Jonathan Tschepe, Chr istian Na vid Na y eri, Chr istian Oliv er
P aschereit
Anal ysis of mo ving model e xperiments in a to wing
tank f or aer odynamic dra g measurement of
high-speed trains
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Tschepe, J ., Na y eri, C. N., & P aschereit, C . O . (2019). Analysis of moving model e xperiments in a towing tank
f or aerodynamic drag measurement of high-speed tr ains. In Exper iments in Fluids (V ol. 60, Issue 6). Spr inger
Science and Business Media LLC . https://doi.org/10.1007/s00348- 019- 2748- 8.
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Analysis of Movi ng Model Experiments in a
Towing Tank for Aer odynamic Drag
Measurement of High-Sp eed Trains
Jonathan Tschepe, Chris tian Navid Nayeri and Christian Oliver Paschereit
J. Tschepe 1 (  ), C.N. Nayeri 1 , C.O. Paschereit 2
[email protected]
1 Berliner Institut für Technologietransfer (BIT GmbH), Berlin, Germany
2 Chair of Fluid D ynamics, Hermann-Föttinger-Ins titut, Technische Universität Berli n,
Germany

Abstract The present study assesses the applicability of towing tank experim ents
using a m oving model for the investigation of the aerodynam ics of long land-borne
heavy vehicles such as buses, trucks, and trains. Based on experim ents with a 1:22
scaled model of a high-speed train the influence of various conditions rele van t for
the transferabili ty of the results obtained i n water to air ar e analysed exem plary.
These conditions include surface waves, cavitation and submergence depth. The
experiments were carried ou t in the shallow-water t owing tank of the Technische
Universität Berli n . I t is shown that outside a critical Froude num ber range of about
0.2 < Fr < 1.2 the impact of the surface wave s can be neglected and no cavitation
appears in t he velocity range invest igated. F urthermore, a c orrection m ethod is pro-
posed taking into account the bias through surface waves at small submergence and
thus allowing for a wider Froude number r ange. The data obtained in th e towing
tank is found to be in excellent agreement to othe r investigation methods.
1 Introduction
In order to ac hieve realistic boundary conditions and Reynolds num bers Re ( ≡ uL ref / ν
with u being the flow/vehicle velocity, L ref =3 m / scale and kinematic viscosity ν)
close to full-scale conditions at reasonable scale and velocity for experiments with
downscaled models, towing tanks can be a useful tool for aerodynam ic testing. Es-
pecially for vehicles operating in close proximity to the ground, a realistic flow
simulation underneath t he vehicle can be achieved with relative eas e a nd without

2
requiring further devices , as for example moving belts in a wind tunnel. The ad-
vantages of moving model approaches, especi ally in case of long vehicles such a s
trains, have already been demon strated by to wing tank experiments in the 1970’s
(Neppert and Sanderson 1974, 1976, 1977). However, the se tests were associated
with drawbacks (m ainly costs) that led to the developm ent of moving m odel facili-
ties operating in air (Baker 1986, Pope 1991) . Especially when conside ring Mach
number dep endent tunnel-effects, which have been studied intensively using mov-
ing m od els (Howe et al. 2003, Heine and Ehrenfried 2012, Zhang et al. 2017), air
appeared to be the m ore practical working fluid. Also in the field of road vehicle
aerodynamics, despite the at tempts to draw more attention to towing tank testing by
Erickson (1986) and Gad- el -Hak (1987), only few such investigations have been
performed over the last decades (Aoki et al. 1992, Larsson et al. 1989, Stephens et
al. 2016, Schmidt et al. 2017) . This m ainly resulted from requirem ents on appropri-
ate test obj ects and measurement techniques, which for a long tim e wo uld have
caused prohibit ively high costs. Howe ver, recent technological developments allow
for water resistant materials and measurement techniques at co m par atively low
price , remedying the drawback of higher costs compared to wind tunnel testing.
Furthermore, moving a m odel through a fluid generally requires m uch l ess power
than driving a large volum e of fluid past a stati onary object, m aking the towing ta nk
the more economical tool in term s of operational costs. A f urther adva ntage of tow-
ing tank tests i s the lower velocity required to achieve sim ilar Reynolds numbers
and hence flow phenom ena to air (Fig. 1). This significantly reduces the effort for
time reso lved measurements and visualization techniques such as P article Image
Velocimetry (PIV), as demonstrat ed by Schmidt et al. (2017 ), Jönsson et al . (2012 ,
2014), and Stephe ns et al. (2016).

Fig . 1: Required velocity u (solid lines) and resulting drag force F d (dashed lines) for
different Reynolds numbers Re at same model scale in air and water ( exemplari ly for a 1:22
scale model with drag coefficient

1

c d =0.5)

1

𝑐 𝑑 = 𝐹 𝑑
𝜌
2 𝑢 2 𝐴 𝑟𝑒𝑓 with A ref = 10 m² / (scale)² a nd fluid density ρ

0
50
100
150
200
250
300
350
0.1
1
10
100
0 0.2 0.4 0.6 0.8 1
F d [N]
u [m/s]
Re x10 6
air
water

