Based on the hydroplaning profile shown in Figure 3.1 and the solver algorithms described in the prior sections, it is first sought if a two-dimensional approach of a typical hydroplaning region is representative of the entire hydroplaning phenomenon.
3.7.1 Geometry of Model
The geometry of the three-dimensional model, which the proposed two-dimensional form is based on, is as shown in Figure 3.5. The tire deformation model shown in the bottom
verification of his two-dimensional model.
Studies conducted by Dreher and Horne (1966) indicated that when the ground speed increases above the critical hydroplaning speed, the angle between the bow wave and runway decreased progressively until at some high ground speed, the bow wave completely disappeared. However in this study, the emphasis is on incipient hydroplaning and not on speeds exceeding the critical hydroplaning speed. Thus, the formation of bow wave is expected.
Therefore, it is necessary that the model must include the ability of surface tracking of the free surface of water and the formation of bow wave. In order to do so, a two-phase flow comprising a layer of air jet and a layer of water jet has to be considered. This is indicated in Figure 3.2(b) and Figure 3.5, in which air and water each has its own inlets and outlets for.
A two-dimensional model is also analyzed in this study. In this model, the centre-line profile of the hydroplaning region is used and is shown in Figure 3.6. This model has a geometry that is essentially a simplification of the proposed three-dimensional model.
3.7.2 Boundary Conditions
As shown in Figure 3.6, the upstream boundary conditions consist of a pair of inlets, namely a velocity inlet of 5.08 mm (0.2 in.) thick for water and a velocity inlet of 50.8 mm (2 in.) thick of air. The choice of the water film thickness is based on the experimental and numerical conditions used by Browne (1971). In this case, a uniform velocity profile is used.
The pavement surface is modeled as a moving wall. The speed of the air, water and the pavement surface are kept as 15.3 m/s (34.4 mph) in order to be consistent with Browne’s work. The assumption of a moving wall to model the pavement surface is different from the experimental conditions assumed by Browne (1971) that the pavement surface is replaced by an imaginary plane symmetrical about the center-line. This is because in his experiment, he had fabricated two plates and directed a jet of 10.16 mm (0.4 in.) towards the plates separated by a gap of 0.254 mm (0.01 in.). Hence in the verification of the model, the choice of a
laminar flow model are also tested. The inlet is placed at a distance of 100 mm away from the leading edge of the wheel (approximately 40 times the thickness of the hydroplaning region, or 400 times the smallest film thickness) so as to allow for any possible formation of bow wave.
The side edges and the trailing edge are modeled as pressure outlets with the pressure set as 0 kPa (i.e. atmospheric pressure). These are consistent with measurements made by prior experimental research by Horne and Joyner (1965) and the boundary conditions used in the numerical research by Browne (1971). Similarly, the top boundary is set as a pressure outlet at atmospheric pressure and the top boundary is placed at a distance of 25.4 mm (or one time the thickness of plate, or 100 times the smallest film thickness). It is noted that the side edge pressure outlet is not used in the two-dimensional model, but has to be used and tested in the three-dimensional model. For the boundary conditions, there is a need to conduct simulations on the effect of the boundary distance from the wheel model to test if there is any convergence in the ground hydrodynamic pressure. It is noted that the centre line of the wheel can be treated as a plane of symmetry. A summary of the boundary conditions used by Browne and this study is shown in Table 3.2.
3.7.3 Material Properties
The properties of water and air at 20oC are used in this study. The density, dynamic viscosity and kinematic viscosity of water at 20oC are 998.2 kg/m3, 1.002 x 10-3 Ns/m2 and 1.004 x 10-6 m2/s respectively (Chemical Rubber Company, 1988). The density, dynamic viscosity and kinematic viscosity of air at standard atmospheric pressure and 20oC are 1.204 kg/m3, 1.82 x 10-5 Ns/m2 and 1.51 x 10-5 m2/s respectively (Blevins, 1984).
3.7.4 Description of Mesh used in the Analysis
As explained in the earlier sections, the pre-processor GAMBIT is used to generate the meshes for the fluids. In the simulation using the two-dimensional hydroplaning model,
method is employed in the analysis, only 4-node quadrilateral mesh elements are allowed.
FLUENT (2005) recommends the use of at least 5 mesh elements for channel and pipe flows. In the simulation in this research, ten 4-node quadrilateral mesh elements are used for the smallest channel in the model, i.e. the hydroplaning region. The optimal number of mesh elements needed to give a converged solution can be tested through a mesh sensitivity analysis.
Figure 3.7 shows the mesh set-up of the two-dimensional hydroplaning model. There are 14,375 elements in the model.
3.7.5 Simulation Results Based on the Proposed Two-Dimensional Model
The simulation is performed on SUN BLADE 1000 workstations which have single 900MHz UltraSPARC-III processor and 1 or 2 G-bytes memory each. The fluid models used in the analysis include the usual Navier-Stokes equations, the k-ε turbulence model and the VOF multiphase model as described in the earlier parts of the chapter.
Based on the specified geometry, boundary conditions and initial conditions, the steady-state volume fraction plot is shown in Figure 3.8. It is observed that a bow-wave is formed, which is expected and is observed in experiments conducted by Browne (1971).
Figure 3.9(a) and Figure 3.9(b) show the velocity vectors under the wheel in the moving reference frame (i.e. the model) and the stationary observer reference frame (i.e. the reality) respectively.
Figure 3.10 indicates the contours of the hydrodynamic pressure in the model and it is seen that the pressure near the boundaries are at near zero pressure (i.e. atmospheric pressure), thereby indicating the suitability of the choice of the boundary conditions. This will be further verified in the later parts of this sub-section.
