1. 1
F2010-C-098
From Soiling to Stone Chipping –
Simulation of Particle Trajectories and Impact in
Time Averaged and Transient Flow Fields around Vehicles.
1
Grün, Norbert*, 1
Schönberger, Andreas, 2
Schulz, Martin
1
BMW Group, Germany, 2
science+computing AG, Germany
KEYWORDS – Aerodynamics, CFD, Simulation, Soiling, Stone Chipping
ABSTRACT
This paper describes the extension of the soiling simulation module in PowerVIZ, the visua-
lizer of the CFD-tool PowerFLOW®
, to enable realistic modeling of stone chipping. To start
with, the basics of computing the movement of particles in a given flow field are explained
briefly. The status is demonstrated on some examples for soiling and simulating a windshield
washer system. The bulk of the paper focuses on the extensions necessary to apply this me-
thod to stone chipping.
First of all this requires the inclusion of reflecting surfaces with a scatter on the outgoing di-
rection as well as a surface normal momentum loss due to imperfect elastic collision. Further,
the momentum transfer from surfaces moving relative to the vehicle body, like the floor and
rotating wheels, has to be accounted for.
Of paramount importance for the practical application is the knowledge of realistic initial
conditions. For this purpose the ejection of stones from the front wheels was recorded with
high-speed cameras and evaluated in terms of velocity magnitude, direction and scatter.
The damage probability is judged by the distribution of the locally surface normal momentum
of particles at the instance of their impact. First results of identifying critical regions are con-
firmed by field tests.
INTRODUCTION
For long term quality of an automobile it is crucial to protect critical locations from the im-
pact of gravel for several reasons. Obviously stone chips are annoying on the visible, painted
parts of the vehicle body but more dangerous are damages to coatings on structural parts
which in the long term may lead to corrosion and subsequently to potential failures.
Traditionally the critical regions could not be identified reliably before roadworthy prototypes
were available and hence too late for effective and low-cost remedy.
1. FLOW FIELD
Nowadays, computational fluid dynamics (CFD) is well established as a simulation tool in the
aerodynamic development process. At the BMW group a commercial Lattice-Boltzmann code
(PowerFLOW®
) is used to simulate the flow around (and through) passenger cars and motor-
cycles.

2. 2
Details of its deployment have been published in [1-4]. These flow fields are characterized by
massive separation with high total pressure loss and strong vortical phenomena. Since even at
moderate speeds typical Reynolds-numbers
(1)

fLv Re
Re



with the vehicle length as reference length LRef, are in the order of 105
, there is hardly any
region of laminar flow. Further, these bodies generally produce a highly transient behavior of
the flow and thus are simulated in time-dependent mode. Nevertheless, results can be aver-
aged over selected time intervals subsequently. An impression of the complexity of these
flows is given in Fig.1.
Figure 1: Time averaged flow field around a passenger car and a motorcycle
(Streamlines, near-surface velocity magnitude and isosurface of total pressure pt=pt∞-q∞)
2. PARTICLE MOTION
Strictly speaking the inclusion of solid particles constitutes a two-phase flow. However, when
looking at soiling and stone chipping the fraction of solids is so low that their influence on the
air flow can be neglected. Therefore the simulation of particles is conducted in a given flow
field as a postprocessing step without feedback to the air flow and also disregarding particle-
particle collisions.
2.1 Particle Properties
In reality the particles considered here have an arbitrary shape,
which means that the air flow will impose aerodynamic forces
and moments in all directions. Since it is not feasible to account
for this, it is assumed that they have a spherical shape. Particle
mass vs. diameter for water, dry sand and stone is depicted in
Fig. 2. Despite the apparently tiny mass of pebbles, for instance
m 0.1g with D=4mm diameter, we will see later that this still
gives them enough momentum to overcome some, but not all of
the aerodynamic drag.
Figure 2: Mass of spherical particles

