mainly related to poor grid size for this case. Furthermore, it does not have a significant influence on the fuel vapor penetration. For all the tested configurations, the employed spray sub-models are not able to describe in detail the first part of the spray penetration history (<0.3 ms) characterized by the breakup of the first injected droplets. However, this part does not play a significant influence on the computed vapor penetration since it is very short and influenced by the injector opening time, which was only 292 T. Lucchini et al. / International Journal of Heat and Fluid Flow 32 (2011) 285–297 Fig. 9. Influence of the mesh resolution on the computed spray shape at 0.4 ms. Fig. 11 shows that the computed mixture fraction distribution by the 0.25 mm mesh is narrower than the experimental one. This is due to the higher liquid spray penetration provided by this mesh, resulting in a lower spray cone angle and a denser vapor region Time = 0.90 ms ASOI, d = 20 mm 0.4 Exp, 20 mm ALMR, 2 lev. ALMR, 3 lev. ALMR, 4 lev. 0.3 Mixture fraction known to be lower than 0.1 ms. For this reason, only the agreement with the steady liquid length was considered. For what concerns the fuel vapor penetration, the computations show a very limited dependency on the grid size except for the 2 mm mesh as it can be seen in Fig. 10. Only the first part of the curves is slightly influenced by the different predicted liquid lengths. Results obtained for vapor penetrations can also be explained by the fact that the momentum exchange between the liquid and the gas phase is mainly related to the injected fuel flow rate. Concerning the 2 mm mesh, the use of high-order numerical schemes for the momentum and species convection can be useful to improve the quality of the results (Jasak, 1996). The main purpose of the mesh size investigation was to determine the best mesh that is able to correctly reproduce both the auto-ignition and the mixing-controlled combustion phases. For this reason, if only the spray and vapor tip penetrations were analyzed, results of this investigation might be misleading, since these data are not completely representative of the fuel–air mixing process. Hence, computed radial values of mixture fraction at three different distances from the injector were compared with experimental data at 0.9 ms (Idicheria and Pickett, 2007). Figs. 11–13 show the comparison at 20 mm, 30 mm and 40 mm distances from the injector for the three different meshes which were evaluated. For all of them, the overall result can be considered rather satisfactory since they can reproduce the mixture fraction distribution for any location. At 20 mm from the injector, 0.2 0.1 0 -4 -2 4 Time = 0.90 ms ASOI, d = 30 mm 0.2 Experimental Exp, 30 mm Size = 0.25 mm 80 Size = 0.50 mm 60 Size = 2.00 mm 40 20 0 0.5 ALMR, 2 lev. ALMR, 3 lev. ALMR, 4 lev. 0.15 Size = 1.00 mm Mixture fraction Vapor Penetration [mm] 2 Fig. 11. Comparison between experimental and computed mixture fraction distributions at 20 mm from the injector. 100 0 0 Distance from the injector axis [mm] 1 1.5 Time ASOI [ms] Fig. 10. Comparison between experimental and computed fuel vapor penetrations for the four tested grid resolutions. 0.1 0.05 0 -4 -2 0 2 4 Distance from the injector axis [mm] Fig. 12. Comparison between experimental and computed mixture fraction distributions at 30 mm from the injector. T. Lucchini et al. / International Journal of Heat and Fluid Flow 32 (2011) 285–297 Time = 0.90 ms ASOI, d = 40 mm 0.2 Exp, 40 mm ALMR, 2 lev. ALMR, 3 lev. ALMR, 4 lev. Mixture fraction 0.15 0.1 0.05 0 -4 -2 0 2 4 Distance from the injector axis [mm] Fig. 13. Comparison between experimental and computed mixture fraction distributions at 40 mm from the injector. close to the spray axis. The agreement for the 0.5 mm mesh is rather good even if it is slightly overestimated in the spray axis region. The 1 mm mesh seems to have a higher diffusivity with respect to the other meshes and also underestimates the maximum mixture fraction value because of the lower spray penetration. Fig. 12 illustrates the mixture fraction distribution at 30 mm from the injector. The computed results confirm the same trend obtained for the previous location, with the 0.5 and 1 mm meshes providing the closest agreement with experimental data. Finally, at the farthest distance from the injector (40 mm), it is possible to see in Fig. 13 that the best results are now provided by the 0.25 mm and the 0.5 mm meshes, while numerical diffusivity along the spray axis is the reason why the coarsest mesh size (1 mm) presents a higher vapor penetration and a narrower radial distribution. Once this detailed investigation was performed, 0.5 mm was identified as the best mesh size since it predicts rather well both the liquid and vapor penetration histories. Furthermore, this mesh provides the best description of the fuel–air