Smoothed particle hydrodynamics based numerical study of hydroplaning considering permeability characteristics of runway surface

Abstract

The presence of water films on a runway surface presents a risk to the landing of aircraft. The tire of the aircraft is separated from the runway due to a hydrodynamic force exerted through the water film, a phenomenon called hydroplaning. Although a lot of numerical investigations into hydroplaning have been conducted, only a few have considered the impact of the runway permeability. Hence, computational problems, such as excessive distortion and computing efficiency decay, may arise with such numerical models when dealing with the thin water film. This paper presents a numerical model comprising of the tire, water film, and the interaction with the runway, applying a mathematical model using the smoothed particle hydrodynamics and finite element (SPH-FE) algorithm. The material properties and geometric features of the tire model were included in the model framework and water film thicknesses from 0.75 mm to 7.5 mm were used in the numerical simulation. Furthermore, this work investigated the impacts of both surface texture and the runway permeability. The interaction between tire rubber and the rough runway was analyzed in terms of frictional force between the two bodies. The SPH-FE model was validated with an empirical equation proposed by the National Aeronautics and Space Administration (NASA). Then the computational efficiency of the model was compared with the traditional coupled Eulerian-Lagrangian (CEL) algorithm. Based on the SPH-FE model, four types of the runway (Flat, SMA-13, AC-13, and OGFC-13) were discussed. The simulation of the asphalt runway shows that the SMA-13, AC-13, and OGFC-13 do not present a hydroplaning risk when the runway permeability coefficient exceeds 6%.

Graphical abstract

Keywords

SPH-FE approach / hydroplaning / numerical simulation / surface texture / runway surface reconstruction

1 Introduction

A runway surface covered with water, may cause the tire of a landing aircract to skid, because skid resistance has been lost. This is referred to as hydroplaning, which is a state in which the tire is separated from the runway by a water film. Generally, to prevent tire skidding at low velocity, the standard practice is to remove any standing or residual water from the tire grooves, sides of the tire, and the runway surface. The velocity differential between the standing water and the running tires creates a hydrodynamic pressure on the tread. Aircraft landing speed can be up to 300 km/h In wet weather conditions, some water may form a layer between the tires and the runway, preventing direct surface-to-surface contact. This may cause the landing aircraft to lose skid resistance and control. The hydrodynamic force, which drives the hydroplaning mechanism, is the sum of vertical hydrodynamic pressure imposed on the tire surface [1]. The critical speed that enables the fluid force to separate the tire and runway is defined as the hydroplaning speed. Therefore, understanding hydroplaning is vital for airport management and aircraft manufacturing companies for improvement of landing safety on a wet runway. To investigate the hydroplaning mechanism, this study carried out experiments and simulations.

Aircraft landing accidents due to runway excursions are very serious. The International Civil Aviation Organization annual report indicated that hydroplaning on a wet runway is the major cause of runway excursions. However, in hydroplaning the tire is affected by many factors, including the runway conditions (such as texture depth, transverse slope, and gradation of the asphalt mixture), tire conditions (such as tire load, inflation pressure, and slip ratio), and water conditions (such as water film thickness, rain density, and ambient temperature). The USA National Aeronautics and Space Administration (NASA) conducted the first experiments with aircrafts, trucks, and cars to predict the vehicle speed above which hydroplaning takes place, the hydroplane speed. With these efforts, Dreher and Horne [2] developed a relationship between the hydroplane speed and the inflation pressure. Since this development, the advancement in testing methods has enabled a better understanding of the hydroplaning mechanism. Gallaway et al. [3] investigated the relationship between hydroplaning speed and many factors, based on vehicle field tests. The formula for predicting hydroplaning speed was developed taking into account a range of factors, including surface texture, water film depth, and groove depth. Beautru et al. [4] discussed the effects of a thin water film (in the range of 0 to 1 mm) on skid resistance performance. That study found that laboratory testing revealed that the skid resistance declined by 30% to 40% due to the water film lubrication. Some studies have suggested that modeling of water film depth can be applied to reduce hydroplaning risk for aircrafts and vehicles. Huebner et al. [5] developed a model for computing water film thickness considering rain density, surface texture, and drainage. Kogbara et al. [6] stated that some tread patterns are more sensitive to water film depth. The study concluded that the loss of skid resistance reaches 60% in wet conditions and under high velocity, due to the depth of water film and the velocity of tire.