3. 3
The assumption of spherical particles eliminates aerodynamic moments and the only remain-
ing force component is the drag in the direction vr of the resulting oncoming flow when the
particle is moving through a given flow field, see Fig.3 below.
Figure 3: Resulting oncoming flow vr of a particle moving with velocity vP through a given flow field vAir
Numerous publications have dealt with the drag coefficient of spheres
(2)
  222
8
2/1 



v
F
DAv
F
C DD
D

as a function of Reynolds number Re = v· D/ . Here we use an approximation from [5]
(3)   16.1
687.0
Re425001
42.0
Re15.01
Re
24


DC
which appears to be valid up to the critical Reynolds numbers of about 3·105
(Fig.4).
However, as can be seen in Fig.5, exceeding this limit requires either diameters and/or veloci-
ties which are beyond the scope of this application.
Figure 4: Drag coefficient of a sphere Figure 5: Reynolds number vs. velocity
vs. Reynolds number for different sphere diameters
2.2 Equation of Motion
The CFD simulation takes place in a vehicle fixed frame of reference where the vehicle at rest
is exposed to an airstream of velocity v∞ like in a wind tunnel. Geometries moving in this
frame relative to the vehicle might be rotating wheels and the floor. The momentum of spher-
ical particles is only affected by gravity and the aerodynamic drag caused by the resulting
oncoming flow vr as already depicted in Fig.3.

4. 4
With these simplifications the equation of motion reads
(4) 






r
r
rD
P
v
v
vDCgm
dt
vd
m



22
2
1
4


and after inserting the particle mass ( m=P·/6·D³ ) the acceleration is
(5) 






r
r
r
D
P
P
v
v
v
D
C
g
dt
vd



2
4
3


The numerical integration of a particle’s path must find the balance between calculation speed
and precision. Adaptive embedded Runge-Kutta tracers have been evaluated in depth in nu-
merical analysis and therefore are the preferred choice [6]. Thus, a Runge-Kutta integration
scheme of order 4(3) is fully sufficient for locally refined Cartesian grids with basically trili-
near interpolation. The minimum and maximum integration step size depends on a heuristic
taking into account the range of cell sizes and velocities within the fluid flow. The given max-
imum error controls the step size choice to compute the particle's position after a determined
period of time.
2.3 Particle Reflection
The reflection of particles from the vehicle surface or the floor is governed by the physics of
solid body collision, i.e. by conservation of momentum and energy. However, for two reasons
we have a special case here. Firstly, due to the assumption of spherical particles we ignore
any rotational effects. Secondly, the mass ratio mParticle / mVehicle  0 implicates a complete
transfer of the (surface normal) velocity from the vehicle to the particle, provided that the
material pairing would produce a perfectly elastic collision. In reality a certain fraction of
kinetic energy will not be recovered due to deformation of either body, described by a coeffi-
cient of restitution, sometimes denoted as CoR.
(6) innoutn
inn
outn
vv
v
v
,,
,
,


 
This coefficient can be obtained experimentally quite simple by drop tests from the ratio of
rebound height h to drop height H.
(7)
H
h

Depending on the materials this value will be between =0 for perfectly inelastic and =1 for
perfectly elastic collision.
If the surface on which a particle impinges is moving in the frame of reference, for instance
the floor underneath a car or the tires and rims of rotating wheels, part of its tangential mo-
mentum may be transferred to the particle, depending on the surface properties. This effect
can be described by a coefficient of kinetic friction (CoF) defined as
(8)  intsrfintoutt
intsrf
intoutt
vvvv
vv
vv
,,,
,
,,



 

5. 5
Note that the particle may also be decelerated if vsrf < vt,in. Obtaining values for the coefficient
φ requires much more intricate tests than for  which are still outstanding.
Figure 6: Velocity components before and after collision
With the above derived velocity components after collision the angle out at which a particle
is reflected from the surface is (Fig.6).
(9)
  )tan(
1/1
)tan(
,
in
intsrf
out
vv



 