3
At the same time, the signifi cantly higher density of water le ads to about f our tim es
higher forces for the same Reynolds number compared to air (Fig. 1) . This facili-
tates an accu rate dr ag measurem ent on the one hand bu t requires resilient models
on the other hand.
On strai ght and level ground (without c urvature or slope) , the running resistance
of a moving vehicle is defined as
𝐹 𝑟𝑢𝑛 = 𝐶 1 + 𝐶 2 ⋅ 𝑢 + 𝐶 3 ⋅ 𝑢 2 (1 )
(CEN 2013, Mayer et al. 2002, Tschepe a nd Nayeri 2018a), also known as the
Davis eq uation, where C 1 is a ssociated with the rollin g mechanical resistance, C 2
with air momentum losses from cooling systems and addit ional mechanical losses
and C 3 is assumed to represent the aerodynamic resistance:
𝐶 3 = 𝑐 𝑑 𝜌
2 𝐴 𝑟𝑒𝑓 . ( 2)
In order t o se parate the aer odynamic dr ag from the other a cting forces, C 1 and
C 2 must be either known in advance or properly determined by the measurements.
This ofte n implies significant un certai nties (Somaschini et al. 2018). The high rati o
of aerodynam ic- to rolling resist ance in th e towing tank makes such attempts un-
necessary because the aerodynamic resistance constitutes about 99% of the running
resistance, Fig. 2 . This repre sents another im portant advantage of the towing tank.
For a rather slender body like a high-speed train with significant frictional drag,
the ove rall drag coefficient is expected to vary slowly with the Reynol ds number
(Brockie and Baker 1990). In case of drag determination by coasting tests, this im-
plies that there might be a small aerodynamic component contained within the linear
velocity term , w hich is not covere d by the common testing proce dure (CEN 2013).
In case of constant speed m easurement, as performed in the towing tank, the Reyn-
olds number im pact is directly captured by the m easured drag coefficient.

Fig . 2: Resistance percentages of the 3-car ICE/V train (c d =0.5 , C 2 =0 ) at dif ferent scale
and fluid with C 1 according to Rosenberger and Herz og (1993) and Y ang et a l. (2017) , as
well as a for a truck (c d =0.6, C 2 =0) with C 1 according to Hucho (2011)
However, when switching from air to water (at least when op erating with a free
water surface) an additional force, the resistance associated with the format ion of

1:22 v ehicle mo del in air , Re~0.8∙10 6
( C 1 from Yang)
1 :2 2 m o d e ll in w a te r ( p rese n t stu d y )
1:1 train in air , Re~14∙ 10 6
( C 1 from Rosenberger)
Running resistan ce (on straight and lev el track/road)

0% 20% 40% 60% 80% 100%
Rolling mechanical resistance Drag

1:22 v ehicle mo del in w ater , Re~0.8∙ 10 6
(present study)
1:1 truck in air , Re~5.6∙ 10 6
( C 1 from Hucho)

4
surface waves, has to be taken into account . In order to transfer the results from
water to air and from model-scale to full-scale, in addition to Reynolds- and Mac h-
number sim ilarity the Froude number (impact of surface wav es) a nd the onset of
cavitation have to be c onsidered as well. T he present pa per gives a n overview of
the relevant qu antities in the water tank. Several water depth to model height rati os
ranging from two to above five were st udied. Finally, the obtained results an d their
impact on drag m easurement of l ong land-borne vehicles in the towing tank will be
presented and discussed.
1.1 Wave resistance
The resistance of an obj ect moving close to or on the inte rface of tw o fl uids with
different densities such as a boat or a submarine in water is affected by waves cre-
ated at the interface, i.e. the water surface. On the one h and, the formation of these
waves requires energy, which is detracted from the driving power of the vessel . T he
mean wave energy E wave depends on the surface elevation ζ , the wa velength λ , grav-
ity g , fluid density ρ and wave widt h b (Clauss et al. 1992):
𝐸 𝑤𝑎𝑣𝑒 = 𝜌
2 𝑔𝑏𝜆 𝜁 2 (3)
Especially for submerged vessels, which generate small surface wa ve amplit ud es ζ ,
this part becomes comparably small and ca n be neglect ed. On the other hand, the
created wave pattern generates a pressure distribution around t he vessel, causing a
horizontal buoyancy force against the direction of m otion . These effects have been
investigated intensely for subm erged vessels (e.g., Gertler 1950, Wigley 1953,
Mansoorzadeh and Javanmard 2014, W ilson-Haffende n et al. 2010, Jagadeesh and
Murali 2010, Molland et al. 2011). The studies indicate that two parameters ( aside
from the shape of the body) predominantly influence this additional drag: the sub-
mergence dept h h and the Froude number Fr :
𝐹 𝑟 𝐿 = 𝑢
√ 𝑔𝐿 . (4)
Figure 3 presents the wave-drag coefficient of submerged streamlined bodies as a
function of Froude number and submergence ratio h/L as collected by Hoerner
(1965) . It can be seen that the wave -drag coefficient is very sensitive to the Froude
number. A drag maximum exists independent of the submergence h/L in the range
of Fr L =0.5 (Fig. 3, right plot) because t he wa velength the n is about twice t he m odel
length and wave crest and trough coincid e with the body’s bow and stern . Therefore,
the pressure difference and hence horizontal buoyancy becomes maximal. This
Froude number is referred to as critical Froude number Fr L,crit . Furthermore, Fig. 3
(left plot) sho ws that the wave-dra g decreases with increa sing distance to the water
surface h and at ratios above h/L =0.5 the surface wave impact almost vanish es . A t
fixed submergence depth h/d, an increase of the body length L by a factor of n de-
creases the wave drag coefficient by the same factor (Fig. 3, left plot). Therefore ,

5
for a given Froude number an elongation of the in vestigated body reduce s the wave
drag coefficient. For the assessment of wave resistance paramet ers, this makes the
ratio of h/d m or e relevant tha n the ratio of h/L .