mixture formation process, both at the periphery of the vapor region where auto-ignition takes place and also close to the injector where soot emissions are formed. Since auto-ignition takes place at different times depending on the operating conditions, it is of relevant interest to verify if the model is able to describe the evolution of the fuel vapor distribution during the whole simulation. For this reason, Fig. 14 compares the experimental boundary of vapor penetration with the computed fuel vapor distribution. Experimental data were extracted from a single injection event while the computed data represent the positions of the cell centroids having mixture fraction values higher than 104. The comparison was performed for six different times after SOI (0.3, 0.5, 0.7, 0.9, 1.1 and 1.3 ms) and, for all the tested cases, the model reproduces correctly the entire evolution of the fuel vapor region in terms of vapor boundary, presence of fuel close to the injector and vapor tip penetration. All these aspects are of great importance when the combustion process and the pollutant formation have to be predicted in a Diesel engine. For the 0.5 mm mesh, the influence of the computed results on the ambient diffusivity was also verified in Fig. 15. This was done by keeping the turbulence intensity constant (0.7 m/s, from experimental data) and changing the integral length by three orders of magnitude (2–200 mm). Fig. 15 shows that computed results are almost independent on the initial ambient diffusivity. This is very important because these initial values and distributions are not ex- 293 actly known at the beginning of the engine simulations, even if a detailed calculation of the gas exchange phase is performed (Van den Heuvel et al., 2004). The use of a liquid-core based atomization model smooths the interactions between the spray and the gas in the vicinity of the injector, it avoids very high gas velocities in that region and a consequent momentum diffusion, which mainly depends on the initial ambient diffusivity in the first part of the simulation. In this way, the proposed approach for Lagrangian spray modeling removes the source of uncertainty related to the initial ambient diffusivity and the spray evolution depends only on the ambient conditions (pressure, temperature), the injection strategy and the existing air flows. As a consequence of these results, the effects of the turbulence length-scale limiter cannot be really appreciated in this case, as it can be seen in Fig. 16. The same result was also obtained with the other meshes which were tested. A possible explanation is related to the use of the limiter only in the liquid-core region, which is very short (4 mm), and to the fact that there is a limited amount of momentum exchange between the spray and the gas phase. Numerical investigations performed in this work at constantvolume conditions allowed to draw the following guidelines for the simulation of the fuel–air mixture formation in Diesel engines.  The best results are provided by a mesh size which is five times greater than the nozzle diameter. This allows to predict correctly both the liquid and the vapor penetrations and the mixture fraction distribution in the domain. Such mesh size represents a rather good compromise for the correct description of the momentum exchange between the liquid and the gas phase.  For evaporating sprays, the grid dependency of the liquid penetration has limited effects on the vapor penetration. In particular, mesh sizes 2–10 times greater than the nozzle diameter provide satisfactory results in terms of vapor penetration and equivalence ratio distribution. This generally happens in IC engine simulations, where calculations are usually performed in deforming meshes and a fixed size cannot be easily used to discretize the complex geometry of the piston bowls.  When a detailed atomization model is used, the main uncertainties related to the secondary drop size and the penetration of the liquid core can be removed. This aspect certainly reduces the grid dependency of the simulations.  Further verifications have to be performed with liquid sprays whose properties are more similar to the real Diesel fuel, which is less evaporating and has a higher penetration. 