Table 3.3 shows the inflow and outflow properties of the two fluids used in the study, namely air and water using the laminar model setup. It is noted that some recirculation will occur and this is mainly due to the formation of the bow-wave and this forces the air to go into
that in the solution scheme of the VOF method, air is used as a primary phase for computational convergence and water is used as a secondary phase. In this case it is noted that 98.83% of the water is lost as splash.
The ground hydrodynamic pressure distribution is shown in Figure 3.11 and the average ground hydrodynamic pressure under the hydroplaning wheel is found to be 121.1 kPa.
This value ought to be equivalent to the tire pressure of the wheel. But at this point, the obtained average ground hydrodynamic pressure serves to act as a verification of the model in terms of mesh quality and the choice of the boundary conditions. The aptness of the model in simulating hydroplaning will be further discussed in the later parts of the section.
3.7.6 Mesh Sensitivity Analysis
In order to ensure that the solution obtained is numerically accurate, grid independence tests have to be conducted. Various finite volume mesh sizes were examined to obtain the optimal mesh quality to be used. In this study, the mesh design in the hydroplaning region is of utmost importance since it is the thinnest flow channel expected in the model. As such, the number of finite volume cells in the depth of the hydroplaning region is a key aspect in mesh design. Four different mesh designs are tested and the steady-state volume fraction plot is shown in Figure 3.12. The aspect ratio (maximum edge length divided by minimum edge length) of the different mesh designs are kept constant at 7.5 for each of the four cases tested.
It can be seen that the plots show similar fluid behaviors with the exception of the use of coarse mesh (i.e. the mesh using 5 quadrilateral mesh elements within the smallest hydroplaning channel). A key indicator of mesh convergence is the average ground hydrodynamic pressure as this parameter is used in the definition of hydroplaning. Figure 3.13 shows the ground hydrodynamic pressure distribution under the wheel for the various meshes and it can be observed that there are little variations among the various pressure profiles except for the one with 5 mesh elements, thereby indicating grid independence. Figure 3.14 and Table 3.4 show the effect of the mesh design on the average ground hydrodynamic pressure. It can be
relatively accurate solution to ensure grid independence and thus this mesh design is used in subsequent two-dimensional analyses.
3.7.7 Effect of Boundary Conditions
The effect of the boundary conditions has to be studied to ensure that the distances of the boundaries, especially the locations of the velocity inlets and the pressure outlets are sufficiently far away to ensure numerical accuracy of the model in terms of the key indicator of hydroplaning, i.e. the average ground hydrodynamic pressure under the wheel. The boundary locations of the various models tested are shown in Table 3.5. The choice of the locations is based on the consideration of the aptness of the location of the boundaries and the computational efficiency of the analysis. Model A is essentially a modification of the model used by Browne (1971). The choice of these boundary conditions are justified through prior experimental and numerical research as explained in the earlier sections of the chapter. The computational constraint is perhaps the sole consideration in the testing of the effects of the boundary conditions. This is because the two-dimensional model proposed would be used in the three-dimensional form of the proposed model. The number of mesh elements of the three- dimensional form of model B and D easily exceeds one million mesh elements while that of models C and E easily exceeds two-million mesh elements. The current computational capabilities of the workstations even with the use of 16 parallel processors, would find the three-dimensional forms of models C and E computationally demanding to solve.
The steady-state volume fraction plots of the various models are shown in Figure 3.15 and it can be seen that the plots exhibit similar fluid behaviors. Table 3.6 shows the effect of the location of the boundary conditions on the average ground hydrodynamic pressure. It can be seen that the average ground hydrodynamic pressure are similar with an error of less than 1%, thereby indicating that the effect of the locations of the boundary conditions considered are insignificant. This indicates that the proposed model with the boundary conditions used in
3.7.8 Analysis of Results and Suitability for Hydroplaning Simulation
From the ground hydrodynamic pressure distribution shown in Figure 3.11, the average ground hydrodynamic pressure under the hydroplaning wheel and the tire inflation pressure are both found to be 121.1 kPa. The ratio of tire pressure to 0.5ρU2 is found to be 1.027 which is much larger than the expected NASA hydroplaning equation value of 0.644.
This means that the proposed two-dimensional model using a turbulent flow model assumption is not an adequate model in simulating hydroplaning. Furthermore, the model shows that 98.83% of the water is lost as splash. This is a reflection of the inadequacy of the model because one important component of outflow, the in-plane and out-of-plane outflows are not modeled. This has resulted in an excessively high hydrodynamic pressure being developed under the wheel and an extremely high percentage of splashes. Browne (1971) indicated in his model that close to 6% of the water would pass through the imprint and approximately 55%
lost as splash. The proposed two-dimensional model predicts that hydroplaning would occur at a speed of 68.6 km/h (42.8 mph) compared to 87.3 km/h (54.4 mph) predicted by the NASA hydroplaning equation. The two-dimensional model is overly conservative in the prediction of hydroplaning speed.
To further substantiate the point, the model is re-run using the plane of symmetry as the pavement surface model. It is noted that the experimental data points for the hydrodynamic pressure do not fit well to the ground hydrodynamic pressure profile obtained from the simulation as shown in Figure 3.16. In fact, this model would over-predict the pressure. The average ground hydrodynamic pressure is 103.0 kPa, yielding a ratio of tire pressure to 0.5ρU2 of 0.87. This is still considerably larger than Browne’s value of 0.56, showing the inappropriateness of the two-dimensional model.