Validation of an Automatic Mesh Generation Technique in Engine Simulations

Transcription

1 International Multidimensional Engine Modeling User's Group Meeting April,, Detroit, Michigan Validation of an Automatic Mesh Generation Technique in Engine s Abstract Long Liang, Anthony Shelburn, Cheng Wang, Devin Hodgson, Ellen Meeks Reaction Design FORTÉ CFD software is designed for multidimensional reactive flow simulations, especially for engine applications. Such applications intrinsically contain moving boundaries and often contain complex geometries, such as poppet valves. This makes mesh generation a challenging task. The latest FORTÉ software features automatic mesh generation that is based on an immersed boundary method. With this technique, computational meshes are generated on the fly based on the instantaneous shape of the computational domain. Dynamic mesh refinement can be used to create higher mesh resolution at system boundaries or in regions of interest at certain periods during the simulation. The flow and physical-model equations are solved using a perfectly orthogonal Cartesian grid in the interior of the computational domain. Special algorithms have been developed to handle the solution in immersed-boundary cells at open boundaries, as well as static or moving wall boundaries. This paper presents the validation of this automatic meshing technique in both diesel engine and spark ignition engine simulations.. Introduction The numerical method used for solving the basic fluid dynamics equations in FORTÉ is similar to that used by KIVA-V []. Specifically, the state variables and velocities are discretized using a staggered scheme; the solution procedure follows the ALE (Arbitrary Lagrangian Eulerian) method; the solution of the pressure-linked equations is patterned after the SIMPLE method; and the convection terms are solved using a quasi-second-order upwind differencing method. The need to generate a body-fitted, block-structured mesh has been a major limitation of the KIVA-V code when applied to complex engine geometries. Not only is the initial mesh creation time-consuming and expertise-intensive, but determining a robust meshmoving control algorithm often requires tedious trial-and-error iterations. In addition, highly distorted cells can introduce inaccuracy and convergence issues into the numerical solution. In FORTÉ, we have developed an automatic mesh generation technique that is based on immersed boundary method and is compatible with the staggered discretization scheme. The automatic meshing technique in FORTÉ starts with a CAD-generated geometry file (such as a file in STL format), which defines the physical boundary surfaces of the computational domain. This imported geometry serves as the starting surface mesh, which can be transformed using several basic geometrymanipulation functions. From this surface mesh, a Cartesian computational mesh is created to cover the physical domain, based on user-specified cell size constraints. Different cell sizes can be specified at different boundary surfaces to provide appropriate resolution for local geometry structures. Subsequently, the Cartesian cells whose centers lie outside of the physical boundary will be eliminated from the Cartesian mesh. The resulting mesh serves as the background framework of the flow solution. In cases that have moving boundaries, the surface mesh (which represents the physical boundary of the domain) is adjusted according to the moving surface velocities for each timestep. Correspondingly, cells are activated or deactivated in the Cartesian mesh depending on how the Cartesian mesh intersects the physical boundary. The general methodology and preliminary results of using this automatic meshing technique have been reported previously []. In this work we discuss several important technical aspects and verification results. The technical topics to be described include ) formulation of the boundary control volume for maintaining mass and energy conservation; ) application of forcing term at moving wall boundaries; and ) mesh refinement capability. These improvements will then be demonstrated by several unit tests and engine simulations.