Although experiments have revealed how hydroplaning occurs, and many factors affect the phenomenon. The cross-influence of these factors makes analysis challenging. Furthermore, experiments are expensive and time-consumingAdvances in numerical tools provide an alternative approach to research. Fluid-structure interaction (FSI) becomes increasingly challenging in numerical simulations due to the interaction between runway, tire, and water at high speeds and under heavy load and inflation pressure. Fwa and Ong [7] developed a finite element (FE) model for predicting hydroplaning speed on a flat runway. The FE model was validated using experimental data and provided a good prediction for hydroplaning. Ong and Fwa [8] proposed a method for calculating braking distance by FE analysis on a wet runway. Based on simulations, braking distances on a wet runway are 1.5 times longer than on a dry runway. Kumar et al. [9] improved the numerical simulation by considering the influence of the surface texture through a rough runway model, using a coupled Eulerian–Lagrangian (CEL) algorithm to analyze the interactions of water with runway and tire within Eulerian domains. Zhu et al. [10] used optical scanning to describe hydroplaning in a textured runway model. The deterioration of surface texture was included as a long-term effect of loads in the FE analysis.

These numerical simulations provide significant progress. However, it is necessary to incorporate macroscopic and microscopic views in hydroplaning simulations using FE-based numerical tools. The diameter of an aircraft tire is about 1 m, while the depth of surface texture is about 1 mm, so hydroplaning simulation must be performed on a large range of element sizes. It is important to note that the deformation of a large element easily distorts the small element. On the other hand, the computational time depends on the smallest element size. Moreover, the hydrodynamic pressure might decrease due to the discharge of water through asphalt runway voids. Considering these limitations, smoothed particle hydrodynamics (SPH) is a promising approach for FSI simulations. Zhang et al. [11] built an SPH-FE model to simulate the splashing of water as an aircraft lands on a wet runway. Hermange et al. [12] used an SPH approach to analyze the fluid forces on a rough runway model with a water film. However, the effects of permeability characteristics of runways on aircraft hydroplaning have not been studied, to our knowledge.

As part of this study, an SPH-FE approach was developed that included a tire model, a rough runway model, a water model, and an interaction model. We also proposed a runway model that takes permeability into account to simulate drainage from a runway. By using FE-SPH algorithms on fluid-solid interaction problems, it is possible to simulate the hydroplaning phenomenon on thin water films.

2 Finite element model

To study hydroplaning, the FE model needs four sub-models, dealing with the tire, runway, interactions, and water models. As shown in Fig.1, the tire and the runway models are FE-based models, the water model is SPH-based, and the interaction model is developed to simulate the frictional property between the tire and the runway. By applying a tire geometric model and material properties to a composite material that consists of rubber and cord layer materials, a tire model is developed. Building on a two dimensional (2D) model of an aircraft tire cross-section, a three dimensional (3D) geometric model of the tire is constructed. By scanning the profiles of asphalt mixtures, the runway textures can be determined. As a result, the permeability coefficient of the asphalt mixture can be calculated to determine its drainage capacity. Based on the FE model of the tire and runway, the interaction between the tire and runway can also be simulated. Here, the power spectrum density (PSD) and viscoelastic property of rubber are used to construct the interaction model. The SPH numerical technique is applied to the water model. Water depth is determined by the size of the SPH inlet, which generate the SPH particles in front of the tire. As a means to save computing resources, an SPH outlet that delete the SPH particles after computation, is created to remove the SPH particles that are no longer in contact with the tire. Following this, the hydroplaning model was validated against experimental results. Details of the hydroplaning modeling are described below.

