American Journal of Fluid Dynamics 2013, 3(4): 119-134 [610374]

American Journal of Fluid Dynamics 2013, 3(4): 119-134
DOI: 1 0.5923/j.aj fd.20130304.04
Methods of Calculating Aerodynamic Force on a Vehicle
Subject to Turbulent Crosswinds
Abdessalem Bouferrouk
College of Engineering, Mathematics & Physical Sciences, University of Exeter, Cornwall Campus, Cornwall, TR10 9EZ, UK

Abstrac t This study com pares two m ethods for calculating the unsteady aerodynam ic side force on a high speed train
running in a turbulent crosswind at discrete points along a straight track: the m ethod of aerodynam ic weighting function, and
that of the quasi -steady theory. Both m ethods are concerned with tim e series estim ation of aerodynamic loading based on
ex perimentally measured crosswind data. By varying the m ean wind speed, the train spe ed, and the distance between the
sim ulation points it is shown that the aerodynamic weighting function approach m ay be m ore appropriate for estim ating the
unstea dy force on a high speed train. The advantage of the weighting function is consideration for th e loss of turbulent
velocity cross -correlations over the surface of the vehicle in contrast to the quasi -steady approach.
Ke ywo rds Aerodynam ic admittance function, Aerodynamic weighting function, Quasi -steady approach, Spectral
approach, Crosswind, Unste ady force, Spectra

1. Introduction
The study of a erodynamics will continue to be central to
the developm ent of ground vehicle such as cars, tr ains, and
other hum an powered vehicles. This is driven by the need for
im proved efficiency in term s of reduc ed har mfu l emissions,
reduc ed fuel consum p tion, increased range and alleviating
stability and safety problem s. The focus of this paper is on
application o f numerical aerodynam ics to predict forces on a
high speed train running in a crosswind.
Time -dom ain approaches are powerful tools for
calculating unsteady aerodynam ic forces and moments on
ground vehicle s running in a turbulent flow. Because high
speed tr ains are constructed with light weight materials to
give them higher accelerations and to m inimise the power needed to overcom e frictional and gravity forces [1], they are
susceptible to the risk of overturning, particularly at locations with high crosswind s, such as em bank ments and
open bridges due to local spe eding effects. There are also a
num ber of other potential effects – for exam ple turbulent
crosswind s can cause dewirement problems with large scale
pantograph and contact wire dis placements [2]. It is thus
im portant for train designers and operators to ensure that aerodynam ic loadings on trains do not infringe their safety
limits. Therefore, accurate prediction of unsteady loads on
high speed trains in crosswinds is required.
The Quasi -Steady (QS) app roach for aerodynam ic force

* Corresponding author:
A.Bouf [anonimizat] (Abdessalem Bouf errouk)
Published online at http://journal.sapub.org/ aj fd
Copyright © 2013 Scientif ic & Academic Publishing. All Rights Reserved calculation is widely used in predicting the response of
structures to turbulent wind. The popular use of the QS
theory is due to its sim plicity and ease of use [3]. Using QS
theory, the unsteady aerodynam ic force F(t) on a high speed
train m oving at velocity V in a crosswind is given a s
()
()( )2 2
22 21() ' () ' ()2
112 ' () ' ()22F
FFFt F F t A U u t V C
A U V C A Uu t u t Cρ
ρρ= += + +
= ++ + (1)
where F is the mean force, ()'Ft is the fluctuating force
com ponent, ρ is the air density , A is a reference area, FC
is the mean (t ime -averaged) force coefficient, U is the mean
crosswind speed and ()'ut is the fluctu ating velocity. The
quasi -steady force as defined in (1) incorporates second
order term s associated with the turbulent velocity ()'ut , i.e.
a non -linear quasi -steady ap proxim ation. The form of the QS
approach as in (1) is a variation of the linearised QS theory
where second order effects are not accounted for, i.e.
()F F Ct AUu CV UA tFFtF )('21)(' )(2 2ρ ρ ++ =+= (2)
It was shown in [5] that the nonlinear quasi -steady
approach produced tim e series of wind -induced shear force
coefficients that were more accurate compared with measured values, in contrast to results obtained using the
linearised quasi -steady such as in the study of Letchford et
al.[5]. From (1) and (2), the unsteady force F(t) follows the
history of the instantaneous turbulent velocity ()'ut , i.e.
they are fully correlated. Thus, the QS approach consider s
the fluctuations due to instantaneous turbulence and neglects

