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FLOW AROUND SHIPS SAILING IN SHALLOW WATER—EXPERIMENTAL AND NUMERICAL RESULTS 968
Flow Around Ships Sailing in Shallow Water—Experimental and
Numerical Results
X.-N.Chen, A.Gronarz, S.List
(Versuchsanstalt für Binnenschiffbau e.V. Duisburg)
N.Stuntz
(Gerhard-Mercator-Universität Duisburg)
ABSTRACT
In this paper the results of the investigation of the flow around the models of two ships in shallow water without
propeller, rudder and appendices are presented. The intention is—based on reliable experimental data—to compare
computational results from different methods, obtained on different grids, and see how these methods perform. The
experimental measurements comprise the velocity field at several stations, the free surface deformation, the hull pressure
distribution and (not presented in this contribution) the bottom pressure. Three different computational methods were
applied: A standard RANSE-solver on two different grids (one block-structured hexahedral, one unstructured tetrahedral),
a RANSE-solver using a fictitious compressibility and thus a flux-difference-splitting technique and a potential theoretic
method devised for the transcritical regime (solitons!), where the hull is described merely by its cross-sectional area.
INTRODUCTION
In this paper the special characteristics of the flow around ships sailing in shallow water are demonstrated for two
hull forms.
Though there is no well defined limit for the water depth h dividing shallow from deep water, the behaviour of
waves is well known to depend on the depth h. The behaviour of vessels sailing in shallow water may be characterized by
one cinematical parameter, the dimensionless number Fnh, the depth Froude number, defined as:
with V the speed of the ship and g the gravitational acceleration. Fnh clearly rules the wave resistance of the ship, by
representing the ratio of V to the so called critical velocity leading thus to a division of the whole velocity range
into a subcritical and a supercritical region depending on Fnh being smaller or larger than 1, in similarity with what
happens at high velocities in air, where the velocity of sound is the critical velocity, separating the subsonic from the
supersonic range. The velocity range of V with values of 0.9

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FLOW AROUND SHIPS SAILING IN SHALLOW WATER—EXPERIMENTAL AND NUMERICAL RESULTS 969
fluid dynamics (CFD) backs the department of experimental fluid dynamics (EFD).
Fig. 1: Lines Plan—Shallow river vessel (Ship A)
Fig. 2: Lines Plan—Inland waterway ship (Ship B)
The two selected examples of ship types are typical for the investigative performance of the VBD in both fields EFD
and CFD. In both cases the emphasis is on the sailing on shallow water, and as the topic here is flow around the ship, no
allusion will be made to trim, sinkage or resistance values obtained.
MODELS INVESTIGATED
Shallow river vessel
The model investigated (Fig. 1) is a typical inland waterway ship for extreme shallow water. The stern is, as can be
seen from the lines plan, designed to allow the installation of a special flat thruster system (Schottel Pump Jet), flush with
the hull surface. Its scale was 1:12. The model was fitted with 120 pressure taps at 12 different stations. In this paper we
will refer to it as Ship A. The main particulars are:
Length betw. perp. [m] 82.0
Lpp
Beam [m] 9.5
B
Draft [m] 1.5
T
[m3]
Displacement 1054
Inland waterway ship
This is a single screw ship with the typical form of inland water vessels. As can be seen from Fig. 2 it has a
characteristic tunneled stern, designed to ensure good water supply to the propeller. In these ships the propellers have
rather small diameters and are highly loaded. Though only results for the scale 1:12.08 are presented, a series of geosim
models was investigated with the scales 1:12.08, 1:14, 1:17.28 and 1:21.54. In this paper we will refer to it as Ship B. The
main particulars are:
Length betw. perp. [m] 110.00
Lpp
Beam [m] 11.45
B
Draft [m] 3.00
T
[m3]
Displacement 3312
EXPERIMENTS
The coordinate system used is righthanded: x-axis oriented in direction of ship motion, y-axis at right angles pointing
to starboard, z-axis orthogonally upward, the origin is at the intersection of aft perpendicular with base-line.