Fig . 3 : Wave-drag coefficient c d,wave of submerged streamlined bodies as a function of
Froude number and submergence ratio h/L (Hoerner 1965)

Fig . 4: Wave-drag coefficient of a floating body as a function of Froude number for dif-
ferent water depths D /L reconstructed from Molland et al. (2011) (dashed lines indicate
Fr D =1)
Molland et al. (2011) a dd itionally took into account the influence of the water
depth D for sh all ow water conditi on s (Fig. 4). With decreasing depth, the m aximum
wave resistance shifts towards small er Froude numbers (decreasin g Fr L,crit ) while
its magnitude is increased significantly (Havelock 1908). This is due to the depend-
ency of the wave dr ag on the surface wave patterns. A very detailed description of
these principles is given by Larsson and Raven (2010). Depending on the water
depth D and the wavelength λ, three different regimes can be defined (Hensen 1955,
Molland et al. 2011):
1. D > λ /2: deep- water regime
2. λ /20 < D < λ / 2: intermediate or transitional re gime
3. D < λ /20: s hallow-water regime.
In a towing tank, experiments with land-borne vehicles will most likely be per-
formed in the transitional or shall ow -water regime . According to linear wave theory
(Airy 1845), the phase velocity c of the waves in ge neral follows

0
5
10
15
20
0 0.2 0.4 0.6 0.8 1
c d,wave /( d/L)²
h/L
Theory, Fr _ L = 0 . 5
Exp., Fr_L = 0 .4

0
5
10
15
20
0 0.5 1 1.5 2
c d,wave /( d/L)²
Fr L
h/L=0.2
h/L=0.1
h/L=0.05

d
h
L
Fr L =0.5
Fr L =0.4

-1
0
1
2
3
4
5
6
0 0.5 1
c d,wave / max [c d,wave,deep ]
Fr L
D/L=0.13
D/L=0.25
Deep water
0
D
L

6
𝑐 = √ 𝑔𝜆
2𝜋 tanh ( 2 𝜋𝐷
𝜆 ) . (5)
In the deep-water ra ng e, the phase velocity becomes independent of the depth
𝑐 = √ 𝑔𝜆
2𝜋 , (6)
while i n the shallow-water range the depth becom es the e xclusive im pact factor:
𝑐 = √ 𝑔𝐷 . (7)
This velocity changes as well affect the generated w ave patterns . Generally, t he
Froude length number as presented in (4) can be utilized to compare wave patterns
generated by a body of length L . However, the limited wave speed in shallow-water
(7) can have a significant impact on the wave pattern as well (Inui 1954, Larsson
and Raven 2010, Molland et al. 2011). The ratio between the velocity of the inves-
tigated body and the m aximum wave speed is given by the Froude depth number:
𝐹 𝑟 𝐷 = 𝑢
√ 𝑔𝐷 . (8)
Evidently, the body’s wave pattern in shallow water no t only depend s on its
Froude length nu m ber but al so on its Froude depth number, wh ich modifies the
wavelengths and thus the interference of wave components. This leads to an in-
crease of wave drag and a shift of the drag maximum towards lower Froude lengt h
numbers as observed in Fig. 4. Based on t he value of Fr D different flow regi mes can
be distinguished, comparable to the Ma ch -num ber in air, with a subcritical range
for Fr D < 0.9, a not precisely bounded transcriti cal range around Fr D = 1, and a su-
percritical range for Fr D > 1 (Larsson and Raven 2010) . Because a g ravity wave
cannot travel at 𝑐 > √ 𝑔𝐷 , above Fr D =1 the transverse wave system is left behind
and only divergent waves are present (Fig. 5). The radical change in the diverging
wave angle and t he general wav e pattern is accompanied with a wave drag m axi-
mum around Fr D =1 (Larsson and Raven 2010, Molland et al. 2011). After wards the
wave dr ag decreases again. In order to avoid a substantial impact by the generated
surface waves, investigatio ns in the subcritical range with Fr D << 1 are recom -
mended (Aoki et al. 1992, Hucho 2011). This however requires either a very deep
submergence or a very low velocity. While the fulfilment of the first is restricted by
practical limitati ons , the latter stays in conflict with the demand for testing at high
Reynolds num bers. Therefore, in the current study, in vestigations have been per-
formed in a wide range of Froude num bers (including the superc ritical Fr oude depth
number range) and the impact of waves has been analysed. Another approach to
reduce the impact of surface waves is the use of a skimming plate as presented by
Stephens et al. (2016). However, this might require the consideration of blockage
effects and was not c onsidered in the current inve stigations.

7

Fig . 5: Change of wave pattern around a ship hull with varying Fr D according to Larsson
and Raven (2010) as well as Molland et al. (2011).
1.2 Cavitation
The second difference when using a towing tank instead of a wind tunnel is the
maximum applicable free st ream velocity. For most applicati ons in vehicle aerody-
namics the flow can be considere d as inc ompressible ( if Ma< 0.25-0.3, Hucho
2011 ). Therefore, the speed limit in the wind tunnel is given by the on set of com-
pressibility effects . Due to the high bulk modulus of water, under normal conditions
no compressible effects appea r in the towing tank. Here , the restriction for the max-
imum free st ream (or vehicle dri ving) velocity is set by the incipience of cavitation
(which does not appear in air) , which can be e stimated by the incipient cavitation
number (Hoerner 1965): 𝜎 𝑖 = 𝑝 𝑎𝑚 𝑏 − 𝑝 𝑣𝑎𝑝
𝜌
2 𝑢 2 , (9)
with ambient pressu re p and vapour pressure p vap . If the absolute value of the m ini-
mum pressure coefficie nt exceeds the incipient cavitation number ( |𝐶 𝑝,min | > 𝜎 𝑖 )
cavitation might occur. The minimum pressure coefficient of a bluff body is about
-2 ≥ C p,min ≥ - 3 (Hucho 2011). At a water depth of a bout D =1 m, this sets an upper
velocity lim it under normal conditi ons ( T 0 =20°C, p 0 = 1 bar) of about u w ater ≈ 8 m/s
for non-cavitating flow. The requirement of Ma<0.3 gives a maxim um velocity of
u air ≈ 10 0 m/s for incompressible flow in air . Hence, at same model scale the maxi-
mum achievable Reynol ds number in air and water is a bout the same (Fig. 1 ). How-
ever, the much lower required veloc ity in water facilitat es the use of on-board meas-
urement technique, tim e resolved measurement techniques, and reduces the
requirements regardin g track and vehicle.