120 Abdessalem Bouferrouk: Methods of Calculating Aerodynamic Force on a Vehicle Subject to Turbulent Crosswinds

the unsteady m em ory effects of preceding turbulent
velocities.
However, unsteady events like flow separation does not
have an instantaneous effect but develop i ts influence on the
body surface over a period of t ime. Tielman [8] discussed
how the quasi -steady theory m ay only be applicable for the
prediction of aerodynam ic forces in stagnation regions,
while failing to predict these forces in separated regions. It
was shown previously that ‘building -generated ’ turbulence
plays an im portant role in the induced pressure forces, e.g. Cook [6], Letchford et al. [5] and Simiu et a l. [7]. The
im plication is that not all fluctuations of upstream flow are transm itted to the bui lding. F or a train passing through a
crosswind and for which the forces are simulated at given
points along a track, the effect of a turbulent crosswind at one point would still exist at neighbouring points. Preceding
turbulence becom es m ore im portant if the simulat ion points
are more condensed due to increased spatial correlation.
Another drawback of the QS m odel is its overestim ation
of the unsteady forces . In a study on long span bridges [3] it
was found that the QS theory was only valid at very
high-reduced velocities for which the frequency -dependent
flu id me mory effects are insignificant. In another study o n
the effects of crosswinds on a vehicle passing through the wake of a bridge tower, Charuvisit et al. [9] pointed out that
the conventional quasi -steady m ethod gives larger
overestimation for many cases. Letchford et al. [5] applied
the quasi -steady theory approach to com pute pressure
distributions on a building and found deviations away from
the theoretical prediction in separated flow regions as a r esult
of ignoring the build ing-generated vortices. Clearly, the
m ore aerodynamic information about the flow behaviour
around the vehicle is taken into account, the more accurate is the prediction of unsteady forces. Despite the se limitations,
the QS approa ch rem ains a sim ple m ethod for quick
calculations of the effects of winds on vehicles and other structures.
An alternative approach for calculating the unsteady
crosswind forces on a vehicle is via the aerodynamic admittance function. This m ethod is not new, dating back to
the early works of Sears [10] using thin airfoil theory.
Aerodynam ic admittance functions, relating the lifting force
on a streamlined section to the vertical fluctuating
com ponent, were developed through the so-called Sears
function [10]. Extending the idea from aeronautics to wind
engineering, Davenport [11] introduced adm ittance functions
that relate the wind fluctuation to the wind -induced pressure
on structures in the frequency dom ain. In Davenport’s form ulation, the rol e of the aerodynamic admittance function
is to account for the lack of correlation between the velocity fluctuations in the fl ow region adjacent to the body. Ho wever,
for a bluff body such as a high speed train, the theory of thin sections is invalid owing to characteristic differences in the
flow behaviour between thin and thick sections. For the latter, the aerodynamic behaviour is affected to a large degree by
the boundary layer separation, typically involving
large-scale strong vortex shedding. For a train with mu ltip le car units, parts of the vehicle will be subjected to a strong wake generated from the leading car. A new fluctuating force
com ponent, often referred to as self -buffeting, is induced as a
result of sig nature turbulence (e.g. [12]). In this case, the
aerodynam ic admittance function should take full a ccount of
the unsteady effects associa ted with signature turbulence.
This has led to new definitions of the aerodynamic admittance functions using both com putational and ex perimental tools (e.g. [12]) in an attem pt to produce
accurate form ulations. Com putationally, various approaches have been proposed with varying degrees of com plexity in
their application; see for instance the works of Scanlan [13]
and Hatanaka et al. [14]. Such analysis is, however , beyond
the scope of this study.
This paper is concerned with numerica l estimat ion of
aerodynam ic adm ittance function based on data from
ex periments on train m odels . Aerodynamic admittance
functions can be defined, in the frequency dom ain, to relate
the spectrum of the inflow velocity fluctuations to the that of
the m easured forces ex perienced by a m odel train in a wind
tunnel [15] or in a full scale ex periment [16]. As noted by
Bake r [17], due to difficulty in com puting am plitude and
phase parameters of an admittance function in the frequency dom ain, doing the analysis in the time-dom ain is m ore
convenient. The equivalent ex pression of aerodynamic
admittance function in the tim e -domain is called the
aerodynam ic Weighting Func tion (WF). Using the WF,
Bake r [17] showed that the fluctuating force m ay be obtained
through a sim ple convolution of the turbulent wind and an ex perimentally deter mined WF. The use of this m ethod has
been highlighted in m any studies e.g.[18]. Using the
definition of the relative wind spee d as
()()2 2'RU U ut V= ++ (3)
The unsteady aerodynam ic force using the WF is [17]
()2
'
01'( )2
() ( )FR
FR F RF t F F t AC U
AC U h U t dρ
ρ τ ττ∞= +=
+−∫ (4)
where '
RU= R RUU− , τ is a time lag, and h F (τ) is the
aerodynam ic weighting function of the force. Com pared
with equations (1) and (2), the m ain difference in ( 4) is the
introduction of an integral ex pression for the fluctuating force com ponent that involves h
F (τ) and a time lagτ. It can
be seen from ( 4) that the unsteady force histories on a train
due to crosswind can be determ ined from the velocity tim e
histori es if the appropriate force coefficients and weighting
functions are known. The WF a llo ws the total force F(t) to be
related not only to the instantaneous turbulence but also to
preceding turbulent velocities within a tim e lag τ. Unsteady
crosswind forces depend on m any parameters like the m ean wind speed, the vehicle speed and the turbulent wind. For the
WF approach, the aerodynam ic weighting function and the
time lag are also important.

