Experimental and Numerical Investigations on Flow ... · The viscous flow on ship hulls at 0°...

Experimental and Numerical Investigations on Flow Characteristics of

the KVLCC2 at 30° Drift Angle

Moustafa Abdel-Maksoud1, Volker Müller1, Tao Xing2, Serge Toxopeus3, Frederick Stern4, *Kristian Petterson5,

Magnus Tormalm5, Sungeun Kim6, Shawn Aram6, Uwe Gietz1, Patrick Schiller1, Thomas Rung1

1 Hamburg University of Technology, Germany, 2 University of Idaho, USA, 3 MARIN, Netherlands, 4 University of Iowa, USA,

5 Swedish Defence Research Agency (FOI), Sweden, 6 NSWCCD West Bethesda, USA.

Investigations of flow characteristics around ship hulls at large drift angle are very important for understanding the

motion behavior of ships during maneuvers. At large drift angles, the flow is dominated by strong vortical structures

and complex three-dimensional separations. An accurate prediction of these flow structures is still a challenge for

modern computational fluid dynamics (CFD) solvers. Hull forms with high block coefficients are blunt and have strong

curvatures, which leads to large area flow separations over smooth surfaces. These areas are sensitive to the relative

angle between the flow and the ship motion direction. The paper is concerned with a collaborative computational study

of the flow behavior around a double model of KVLCC2 at 30 degrees drift angle and Fr=0 condition, including

analysis of numerical methods, turbulence modeling and grid resolution, and their effects on the mean flow and

separation onset as well as formation of the vortical structures. This research is an outcome of a multi-year

collaboration of five research partners from four countries. The overall approach adopted for the present study

combines the advantages of CFD and EFD with the ultimate goal of capturing the salient details of the flow around

the bluff hull form. The experiments were performed at the low - speed wind tunnel of the Hamburg University of

Technology (TUHH). The main features of the global and local flow were captured in the experimental study. To

determine the global flow characteristics, two different flow visualization techniques were used. The first one is a smoke

test, which allows the visualization of vortex structures in vicinity of the ship model. The second test is a classic oil film

method, which yields the direction of the limiting wall streamlines on the surface of the model. The analysis of the

experimental results helped identify the separation zones on the ship model. To resolve the local flow-fields, LDA and

PIV measurements were carried out in a selected number of measuring sections. Subsequently, the EFD and CFD

results for the global and local flow structures were compared and analyzed. The numerical simulations were carried

out by 5 institutions: Iowa Institute of Hydraulic Research of the University of Iowa (IIHR), USA, Maritime Research

Institute Netherlands (MARIN), The Netherlands, Hamburg University of Technology (TUHH), Germany, Naval

Surface Warfare Center, Carderock Division (NSWCCD) West Bethesda, USA and Swedish Defense Research Agency

(FOI), Sweden. For the comparison with the experimental results, seven submissions of steady and unsteady CFD

results are included in the present study. The participating codes include CFDShip-Iowa, ReFRESCO, FreSCo+, Edge,

OpenFOAM (FOI) and NavyFoam. The size of the computational grids varies between 11 and 202 million control

volumes or nodes. The influence of turbulence modeling on the predicted flow is studied by a wide variety of models

such as isotropic eddy viscosity models of k-family, Explicit Algebraic Reynolds Stress Model (EARSM), hybrid

RANS-LES (DES), and LES. Despite notable differences in the grid resolutions, numerical methods, and turbulence

models, the global features of the flow are closely captured by the computations. Noticeable differences among the

computations are found in the details of the local flow such as the vortex strength and the location and extent of the

flow separations.

KEY WORDS: KVLCC2, Verification and Validation,

Turbulence models.

INTRODUCTION Investigations of flow characteristics around ship hulls at high

drift angles are very important for understanding the motion

behavior of ships during maneuvers. At high drift angles, the flow

is strongly dominated by vortical structures and large separation

areas. An accurate computation of these flow structures is still a

challenge for advanced computational fluid dynamics (CFD)

solvers. Hull forms with high block coefficients are blunt and

have a smooth surface, which leads to large area flow separation.

This area is sensitive to the relative angle between the flow and

the ship motion direction.

The viscous flow on ship hulls at 0° drift angle has been

extensively investigated in the last two decades. At the CFD

Workshop Gothenburg 2010 [18], various research groups

studied the flow on the tanker hull form KVLCC2 with large

block coefficient CB = 0.81 at calm water straight-ahead condition

with Fr = 0.142. Special attention was given to the wake field and

the flow structure in the stern region, especially 3D separation at

the stern, which was characterized by 2 co-rotating axial vortices

with hook-shaped axial-velocity contours in the nominal wake

plane (symmetric with respect to center plane).

The results presented by Bhushan et al. [2] show that when using

sufficiently fine grids, both URANS- and DES-based numerical

methods are able to provide good predictions of the mean flow on

Abdel-Maksoud Experimental and Numerical Investigations of the KVLCC2 at 30° Drift Angle 2

KVLCC2. Compared with experimental wind-tunnel data,

differences can be seen in vortex strength and the turbulence

variables. According to the presented results in Bhushan et al. [2],

it can be concluded that anisotropic URANS models deliver more

accurate results than the isotropic models for the prediction of

onset and formation of vortical structures, but are too dissipative

even for fine grids. Hybrid RANS/LES models are promising in

providing the details of the flow topology, but show modeled

stress reduction and grid-induced separation in the boundary layer

issues for bluff bodies.

The calm water static and dynamic maneuvering conditions of

KVLCC2 were the topics of study at the Maneuvering Simulation

Workshops SIMMAN 2008 [32] and SIMMAN 2014 [33]. At the

2014 workshop, the X- and Y-force and the yaw moment acting

on the ship model with rudder at a static drift angle β = 12 degrees

and a rudder angle δ = 0 degrees were compared with

experimental data D. The comparison of the mean errors E for

surge force (X) are relatively high at about 44.39%D, and for

sway force (Y) and yaw moment (N) are relatively small at about

3.43 and 2.09%D, respectively.

Vortical and turbulent structures for KVLCC2 (Fr = 0) at β = 0,

12 and 30 degrees were investigated using DES and URANS for

KVLCC2 by Xing et al. [45]. One shear layer, one Karman-like

Vortex shedding and three helical mode instabilities were

identified. The After-Body Side Vortex, Fore-Body Side Vortex,

and After-Body Bilge Vortex exhibit all characteristics for helical

instability. For analyzing the helical instability in the wake of the

vortex, the Strouhal number, based on the ship’s length or the

distance along the vortex core, was considered. The turbulent

kinetic energy (TKE) peaks near the separation point at the bow

and on the vortex core of all vortices. For the After-Body Side

Vortex, TKE reaches the local maximum right after the helical

instability and intensifies along the vortex core further

downstream. In addition to the three steady vortices previously

identified, the simulations at β = 12 and 30 degree show unsteady

Aft-Body Hairpin and After-Body Side Vortices.

The objective of the present study is to verify and validate the

CFD for KVLCC2 at the β = 30 degree condition, which includes

the analysis of numerical methods, turbulence modeling and grid

resolution, mean flow and onset as well as formation of vortex

structures. This research is a collaborative of five research

partners from Germany, Netherlands, Sweden and USA. The

present paper documents the CFD results for KVLCC2 in deep

water without consideration of the free surface effect. The

experiments are performed at TUHH’s low-speed wind tunnel

and the CFD studies are carried out at 5 institutions with different

numerical methods, turbulence models and grid resolutions. The

CFD and EFD studies are consolidated such that the global flow

visualization in CFD complements EFD for identifying the

vortical structures; the EFD provides global and local

measurements for CFD validation; and once validated, the CFD

then fills in the sparse EFD, thus enabling detailed diagnostics of

the flow field.

As a successful validation requires precise planning of

experimental investigations, the published CFD results by Xing

et al. [45] using the CFDShip-Iowa code are evaluated to identify

the main flow characteristics and to localize the position of the

each vortex. After analyzing the CFD data, the experimental

setup is designed and the locations of the measurement planes are

determined. The experiments are designed to capture the global

and the local flow properties. In order to determine the global

flow characteristics, two different flow visualization experiments

are carried out. The first one is a smoke test, which visualizes

vortex structures in the flow field in the region close to the ship

model. The second test is a classic oil film method, which yields

the direction of the wall streamlines on the surface of the model.

The analysis of the experimental results provides the separation

zones on the ship model. To determine the local flow properties,

Laser Doppler Anemometry (LDA) and Particle Image

Velocimetry (PIV) measurements are carried out in the

previously defined measuring planes. Subsequently, the EFD and

CFD results for the global and local flow structures are compared

and analyzed.

The present study includes a summary of an experimental

investigation for the ship hull KVLCC2 at 30 degree angle of

attack in the TUHH wind tunnel and a comparison between the

measured and CFD results. The numerical calculations are carried

out at Iowa Institute of Hydraulic Research (IIHR) of the

University of Iowa, USA, Maritime Research Institute

Netherlands (MARIN), The Netherlands, Hamburg University of

Technology (TUHH), Germany, Naval Surface Warfare Center,

Carderock Division (NSWCCD), USA and Swedish Defence

Research Agency (FOI), Sweden.

OVERVIEW OF EXPERIMENTAL VALIDATION

VARIABLES AND CFD SIMULATIONS The experimental investigations focus on the global and local

flow structures on the ship hull KVLCC2 at 30 degree drift angle.

The global flow structure is captured by applying two flow

visualization tests.

Table 1: Overview of performed wind tunnel experiments at

TUHH.

Experiment

Wind

tunnel

velocity

Measured variables / aim

PIV 27 m/s Velocity components, Ux, Uy, Uz

Vorticity component ωx

LDA 27 m/s

Tangential velocity

component Ut, and its turbulence

degree TUt.

Smoke test 12 m/s Vortex visualizations

oil film test 27 m/s Limiting streamline visualization

The local flow structure is obtained from velocity measurements

at pre-defined cross sections. Flow velocities are measured

mainly by PIV technique. Additional LDA measurements are

conducted to get a reliable resolution close to the hull. An

overview of the available model test data is given in Table 1.

The numerical simulations are carried out by 5 institutions. For

the comparison with the experimental results, 7 submissions are

included in the present study. The codes involved are CFDShip-

Iowa (IIHR), ReFRESCO (MARIN), FreSCo+ (TUHH), Edge

(FOI), OpenFOAM (FOI) and NavyFOAM (NSWCCD). The

finest grid of each submission varies between 11 and 77 million


control volumes. The influence of the turbulence on the flow is

investigated with a wide variety of models such as Explicit

Algebraic Reynolds Stress Model (EARSM), Detached Eddy

Simulation (DES), k-ω- SST (1994 and 2003 versions), EARSM

in combination with Hellsten k-ω, Large Eddy Simulation (LES),

RANS-LES and Mixed Model (MM). Computations are carried

out either as steady or unsteady condition. An overview of the

submissions is given in Table 3.

EXPERIMENTAL INVESTIGATION The experimental investigation for the very large crude oil carrier

hull form (KVLCC2) is carried out at the TUHH’s Institute for

Fluid Dynamics and Ship Theory (FDS). A double model of the

KVLCC2 is installed in the low speed wind tunnel. The free

stream velocity is fixed at 27 m/s. Dimensions of the model are:

LOA = 1.6m, Lpp = 1.535m, B = 0.279m, 2 x T = 0.206m, and

the corresponding Reynolds numbers based on the model length

is 2.74 × 106.

The measurements consist of smoke tests for the flow

visualization as well as the spatial distributions of the velocity

components at predefined planes in the bow, midship, stern

regions and behind the model. The measurements focus on the

longitudinal vortices formed on the hull and in the near-wake

region.

Wind Tunnel The wind tunnel measurements are performed using the 5.5m

long open test section with a cross-sectional area of 6m2. The

degree of the free stream turbulence is less than 0.3%. The test

section allows manual and optical access from the top and the two

lateral sides. Positioning the intrusive and non-intrusive

measurement sensors is supported by a multiple-axes traversing

system mounted to the lateral sides of the section. This two-

component traverse system is used to move the complete PIV or

LDA system for measuring the flow velocity at various planes

and positions.

Ship Model The MOERI tanker KVLCC2 without any appendages is

considered in the CFD and experimental investigations [32]. A

double-body model of the underwater hull of KVLCC2 is used

for the experimental investigations in the wind tunnel. Compared

with the original draft of the ship, the draft of the investigated

model is slightly increased in order to avoid a strong discontinuity

of the hull near the forward perpendicular. The CAD-data of the

hull are mirrored about the waterline, which is located at 0.68m

above the design waterline. The main dimensions of the ship and

model are given in Table 2 and Figure 1.

Experimental Setup The model is suspended at 30 degree drift angle inside the test

section by 8 wires, each 1.0mm in diameter. The forward and the

aft 4 wires are fixed at x/Lpp = 0.085 (frame 18) and at x/Lpp =

0.908 (frame 2), respectively, see Figure 2. A wind tunnel-based

coordinate system is applied. The origin is located at the

intersection of the waterline and forward perpendicular. The x-

axis coincides with the main flow direction. The y-axis is directed

toward the top. The z-axis points toward the sidewall of the

tunnel, at the opposite side of the PIV system, as required by a

left-handed coordinate system (shown in Figure 2).

Figure 2: Model and coordinate system.

At the forward perpendicular the model is equipped with a strip

of sand similar material to ensure the boundary layer transition at

this location. The blockage factor (ratio between the projected

areas of the double model and the cross section of the measuring

section) is about 0.026. The effect of blockage is not

homogeneous along the model. The distance between the model

and the sidewalls of the test section changes due to the drift angle

along the model. In the forward region of the model, the leeward

side is close to the wind tunnel ceiling. In the aft region, the

windward side comes close to the wind tunnel bottom. The

sidewalls of the test section are open and the ceiling and the

bottom are closed during the tests. Due to these special blockage

conditions, the velocity deviations in the far field of the model are

about 1% of the adjusted inflow wind speed.

Figure 1: Main model dimensions and Body plan of KVLCC2.

Table 2: Main dimensions.

Ship Model

Length over all LOA [m] 333.6 1.600

Length between perpendiculars Lpp [m] 320 1.535

Breadth (waterline) B [m] 58 0.279

Draft (original) T [m] 20.8 0.099

Draft (modified) D [m] 21.4755 0.103

Block coefficient CB [-] 0.8098 0.8098

Scale ratio [-] 1 208.5

Measuring Planes for the Velocity Measurements


Figure 3: Vortex system of KVLCC2 (isosurface of Q=200

colored by helicity) bottom view at β=30° [45] (top).

