A Generalized Adaptive Collision Mesh for Multiple Injector Orifices Shuhai Hou, Sasanka Are, David P. Schmidt University of Massachusetts-Amherst ABSTRACT An algorithm for creating a generalized adaptive collision mesh has been proposed. Contrary to the common practice of using the fixed gas phase mesh to calculate droplet collision incidence, the proposed algorithm generates a collision mesh independent of the gas phase mesh. The collision mesh adaptively encompasses all the droplets in the computational domain at each time step. Schmidt and Rutland used such a mesh, but it was limited to a single spray. A new, more general algorithm has been developed. Calculations of multi-hole injectors have been performed to show that the mesh is capable of producing higher spatial resolution than the fixed gas phase mesh without incurring significant CPU cost. At the same time, the mesh algorithm maintains an adequate statistical sample of parcels in the collision cells. The continual random rotation of the mesh prevents collision from causing grid dependency in the spray shape. In addition to common multi-hole diesel injectors, the mesh algorithm is especially useful in simulating MEMS nozzles that are characterized by many closely spaced orifices. Due to the extremely small spacing between the orifices, inter-spray collisions are expected to play a significant role on spray behavior, requiring accurate calculation of droplet collision. INTRODUCTION It is intuitive to use the fixed gas phase mesh to calculate droplet collision. However, this approach suffers from two important weaknesses. First, the gas phase mesh is usually too coarse to evaluate collisions in the vicinity of the injector, where the droplet number density and collision frequency are very high. The gas phase mesh tends to produce large spatial discretization errors and under-resolves this dense spray area, aggravating grid dependency. Secondly, grid dependency is particularly severe when using a Cartesian mesh, as shown in Figure 1. The computed spray shape can be turned into a four-leafed clover by the grid dependency [1]. Since only parcels located in the same cell are allowed to collide, and the incidence of collision is proportional to the relative velocity between the pair parcels, the parcels whose trajectories are 90 degrees apart are most likely to collide because they have the greatest relative velocity. Parcels very close to each other, but in different quadrants, do not collide because their trajectories are almost aligned with the cell boundaries and more likely not in the same cell. Figure 1 A solid-cone spray directed towards the viewer. Figure 2 A cylindrical collision mesh superimposed on a spray. A Calculation made on a Cartesian mesh. Cartesian mesh gas phase mesh is partially cut away for visibility. This type of mesh has been used in past modeling of a single spray [1]. GENERALIZED ADAPTIVE COLLISION MESH In addressing the above shortcomings, Schmidt and Rutland took advantage of the fact that droplet collisions need not be directly dependent on the gas phase mesh [1]. They proposed a method for creating a collision mesh independent of the gas
phase. An example is shown in Figure 2. The collision mesh was based on a polar coordinate system aligned with the injection axis. The mesh would automatically form around an injection with any orientation. The angular orientation was chosen randomly every time step to produce a stochastic mesh. This method significantly improved spatial resolution and reduced grid dependency. It was successfully applied to spray bomb and engine simulations. However, this method is only applicable for a single spray because for multiple orifice nozzle sprays there is no single injection axis. Inspired by Schmidt and Rutland s ideas, an algorithm for creating a generalized, adaptive collision mesh is proposed. The collision mesh is independent of the gas phase mesh and created automatically at each time step. The new method can be used in multiple sprays with any configuration of orifices. It is based on the following steps: 1. Rotate the x, y, z axes by three random angles to generate a new coordinate system. The collision mesh will be oriented with this random coordinate system. This step is important to suppress the artifacts caused by Cartesian mesh. The randomness ensures that the collision mesh will be entirely different every time step. 2. Create a regular, three-dimensional, hexahedron encompassing all the parcels. 3. Check if the number of parcels in the hexahedron is below the maximum allowed number. If so, the mesh is completed. If not, continue to the next step. The maximum allowed number is used to make sure that a statistically adequate sample of parcels is used for the collision cells, and at the same time to control the resolution. 4. Halve the hexahedron in the x, y and z directions to make eight smaller hexahedra. 5. Check each hexahedron to see if the number of parcels in it is below the maximum allowed number. If so for all the interior hexahedra, the mesh refinement is completed. If not, repeat steps 4 and 5 for all the hexahedra in which the number of parcels exceeds the maximum allowed number. Though the creation of the collision mesh requires significant programming effort, its CPU cost is negligible. Typical runs show the collision meshing takes less than 0.1% of the total CPU time. Together with the NTC method, droplet collision represents less than 7% of the total CPU cost of the code when using about fifty thousand parcels. Figure 3 shows a picture of the collision mesh. It can be seen that the collision cells are smaller in the denser areas. Figure 4 shows its effectiveness in avoiding the grid dependency caused by a Cartesian mesh and traditional collision algorithm. Due to the stochastic nature of the mesh, it is entirely different every time step. Grid artifacts, which appear when the parcels are artificially forbidden to collide with near-neighbors, are prevented. Instead, the stochastic grouping means the parcels that are close to each other are likely to end up in the same collision cell during most time steps. The value of this approach is that the collision mesh is not tied to any one geometric configuration. Figures 5a and 5b show how the collision mesh can also be applied to a more typical spray configuration from a multi-hole diesel fuel injector. The mesh forms around any number of sprays in any orientation. Figure 3 A picture of the generalized adaptive collision mesh. Figure 4 Same case as in Figure 1, but with the generalized adaptive collision mesh.