2.1 Aircraft tire model

An appropriately designed aircraft tire can improve landing safety. In particular, aircraft tire treads are designed with circumferential grooves to drain water. In this paper, the main wheel of the aircraft model is the Airbus A320, as it is one of the most delivered aircrafts in the world [13]. The main tire used in the model was a 46 × 17.0 R20 type, and had a diameter of 116.84 cm and a tread width of 43.24 cm. There are four circumferential grooves with depth and width of 10 and 9 mm, respectively, on the tread pattern. In the tire model, a standard inflation pressure of 1.147 MPa was applied on the tire’s inner surface. Additionally, a landing load of 66 kN was applied to the center of the tire model and transferred to the rim. Tab.1 presents the detailed information of the A320 main tire.

Tire tread blocks influence a tire’s major performances, such as rolling resistance, grip force, contact pressure, and footprint. Appropriate computational resources were precisely assigned to different tire parts to improve numerical prediction [14]. The tire tread meshes, with a size of 2 mm, capture the surface texture; the tire body is less critical for hydroplaning simulations than the tread. Therefore, a 2D FE model of the aircraft tire cross-section was divided into 60 segments for the creation of a 3D tire model. This was followed by an assembly of the tread and the tire body model with a tie constraint. The FE model was developed as shown in Fig.2. An aircraft tire is composed of rubber material for providing abrasion resistance and shock absorption and cord layers prevent excessive deformation. Material with high stiffness is used for cord layers, such as steel wires and fabric wires. In the tire geometric model, structure was considered to include a composite material of rubber and a cord layer. A Yeoh constitutive model was used for the FE simulation to simulate rubber’s hyperelastic properties. Tab.2 and Tab.3 list the parameters of rubber and cord layer materials, where Yeoh model was used to represent the hyperelastic property of rubber materials.

2.2 Runway surface model

As a result of an aircraft tire’s pressure on the runway, water escapes from underneath the tire. Due to the presence of air voids in the asphalt mixture, water will flow across the pavement structure. In this study, the effect of runway surface texture on the mechanism of hydroplaning was investigated using three asphalt mixture specimens, namely asphalt concrete (AC-13), Stone mastic asphalt (SMA-13), and open-graded friction course (OGFC-13), with different gradations. These three asphalt mixtures are the most commonly used for asphalt pavements, with SMA-13 mainly being used for airport runways and OGFC-13 having the best textural characteristics and greatest permeability. The gradations and basic properties of the asphalt mixture are shown in Tab.4 and Tab.5. The Mean Profile Depth (MPD) and Root Mean Square (RMS) are indicators of the roughness of the runway surface. Tab.5 indicates that the order of the roughness for the three asphalt mixtures is OGFC, SMA, and AC. The drainage capacity of the corresponding runway models can be implied by this order. The asphalt mixtures were compacted into a 30 cm × 30 cm × 5 cm board by the wheel grinding method. The runway surface texture was scanned by optical scanning and data were transferred into a runway model. Fig.3 shows the process of developing the runway model, the asphalt mixture board, the optical scanning results and the reconstruction model of asphalt mixture, separately.

In addition to modeling the runway, the drainage capacity of the runway should also be considered to improve the simulation results. As this study used the SPH technique for the FSI problem, an outlet plane was set so that the SPH particles would exit the simulation when they contacted the outlet plane, thus making the fluid flow away through the runway pores. In the runway model, specific regions were designated for SPH particles to disengage from calculations upon contact. This was implemented to simulate road surfaces with varying permeability, thereby modeling different water drainage capacities. As shown in Fig.4, we calculated the projected area of an air void on the runway when the outlet plane was set. As presented in Tab.3, the permeability coefficients of AC, OGFC, and SMA runways were 4.9%, 3.7%, and 20.5%, respectively. It can be seen that the OGFC runway had the highest permeability coefficient and best pore connectivity. Also, we observed that the OGFC model had the best drainage capacity, followed by the AC model, while SMA runway had the worst drainage capacity.