American Journal of Fluid Dynamics 2013, 3(4): 119-134 121

In this paper, the QS and WF approaches are com pared when com puting the unsteady side force on a high speed train at
discrete sim ulation points in the presence of a tur bulent crosswind. The com parisons are based on the variation of three m ain
parameters: the m ean crosswind U , the train speed V , and the sim ulation distance between the sim ulation points. The aims are
twofold: 1) to show which approach is m ore accurate and 2) to investig ate the range of applicability. In order to increase the
accuracy of simulati on for the unsteady crosswind force on a high speed train, a simple, practical, and accurate representation
of the wind load is the prim ary m otivation of this research work.
2. Methodology
2.1. The Simul ated Problem

Fi gure 1. Coo rdinat e sy st em, th e trai n/w ind vel oci ties , a nd trac k defi ni t io n
In order to com pare the two m ethods, the side force is
com puted on the leading car of a UK high speed train, the
Class 365 Electrical Multiple Unit (EMU), running along a 1
km s trai ght track subjected to an unsteady crosswind norm al
to th e train’s direction of m otion. The side force is estimated
at discrete points along the track, the num ber of which is
controlled by the separation distance x∆. Figure 1
illustrates the sim ulated problem where the leading car of
length L m oving at velocity V is subjected to a crosswind
)('tuU uw+= .
The resultant relative velocity that im pinges on the train is
2 2V u U w R+= , β is the relative wind angle or yaw
angle, i.e. β= tan−1 (V/wu(t)).
2.2. Outline of the Numerical Method
The num erical procedure to com pute the side force
consists of the following:
(i) G eneration of a turbulent crosswind velocity tim e
history via a spectral approach
(ii) Experimental determination of mean force coefficient
as a function of relative wind angle
(iii) Calculation of the aerodynamic weighting function of
the train
(iv) Convolution of the weighting function with the
fluctuating velocity gives the crosswind force on the train;
this is then added to the mean force to get the total unsteady
force (v) For a range of param eters, the two approaches are
compared e.g. in terms of mean force value, standard deviation, ratio of standard deviati on to m ean, and force
spectra.
Regarding point (v), since the m agnitude of unsteady
force is a function of relative velocity then it is natural to be
interested in studying how the force varies with m ean wind
speed, train speed, and gust distribution. But one essential
fact about the force along the track is that the larger the
num ber of sim ulation points along the track, the m ore
inform ation it is possible to gather about the force history .
When the di stance between the sim ulation points along the
track is reduced, the fluctuating velocities become spatially more co rrelated, so that the unsteady force will exhibit
increased sensitivity to the spatial dis tribution of velocity.
This is why it is convenient to com pare the two approaches
with varying distances
x∆. The distance x∆ is a key
parameter for co mparison because the effect of the WF
integral in ( 4) becomes more significant with decreasing this
distance as x∆ also controls the tim e lag. In practice ,
frequencies associated with unsteady flow at certain distances
x∆ may coincide with the natural frequencies of
the ve hicle (e.g. its suspension) and so under certain
conditions these could be ex cited, resulting in potential
detriment to the vehicle safety.
2.3. Simulation of Turbulent Crosswind
The crosswind flow past the train is assumed
two-dim ensional, incom pressibl e, isotherm al and turbulent

122 Abdessalem Bouferrouk: Methods of Calculating Aerodynamic Force on a Vehicle Subject to Turbulent Crosswinds

in the wh ole flow dom ain. The turbulent fluctuation u’(t) is
specified by the longitudinal velocity fluctuations at the
sim ulation points along the track. The spectral approach is
used to provide a m eans of sim ulating a turbulen t crosswind
field as a stationary random process [18]. The simplicity of
the spectral approach is an advantage of tim e -dom ain
s imu lations. The turbulent velocities are predicted at every
x∆. Full details of the sim ulation procedure can be found
in[18]. For t he crosswind velocity tim e series sim ulation,
each tim e history at a given point was generated with a
sam pling rate of 0.167 seconds with a sim ula tion period of ~
8533 seconds. All realisations were equally spaced in time.
The outcom e of the spectral appro ach is a time series of a
mu lti -point correlated crosswind velocity fiel d. It is
sufficient to note that in all cases the time series are appropriately correlated to one another and have coherence
properties as described by Davenport’s coherence
function [18]
()