Ship A
The wave pattern on the water surface was recorded by a series of wave probes located at appropriate y-positions and
standard pressure sensors were used to obtain the hull-pressure distribution for the cases
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FLOW AROUND SHIPS SAILING IN SHALLOW WATER—EXPERIMENTAL AND NUMERICAL RESULTS 970
h/T Fn Fnh Rn
3.95*106
1.2 0.078 0.529
4.94*106
2.0 0.098 0.512
The velocity field was only measured at h/T=2.0 with a 3-component Laser-Doppler-Velocimetry (LDV) system,
specifically designed to be used in shallow water tests.
Ship B
Here we will deal only with the model at the scale 1:12.08, comparing some results obtained in the shallow-water
towing-tank with computational results. Results are presented for the h/T values
h/T Fn Fnh Rn
1.19*107
1.5 0.138 0.685
1.44*107
3.0 0.180 0.681
Deformation of the free surface and hull pressure were determined as in Ship A, while for the survey of the velocity
field (5 sections in the stern region) a spherical 5-hole pressure probe was used.
CFD-METHODS APPLIED
RANSE-Solver CFX-5 from AEA Technology (Methods 1 & 2)
This solver applies the finite volume-element method. Several turbulence models of the scalar eddy-viscosity type or
a Reynolds-stress model may be chosen. Here the choice between blockstructured hexahedral (Method 1) as well as
completely unstructured grids with tetrahedral meshes (Method 2) is an appealing feature. A mesh refinement in
(eventually nested) regions is easily performed for the unstructured grid. For the detailed characteristics and features
available consult the specification sheet provided by AEA Technology (Grotjans & Menter, 1998; N.N., 1999).
RANSE-Solver based on the principle of artificial compressibility (Method 3)
This is a finite volume method using a structured grid. Introducing an artificial compressibility a coupling of the so
obtained continuity equation to the momentum equations is achieved, allowing an efficient Roe flux-difference-splitting
technique. The method includes the computation of the free water surface considered as the boundary between water and
air and described by a so called level set function satisfying an additional equation (derived from the boundary condition)
solved together with the above mentioned equations. The turbulence model is the standard k-ε model (Stuntz, 1999).
For the three RANSE-methods the following boundary conditions have been used:
• Bottom and wall: no slip, moving boundary (u=−Va), velocity normal to boundary is zero
• Ship: no slip, stationary wall, velocity normal and tangential to boundary is zero
• Water surface, center plane: symmetry, velocity normal to boundary is zero, viscous forces parallel to boundary
are zero, Method 3: boundary condition according to free surface flow
• Inlet: velocity Va normal to inlet, turbulence intensity=0.05, turbulent to molecular viscosity ratio=10
• Outlet: constant pressure, Method 3: non reflecting boundary condition (free surface flow)
Shallow water potential-theoretic transcritical treatment (Method 4)
Applying shallow-water-wave theory and a far-field/near-field treatment with appropriate matching, an ingenious
potential theoretic method was developed by Chen and Sharma (1994) and reimproved by Chen (1999) yielding a detailed
description of the deformation of the free surface. An equation of a Kadomtsev-Petviashvili type for a depth averaged
potential is derived for the far field and an improved slender body theory for the near field. Though the method was
developed originally to cope with the instationary case of the transcritical regime (including the solitons), it is applicable
over a wide range of Fnh numbers. The numerical solution is obtained by applying a finite difference method. The
topography of the waterway bottom may vary in direction perpendicular to the direction of ship motion and the hull is
represented merely by its cross-sectional area distribution and the local beam of the water line.