2.3 Water film model

The most commonly used algorithms to solve FSI problems in hydroplaning simulations are mesh-based algorithms, like computation fluid dynamics and CEL algorithm. When considering runway textures or pores during hydroplaning simulations, then a small Eulerian mesh is required, which means this class of algorithms has some major drawbacks. This results in extremely long computing times, along with excessive deformation during simulations. Additionally, these algorithms have to set up the Eulerian domain in advance, and it is difficult to divide the Eulerian domain based on runway texture and permeability. As a result, most numerical simulations using such algorithms would simplify the runway model as much as possible [16,17]. In this study, hydroplaning simulations using the SPH algorithm were meshless algorithms and therefore did not suffer from the drawbacks of mesh-based algorithms. Additionally, the minimum diameter of SPH particles in the simulation was 0.25 mm, meeting the runway texture and permeability coefficient simulation requirements [12]. The density of water, the speed of sound in water, the material parameter of water after impact and the slope of the Us-Up curve are needed for simulation, and were set to 990.20 kg/m³, 1480 m/s, 1.92 and 1.2, respectively.

Due to the use of discrete particles to solve the continuity equation in the fluid domain, the SPH algorithm does not present a convergence problem in thin water film simulation, unlike mesh-based methods. The movement particle of interest as determined by the kernel function that counted all the particles in a certain range. In Fig.5, a water SPH particle (the green center of the circle) was influenced by all the SPH particles in the circle. However, the force between two particles was determined by the weighting function W, which took into account the particle positon r and smoothing lengthe h.

When the SPH technique was used for hydroplaning or splash simulation, the water film was typically converted into SPH particles on a section of the runway, and then the tires would pass through the water film [11,12]. Considering computing resources, only a small range of runway was coverd with water. Moreover, the water film left after the tire passed by did not contribute to the simulation, yet it could still consume computational resources. The inlets and outlets for SPH particles are shown in Fig.6. It should be noted that although there is an overlapping relationship between the SPH outlet and the tire, the two objects did not come into contact in the simulation, so the outlet did not affect the simulation results. The inlet enabled the hydroplaning simulation to generate new SPH particles continuously, while the outlet allowed particles that are not contributing to the hydroplaning simulation to terminate the analysis. The SPH particles with a diameter of 0.75 mm was used in the simulations.The height of the SPH inlet determined the thickness of the water film, and the speed of the water flow and the runway determined the landing speed.

2.4 Contact model between tread rubber and rough surface

The numerical analysis was used to simulate the tire hydroplaning behavior based on the interactions between the tire and the rough runway surface. In analyzing the contact mechanism between the rubber tire and the rough runway, a common view is that the rubber is excited by the runway asperities leading to energy dissipation and contributing to friction force [18]. The friction effect relies on the roughness of the runway surface and the material properties of the tire tread rubber. In predicting the friction coefficient between rubber and runway, Persson’s friction model [19] is often used. By using this model, the PSD of the runway surface was calculated. Equation (1) explains the PSD as a Fourier transform of the runway morphology. The PSD reflected the excitation frequencies for the maximum and the minimum frequencies. As shown in Fig.7, the PSDs of AC, SMA, and OGFC runway were obtained by this method. The curves for OGFC and SMA are higher than that for AC, indicating that the former two runways were, overall, rougher than the AC runway. The same roughness order can be found in the RMS and MPD. The SMA curve is longer than those of AC and OGFC in terms of the maximum cutoff frequency. This indicates that the SMA runway had more abundant microtextures compared with AC and OGFC runways. The reason might be that the proportion of fine aggregate with a particle size of 0.075 mm in SMA is 10.3%, while in AC and OGFC, these proportions are are 5.9% and 5.3%, respectively.