∆−=UC Cohjm
z jmπωω2exp (5)
where ω is the angular frequency, C Z is a constant decay
factor taken as 10, and jm∆ is the distance between points jand m, i.e. jm∆ = x∆. The m ean crosswind speed is
m easured, conventionally, at height z = 3 m above the
ground. The turbulent wind also conform s to Kaim al’s
empirical wind spectrum [11] as given by
()2
5/3*20011,221 502uzSzU z
Uωπ ω
π=
+ (6)
where *u is the shear velocity defined as
0*log( / )KU
zzu= (7)
The shear velocity depends on the ground roughness z 0
(assumed z 0 = 0.03 m for open country terrain). A sam ple
time-series of crosswind velocity )('tuU uw+= is
given in Figure 2 (a) for U = 22 m/s . The actual wind as seen
by the m oving train is the vector addition of the wind tim e
series uw and the train velocity V . Figure 2 (b) sh ows fair
agreem ent between the target longitudinal Kaim al’s
spectrum [19] and the one sim u lated via the spectral method.

(a)

(b)
Fi gure 2. Sp ect ral simulat ion result s : (a ) c rossw ind vel oci ty, (b) s pec tra 0 5 10 15 20 25 30 35 40 4515202530
Time (s)Absolute velocity, u (m/s)
10-310-210-110010110210-1010-5100105
Frequency (Hz)Power spectrum (m2/s)

Kaimal's spectrum
Simulated spectrum

American Journal of Fluid Dynamics 2013, 3(4): 119-134 123

2.4. Prediction of Unsteady Aerodynamic Forces
In reality, the behaviour of the wind profile around a
m oving vehicle is not sim ple. The resultant velocity field
from a m oving vehicle and an atm ospheric boundary laye r
profile results in a skewed boundary layer profile as
ex plained by Hucho and Sovr an[20]. Due to the nature of
wind, one ex pects the response of the train due to crosswind
to have two com ponents: 1) a quasi -static response as a result
of the mean velocity and low frequency fluctuations, and 2) a
response due to high frequency wind fluctuations that are the
source of dynam ic excitations. To determine the unsteady
forces on a train due to crosswind either through the QS or the WF theory, two pieces of inform ation are required :
records of wind speeds, and variation of force coefficient as a function of yaw angle. While the wind history is obtained from the spectral simulation, the force coefficients were obtained from wind tunnel tests of a Class 365 scaled m o del,
according to ex perimental details in [21]. The force
calculation using the WF is as follows. The m ethod builds on
the one developed by Baker [17] for a m oving vehicle, which
requires determination of an aerodynamic admittance function from e x perimental data. Based on a linearised
analysis in the frequency dom ain, Baker [1] expressed thi s
function in term s of m easured quantities as
()()
()()nSU ACnSnX
U FF
F 2 22 4
ρ=
(8)
where ()2nXF is the aerodynamic admittance function,
and n is frequency in Hz. In (8), ()nSF and ()nSU are the
power spectra of the force being considered and longitudin al
wind velocity, respectively. Clearly, the admittance func tion
is frequency -dependent. In the time -dom ain, the weighting
function appearing in the unsteady force equation ( 4) is the
Fourier transform of the tra nsfer function in ( 8). Thus, if the
admittance is known, the weighting function can be found as
a function of train geom etry, yaw angle, etc. This is not
usually the case, however, as wind velocities and forces are not usually m easured sufficiently close t ogether in tim e to
enable ()2nXF to be determined[17]. Therefore, the
following approach is adopted. From wind tunnel tests, the force admittance functions as defined in ( 8) are plotted
against the yaw angle , as illustrated in Figure 3.
The immediate observation is that all admittance functions
drop off at high frequencies, roughly to about a tenth of their
value at lower frequencies. The decrease of aerodynamic
admittance with increasing frequency is ex pected because at
higher frequencie s the sm aller turbulent eddies have shorter
wavelengths and thus have a m ore rapid loss of coherence
than fo r the larger eddies[7]. These admittances are then
transform ed into the tim e dom ain to yield equivalent
aerodynam ic weight ing functions. By fitting curves to data
in Figure 3, it can be shown that the weighting function for the Class 365 could be written as
()




τ

π=ττ
π−LUn
F
LUn h'2 22exp '2)( (9)
where ''LnnU= , L is the vehicle’s length (~ 20 m), and τ
the weighting function tim e lag appearing in ( 4). The
weighting functions are thus found from the m easured
admittance functions , thereby enabling calculation of the
unsteady force in ( 4). Now that the weighting functions are
com puted, the unsteady forces are obtained by adding the averaged and fluctuating parts as defined in ( 4). The
fluctuating part F’(t) in (4) can be com puted either in the
tim e dom ain by ex pressing the integral as a convolution
)(τFh * )('tu (since the system is causal), or in the
frequency dom ain by taking a Fourier transform for F’(t)
which changes the integral to a product o f the Fourier
transforms of )(τFh and )('tu , and then taking an
inverse Fourier transform of the result.