Regions of the experimental investigations (bottom).

Important regions for the local flow measurement are defined

based on the results of the numerical simulations published by

Xing et al. [45], see Figure 3. The results show that the flow is

dominated by three vortices. These are the Fore-Body Side

Vortex (FSV), After-Body Side Vortex (ASV) and After-Body

Bilge Vortex (ABV).

In order to capture the vortical flow structure, three regions for

the velocity measurements are defined. The flow in each region

is measured in vertical planes (perpendicular to the inflow

velocity). The three regions include the FSV, ASV and ABV, see

Figure 3. The marked planes at Figure 3 indicate width and height

of the measured regions. Their extent from the water plane of the

model (z = 0) is 200mm outwards.

A comparison between the experimental data and simulation

results is given in Section Results Planar Plots exemplary for the

section ASV-11.

Measuring Techniques PIV System

The spatial distribution of the velocity components in different

planes is measured by a modular commercial 2D-3C-PIV system

from TSI Inc.. The stereo PIV system (S-PIV) consists of a pulsed

laser, a light sheet optic, two cameras and a computer with

software to control image generation and processing.

The light sheet is generated by a 200 mJ two-head Nd-YAG laser

(Quantel Big Sky). Scattered light is received by two PowerView

4M (2048x2048 pixel, 12bit, monochrome) cameras equipped

with 105mm/F2.8 lenses; their baseline is located approximately

1.7m from the middle of the test section. The optical axes of the

lenses are inclined against the light sheet at angles of 62.5° and

41°.

For the PIV measurements, particles of an average diameter of

about 1 μm are generated as the seeding. The Laskin type droplet

generator uses dioctyl sehacate (DOS). The generator is placed

downstream of the test section. The fog generated spreads

through the wind tunnel at the closed loop operational mode and

leads to a global seeding. Therefore, any influence due to

turbulence of the generated fog is negligible.

LDA System

Additional velocity measurements are carried out using a one-

component LDA-system. The power of the Nd-Yag laser is 200

mW and the wavelength equals 532 nm. The tangential velocity

component to the hull sidewall Ut is measured at the planes in the

FSV and ASV regions. The velocity component is parallel to the

centerline of the model; this means the angle between the

measured velocity component and the free stream velocity is

equal to the drift angle (30 deg). The estimated uncertainty of the

measured velocity component is ± 0.20 m/s. The turbulence

degree is evaluated for this velocity component

Measuring Uncertainties

The PIV system is mounted on a crossbar, 2D automated traverse

system. The uncertainty of the positioning is ±0.1mm. The

estimated uncertainties of the three velocity components are Uw

= ± 0.08m/s, Uv = ± 0.06m/s and Uu = ± 0.33m/s. The relative

uncertainties of the velocity components to the free stream

velocity are Uw = ± 0.30%, Uv = ±0.22% and Uu = ± 1.22%. The

uncertainty of the 2D automated traverser system is estimated

based on a high number of repetition tests. The uncertainties of

the measured data by the PIV system are determined by the

dimensions of the measuring planes and the resolution of the

cameras as well as the selected time shift between two images.

Each measuring plane has about 50% overlap area with its

neighbors.

Flow Visualization

The smoke (vaporized oil) is injected to the wind flow near the

bow of the double model at different vertical and lateral positions.

The smoke distribution is recorded using a moving laser light

sheet for a sectional illumination. Stills (at the positions of the

PIV measuring planes) and movies are taken.

For visualizing the wall streamlines at the hull surface, a classic

oil film method is employed. The hull is coated uniformly with

oil mixed with sooty particles. As the wind tunnel starts, the oil

film on the surface moves in the direction of local shear forces.

After an order of 10’s of seconds, a nearly stationary particle

distribution is reached which appears as a sketch of the wall

streamlines. The evolving streamline pattern is recorded with a

Canon EOS 70D camera. The same camera is used to record the

smoke distribution. Stills are taken after shutting down the wind

tunnel to capture the final oil patterns reflecting the spatial

distribution of wall shear stress.

Experimental Results at Measuring Sections The experimental results include the flow visualization pictures

obtained by the smoke and classic oil film tests as well as the

velocity components in the measuring planes Ux, Uy, Uz in the

wind tunnel coordinate system and tangential velocity component

parallel to the hull surface Ut. The velocity components Uy, Uz

are used to calculate the longitudinal vorticity ωx. The fluctuation

of the velocity component (turbulence degree in [%]) is evaluated

as follows:

FSV

ASV

ABV

1 32 4

1 32 4 5 6 7

1 3 5 7 9

11 14


TUt= 100 × √(Ut

− Ut)2

/Ut,

where Ut is the instantaneous Ut.value and Ut its mean value.

The measured velocity components are normalized by the inflow

velocity. The analyses of the velocity vectors in the measuring

planes allow for identifying the particular vortex. The velocity

vectors can be used to identify the location and the strength of the

different vortices in the flow field. The results of the experimental

study will be discussed in detail in the results section.

CFD COMPUTATIONS

CFDShip-Iowa Setup The general-purpose CFDShip-Iowa-V.4 [3] solves the unsteady

RANS or DES equations in the liquid phase of a free surface flow.

The free surface can be captured using a single-phase level set

method, and the turbulence is modeled by isotropic or anisotropic

turbulence models. Numerical methods include advanced

iterative solvers, second and higher order finite difference

schemes with conservative formulations, parallelization based on

a domain decomposition approach using the message-passing

interface (MPI), and dynamic overset grids for local grid

refinement and large-amplitude motions.

Turbulence Models

The Algebraic Reynolds Stress (ARS) model [41] is based on a

modified version of Menter’s k- / k- turbulence model as the

scale determining model, and an explicit ARS model as the

constitutive relation in place of the Boussinesq hypothesis. The

ARS model is extended to ARS-DES models in CFDShip-Iowa-

V.4. The readers can refer to Xing et al. [45] for details of this

model.

Geometry, Boundary Conditions, and Simulation Domain

The domain extends (-2Lpp, 2Lpp) in the streamwise direction

(𝑥), (-1.5Lpp, 1.5Lpp) in the transverse direction (𝑦), and (-

1.2Lpp, -0.1Lpp) in the vertical direction (𝑧). The negative 𝑧

ensures that the entire ship hull is submerged in the water without

solving the level set transport equation. The top boundary is

specified as a “symmetry” boundary to mimic the double-body

model in the experiment. Body-fitted “O” type grids are

generated for ship hull and rectangular background grids are used

for specifying boundary conditions away from the ship hull, with

clustered grid near the symmetry boundary to resolve the flow

near the ship hull. As required by the turbulence models, 𝑦1+ <

1.2 is enforced for the first grid point away from the ship hull for

all the grids. The total number of grid points is 13 million, which

is much finer than Grid 3 used in [46]. The use of a 13-million

grid and DES model resolves 87% of the total TKE in the LES

region. An overview of the generated grid for the applied CFD

solvers is given in Table 4.

FreSCo+ and ReFRESCO FreSCo+ and ReFRESCO both originate from the FreSCo code

[40] that was developed within the VIRTUE EU Project together

with TUHH, Hamburg Ship Model Basin (HSVA) and MARIN.

After VIRTUE, two separate developments were continued:

FreSCo+ [30] by HSVA and the Institute for Fluid Dynamics and

Ship Theory (FDS) at TUHH; and ReFRESCO by MARIN.

These codes are generally similar and are described below.

The codes solve the multi-phase unsteady incompressible RANS

equations, complemented with turbulence models [44] and

volume-fraction transport equations for each phase.

The equations are discretized using a finite volume approach. The

procedure uses a segregated algorithm based on the strong

conservation form of the momentum equations. In ReFRESCO,

the mass and momentum equations can be solved in a coupled

manner as well.

A cell-centered, collocated storage arrangement for all transport

properties is employed. Structured and unstructured grids, based

on arbitrary polyhedral cells or local grid refinement with

hanging nodes, can be used. The implicit numerical

approximation is second-order accurate in space and time.

Integrals are approximated using the conventional mid-point rule.

The solution is iterated to convergence using a pressure-

correction scheme. Various turbulence-closure models are

available with respect to statistical (RANS) approaches. In both

codes scale-resolving (LES, DES) approaches are available as

well [28].

For ReFRESCO, several code verification studies have been

performed [6][7]. For maneuvering applications, several solution

verification and validation studies [35][36][37][38] have been

completed.

FreSCo+ Computational Setup

For the numerical calculation only one half of the double body

model is used in order to reduce the computational effort.

Therefore, the cutting plane is considered as a symmetry plane.

The size of the computational domain is 6L×6L×1.5L (L x B x

H). The hull has the same dimensions as in the experiment. The

walls of measuring section of the wind tunnel are not included in

the calculation.

Three unstructured grids are generated with the software

HEXPRESS. The grids have a refinement around the hull and in

the wake area, where the vortex structure will be formed. The drift

angle of β = 30° is taken into account. The typical cell size in

this region is reduced in two steps by a factor of √2. The detailed

values for the cell size and the number of cells of the grids can be

seen in Table 4. The y+value is kept constant for all grids and its

maximum value is y+= 1.2. In fact, the converged solution using

the same y+ for all grids may be different from the solution when

also refining the y+ upon grid refinement. Furthermore, y+=1.2

may be too large for SST, due to the very large gradients of ω

close to the wall, see [15].

In contrast to the wind tunnel experiment, in the numerical

calculation the fluid properties (density and viscosity) of water

are used. To achieve the same Reynolds number as in the

experiment the free stream velocity in the computation is set to

u = 1.785m/s. The calculations are performed with standard

k − ω and with the SST k − ω turbulence model of Menter 2003.


Unless otherwise noted, the results are presented for the finest

grid and Menter’s SST k − ω turbulence model.

An unsteady computation is also performed on the finest grid with

Menter’s SST k − ω turbulence model. The applied time step in

this calculation is dt=0.002s and the total simulation time is 12s.

The residuals (L1 norm) of unsteady calculation are three orders

of magnitude smaller than those of the steady calculation.

ReFRESCO Computational Setup

Multiblock structured O–O grids are used for this study for the

best performance of ReFRESCO. Grid points have been clustered

toward the hull surface and bottom to ensure proper capturing of

the boundary layers. The far field boundary is generated as a

cylindrical surface, to facilitate the use of a single grid for all

computations. The grids have been used previously for other

studies in which various water depths were studied [36][38]. The

diameter of the domain is 4Lpp and the bottom is located at about

2Lpp below the top surface. For all cases the y+ values in the first

cell from the wall are smaller than 1 for the finest grid, such that

the equations are integrated down to the wall. Grids have been

generated with GridPro. Based on these grids, geometrically

similar grids are generated by coarsening the finest grid in all

directions in order to assess the discretization errors and to

accelerate the iterative procedures by using coarse grid solutions

as initial flow fields for fine grid computations. Seven grids are

obtained, ranging from 121×103 to 12721×103 cells. Unless

otherwise noted, the results are plotted for the finest grid.

To simplify the calculations, symmetry boundary conditions are

applied on the undisturbed water surface. On the hull surface, no-

slip and impermeability boundary conditions are used. For all

calculations, the boundary condition on the bottom surface is set

to moving-wall/fixed slip (V=V∞, with V∞ the inflow velocity).

Since the computations were carried out prior to the wind tunnel

tests, some differences between the computational settings and

the wind tunnel setup are present. First, the loading condition of

the KVLCC2 corresponds to the normal loading condition (i.e.

T=20.8m). This means that the submergence of the hull is slightly

less than half the height of the wind tunnel model. Second, the

Reynolds number is slightly higher than that reached in the wind

tunnel: in the computations, the Reynolds number corresponds to

3.7 million. Third, the walls of the wind tunnel have not been

modeled in the grid and therefore blockage effects of the wind

tunnel are neglected. Computations have been conducted using

the 1994 [23] and 2003 [24] versions of Menter’s k-ω SST model.

The Spalart correction (proposed by Dacles-Mariani et al. [5]) of

the streamwise vorticity has been applied.

Edge FOI Edge Solver and Setup

Edge [8] is a three-dimensional Reynolds-averaged Navier-

Stokes (RANS) solver for compressible flows on unstructured

grids. The solver has an edge-based formulation and uses node-

centered finite volume techniques. The edge-based formulation

makes it possible to compute any arbitrary n-faced polyhedral

elements. The current version handles two types of surface

elements and four types of volume elements. The control volumes

are non-overlapping and are formed by a dual grid that is

computed by the pre-processor from the control surfaces for each

edge of the primary input mesh. Edge can be used for both steady

state and time accurate calculations. Time accurate computations

can be performed using a semi implicit, dual time stepping

scheme which exploits convergence acceleration technique via a

steady state from inner iteration procedure. A large number of

turbulence models are available that are categorized into three

different groups: RANS, Detached Eddy Simulation (DES) or

hybrid RANS-LES, as well as Large Eddy Simulation (LES).

Edge is the only compressible solver in this investigation and is

primarily developed by FOI with contributions from universities,

research institute and industry including Saab Aerospace and the

Royal Institute of Technology (KTH). More information is

available from the Edge homepage www.foi.se/edge.

For the KVLCC2 case, both time-accurate RANS (URANS) as

well as hybrid RANS-LES computations are run with Edge

version 5.2. The turbulence model selected for the URANS case

is the Wallin and Johansson [41] Explicit Algebraic Reynolds

Stress Model (EARSM) with the Hellsten k-ω model [11]. The

ship hull surface is modeled as an adiabatic weak wall boundary

condition with fully turbulent flow. A symmetry condition is

applied on the water surface to mimic the wind tunnel setup with

the double hull. All other surfaces are modeled as a far-field

boundary with a weak characteristic formulation. Standard day

conditions (NASA definition) are prescribed with a static

pressure of 101325 Pa, static temperature of 288 K and a free-

stream flow velocities of 27m/s with a drift angle of 30°. Thus,

no effect of wind tunnel walls is accounted for in the FOI Edge

computations. The low Mach number of 0.08 is a challenge for

any compressible solver and Edge has a pre-conditioning option

to overcome this. It is used to reduce the effect of too high

numerical dissipation. However, this was not applied in the

current investigation, which may have a negative effect of the

intensity and extend of the vortical flow. Otherwise, Edge default

input values are used.