(1)C\left(q\right)=\frac{1}{{\left(2\text{π}\right)}^2}\int {\mathrm{d}}^{2}x⟨h\left(\overline{x}\right)h\left(0\right)⟩{\mathrm{e}}^{\overline{iqx}},

where C\left(q\right) denotes the PSD of the runway surface, h\left(\overline{x}\right) is the height of an asperity, q represents the wave vector of asperities.

Furthermore, the calculation of the kinematic friction coefficient also requires the material viscoelastic performance of the tread rubber [20]. Persson proposed that the kinematic friction coefficient between the rubber material and the rough runway surface can be calculated by Eqs. (2) and (3) [21].

(2)\mu =\frac{1}{2}{\int }_{q_{\mathrm{L}}}^{q_1}{q}^{3}C\left(q\right)P\left(q\right)\mathrm{d}q{\int }_0^{2\text{π}}\cos \varphi \mathrm{I}\mathrm{m}\frac{E\left(q_0\zeta \cos \varphi \right)}{\left(1-v^2\right){\sigma }_0}\mathrm{d}\varphi ,

(3)P\left(q\right)\approx \frac{4\left(1-{\upsilon }^2\right)}{q_0{h}_0}{\left(\frac{1-H}{\text{π}\alpha H}\right)}^{\frac{1}{2}}\frac{{\sigma }_0}{E}{\left(\frac{q}{q_0}\right)}^{H-1},

where \mu represents the kinematic friction coefficient of tread rubber, q_1 and q_0 are the maximum and minimum excitation frequencies of the rough surface, and are affected by the minimum and maximum asperities, respectively, q_{\mathrm{L}} is the excitation frequency that can be determined by the footprint length, E is the imaginary part of the viscoelastic performance of the tread, \zeta is the magnification coefficient, v is the sliding speed of the rubber material, {\sigma }_0 represents the mean contact stress, h_0 denotes the mean excitation depth of the rubber material, H is the Hurst exponent that is determined from the surface roughness.

As shown in Fig.8, the kinematic friction coefficients of the tread rubber on the AC, OGFC, and SMA runway surfaces were calculated based on Eqs. (1)–(3). The figure shows that the magnitudes of the friction coefficients of the three runways coincided with their roughness. At high speeds (above 60 km/h), the OGFC runway maintained a friction coefficient of about 0.4, while the friction coefficients of SMA and AC runway were about 0.38 and 0.3, respectively. The friction coefficient is largely responsible for braking performance during high-speed landings.

3 Computational efficiency and accuracy

3.1 Computational efficiency of the finite element and smoothed particle hydrodynamics model

The hydroplaning model computed by the SPH algorithm and the CEL algorithm were developed separately to compare their computational efficiency. Both of them used the same tire and road models. The difference was that the Eulerian domain and SPH particles were built independently in SPH and CEL, for FSI simulation. As shown in Tab.6, the CEL algorithm has approximately 240000 more elements than there are in the SPH-based model. Since SPH only performs FSI simulation through particles, it has many more nodes than the CEL algorithm. It should be noted that the size of the minimum Eulerian element was 2 mm, while the diameter of the SPH particle was 0.75 mm. This implies that the SPH model can better simulate the interaction between water and runway. Moreover, the processing times of SPH and CEL models were 11.8 and 13.9 h, respectively. The computational efficiency of the SPH model was 15% higher than that of the CEL model.

3.2 Validation of the hydroplaning model

The first empirical equation to explain the hydroplaning phenomenon was proposed by NASA. based on numerous experiments, with the resulting empirical relation presented in Eq. (4). The NASA equation describes the relationship between the critical hydroplaning (v_{\mathrm{h}}) speed and the inflation pressure (p) [22]. The maximum load on a single tire was about 10 kN, and the water film thickness was 7.62 mm, in the NASA experiments. The same boundary conditions are used in our numerical simulations. As shown in Fig.9, the maximum error between the numerical simulations and the experiments is about 7.2%, and the average error is about 5.4%. The results show that the model is reliable at predicting hydroplaning behavior.