2 . Methodology The approaches to handling immersed boundary cells of a Cartesian grid can be largely classified into two groups in the literature: cut-cell approach and ghost-cell approach [-6]. In a cut-cell approach, the Cartesian boundary cells are reshaped to fit the local geometry, and the flow solution numerical algorithms are modified to handle the irregular shaped cut cell faces and volumes. The cut-cell approach tries to reconstruct the physical computational domain for each timestep and thus it tracks the mass and energy contained in the physical domain accurately. However, implementation of an exact cut-cell approach for D problems is complicated and the cell reshaping may result in tiny cells. It also imposes stringent requirement for the quality of the STL surface mesh. In a ghost-cell approach, the boundary cells keep their hexahedron shape. The variables at the ghost cell centers or ghost vertices are defined through interpolation. Thus the flow equations are solved on the Cartesian grid directly. In this approach, the variable interpolation must be precise enough to maintain strict mass and energy conservation inside the system boundary. The immersed boundary method used in FORTÉ combines features of both ghost-cell and cut-cell approaches. Specifically, the momentum equation is solved on the Cartesian grid directly. Since the momentum equation involves only intensive variables (such as pressure, gas density, specific energy) instead of extensive variables (mass and total energy) and there is no mass crossing cell or system boundaries during this step, it is not critical to track the location of the system boundary precisely and solve the equation on reshaped cut cells. However, when solving terms that involve mass or energy transfer across the system boundary, the cut-cell approach is employed. Next, we briefly discuss a few technical topics involved in this hybrid approach used in FORTÉ.. Formulation of boundary control volumes In FORTÉ, the Cartesian cells whose cell centers lie inside the system boundary are defined as fluid cells, and the fluid cell domain is wrapped by a layer of ghost cells. When the flow solution involves mass or energy transfer across the system boundaries or cell boundaries, a cut-cell boundary control volume is formulated and tracked for each boundary cell. This is done by splitting the physical volume contained in ghost cells into their neighboring fluid cells. In this way, the total volume of the system can be recovered by the cut-cell control volumes of all the boundary cells plus the volumes of the interior fluid cells. In the example shown in Fig., the cut-cell control g Interior Fluid 4 g Figure. Cut-cell boundary control volumes volumes for fluid cells -6 correspond to the respective volumes encompassed by the red segments. The physical volume contained in ghost cell g is shared by the control volumes of cells and. Similarly, the physical boundary area contained in the ghost cells are also shared by their neighboring fluid cells. Using the wall heat transfer calculation as an example, the heat transfer amount across each cut-cell control volume is calculated as the local heat flux times the physical wall area contained in the control volume. Then this delta of energy is added to the total energy contained in the control volume (instead of the Cartesian boundary cell). This in turn changes the specific internal energy of the gas inside this control volume. The calculation of mass and energy exchange between a vaporizing spray particle and the gas in a boundary cell is another example.. Forcing term at moving wall boundary At moving wall boundaries, the transport equations are solved using the ALE procedure. During the Lagrangian stage, the fluid cells float with the fluid and consequently change their location, shape and volume. The moving wall cell faces of the Cartesian grid also move to a new location according to the moving velocity of the wall boundary. The motion of the moving walls imposes a pushing or pulling force on the interior fluid. In the method used by FORTÉ, the Cartesian boundary cells at moving wall boundary will resume their hexahedral shape after each time step, meaning that the wall boundary will always impose the force at the Cartesian wall boundary. The purpose for doing this is to avoid changing the boundary cell volumes continuously and thus greatly simplify the numerical algorithm of the ALE procedure. However, since the pushing or pulling force is not applied at the exact system boundary location, the force (and consequently the moving velocity) of the Cartesian moving cell faces must be adjusted continuously in order 5 6

