Numerical Issues in Droplet Collision Modeling David P. Schmidt* , University of Massachusetts-Amherst Christopher J. Rutland, University of Wisconsin-Madison
Abstract A faster, more accurate replacement for existing collision algorithms has been developed. The method, called the NTC algorithm, is not grid dependent, and is much faster than older algorithms. Calculations with sixty thousand parcels required only a few CPU minutes. However, there is a significant need to develop mesh-independent momentum coupling between the gas and spray, so that the collision algorithm’s full accuracy can be fully realized.
Introduction Droplet collisions have proven very difficult to calculate. In addition to the complex physics, droplet collision has presented several numerical difficulties. These difficulties include high computational cost and large numerical errors. In light of the large role that droplet collisions can have in determining drop size, this situation is most unfortunate. In response to the need for faster and more accurate collision calculations, a new algorithm has been developed for calculating the incidence of droplet collision. The most widely used approach currently is O'Rourke's algorithm, as implemented in Kiva (Amsden, l989). O'Rourke's algorithm has had remarkable success and is based on sound reasoning. O'Rourke's algorithm is consistent with the stochastic nature of spray simulations, where only a sub-sample of droplets is tracked. The tracked drops represent parcels of varying numbers of drops. The size, number, and velocities of parcels determine the probability of colliding with other parcels. Collision partners are chosen stochastically. Only parcels within the same gas phase cell are permitted to collide. Schmidt and Rutland (2000) showed that this approach is second order accurate in space, but fails dramatically with typical mesh resolution. The problems of mesh dependency are particularly severe when using a Cartesian mesh, as shown in Fig. 1. The grid dependency can clearly be seen in the "clover leaf" pattern that forms from what should be an axisymmetric calculation. The grid dependency also causes large quantitative fluctuations in the predicted drop size.
Figure 1. Simulation of a hollow-cone spray using the standard Kiva3V Release 2 code with collision (O'Rourke's) turned on and break -up and turbulence turned off. Simulation is done with the injector on a vertex of a Cartesian mesh. Injection is directed towards the viewer, and the drops are colored by diameter. *
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The new algorithm, first reported in Schmidt and Rutland, (2000) is based on the No Time Counter (NTC) method used in gas dynamics for Direct Simulation Monte Carlo (DSMC) calculations. However, Schmidt and Rutland rederived the algorithm from first principles so that it could be applied to sprays, where the number of droplets per parcel is variable. Like O'Rourke's algorithm, the NTC algorithm is first order accurate in time and second order accurate in space. The algorithm was tested against analytical solutions and found to converge to the exact answer. There are two significant improvements with the NTC algorithm. The first improvement is speed. The NTC algorithm achieves its results without consuming significant amounts of CPU time. For example, the calculation of collision between sixty thousand parcels requires less than three minutes of CPU time for twenty milliseconds of simulated injection duration. The second improvement is the use of a fully automatically generated collision mesh. There is no direct connection between the gas flow field and the incidence of droplet collisions. Consequently, it is possible to generate a collision mesh that is optimized for accuracy. Like O'Rourke's algorithm, The collision mesh achieves very high spatial resolution without incurring significant CPU cost. The mesh is cylindrical, oriented around the injection axis, and sized just large enough to contain all of the parcels. The mesh resolution is then set as fine as possible while maintaining a significant sample size of drops in each collision cell. This approach allows the code to achieve millimeter-level resolution of the spray while only costing the user a few CPU minutes. Because the algorithm is both convergent and consistent, the user can be assured of getting very accurate results.
Methodology The NTC method involves stochastic sub-sampling of the parcels within each drop. This results in much faster collision calculations. Unlike O'Rourke's method, which incurs a cost that increases with the square of the number of parcels, the NTC method has a linear cost. Also, O'Rourke's method assumes that multiple collisions can occur between parcels and that this process is governed by a Poisson distribution. The Poisson distribution is not correct unless collision has no consequences for the parcels. Since collision changes parcels’ velocities, size, and number, the method of repeated sampling used by the NTC method generates more accurate answers. The NTC method is derived, without assump tions, from the basic probability model for stochastic collision. The basic probability model requires that the cell size is sufficiently small that spatial variations can be neglected. For a detailed derivation, see Schmidt and Rutland (2000). Only a simple description of the actual implementation will be given here. For N parcels, there are N2 possible collision partners. O’Rourke’s scheme scans through all the possibilities and eliminates parcels that are not in the same cell. For parcels in the same cell, there is a probability of collision based on size, relative velocity, and the number of drops in the parcel. An example of twenty-one probabilities is shown in Figure 2. The cost of this method is proportional to N2 . To run with ten thousand parcels, for example, requires an effort proportional to one hundred million possible collision partners.