The RANS case is initially run using a steady state approach with

local time stepping. However, due to oscillations in the solution,

a switch is made to a time-accurate RANS approach. The dual-

time scheme is set up using an implicit time step of 50 μs and 40

inner iterations. Initialization is done from the steady state RANS

and the solution is stabilized after 100 time steps but is continued

up to 1000 steps to fully develop force and moments. Mean-value

sampling starts after 300 time steps. For the hybrid RANS-LES

approach, the HYB0 method developed by Shia-Hui Peng

[26][27] at FOI is selected. The same dual-time scheme used in

the URANS case is used for the HYB0 case.

The computation is run with a three level multigrid cycle which

took approximately 3.5s/it on 128 cores on FOI’s J29 Linux

cluster. Also the HYB0 case is initialized from the RANS

solution and run for 1000 time steps before the mean-value

sampling starts for another 1000 time steps. The total

accumulated time for the HYB0 case is 0.1 s corresponding to

1.75 flow passes of the hull.


FOI Edge Mesh

The KVLCC2 mesh used for the Edge computations is a ship-

oriented, fully symmetric mesh. The unstructured mesh is

generated on the half-breath geometry and mirrored in the ship’s

symmetry plane to get a symmetric node distribution and scaled

to the wind tunnel model dimensions. FOI followed a two-step

approach to create the grid. In the first step, the ANSYS® ICEM

CFD™ was used to generate a patch dependent surface mesh and

from that make an initial unstructured volume mesh with the

Delaunay advancing front method. The far field was modeled as

a box, 3.75Lpp ahead, to the side and down of the model. The box

behind the ship is extended to 12.5Lpp. In the second step, FOI’s

in-house mesh generator TRITET [39] is used to build prismatic

boundary layers normal to the ship wall. A maximum of 34 layers

are added, layer by layer, with a first cell spacing of 5μm and a

prism expansion ratio of 1.22-1.25. The initial ICEM volume

mesh is used by TRITET to estimate cell sizes in the inner

domain. Any large volumetric steps are smoothed out in an

iterative process. The final tetrahedral volume mesh is built by

TRITET, starting from the outer boundary layer, with an

advancing front technique. The size and data on the FOI Edge

mesh is provided in Table 4.

FOI OpenFOAM OpenFOAM (Open Field Operation and Manipulation) is an

open-source CFD framework, licensed under the GNU General

Public License (GPL). OpenFOAM comes with basic meshing

tools, various solvers and utilities to import meshes and export

data for post processing. One of the strengths of OpenFOAM is

that customized solvers can be written or modified quickly

because of its modular design and the usage of advanced C++

features. The concept of object-oriented programming allows

OpenFOAM solvers to be written as a hierarchy of classes and

functions that are related to datasets and operations. These classes

and functions can be deemed as e.g. variable fields, mesh,

boundary conditions, numerical operators, etc.

In this study, the incompressible LES solver developed by FOI is

used [42]. LES is based on the idea of separating scales and

dividing the flow into two regimes by means of a low-pass filter

applied to the Navier-Stokes Equations (NSE) with a cut-off

length based on the grid size ∆. The first regime, composed of the

large-scale eddies and properly resolved on the grid, is computed

using a space-time accurate algorithm. The other regime,

containing the small unresolved eddies and ranges from the filter

cut-off down to the Kolmogorov scales, results in an additional

subgrid term, ∇ ∙ B, in the LES momentum equation that must be

modeled, [31], where B = (v ⊗ v − v ⊗ v) is the sub-grid stress

tensor, with v being the velocity and the overbars denoting the

low-pass filtering.

The direct simulation of the large, energy-containing eddies

(being geometry and flow dependent) gives LES much more

generality than RANS, in which the full spectrum of the

turbulence is modeled. The physics of the small-scale, unresolved

flow, or rather the effects of this on the resolved flow, have to be

represented by sub-grid models. However, since only the small-

scale flow is subject to modeling, the assumption is that more

universal models can be adopted. A common criterion for LES is

that at least the energy of the flow is resolved, see [29]. For the

LES computations reported herein, approximately 95% of the

turbulent kinetic energy of the flow is resolved, and accordingly

these simulations can be considered trustworthy.

OpenFOAM Computational Setup

Regarding the sub-grid modeling in LES, it is common to

differentiate between functional and structural models [31].

Functional models aim at reproducing the effects of the small,

unresolved flow on the resolved flow [21][34]; whereas structural

models aim at estimating the sub-grid flow physics based on the

nature of the resolved flow physics and appropriate scaling laws

[1][19][22]. In this study the Mixed Model (MM) is used, which

is composed of one scale similarity term and one eddy viscosity

term [1]. The model coefficients are obtained by integrating the

energy spectra [31], or through a dynamic procedure [10][16].

The longitudinal vortices with a characteristic length and spacing

dominate dynamically the eddy scales and the flow close to solid

wall. The longitudinal vortices have W-shaped vortex structures

that interact with the bulk flow. The scales dominating the near-

wall flow scale with the Re number are typically much smaller

than those of the free flow. Simulating wall-bounded flows with

LES is thus a challenge, since either the near-wall flow structures

need to be resolved, i.e. wall-resolved LES, or the influence of

the near-wall flow needs to be modeled, i.e. wall-modeled LES

[14].

The preferred model for complex geometries is based on the LES

boundary-layer equations, the solution of which, μt and νy+ is

employed to modify the viscosity. The superscript + denotes

viscous scaling, and ν is close to the wall so that νeff = ν + νk = ρτω/du/dy = ρμτ y+/ν+ [9]. This model can be combined

with any other sub-grid model such as the MM used here.

OpenFOAM is based on an unstructured collocated finite volume

method based on Gauss’s theorem together with a multi-step

time-integration method [17]. For LES, the time-integration is

performed using a semi-implicit second-order two-point

backward-differencing scheme. Convective fluxes are

reconstructed using multidimensional cell-limited linear

interpolation, whereas diffusive fluxes are reconstructed using a

combination of central difference approximations and gradient-

face interpolation to minimize the non-orthogonality error. The

Pressure Implicit with Splitting of Operators (PISO) method is

used to discretize the pressure-velocity coupling [13]. The

scheme is second-order accurate in space and time, and the

equations are solved sequentially, with iteration over the explicit

source terms, with a Courant number of approximately 0.4.

FOI OpenFOAM Mesh

The OpenFOAM mesh is generated using ANSYS®ICEM

CFD™. To account for the drift angle, the ship hull is rotated

around the forward perpendicular (FPP), keeping the outer free

stream box fixed. The boundaries of the free stream are located

three hull lengths upstream, below and to the sides. The out-flow

boundary is located ten hull lengths downstream. The wake in the

mesh is adapted to the drift angle using density boxes in the

expected wake region. The original full-scale KVLCC2 hull

geometry, with an Lpp of 320m, is obtained from the SIMMAN


2014 web site [33]. The final mesh is scaled to the TUHH wind

tunnel model dimension with an Lpp of 1.535m.

The mesh generation process starts with the creation of a patch-

dependent triangular surface mesh on the hull. Careful attention

is taken to ensure a smooth transition of cell sizes between and

inside each patch. A global cell size of 1 is specified. The bow

cell size of 0.2 is blended with an exponential expansion ratio of

1.07 into a mid-ship size of 0.5 down to the smallest size of 0.05

at the stern. The waterline boundary is split into patches to match

the density box in the wake region. The dimensions of the wake

density box are set individually for each drift angle and locally

aligned with the expected flow one hull-length downstream of the

stern with a similar draught and breadth as the hull. The

expansion ratio is set at 1.07 away from the hull towards the far

field with a maximum size of 20. An intermediate tetrahedral

volume mesh is generated using the Delaunay advancing-front

method with a spacing-scaling factor of 1.15. From this mesh, 10

prismatic layers are added to the hull surface mesh with an

expansion ratio set at 1.15 without specifying initial or total

height. ICEM CFD thereby adapts each layer to the local cell size

with thinner layers where cell sizes are small. The final

tetrahedral volume is again generated using the Delaunay

advancing-front method but this time with the expansion ratio

lowered to 1.07.

At the inflow boundary, Dirichlet conditions are used for the

velocity and sub-grid kinetic energy, whereas a zero Neumann

condition is used for the pressure. At the outlet, zero Neumann

conditions are used for the velocity and sub-grid kinetic energy,

whereas a Dirichlet condition is used for pressure. No-slip

conditions are used at the hull surface and free stream conditions

are used at the remaining boundaries.

NavyFOAM NavyFOAM employs a cell-centered finite-volume spatial

discretization which permits use of arbitrarily unstructured

polyhedral meshes including hexahedron, tetrahedron, prism and

wedge to name just a few. The linear gradients of the flow fields

can be computed using either the Green-Gauss theorem or least-

square fitting of the neighboring cell data. A large number of

convection discretization schemes are available, including high-

order upwind-biased schemes. Among them, the second-order

upwind scheme is both sufficiently accurate and robust, being

adequate for Reynolds-averaged Navier-Stokes (RANS)

computations of industrial flows. The diffusion terms are

discretized using a second-order central-difference scheme. All

the spatial discretization schemes, including the flux interpolation

schemes, are formally second-order accurate for arbitrary

polyhedral grids. An implicit, segregated, iterative projection

algorithm is employed for time-advancement of solutions. The

system of linear equations resulting from the discretized

governing equations are solved using a choice from the iterative

solvers such as Gauss-Seidel, diagonal incomplete Cholesky

(DIC), preconditioned conjugate gradient (PCG) or generalized

geometric multigrid (GAMG) methods.

NavyFOAM offers a suite of RANS turbulence models including

the k-ε and k-ω families of linear and non-linear eddy-viscosity

models and Reynolds-Stress Transport Model (RSTM).

NavyFOAM also has several subgrid-scale (SGS) turbulence

models for large eddy simulation (LES) and hybrid RANS-LES

models. NavyFOAM offers an adaptive, two-layer wall function

that can handle both near-wall-resolving (y+ = 1) and near-wall-

modeling grids.

NavyFOAM Computational Setup The computational domain with the size of

[5.69Lpp×0.98Lpp×1.3Lpp] is bounded by an upstream inlet, the

tunnel wall, a downstream exit and symmetry boundary

conditions at the symmetry plane of the double hull model. Note

that we have decided to include the tunnel wall in light of

potential blockage effects. Only a half of the full domain is

modeled, assuming that the flow remains symmetrical with

respect to the symmetry plane of the double-body. A hex-

dominant unstructured grid with 77 million volume elements is

used for the computation. We deliberately choose to employ the

wall function approach to evaluate its efficacy in resolving the

turbulent boundary layer and three-dimensional flow separation.

The y+ at the centers of the wall-adjacent elements of the grid

ranges between 40 and 100. The effects of the tunnel wall-shear

are included using a fairly coarse near-wall resolution. A block of

grid with 6 levels of refinement is embedded around the hull to

resolve the hull boundary layer, cross-flow separation, free vortex

layer, and multiple vortices in the near-body region. A steady

RANS solver in NavyFOAM is used for the computation using

the high Reynolds number version of Wilcox’s k-ω model [43].

ANALYZING METHODS

Forces In order to analyze the hydrodynamic forces acting on the ship

body, the following normalizations are used for the forces in x-

and y-direction and the yaw moment around the midship:

𝑋′ =𝐹𝑥

12𝜌∙𝑈2∙𝐿∙𝑇

, 𝑌′ =𝐹𝑦

12𝜌∙𝑈2∙𝐿∙𝑇

, 𝑁′ =𝑁𝑧

12𝜌∙𝑈2∙𝐿2∙𝑇

U is the inflow velocity. The forces (𝑋′ and 𝑌′) and the yaw

moment (𝑁′) are given in ship coordinate system.

Procedure of Vortex Core Analysis For the three main vortices FSV, ASV and ABV, a detailed vortex

core analysis is carried out. The aim of the analysis is to calculate

some reference parameters in order to carry out an accurate

comparison between the measured and calculated results as well

as between the different numerical methods. The reference

parameters for the comparison with measured values are the

location of the vortex core, the axial vorticity component and the

axial velocity component. These values can be obtained from the

wind tunnel measurement data as well as the CFD results. The

comparison of the vortex parameters includes some parameters

that are determined by analysis of the CFD results such as: Q-

value, the pressure and the TKE at the core of each vortex. The

results of CFDShip-Iowa, FreSCo+, ReFRESCO, Edge-EARSM

and Edge-HYB0 are considered in the analysis. Identifying the

position of the vortex core can be accomplished by analyzing the

location of the maximum axial vorticity component or the

maximum of Q-value. The location of the axial vorticity

component and the maximum of Q-value are evaluated in


different sections perpendicular to the inflow direction for the

different vortices. In most cases, the determined core positions

based on maximum ωx or Q are identical or the differences are

negligible. To verify this fact the coordinates of the core position

of the vortices FSV, ASV and ABV are identified in the results

of FreSCo+ according to the two different criteria. In the

presented analysis results the location of the vortex core in the

EFD results and in the CFDShip-Iowa are determined based on

the axial vorticity component, while the maximum of Q-value

criterion is considered by the evaluation of the results of FreSCo+,

ReFRESCO, Edge-EARSM and Edge-HYB0.

The above-mentioned parameters of the core vortex analysis are

made dimensionless by using the inflow velocity and model

length as follows:

𝑦𝑐𝑜𝑟𝑒: core y-location, normalized by 1/𝐿𝑝𝑝, measured from the center line

of the double body,

𝑧𝑐𝑜𝑟𝑒: core z-location, normalized by 1/𝐿𝑝𝑝, measured from the symmetry

plane of the double body,

𝑄𝑝𝑒𝑎𝑘: maximum Q-value, normalized by 𝐿𝑝𝑝2/𝑈∞2 ,

𝜔𝑥,𝑐𝑜𝑟𝑒: core axial vorticity, normalized by 𝐿𝑝𝑝/𝑈∞ ,

𝑈𝑥,𝑐𝑜𝑟𝑒: core axial velocity, normalized by 1/𝑈∞ ,

𝑐𝑝𝑐𝑜𝑟𝑒: core pressure, corrected for free stream pressure and normalized by

1/(0.5 ∙ 𝜌 ∙ 𝑈∞2 ),

𝑇𝐾𝐸: turbulent kinetic energy, normalized by 1/𝑈∞2 .

The core vortex parameters are plotted over x/Lpp in all

corresponding diagrams, where x points in the wind tunnel main

flow direction and the origin is located at the forward

perpendicular. Figure 11 show the results of the FSV, the ASV

and the ABV. The location of the measuring sections FSV-1,

FSV-2, ASV-1, ASV-6, ASV11, ABV-1 and ABV-2 are marked

in Figure 11. The distribution of axial vorticity and distribution

of axial velocity in the selected measuring sections are presented

in Section: Results.