T ransverse waves Divergent w aves
α
α
Sub-critical Fr D < 1 Super-critical Fr D > 1
α = 19.46 α = asin(1/ Fr D )
T ranscritical Fr D ~ 1
α = asin(1/ Fr D )
α

8
2 Experimental setup
The experimental setup has been implem ented in t he 8 m wide and 120 m long shal-
low-water ta nk of t he Technische Universität Berlin (TUB) . The maxim u m water
level was set to approximat ely 1 m. Figure 6 shows the principle of the test rig: The
vehicle was pulled along a track by a 1.5 mm dia meter towing rope, which wa s
placed inside t he track bed and c onnected to a winch. The rope wa s directly attached
to a HBM S9M one component 2 kN force sensor (accuracy class 0.02) inside the
model, as shown in Fig. 7. Th us , the running resistance could directly be deter-
mined . The speed of t he winch was controlled by a LabVIEW base d computer rou-
tine, which allowed for the generation of arbitrary velocity profiles (Fig. 9). The
track was 64 m long i n t otal, extendable up to the tank length of 120 m for future
experiments. About 10 m we re required for acceleration of the model and 14 - 20 m
for deceleration.
Different sensors beside the track were used for the measurement of surface
waves, the pr esence of cavitation, trackside loads , and the velocity of the model
(Fig. 6). For the latter, a reflective pattern on the roof of the model combined with
a light-gate (I DEC SA1E-L PP3, 0.25 m s switching time) ab ove the track, as well
as a shaft encoder (Leine&Li nde RHI503, 1024 ppr) at the winch and an accelera-
tion sensor (Analog devices ADXL345 3-component sensor, 1.5kHz maximum
sampling rate, resolution of 0.038 m/s² ) inside the model were utilized. The velocity
detected by the light gate and the shaft encoder showed very good agreement with
deviations below 0.2% (Fig. 9) . The use of these different methods rendered the
possibility of very accurate determination of the model velocity. Furthermore, a
comparison of the different results was used to prove that no slip occurred at the
drive shaft. The velocity calc ulated from the integrated acceleration appea red to be
less accurate, due to insufficient resolution of the sensor and flexible mounting in-
side the model.

9

Fig . 6 : Principle and dimensions of the towing setup imple mented in the shallow-water
tank (upper picture); lower picture: external sensor positions: 1) hydrophone, wave sensor,
and light gate; 2/3) wave sensors; 4) pressure probe for measurement of head pressu re pulse,
y HPP =2.5 m and z HPP =1.8 m (at full-scale) as defined in CEN (2013), h/H=1
The wave height was measure d by resistance wave level sensors (0.5 mm accu-
racy ). Regardi ng wave measurement, the test stand offers the advantage of no water
surface piercing struts influencing the wave patterns. A Brüel and Kj aer miniature
Type 8103 hydroph one with a voltage sensitivity of −211 dB re 1 V/μPa over a
frequency band of 0.1 Hz to 180 kHz and a frequency response of ±1 dB at 4 kHz
to 200 kHz was used with 10 kHz sam pling rate for the dete ction of cavitation. Dif-
ferential pressure sensors inside t he train and along the track (Honeywell 26PC se-
ries with ±1 psi and ±5 psi range) were used for s urface pressure measurements and
the investigation of aerodyna mic loads beside the track. T he water temperature was
measured us ing a Pt100 temperature senso r to determine density and viscosity of
the surrounding fluid. The drag coefficient was determined from the mean drag
force during the constant velocity per iod (cf. Fig. 9), averaged over at least two runs.
For the m aximum velocity, t his co rresponded t o a minimal averaging time of 7 sec-
onds.
The investigated train model wa s a 1: 22 sc ale m odel ( L = 2.99 m , H = 0.17 m )
of the 3-car I nterCityExperimental (or ICE/V), manufactured as one solid body. The
model has bee n constructed modularly from synthetic materia l elements m ounted
on an aluminium core beam. That way the model was kept at the minimum required
weight to stay on the track safely and allowed for a simple change of geometrical

internal sensors and data logger
64 m
acceleration 10 m measurement sec tion 40 m braking 14 m
max. 1.1 m
drive shaft
tension weight
towing rope
external
sensors
2 3
4
H
h
z HPP
y HPP
y wave2 =1.7H
y wave3 =2.9H
1
top of rail
y wall =23H
z
y
D
z
x

10
configurations. In the pr esent study, a simple geometry variant without roof ele-
ments and with simplified bogies, as well as a complex variant equipped with ge-
neric roof elements and detailed bogies has been investigated. For both variants,
only the 2 nd and 5 th bogie (Fig. 8) were realized as rolling bogies. All other bogies
were equipped with wheels cut 2 mm above the rail to avoid rail contac t. Thus, the
mechanical system was kept simple and rolling resistance was reduce d. A more de-
tailed description of the model can be found in Tschepe et al. (2017, 2018a, and
2018b).

Fig. 7: Connection of towing rope to force sensor in the middle car

Fig . 8: Flying (left) and rolling (right) bogies of end cars (complex g eometry variant)

Fig . 9 : Measured force and velocity signal (u acc =velocity from integrated acceleration
signal, u winch =velocity from shaft encoder, u LG =velocity from light gate pattern ; the period
used for data evaluation is indicated by grey shading)

force sensor data logger box
data cable
rail towing rope towing connection ballast bed

100
u [m/s]
F tot [N]
u acc
u winch
u LG
F tot

11
The ballast an d rail design was inspire d by the single-track ballast and rail setup
(STBR) required for wind tunnel investigations of crosswinds CEN (2010). Due to
the shape of the aluminium elements used, the total height as well as the lateral slant
differed marginall y from the STBR norm configurati on.
The in vestigations were perform ed at a velocity range of u =1 -7 m/s at three dif-
ferent submergence depths, Table 1 (cf. Fig. 6 ).

