It has been shown in Section 3.7 of Chapter 3 that a two-dimensional model is inapt to model hydroplaning and thus the three-dimensional model of the hydroplaning profile shown in Figure 4.1 is used in the hydroplaning simulation. This section describes the main features of the 3D model and its verification.
4.4.1 Geometry of Model and Selection of Boundary Conditions
This proposed three-dimensional model has geometry as depicted in Figure 4.2. The boundary conditions and the initial conditions adopted, as summarized in Table 4.1, are similar to those as described in Chapter 3. As shown in Figure 4.2, the upstream boundary conditions consists of a pair of inlets, namely a velocity inlet of 7.62 mm (0.3 in.) thick for water and a velocity inlet of 76.2 mm (3 in.) thick of air. A uniform velocity profile is used. The pavement surface is modeled as a moving smooth plane wall with no microtexture. The speed of air, water and the pavement surface are kept as 96.5 km/h (60 mph) in order to be consistent with Horne and Joyner’s (1965) experiments. 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,
formation of bow wave.
The side edges and the trailing edge of the model are modeled as pressure outlets with the pressure set as 0 kPa (i.e. atmospheric pressure). The top boundary is set as a pressure outlet at atmospheric pressure and is placed at a distance of 25.4 mm (equal one time the thickness of plate, or 100 times the smallest film thickness, whichever is larger). On the boundary conditions, there is a need to conduct simulations on the effect of the distance of the boundaries 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.
4.4.2 Description of Mesh used in the Analysis
The pre-processor GAMBIT is used to generate the finite volume mesh for the fluids (Fluent Inc., 2004). In this simulation, ten 8-node hexahedral 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 sufficiently accurate solution can be tested through a mesh sensitivity analysis.
Figure 4.3 shows the mesh design of the three-dimensional hydroplaning model. There are altogether 463,300 mesh elements in the proposed model.
4.4.3 Simulation Results
The simulation is performed either on the 3 or 8 parallel CPUs (depending on availability) of the COMPAQ GS320 alpha server, which is configured with 22 EV67 731 MHz Alpha 21264 CPUs and 11 GB of memory. The computational time needed for the simulation ranges from 36 CPU-hours for a 0.5 million elements model to 150 CPU-hours for a 1.7 million elements model. Based on the specified geometry, boundary conditions and initial conditions, the steady-state phase plot along the plane of symmetry is shown in Figure 4.4. It is observed that a bow-wave forms at the front of the wheel and the splash is observed as shown
(i.e. the model) and the stationary observer reference frame (i.e. the reality). It is observed that the velocities near the wheel are near-zero in Figure 4.5(a). This means that in the actual reference frame under the hydroplaning wheel, there is a thin film of lubricant moving at near the vehicle speed along with the sliding wheel as shown in Figure 4.5(b).
Figure 4.6 indicates the contours of the hydrodynamic pressure in the model and it is seen that 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 4.2 shows the inflow and outflow properties of the two fluids in the system, namely air and water using the turbulent model setup. Also, the conservation of mass is obeyed as 99.97% of the air and 99.95% of the water is conserved. 73.0% of the water is lost as splash.
The ground hydrodynamic pressure distribution under the centre-line of the wheel is shown in Figure 4.7 and selected profiles along lines in the wheel direction are shown in Figure 4.8. The locations of the planes labeled as I = 1, I = 2, I = 3 and I = 4 are shown in the table in Figure 4.8. The ground hydrodynamic pressure distribution under the entire hydroplaning wheel is shown in Figure 4.9. The average ground hydrodynamic pressure under the hydroplaning wheel is found to be 228.5 kPa. This ground hydrodynamic pressure exceeds the tire pressure of 186.6 kPa, implying that hydroplaning has already occurred and is indeed the case.
4.4.4 Mesh Sensitivity Analysis
In order to ensure that the solution obtained is numerically accurate, grid independence tests were conducted. Different mesh densities were examined to obtain the optimal mesh design. The number of hexahedral elements in the width of the hydroplaning region is a key aspect of mesh design. Four different mesh designs were tested and the steady-state volume fraction plot is shown in Figure 4.10. 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
mesh convergence is the average ground hydrodynamic pressure as this parameter is used in the definition of hydroplaning. Figure 4.11 shows the ground hydrodynamic pressure distribution along the centre line under the wheel for the various mesh designs and it can be observed that there are little variations between the various pressure profiles, thereby indicating grid independency except for the coarsest mesh (i.e. 5 hexahedral cells in the smallest channel). Figure 4.12 and Table 4.3 shows the effect of the mesh design on the average ground hydrodynamic pressure. It can be seen that using 10 mesh elements within the hydroplaning regions is sufficient to render a relatively accurate solution and thus this mesh design is used in the subsequent three-dimensional analyses.
