The prior section shows that the two-dimensional model is inadequate to simulate hydroplaning accurately. This section presents the development of a three-dimensional model
Section 3.3.
3.8.1 Geometry of Model and Selection of Boundary Conditions
This proposed three-dimensional model uses the tire deformation profile shown in Figure 3.1 which is based on Browne’s (1971) experiment and a geometry shown in Figure 3.5.
The boundary conditions and the initial conditions adopted are as described in Section 3.7.2 for the two-dimensional model.
3.8.2 Description of Mesh used for 3-D simulation
The pre-processor GAMBIT is used to generate the finite volume mesh for the fluids.
In the simulation using the three-dimensional hydroplaning model, hexahedral and wedge elements are used to depict each finite volume. Since the finite volume method is employed in the analysis, only 6-node wedge elements and 8-node hexahedral elements are allowed.
FLUENT (Fluent Inc., 2005) recommends the use of at least 5 mesh elements for channel and pipe flows and in this simulation, ten 8-nodes 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 converged solution can be tested through a mesh sensitivity analysis. Figure 3.17 shows the mesh design of the three-dimensional hydroplaning model. There are 394,900 mesh elements in the proposed model.
3.8.3 Simulation Results Based on Proposed 3-D Model
The simulation is performed either on the 3 or 8 parallel CPUs available in 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, thereby warranting the need for parallel processing. The fluid models used in the
the steady-state phase plot along the plane of symmetry is shown in Figure 3.18. It is observed that a bow-wave is formed, which is expected and is observed in experiments conducted by Browne (1971). Figure 3.19 shows 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). It is observed that the velocities near the wheel are near-zero, indicating that in the actual reference frame, there is a thin film of lubricant under the hydroplaning wheel moving at near the vehicle speed along with the sliding wheel.
Table 3.7 shows the inflow and outflow properties of the two fluids used in the study, namely air and water using the laminar model setup. Also, the conservation of mass is obeyed as 99.93% of the air and 99.60% of the water is conserved. In this case, 25.28% of the water is lost as splash.
Figure 3.20 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 latter parts of this sub-section. The ground hydrodynamic pressure distribution under the centre-line of the wheel is shown in Figure 3.21 and selected profiles along lines in the wheel direction are shown in Figure 3.22. The average ground hydrodynamic pressure under the hydroplaning wheel is found to be 72.5 kPa. This value would serve to act as a verification of the model in terms of mesh quality and the choice of the boundary conditions.
This is smaller than the average ground hydrodynamic pressure of 121.1 kPa obtained from the two-dimensional analyses, as shown in Figure 3.11. This is expected since the two- dimensional analyses ignore the side flow of the water impinging on the wheel as opposed to the three-dimensional analyses of fluid flow shown in this section.
3.8.4 Mesh Sensitivity Analysis
In order to ensure that the solution obtained is numerically accurate, grid independence tests have been conducted. Different mesh densities were examined to obtain the optimal mesh
aspect of mesh design. Five different mesh designs are tested and the steady-state volume fraction plot is shown in Figure 3.23. 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 exhibit similar fluid behaviors. A key indicator of mesh convergence is the average ground hydrodynamic pressure as this parameter is used in the definition of hydroplaning. Figure 3.24 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 except for the mesh with 5 elements, thereby indicating grid independence for the for designs with 10 or more elements. Figure 3.25 and Table 3.8 show 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 subsequent three-dimensional analyses in this research.
3.8.5 Effect of Boundary Conditions
Similar to that performed in the two-dimensional analyses, 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 3.9. 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.
Models A to E are essentially the three-dimensional form of the two-dimensional models highlighted in the prior analyses. Model F is added to test the effect of varying the location of the pressure outlet from the side of the hydroplaning wheel.
The steady-state volume fraction plots of the various models are shown in Figure 3.26
be seen that the average ground hydrodynamic pressure are similar with an error of less than 2%, 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 the prior sections is adequate to achieve the intended numerical accuracy.