Table 1: Investigated submergence depths

h/H

h/L

D/L

1

0.06

0.14

2.5

0.15

0.23

4.5

0.26

0.34

In order to reduce m easurement time (before each run a wa iting ti me of about 2 0
minutes wa s applied for the water to come to rest), runs with lower velocities (nom-
inal speeds of 2&3 m/s and 1&4 m/s) were combined into one test run. Conse-
quently, the external sensors be side t he trac k, placed at t he rear part of the measure-
ment section (Fig. 6), were passed with a minimal velocity of u = 3 m/s (Table 2 ).
Runs at a very low velocity ( u ≈ 0.15 m/s) were used t o determine the rolling re-
sistance, assuming that the rolling resistance is independent of the velocity as sug-
gested by (1). Since the drag force was sti ll noticeable e ven at such low velocity,
the rolling resi stance could not be dete rmined more accurate tha n F R = 2±0.5 N. The
impact of this uncertainty to the m easurement results will be discussed in the fol-
lowing section.

Table 2 : Investigated velocities and corresponding Froude/ Reynolds numbers (grey
shaded runs are not captured by external sensors)

u [m/s]

Fr L

Re (x10 6 )

1.06

0.20

0.12

1.95

0.36

0.24

3.02

0.56

0.36

4.10

0.76

0.49

5.00

0.92

0.60

6.03

1.11

0.73

6.92

1.28

0.83

12

Fig . 10: 1:22 scale 3-car ICE /V model (simple configuration) on test track in empty tow-
ing tank
3 Results
In this section, the res ults of wave- and cavitation measurements are presented. The
impact of waves regarding bot h drag determination a nd trackside load measurem ent
as well as possible correction m ethods will be discussed. Unless stated otherwise,
investigations were performed usi ng t he simple train ge ometry. The submergenc e
depth will be normalized by the veh icle’s height instea d of its len gth, because this
ratio appears t o be the more important param eter here as mentioned a bove.
3.1 Surface wave impact
Figure 11 s hows the drag c oefficient (with the rolling resistance F R =2 N subtracted
from the running resist ance) as a functio n of the Froude number, norm alized by the
drag at highest Froude num ber , c d,0 ≈ 0.46 . The impact of the submergence/water
depth as mentioned above becom es apparent clearly . The wave drag maximum ap-
pears little below Fr D = 1 and is strongly increased for decreasing submergence ra-
tios h/H (Fig. 11 a) . In the supercriti cal Froude depth number range, Fr D > 1, the
wave drag is strongly reduced and vanishes from a certain point on , depending on
the submergenc e depth h/H . Referring to the Froude length number (Fig. 11b) , the
regime of negligible wave impact with rest ricted depth seem s to start at e ven lower
Froude length numbers (around Fr L = 1) c ompared to unrestricted water depth
(Fig. 11 c) . Figure 11c shows the impact of wave drag for a streamlined bod y with
same dia met er to length ratio as the investigated train model (d /L=H/L ≈ 0.057) using
c d,0 = 0.055 (Hoerner 1965). The ratio of wave drag to total drag for both bodies
agrees quite well, because the streamli ned body creates both less surface waves and

13
lower aerodyn a m ic resistance . Hence, the impact of the wave drag on the determi-
nation of the aerodynam ic drag coefficient c d,0 is assumed to be independent of the
geometry of the inve stigated body.

Fig . 11: Drag coefficient as a function of the Froude number for dif ferent submergence
depths h/L ve rsus a) Froude depth number Fr D and b) Froude length number Fr L (dashed
lines indicate Fr D =1 ). c) drag coe fficient of ellipsoid in unrestricted water according to
Hoerner (1965) using d/L=0.057 and c d,0 = 0.055, h/H=h/d+0.5 (cf. Fig. 3)
In order to analyse the wave drag in more detail, the surface wave height has
been in vestigated. Figure 12 shows the elevati on of the water surface ζ m easur ed a t
different water levels at wave sensor 1 (Fig. 6) . At h/H=1, an increase of the wave
trough with increasing Froude number can be observed while the wav e crest is
slightly decreasing. W ith increasi ng water level the wave a mplitude generally de-
creases and the amplit udes of crest and trough at different F roude num bers are con-
verging (except for the lowest Froude number Fr L =0.56), while the wavelength in-
creases with the Froude number. The impact of the water level on the waves phase
velocity can be estimat ed by observing the distance behind the model until the lat-
erally propagating waves are reflected at the walls of t he t owing tank and super po se
again in the middle (i ndicated by dashed lines in Fig. 12). This distance can also be
calculated by using (7 ), see Fig. 13. It can be observed, that there is a n offset be-
tween t heoretical and experim ental data that is m ore distinct for higher wate r levels
(Fig. 13, left figure), indicating that the phase velocity is above shallow water ce -
lerity. Good agreem ent between theoretical and experim ental data can be achieved
by applying an iteratively achieved correction factor of n to (7), as shown in Fig. 13
(right figure): n ( h/H =1)=1.03, n ( h/H =2.5)=1.07, n ( h/H =4.5)= 1.13. This confirms
the assumption of tra nsitional water dept h for the experiments carried out . The dis-
tance until t he reflection peak and thereby induced pressures becomes rel evant in
case of pressure m easurements in the vehicle ’s wake .