3 for the interior fluid to feel the precise compression or expansion effect due to wall motion. This moving velocity correction will be demonstrated using a unit compression-expansion test case below.. Mesh refinement The Cartesian mesh in FORTÉ follows an octree structure, in which all cells are perfect cubes. When a cell is refined, it will be split into eight smaller cubes evenly. Engine geometries inevitably contain small structures, such as the crevice between valve seats and the cylinder head. Such small fluid passages have to be resolved using very small cells, and it is impractical to use such small cells throughout the computational domain. This makes mesh refinement a necessary capability. In FORTÉ, mesh refinement can be applied on boundary surfaces or in specified regions. Different cell sizes can be specified for different refinement controls, and the refinement can be applied dynamically for certain time or crank angle intervals. As mentioned earlier, FORTÉ uses the staggered discretization scheme in which velocities are defined at vertices. To handle a refined mesh, the momentum control volumes of "hanging" nodes have been properly formulated to avoid "checker-board" patterns of velocity and pressure, which are often seen in CFD applications. In solving the momentum equation, pressure gradient are applied at the corners of each momentum control volume. Thus, pressures need to be interpolated at certain locations inside a bigger cell. Solving the momentum equation on a staggered and refined mesh is more complicated than doing so on a collocated mesh (in which both state variables and velocities are defined at cell centers), but the use of a staggered scheme can avoid the "checker-board" pattern without resorting to any numerical damping (called node coupling) method. The refinement capability will be demonstrated using a flow-through-pipe test case.. Validation Using Unit Tests. Heat transfer in a compression-expansion test The compression and expansion of gas mixture inside the cylinder is a fundamental process in engine operations. In a previous paper [], we demonstrated the effectiveness of formulating cut-cell boundary control volumes by simulating a wall heat transfer problem for a constant volume vessel. In the present work, wall heat transfer is tested in a compression-expansion environment. This test can help validate the technical treatment discussed in sections. and.. The geometry used is a simple cylinder (bore diameter = 6 cm, stroke = 7.5 cm, squish height =.5 cm), which gives a compression ratio (CR) 6:. The calculation is carried out from -5 ATDC to 5 ATDC at 5 RPM. The initial temperature and pressure are 4 K and bar, respectively, and the wall temperature is K. The initial swirl ratio is.5. The results using immersed boundary are benchmarked against results generated using a KIVAV compatible body-fitted mesh. The average grid size of the body-fitted case is cm; the global grid size of the immersed boundary mesh is cm, and the wall boundaries are refined using / size cells. In Fig., the histories of in-cylinder averaged pressures and heat transfer rates are compared between the immersed boundary case and body-fitted case. Very good agreements have been obtained for both parameters. Figure shows the temperature contours on a horizontal cut plane (z=7.5 cm) at - ATDC during the compression. The contours show very similar pattern of temperature stratification. It is worth noting that the immersed boundary method does a better job because the temperature distribution is more axisymmetric. By contrast, the body-fitted case shows some non-axisymmetric pattern due to cell-size variation along the boundary Body-fitted Auto Mesh Wall Heat Transfer Rate (J/deg).6.4. Body-fitted Auto Mesh (a) In-cylinder pressures (b) Wall heat transfer rates Figure Pressure and wall heat transfer rate curves of body-fitted and auto mesh calculations

4 (a) Auto Mesh (b) Body-fitted Figure Computed temperature contours on a horizontal cut plane at z = 7.5 cm and - ATDC. Mesh refinement tests A simple flow-through-square-pipe problem was used to test the accuracy of mesh refinement treatment. The width, height, and length of the square pipe are 5cm 5cm cm, respectively. The geometry is tilted by a 45 angle relative to the horizontal location. The lower left face is the inlet and the upper right face is the outlet. A fixed-velocity inlet boundary condition and an outflow outlet boundary condition are specified. The inlet velocity is m/s and its direction is normal to the inlet face. Freely slipping and adiabatic wall boundary conditions are specified at the pipe walls. In this test, we should expect to see uniform velocities at any cross-sectional plane of the pipe. Two refinement arrangements are tested, both using global grid size =.5 cm. In the first test, a spherical region at the center of the pipe is refined by two levels. In the second test, all the boundaries are refined using / size cells. Figures 4 and 5 show that uniform and smooth velocity patterns could be preserved using both refinement arrangements. This demonstrates the effectiveness of the refinement treatment. (a) t = ms (b) t =.5 ms (c) t = 4 ms Figure 4 Computed velocity contours on a vertical cut plane using interior refinement (a) t = ms (b) t =.5 ms (c) t = 4 ms Figure 5 Computed velocity contours on a vertical cut plane using boundary refinement 4