EFD RESULTS
Ship A
The pressure distribution on the hull is shown in dimensional form by colours and isolines on the hull surface. The
location of the pressure taps is shown (red dots), Figs. 6 and 9. The results (Gronarz et al,
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FLOW AROUND SHIPS SAILING IN SHALLOW WATER—EXPERIMENTAL AND NUMERICAL RESULTS 971
Fig. 5: Computational grid (Method 2), h/T=2.0
Fig. 6: Hull pressure distribution. Experimental results h/T=2.0, Fn=0.098, Fnh=0.512, Rn=4.94*106
Fig. 7: Hull pressure distribution. Computational results
Fig. 9: Hull pressure distribution. Experimental results at
(Method 1) h/T=2.0, Fn=0.098, Fnh=0.512, Rn=4.94*106
h/T=1.2, Fn=0.078, Fnh=0.529, Rn=3.95*106
Fig. 8: Hull pressure distribution. Computational results
Fig. 10: Hull pressure distribution. Computational results
(Method 2) at h/T=2.0, Fn=0.098, Fnh=0.512, Rn=4.94*106
(Method 1) h/T=1.2, Fn=0.078, Fnh=0.529, Rn=3.95*106
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FLOW AROUND SHIPS SAILING IN SHALLOW WATER—EXPERIMENTAL AND NUMERICAL RESULTS 972
Fig. 11: Velocity distribution (stern section x/Lpp=0.1). LDV-measurements at h/T=2.0, Fn=0.098, Fnh=0.512, Rn=4.94*106
Fig. 12: Velocity distribution (stern section x/Lpp=0.1). Computational results (Method 1) at h/T=2.0, Fn=0.098, Fnh=0.512,
Rn=4.94*106
Fig. 13: Velocity distribution (stern section x/Lpp=0.1). Computational results (Method 2) at h/T=2.0, Fn=0.098, Fnh=0.512,
Rn=4.94*106
Fig. 14: Velocity distribution (stern section x/Lpp=0.0). LDV-measurements at h/T=2.0, Fn=0.098, Fnh=0.512, Rn=4.94*106
Fig. 15: Velocity distribution (stern section x/Lpp=0.0). Computational results (Method 1) at h/T=2.0, Fn=0.098, Fnh=0.512,
Rn=4.94*106
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Fig. 16: Velocity distribution (stern section x/Lpp=0.0). Computational results (Method 2) at h/T=2.0, Fn=0.098, Fnh=0.512,
Rn=4.94*106
The colour of the velocity vectors of the computational results for Ship A indicates the absolute value of the velocities,
with warm for high and cold for low speed.

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FLOW AROUND SHIPS SAILING IN SHALLOW WATER—EXPERIMENTAL AND NUMERICAL RESULTS 973
Fig. 17: Velocity distribution (stern section x/Lpp=−0.1). LDV-measurements at h/T=2.0, Fn=0.098, Fnh=0.512, Rn=4.94*106
Fig. 18: Velocity distribution (stern section x/Lpp=−0.1). Computational results (Method 1) at h/T=2.0, Fn=0.098, Fnh=0.512,
Rn=4.94*106
Fig. 19: Velocity distribution (stern section x/Lpp=−0.1). Computational results (Method 2) at h/T=2.0, Fn=0.098, Fnh=0.512,
Rn=4.94*106
Fig. 20: Velocity distribution (stern region). LDV-measurements at h/T=2.0, Fn=0.098, Fnh=0.512, Rn=4.94*106
Fig. 21: Water surface deformation. Experimental results at h/T=2.0, Fn=0.098, Fnh=0.512, Rn=4.94*106
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Fig. 22: Water surface deformation. Computational results (Method 4) at h/T=2.0, Fn=0.098, Fnh=0.512

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FLOW AROUND SHIPS SAILING IN SHALLOW WATER—EXPERIMENTAL AND NUMERICAL RESULTS 974
Ship B
Overview of figure numbers showing results
x/Lpp EFD CFD-Method
h/T
Exp. 1 2 3 4
hull presssure 1.5 23 24 25
surface deformation 1.5 29 30
31
51 51
53
velocity distribution 1.5 −.1098 33 34 35
velocity distribution 1.5 .0274 36 37 38 45
velocity distribution 1.5 .0640 39 40 41 46
velocity distribution 1.5 .1006 42 43 44 47
streamlines 48 49
surface deformation 3 52 52
(Remark: The calculations with Method 3 have been carried out at h/T=2.0 instead of 1.5)
The computational grid for the Methods 1 and 2 (Fig. 26 and 27 on the next page) extended in x-direction from 1.5
Lpp in front to 0.5 Lpp behind the ship, from the water plane to the bottom and from the center plane to a channel width of
4.9 m (half breadth of the VBD towing tank).
The calculations with Method 3 used a structured grid (Fig. 28 on the next page) which extends in x-direction from
0.82 Lpp in front to 1.82 Lpp behind the ship, from the water plane to the bottom and from the center plane to the channel
wall. The structured grid used consisted of 160 elements in longitudinal, 50 in transverse and 33 in vertical direction.