3.8.6 Analysis of Results and Suitability for Hydroplaning Simulation
Modeling after Browne’s (1971) experimental set-up, the ground hydrodynamic pressure distributions obtained from the simulations are shown in Figure 3.21 and Figure 3.22 and the average ground hydrodynamic pressure under the hydroplaning wheel is found to be 72.5 kPa. This value should theoretically be equal to the tire inflation pressure in order for hydroplaning to occur. The average ground hydrodynamic pressure is therefore used to evaluate the ratio of tire pressure to 0.5ρU2. This ratio is found to be 0.620 which is close to the expected NASA hydroplaning equation value of 0.644 with a percentage difference of 3.7%. This difference is acceptable, considering the fact that the NASA hydroplaning equation is empirically derived from a wide variety of tires operating on flooded pavements (Horne and Joyner, 1965). This means that the proposed three-dimensional model using a turbulent flow model is an acceptable model in simulating hydroplaning. Furthermore, the model shows that 95% of the water is lost either as side-flow or splash and will not leave through the trailing edge of the wheel. This corresponds to Browne’s numerical research which claimed that approximately 94% of the water will not leave through the trailing edge of the wheel.
Comparing the hydroplaning profile with experimental data from past research (Horne and Dreher, 1963; Horne and Joyner, 1965; and Browne, 1971) indicates that the hydrodynamic pressure profile obtained from the simulations (as shown in Figure 3.27) is similar to those works. These evidences highlight the appropriateness of the model in hydroplaning simulation.
It is noted that the proposed model is not entirely consistent with the experimental model used in Browne’s research (Browne, 1971) as highlighted in Table 3.2. In order to assess the validity of the model, it is sought if using a plane of symmetry as the boundary
used in his numerical studies) would yield compatible results to Browne’s experimental data.
In this case, 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 fit rather well to the ground hydrodynamic pressure profile obtained from the simulation as shown in Figure 3.28.
The average ground hydrodynamic pressure is 67.0 kPa, yielding a ratio of tire pressure to 0.5ρU2 of 0.57. This corresponds extremely well with Browne’s value of 0.56, showing the appropriateness of the model and the fundamental governing equations behind the hydroplaning theory. However, this also highlights that Browne’s research could not accurately predict hydroplaning primarily because of a different choice in the boundary conditions and flow model.
Another option to be considered is the use of laminar flow model instead of the turbulence flow model. This is because of the fact that hydrodynamic lubrication theory invariably makes use of Reynolds assumptions as shown in Table 3.11. It is noted that laminar flow is one of the many critical assumptions made and may not be valid under the hydroplaning condition. The assumptions made are appropriate for viscous hydroplaning which is essentially a low speed phenomenon. However, no research has been conducted to assess the validity of this assumption in the case of hydroplaning under the high speed scenario.
It is highlighted here that Reynolds lubrication equation is in fact a special case of the set of Navier-Stokes equations and can be derived from the Navier-Stokes equations by using the Reynolds assumptions. As such, simulations using the 3D model are run using the laminar flow assumption to assess the suitability in hydroplaning simulation. The ground hydrodynamic pressure profile along the centerline of the wheel is shown in Figure 3.30. The average ground hydrodynamic pressure is 67.0 kPa, yielding a ratio of tire pressure to 0.5ρU2 of 0.57, which is close to Browne’s value of 0.56. This highlights that using a laminar model such as the one proposed by Browne is inappropriate. This value, as compared to the proposed 3-D model using the turbulent flow model, provides a poor approximation to the ratio
invalid. As such modifications have to be made to consider the effects of turbulence (Stachowiak and Batchelor, 2001). Since the simulation considers the full set of Navier-Stokes equations of which the lubrication theory is a special situation, it is deemed that the proposed 3-D model is theoretically sound in hydroplaning simulation. This is supported in the current study through evidences in the experimental and numerical verification as highlighted in this chapter.