1
1.2
1.4
1.6
1.8
2
0 0.5 1 1.5
Fr L

1
1.2
1.4
1.6
1.8
2
0 1 2 3 4
c d /c d,0
Fr D
D / L= 0. 34 D / L= 0. 23 D/L=0 . 14

h/H=4.5

1
1.2
1.4
1.6
1.8
2
0 0.5 1 1.5
c d /c d,0
Fr L
h/ L= 0. 2
h/ L= 0. 1
h/ L= 0. 0 5

h/H=2.5 h/H=1
a) b) c)
h/H=4
h/H=2.3
h/H=1.4
h
L
H h
H

14

Fig . 12 : Surface wave amplitude during and after train passage at different submergence
depths (grey area marks locati on of the train; dashed lines indicate superposition of lateral
reflected waves at respective colour)

h/H=1
h/H=2.5
h/H=4.5
a)
b)
c)

15

Fig . 13: Distance until superposition of lateral reflected waves in the towing tank. Theo-
retical values for shallow water phase velocity (left figure) and adapted velocity (right figure)
From the surface elevati on the pressure distribution along t he model induced by
the waves in transitional water can be calculated by (10) (Clauss et al. 1992) with
z= -h .
𝑝 𝑤𝑎𝑣𝑒 ( 𝑥 , 𝑧 ) = 𝜌𝑔 ζ ( x ) cosh [2𝜋 ( 𝑧 + 𝐷 )
𝜆 ]
cosh [2𝜋 𝐷
𝜆 ] ( 10 )
Normalized by the dynam ic pressure
𝑝 𝑑𝑦𝑛 = 𝜌
2 𝑢 2 ( 11 )
a wave pressure coe fficient can be defined
𝐶 𝑝,𝑤𝑎𝑣𝑒 = 𝑝 𝑤𝑎𝑣𝑒
𝑝 𝑑𝑦𝑛 . ( 12 )
Figure 14 shows the wa ve pressure c oefficient plotted along the train le ngth. An
improved impression about the dimensions is given by the c omparison with the train
induced pr essure signature in Fig. 19. The pr essure distribution as expected in a
wind tunnel with open (according to Hucho 2011) a nd closed (acc ording to Barlo w
1999) test section

2

and equivalent cross-section is shown in Fig. 14 as well. It can
be seen that at highest water level h/H =4.5 only the measurements at lowest vel oc-
ities/Froude numbers show a significa ntly higher pressure gradient than ob served
in an open test section wind tunnel. At the critical Froude number Fr L =0.56 the
pressure gradient is c omparable to the one obtained in a closed test section wind

2

According to Barlow, the pressure gradient in a closed square jet with width B
can be calculated depending on the distance Δ L using c p ( Δ L) =- k Δ L /B . The factor k
has been observed in the range of 0.016-0.04. For the plot shown in Fig. 14, B= √ 8
(same cross-section area as in the towing tank) and k =0.016 were used. The choice
of the lower lim it k -value considers the norm ally optim ized shape of the test sect ion
regarding boundary layer growth. However, the appearance of blocka ge effec ts bot h
in closed and op en test sections might impose additional pressure gradients that
were not considere d here.

16
tunnel with similar cross-section. Therefore, a correction approach similar as ap-
plied in open and closed wind tunnels (Wickern 2001) is aspired , focusing on the
pressure gradient. Du e to the rather smal l wav es, an im pac t on flow separation or
transition is not consi dered as critic al. Since the train m odel has a rather constant
cross-section, for drag calcula tion m ainly the pressure difference between head and
tail is of interest. Hence, the wave drag is calculated by the difference of wa ve pres-
sure at the head and tail, each averaged over the head length indicated by grey
shaded areas in Fig. 14 . The wave drag then can be evaluated usin g
𝑐 𝑑 ,𝑤𝑎𝑣𝑒 = 𝛥𝐶 𝑝 ,𝑤𝑎𝑣𝑒 = 𝐶 𝑝,𝑤𝑎𝑣𝑒 ( 𝑓𝑟𝑜𝑛𝑡 ) − 𝐶 𝑝,𝑤𝑎𝑣𝑒 ( 𝑡𝑎𝑖𝑙 ) ( 13 )
and 𝐹 𝑤𝑎𝑣𝑒 = 𝑝 𝑑𝑦𝑛 𝑐 𝑑 ,𝑤𝑎𝑣𝑒 𝐴 . ( 14 )

Fig . 14 : Wave pressure coefficien t along the train for different sub mergence depths (grey
shading indicates head length)
Figure 15 compares the wave drag calculated by (13) respectively (14) and the
wave drag that is ob tained when s ubtracting t he un affecte d drag coefficient c d,0 f ro m
the measured dr ag coefficient c d . It can be seen that the results agree quite well,
except for t he lowest Froude number, where the wave drag calculated from wave
height measurement, especially at deeper submergence, is much lower than the
measured drag increase. This m ight be due to additio nal Reynolds number effects
that occur at lower velocities (cf. Fig. 19) or energy losses due to the wave genera-
tion as described by ( 3).

open wind tunn el

17

Fig . 15: Comparison of wave drag coefficient calculated using different methods
The im pact of rolling resistan ce and wave resi st ance on the dr ag coefficient is
shown in Fig. 16. Wh ile at the highest velocity investigated t he wave drag at deepest
submergence and the rolling resistance each contribute less than 1% to the measured
running resistance, at lower velocities the proportion of these forces increases and
hence, needs to be subtracte d for a proper aerodynamic drag determination (Fig. 16
and 19 ). Interesti ngly, Fig. 16 also s hows that for the critical Froude number (equiv -
alent to Re=0.36) the wave resistance cannot be com pensated for by the proposed
correction method. Furthe rmore, the reproduction uncertainty at this point is sub-
stantially higher than for the other velocities investigat ed , which could be a ttributed
to m ore complex a nd sensitive wave structures at this velocity. However, above t hat
critical Froude number it can be seen that if the measured running resistanc e is cor-
rected for rolling and wave resistance, the obtained drag coefficient agrees reason-
ably well for all submergence depths investigated. This sh ows that reliable drag
measurement is possible even at very low submergence depths when apply ing the
proposed surface wave correction. For Re>0.5 ∙ 10 6 the drag coefficient appears to
be nearly independe nt of the Reynolds number which agrees well with the results
of previous studies (Willem se n 1997, Kwon et al. 2001). The excellent agreem ent
between the results at different boundary conditions, i.e. Reynolds numbers and
submergence de pths, shown in Fig. 16 under lines the reliability of the m ethod.

