## 2 Friction Phenomenon
### Sliding Friction
#### 2.2.2 Dynamic Hydroplaning  
In contrast to viscous hydroplaning, dynamic hydroplaning is the result of the hydrodynamic forces developed when a tire rolls over a water-covered surface [26]. It occurs when the amount of fluid encountered by the tire exceeds the combined drainage capacity of the tire tread pattern and the pavement macrotexture. If the aircraft speed is sufficiently high, the uplift force developed in the fluid film is comparable with the tire inflation pressure. This causes the tire surface to buckle and produce a large region of fluid capable of supporting the loaded tire. In these conditions, a loss of contact between the tire and the pavement occurs, resulting in an extremely low coefficient of friction [27]. Dynamic hydroplaning modeling has been studied in several studies [23]. A contact model between the tire and the surface, including water film between locked tires and the surface, has been developed [28]. The water film depth was calculated at different distances from the center line, as well as the friction coefficient at different groove depths and water film thickness conditions. It was found that the water film increased, as the distance to the center line increased, which can increase the safety risk of landing when an aircraft wanders away from the center of the runway. At the same time, surface groove depth was found to be important, due to its influence on the water film thickness and the hydrodynamic pressure under the tire. In a similar study [29], a hydroplaning model was developed and was used to evaluate the minimal maintenance level of groove depth according to Federal Aviation Administration (FAA) regulations. According to the study, there is a high risk of hydroplaningwhen the groove width and depth are less than 3 mm. Below this threshold, the rate of deterioration of skid resistance increased rapidly, which verifies the appropriateness of the groove maintenance threshold set by the FAA regulations [30]. Hydroplaning speed also depends on the tire groove pattern. Modeling results show that tires with circumferential non-interconnected grooves are more susceptible to hydroplaning. It has been shown that the combination of circumferential and transverse tire groves provides the highest hydroplaning speed when other factors are constant [31]. The same results appear in the case of runway grooving, with transverse groves consistently producing higher hydroplaning speeds than longitudinal grooving [32]. Aircraft tires, however, can only use a longitudinal groove pattern due to intense wear during landing [33]. This means that the provision of surface macrotexture is more important for runways than it is for roads. For practical hydroplaning speed estimation, Equation (2) is commonly used [34]. This equation was established by NASA in the 1960s, during tests on a flooded runway. Later, in the 1970s, NASA modified an equation after investigations of aircraft hydroplaning accidents, suggesting that the spin-up hydroplaning speed for a non-rotating tire, as occurs at aircraft touchdown, might be lower than implied by Equation (2). As a consequence, estimated hydroplaning speed was reduced by 15%, as detailed in Equation (3) [35; 36]. Figure 3: Water film between tire rubber and the sinusoidal smooth asperity of the surface [24]. \[v_{P}=9\sqrt{P} \tag{2}\]  
\[v_{P}=7.7\sqrt{P} \tag{3}\]  
where \(v_{P}\) is the ground speed in knots and \(P\) is the tire pressure in lb/in\({}^{2}\). Equations (2) and (3), however, do not include water film thickness. It is well known that hydroplaning speed can change with a change in film thickness [37]. The analysis, however, is complex due to variation in surface height and the small size of irregularities in the geometry. Consequently, the tire does not fully come into contact over the entire surface of the pavement, because it is not flexible enough to conform to all the small depressions [38; 39]. Loss of contact between some parts of the tire and the pavement surface occurs during normal braking in wet conditions as well [34].