Additionally, a verification and validation (V&V) analysis [4] for

the three main vortices (FSV, ASV, ABV) is carried out. The

errors are only calculated for the measuring sections were both

CFD and EFD data are available. The error (E) between CFD (S)

and EFD (D) is expressed by following formula:

|E(%D)| = |S−D

D× 100|.

Here, the absolute formulation of the error is used because an

averaged value is considered for the V&V analysis. Signed values

might lead to a too small error. The values can be found in Table

6, Table 7 and Table 8.

Planar Plots Further comparison of the measured and calculated results is

given in Figure 12. The figure includes the following values: 𝑈𝑥,

𝑈𝑦 , 𝑈𝑧, 𝑈𝑡, 𝜔𝑥, 𝑇𝑈𝑡 and TKE.

The values are normalized as mentioned before. It should be

noted that figures with TUt exist only for the wind tunnel

measurements. The TKE data is only available for the simulations

of FOI-Edge, TUHH and MARIN.

RESULTS

Forces and Yaw Moment An overview of normalized force and yaw moment coefficients

obtained by the numerical methods is given in Table 5. Due to a

lack of data for the force measurements in the wind tunnel, the

RMS-values of all CFD simulations are used for further

comparison. Only the computations with the finest grid and

different turbulence models of each code are considered. The

corresponding simulations are marked in Table 5. For the global

forces and moments comparison, the deviations of CFD results

from the RMS value are calculated as follows:

𝐷(%𝑅𝑀𝑆) =𝑆−𝑅𝑀𝑆

𝑅𝑀𝑆× 100.

The calculated X-force by CFDShip-Iowa using ARS-DES is

higher than other applied turbulence modeling techniques. The

simulation results of Edge-EARSM also deliver high X-force.

The lowest value is obtained by OpenFOAM-LES. The

calculated values of X-force using two-equation-based turbulence

modeling are close to each other.

The calculated Y-force by CFDShip-Iowa using ARS-DES is

higher than other applied turbulence modeling techniques. The

lowest value is obtained by Edge-HYB0. The calculated values

of Y-force using two-equation-based turbulence modeling are

also close to each other.

The calculated N-moment by CFDShip-Iowa using ARS-DES is

also higher than other applied turbulence modeling techniques.

The lowest value is obtained by OpenFOAM-LES.

Although extensive V&V studies are not carried out, some

comments are given on the presented results. The ARS-DES (13

M nodes) model delivers the highest and the LES-MM (36.5 M

nodes) the lowest value. The results of the two-equation

turbulence models such as standard k-ω or SST k-ω show a

decrease in magnitude of the calculated X-force coefficient when

increasing the grid resolution, see Table 5. The percental standard

deviation with respect to the RMS value (SD%RMS) is 72% for

the X-force, 8.3% for the Y-force and 11.6% the N-moment.

Vortical Structures and Vortex Core Analysis Figure 4 shows the visualized flow on the model at different

stations. The three main vortices, Fore-Body Bilge Vortex (FBV),

Fore-Body Side Vortex (FSV) and After-Body Side Vortex

(ASV), can be clearly identified. Where the Fore- and the After-

Body Side Vortices are elongated (the axis parallel to the ship

bottom is longer than the axis perpendicular to it), the cross

section of the Fore-Body Bilge Vortex has nearly a circular shape.

As mentioned before, the investigation is carried out for a double

model. Therefore, the FSV exists in Figure 4 twice: one

developed due to the interaction of the flow with the front half of

the double model and other FSV by the interaction with the

second half of the model. Both FSVs induce a transverse velocity

component in symmetry plane, which is directed to the model

sidewall, as can be clearly seen in the plane shown in Figure 4. It

is difficult to achieve an exact symmetry condition between the

ship and inflow in the experimental setup in the wind tunnel,

therefore it can be seen that the induced flow by the FSVs is not

fully symmetric.


Figure 4: Visualization of the Fore-Body Side Vortex (FSV) in

the wind tunnel, blue arrows indicate the flow /

rotation directions.

The results of visualization of the flow in the wind tunnel are

documented in black white photos. The grey color in the photos

can be analyzed and assigned to a grey scalar. This scalar can be

translated to colored scalar that shows the vortical structure, see

Figure 4. Unfortunately, it is not possible to define the

corresponding Q-value for each color. Therefore, only the main

shape of the vortex structure can be compared irrespective of the

absolute Q-value.

An overview of the vortical structure is given in Figure 5. The Q-

criterion is applied to identify the vortical structures [12]. The

criterion is based on the second invariant of velocity gradient

tensor ∇u. Q is made non-dimensional using Q´=Q*L2/V2. For

simplicity the dash over the non-dimensional quantity Q´ will be

omitted. However, the Q-criterion cannot be used to visualize the

orientation of the detected vortex. To overcome this weakness,

the normalized helicity density [20] is proposed as follows:

𝐻𝑛 =𝑈∙𝜔

|𝑈|∙|𝜔|

where U and ω are the velocity and vorticity vector at the same

point. The normalized helicity density Hn will be applied to color

code the Q-isosurface in the current study. It represents the

directional cosine between the vorticity vector and the velocity

vector, −1 ≤ Hn ≤ 1. The sign of Hn indicates the direction of

swirl of the vortex relative to the streamwise velocity component.

The flow structure around the model can be characterized in

Figure 5 (Q=200) by the following vortices:

1. Fore-Body Bilge Vortex (FBV), Leeward

2. Fore-Body Side Vortex (FSV), Leeward

3. After-Body Side Vortex (ASV), Windward

4. After-Body Bilge Vortex (ABV), Leeward

5. Stern Vortex (SV), Leeward

6. Aft-Body Hairpin Vortex (ABHPV), Leeward

7. Kelvin-Helmholtz Vortices (KH), Leeward

8. Karman-like Vortices (KL), Leeward

The nomenclature of the above vortices follows the suggestion

given by Xing et al. [45].

Figure 5: Predicted vortical flow structure for Q=200,

analyzed based on the FreSCo+ results:

In general, the flow field is dominated by the strong interaction

that takes place between the different vortices, see Figure 8 and

Figure 9. The comparison between the instantaneous and the

mean simulation results facilitates a deep insight into the detail of

this interaction, particularly in the stern region. Figure 8 shows

the structure of Q=100 based on the mean values of the final 1000

iterations of the Hybrid RANS-LES (HYB0) and the LES-MM

simulations. The instantaneous structure of Q=100 is shown in

Figure 9. In the present study, the structure of different vortices

will be discussed focusing on the FSV, ASV and ABV vortices.

FSV, ASV and ABV show open-type cross-flow separation from

the hull. The vortices separate as a circular type, undergo helical

mode instability, and transform into spiral vortices, as shown in

Fig. 3(top). As the vortices progress, the vortex core axial velocity

and vorticity decreases, and pressure and TKE increases. The

swirl ratio S =𝑈𝜃/𝑈𝑥, where 𝑈𝜃is the tangential velocity and 𝑈𝑥

is axial velocity, is S~0.5 (CFDShip-Iowa) and S~0.6 (FreSCo+)

at the inception of the spiral streamline. The dominant frequency

f along the vortex core, evaluated using pressure fluctuation in the

vortex core, show that the St =fL/U0 decreases with the

progression, where U0 is free-stream velocity and L is ship length.

The scaling of the dominant frequency using the distance from

the origin of spiral streamline location X shows for instability

analysis of the CFDShip-Iowa result that St decreases with the

progression of the vortices, and is StX = 1.2 ~ 1.3 for FSV, StX =

1.35 ~ 1.45 for ASV and StX = 1.8 ~ 2.25 for ABV. Xing et al.

[45] referred to the transformation of circular to spiral vortices as

vortex breakdown.

Figure 6 shows streamlines in the vortex core and the location of

helical instability inception (black points) for the FreSCo+ results.

ASV

FBV FSVFSV induced

flow


Figure 6: Predicted vortical flow structure for Q=200 with

vortex core streamlines and location of helical

instability inception (black points), based on the

FreSCo+ results.

Spiral vortices have been reported for swirling jets [47], wing tip

vortices [25] and delta wing leading edge vortices [50]. In these

cases, circular vortices exhibit vortex breakdown followed by

helical mode instability and transformation into spiral vortices.

The breakdown is characterized by a sudden expansion of the

vortex core and sharp gradients in vortex core variables, i.e., drop

in axial velocity and vorticity, and increase in pressure and TKE

[25]. The initiation of the vortex breakdown occurs if the swirl

ratio is higher than the critical value Sc. The critical value

depends on the flow type, Re, inflow condition, swirl angle with

respect to jet axis etc. [48]. Spall et al. [51] showed that Sc

decreases with Re, i.e., Sc = 2 to 3 for Re < 100, and 1.6 for large

Re > 104 for wing tip vortices. For delta wings, Sc ~ 1 for range

of Re, as shown by Greenwell [48] for a range of delta wing

vortex configurations. For delta wings, the helical mode

instability frequency is inversely proportional to the distance

from the breakdown location. Thus, StX is constant along the core.

Gursul [49] reported that StX ranges from 0.75 to 1.35 for large

sweepback delta wings at different flow conditions.

The vortices for KVLCC2 at β = 30 have similarities with the

swirling jet or delta wing vortices as they show helical mode

instability, spiral streamlines/vortices, high swirl at instability

inception, and Stx range is consistent with those of delta-wings.

However, the swirl ratio is lower and large differences in

transition process from circular to spiral vortices. For swirling jet

or delta wing vortices, the transition process is abrupt with large

change in vortex size and gradients of the vortex core variables,

i.e., vortex breakdown. For ship flows, the onset/transition

process is abrupt but without large change in vortex size and

differences in gradients and trends of the vortex core variables.

Nonetheless, we retain the terminology vortex breakdown for this

process. The differences between ship and aero flows likely due

to global geometry and pressure gradients and the interaction of

the ship vortices with the boundary layer and other vortices.

Fore-Body Side Vortex (FSV)

As can be seen in Figure 8 and Figure 9 the FSV is developed on

the leeward side of the forward part of the ship hull. The distance

from the vortex center to the sidewall as well as to the hull bottom

grows with increasing x/Lpp-value, see Figure 11. In the stern

region, the coordinates of the core center follow the contour of

ship sidewalls. The measured y- and z-coordinates of the vortex

center are included in Figure 11. It can be seen that the calculated

coordinates of the vortex center agree well with the measured data

in the range of x/Lpp 0.25 to 0.7. Unfortunately, no experimental

results are available after x/Lpp=0.7. The calculated y-

coordinates by the different codes in the x/Lpp range between 0.7

and 1.0 and show some differences, see Figure 11. The obtained

y-coordinates in this range by the Edge Hybrid RANS-LES

HYB0 simulation are smaller than the values calculated by RANS

simulations using two-equation models.

The instantaneous pattern of the FSV is shown in Figure 9: the

width of the vortex pattern obtained by FOI OpenFOAM

increases with the growing run length of the vortex. In all

numerical simulations the diameter of the vortex increases along

its run length. FreSCo+ results show a high increase of the vortex

diameter, see Figure 8. The limited grid resolution in this region

may be responsible for the increase of the vortex diameter.

In Figure 8 a vortex sheet that surrounds the FSV can be clearly

seen. Most of the CFD computations predict these secondary

vortices, but the shape of the vortical structure shows different

characteristics. NavyFOAM and the time-averaged FOI

OpenFOAM predict a vortex sheet that spirals around the FSV

vortex axis and the spiraling vortex lines are inclined to the FSV.

Nelson and Pelletier [25] have experimentally investigated the

behavior of similar vortex structure on the leading edge of delta

foil and have reported that vortex sheet induces an axial flow in

the downstream direction of the vortex. This observed flow

behavior is confirmed by the measured data for the longitudinal

velocity component in the vortex core, as can be seen in Figure

11. This tendency is captured well by the simulation results of

CFDShip-Iowa using ARS-DES for a certain range of x/Lpp. The

flow acceleration is not captured by the other codes.

The time-averaged results of FOI Edge-HYB0 and ARS-DES

CFDShip-Iowa also show the secondary vortices structure, but

the spiraling vortex lines around the FSV are less pronounced.

The vortex sheet can be also be seen in the results of FreSCo+ and

ReFRESCO. The structure obtained by FreSCo+ also shows

spiraling vortex lines that are inclined relative to the FSV, but the

inclination direction is not in accordance with the results obtained


by the other codes. Further computations are carried out to

analyze this problem. The shape of the vortex sheet calculated by

FreSCo+ using steady case has no physical meaning: as the

computation switched from steady to unsteady, the shape of the

vortex sheet was changed. The new results are similar to

ReFRESCO results. This means the vortex sheet around the FSV

exists and has a smooth surface, see Figure 8.

The secondary vortices structure and the spiraling vortex lines

around the FSV are also predicted by the NavyFOAM using the

k-ω model (Wilcox 2008 [43]) in combination with wall function

(y+ between 40 and 100). The NavyFOAM computation is

carried out on a quite fine grid using an isotropic turbulence

model. This secondary vortex structure is also visible in the

results using the anisotropic turbulence model such as Edge-

HYB0 or ARS-DES CFDShip-Iowa as well.

The Q-values show that two peaks exist, see Figure 11. The first

one takes place between x/Lpp 0.25 to 0.3 and the second one in

the x/Lpp region between 0.95 and 1. The first peak is much

larger than the second one. Between the two peaks there is a

continuous reduction of the Q-value. The second peak may be

induced by other vortices existent in the after-body region. The

results obtained by FreSCo+, ReFRESCO, Edge-EARSM and

Edge-HYB0 show the same tendency over a wide range. At x/Lpp

>0.9 the Q-values by FreSCo+ start to decrease faster than the

other method; the reason may be the limited resolution of the used

grid in this region. The calculated values by Edge-HYB0 are in

general lower than the result of the FreSCo+ and ReFRESCO but

they follow the same tendency. The Q-values along the model by

Edge-EARSM are much lower than the above-mentioned three

codes. Behind the ship model the Q-values by Edge-EARSM are

low but it is still in the range obtained by Edge-HYB0 and

FreSCo+.

The variation of the longitudinal vorticity (ωx) along the non-

dimensional x-axis is presented in Figure 11. While all numerical

simulation methods show a continuous reduction of ωx in the

x/Lpp range higher than 0.3, the measured ωx show an increase of

the in the range between x/Lpp=0.3 and 0.4 and after the position

of x/Lpp=0.45 the measured ωx value remains constant until

x/Lpp=0.7.

