PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

CFD modeling of pressure variation in a road cavity with volume
variation
Marianne BOU LEBA BASSIL1; Julien CESBRON2; Philippe KLEIN3

3

1

IFSTTAR, CEREMA, UMRAE, F-44344 Bouguenais, France

2

IFSTTAR, CEREMA, UMRAE, F-44344 Bouguenais, France

Univ Lyon, IFSTTAR, CEREMA, UMRAE, F-69675, Lyon, France

ABSTRACTThis paper deals with the variation of air volume at the contact interface between a rolling tyre and a roadcavity leading to aerodynamic phenomena such as air-pumping. A numerical model was developed (in 2D
and 3D) based on Computational Fluid Dynamics, using the Fluent solver, coupled with a displacement
function of the bottom of the cavity representing its volume variation. Results showed that the maximum
pressure reached at the bottom of the cavity during its complete closure increased as its volume decreased.
The pressure variation was related to the volume variation by a relationship close to Laplace law for perfectgases. In addition, pressure oscillations after the opening of the cavity increased in amplitude and frequency
as its volume decreased. Moreover, in the case of the experimental setup of Hamet (1990), introducing thevolume variation estimated by Laplace law in the 3D model has shown better agreement between thecalculated pressure and the measured pressure in comparison with the model neglecting this volume variation.
Future work would therefore be to obtain the actual volume variation caused by the penetration of the tyretread inside the cavity. This could be done either experimentally or numerically using a tyre/road contact
model.
Keywords: Tyre/road noise, Air-pumping, Computational Fluid Dynamics

1. INTRODUCTION
Several generation mechanisms are responsible for tyre/road noise emission. Among these,
air-pumping is an aerodynamic source that contributes to rolling noise in the medium and highfrequency range. It is usually meant by this term a series of rapid air compression and release at thetyre/road interface and the possible associated resonances.
Air-pumping phenomenon was often attributed to variation of air volumes due to the rubber
deformation in the contact patch caused by road asperities indentation (1) or tyre grooves squeezing(2–4). Recent works based on Computational Fluid Dynamics (CFD) have shown that the effect ofviscous boundary layer at the tyre and road surfaces also contributes to the generation of pressure
fluctuations that are likely to produce noise emission (5,6).
This paper deals with the rather simple geometrical case of a slick tyre rolling on a cylindricalcavity inserted in a smooth road surface. This case was experimentally addressed in Hamet (7,8) andshowed a three-phase process: first a gradual air pressure increase in the cavity as the tyre approaches,then a constant overpressure stage during the full cavity closure by the tyre, and finally a pressure
release with an Helmholtz resonance when the tyre leaves the cavity. This case was investigated byConte (5,9) with CFD simulations without considering the tyre rubber penetration in the cavity.
Although the mechanism at stake was qualitatively reproduced, there was a significant
underestimation of the overpressure inside the cavity compared to experimental results.
In this paper both viscous layer effect and volume variation due to the tyre rubber penetration in the1marianne.bassil@ifsttar.fr
julien.cesbron@ifsttar.fr
3 philippe.klein@ifsttar.fr
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cavity during its closure are considered in CFD simulations. The volume variation is introduced in a
simplified manner. Its influence is highlighted regarding the overpressure reached in the cavity. The
resulting pressure waves travelling to the front and to the rear of the tyre are compared too.

2. RESULTS OF CFD MODEL WITHOUT VOLUME VARIATION
2.1 Context
Solving fluid dynamics problem usually requires the calculation of various flow properties such as
velocity, density, pressure, and temperature as functions of space and time. The equations that govern
the motion of a fluid are the Navier-Stokes equations and its derivatives. These are nonlinear
differential equations describing the movement of fluids. They are obtained by applying conservation
laws such as the conservation of mass (also called the continuity equation), the conservation of
momentum (Newton's second law), and the conservation of energy. These equations, when they are notsimplified, do not have analytical solution and are therefore only solvable with numerical simulations
based on Computational Fluid Dynamics (CFD). The ANSYS Fluent software is based on the finite
volume method to solve the Navier-Stokes equations governing the fluid flow. The model chosen for
the numerical simulation is the one developed during the work of (5,10) (2D model) and (9) (3D
model) simulating the passage of a smooth tyre on a cylindrical cavity. In the case of this air-pumping
study, the numerical resolution of the problem enables the calculation of the dynamic pressure of the
air at the bottom of the cavity during rolling. The same geometry and mesh were used but someparameters were modified in the method for solving the problem in Fluent.
2.2