(4)v_{\mathrm{h}}=6.36\sqrt{p}.

4 Results and discussion

4.1 Contact force of aircraft tire

To study the hydroplaning risk when aircrafts land on runways covered with different water film thicknesses, the SPH inlet was set to sizes of 0.75, 1.5, 2.25, 3, 3.75, 4.5, 5.25, 6, and 6.75 mm, representing different thickness of water films. The tire pressure and load were 1.17 MPa and 66 kN, respectively. Fig.10 shows the contact force of the simulations, where tire skid about 300 mm distance on runway model. First, when contact between the tire and the water film first occurs (initial contact), the contact force is not stable. While the tire slides for a distance of about 170 mm, the contact force gradually stabilizes. The distance covered is approximately equal to the length of the tire footprint. Secondly, the force fluctuations during the initial contact stage become more pronounced at higher landing speeds. In addition, the 30 and 80 km/h landing speed curves are already stable when the skidding distance exceeds 170 mm, while the 130 km/h landing speed curve becomes stable when the distance exceeds approximately 200 mm. It is important to select the time, and not only the distance, to reach stable contact for different landing speeds. The average force during the stable stages is used to investigate the hydroplaning phenomenon on the wet runway.

4.2 Effect of surface texture

Although an asphalt runway mainly uses an SMA-13 gradation asphalt mixture, according to the specification, the mechanism of hydroplaning on different surface textures remains relevant. Therefore, four different runway surface models were established including flat and rough runways with gradations of SMA-13, AC-13, and OGFC-13. Using the SPH model, 10 different water film thicknesses, 0.75, 1.5, 2.25, 3, 3.75, 4.5, 5.25, 6, 6.75, and 7.5 mm, were established. The aircraft tire load was 66000 kN, the tire pressure was 1.14 MPa, and the slip ratio was 20%. From the simulation, it could be observed that the contact area decreased to 24800 mm² when landing on a runway with a thin water Film. This indicated that a full hydroplaning did not occur. Fig.11 shows the simulation results relating to the contact area between the runway surface and the tire under a thin water film. When the water film thickness was less than 2.25 mm, full hydroplaning was not observed on flat, SMA, or AC runways. A similar phenomenon could be found in the OGFC runway when the water film thickness was less than 3.75 mm. The reason for this is that the OGFC asphalt mixture has a better surface texture that relieves hydrodynamic pressure during landing.

Fig.11(a) shows that at a low speed (30 km/h) a larger contact area between the flat runway and the tire is kept. However, the contact area decreases as the landing speed increases, and the rate at which the contact area decreases is related to the depth of the water film. Specifically, the contact area of a film thickness of 2.25 mm was 57000 mm² at a speed of 30 km/h As the landing speed increased to 280 km/h, the contact area decreased to 20000 mm². At low landing speeds, the contact between the tire and the runway surface is very tight, so the water is evacuated through the sides of the tire. However, this did not provide enough room for water drainage. Therefore, the contact area decreased rapidly due to the rapid increase of the hydrodynamic force. As shown in Fig.11(b), the contact area of the 0.75 mm curve was 60000 mm² at a landing speed of 30 km/h, while the contact areas of the 1.5 mm and 2.25 mm curves, at the same speed, were 48000 and 47000 mm², respectively. This means that a water layer was trapped between the runway and tire at low speeds when the depth of the water film exceeded the depth of the runway’s mean texture. However, as the speed increased to 280 km/h, the contact area of the 2.25 mm curve decreased to 25000 mm². This suggested that while the contact area of a rough runway at low speeds was smaller than that of a flat runway, the rate of decrease of contact area for a rough runway was also smaller than that for a flat runway. Therefore, the contact area for the rough runway was higher than the flat runway at high landing speeds. As illustrated in Fig.11(c), the reduction in contact area on the SMA runway was significantly less than on the smooth and AC surfaces. Specifically, at a speed of 30 km/h, the contact area for the 2.25 mm curve measured approximately 58000 mm². However, with an increase in speed to 280 km/h, the contact area decreased to 53000 mm². The MTD of the SMA runway was higher than that of the AC runway. Therefore, the contact area and the decreasing rate were higher than those of the AC runway.