In order to understand the reason for the different behaviors of the

measured and calculated ωx, it is important to consider the

characteristics of the flow surrounding the FSV. This vortex

arises in the boundary layer of the model, leaves the boundary

layer very fast and progresses in the outer flow region. The

influence of shear flow and the associated diffusion play a minor

role in the outer flow region. Therefore, the main part of the FSV

can be considered as an isolated vortex with negligible interaction

with the surrounding outer flow and these circumferences allow

the vortex to keep its strength over quite a long distance. The

measured vorticity increases after passing the aft shoulder; this

may be due to the interaction with other vortices in this region.

The calculated axial vorticity in all simulations shows a contrary

tendency to the measured results. The calculated ωx decays along

the x/Lpp. This fact is valid for simulations based on the isotropic

as well as the anisotropic turbulence modeling. As mentioned

before, while the measured results show a concentrated vortex in

the outer flow, which keeps its strength for a long distance, the

simulations cannot reproduce this tendency.

The helical instability according to the prediction presented in

Xing et al. [45] takes place at a position x/Lpp=0.32 close to FSV-

1 section. It can be seen that there is a steep reduction of the ωx

calculated by CFDShip-Iowa at this location. A similar behavior

can be seen in the results of Edge-EARSM, but this reduction

exists much earlier at the position x/Lpp=0.22.

The comparisons between the calculated pressure coefficient and

TKE are presented in Figure 11. With exception of CFDShip-

Iowa code the other applied methods predict the minimum

pressure coefficient at nearly x/Lpp=0.32, which agrees with the

predicted position of the helical instability by in Xing et al. [45].

The absolute value of the minimum pressure varies over a wide

range; the lowest is about -1.65 predicted by FreSCo+ and the

highest is -1.0 by Edge-EARSM. The results of the other codes

lie within this range. The CFDShip-Iowa results follow the same

tendency, but they have two peaks located between 0.4 and 0.6

that cannot be seen in the results of the other codes. The

calculated pressure coefficients by ReFRESCO, Edge-EARSM

and Edge-HYB0 in the x/Lpp range higher than 0.8 show two

peaks. The location of the first peak calculated by ReFRESCO

agrees well with FreSCo+ results. The comparison of the TKE

values shows that they are nearly close to each other at the

beginning of the FSV. With increasing x/Lpp value, the

calculated TKEs by CFDShip-Iowa and ReFRESCO decrease

while the TKE values by FreSCo+ and Edge-HYB0 show a

continuous increase. FreSCo+, Edge-EARSM and Edge-HYB0

predict a continuous increase of TKE after x/Lpp=0.7. In this

location the vortex changes its y-coordinates and moves in the

direction of the sidewall of the hull. In the stern region, the

calculated TKE values show a maximum value followed by a

steep reduction. The predicted location of the maximum TKE

value by the different codes varies between x/Lpp=0.9 and 1.1.

The calculated maximum in the stern region shows a wide

variation. The lowest value is obtained by ReFRESCO and the

second highest value is calculated by FreSCo+, although both

codes use the same turbulence model. The highest TKE value is

obtained by Edge-HYB0 at x/Lpp=1.07.

After-Body Side Vortex (ASV)

The ASV dominates the flow on the model bottom beginning at

the windward forward shoulder along the chine. At circa

x/Lpp=0.55 the vortex changes its direction and follows more or

less the free stream direction. This effect can be clearly seen in

Figure 8 and Figure 11, which show the y-position of the ASV

core along the hull. Considering Figure 8, where the vortices have

been visualized by the Q-criterion (Q=100), the ASV seems to be

most clearly pronounced in the results of NavyFOAM and

FreSCo+ followed by CFDShip-Iowa, ReFRESCO and FOI Edge.

In the FOI OpenFOAM results the vortex is the least pronounced.

This applies to both the averaged and the instantaneous results. It

is important to notice that on the windward side of the ASV a

smaller contra-rotating vortex exists, which seems to turn around

the ASV. This vortex becomes visible in the region of

x/Lpp=0.55, where the ASV changes its direction. The


NavyFOAM results show that ASV, after a short distance, splits

into two parallel running vortices, see Figure 8. Similar flow

behavior can be seen in the FreSCo+ results. The split vortices are

also clearly visible in the planar data comparison. Due to the

complexity of the ASV system, it forms different separation and

reattachment lines on the hull bottom, as discussed in Section

Results: Limiting Streamlines.

Figure 11 shows the variation of ASV core position beside and

below the hull. All CFD simulations predict the y-position very

well over the whole x-range considered by the wind tunnel

measurement. After x/Lpp=0.95, the numerical results show two

different tendencies. While the results of ReFRESCO, Edge-

EARSM and Edge-HYB0 predict a direction change of the vortex

core, in the results of FreSCo+ and CFDShip-Iowa the ASV keeps

its direction. Unfortunately, no measurement data is available in

this region.

The predicted z-position by the different solvers for the ASV is

more or less the same until x/Lpp=0.7. After this position, it can

be seen that different trends are predicted. While the results of

ReFRESCO, Edge-EARSM and Edge-HYB0 predict a steep

moving of the vortex core upward, a moderate raise of the vortex

core can be seen in the results of FreSCo+ and CFDShip-Iowa.

At the beginning, the vortex stays very close to the bilge region

and starts to move away when it comes near the aft shoulder

(x/Lpp=0.55), where it changes its y-direction and crosses the

hull. The deepest point is then reached at about x/Lpp=0.65-0.7,

where ASV separates from the hull. The lowest position predicted

by FreSCo+ is in good agreement with the measurement. The

CFDShip-Iowa predicts the lowest position only after the vortex

has separated from the hull. Before the separation point, the

location of the vortex core stays closer to the hull than the

measured values.

The calculated Q-values in the vortex core along the hull are

presented in Figure 11. The results of ReFRESCO and FreSCo+

have a similar tendency regarding the rate of the reduction of Q-

values over x/Lpp. While the results of ReFRESCO show an

oscillating behavior, the FreSCo+ results are smooth. The

predicted Q-values by Edge-EARSM and Edge-HYB0 are

similar, and behind x/Lpp=0.7, the gradient of the reduction of Q-

values over x/Lpp is much steeper compared with results of

ReFRESCO and FreSCo+.

The vorticity (ωx) in the vortex core along the hull is depicted in

Figure 11. As the FSV is strongly influenced by the shear flow in

the boundary layer region on the ship bottom. The calculated

values show a continuous decrease of the vorticity over x/Lpp.

The measured results confirm the calculated tendency shown by

the applied numerical methods. The absolute values of

computation results are in good agreement with the measurement.

The CFDShip-Iowa and Edge-HYB0 predict a steep decrease of

ωx at x/Lpp=0.55. This local strong reduction cannot be seen in

the results of Edge-EARSM, ReFRESCO and FreSCo+, which

use the two-equation turbulence model.

The identified position of helical instability by Xing et al. [45] is

x/Lpp=0.72, which is close to the position where the vortex has a

strong change of its y-direction. The measured data also show a

local reduction of ωx in this region.

Figure 11 shows the comparison of the axial velocity component.

It can be noticed that the calculated values agree well and they

follow the measured values. However, the measured axial

velocity component is about 0.2 higher than the calculated values.

The reason for this difference is not clear; it may be due to the

fact that the accuracy of the PIV measurement of the axial

velocity component very close to the ship bottom suffers from the

surface reflection in this region.

Figure 11 includes the comparison for the pressure coefficient

and the TKE, respectively. While CFDShip-Iowa, Edge-HYB0

and Edge-EARSM show a strong variation of the pressure

gradient along the vortex core, the results of ReFRESCO and

FreSCo+ show more or less a continuous decrease of the pressure

coefficient. In the region of x/Lpp=0.4, the predicted pressure

coefficient varies a lot: FreSCo+ delivers the highest and Edge-

HYB0 the lowest one. The results of Edge-EARSM are located

in between, as expected.

The calculated level of the TKE by the applied methods at the

beginning of the ASV is different. The TKE obtained by FreSCo+

is much higher than the ReFRESCO values, although the same

turbulence model is applied in the both simulations. The

calculated maximum TKE values by Edge-EARSM are higher

than FreSCo+. The calculated results by ReFRESCO and Edge-

HYB0 are nearly in the same range and much lower than the

Edge-EARSM and FreSCo+ results. The calculated TKE by

CFDShip-Iowa at x/Lpp=0.6 are close to the results of Edge-

EARSM and FreSCo+, and after this position the TKE falls down

to the level obtained by ReFRESCO and Edge-HYB0. The results

of all methods show an increase of the TKE nearly at the same

position, around x/Lpp=0.88. Comparing Figure 11, it can be seen

that after the identified position of helical instability by Xing et

al. [45], a strong local increase of the pressure coefficient and the

TKE take place. Both are an indication for helical instability.

After-Body Bilge Vortex (ABV)

The ABV is initiated by the secondary separation line on the

leeward side of the aft ship near the hull bottom. It separates from

the hull shortly before the end of the skeg. The ABV is clearly

visible in all simulation. This is valid for the steady and averaged

unsteady simulations. The vortex follows the free steam direction

after it leaves the hull and remains visible for a long distance

behind it, see Figure 8 and Figure 9.

It can be seen in Figure 11 that over quite a range of x/Lpp the y-

coordinates of ABV core have a constant angle relative to the

centerline of the ship. This tendency is captured well by the

applied methods with the exception of the Edge-EARSM code. A

good agreement between the calculated z-coordinates and the

measured values has also been achieved for the z- coordinates;

the differences are located in the range of x/Lpp=± 0.03, see

Figure 11.

The variation of the Q-value in the vortex core along the non-

dimensional x-axis is presented in Figure 11. The results of

ReFRESCO and FreSCo+ show the same tendency that the Q

decreases over x/Lpp. The results of Edge-EARSM and Edge-


HYB0 also follow the same tendency but with reduced Q-values.

Edge-EARSM predicts the lowest Q-values. The predicted

gradient of Q by all applied methods over x/Lpp is nearly the

same.

The variation of the ωx along the non-dimensional x-axis is

presented in Figure 11. The results in the x/Lpp region higher than

0.8 show the same tendency, namely, that the vorticity decreases

over x/Lpp. Edge-EARSM predicts the lowest ωx-values. The

ABV is strongly influenced by the shear flow in the wake region

behind the ship. The measured results not only confirm the

calculated tendency, but the absolute values of the measurement

and the computations are also in good agreement. The identified

position of helical instability by Xing et al. [45] is characterized

by a strong reduction of the ωx.

Figure 11 shows the comparison of the axial velocity component.

The calculated values of FreSCo+, Edge-HYB0 and Edge-

EARSM agree with the measured values. While the ReFRESCO

results over quite a range are also in good agreement with the

measured values, CFDShip-Iowa overpredicts the axial velocity

component of the ABV over the range.

Figure 11 includes the comparison of the pressure coefficient and

the TKE, respectively. The calculated pressure coefficients by

ReFRESCO, Edge-HYB0 and FreSCo+ are in the same range,

although Edge-HYB0 predicts a much higher pressure coefficient

at or near x/Lpp=0.9. The maximum pressure coefficient by

Edge-EARSM is much lower than those by other applied

methods. The results of CFDShip-Iowa show a steep increase of

the pressure coefficient at or near x/Lpp=1.05; all other codes

predict a gradual pressure increase. The pressure level far behind

the ship hull is nearly the same in all applied methods.

Compared with CFDShip-Iowa, the calculated mean values of the

TKE by Edge-HYB0 are quite high. The ReFRESCO results do

not show any high gradient of the TKE. In contrast, computations

using anisotropic models such as CFDShip-Iowa and Edge-

HYB0 show high fluctuations. The reason for the strong

reduction of TKE calculated by CFDShip-Iowa at x/Lpp=1.1 is

not clear. The TKE predicted by CFDShip-Iowa shows an

increase at x/Lpp=1.05. At this position a high local pressure

increase is predicted by the same method, compare Figure 11. In

Figure 11, it can be seen that at x/Lpp=1.1 a reduction of ωx takes

place. All these parameters can be interpreted as an indication for

helical instability at this location.

Comparisons of global vortex characteristics between EFD and

CFD

A summary of the comparison between the EFD and the CFD

results is presented in Table 6, Table 7 and Table 8. The tables

include the percentage deviation between EFD and the CFD

regarding the coordinates of the vortex core, the longitudinal

velocity and vorticity. The results of ReFRESCO (SST 2003,

12.7M), FreSCo+ (SST k-ω 2003, 11.0M), CFDShip-Iowa (ARS-

DES, 13.0M), Edge (ERSM, 17.4M) and Edge (HYB0, 17.4M)

are compared. The estimated mean uncertainty of the measured

values is 5.5%.

For the FSV, it can be seen that the accuracy of the predicted y-

and z- coordinates is higher compared with the other values. The

deviation of the predicted longitudinal velocity and the vorticity

are high near the onset, see the results at x/Lpp=0.3153. The

overall deviation lies in the range between 13% and 27%, see

Table 8. The level of deviations is higher for the ASV and ABV

(20.8% to 42%), see Table 6, Table 7 and Table 8. The reason

may be that ABV and a considerable part of ASV are located on

the after-body of the ship, where a strong shear flow and

interaction between the different vortices take place.

Limiting Streamlines The limiting streamlines will be discussed in this section. In the

figures belonging to this section, different separation /

reattachment lines / areas are marked as followed:

SL 1 = Separation Line FBV,

SL 2-1 = Separation Line FSV, the forward part of the second separation line about 0.2L- 0.62L, on ship bottom,

SL 3-1 = Separation Line ASV, separation line close to the bilge radius,

SL 3-2 = Separation Line ASV, separation line close on the ship bottom,

RL 3 = Reattachment Line ASV,

SL 4-1 = Separation Line ABV, the after part of the second separation line about 0.62L- 0.95L, close to the bilge radius,

SL 4-2 = Separation Line ABV, the after part of the third separation line till about 0.65L- 0.95L, on leeward ship side,

RL 4 = Reattachment Line ABV, Figure 10,

SL 5 = Separation Line SV,

SA 6 = Separation Area ABHPV.

They are only marked as far as they are clearly identifiable in the

corresponding views. The numbering of the different vortices

corresponds to the numbering in Section: Vortical Structures. For

the CFD results for which a vortex core analysis is available, the

core location of this vortex is marked with a blue dashed line in

the figures (Figure 10).