The volume meshes for calculations carried out with the 3 RANSE-methods had the extensions
Method Vertices Elements
h/T
1 (Hex) 1.5 843.867 780.606
2 (Tet) 1.5 217.618 986.935
3 (Compr) 2.0 279.174 264.000
The resolution used for the calculations with Method 4 was 60 sections for the model, 420 cuts in front and 840
behind the ship. The tank width (9.81 m) was divided into 61 intervals.
Fig. 23: Hull pressure distribution. Experimental results at h/T=1.5, Fn=0.138, Fnh=0.685, Rn=1.19*107
Fig. 24: Hull pressure distribution. Computational results (Method 2) at h/T=1.5, Fn=0.138, Fnh=0.685, Rn=1.19*107
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Fig. 25: Hull pressure distribution. Computational results (Method 3) at h/T=2.0, Fn=0.138, Fnh=0.546, Rn=1.19*107

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FLOW AROUND SHIPS SAILING IN SHALLOW WATER—EXPERIMENTAL AND NUMERICAL RESULTS 975
Fig. 26: Computational grid (Method 1, hexahedral, blockstructured), h/T=1.5
Fig. 27: Computational grid (Method 2, tetrahedral), h/T=1.5
Fig. 28: Computational grid (Method 3, structured, hexahedral), h/T=2.0
Fig. 29: Water surface deformation ζ.
Experimental results at h/T=1.5, Fn=0.138, Fnh=0.685, Rn=1.19*107
Fig. 30: Water surface deformation ζ.
Computational results (Method 4) at h/T=1.5, Fn=0.138, Fnh=0.685
Fig. 31: Water surface deformation ζ.
Computational results (Method 3) at h/T=2.0, Fn=0.138, Fnh=0.5458, Rn=1.19*107
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Fig. 32: Selected Sections for the measurement and computational of the velocity distribution

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FLOW AROUND SHIPS SAILING IN SHALLOW WATER—EXPERIMENTAL AND NUMERICAL RESULTS 976
Fig. 33: Velocity distribution (behind stern, x/Lpp=−0.1098).
Computational results (Method 1) at h/T=1.5, Fn=0.138, Fnh=0.685, Rn=1.19*107
Fig. 34: Velocity distribution (behind stern, x/Lpp=−0.1098).
Computational results (Method 2) at h/T=1.5, Fn=0.138, Fnh=0.685, Rn=1.19*107
Fig. 37: Velocity distribution (stern section x/Lpp=0.0274).
Fig. 35: Velocity distribution (behind stern, x/Lpp=
Computational results (Method 2) at h/T=1.5, Fn=0.138,
−0.1098).
Fnh=0.685, Rn=1.19*107
Experimental results (5-hole-pressure probe) at h/T=1.5,
Fn=0.138, Fnh=0.685, Rn=1.19*107
Fig. 38: Velocity distribution (stern section x/Lpp=0.0274).
Experimental results (5-hole-pressure probe) at h/T=1.5,
Fig. 36: Velocity distribution (stern section x/Lpp=0.0274).
Fn=0.138, Fnh=0.685, Rn=1.19*107
Computational results (Method 1) at h/T=1.5, Fn=0.138,
Fnh=0.685, Rn=1.19*107
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FLOW AROUND SHIPS SAILING IN SHALLOW WATER—EXPERIMENTAL AND NUMERICAL RESULTS 977
Fig. 39: Velocity distribution (stern section x/Lpp=0.064).
Computational results (Method 1) at h/T=1.5, Fn=0.138, Fnh=0.685, Rn=1.19*107
Fig. 40: Velocity distribution (stern section x/Lpp=0.064).
Computational results (Method 2) at h/T=1.5, Fn=0.138, Fnh=0.685, Rn=1.19*107
Fig. 43: Velocity distribution (stern section x/Lpp=0.1006).
Fig. 41: Velocity distribution (stern section x/Lpp=0.064).
Computational results (Method 2) at h/T=1.5, Fn=0.138,
Experimental results (5-hole-pressure probe) at h/T=1.5,
Fnh=0.685, Rn=1.19*107
Fn=0.138, Fnh=0.685, Rn=1.19*107
Fig. 42: Velocity distribution (stern section x/Lpp=0.1006).