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Pair Index Figure 2. A plot of the chance of collision for all possible collision pairs in a given cell.
The NTC method first sorts the parcels into groups that reside in the same cell. This only requires 2N operations. Next, the NTC method picks a stochastic sub-sample from all the possible pairs in a cell. The number of picked pairs does not affect the final average answer, so long as the number meets constraints derived in Schmidt and Rutland (2000). The number of pairs in the sub-sample would be a fraction of the twenty-one pairs shown in Figure 2. The probabilities for the sub-sample pairs are multiplied by the reciprocal of this fraction, increasing the probability of collision. An example of sub-sampled pairs is shown in Figure 3. Sampling is done with replacement so that multiple collisions for a pair can be correctly calculated. The resulting method incurs a cost that is linearly proportional to the number of parcels, as opposed to the Nsquared cost of many existing methods.
Chance of collision
The other critical part of the NTC collision algorithm is the use of a collision mesh. O’Rourke’s collision model is griddependent because of a lack of spatial resolution. The incidence of collision is calculated using the gas-phase mesh, which is not sufficiently refined for collision calculations. However, there is no requirement to use the gas-phase mesh for collision calculations. The NTC implementation automatically creates a cylindrical mesh around the axis of injection. The mesh resolution is set as fine as possible, while keeping a sufficient sample size in the cells near the spray center. The more parcels that are used in the calculation, the finer the collision mesh. A sketch of a very coarse example is shown in Figure 4. In practice, the collision mesh resolution is on the order of ten microns. The extent of the mesh is just sufficient to include the entire spray. The special mesh is essential, because it provides high-resolution of the spray for calculating collision.
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Figure 3. A sub-sample of possible collision partners drawn from the example in Fig. 2. The probabilities have been multiplied by the number of possible pairs divided by the sub-sample size.
Figure 4. A coarse collision mesh, showing its orientation around the spray.
Results The NTC method has been tested against several analytical solutions for particle collision. The scheme was shown to be first order accurate in time and second order accurate in space. For example, the code was tested using the spatially varying problem of a planar spray oriented along the x-axis. The particles were initialized using known probability density functions for position, velocity, number, and diameter. A picture of this test problem is shown in Figure 5. The spatial accuracy of the algorithm was measured by forcing the collision mesh to be increasingly fine. The error was defined as the difference between the analytical solution and the prediction of the NTC algorithm. Note that with a very fine mesh, the error is less than 1%.
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Figure 5. A test case used to determine the accuracy of the NTC method. Parcels are distributed according to prescribed probabilities so that an analytical solution for the incidence of collision is possible. Note the strong spatial variation in the parcel density.
Figure 6. The spatial accuracy of the NTC method is second order. With twenty-five cells in the collision mesh in the y direction, the error is less than 1%.
Next, the NTC algorithm was installed in Kiva. First, a diesel-type spray was simulated in Kiva with three different meshes. The domain was a cylinder with a diameter of 5cm and a height of 20cm. The cylinder was divided into Cartesian meshes, with the resolutions shown in Table 1. All the optional models were turned off, with the exception of collision. The test case is not intended to generate physically reasonable results, but to show clearly the numerical behavior of the collision model. As a baseline, the original Kiva3V Release 2 code was run with eight thousand parcels, and the results are shown in Fig. 7. The average drop size is very sensitive to the mesh resolution. The predicted drop size increased by 60% from the coarsest mesh to the finest. This difference of 40 microns is unacceptable for a modern CFD code. Next, the NTC collision model was used, with the results shown in Fig. 8. The predicted mesh size is much less mesh dependent.