Figure 7: Visualization of limiting streamlines on the wind

tunnel model bow and stern section.

These measured data of the wind tunnel oil film test are presented

in Figure 7. In the central region of the hull, unfortunately, no

0.2 0.30.1

FSV - 1

primary separation lineprimary reattachment line

SL 1SL 2-1

SL 3-1

RL 3

0.7 0.8 0.9

ASV - 1

ASV - 11

secondary separation lineprimary reattachment line

primary

separation lineASV - 6

separated flow

region

SL 5

SL 3-1

RL 3

SL 4-1 SA 6

third separation lineSL 4-2

SL 3-2

RL 4


results are present. The Figures show the limiting streamlines on

the model hull obtained by the oil film test. In order to identify

the position of the different vortical structures, the dimensionless

positions along the longitudinal axis of the model are plotted in

the figures, where 0L and 1L are the locations of the forward and

after perpendicular, respectively. Figure 7 shows at the top the

limiting streamlines on the fore-body. The locations of the FBV

and the FSV can be easily identified; the streamlines in these

areas are light grey. Dark grey indicates the flow separation area,

as the oil film keeps its position due to negligible flow speed in

this area. The dashed line shows the positions of appearance and

development of the ASV. Behind this region, the primary lines

for separation and reattachment can be identified. Figure 7 at the

bottom shows the separation area and reattachment lines on the

after-part bottom.

In order to analyze the characteristics of the flow, the measured

and calculated limiting streamlines directions on the model hull

are compared. The limiting streamlines predicted by CFD codes

can be seen in Figure 10. These pictures identify the separation

and reattachment lines. The different limiting streamlines for the

FSV, ASV and ABV will be discussed in detail in the following.

Separation Lines FSV, SL 2-1

Shortly after disappearance of the SL 1, the separation line of the

FSV (SL 2-1) starts on the ship bottom. It is the forward part of

the secondary separation line in the region between 0.2L-0.62L.

It proceeds directly on the surface of the hull in the region located

between the bilge radius and the hull bottom and ends at the after

shoulder. At the beginning of SL 2-1, the FSV separates from the

hull. On its way to the aft ship, the SL 2-1 adds energy to the FSV.

The SL 2-1 is predicted by all CFD solvers similarly and agrees

very well with the wind tunnel tests. This is not surprising since

this separation line develops at a striking geometric location on

the hull.

Separation Line ASV, SL 3-1, SL 3-2

The ASV separation consists of two separation lines (SL 3-1 and

SL 3-2), which can be clearly identified in all simulation results

(Figure 10). In the wind tunnel oil film test (Figure 7 bottom),

both the SL 3-1 and the SL 3-2 are present but are hard to identify

due to reflection on the model surface. The SL 3-1, also called

primary separation line, begins near the forward shoulder at about

0.22-0.25L and develops directly on the transition from windward

hull side to the bilge radius parallel to the chine. At the aft

shoulder (circa 0.7L), the separation line changes its direction and

follows more or less the free stream direction. It crosses the hull

and ends at 0.86L at the secondary separation line or SL 4-1,

respectively. It is predicted by all CFD results in the same manner

and fits well to the measurements. Only the starting point differs

a little bit.

The SL 3-2 lies more or less centrally between the SL 3-1 and the

reattachment line RL 3 (see next section). Its starting point is

predicted in the simulations quite differently. In the FreSCo+ and

FOI Edge (EARSM) results, it starts quite early at 0.4L and

before, whereas it appears rather late, at 0.65L, in the CFDShip-

Iowa results. The other results lie within this range. The

development is then similar to the SL 3-1 and it also ends at the

secondary separation line (SL 4-1), but a little earlier at 0.83L. In

the measurements, the SL 3-2 lies closer to the SL 3-1 than in the

numerical simulations and seems to start rather late as in the

CFDShip-Iowa results. Another small reattachment line lying

between these two separation lines can be found in all simulation

results. This complex structure of separation and reattachment

lines is due to the complex interaction between the large and small

vortex structures generated in this area. In Figure 10, where the

core location of the ASV is included, it can be seen that the ASV

lies very close to SL 3-2 and has more or less the same shape. In

the results of FreSCo+, ReFRESCO and both of FOI Edge

(EARSM and HYB0), the ASV is located on the leeward side of

SL 3-2, whereas it is on the windward side in the CFDShip-Iowa

results.

Reattachment Line ASV, RL 3

The reattachment line RL 3 starts near the front shoulder at 0.22L

on the windward side of the model and moves along the model

bottom toward to the centerline of the model. It is caused by the

ASV when its rotation ensures that the flow is again directed

orthogonal to the hull bottom and flows onto it. It ends at the

secondary separation line on the leeward side of the hull bottom

at about 0.83L. The shape of the reattachment line differs in the

numerical results. FreSCo+, CFDShip-Iowa, FOI Edge (EARSM)

and NavyFOAM predict it as a straight line, ReFRESCO also has

a straight line but with an earlier ending, whereas it has an s-shape

in the FOI Edge (HYB0) and FOI Open FOAM results, which

might be due to the different turbulence modeling. So far it is

visible in the measurements it seems to be a straight line.

Separation Line ABV, SL 4-1, SL 4-2

The separation line of the ABV (SL 4-1) is the rear part (0.62L-

0.95L) of the secondary separation line which starts at the fore-

body and extends till the end of the hull bottom along the chine.

It proceeds directly on the transition from the hull bottom to the

bilge radius and ends at the aft position of the flat bottom. The

main separation causal for the ABV takes place from 0.86L till

the end of hull bottom (0.95L). This is the part after the separation

lines of the ASV (SL 3-1 and SL 3-2) cross the secondary

separation line. However, as can be partially seen on the pictures

of the vortex structures (Figure 8), the ABV is additionally

initiated till the secondary separation line passes 0.65L, the rear

shoulder. The SL 4-1 is predicted by all CFD codes similarly and

agrees well with the wind tunnel tests. This is due to the fact that

this separation line develops at a striking geometric location on

the hull.

The SL 4-2 is the rear part of the third separation line and lies on

the leeward hull side at about 0.65L-0.95L. This separation line

does not lead to a recognizable vortex. When the third separation

line comes to the aft ship behind the rear shoulder, it moves far

away from the ship bottom due to the influence of the hull shape.

The predictions of the SL 4-2 differ among different CFD codes

(Figure 10) and are hardly identifiable in some results. The

predictions are in agreement with the wind tunnel oil film test, see

Figure 7 bottom. An exception is the result of FOI Edge

(EARSM), where the third separation line is pushed towards the

outside and the secondary reattachment line becomes clearly

visible at this place.

Reattachment Line ABV, RL 4


The RL 4 is the rear part of the secondary reattachment line. It

starts at the rear shoulder at about 0.6L but shortly after that the

location of RL 4 is hard to identify and it is hard to define where

it really ends. This is also true for the wind tunnel oil film test

results, see Figure 7 bottom. In the results of FOI Edge (EARSM)

in Figure 10, this reattachment line is clearly visible. With some

imagination it can also be found in the other CFD results.

Planar Plots A further comparison of the measured and calculated results is

given in Figure 12. The results are presented exemplary for the

section ASV-11. In each figure, the results contributed by the

different CFD codes are placed.

In general, it can be noted that the CFD results are able to capture

the main flow features seen in the experimental results. The CFD

results seem to be quite similar at first sight, but there are some

important differences in the three main different vortex regions

that will be described and discussed below.

After-Body Side Vortex (ASV)

The ASV-11 is located at x/Lpp=0.7030. The corresponding

results can be found in Figure 12. The wind tunnel measurements

in this section cover both the ASV and the FSV.

In this plane, the ASV is located nearly close to the centerline

(slightly more leeward) of the hull bottom. The distance between

the ASV center and the wall surface of the model bottom varies

among the simulation results. Additionally, the calculated

strength of the ASV shows noticeable differences. Here, the

ReFRESCO, FOI Edge (HYB0) and NavyFOAM show the

highest ωx values. The NavyFOAM results again illustrate a

multiple core vortex system, which is no longer the case in the

FreSCo+ results.

The FSV is located leeward of the hull and its distance to the hull

is the same as the previous section. The core position of the vortex

in all CFD results agrees very well with the experiment. All CFD

codes underpredict x. The FOI OpenFOAM results show the

highest and the FOI Edge (EARSM) the lowest ωx values.

Concerning the TKE data, it can be stated that the TKE values

are, in general, lower in the ReFRESCO results compared to the

FreSCo+ results. Figure 12 shows the distribution of Tut on the

leeward side of the hull. It can be seen that Tut values are high in

the boundary layer and in the ASV region. The pattern of the FSV

becomes clearly visible TKE and Tut plots.

CONCLUSION Experimental Investigation

An extensive experimental study has been conducted by the

Hamburg University of Technology (TUHH) for a detailed

validation of CFD predictions for flow over a double model of the

MOERI tanker (KVLCC2) at static drift angle of 30 degrees. The

aim of the test case is determining the most important features of

the flow on ships with high block coefficient during maneuvers.

The flow at high static drift angle is very complex and dominated

by multiple vortices of various origins emanating along the hull.

The uncertainties were estimated according to the International

Towing Tank Conference (ITTC) Recommended Procedures and

Guidelines 7.5-01-03-03 “Uncertainty Analysis – An Example

for PIV Measurement” (ITTC 2008). The estimated uncertainties

for three velocity components are Uz = ± 0.08m/s, Uy = ± 0.06m/s

and Ux = ± 0.33m/s. The relative uncertainties of the velocity

components to the free stream velocity are Uz = ± 0.30%, Uy =

±0.22% and Ux = ± 1.22%. The uncertainty of the longitudinal

vorticity ωx is calculated based on the uncertainties regarding the

velocity components and the measuring positions. The estimated

uncertainty of ωx is ±5.32 Lpp/U∞.

The uncertainty of the measured velocity component parallel to

the inflow Ut using PIV-technique is ± 0.536m/s and 1.98%

relative to the free stream velocity. The absolute and the relative

uncertainty of the measured results using LDA-techniques for the

Ut are ± 0.2m/s and: ±0.74%, respectively. The uncertainties can

be locally higher in some areas due to the reflection of the model

walls.

To capture the global flow structure, flow visualization tests were

conducted in the wind tunnel. The tests include a smoke test for

visualizing the vortical flow structure around the model and a

classic oil film method for identifying the limiting streamlines

and separation areas on the model surface.

The results show that most of the vortices are displaced far from

the hull once they are generated. Thus, the evolution of the mean

flow and turbulence in their cores tends to occur in a free shear

regime. But there are also vortices which progress close to the

hull surface and, thus, it is expected that these have strong

interactions with the boundary layer flow of the hull.

In the results of the planer velocity, the main vortices can be

clearly identified as FSV, ASV, ABV and SV. In the smoke tests,

FSV, ASV and ABV could be visualized. The results of the oil

film method in the stem and stern regions give a clear indication

for the influence of the FBV and the SV on the limiting

streamlines. In the stern region, the separation area ABHV can be

clearly seen. Due to the strong reflection of the laser beam in

certain locations, the quality of the measured results is limited,

but in general, the majority of the experimental data seem

reasonable and reliable.

The measured vorticity and axial velocity component of the FSV

show different characteristics compared with the corresponding

values of the other vortices, namely vorticity and axial velocity

component of the FSV increase downstream while these values

are nearly constant or decrease for other vortices. The increase of

the vorticity and axial velocity component of FSV may be a result

of the secondary vortex structure around the FSV. This structure

can also be noticed around the leading edge vortex of delta foil.

CFD Computations

There were 7 CFD submissions from five different institutions.

Different turbulence models were used, ranging from different k-

ω models to DES, LES modeling to various combinations of

these. In most submissions, the boundary layer on the hull was

resolved down to the surface, but also wall functions were used.

Analyses of the vortex core data were carried out for all

submissions with the exception of NavyFOAM and FOI-

OpenFOAM. The limiting streamlines, onset and separation were

analyzed for all submissions.


The resolution of the grids used by different groups ranged from

11M to 77M. NavyFOAM simulations used the finest grid,

whereas FreSCo+ simulations used the coarsest grid. The grids

used for most of the simulations were not finer than 18M, except

for those used by FOI OpenFOAM and NavyFOAM. One of the

important conclusions drawn from this study is that fine grids in

the vortices regions are mandatory to accurately capture the onset

and progression of the vortices. Numerical dissipation can

completely change the strength of the vortices and therefore the

interaction between them.

In general, a good agreement is achieved among the CFD

methods applied for the prediction of the dominant vortical

structures, but less agreement is achieved for the vortical

structures on the leeward side of the ship hull. This applies mainly

to the axial vorticity and axial velocity of the FSV, which show a

decrease in all CFD methods (an exception is the axial velocity

prediction by CFDShip-Iowa), whereas it has a constant or

increasing level in a certain range of x/LPP. Also, some methods

predict secondary vortices for the ASV vortex system. The

OpenFOAM-LES results show significant small-scale resolved

turbulent vortical structures. The separation pattern and topology

were consistent with the available experimental data.

Forces and Yaw Moment

Many of the CFD submissions include the forces and yaw

moment for different grid resolutions. Grid dependency studies

have been carried out by CFDShip-Iowa, ReFRESCO and

FreSCo+ with four, seven and three grids, respectively. The

calculated forces and yaw moment on the seven grids considered

by CFDShip-Iowa and on the three grids used by FreSCo+ are

given in Table 5. The results of ReFRESCO are given only for

the final grid. Only CFDShip-Iowa has applied quantitative

solution verification including forces (X, Y) and moment (N) on

two grid-triplet studies with a refinement ratio √2. The results of

the verification study show that a monotonic convergence is

achieved for X on (1, 2, 3), whereas oscillatory convergence is

achieved for Y on (1, 2, 3) and (2, 3, 4) and for N on (1, 2, 3). The

non-smooth grid convergence is due to different flow physics

between coarse and fine grids and the lack of solution verification

methods for DES where the numerical and modeling errors are

strongly coupled.

Table 5 shows that the X and Y forces calculated by FreSCo+ have

a monotonic convergence behavior, whereas for N shows an

oscillatory behavior.

The variation range of the computed X force by different codes is

quite large. It varies between -0.03× 10−2 and -0.581× 10−2.