CFD model

2.2.1 Configuration
A sketch of the general concept of modeling is represented Figure 1: a slick tyre with a diameter of631.4 mm rolls on a cylindrical cavity inserted in a smooth road surface with a diameter of 15 mm anda depth of 30 mm at a speed of 22.2 m/s (80 km/h). The chosen parameters enable to compare theresults with the measurements reported in Hamet (7) for the same dimensions.

Figure 1 – General concept of the CFD model of a rolling tyre on a cylindrical cavity
Figure 2a shows the 2D geometry of the model and its mesh configuration. The fluid domain aroundthe tyre is formed by a half-disc of 2.5 m radius divided into sub-domains according to their physicalimportance with respect to the flow: the most critical zones are the regions in front and behind thecontact zone and the cavity. These areas are meshed finer than the others. The mesh becomes all themore coarse as one moves away from the contact zone.
The 3D geometry of the model and its mesh configuration are represented in Figure 2b. We consider
half the field of computation by introducing a condition of symmetry in the vertical plane cutting thetyre and the cavity in order to reduce the computational cost. The area around the tyre is thus
constituted by a quarter sphere of 1 m radius. This sphere is then divided into several subdomains forthe same mesh criterion as for the 2D case. The static deformation of the tyre was considered for both
2D and 3D cases. The corresponding contact length is 92 mm in the rolling direction.

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a)

b)

Figure 2 – a) 2D meshed geometry; b) 3D meshed geometry
2.2.2 Method of resolution
The reference frame is chosen at the center of the wheel and therefore the tyre is considered torotate with an angular velocity 𝜔 of 71.7 rad/s while the cavity and the roadway are in translationalmotion with a velocity 𝑣 of 22.2 m/s. The sliding mesh technique at the cavity/road interface has been
adopted. The airflow on the wheel is not considered in this case due to its low contribution to theoverpressure generated in the cylindrical cavity and its negligible effect on the main mechanism
studied (5). Thus, airflows are generated only by the surface displacement and the development ofcorresponding boundary layers due to air viscosity.
The boundary limiting the artificially truncated domain leads to reflection of outgoing waves.
Non-reflective boundary conditions are imposed at the disk (2D case) or the sphere (3D case)boundary to control spurious wave reflections.
The considered flow is turbulent and controlled by the boundary layer phenomenon. A near-walltreatment is therefore necessary to solve the flow in the inner zone of the boundary layer. The 𝑘 − 𝜔
SST turbulence model is used to solve the equations, this model being the most appropriate for thistype of flow.
Since the flow is unsteady and considered compressible for enabling the evaluation of pressure
variations, furthermore a case of rotation, the resolution of the problem is based on the PISO scheme(implicit sequential method). The pressure-based solver is used to solve the problem in two steps. In afirst step, the cavity is fixed and the solution of a stationary flow is computed. This latter is used in asecond step as the initial condition to solve the unsteady problem with the cavity in translationalmotion. The cavity being at a distance of 0.2 m before the center of the contact zone, the simulation
duration is 0.018 s in order to have the cavity at a symmetrical distance from the beginning of thesimulation. The time step is ∆𝑡 = 2.5e -6 s for 2D simulations and 5e -6 s for 3D simulations (a coarser
spatial resolution is used in 3D).
2.2.3 Results and comparison to experimental results
Figure 3a shows the air pressure calculated in 2D and 3D and Figure 3b shows the measured airpressure at the bottom of the cavity reported in Hamet (7). The average maximum pressure measured isapproximately 1740 Pa while those calculated in 2D and 3D are respectively 828 Pa and 969 Pa. The
pressure difference ∆𝑃 between measured and calculated values is then 912 Pa in 2D and 771 Pa in 3D.
This difference can be explained in several ways. Firstly, the airflow around the tyre due to its
displacement was estimated by Conte (11) and can produce an overpressure around 200 Pa which ishere not taken into account in the simulation. Secondly, the dynamic deformation of the tyre tread andits penetration into the cavity are neglected. In the following, this paper focuses on the last
phenomenon. Indeed, the penetration of the tread in the cavity causes a slight reduction in the volumeof the cavity, which in turn increases the maximum pressure reached at the bottom of the cavity when
it is completely closed. In the next section, the volume variation is introduced in the CFD model (2D
and 3D simulations) to evaluate its influence on the variation of the overpressure at the bottom of thecavity.