Fi gure 3. Side force admittance fu nction for a 1/30th scale model of the Class 365 EMU [21]

124 Abdessalem Bouferrouk: Methods of Calculating Aerodynamic Force on a Vehicle Subject to Turbulent Crosswinds

2.5. Discussion of the Aerodynamic Weighting Function

Fi gure 4. Side force weight ing fu n ct io n fo r different sep arat io n dist an ces x∆, U = 25 m /s, V = 80 m/s
The weighting function represents the contribution of
preceding turbulent crosswind velocities to the c urrent value
of unsteady force. There is a higher weighting for points
closer to the current sim ulation point ()'ut , which then
reduces for points far apart, showing the physical effects of
spatial corre lation. This is in contrast to a m oving average
filter for which equal weighting is usually applied. In a
discrete formulation the side force F S may be written as
τττ ρ ρ ∆− + = ∑) ()(
21 ' 2tU h UAC UAC FR F R S R S S (10)
Fro m ( 10), it is seen that the effects of the weighting
function, and thus the fluctuating force, becom e mo re
important if the integration time step τ∆ is very small. The
τ∆ can be reduced making x∆ s maller, imp lying a la rger
num ber of sim ulation points. This is, however, num erically
m ore ex pensive due to the need to simulate a larger number
of turbulent velocit ies via the spectral approach. Another
important characteristic time scale is the time period of the
weighting function hτ after which t he weighting function
is zero. The weighting function m ay be regarded as a low
pass filter which sm oothes high frequency, short period
fluctuations of periods less or equal than hτ, and
dim inishes any fluctuations for tim e periods where the
weighting function falls to zero as tim e increases beyond
hτ. The tim e step in the integration of the weighting
function is the same as the ti me step for force calculations.
Hence, the train speed V also affects the size of τ∆ in
addition to x∆ since roughly Vx/∆=∆τ . The mean crosswind speed U is also important since it ap pears in
Kaimal’s spectrum definition. Sam ple form s of the side force
weighting function hS (t) for different separation distances at
constant V and U are shown in Figure 4. For the cases shown,
although the weighting function’s characteristic time scale i s
the same for all separation distances (~ 1.2 s) the filtering
effects are more pronounced with smaller distances x∆.
From force definitions, the form of weighting function as
seen in Figure 4 will affect the resulting unsteady
aerody namic force. It is no ted in passing that the WF
approach applied to a high speed m odel was used successfully by Ding [18] to reconstruct som e ex perim ental
data.
3. Study Cases
In order to investigate the sensitivity of the two force
calculation approaches, a matrix of three parameters of mean
wind speed U , train speed V , and separation distance x∆
was used in this study as summ arised in Table 1. Fro m Table
1, the implication is that that 25 simulation cases are required
because the turbulent velo city field depends on both the
m ean crosswind speed (through Kaim al’s spectrum ) and the
separation distance (through Davenport’s coherence
function), but is independent of train speed. To obtain the
unsteady force for the full matrix of these parameters a total
of 125 force com putations are required. However, only s ome
selected cas es will be studies and discussed. The com parison
between the two approaches in each case is considered in three ways: standard deviation of the force (and its ratio to
the mean fo rce), the force power spectra, and variances. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 200.511.52
Travelling time (s)Side force weighting function hS (τ)

∆ x = 10m
∆ x = 20 m
∆ x = 40 m
∆ x = 5m
∆ x = 1m

American Journal of Fluid Dynamics 2013, 3(4): 119-134 125

T able 1. Matrix of P aramet ers for U ns tead y Force Calculations with QS
and WF A pproac hes
Mean wind speed (m/s)
T rain speed (m/s) 10 15 20 25 30 x∆ (m )
1 x x x x x 1
20 x x x x x 5
40 x x x x x 10
60 x x x x x 20
80 x x x x x 40
3.1. Comparison of Unsteady Force Time Histories
In order to illustrate the effects of the WF on the unsteady
force com pared with the QS theory, two time histories of
side force acting on the Class 365 EMU train are plotted
together in Figure 5 (b) in response to the crosswind velocity
history (for U = 25 m/s and x∆= 10 m) of Figure 5 (a),
illustrating how the WF filters the high frequency content. The force signal due to the WF is also slightly delayed
com pared with the QS force signal which follows the
velocity history ex actly. No vehicle suspension effe cts were
included when com puting the unsteady forces. From the
unsteady force definitions for the QS and WF approaches, the m ain difference is how the fluctuating part of the force is
accounted for. Since the force standard deviation, F
std, is a
measure of the spread of data from a mean value it is
appropriate to use it as a param eter to distinguish the effects
of turbulent fluctuation s for each calculation method. With
reference to Table 1, for each train speed, five standard
deviations are com puted corres ponding to f ive m ean wind
speeds. Each value of F std is com puted from the total force
time history as the train travels a 1 km track. Results for side
force standard deviation at different mean wind speeds are
shown in Figure 6 for a train with V = 80 m/s and x∆= 5 m.