Although a systematic verification study for all applied methods

is not conducted, the X-force obtained by OpenFOAM using

LES-MM is the lowest and by CFDShip-Iowa using ARS-DES is

the highest. Compared with X, the range of variation of the Y

force is limited. The lowest and the highest values are 25.2× 10−2

and 31.7× 10−2. The lowest and highest values are obtained

again by OpenFOAM-LES-MM and CFDShip-Iowa-ARS-DES,

respectively. A similar tendency can be noticed for the yaw

moment N. The lowest value 5.09 × 10−2 is obtained by

OpenFOAM-LES-MM and the highest 7.27 × 10−2 is calculated

by CFDShip-Iowa-ARS-DES.

The wide range of the calculated X force (E= -90.35% to +

86.98%) shows that this component is more sensitive to the

applied turbulence model than the Y force. The E range for Y is -

15.18% to +13.93%, and the E range for N = -21.87% to

+11.63%. The calculated force components obtained by the k-ω

model are close to each other although a large variation of the grid

resolution exists. As the Y force is less sensitive to the applied

turbulence model, the predicted yaw moment also shows less

scattering.

An overview of normalized force and yaw moment coefficients

is given in Table 5. Although extensive V&V studies are not

carried out, some comments are given on the presented results.

The ARS-DES (13M nodes) model predicts the highest and the

LES-MM (36.5M nodes) the lowest value. The results of two-

equation turbulence models, such as standard k-ω or SST k-ω,

show a decrease of the calculated X-force coefficient when

increasing the grid resolution, see Table 5.

Vortical Structures:

As shown in Figure 8, all the solvers are more or less consistent

in the prediction of vortical structures. CFDShip-Iowa,

ReFRESCO and FOI Edge predict very similar results, whereas

FreSCo+ (steady), FOI OpenFOAM and NavyFOAM predict

many secondary vortices for FSV. FOI OpenFOAM and

NavyFOAM show secondary vortices for ASV. The nature of

FSV secondary vortices predicted by FreSCo+ and FOI-

OF/NavyFOAM is different. As shown in Figure 8, the shape of

the secondary vortices calculated by FreSCo+ is completely

changed when the computation was carried out as an unsteady

case. The instantaneous shape of the secondary vortices from

FreSCo+ is very similar to the ReFRESCO results. FOI hybrid

RANS-LES model shows very low resolved turbulence levels in

instantaneous plot. It seems like HYB0 suffers from turbulence

trigger issues similar to DES, as observed in straight-ahead 5415

case. The results of FOI OpenFOAM LES show significant small-

scale resolved turbulent vortical structures.

Vortex Core Analysis:

Figure 11 show that the CFD codes predict the FSV core location

well. This is consistent with 5415 results. The FSV core x-

vorticity is underpredicted by 50% by CFD for x/L > 0.6, see

Figure 11. All CFD simulations show a similar trend. EFD also

shows an increasing trend in axial velocity for x/L between 0.7-

0.8. The increase in the core could be due to either the influence

of the secondary vortices or the interaction between the FSV and

the ASV.

The FSV x-velocity shows large errors similar to x-vorticity for

x/L > 0.6. The EFD data shows a sharp flow deceleration for x/L

between 0.7 and 0.8. Combined with the x- vorticity, there

appears to be a strong interaction between the FSV and the ASV

in the experiments, which CFD do not predict. The influence of

the secondary vortices may also play an important role in this

context.

Overall, all codes predict the FSV reasonably well in comparison

with the experiments; CFDShip-Iowa predicts most accurate core


axial velocity, i.e., longitudinal velocity component in the vortex

core increases, which is not captured by other codes; all codes

predict decreasing axial vorticity, whereas experiments showed

increasing trends.

As shown in Figures 11, y- and the z-locations of ASV are

predicted well. The z-location shows wider variation, especially

for x/L between 0.6 and 0.7. In the experiments, the vortex

sharply moves upwards around x/L = 0.7, which is not predicted

in any CFD results. This is probably due to FSV-ASV interaction,

which is missing in CFD predictions.

For the axial vorticity component ωx, CFD predictions are,

overall, in good agreement with the experiments. CFDShip-Iowa

underpredicts ωx. RANS predictions seem to be best for ωx. All

CFD underpredict experimental core axial velocity by 20-40%.

Because the ASV is much closer to the model surface than the

other vortices, the quality of the experimental results may suffer

from the wall reflection in this region.

ABV predictions using URANS are significantly better than those

for the sonar dome tip vortex (SDTV) of the combatant 5415 at β

= 20° predictions. Overall, all codes predict ABV reasonably well

in comparison with the experiments; CFDShip-Iowa and

ReFRESCO over-predict core axial velocity compared to other

codes, see Figure 11.

In general, all CFD predictions compare reasonably with EFD

data during the onset and early stage of the progression of the

vortices. However, details of the simulated flow are different

from measurements further downstream. The core coordinates of

the main vortices are predicted well by the CFD methods

considered in the vortex core analysis. The same is valid for the

axial vorticity component, with the exception of the FSV. This

vortex arises in the boundary layer of the model, leaves the

boundary layer very fast and progresses in the outer flow region.

Therefore, a considerable part of the FSV can be considered as an

isolated vortex with negligible interaction with the surrounding

outer flow and these circumferences allow the vortex to keep its

strength over a quite long distance.

Turbulence Model

From the planar solution it becomes obvious that the predictions

for the FSV and the ABV are quite similar in both URANS and

hybrid RANS/LES or LES. NavyFOAM results show multiple

vortex separations for ASV, which probably are caused by the use

of a different turbulence model with wall functions. The

applicability of the wall functions for separated flows under

adverse pressure gradients as in the case of the flows on KVLCC2

at large drift angles needs further validation.

Prediction of significant small-scale resolved turbulent structures

and reasonable mean-flow predictions by FOI-OpenFOAM with

wall-modeled LES suggests that LES simulations are reliable in

free shear regions away from the hull. Low resolved turbulence

level in FOI hybrid RANS-LES suggests that HYB0 suffers from

the turbulence trigger issue similar to DES, as observed in the

straight-ahead 5415 case.

The effect of using steady state calculations for an unsteady flow

is investigated by FreSCo+. The comparison between the steady

and unsteady calculations shows a considerable influence on the

structure of the secondary vortices around the FSV and negligible

changes of the predicted characteristics of the main vortices. The

reason for this could be caused by insufficient iterative

convergence in the steady simulation, which is four orders of

magnitude higher.

With the current grids and solvers, there does not appear to be one

turbulence model that can accurately predict all aspects of the

flow. DES underpredicts the TKE levels. The calculated

separation areas by using NavyFOAM and Edge-HYB0 are

significantly larger than the area observed in the model tests.

The results show that the numerical dissipation of vortical flows

can be a weak point in the simulations. This problem can be

mitigated by using high quality grids with a strong local

refinement in the regions of the expected main vortices. Besides

the need of accurate turbulence closure models, it may be useful

to develop numerical methods that are able to solve the

momentum conservation equations based on velocity-vorticity

formulation.

Surface Streamlines

Vortices show open-type separation and reattachment, and are

identified by converging and diverging streamlines. All solvers

agree very well for the prediction of separation lines SL 1, SL 2-

1, SL 3-1 and SL 3-2; and reattachment lines RL 3 and RL 4.

SL 5 (SV separation) is predicted well by all solvers except FOI

Edge HYB0 and FOI OpenFOAM-LES. The latter solvers show

the presence of nodes, suggesting the formation of an additional

closed-type separation, which is physically possible. FOI

OpenFOAM-LES and NavyFOAM predictions show complex

flow streamlines between RL 3, SL 3-1 and SL 3-2, related to the

separation and reattachment of ASV. This correlates with the

ASV secondary vortices predicted by the solver in Figure 8. ASV

secondary vortices could be predicted due to wall modeling used

by both FOI-LES and NavyFOAM.

FreSCo+, CFDShip-Iowa and ReFRESCO predict mostly similar

results as all use near wall modeling. FOI OpenFOAM-LES and

NavyFOAM show complex surface streamlines on after-body

windward side likely due to effects of wall functions.

Future work should also focus on:

(a) Improving the accuracy of computing the leading edge

vortex; special attention should be given to vorticity transport

and reducing the numerical dissipation,

(b) Development of numerical methods that are able to solve the

momentum conservation equations based on velocity-

vorticity formulation,

(c) Investigation of RANS-LES transition modeling for hybrid

RANS-LES calculations,

(d) Additional experimental test of the present test case for

confirmation of current EFD findings,

(e) Development of collaborative experimental programs for

generating detailed validation data, especially velocity and

turbulence fields on ship hulls in unsteady motion,

(f) Validation by using experimental data including free surface

effects, if available.


ACKNOWLEDGMENTS The U.S. Office of Naval Research under Grant N00014-10-1-

0017 sponsored the research at the University of Iowa under the

administration of Drs Thomas Fu and Ki-Han Kim. Drs Patrick

Purtell and Ki-Han Kim provided guidance to the international

collaboration over the course of this study and Dr Kim graciously

aided in editing. Dr Stephan Hitzel, Airbus, provided consultation

on the assessment of the ship-flow vortex breakdown and helical

mode instability/spiral vortices.

REFERENCES [1] Bensow, R., & Fureby, C. (2007). On the Justification and

Extension of Mixed Models in LES. J. Turb., 8 N54:1.

[2] Bhushan, S. T. (2014). “Chapter 7: Post Workshop

Computations and Analysis for KVLCC2 and 5415,”

Numerical Ship Hydrodynamics: An Assessment of the

Gothenburg 2010 Workshop. Springer, ISBN: 978-94-007-

7188-8.

[3] Carrica, P., Wilson, R., & Stern, F. (2007). An unsteady

single-phase level set method for viscous free surface flows.

Int. J. Numer. Methods Fluids 53 (2), 229–256.

[4] Committee, A. V. (2009). Standard for Verification and

Validation in Computational Fluid Dynamics and Heat

Transfer. ASME.

[5] Dacles-Mariani, J., Zilliac, G. G., Chow, J. S., & Bradshaw,

P. (1995). Numerical/experimental study of a wingtip

vortex in the near field. AIAA Journal, Vol. 33, pp. 1561–

1568.

[6] Eça, L., & Hoekstra, M. (2009). Evaluation of numerical

error estimation based on grid refinement studies with the

method of the manufactured solutions. Computers & Fluids,

Vol 38, No. 8. pp1580–1591.

[7] Eça, L., Hoekstra, M., & Vaz, G. (2012). On the Use of

Manufactured Solutions for Code Verification of RANS

Solvers Based on Eddy-viscosity Models. Las Vegas, NV:

ASME Verification and Validation Symposium.

[8] Eliasson, P. (2002). Edge, a Navier-Stokes solver for

unstructured grids. Finite Volumes for Complex

Applications III, ISBN 1 9039 9634 1, pp. 527–534.

[9] Fureby, C. (2007). On LES and DES of Wall Bounded

Flows. Ercoftac Bulletin, 72.

[10] Germano, M., Piomelli, U., Moin, P., & Cabot, W.

(1991). A Dynamic Sub Grid Scale Eddy Viscosity Model.

Phys. Fluids, A 3:1760.

[11] Hellsten, A. (2005). New Advanced k-ω Turbulence

Model for High-Lift Aerodynamics. AIAA Journal, Vol.

43, No. 9, pp. 1857–1869.

[12] Hunt, J., Wray, A., & Moin, P. (1988). Eddies, Stream,

and Convergence Zones in Turbulent Flows. Cent. Turbul.

Res.

[13] Issa, R. (1986). Solution of the Implicitly Discretized

Fluid Flow Equations by Operator Splitting. Journal of

Computational Physics, 62:40.

[14] Jiminez, J. (2013). Near Wall Turbulence. Phys. Fluids,

101302.

[15] Kerkvliet, M. (2013). Influence on the Numerical

Uncertainty of a Generic Submarine Model by Changing

the Wall-Normal Distribution of the Wall-Bounded Grid

Cells. 16th Numerical Towing Tank Symposium (NuTTS),

pp 78–83.

[16] Kim, W.-W., & Menon, S. (1995). A New Dynamic One

Equation Subgrid-scale Model for Large Eddy Simulations.

AIAA 95-0356. American Institute of Aeronautics and

Astronautics.

[17] Lambert, J. (1973). Computational Methods in Ordinary

Differential Equations. Wiley, New York.

[18] Larsson, L. S. (2014). Numerical Ship Hydrodynamics:

An Assessment of the Gothenburg 2010 Workshop.

Springer, ISBN: 978-94-007-7188-8.

[19] Layton, W. (2001). Analysis of a Scale-Similarity Model

of the Motion of Large Eddies in Turbulent Flows. J. Math

An. and Appl., 264:546.

[20] Levy, Y., Degani, D., & Seginer, A. (1990). Graphical

visualization of vortical flows by means of helicity. AIAA

J. 28 (8), 1347–1352.

[21] Li, Y., Chevillard, L., Eyink, G., & Meneveau, C. (2009).

Matrix Exponential-based Closures for the Turbulent

Subgrid-scale Stress Tensor. Phys. Rev., 79.

[22] Liu, S., Meneveau, C., & Katz, J. (1994). On the

Properties of Similarity Subgrid Scale Models as Deduced

from Measurements in a Turbulent Jet. J. Fluid Mech.,

275:83.

[23] Menter, F. R. (1994). Two-equation Eddy-viscosity

turbulence models for Engineering Applications. AIAA

Journal, Vol. 32, pp. 1598–1605.

[24] Menter, F. R., Kuntz, M., & Langtry, R. (2003). Ten

Years of Industrial Experience with the SST Turbulence

Model,. Turbulence, Heat and Mass Transfer 4, ed: K.

Hanjalic, Y. Nagano, and M. Tummers, Begell House, Inc.,

2003, pp. 625–632.

[25] Nelson, R., & Pelletier, A. (2003). The unsteady

aerodynamics of slender wings and aircraft undergoing

large amplitude maneuvers. Progress in Aerospace Sciences

39, 185–248.

[26] Peng, S.-H. (2006). Algebraic Hybrid RANS-LES

Modelling Applied to Incompressible and Compressible

Turbulent Flows. AIAA-Paper 2006-3910.

[27] Peng, S.-H. (2005). Hybrid RANS-LES modelling based

on zero- and one-equation models for turbulent flow

simulation. In Proceedings of 4th Int. Symp. Turb. and

Shear Flow Phenomena, volume 3, pp 1159–1164.

[28] Pereira, F. S., Vaz, G., & Eça, L. (2014). On the use of

Hybrid Turbulence Models. 17th Numerical Towing Tank

Symposium (NuTTS), Marstrand, Sweden, pp. 135–140.