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a)
b)
Figure 3 – a) Calculated air pressure at the bottom of the cavity; b) Measured air pressure (7)3. CFD MODEL WITH VOLUME VARIATION
3.1 Calculation of the volume variation
The volume variation is introduced in a simplified way, which is referred as the piston method, by
acting on the depth of the cavity (Figure 4). It is thus considered that the bottom of the cavity moves
vertically upwards as the cavity closes to reach a maximum value of displacement considered constantduring the course of the contact zone (plateau phase) and moves down symmetrically when it opens.

Figure 4 – Sketch of the piston displacement method to describe volume variation
In order to represent the displacement of the bottom of the cavity as a function of time,
noted 𝑦(𝑡) , a Tukey window function 𝑤(𝑡) (tapered cosine window) is used, which
satisfactorily fits the compression, plateau and expansion phases observed in the experimental results.
(𝑡) is obtained via the maximum displacement 𝑑 of the cavity bottom and the function 𝑤(𝑡)𝑦by the following equation:𝑦(𝑡) = 𝑑 𝑤(𝑡)(1)
The mathematical formula of the Tukey function 𝑤(𝑡) is as follows:
𝑤(𝑡) ={1 + cos(2𝜋[𝑡 − 𝑟⁄2]⁄𝑟)}⁄2 ,1,{1 + cos(2𝜋[𝑡 − 1 + 𝑟⁄2]⁄𝑟)}⁄2 ,0 ≤ 𝑡 < 𝑟 ⁄2
𝑟 ⁄2 ≤ 𝑡 < 1 − 𝑟 ⁄2
1 − 𝑟 ⁄2 ≤ 𝑡 < 1(2)where 𝑡 is the time vector at 𝑁 linearly spaced points. The parameter 𝑟 is the ratio between thelength of the tapered cosine section and the total length of the window with 0 ≤ 𝑟 ≤ 1.
The parametric study is performed in 2D, which is less time-consuming. The displacement of thepiston is adapted to the numerical model regarding the time discretization, start time, duration of eachphase, etc. as shown in Figure 5. In this case, 𝑟 = 0.279 and 𝑁 = 1927 points.

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The percentage of volume decrease (100 ∆𝑉 ⁄𝑉 ) of the cavity during the compression phase wasvaried according to the values [0; -0.2; -0.4; -0.6; -0.8; -1; -2; -3; -4; 5; -10; -15; -20; -25] (in %), whichcorresponds to a maximum displacement of the cavity bottom 𝑑 varying from 0 to 7.5 mm. 𝑉 is theinitial reference volume of the cavity (∆𝑉 ⁄𝑉 = 0 and cavity depth 𝐿 = 30 mm).