0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.5 0.7 0.9 1.1 1. 3
c d,wave /c d,0
Fr L
c dw ave= c d-c d, 0,
h/ H = 1
c dw av e= c d-c d, 0,
h/ H =2 . 5
c dw av e= c d-c d, 0,
h/ H =4 . 5
c dw ave= D C p, h/ H = 1
h/ H =2 . 5
h/ H =4 . 5
c d,wave = Δ C p ,w ave
c d,wave = c d -c d,0
h/H=1
h/H=2.5
h/H=4.5

18

Fig . 16 : Proportion of wave and rolling resistance to fluidic force over Reynolds number
(left plot); drag coefficient (with F roll =2 N subtracted) for different water levels as a function
of Reynolds number with (solid lines) and without (dashed lines) wave drag subtracted and
reproduction error bars (right plot)

Fig 17: Measured running resistance (exp) with wave resistance subtracte d and least
squares fit for simple ( SG) and complex (CG) geometry varian t w ith all coefficien ts variable
(dashed/dotted lines) and C 1 =F roll =2 N, C 2 =0, and C 3 =c d ρ /2 A ref (solid lines)

For a more detailed evaluat ion of the impact of the rolling resistance, the meas-
ured f orce data at different velocities (for measurem ents with u ≥ 4 m/s in order to
subtract the wave resistance) is a nalysed ac cording to (1) usin g a least squares fit
method (Fi g. 17). A comparis on is made to the coefficients obtained when us ing t he
data as presented in Fig. 19 (using F roll =2 N a nd c d =mean[ c d ( u )] ). It can be seen
that the difference in the rolling resistance is negligible (compared to the overall
forces). The C 3 co efficient of both m ethods is almost identical to a differe nce below
0.2% for the sim ple and about 4% for the c omplex geometry varia nt. For the latter,
this discrepancy results from the m ore distinct Reynolds number dependency of the

x10 6
c d - Δ C p,wave
c d
h/H=1
h/H=2.5
h/H=4.5
x10 6

19
drag coefficient , which in terms of the Davis formula is expressed by an increased
C 2 coefficient (Fig 17). It can be concluded that the rolling resistance only contrib-
utes very little to the C 2 coeffici ent. Hence, the assumption of a speed independent
rolling resistan ce appears to be justified.

In order to evaluate the accuracy of the experim ental method and to validate nu-
merical si mulations (Tschepe et al. 2018b) different sta ges of geometric complexity
we re investigated. Figure 18 shows that the waves generated at the surface remain
the same if elements are applied to the roof, implying that small geometry changes
do not af fect the wave pattern, even if those components significantly increase drag
(Tschepe and Nayeri 2018a). Figure 19 illustrates the impact of the rolling re-
sistance. It can be seen that the uncertainty band of the rolling resistance has no
significant effect on the drag coefficient.

Fig . 18: Wave pressure coefficient at h/H=4.5 for the model without (simple) and with
(complex) roof elements applied as shown
In Tschepe et al. (2018b), t he dat a obtained in the towing tank are compared to
CFD and full-scal e results. For the simple configuration (not investigated at full-
scale) the difference to CFD is about 3%. For the complex configuration, less than
2% difference to the CFD a nd less than 8% difference to the full-scale res ults were
found. The latter might result from Reynolds number effects and the m odelling of
the roof elem ents. Nevert heless, these comparisons highlight the potential of towing
tank experiments for dr ag determ ination of long vehicles.

simple
complex
roof elements

20

Fig . 19: Drag coefficient

3

for models of different detail complexity and varying r olling
resistance as a function of Reynolds number, h/H=4.5
The influence of surface waves can as well be observed in the pressure signature
of the train (Fig. 20), m easured at position 4 (Fig . 6) . For the lowest Froude number
the wave pressure is in the order of 10% of the train induced head suction peak (at
x/L ≈ 0.04), while for the highest Froude number investigated this reduces to about
1%. If the pres sure generated by the surface waves is subt racted from the measured
pressure signature, the data at all Froude numbers qualitatively agrees well with the
corresponding ref erence data, which underlines that the proposed correction method
is reasonable. Some significant discrepa n cies can still be observed. These are due
to oscillations of the pressure probe ( which diam eter appea red t o be insufficient) ,
caused by the pressure fluctuations during the train passage . This results in multiple
additional pressure peaks at lower velocities and a significant increase of the head
and tail peak at high velocities. Hence, future investigatio ns of trackside loads re-
quire more robust m easurement equipment.
In Fig. 20 e) a comparison of the pressure signature to full-scale data given by
Baker et al. (2013) for the very sim ilarly shaped ICE2 is m ade. The towing tank
data is averaged over different Reynolds numbers to lower the impact of t he velocity
dependent probe oscillations (Fig. 20a- d) . All data qualitatively agrees well, despite
the pressure rise at 3 ≤ x ≤ 10 obtained in the towing ta nk which is due to t he probe
oscillations. The inter-car gap peak of the ICE2 can be seen at about x=26 m. The
ICE/V dr iving car only has a length of about 20 m. Therefore, the inter-car gap peak
appears at about x=20 m. The peak- to -peak val ue of the nose pressure fr om the ex-
periments exceeds the full-scale data by a bout 8%. Considering the slightly different
measurement height an d probably some differences in the t rack bed, this a ppears to
be a good agreem ent.

3

I n Tschepe et al. (2018), though stated otherwise , the drag coefficient with
F roll =0 is shown!