As shown in Fig.11(d), the aircraft tire was not hydroplaning on the OGFC runway until the water film depth exceeded 3.75 mm. The contact area of the OGFC runway at a speed of 30 km/h was the lowest out of the four runway types. This was because the OGFC mixture had the highest MTD and contained a small proportion of fine aggregates. Therefore, the contact area between the OGFC runway and the tire was very small even at low speeds. However, the decreasing rate of the contact area as landing speed increased was also the lowest, due to the high MTD.

Fig.12 shows that the OGFC runway has the highest hydroplaning speed out of all the runways. An Aircraft will not have a hydroplaning risk when landing on the OGFC runway when the water film thickness is between 3 and 3.75 mm. The results suggest that tires landing with a water film depth above 3 mm on the SMA and AC runway may be at risk for hydroplaning. However, the hydroplaning speeds of the SMA runway are higher than those of the AC runway. When the water film thickness exceeds 4.5 mm, the hydroplaning speed on the OGFC roadway is higher than those of the AC, SMA, and flat runways. This is because the MTD of OGFC is the highest. On the flat runway, the hydroplaning speed is the lowest because the surface does not have texture and water cannot drain away from it. These results indicate that as the roughness of the runway increases so the risk of hydroplaning reduces.

4.3 Tire stress during hydroplaning

To analyze the deformation and stress of an aircraft tire footprint, the contact force and contact area of tread blocks are obtained separately. The tread blocks are shown in Fig.13. An OGFC runway model and the water film model, with depths of 0.75, 1.5, 2.25, and 3 mm were built. The load of the aircraft tire was 66000 kN and the tire pressure was 1.14 MPa. The slip ratio was set to 20%.

The contact forces at the different tread bands were shown in Fig.14. It can be seen that the contact force at tread block 3 (the middle part of the tread pattern) is the highest. This indicates that tread block 3 bears the major part of the load. Moreover, the reduction in contact force of tread block 3 is the highest, due to increasing landing speed. It thus shows that it was significantly affected by the hydrodynamic force. A better-designed tread block at this location will increase landing safety. Additionally, the contact force decreased in a nonlinear manner. Fig.14(c) shows that contact force decreased significantly when speed increased from 180 to 230 km/h compared to the change during increase from 30 to 180 km/h. Accordingly, if the drainage of the front part of the runway where the aircraft first landed was increased, the landing safety could be significantly improved. The structural strength of the runway would not be affected.

The contact areas of the different tread brands were shown in Fig.15. The contact areas of tread blocks 1, 2, 4, and 5 hardly changed with an increase in speed. This indicates that the contact area for these tread blocks was not affected by water films. However, the declining trend of tread block 3’s contact area is consistent with a decreasing force. This indicates that during hydroplaning, the water film first wedges into the middle part of the footprint rather than its sides.

4.4 Effect of permeability coefficient

Apart from surface texture, air voids of asphalt mixtures also contribute to the runway's permeability coefficient. We set up five different permeability coefficients of 0, 2%, 4%, 6%, and 8% for the SMA, AC, and OGFC runway models, to investigate the effect of the air void on hydroplaning. It should be noted that the permeability coefficient is the proportion of the discharged water from the runway.The water film thickness was set to 6 mm in the simulation. The tire load was 66000 kN, the tire pressure of the wheels was 1.14 MPa, and the slip ratio of the wheels was 20%.

As shown in Fig.16, the aircraft will not hydroplane when landing on a wet runway when the permeability coefficient is above 6%. When the runway permeability coefficient is below 6%, the order of the hydroplaning speeds obtained from the simulations is OGFC > SMA > AC. It is consistent with the order of the MTD for asphalt mixtures.