(a)

(b)
Fi gure 5. (a) simulat ed t urbulent crosswind, (b) Unst eady forces 0 2 4 6 8 10 12 14-6-4-202468101214
Train travel time (s)Turbulent velocity (m/s)
0 2 4 6 8 10 1211.11.21.31.4x 105
Time (s)Side force (N)

WF force
QS force

126 Abdessalem Bouferrouk: Methods of Calculating Aerodynamic Force on a Vehicle Subject to Turbulent Crosswinds

Several observations can be m ade by looking at Figure 6.
In both approaches the force std increases with increasing
m ean wind speed due to increased level of fluctuations, in accordance with Kaimal’s spectrum of lo ngitudinal
turbulence, see (5). However, when the std of both
approaches is compared at the same m ean wind and train
speed, the std is higher in the QS approach than in the WF
m ethod due to the velocity fluctuations in the QS theory being unfiltered as opposed to the WF m ethod in which force
fluctuations are filtered via the WF integral. Although not shown, this trend in force std between the two m ethods was
found for all the train speeds. For the specific example
illustrated in Figure 6 the std for QS theory is almost twice
the std of the WF approach , illustrating the sm oo thing nature
of the WF m ethod.

Fi gure 6. Variat ion of side force standard deviat ion wit h mean crosswind speed between the QS a nd WF a pproac hes

(a) x∆= 1 m 10 12 14 16 18 20 22 24 26 28 30010002000300040005000600070008000
Mean velocity U (m/s)Standard deviation (N)

WF approach
QS approach
10 12 14 16 18 20 22 24 26 28 30010002000300040005000600070008000900010000
Mean velocity U (m/s)Standard deviation (N)

WF approach
QS approach

American Journal of Fluid Dynamics 2013, 3(4): 119-134 127

(b) x∆= 5 m

(c) x∆= 10 m 10 12 14 16 18 20 22 24 26 28 30010002000300040005000600070008000
Mean velocity U (m/s)Standard deviation (N)

WF approach
QS approach
10 12 14 16 18 20 22 24 26 28 300100020003000400050006000
Mean velocity U (m/s)Standard deviation (N)

WF approach
QS approach

128 Abdessalem Bouferrouk: Methods of Calculating Aerodynamic Force on a Vehicle Subject to Turbulent Crosswinds

(d) x∆= 20 m

(e) x∆= 40 m
Fi gure 7. Variat ion in side force st d due to WF and QS as x∆ is varied from 1 m to 40 m for all mea n w ind s peeds ( V = 80 m/s )
If the std of the side force is com pared at a constant train
speed of V = 80 m /s but different separation distances and for
all m ean wind speeds the results are plotted in Figure 7. The
imm ediate observation is that when the separation distance decreases the std of the force increases for the WF, presum ably due to increased n um ber of turbulent velocities
fluctuating off the mean. However, it is lower than the QS approach where it rem ains quite high for all separation
distances. The higher filtering effect of the WF at larger
separation distances is clearly visible, particularl y at
x∆=
40m where the std is negligible. This is caused by a larger
force integration time step at larger x∆ as shown in the
definition of the WF integral of Eq. ( 4) which causes the WF
to becom e zero rather quickly (the WF com pletely dam ps out 10 12 14 16 18 20 22 24 26 28 3001000200030004000500060007000
Mean velocity U (m/s)Standard deviation (N)

WF approach
QS approach
10 12 14 16 18 20 22 24 26 28 300100020003000400050006000700080009000
Mean velocity U (m/s)Standard deviation (N)