Figure 5 – Displacement of the cavity bottom as a function of time
3.2 Implementation of the volume variation into Fluent
Fluent processes the Dynamic Mesh (DM) together with the sliding motion technique in order tohandle the topological changes of the domain during the simulation. There are several methods ofdynamic meshing, whose most used are smoothing (this method moves the border and the inner nodes
to absorb the movement of the moving border) and layering (used to add or remove cell layers). In thepresent model, the smoothing method is used for volume variations up to 2 % and the layering method
is used for higher volume variations. The dynamic mesh is attributed to the bottom of the cavity
considered as the deforming zone.
(𝑡), calculated according to Eq. (1), is assigned to the cavity
The displacement function 𝑦bottom using the UDF (User Defined Function) option in Fluent solver, based on C language. Thisfunction describes the movement of the bottom of the cavity by assigning, for each node 𝑗 of thecavity bottom and each time step, its vertical position 𝑦 according to the following equation:(𝑡)𝑦 (𝑡) = 𝑦 − 𝐿 + 𝑦(3)
where 𝑦 < 0 is the vertical position of the road and 𝐿 is the reference depth of the cavity (𝐿 =30 𝑚𝑚). The average air pressure 𝑝(𝑡) is calculated at the bottom of the cavity. For each percentage
𝑖 of volume variation, 𝑝 is the maximum pressure reached during the contact phase.
3.3

Results

3.3.1 Influence of volume variation
Figure 6 shows the results of the pressure variation as a function of the volume variation withregard to the compression and plateau phases and the expansion phase (pressure oscillations). The
maximum pressure reached at the bottom of the cavity during the plateau phase is greater when thecavity volume variation is considered. The higher the volume decreases the higher the overpressure
increases. Then, the penetration of the tread causing the volume decrease of the cavity has a significant
impact on the overpressure at the bottom of the cavity during the passage of the tyre. The amplitude as
well as the frequency of the oscillations of the plateau increase when the volume decreases. In fact, thecavity is acting like a closed tube during this phase leading to the cavity resonance phenomenon
characterized by a frequency inversely proportional to the depth of the cavity. Similarly, the amplitude
as well as the frequency of the pressure oscillations during the expansion phase increase when thevolume decreases.

510

Figure 6 – Influence of the volume variation on the air pressure at the cavity bottom

P CFD (Pa)3.3.2 Relation between the pressure variation and the volume variation
For each percentage of volume variation, the variation of the maximum pressure ∆𝑃=𝑝 −𝑝
is evaluated, with 𝑝 the reference maximum pressure without volume variation. Figure 7 shows∆𝑃
values as a function of volume variation (lower red curve, “Simulations”).

Figure 7 – Variation of the overpressure ∆𝑃 with the variation of the cavity volume
This result can be compared to the classical Laplace law in thermodynamics, which gives therelation between the pressure and the volume of a perfect gas undergoing an isentropic transformation:
𝑃 𝑉 =𝑃 𝑉(4)
where 𝑃 and 𝑃 are respectively the absolute pressures of the gas after and before the change ofvolume (𝑃 = 𝑝 ,+ 𝑝, 𝑝
is the atmospheric pressure = 101 325 Pa and 𝑝 ,is therelative pressure of the gas), 𝑉 and 𝑉
are respectively the volumes of the gas after and before
the volume change, and 𝛾 =1.4 is the Laplace coefficient. By replacing 𝑉 by 𝑉 = 𝑉 (1 +∆𝑉 ⁄𝑉 ) in Eq. (4), the pressure variation ∆𝑃
calculated by Laplace law can be written:∆𝑃 = 𝑃 − 𝑃 = 𝑝 ,− 𝑝 = 𝑃 (1 + ∆𝑉 ⁄𝑉 ) − 1(5)
The theoretical overpressure ∆𝑃
is compared to the ∆𝑃obtained by the 2D simulations inFigure 7 (upper blue curve, “Laplace Law”). The variation of the maximum pressure reached at thebottom of the cavity follows a law which is close to Laplace law (difference between 6.8 % and 8.9 %).
3.3.3 Comparison with measurement
For very small volume variations with respect to the volume of the cavity, Eq. (5) can be linearized
as follows:∆𝑃= 𝑃 [1 − 𝛾 ∆𝑉 ⁄𝑉 − 1] ≅ −𝛾𝑃 ∆𝑉⁄𝑉
(6)
The latter equation can be used to calculate the cavity volume variation from the pressure
difference found between the measurements and the CFD model with no volume variation. The volume511