Coarse Medium Fine
Cells in X Direction 15 30 45
Cells in Y Direction 15 30 45
Cells in Z direction 20 100 150
Total Cells 4,500 90,000 303,750
Table 1. A description of the meshes used in the tests shown below. The results shown in Figure 8 are a significant improvement over the original Kiva collision algorithm. The results are much less grid-dependent and were generated quickly. The total CPU time required on a desktop PC for the NTC algorithm was about 45 seconds compared to 1,277 seconds for O’Rourke’s method. However, the results could be better. There is still some grid dependency in the NTC results. This grid dependence is not directly a fault of the NTC method, since the NTC algorithm does not rely on the gas phase mesh in any way. The mesh dependence is, instead, an indirect consequence of the mesh sensitivity of the coupling between the spray and gas phase momentum equation. The corrupting influence of the momentum and drag coupling is clearly demonstrated by imposing a velocity of zero on the gas phase. The velocity updates in Kiva were temporarily turned off, so that the code would run with a zero gas phase velocity. This trivial gas phase flowfield was chosen because it is grid independent. Another important feature is that there will be no interpolation error of the local gas velocity to the particle position, because the zero velocity flowfield is perfectly uniform. As the results in Figure 9 show, the NTC method becomes perfectly grid independent. This result is expected, since the gas phase mesh is now completely unconnected from the calculation. However, when the same test is applied using O’Rourke’s collision model, the final drop size varies from 32 to 39 microns, showing moderate grid dependency. As a final test of the NTC algorithm a calculation was performed with varying numbers of parcels. Because the mesh automatically sets its refinement based on the number of parcels, this test simultaneously shows the affect of number of parcels and collision mesh resolution. As can be seen in Figure 10, the predicted drop size is not significantly mesh dependent or sensitive to the number of parcels.
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Figure 7. Calculated average drop size for a diesel-type spray using Kiva3V Release 2 with three different Cartesian meshes. This figure shows the grid dependency of O’Rourke’s collision model. The evaporation, breakup and turbulence models were turned off, and collision was turned on.
Figure 9. The prediction of the NTC collision algorithm when the gas velocity is constrained to zero. The results are perfectly grid independent.
Figure 8. The same test as shown in Figure 7, but with the NTC collision algorithm. Note the reduced mesh dependency.
Figure 10. The prediction of the NTC collision algorithm when the gas velocity is constrained to zero. The results show the effect of simultaneously changing the number of parcels used and the collision mesh resolution.
In order to see the qualitative impact of the NTC algorithm on the structure of hollow cone spray computations, the simulation in Fig. 1 was repeated using the NTC method. The results with the NTC method are shown in Fig. 11. The NTC method has significantly improved the simulation of the hollow-cone spray. The view shows that the spray is basically axisymmetric, unlike the clover-leaf pattern that formed in Fig 1. There are still some visible grid-dependent artifacts due to the discretization error in the momentum coupling between the gas phase and droplets. The figure shows that the parcels prefer to follow grid lines, because the direction of the gas phase velocity is perfectly aligned with parcel velocity. On offaxis trajectories, this alignment is impossible due to the Cartesian mesh. With proper interpolation of gas-phase velocities to the parcel locations, this grid dependency would be reduced.
Figure 11. Simulation of a hollow-cone spray using the standard Kiva3V Release 2 code with the NTC collision algorithm turned on and break -up and turbulence turned off. Simulation is done with the injector on a vertex of a Cartesian mesh. Injection is directed towards the viewer, and the drops are colored by diameter.
Conclusions The NTC collision model has been demonstrated as a faster, mo re accurate successor to current collision algorithms. The method is so fast, that the collision calculation shown in Figure 10 performed with 60,000 parcels took only about two minutes on a desktop computer. As another benefit, the NTC method shows only minimal mesh sensitivity. This small grid dependence is actually not a limitation of the collision algorithm, but is due to the limitations of the gas phase solution. If the gas phase solution were grid-independent, then the collision results would also be grid-independent. The NTC collision algorithm would be a good match with advanced interpolation schemes and momentum coupling techniques, such as Béard et al. (2000). The combination of accurate collision calculations and accurate coupling between the phases will produce results with much less grid dependence.
Acknowledgements We thank Renault SA, who sponsored the original development of the NTC algorithm. We also thank the Convergent Thinking LLC consulting firm, who leant advice and computer resources for the Kiva calculations.
References Amsden, A. A., “Kiva-II: A Computer Program for Chemically Reactive Flows with Sprays,” Los Alamos Report LA -11560MS, May, 1989. Béard, P., J-M Duclos, C. Habchi, G. Bruneaux, K. Mokaddem, and T. Baritaud, “Extension of Lagrangian-Eulerian Spray Modeling: Application to High Pressure Evaporating Diesel Sprays,” SAE Paper 2000-01-1893, 2000. O’Rourke, P.J., Collective Drop Effects on Vaporizing Liquid Sprays, Ph.D. Thesis, Department of Mechanical and Aerospace Engineering, Princeton University, 1981. Schmidt, David P., and C. J. Rutland, “A New Droplet Collision Algorithm,” Journal of Computational Physics, v. 164, pp. 62-80, 2000.