WF approach
QS approach

American Journal of Fluid Dynamics 2013, 3(4): 119-134 129

m ost fluctua tions). Thus , at x∆= 40m the unsteady force is
overly -filtered and becomes a constant value.
3.2. Fre quenc y Anal ysis of Force Histor y
In this section, the unsteady forces due to the WF and QS
approaches are com pared when the train passes through
sinusoidal gusts with single frequencies. In principle, the
power spectra of both approaches should reproduce the m ain frequency of the sinusoidal gust. We consider gust time
histories of the form
()t Atu ω=sin )(' (11)
where A is the gust am plitude, ω is the gust radian
frequency (ω= 2π f, f frequency in Hz), and t is the time.
Since the train travels at different speeds and the forces are
com puted at different separation distances, the sam pling
frequency for the forces will depend on these two param eters.
Hence, on ly specific gust frequencies will be captured by the
train without any aliasing for a given train speed and separation distance. For ex am ple, when the train speed is 80
m/s and x∆ = 10 m , the sam pling f requency of the force is
8 Hz. Thus, by the Nyquist the orem only gust frequencies of
gustf4≤ Hz may be seen by the train. For constant
am plitude gust with three different frequencies (1, 2 and 3
Hz) the unsteady forces and corresponding spectra of both approaches are shown in Figure 8 .
A sinusoidal crosswind gust i s a deterministic model
which is fully correlated along the length of the track. As
ex pected, the sinusoidal crosswind ex citations lead to sinusoidal side force. It is seen that the power spectra produced by both approaches give the correct spectrum for each gust frequency. This indicates that the current m ethod
of force response to sinusoidal gu sts produces accurate
results. However, despite the mean force being the same, the
fluctuating part due to the QS approach is higher than the one
due to the WF m eth od.

(a) 0 2 4 6 8 10 12 140.850.90.9511.051.11.15x 105
Travelling time (s)Side force (N)
WF
QS
0 0.5 1 1.5 2 2.5 3 3.5 410010510101015
Frequency (Hz)Side force power spectrum
WF
QS

130 Abdessalem Bouferrouk: Methods of Calculating Aerodynamic Force on a Vehicle Subject to Turbulent Crosswinds

(b)
0 2 4 6 8 10 12 140.850.90.9511.051.11.15x 105
Travelling time (s)Side force (N)

WF
QS
0 0.5 1 1.5 2 2.5 3 3.5 410210410610810101012
Frequency (Hz)Side force power spectrum

WF
QS
0 2 4 6 8 10 12 140.850.90.9511.051.11.15x 105
Travelling time (s)Side force (N)

WF
QS

American Journal of Fluid Dynamics 2013, 3(4): 119-134 131

(c)
Fi gure 8. Force histories due t o sinusoidal gust s a nd their pow er s pec tra : (a) f =1 Hz, (b) f =2 Hz, (c) f =3 Hz

Fi gure 9. Force p ower s pectra comparison bet ween t he WF and QS approaches (U = 25 m /s , V = 80 m /s ,x∆= 10 m )
The weighting function works in a similar way to a digital
filter in which a weighting is applied to a sequence of
discrete data points in a time series. Similar to a digital filter, temporal fi ltering through the WF has an effect on the
characteristics of the structure of the force sign al such as its
power spectrum . It is the purpose of this section to briefly ex plore the im plications of filtering through the WF on som e
turbulence statistics inc luding varian ce, and on the power
spectra. The fluctuating force com ponent in Eq. ( 4) may be
re-written discretely as
()'
0'( )FRm
j ijR
jF i AC i U hU ρ−
==∑ (12) where )('iF is the filtered force at time i, hj is the
weighting applied to the fluctuating velocity '
Ri jU−
simulated at time ji−, m is the num ber of tim e lags
over which the WF is applied to the values preceding the
current relative velocity '
RjU . Because of filte ring, a loss in
the force m agnitude and its variance is expected. In order to
illustrate the role of filtering due to the weigh ting function as
com pared with the QS approach, the power spectra of the
side force in both approaches are com pared for different
spacing x∆ between the turbulent velocities. One ex pects
the decrease in the slope of the power spectra at high
frequencies to be greater as a resu lt of filtering due to the WF. 0 0.5 1 1.5 2 2.5 3 3.5 410210410610810101012
Frequency (Hz)Side force power spectrum

WF
QS
10-210-1100101101102103104105106107108
Frequency (Hz)Power

Non-Linearised QS
Linearised QS
WF with U

132 Abdessalem Bouferrouk: Methods of Calculating Aerodynamic Force on a Vehicle Subject to Turbulent Crosswinds

Figure 9 shows the difference in slope at high frequencies
between the QS and WF m ethods. As seen from Figure 9, the
WF power spectrum (plotted on a log -log scale) has a roll -off
in the higher frequencies (higher m ean slope) com pared with
the QS spectrum (both linearised and non -linearised forms) .
The spectra are not smooth presumably because of
turbulent noise as part of the spectral sim ulation. At low
frequencies, whilst the linearised QS and the WF meth ods
have nearly the sam e power, the non -linearised QS approach
has a slightly higher power due to the contrib ution from
second order terms. At higher frequencies, however, the
power spectrum from the WF m ethod falls m ore rapidly com pared with both ver sions of the QS ap proach which are
very similar. This illustrates the effects of the WF which
com pletely dam pens out higher frequencies. The contrasting
perform ances of the WF in the tim e and frequency dom ains are typical of som e filters such as the m oving aver age filter
[22] where good perform ance in the time dom ain results in
poor perform ance in the frequency dom ain, and vice versa (e.g. the moving average is ex cellent as a sm oothing filter (time -dom ain) but can be an ex ceptionally bad low-pass
filter (frequen cy dom ain)).
3.3. Variation of Cutt -off Frequency with Separation
Distance
This section ex plores how the cut -off frequency for the
weighting function changes with separation distance when the mean w ind and train speeds are fixed. For three
separation distances of 1 m, 5 m, and 10 m the spectra of side force for both prediction m ethods are shown in Figure 10 for a m ean cross wind of U = 25 m/s and V = 80 m/s.