5 4. Validation in Engine s 4. Diesel engine In a validation test that simulated a diesel engine, only the in-cylinder spray and combustion processes were simulated. The engine modeled is a single-cylinder direct injection optical engine operating at low temperature combustion conditions (CR=6:; bore stroke = ; injection timing = -.5 ATDC) [7]. The simulation is carried out from IVC (-65 ATDC) to EVO (5 ATDC). The global grid size is mm, and the cells along the wall boundaries are refined by / size. The spray breakup models employ gas-jet theory to achieve results that are independent of grid resolution [8-9]. A 5-species n-heptane mechanism is used for chemistry. Using the default values for all model constants, the simulated pressure matches the measured pressure very well, as shown in Fig. 6. Figure 7 shows the simulated spray structures and temperature contours on two cut-planes at selected crank angle locations through the spray and ignition processes Apparent Heat Release Rate (J/deg) Figure 6 In-cylinder pressures and heat release rates of the diesel engine case (a) -7 ATDC (b) - ATDC (c) - 5 ATDC Figure 7 Computed spray structures and temperature contours on cut planes 4. Spark ignition engine A four-valve Ford engine operating in PFI (port fuel injection) mode is simulated to validate the flow and spark ignition and combustion models in the immersed boundary framework. These calculations include the whole engine cycle starting from EVO (6 ATDC). Specifications of the engine and operating conditions include: CR=:; bore stroke = 8.9 cm 7.95 cm; RPM = 5; manifold pressure =.65 bar []. Conditions with a spark timing sweep are modeled. The global grid size of the immersed boundary meshes is set to mm. Dynamic refinement controls are applied at boundary surfaces. For example, cells at the valve surfaces are refined to /4 size when the valves are lifted. The DPIK (discrete particle ignition kernel) model is used to track the flame location of the ignition kernel and the G-equation model is used to simulate the subsequent flame propagation []. Figure 8 shows computed temperature contours on a vertical cut plane across one pair of valves during the exhaust and intake strokes. The four crank angle locations correspond to exhaust valve opening, valve overlap, intake valve opening, and intake valve closing, respectively. Figure 9 shows the temperature contours on a vertical cut plane across the spark plug during ignition and flame propagation. The ignition kernel flame particles are shown on the Fig. 9(a). It can be seen in Fig. 9(c) that /4 level refinement is used at the boundaries in order to resolve the narrow squish region. 5

6 (a) CA = ATDC (b) CA = 4 ATDC (c) CA = 5 ATDC (d) CA = 6 ATDC Figure 8 Computed temperature contours on a vertical cut plane (a) CA = 69 ATDC (b) CA = 7 ATDC (c) CA = 75 ATDC (d) CA = 76 ATDC Figure 9 Temperature contours during spark ignition and flame propagation (spark timing = 68 ATDC) Simulated in-cylinder pressures are compared to the measured data [] in Fig.. The agreement is very good during the exhaust, intake and early compression strokes. Reasonable agreements are also obtained for the ignition and combustion stages and expansion strokes. The trend of peak pressure as function of spark timing is well captured. The normalized NOx emissions are compared between simulation and experiment in Fig.. Again, the trend of NOx as function of spark timing is very well reproduced in the simulation. 6

7 .5 Spark Timing = 676 ATDC.5 Spark Timing = 68 ATDC Spark Timing = 684 ATDC.5 Spark Timing = 688 ATDC NOx (ppm) normalized to Maximum Figure Computed and measured in-cylinder pressures Spark Timing (CA ATDC) (Normalized) Model (Normalized) Figure Computed and measured NOx emissions at EVO Summary An automatic mesh generation technique based on immersed-boundary theory is described. Several key technical aspects of the implementation are discussed. The technique is validated using a few unit tests and both diesel and spark ignition engine simulations. The validation shows that the immersed-boundary method is an effective technique for achieving accuracy in engine CFD simulations without the burden of manual mesh generation. References. A.A. Amsden, et al. (989) KIVA-II manual, LA-56-MS.. L. Liang, et al. () st International Multidimensional Engine Modeling User's Group Meeting.. R. Mittal, et al. (5) Annu. Rev. Fluid Mech., 7: T. Ye, et al. (999) J. Comp. Phys., 56: Y.-H. Tseng, et al. () J. Comp. Phys., 9: J. Yang, et al. (6) J. Comp. Phys., 5: M.P.B. Musculus, (4) SAE N. Abani et al. (8) SAE Y. Wang et al. () SAE R. H. Muñoz, et al. (5) SAE L. Liang, et al. (7) SAE