Fig. 44: Velocity distribution (stern section x/Lpp=0.1006).
Computational results (Method 1) at h/T=1.5, Fn=0.138,
Experimental results (5-hole-pressure probe) at h/T=1.5,
Fnh=0.685, Rn=1.19*107
Fn=0.138, Fnh=0.685, Rn=1.19*107
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FLOW AROUND SHIPS SAILING IN SHALLOW WATER—EXPERIMENTAL AND NUMERICAL RESULTS 978
Fig. 45: Velocity distribution (stern section x/Lpp=0.0274).
Computational results (Method 3) at h/T=2.0, Fn=0.138, Fnh=0.546, Rn=1.19*107
Fig. 46: Velocity distribution (stern section x/Lpp=0.064).
Computational results (Method 3) at h/T=2.0, Fn=0.138, Fnh=0.546, Rn=1.19*107
Fig. 47: Velocity distribution (stern section x/Lpp=0.0274).
Computational results (Method 3) at h/T=2.0, Fn=0.138, Fnh=0.546, Rn=1.19*107
Fig. 48: Streamline visualisation by ribbons.
Computational results (Method 2) h/T=1.5, Fn=0.138, Fnh=0.685, Rn=1.19*107
Fig. 49: Streamline visualisation by ribbons.
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Computational results (Method 3) at h/T=2.0, Fn=0.138, Fnh=0.546, Rn=1.19*107

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FLOW AROUND SHIPS SAILING IN SHALLOW WATER—EXPERIMENTAL AND NUMERICAL RESULTS 979
Fig. 50: Dependence from the water depth: Velocity distribution (stern section x/Lpp=0.0274).
Computational results (Method 1) at Fn=0.0426, Rn=3.64*106
Fig. 51: Comparison of measured (solid line) and Fig. 52: Comparison of measured (solid line) and
computed (dashed line, Method 4) wave cuts for different computed (dashed line, Method 4) wave cuts for different
y/h at h/T=1.5, Fn=0.1406, Fnh=0.6302, Rn=1.097*107 y/h at h/T=3.0, Fn=0.1803, Fnh=0.6805, Rn=1.440*107
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FLOW AROUND SHIPS SAILING IN SHALLOW WATER—EXPERIMENTAL AND NUMERICAL RESULTS 980
Fig. 53: Computed 3-D-wave pattern showing reflection on tank walls (Method 4) at h/T=1.5, Fn=0.1406, Fnh=0.6302,
[Domain comprising the full tank width, vertical scale exaggerated 8 times]
Fig. 54: Computed 3-D-wave pattern showing detaching solitons (Method 4) at h/T=4, Fn=0.353, Fnh=1.012, [Domain
comprising the full tank width, vertical scale exaggerated 8 times]
COMPARISON OF RESULTS
The correspondence between hull pressure distributions obtained by the different computational methods and
between computation and experiment is disappointing (Figs. 6–8 and 9–10 and 23–25). This applies to both hull forms
and to any of the h/T ratios presented. The computed distributions show a broader range of variability than the
experimental distributions suggesting that the (three-dimensional and probably rather thick) boundary layer in the stern
region has a certain equalizing effect and is probably not well reproduced in the computation. A pressure variation across
the layer (contrary to the assumption for thin boundary layers) strongly dependent on the location on the hull is possibly
present too but not reproduced in the computation. For Ship B in addition the experimental values as shown by the color
distribution are misleading, as pressure taps were not located precisely on the sharp edges bordering the tunnels but on the
sides of these edges avoiding thus likely cusps in the real pressure distribution, the interpolating postprocessing routine
producing rather a smoothed distribution.
Advancing to the distribution of the velocity as a vector field we are nevertheless faced with a better correspondence.