Figure 5a and 5b Six sprays from a traditional orifice configuration are shown at the left. On the right, the collision mesh at one moment in time is shown superimposed around the spray. Part of the collision mesh and spray are cut away for clarity. NUMERICAL TESTS OF THE GENERALIZED ADAPTIVE COLLISION MESH In order to verify the characteristics of the generalized adaptive collision mesh, calculations were applied to a 5-hole MEMS nozzle sprays under typical diesel conditions. A three-dimensional, transient, CFD code named MOSES (MOdifiable Source Engine Simulation) was used for the present calculations. MOSES is currently under development by Convergent Thinking LLC in collaboration with the University of Massachusetts-Amherst. Numerical Schemes and Models A finite difference approach with a regular block-structured mesh is employed. With minor variations, the PISO method of Issa is used for solving the gas phase conservation equations for compressible flow [2]. All spatial derivatives are second-order central-difference, with the exception of the convection terms. For convection, since the main goal is to investigate the collision mesh, first-order upwinding is used for simplicity and efficiency. Turbulence is represented using the RNG k-ε model. All boundary conditions are no-slip walls. A semi-implicit Euler method is used for solving the ODE governing parcel trajectory. This method is exact with a constant ambient gas field. A predictor-corrector scheme is used to account for the variable gas velocity and density, with distinct time steps for individual parcels to control the spatial accuracy of the scheme. Parcels are sub-stepped with a time step approximately equal to the time required to traverse a cell. Tri-linear interpolation is used for gas to liquid coupling. Sirignano s interpolation method is used for liquid to gas coupling [3]. This method is second order accurate in space. The Enhanced Taylor Analogy Breakup model is used to model jet and droplet breakup [4]. For droplet turbulent dispersion, a time-correlated stochastic droplet dispersion model based on the k-ε model is used [5]. The NTC collision algorithm is used to calculate collision incidence [1]. The NTC algorithm is a popular method in calculating molecular collisions in gas dynamics. This algorithm has a computational cost linearly proportional to the total number of parcels in the cell. However, it had only been applied to the cases where the number of molecules in each parcel was constant. Schmidt and Rutland extended it to spray droplet collisions where each parcel contains varying number of droplets. Like O Rourke s collision algorithm, the NTC algorithm is first order accurate in time and second order accurate in space. Unlike O Rourke s collision algorithm, in which all the possible collision pairs are considered, the NTC algorithm only considers a randomly chosen subset of candidate pairs after scaling up the collision probability by the estimated maximum probability, reducing the computational cost. Thus more computational parcels can be used, allocating statistically enough parcels for each orifice in the MEMS nozzles. Detailed derivation and testing of the NTC algorithm can be found in reference [1]. O Rourke s model for collision outcomes is used to calculate the outcome of collision [6]. Post and Abraham have demonstrated the need for maintaining an adequate number of parcels per cell in collision calculation [7]. In the extreme, if there are less than two parcels in a cell, collision is totally suppressed. In order to observe this effect with the NTC algorithm, a spatially homogenous test case was run using the three-dimensional demonstration code of Garcia [8]. This is a freely-available program for demonstrating collision calculations. The number of parcels per cell was varied to see how this changed the predicted number of collisions over a short time interval. It was verified that the NTC collision algorithm is sufficiently accurate when at least five parcels are in a cell.