0.45
0.475
0.5
0.525
0.55
0.575
0.3 0.5 0.7 0.9
c d - Δ C p,wave
Re x10 6
F_roll = 0 N F_roll = 1.5 N F_roll = 2 N F_roll = 2.5 N

complex
simple

21

Fig . 20 a)-d): Pressure level measured at position 4 for the simple geometry at h/H=4.5
at different Reynolds/Froude numbers compared to CFD (PANS) results Tsc hepe et al.
(2018b); e): Nose pressure meas urements of moving model (MM) compared to CFD and full-
scale (FS) data (Baker et al., 2013)

a) b)
c) d)
e)

22
3.2 Cavitation impact
As mentioned above, the onset of cavit ation can be estim ated by the incip ient cavi-
tation number, (9). Since the ambient pressure depends on the wate r depth, the in-
cipient cavitation num ber increases with inc reasing depth, Fig. 21 (left plot). The
minimum pr essure coefficient of the investi gated t rain is a bout C p ≈ -1 (Tschepe et
al. 2018b, Fischer et al. 2018). With the m aximum velocit y being about u =7 m /s,
cavitation appears to be very unlikely. The presence of cavitat ion ca n be detect ed
experimental ly by acoustic measurem ents, indicated by a drastically noise increase
for freque ncies f ≥ 3kHz (Zhang et al. 2002, Brennen 2005, Schmidt et al. 2017) .
Figure 21 shows that this has not been detected in the c urrent measurement results ,
confirming that no cavitat ion occurs. The results are shown for the lowest water
depth h/ H =1. According to Fig. 21 (left plot ), higher water levels ap pear to be even
less critical towards t he onset of cavitation.

Fig . 21: Incipient cav itation number for di fferent water depths and velocities ( left f igure);
Frequency spectr a of hydrophone measurements duri ng train passage for h/H=1 (right)
4 Conclusion
In the paper advantages of wa ter based moving model facilit ies, focusing on drag
measurement of trains , were discussed. It wa s shown, that the significant change in
the ratio of fluid dynamic drag to rolling resistance in the water tank allows for an
accurate determ ination of the total drag coefficient . The utilization of a semi self-
sufficient model enables undisturbed m easu rements of aerodynamic quantities.
Limitations concerning the boundary conditions, such as maximum speed or mini-
mum submergence depth, a re pose d by the onset of ca vitation a nd the generation of
surface waves. However, a s ufficiently wide range of t hese param eters for undis-
turbed m easurements of long and rather sm ooth vehicles like high-spee d trains was
found. In comm on literature, investigations o f land -borne vehicles are proposed to
be performed under subcritical Froude number conditions to avo id wave impact

1
2
3
4
5
6 7 8 9 10
σ i
v [m /s]
h/H=1
h/H=2.5
h/H=4.5

risk of cav itation
range of current
investigations
u [m/s]
σ i
PSD [V²/Hz]
f [Hz]
3 kHz
frequency of the w hee ls

23
(Hucho 2011, Aoki 1992). Howe ver, the current resu lts show that for geom etries
similar to the one investigated here on ly the critical range of about 0.2 < Fr L < 1 has
to be avoided and supercritical conditions with Fr L ≥ 1 allow for accurate measure-
ments as well. It was shown, that for submergence depths of about h/H ≥ 4.5 the
wave drag becomes su fficientl y small to be neglected ( ≤1 % of the total drag for
Fr L ≥ 1). Furtherm ore, a correction method using measured surface wave heights
was introduced. By applying this method t o the drag m easurement results, the wave
drag related uncertainty in the aerodynamic drag coefficient is further reduced and
be comes about 0.5%, even for m uch lower submergence depths and Froude num-
bers. A practical minim um of the submergence depth in com bination with the sug-
gested correction method can be assumed to be about h/H = 2-3 and the minim um
Froude number of ab out Fr L = 0.75. For lower Froude numbers or in cases of blunt
geometries with probably m or e distinct wave gen eration, the application of a skim
plate close to the water surface, as described by Nepper t (1981) or Stephens et al.
(2016), should be investigate d. The advantage of the test facility presented in this
paper, allowi ng for th e investi gation of surface waves without the interference of
surface piercing support struts, should be used for further analysis of the waves
generated by di fferent geometries, such as trucks, buses a nd cars.
Theoretical and experimental approaches show that the investigated velocity
range is uncritical regarding the onset of cavitation at all submergence depths in -
vestigated, at least for geometries similar to the investigated one. Therefore , the
results ca n be t ransferred to air without limitation, as long as the corres ponding air-
flow ca n be conside red as incompressible. T his is a valid as sumption for free stream
Mach numbers below Ma<0.25-0.3, corresponding to a vehicle speed of a bout
300 km/h in the open air.
A compariso n of the results to other investi gation methods showed excellent
agreement. Also the high level of agreement between the measurem ents for differ-
ent conditions (i.e. water heights/submergence depths ) proves the viab ility of the
method. Therefore, the towing tank can be considered as a very promising facility
for drag measurements of long land-borne vehicles. Furthe rmore , it allows for the
investigation of tra nsient e ffects such as vehicle e ncounters and passings with re-
gard to aerodynamic loads on the vehicle and its surroundings (e.g., slipstreams,
head pressure pulse, etc.). Therefore, a high number of different situations and quan-
tities c an be investigated with a single model, significantly reducing the costs for
research and de velopment.

Acknowledgments The resea rch presented was supported by the ZIM program
and t he BIT GmbH. The project was funded under grant number EP 141376 from
the “Zentrales Innovationsprogramm Mittelst and (ZIM)” of the Fe deral Mi nistry of
Economy and Energy, following a decision of the Germ an Bundestag.

Conflict of interests The a uthors declare that they have n o conflict of interest.

24
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