For Ship A (velocity measured by LDV at the h/T ratio 2) this is fair at the section x/Lpp=0.1 while at the section x/Lpp=0.0
there seems to be a suspicious deviation from symmetry in the experimental values as displayed by the arrows
representing the velocity vectors in the plane y=0. For Ship B (velocity measured by five-hole pressure probe at h/T=1.5)
the computed distributions of the velocity obtained by Methods 1 and 2 in the four selected transverse planes (Fig. 32)
show the characteristic vortex well known from the experiment, though its location appears slightly altered, the
correspondence being somewhat better for small x/Lpp (Figs. 33–35, Figs. 36–38, Figs, 39–41 and Figs. 42–44). Due to
difficulties in the generation of the structured grid for the Method 3 here an h/T ratio of 2 and not of 1.5 was chosen. The
results are displayed in the Figs. 45–47.
The change of velocity field with decreasing water depth presented in Fig. 50 (computational results only) gives a
good impression of the so called shallow water effect. It can be seen, that the vortex core is displaced outward with
decreasing water depth and the influence of the hull pressure field influences the bottom pressure more and more.
The deformation of the free surface is not well reproduced by the computational methods, neither for Ship A, where
only results from the potential theoretic Method 4 (Fig. 22) may be compared with the experimental results (Fig. 21), nor
for Ship B, where results from Method 3 (Fig. 31) and from Method 4 (Fig. 30) should match the experimental pattern
(Fig. 29). The comparison of wave cuts for Ship B shown in Fig. 51 (h/T=1.5) and Fig. 52 (h/T=3) gives an impression of
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the quality attained by Method 4.
The 3-D wave pattern shown in Fig. 53 demonstrates the ability to reproduce wave reflections from the side walls,
while in Fig. 54 the generation of solitary waves detaching from the bow at a speed near the critical Froude number has
been computed.

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FLOW AROUND SHIPS SAILING IN SHALLOW WATER—EXPERIMENTAL AND NUMERICAL RESULTS 981
CONCLUSION
As a towing tank institution the VBD has a paramount interest in deliver reliable data to its customers. This applies
nowadays as well to CFD results as it has been always expected for the EFD results. Independence of the computational
grid used is one of the first achievements to be reached. We see, that for us, there remains some work to be done.
Realistic results of the flow around ships in shallow water will only be obtained if the local surface deformation and
the squat are appropriately worked out. In addition to model scales full scale cases should be computed too. In the future
the influence of the turbulence model applied has to be clarified. So far the numerical treatment corresponds to the
situation in the standard resistance test. The dependence of the flow around the ship on different canal widths will have to
be one of the next steps. In the future the appropriately simplified simulation of the propulsor will be integrated into the
computational treatment as well as rudder and appendages thus corresponding to the situation in the self-propelled-test.
This will finally lead us to more thorough investigations of the instationary flow and the manouevring of ships in
restricted water.
ACKNOWLEDGEMENTS
We acknowledge the dedicated work of all the members of the VBD engaged with those projects which lead to the
results (EFD and CFD) presented in this paper. Prof. E.Müller and Dr. J.Kux contributed by their guidance in the
preparation and formulation of this contribution. Thanks go to the institutions from the state Nordrhein-Westfalen and the
Federal Republic Germany which funded the research projects involved.
REFERENCES
Chen, X.-N. (1999): Hydrodynamics of Wave-Making in Shallow Water, Dissertation, Faculty of Mathematics, University of Stuttgart
Chen, X.-N. and Sharma, S.D. (1994): Nonlinear theory of asymmetric motion of a slender ship in a shallow channel. 20th Symp. on Naval
Hydrodynamics (ONR), Santa Barbara, California, National Academy Press, Washington, D.C.
Gronarz, A., Grollius, W., Rieck, K. and Pagel, W. (1995): Experimentelle und theoretisch-numerische Strömungsuntersuchungen an Binnenschiffen.
VBD-report no. 1366
Bet, F., Stuntz, N., Hänel, D. and Sharma, S.D. (1999): Numerical Simulation of Ship Flow in Restricted Water. 7th International Conference on
Numerical Ship Hydrodynamics, Nantes
Grotjans and Menter (1998): Wall Functions for General Application CFD Codes, ECCOMAS 98, Fourth European Computational Fluid Dynamics
Conference, Athen
N.N. (1999): Using CFX-5 for Unix & Windows NT, User manual from AEA Technologies, Harwell, UK
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FLOW AROUND SHIPS SAILING IN SHALLOW WATER—EXPERIMENTAL AND NUMERICAL RESULTS
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