5.2. Fuel–air mixture formation in an optical engine Experiments conduced in an optical engine with laser-based imaging technology were used to validate the proposed methodology for evaporating sprays (Genzale et al., 2009). The optical engine is equipped with an extended piston bowl and a flat piston crown window. Fuel is delivered by a Cummins XPI common-rail injection system through an 8-hole, 0.196 mm orifice. The engine runs at 1200 RPM and is operated at a late-injection condition, with a single fuel injection event starting at 0° after top dead center (ATDC). The injection duration is 6.75 crank angle degrees (CAD) and 56 mg of fuel were delivered (Genzale et al., 2009). To measure fuel–vapor concentrations, the engine was supplied with 100% N2 intake stream to avoid combustion. The fuel used both in experiments and simulations was a gasoline primary reference fuel (PRF) blend of 71% n-heptane and 29% iso-octane to which 1% toluene was added as a tracer to measure fuel vapor concentration. The fuel–air mixing process was studied using three different planar-laser-induced fluorescence (PLIF) diagnostics, in particular toluene-tracer fluorescence (T-PLIF) was used to measure 294 T. Lucchini et al. / International Journal of Heat and Fluid Flow 32 (2011) 285–297 30 30 Time = 0.5 ms ASOI (a) 20 Distance from the injector [mm] Distance from the injector [mm] Time = 0.3 ms ASOI 10 0 -10 -20 Exp. Calc. -30 0 20 40 10 0 -10 -20 Exp. Calc. -30 60 0 Distance from the injector [mm] Time = 0.9 ms ASOI (c) Distance from the injector [mm] Distance from the injector [mm] Time = 0.7 ms ASOI 60 10 0 -10 -20 Exp. Calc. 0 20 40 10 0 -10 -20 Exp. Calc. -30 60 (d) 20 0 Distance from the injector [mm] 20 40 60 Distance from the injector [mm] 30 30 Time = 1.1 ms ASOI Time = 1.3 ms ASOI (e) 20 Distance from the injector [mm] Distance from the injector [mm] 40 30 20 10 0 -10 -20 Exp. Calc. -30 20 Distance from the injector [mm] 30 -30 (b) 20 0 20 40 60 Distance from the injector [mm] (f) 20 10 0 -10 -20 Exp. Calc. -30 0 20 40 60 Distance from the injector [mm] Fig. 14. Comparison between experimental (black line) and computed fuel vapor distribution (blue squares) at different times after SOI: (a) 0.3 ms; (b) 0.5 ms; (c) 0.7 ms; (d) 0.9 ms; (d) 1.1 ms; (e) 1.3 ms. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) quantitative fuel–vapor concentrations. More details about the engine geometry and the simulated operating condition are provided in Table 4. Since the injector has eight holes, a 45° sector of the combustion chamber was discretized. In-cylinder conditions were initialized with experimental data of pressure and temperature at the beginning of the compression stroke, while the measured swirl ratio was used to define the starting in-cylinder velocity field according to a wheel-flow profile. To simulate compression, a coarse mesh was used and its size in the piston bowl region is similar to the one of the simulated vessel. The dynamic mesh layering technique was used during compression, to keep an acceptable mesh aspect ratio and to progressively reduce the number of cells when the cylinder volume decreases. At TDC, when injection starts, the mesh has 1600 cells with a mean 4 mm size and it is illustrated in Fig. 17. During the fuel injection process, the grid is deformed and dynamically refined according to the criterion described in Eq. (17). Two levels of refinement were used in this case, according to the results obtained at constant-volume conditions. This value was suggested by the injector diameter value and in this way it T. Lucchini et al. / International Journal of Heat and Fluid Flow 32 (2011) 285–297 295 100 Liq. and vap. penetrations [mm] Li = 200 mm, vap. Li = 200 mm, liq. 80 Li = 20 mm, vap. Li = 20 mm, liq. 60 Li = 2 mm, vap. Li = 2 mm, liq. 40 Fig. 17. Computational mesh of the optical engine at TDC (start of injection). 20 0 0 0.5 1 1.5 Time ASOI [ms] Fig. 15. Influence of the initial ambient diffusivity. 80 Liq. and vap. penetrations [mm] Vap., limiter off Liq., limiter off Vap., limiter on 60 Liq., limiter on Vap., exp. Liq., exp. 40 The computed distributions of equivalence ratio are compared with experimental data in Figs. 19 and 20 for the three different imaging planes located 7 mm, 12 mm and 18 mm below the firedeck. At 7ATDC, immediately after the end of injection, Fig. 9a shows that the proposed model correctly predicts the fuel vapor penetration and describes the formation of a rich fuel core along the spray axis. On the other two planes the agreement between computed and experimental data still remains rather good, showing the presence of a very rich mixture region close to the piston bowl walls which is typical of impinging liquid jets (Bruneaux, 2005). The influence of swirl is also visible in Fig. 9b and c where the fuel vapor distribution is not symmetric with respect to the spray axis. The results obtained for this case are satisfactory and show that the proposed approach can model the fuel/air mixture formation process in Diesel engines. The possibility to use adaptive local mesh refinement also provides a drastic reduction of the computa- 20 0 0 0.5 1 1.5 Time ASOI [ms] Fig. 16. Influence of the limiter length scale on the fuel vapor penetration. Table 4 Geometry and operating conditions of the simulated optical engine (Genzale et al., 2009). Bore Stroke Displacement Squish height Compression ratio Rail pressure Swirl ratio Bowl diameter Bowl depth Injector included angle IVC EVO p@IVC T@IVC 139 