Only in case of a perfectly elastic collision ( =1) and a perfectly smooth surface ( = 0), the
particle would be reflected like light on a mirror with out = in. On all surfaces at rest  = 0
and therefore the denominator cannot vanish.
The influence of particle mass is demonstrated with a generic case in Fig. 7. Stones of equal
diameter but varying density are emitted from the floor upstream of the vehicle at a steep an-
gle with a velocity equal to that of the oncoming flow.
Figure 7: Influence of particle mass (decreasing order: red > yellow > cyan > green)
For very heavy stones (red path) the trajectory is actually a ballistic curve until multiple colli-
sions successively reduce momentum. Reducing the mass more and more (yellow  cyan 
green) demonstrates how the trajectory is flattened right from the start when aerodynamic
drag prevails inertial forces. The path of stones with diameters up to 7mm, mostly responsible

6. 6
for surface damage, would lie between the green and the cyan curves. This emphasizes that
the influence of aerodynamics on stone chipping cannot be neglected and methods using only
ballistics will not produce meaningful results.
2.4 Stochastic Scatter
So far all considerations have assumed ideal conditions. Real-world effects are modeled by
applying a stochastic scatter on various parameters like particle density and diameter, initial
velocity magnitude and direction as well as on the direction of surface reflection. This way
the real behavior of irregularly shaped particles, particularly during reflection, is at least partly
accounted for.
3. EXPERIMENTAL INVESTIGATION OF INITIAL CONDITIONS
Even with accurate trajectory integration the usability in practice depends strongly on realistic
initial conditions. Furthermore, some knowledge of the afore mentioned stochastic is required.
For this purpose the ejection of particles from the wheels was recorded with high-speed cam-
eras at 1000 fps and evaluated in terms of velocity magnitude, direction and scatter.
In earlier work [7] the behavior of dry and wet soiling had been recorded with the camera
mounted on the vehicle while driving on a proving ground (Fig. 8, left). However, when ob-
serving individual pebbles it turned out that the lighting was insufficient. Therefore in [8] and
[9] these tests were conducted with a stationary camera on a crash test facility where high
intensity floodlights were available (Fig. 8, center and right). Gravel with a diameter varying
between 2 and 7 mm was applied on the ground with 200 g/m² which is equivalent to the size
and rate at which a gritter disperses split on the road. With a density between 2600 kg/m³ for
basalt and 2950 kg/m³ for dolomite this means the particle mass spreads between 0.01 g and
0.2 g.
Figure 8: Experimental setups (left: vehicle fixed [7], center and right: ground fixed [8,9])
Different mechanisms of entrainment could be observed. The majority of stones adhere to the
tire due to elastic deformation of the rubber when being overrun and is released when the rub-
ber expands again. If stones get pinched in the tread they separate later at a higher position, in
particular under wet conditions where the water-filled grooves impose a suction effect on
stones. To determine velocity magnitude and direction the high-speed movies (1000 fps) are
analyzed by a tool which identifies and tracks the position of individual stones from frame to
frame.
Other than one might think, the tire does not really “eject” the stones - as long as there is no
noteworthy wheel spin. Monitored in a ground-fixed frame of reference the stones are basical-
ly only lifted in vertical direction without getting a significant longitudinal component.

7. 7
Only the transformation in the vehicle-fixed frame of reference makes their movement look
like a jet. Fig. 9-left indicates the sectors of initial stone trajectories for adhesion (lower) and
pinching (upper). Lateral pinching (Fig. 9-right) is only getting significant when a blocked
wheel is sliding and pushing the gravel.
upper sector : entrained by pinching lateral pinching
lower sector : entrained by adhesion
Figure 9: Sectors of initial stone velocities [9]
Sector angles in Fig.9-left decrease slightly with velocity but there is a considerable spread
due to parameters like tire hardness, temperature and wetness, tread pattern, and sharpness of
stones. All these effects are accounted for by the stochastic scatter mentioned before.
4. SELF SOILING
First validations of self soiling using the time averaged flow field had been conducted in [10].
Fig. 10 compares the sedimentation density [kg/m²] of dust with density ρP=2700 kg/m³ and
diameters DP=2…200μm on the rear end of an SUV.
Figure 10: Comparison of dry dust (ρP=2700 kg/m³, DP=2…200μm) soiling in [10]
left = experiment, right = simulation based on a time averaged flow field
Qualitatively the distribution compares roughly to the experimental result. However, better
results require a deeper knowledge about where and how dust particles have to be emitted to
reproduce reality. Further, this simulation was based on a time averaged flow field which does
not seem to be appropriate with these extremely light particles.