The distribution of water film on the SMA runway was further analyzed through the numerical simulations. Fig.17 shows the water film only within the tire grooves and on the troughs of the surface texture, where the blue area and the gray area represented the water and the runway separately .When the landing speed was 180 km/h, 50% of the front part of the footprint was completely covered by the water film at permeability coefficients of 2%, 4%, and 6%. However, Fig.17(c) shows that when the permeability coefficient was 8%, water discharged effectively from the texture and voids. It suggested that the full tire footprint keep contact with the runway model, as the gray area appeared in all footprint.

The stress distribution of the tire footprint was further analyzed as shown in Fig.18–Fig.21, as the green and red part denotes the higher pressure on tire tread rubber. When a tire lands on a runway model with permeability coefficients of 2%, 4%, and 6%, the rear part of the footprint can remain in contact with the runway surface, while the front part of the footprint is separated from the runway surface because of the hydrodynamic force. Therefore, there is no contact stress at the rear of the tire footprint. In the case of the runway with a permeability coefficient of 8%, the entire footprint of the tire remains in contact with the runway. Furthermore, by increasing the permeability index from 2% to 8%, the maximum contact stress in the footprint area decreases from 2.7 to 1.2 MPa. This indicates that the contact stress of the tire decreased when the runway has an adequate drainage system.

5 Conclusions

In this study, an FE-SPH model for aircraft tire hydroplaning simulation is presented. Based on the materials and cross section of an aircraft tire, a model of an aircraft tire was constructed. Based on the surface scanning and 3D reconstruction, four runway models were constructed, including flat runway models and rough runway models with grades of AC, SMA, and OGFC. The water film models with thicknesses from 0.75 to 7.5 mm were incorporated in the SPH model. An interaction model for the runway and aircraft tire was developed using the runway’s PSD and the viscoelastic property of the rubber tire. The FE-SPH model was validated by reference to the NASA equation. Based on the FE-SPH analysis, the following conclusions were drawn.

1) The computational times of the FE-SPH model and traditional CEL model were 11.8 and 13.9 h, respectively. This indicates that the computational efficiency of the FE-SPH model was 15% higher than that of the CEL model. Moreover, the FE-SPH model could simulate the hydroplaning behavior on a thin water film.

2) There was a decreasing trend of contact area between the runway and tire on the runway model of flat, SMA, and AC surface types as water film thickness increased to values 0.75, 1.5, and 2.25 mm. However, the fact that the contact area was not zero proved that the tire would not completely hydroplane on these runways. Similar results were obtained on the OGFC runway with water film thicknesses of 0.75, 1.5, 2.25, 3, and 3.75 mm. By further increasing the water film thickness, the potential for a hydroplaning risk as the aircraft lands will increase.

3) Simulated results showed a change in contact force and contact area at tread bands. Based on the decreasing contact force with increasing landing speeds, it could be concluded that the largest reduction in contact area occurred when the speed increased from 180 to 230 km/h This trend indicated that hydroplaning should be a concern during this stage of landing on a wet runway. The largest decrease in contact area occurred in the middle of the. There was, however, little change at the sides of the footprint.

4) The air voids in the asphalt mixture provided a means of drainage from the runway surface. A permeability coefficient was proposed in this study to simulate the drainage in the asphalt mixture. The tire would not hydroplane when landing on a wet runway when the permeability coefficient exceeded 4%. However, when the runway permeability was under 6%, the order of hydroplaning speeds obtained from simulations was OGFC > SMA > AC. The observation of water film distribution and contact stress indicates that the front part of the tire was not in direct contact with the runway surface due to the presence of a water filmwhen landing on a runway with permeabilities of 2%, 4%, and 6%. However, when the permeability coefficient increased to 8%, the tire could maintain full contact with the runway surface.

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