Computational Conditions Calculations were made on the 5-orifice MEMS nozzles as used by S. Baik, et al.[9]. Table 1 shows the computational conditions. Computational domain Gas phase mesh resolution 37 37 120mm 1 1 1 mm Parcels used 100,000 Table 1 Computational conditions. RESULTS AND DISCUSSION The results and features of the generalized adaptive collision mesh are presented below. For the sake of clarity, three types of plots regarding the mesh are shown. One is Fraction of Cells versus Average Cell Size, where the Fraction of Cells refers to the percentage of the cells with size less than the indexed values. One is Fraction of Parcels versus Average Cell Size, where the Fraction of Parcels refers to the percentage of the parcels located in the cells with size less than the indexed values. One is Fraction of Parcels versus Number of Parcels in Cells, where Fraction of Parcels refers to the percentage of the parcels located in the cells in which the number of parcels is lager than the indexed values. The average cell size is calculated from the cube root of the cell volume. The term ASI refers to After Start of Injection. Figures 6, 7 and 8 show the results at 0.4ms ASI with the maximum allowed number of parcels (as used in the algorithm description in Step 3) equal to 160. It can be seen from Figure 6 that over 80% of the cells have average size less than 1mm, and over 50% less than 0.46mm. Also it is shown in Figure 7 that over 90% of the parcels are located in the cells with size less than 1mm, and almost 60% in less than 0.46mm. It is clear that the collision mesh produces better resolution than the gas phase mesh does, and thus is expected to produce higher spatial accuracy and to reduce grid dependency. Figure 8 verifies that the present approach maintains an adequate statistical sample size. Almost all parcels are located in the cells with more than five parcels in them, providing a minimum acceptable sample size for the stochastic collision calculation. Figure 6 Distribution of cells about cell size at 0.4ms ASI Figure 7 Distribution of parcels about cell size at 0.4ms ASI. Figures 9, 10 and 11 show the same plots for time at 3.0ms ASI. As we may expect, the spray gets more spread out and dilute, requiring fewer small cells. Figure 9 shows that over 55% of the cells have average size less than 1mm. Figure 10 shows that over 45% of the parcels are located in the cells with size less than 1mm. It is interesting to note that the distribution in Figure 11 differs little from that in Figure 8. In both cases, the vast majority of cells contain many more than five parcels, the minimum acceptable sample size. This observation suggests that the maximum allowed number of parcels could be decreased to get better resolution, still yet maintaining the minimum acceptable sample size in cells. Further calculations have shown that 100 is an optimal value of the maximum number of parcels for Step 3 of the algorithm.
Figure 8 Distribution of parcels about sample size at 0.4ms ASI. Figure 9 Distribution of cells about cell size at 3.0ms ASI. Figure 10 Distribution of parcels about cell size at 3.0ms ASI. Figure 11 Distribution of parcels about sample size at 3.0ms ASI CONCLUSIONS The proposed algorithm of a generalized adaptive collision mesh is demonstrated to achieve good spatial resolution, and at the same time assure statistically adequate sample size. The optimum value of the maximum number of parcels in a cell required by the algorithm is approximately 100. The computational cost was very modest; the liquid phase calculation consumed less then seven percent of the CPU time. The generalized adaptive collision mesh, together with the NTC collision algorithm, provides a useful technique to study the droplet collisions with multiple injector orifices. ACKNOWLEDGEMENTS This work is funded by Caterpillar Inc. and General Motors. Inc. The authors thank Prof. Franz X. Tanner for generously sharing the ETAB source code. They also thank Eric Pomraning and Keith Richards of Convergent Thinking LLC for the use of their gas phase solver. REFERENCES 1. David P. Schmidt, C. J. Rutland, A New Droplet Collision Algorithm, Journal of Computational Physics, v. 164, 2000. 2. R. I. Issa, Solution of the Implicitly Discretised Fluid Flow Equations by Operator-Splitting, Journal of Computational Physics, v. 62, 1985. 3. William A. Sirignano, Fluid Dynamics and Transport of Droplets and Sprays, p.p.182-183, Cambridge University Press, 1999. 4. F. X. Tanner, Liquid Jet Atomization and Droplet Breakup Modeling of Non-Evaporating Diesel Fuel Sprays, SAE 1997 Transactions: Journal of Engines, Vol. 106, Sec. 3, p.p.127-140, 1998. 5. Q. Zhou, M. A. Leschziner, A Lagrangian Particle Dispersion Model Based On A Time-Correlated Stochastic Approach, FED-Vol. 121, Gas-Solid Flows, ASME 1991.
6. P. J. O Rourke, Collective Drop Effects on Vaporizing Liquid Sprays, Ph.D. thesis, Department of Mechanical and Aerospace Engineering, Princeton University, 1981. 7. S. Post and J. Abraham, Modeling Collisions Among Drops with a Non-Uniform Spatial Distribution, ILASS America 2002, Madison, WI. 8. Alejandro L. Garcia, Numerical Methods for Physics, 2 nd ed. Prentice-Hall, 2000. 9. Seunghyun Baik, James P. Blanchard, Michael L. Corradini, Development of Micro-Diesel Injector Nozzles via MEMS Technology and Effects on Spray Characteristics, Ph.D. thesis, Engine Research Center and Engineering Physics, University of Wisconsin Madison, 2001.