8. 8
Of course the difference between a time averaged and a transient flow field can hardly be vi-
sualized in a static image. However, the instantaneous distribution of the velocity magnitude
on the right in Fig.11 indicates the fundamental difference to a long-term time average as de-
picted on the left.
Figure 11: Velocity magnitude in a time averaged (left) vs. transient (right) flow field
Light dust particles are strongly affected by the flow field and therefore their sedimentation
pattern will depend very much on the unsteadiness of the flow field. The comparison in
Fig.12 from [6] suffers from an insufficient number of particles being simulated. In particular
the usage of a transient flow field is not practicable on a workstation in interactive mode due
to the enormous computational effort and would require parallel processing.
Figure 12: Soiling using a time averaged (left) vs. a transient (right) flow field [7]
5. WATER AND SNOW
Currently the capabilities to simulate water (Fig.13) and snow (Fig.14) are limited because the
trajectory integration ends when the particles hit the surface. Major efforts for the future will
be the development of models for the formation, propagation and breakup of water films on
the surface and the accretion of snow. Most likely the latter will affect the local flow field and
hence require a full coupling.
Figure 13: Spray water impingement Figure 14: Snow impingement
(stationary wipers) (without accretion)

9. 9
6. STONE CHIPPING
With the extensions described before it is now possible to simulate the trajectories of stones
including multiple reflections from vehicle and floor. It is assumed that the extent of damage
caused by a stone impinging on the surface depends mostly on its momentum component
normal to the surface. Colorizing hit points according to this quantity allows to identify criti-
cal regions. Fig.15 shows a result after tracing 100.000 particles with D=2…7mm emitted at
the front wheel. Based on such a distribution especially wheel suspension components can be
protected very goal-oriented from damage and subsequent corrosion by coatings or applying
shields. Comparing the condition of high mileage vehicles with simulation results shows a
good correlation and builds confidence in the accuracy of this method.
Figure 15: Impact locations of gravel ejected from the front wheel (colored by surface normal momentum)
CONCLUSION AND OUTLOOK
The integration of particle trajectories in a given flow field around vehicles has been imple-
mented as a postprocessing step without feedback on the flow itself. Particles are assumed to
have spherical shape for which the drag coefficient is known as a function of Reynolds num-
ber. Stochastic scatter applied to various parameters accounts for real-world effects. A key for
realistic results are the initial conditions which have been obtained by intricate experiments.
Especially for light particles there is a significant difference between using a transient or a
time averaged flow field.
To simulate stone chipping it was necessary to include the reflection from the vehicle surface
and the floor. Coefficients of restitution and kinetic friction are used to model the momentum
exchange of different material pairings during collision. Damage probability is assessed by
the normal component of particle momentum at impact. First comparisons with field tests
show a good correlation of predicted critical regions. The ultimate goal is the forecast of abra-
sive wear of coatings and consequential corrosion rates.
Apart from further validation and investigation of realistic initial conditions future work will
also focus on wall film modeling for water management and the accretion of snow.
SUBSCRIPTS
in, out before, after collision
n, t surface normal, tangential direction
P particle property
∞ free stream condition

10. 10
SYMBOLS
A [m²] particle cross section
CD [-] drag coefficient
D [m] particle diameter
FD [N] drag force
g [m/s²] gravitational acceleration
h, H [m] drop test heights
L [m] length
m [kg] mass
p, pt [Pa] static and total pressure
q [Pa] dynamic pressure
Re [-] Reynolds number
 [°] angle between particle trajectory and surface
v [m/s] velocity
 [-] coefficient of restitution (CoR), 0    1
 [-] coefficient of kinetic friction (CoF), 0    1
 [kg/m³] density
 [m²/s] kinematic viscosity
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