4.4.5 Effect of Boundary Conditions
The effect of 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 4.4. The choice of the locations includes the consideration of the aptness of the locations of the boundaries and the computational efficiency of the analysis.
The steady-state volume fraction plots of the various models are shown in Figure 4.13 and it can be seen that the plots show similar fluid behaviors. Table 4.5 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 2%, thereby suggesting that the effect of the locations of the boundary conditions considered are insignificant. This shows that the proposed model with the boundary conditions used in the prior sections is adequate to achieve the intended numerical accuracy.
Comparisons of the proposed model with the numerical research conducted by Browne (1971), and the experimental results from existing literature are presented in this section. Some discrepancies are expected as it is noted in the previous chapter that (i) the boundary conditions used by Browne’s analysis for the pavement surface is inappropriate, and (ii) the proposed model provides a much better recovery factor that is closer to the NASA’s recovery factor of 0.644 as compared to Browne’s.
Figure 4.8 and Figure 4.14 show the ground hydrodynamic pressure distributions for various locations along the wheel for the main hydroplaning region for the proposed model and Browne’s (1971) model respectively. The general trends of results obtained with the two models are similar. It is noted that Browne’s solution shows some regions with excessively high negative hydrodynamic pressure. Compared with experimental results from various tests done by researchers (Horne and Leland, 1962; Horne and Joyner, 1965; Yeager and Tuttle, 1972), excessively high negative hydrodynamic pressures such as that along line I = 4 were not detected and typically, negative hydrodynamic pressures were not found in the main hydroplaning region. Although regions of negative hydrodynamic pressure is not shown in the cut-off planes, they do exist in the proposed model, and in this configuration, the maximum negative hydrodynamic pressure is -38.9 kPa.
The average ground hydrodynamic pressure under the hydroplaning wheel is found to be 228.5 kPa. The average ground hydrodynamic pressure is therefore used to evaluate the recovery factor (i.e. ratio of tire pressure to 0.5ρU2). This factor is found to be 0.636 which is close to the expected NASA hydroplaning value of 0.644 with a percentage difference of 1.2%.
This, compared to the value of 0.56 obtained by Browne, provides a much better modeling of the hydroplaning phenomenon and offers credibility to the NASA hydroplaning equation. It implies that if the point of analysis is that of incipient hydroplaning, the expected tire pressure predicted by the proposed model is equal to that proposed by NASA which is experimentally based. Conversely, if it is known that the tire pressure is 186.6 kPa (which is the case), the model predicts that hydroplaning has already occurred since the uplift force is greater than the
exceeds the tire pressure). This suggests that hydroplaning has already occurred, as noted by Horne and Joyner (1965).
4.4.7 Repeat of Analysis Using NASA Predicted Hydroplaning Speed
The simulations conducted in the preceding section were based on the experimental speed of 96.5 km/h (60 mph). The analyses are repeated using the NASA predicted hydroplaning speed of 87.5 km/h (54.4 mph) in order to verify that the model could closely simulate hydroplaning. In this case the velocities of the air, water and pavement are fixed at the NASA predicted hydroplaning speed of 87.5 km/h. The resulting ground hydrodynamic pressure obtained from the simulation analysis is 184.6 kPa. Figure 4.15 shows the selected ground hydrodynamic pressure profiles along lines in the wheel direction. The tire pressure of the passenger car tire used in this simulation is 186.6 kPa. The computed ground hydrodynamic pressure differs from this tire pressure by 1.1 %. The recovery factor is 0.640 which differs from NASA’s value of 0.644 by 0.6%. Based on this value, the predicted hydroplaning speed of the 186.6 kPa tire pressure passenger car tire is found to be is 87.0 km/h which differs from the NASA predicted hydroplaning speed by a mere 0.3%. This provides affirmative verification of the ability of the proposed model to accurately predict the onset of hydroplaning.