(a) x∆= 1 m

(b) x∆= 5 m
10-210-1100101100101102103104105106107
Frequency (Hz)Side force power

WF
Linearised QS
Non-linearised QS

American Journal of Fluid Dynamics 2013, 3(4): 119-134 133

(c) x∆ = 10 m .
Fi gure 10. Variation of cu t -off frequency with separat ion dist ance x∆
When the separation distance is smallest at 1m, the
behaviour of the power spectrum between the two m ethods
resembles more what is expected: in the lower frequency
range, the power spec tra collapse on a single line. At higher
frequencies, however, the power spectrum due to the WF approach decreases m ore rapidly com pared with the two
versions of the QS m ethod. It is evident that sam pling the
force at smaller separation distances, i.e. at s maller force
integration tim e steps, results in the power spectrum being better defined at a larger frequency range (ex tended range due to ability to resolve higher frequencies according to Nyquist theory) as well im proving the accuracy of the WF at
low frequencies (power spect ra being the same as ex pected).
From the spectra plotted, the energy starts to fall at the cut-off frequency of about 1 Hz for all sam pling distances.
Variation of force variance with separa tion distance
Table 2 shows the decrease in side force variance
()222/msσ due to the WF com pared with the QS approach
for four cases of spacing (for fixed U and V).
Ta b l e 2 . Decrease in force variance due to filtering by the WF for U = 25
m/s V = 80 m/s
S pa cing (m ) QS approach W F a pproa ch % De crease
5 1.2716 x 107 4.9017 x 106 61.45
10 3.5978 x 107 2.0161 x 106 43.96
20 3.0670 x 107 1.7099 x 106 44.24
40 2.2805 x 107 5.6511 x 106 75.21
4. Discussion
The separation distance for the m ulti -point correlated
crosswind velocity field is a crucial ingredient in deciding which approach is the m ost suitable for unsteady force calculation. At larger distances, the WF alm ost filters out all the turbulent fluctu ations, giving a conservative estim ate
of unsteady aerodynamic effects which are otherwise
im portant for dy namic ana lysis. This, however, im proves as
the WF function time lag is decreased, i.e. smaller separation
distances, so that the calculation is sm oot hed over a larger
num ber of pre ceding turbulent velocities. In a dynamic sense,
it im plies that unsteady fluid m em ory effects due to older crosswind velocities on th e train are taken into account. On
the other hand, the QS approach is likely to overestim at e the
unsteady aerodynam ic force as reported in other studies, leading to un necessary design contingencies. The
im provem ent in prediction through the WF approach com es at an increased computational time because the velocity field along the track contains a larger num ber of turbulent
velocities that are simulated via the spectral approach.
Further work is needed to investigate the sensitivity of the
unsteady forces, and thus of the calculation m ethods, to the full range of sim ulation param eters.
5. Conclusions
An im portant conclusion of this paper is that the
com putation of unsteady force histories with large spatial differences between the sim ulation points entails loss of correlation between the velocity perturbations and hence the tem poral flow d evelopment filters out the physical effects of
a correlated crosswind field. The solution is to use a condensed sequence of sim ulation points for tu rbulent
velocity along a track. The reliability of the m ethod of
ex tracting the forces from the turbulent cr osswind is
ex amined via two different approaches: the classical quasi -steady theory and the weighting function m ethod.
While the QS theory produces increasingly higher force fluctuations as the sim ulation distance decreases, the WF provides a sm oother form of a force signal which rem oves 10-210-1100101100102104106108
Frequency (Hz)Power

WF
Non-linearised QS
Linearised QS

134 Abdessalem Bouferrouk: Methods of Calculating Aerodynamic Force on a Vehicle Subject to Turbulent Crosswinds

the short -term high frequency oscillations by applying
different weightings to older veloc ity histories. The ongoing
analysis has illustrated the effects of the weighting function
on the characteristics of the unsteady side force on a high
speed train. Because of its filtering effect, using the WF will
be useful in dam ping out undesirable levels of noise both in numerical as well as physical ex periments. Reliability of the
WF is, however, strictly limited to using very sm all time
ste ps.

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