mm 152 mm 2.34 l 5.5 mm 11.2 1600 bar 0.5 97.08 mm 16 mm 152 165 ATDC 140 ATDC 2.03 bar 351 K is possible to have a minimum mesh size which is fives time higher. The spray evolution, combined with the fuel vapor distribution and the grid structure during expansion are illustrated in Fig. 18a– c: the mesh is refined only where liquid fuel and vapor exist, and in this way it is possible to accurately describe the fuel–air mixing process and to keep an acceptable mesh size. The mesh structure remains always hexahedral, and the diagonal connecting lines which are displayed are only due to the used post-processing tool. Fig. 18. Application of adaptive local mesh refinement to simulate non-reacting and evaporating Diesel sprays in an optical engine. 296 T. Lucchini et al. / International Journal of Heat and Fluid Flow 32 (2011) 285–297 50 40 Fixed mesh (1 mm) CPU Time [h] ALMR 30 20 10 0 0 5 10 15 20 25 30 Crank Angle [deg] Fig. 19. Comparison between computed (left) and experimental (right) mixture fraction distributions at seven ATDC at different distances from the cylinder head plane: (a) 7 mm; (b) 12 mm; (c) 18 mm. Range 0.2 (blue)–3.7 (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 22. Comparison between the computational time required by the 1 mm mesh and the ALMR mesh. Calculations were run on a AMD Athlon 64 X2 2200 MHz PC with 2 Gb RAM using one processor. tional time. Figs. 21 and 22 show that the number of cells required to run a calculation with ALMR are much less than what was required by a uniform 1 mm mesh which was identified as to be the best size for this case. Hence, the computational time is drastically reduced, as it is illustrated in Fig. 22 where a speed-up factor of about 10 was obtained with respect to the conventional uniform mesh. This aspect can be very important when detailed chemistry is necessary to model the combustion process since the direct integration of the chemical system will be performed in a reduced number of cells (D’Errico et al., 2009). 6. Conclusions Fig. 20. Comparison between computed (left) and experimental (right) mixture fraction distributions at 12 ATDC at different distances from the cylinder head plane: (a) 7 mm; (b) 12 mm; (c) 18 mm. Range 0.2 (blue)–1.4 (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 200 Fixed mesh (1 mm) A comprehensive model for Diesel spray calculations has been presented in this study and applied to investigate the numerical and physical interactions which govern the fuel–air mixture formation process. Specific sub-models were developed by the authors to improve the accuracy of the Eulerian–Lagrangian approach used in the simulations. An atomization model was introduced to describe the liquid jet primary breakup, while the interaction between turbulence and spray was dealt by limiting the turbulence length scale to the nozzle diameter where the liquid jet exists. Finally adaptive local mesh refinement techniques were introduced to reduce the computational time and to improve the result accuracy. The proposed approaches were extensively assessed by simulating the fuel injection and evaporation processes in constant volume experiments and in an optical engine. The following conclusions can be drawn: ALMR Mesh Cells 10 3 150 100 50 0 0 5 10 15 20 25 30 Crank Angle [deg] Fig. 21. Computational cells required to simulate fuel–air mixture formation with a constant size mesh (1 mm) and adaptive local mesh refinement.  When evaporating sprays are simulated, the effects of the grid size have to be mainly verified on the fuel vapor penetration and distribution, because these quantities play the major influence both on auto-ignition and on the mixing-controlled combustion phases.  The fuel vapor penetration and distribution show a less significant dependency on the grid than the liquid phase. In general, a mesh size which is 2–10 times the nozzle diameter provides satisfactory results. This is quite important when engines are simulated since deforming meshes are used with the cell size changing during compression and expansion.  The use of a liquid-core based atomization model reduces the dependency of the results on the ambient diffusivity, because it smooths the interaction between the liquid and the gas phases at the nozzle exit, avoiding very high gas velocities in T. Lucchini et al. / International Journal of Heat and Fluid Flow 32 (2011) 285–297 that region and a consequent momentum diffusion, which mainly depends on the initial ambient diffusivity in the first part of the simulation.  Adaptive local mesh refinement (ALMR) represents a very powerful tool for the simulation of high-pressure Diesel sprays. It drastically reduces the computational time and, once an initial coarse mesh is generated, the result accuracy can be easily controlled by changing the level of mesh refinements. 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