variation is then introduced in the CFD model using the piston method described in section (3.2) inorder to compute the resulting air pressure at the bottom of the cavity. According to the section (2.2),
for the 2D case, the pressure difference ∆𝑃 between the measurements and the calculation amounts to
912 Pa. By applying Eq. (6), the volume variation ∆𝑉 ⁄𝑉 to offset ∆𝑃 is estimated to be -0.64%.
This corresponds to a maximum displacement of the cavity bottom equal to 0.192 mm. Figure 8ashows the result of the 2D simulation corresponding to this displacement. A maximum pressure ofabout 1670 Pa is obtained, which is close to the measured value (4% error compared to themeasurements).
The same correction is performed with the 3D geometry for which the dynamic mesh of the piston
is modeled by the layering method, most suitable for the 3D mesh of the cylindrical cavity. From thesection (2.2), the difference between the measurement and the calculation is 771 Pa. By applying
Eq. (6), the corresponding volume variation ∆𝑉 ⁄𝑉 is -0.54%.
This corresponds to a maximum displacement of the cavity bottom equal to 0.162 mm. Figure 8b
shows the result of the 3D simulation corresponding to this displacement. The maximum pressure isabout 1605 Pa, which is also close to the measured value (7.7 % error). Furthermore, by comparing thepressure oscillations of the expansion phase with those reported by Hamet (7), a good agreement isfound in terms of amplitudes and frequencies. Note that volume variation has a weak influence on theoscillations.
V/V 0%
V/V 0.54%

1500

Pressure (Pa)

1000

500

0

-500

-1000

-1500

0.006

0.008

0.01

0.012

0.014

0.016

0.018

Time (s)Pressure at the trailing edge (Pa)

Pressure at the leading edge (Pa)a)b)
Figure 8 – Results of the CFD model with volume variation a) 2D; b) 3D
On the other hand, Figure 9a and Figure 9b show the 3D air pressures calculated respectively at theleading and the trailing edges (located at 20 cm on both sides of the contact zone), compared to thecase without volume variation. There is a decrease in amplitude of the pressure wave generated at theleading edge (more pronounced at the minimum of pressure than at the maximum), with the decreaseof volume. On the other hand, the amplitude of the oscillations at the trailing edge slightly increases(by 17% approximately) with the decrease of volume.

a)b)
Figure 9 – Pressure calculated in 3D: a) at the leading edge and b) at the trailing edge512

4. CONCLUSIONSThis paper highlights the influence of the volume variation of the cavity on the dynamic pressure of
the air at the tyre/road interface during the rolling of a smooth tyre on a cylindrical cavity incorporated
in the pavement. The variation of volume has been introduced in a simplified piston method byconsidering that the bottom of the cavity moves vertically upward as the cavity closes to reach amaximum value of displacement considered constant during the course of the contact. Then, thebottom of the cavity moves down symmetrically when it opens.
A 2D calculation with the commercial code ANSYS Fluent allowed to make a parametric study andto deduce a relation between the variation of the maximum pressure reached at the bottom of the cavity
and the variation of volume of the latter during the contact. The overpressure generated during thecontact increases with the decrease of the volume of the cavity according to a relation close to
Laplace's law indicating an isentropic transformation (reversible and adiabatic). These data were thenused for the comparison between the simulations of a smooth tyre running on a cylindrical cavity andthe experimental results of (7) using the same parameters. The calculations are close to themeasurements when the volume variation of the cavity is introduced into the numerical model. For thisparticular case of a fully closed cavity, although the rubber penetration strongly influences thepressure in the cavity, it seems to have little effect on the noise generated.
The volume variation has been estimated based on Laplace law and an approximate function. It
would therefore be interesting to find the exact volume variation of the cavity instead of estimating it.
This could be done by using two methods. The first one could be a numerical method using atyre/pavement contact model for calculating the penetration of the tread into the cavity during therolling and thus deducing the resulting volume variation. The second one could be an experimental
method, which consists of measuring the penetration of the tyre tread by specific displacement
sensors.

ACKNOWLEDGEMENTS
This work has been jointly funded by the French Region of Pays de la Loire and IFSTTAR.

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