[29] Pope, S. B. (2004). Ten Questions Concerning the Large-

Eddy Simulation of Turbulent Flows. New Journal of

Physics, 6:35.

[30] Rung, T., Wöckner, K., Manzke, M., Stück, A.,

Brunswig, J., & Ulrich, C. (2009). Challenges and

Perspectives for Maritime CFD Applications. Jahrbuch der

Schiffbautechnischen Gesellschaft (Vol. 103).

[31] Sagaut, P. (2001). Large Eddy Simulation for

Incompressible Flows. Springer Verlag.

[32] SIMMAN 2008. (2008). Retrieved from

www.simman2008.dk/KVLCC/tanker.htm


[33] SIMMAN 2014. (2014). Retrieved from

www.simman2014.dk

[34] Smagorinsky, J. (1963). General Circulation Experiments

with the Primitive Equations. Mon. Weather Rev.,91:99.

[35] Toxopeus, S. L. (2011). Practical application of viscous-

flow calculations for the simulation of manoeuvring ships.

Delft University of Technology, Faculty Mechanical,

Maritime and Materials Engineering.

[36] Toxopeus, S. L. (2013). Viscous-Flow Calculations For

KVLCC2 In Deep And Shallow Water. Chapter 9 in Eça,

L.; Oñate, E.; García-Espinosa, J.; Kvamsdal, T. & Bergan,

P. (Eds.) "MARINE 2011, IV International Conference on

Computational Methods in Marine Engineering", Springer,

pp. 151–169.

[37] Toxopeus, S. L., Atsavapranee, P., Wolf, E., Daum, S.,

Pattenden, R. J., Widjaja, R., et al. (2012). Collaborative

CFD Exercise for a Submarine in a Steady Turn. 31st

International Conference on Ocean, Offshore and Arctic

Engineering.

[38] Toxopeus, S. L., Simonsen, C. D., Guilmineau, E.,

Visonneau, M., Xing, T., & Stern, F. (2013). Viscous-Flow

Calculations for KVLCC2 in Manoeuvring Motion in Deep

and Shallow Water. Journal of Marine Science and

Technology.

[39] Tysell, L. (2007). The TRITET Grid Generation System.

Greece: Proceedings of the 10th ISGG Conference on

Numerical Grid Generation, Society of Grid Generation

(ISGG).

[40] Vaz, G., Jaouen, F. A., & Hoekstra, M. (2009). Free-

Surface Viscous Flow Computations. Validation of

URANS Code FreSCo. 28th International Conference on

Ocean, Offshore and Arctic Engineering (OMAE).

[41] Wallin, S., & Johansson, A. V. (2000). An explicit

algebraic Reynolds stress model of incompressible and

compressible flows. J. Fluid Mech., Vol 403, pp. 89–132.

[42] Wikström, N., Svennberg, U., Alin, N., & Fureby, C.

(2004). Large Eddy Simulation of the Flow around an

Inclined Prolate Spheroid. Journal of Turbulence.

[43] Wilcox, D. C. (2008). Formulation of the k-omega

Turbulence Model Revisited. AIAA Journal, Vol. 46, No.

11, pp. 2823–2838.

[44] Wilcox, D. (2004). Turbulence Modelling for CFD. 2nd

edn. DCW Industries Inc.

[45] Xing, T., Bhushan, S., & Stern, F. (2012). Vortical and

Turbulent Structures for KVLCC2 at Drift Angle 0, 12, and

30 Degrees. Ocean Engineering, Vol. 55, 23–43.

[46] Xing, T., Shao, J., & Stern, F. (2007). BKW- RS-DES of

Unsteady Vortical Flow for KVLCC2 at Large Drift

Angles. Ann Arbor, Michigan: 9th International Conference

on Numerical Ship Hydrodynamics.

[47] Gallaire, F and Chomaz, JM. (2003). Mode selection in

swirling jet experiments: a linear stability analysis, J. Fluid

Mech., vol. 494, pp. 223–253.

[48] Greenwell, DI. (2009). Chapter 21 – ‘Engineering’

Models Of Delta Wing Vortex Breakdown And Its Effect

On Aerodynamic Characteristics. RTO-TR-AVT-080.

[49] Gursul, I., (1994). Unsteady flow phenomena over delta

wings at high angle of attack. AIAA J. 32 (2), 225–231.

[50] Gursul, I. (2009). Chapter 6 – Unsteady Aspects Of

Leading-Edge Vortices, RTO-TR-AVT-080.

[51] Spall, RE., Gatski, TB. and Grosch, CE. (1987). A

criterion for vortex breakdown, Physics of Fluids 30, 3434.


Figure 8: Predicted vortical flow structure steady/ time-averaged results, Q=100.

Figure 9: Predicted vortical flow structure, instantaneous results, Q=100.

Figure 10: Predicted limiting streamlines. Blue dashed line indicates the ASV core position.

secondary separation line

primary reattachment lineprimary separation line

RL 3SL 3-1 SL 5

SL 4-1SL 1 SA 6

SL 3-2

0.60.1 0.2 0.50.3 0.4 0.7 0.8 0.9

SL 2-1SL 1

SL 2-1

SL 3-1RL 3

SL 5SL 3-2

0.60.1 0.2 0.50.3 0.4 0.7 0.8 0.9

primary reattachment line

primary separation line


SL 4-1

0.60.1 0.2 0.50.3 0.4 0.7 0.8 0.9

RL 3

SL 3-1

SL 5

SL 4-1SL 2-1

SL 1

SA 6

SL 3-1

Third SL 4-2

separationline

SL 3-2

Primary

separationline (ASV)

Primary

reattachmentline Separation

line (ABHV)

Separation

line (ABV)Secondary

separationline (FSV)

Separation

line (FBV)

0.60.1 0.2 0.50.3 0.4 0.7 0.8 0.9

primary reattachment lineprimary separation line


RL 3SL 3-1 SL 5

SL 4 -1

SL 2-1

SL 1SA 6

third separation linesecondary reattachment line

SL 3-2

SL 4 -2RL 4


0.60.1 0.2 0.50.3 0.4 0.7 0.8 0.9



RL 3SL 3-1

SL 5

SL 4-1

SL 2-1

SL 1 SA 6

third separation line

SL 3-2

SL 4-2

0.60.1 0.2 0.50.3 0.4 0.7 0.8 0.9




RL 3SL 3-1 SL 5

SL 4-1

SL 2-1

SL 1 SA 6

SL 3-2

0.60.1 0.2 0.50.3 0.4 0.7 0.8 0.9




RL 3 SL 3-1 SL 5

SL 4-1

SL 2-1

SL 1 SA 6

SL 3-2

CFDShip-Iowa ReFRESCO

CFDShip-Iowa

FreSCo+, steady FreSCo+, unsteady

NavyFOAM

FreSCo+

ReFRESCO

NavyFOAM

FOI Edge HYB0

FOI OpenFOAM

FOI OpenFOAM FOI Edge HYB0

FOI Edge EARSM

FOI OpenFOAM FOI Edge HYB0


Figure 11: Vortex core analyses: FSV (left), ASV (middle) and ABV (right).

0,00

0,05

0,10

0,15

0,20

0,25

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

Co

re y

/Lp

p L

oca

tio

n

x/Lpp

Variation of FSV Ycore

inception of helical instability (CFDShip-Iowa)

inception of helical instability (FreSCo+)

-0,15

-0,10

-0,05

0,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

Co

re y

/Lp

p L

oca

tio

n

x/Lpp

Variation of ASV Ycore

inception of helical instability


0,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

0,6 0,8 1,0 1,2 1,4 1,6

Co

re y

/Lp

p L

oca

tio

n

x/Lpp

Variation of ABV Ycore



-0,08

-0,07

-0,06

-0,05

-0,04

-0,03

-0,02

-0,01

0,00

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

Co

re z

/Lp

p L

oca

tio

n

x/Lpp

Variation of FSV Zcore



-0,10

-0,09

-0,08

-0,07

-0,06

-0,05

-0,04

-0,03

-0,02

-0,01

0,00

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4C

ore

z/L

pp

Lo

cati

on

x/Lpp

Variation of ASV Zcore


(CFDShip-Iowa)


-0,08

-0,07

-0,06

-0,05

-0,04

-0,03

-0,02

-0,01

0,00

0,6 0,8 1,0 1,2 1,4 1,6

Co

re z

/Lp

p L

oca

tio

n

x/Lpp

Variation of ABV Zcore



10

100

1000

10000

100000

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

Q

x/Lpp

Variation of FSV QPeak


10

100

1000

10000

100000

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

Q

x/Lpp

Variation of ASV QPeak


10

100

1000

10000

100000

0,6 0,8 1,0 1,2 1,4 1,6

Q

x/Lpp

Variation of ABV QPeak


0

100

200

300

400

500

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

Co

re A

xial

Vo

rtic

ity

x/Lpp

Variation of FSV ωx,core



0

100

200

300

400

500

600

700

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

Co

re A

xial

Vo

rtic

ity

x/Lpp

Variation of ASV ωx,core

inception of helical instability (CFDShip-

Iowa)


-100

0

100

200

300

400

500

600

0,6 0,8 1,0 1,2 1,4 1,6C

ore

Axi

al V

ort

icit

y

x/Lpp

Variation of ABV ωx,core



0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

Co

re A

xial

Ve

loci

ty

x/Lpp

Variation of FSV Ux,core



0,0

0,2

0,4

0,6

0,8

1,0

1,2

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

Co

re A

xial

Ve

loci

ty

x/Lpp

Variation of ASV Ux,core


(CFDShip-Iowa)


0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

0,6 0,8 1,0 1,2 1,4 1,6

Co

re A

xial

Ve

loci

ty

x/Lpp

Variation of ABV Ux,core



-1,8

-1,6

-1,4

-1,2

-1,0

-0,8

-0,6

-0,4

-0,2

0,0

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

Co

re P

ress

ure

x/Lpp

Variation of FSV cpcore



-1,6

-1,4

-1,2

-1,0

-0,8

-0,6

-0,4

-0,2

0,0

0,2

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

Co

re P

ress

ure

x/Lpp

Variation of ASV cpcore


(CFDShip-Iowa)inception of helical instability (FreSCo+)

-1,8

-1,6

-1,4

-1,2

-1,0

-0,8

-0,6

-0,4

-0,2

0,0

0,6 0,8 1,0 1,2 1,4 1,6

Co

re P

ress

ure

x/Lpp

Variation of ABV cpcore



0,000

0,005

0,010

0,015

0,020

0,025

0,030

0,035

0,040

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

Co

re T

KE

x/Lpp

Variation of FSV TKEcore



0,000

0,005

0,010

0,015

0,020

0,025

0,030

0,035

0,040

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

Co

re T

KE

x/Lpp

Variation of ASV TKEcore


(CFDShip-Iowa)


0,000

0,005

0,010

0,015

0,020

0,025

0,030

0,035

0,040

0,045

0,6 0,8 1,0 1,2 1,4 1,6

Co

re T

KE

x/Lpp

Variation of ABV TKEcore



0,00

0,10

0,20

0,30

0,40

0,50

0,60

0,70

0,80

-0,6 1,4

Co

re y

/Lp

p L

oca

tio

n

x/Lpp

ReFRESCO, 12.7M, SST2003 FreSCo+, 11.0M, SST k-ω 2003 FreSCo+, ω based identification CFDShip-Iowa, 13.0M, ARS-DES

EFD, PIV Edge, 17.4M, EARSM Edge, 17.4M, HYB0


Figure 12: Planar plots ASV-region section 11, : 𝑈𝑥, 𝑈𝑦 , 𝑈𝑧, 𝜔𝑥, 𝑇𝑈𝑡 and TKE.

𝑈𝑥

𝑈𝑦

𝑈𝑧

𝜔𝑥

𝑇𝑈𝑡, TKE

EFD: TUHH

EFD: TUHH

EFD: TUHH

EFD: TUHH

EFD: TUHH

FreSCo+

FreSCo+

FreSCo+

FreSCo+

FreSCo+

ReFRESCO

ReFRESCO

ReFRESCO

ReFRESCO

ReFRESCO

CFDShip-Iowa

CFDShip-Iowa

CFDShip-Iowa

CFDShip-Iowa

OpenFOAM

OpenFOAM

OpenFOAM

OpenFOAM

NavyFOAM

NavyFOAM

NavyFOAM

NavyFOAM

Edge, EARSM

Edge, EARSM

Edge, EARSM

Edge, EARSM

Edge, EARSM Edge, HYB0

Edge, HYB0

Edge, HYB0

Edge, HYB0

Edge, HYB0


Table 3: Main characteristics of the CFD codes

Organization / code

name

Modeling Grid characteristics

Turbulence Wall boundary

Steady

/unsteady

calculation

Type/Size Resolution

Iowa Institute of Hydraulic Research of the University of Iowa (IIHR) CFDShip-Iowa

EARSM

DES

low Re near wall

turbulence model,

no slip condition

unsteady

overset,

multiblock

structured;

0.6M,1.6M,

4.6M and 12.9

M points

grid refinement

block around the

ship hull

Maritime Research Institute Netherlands (MARIN) ReFRESCO

k- SST

(1994 and

2003

versions)

low Re near wall

turbulence model,

no slip condition

steady

structured

0.12M, 1.59M,

2.27M, 3.34M,

5.38M, 8.45M

and 12.72M

multiblock

structured O–O

Hamburg University of Technology (TUHH) FreSCo+

k- SST

(2003)

low Re near wall

turbulence model,

no slip condition

steady /

unsteady

unstructured

2.97M, 4.15M

and 11M

grid refinement

around the ship

hull and in the

wake area

Swedish Defence Research Agency (FOI) Edge RANS-Simulation

EARSM in

combination

with

Hellsten k-ω

low Re near wall

turbulence model,

no slip condition

unsteady unstructured,

17.4M

grid refinement

around the ship

hull and in the

wake area

Swedish Defence Research Agency (FOI) Edge RANS-LES-Simulation

RANS-LES

HYB0, algebraic

mixing-length in

combination with

the SGS model by

Smagorinsky

(1963).

unsteady unstructured,

17.4M

grid refinement

around the ship

hull and in the

wake area

Swedish Defence Research Agency (FOI) OpenFOAM

LES, Mixed

Model

LES boundary-

layer equations unsteady

unstructured,

36.5M

grid refinement

around the ship

hull and in the

wake area

aval Surface Warfare Center Carderock Division (NSWCCD) NavyFOAM

k- model

(Wilcox,

2008)

wall function

condition

y+ = 40-100

steady unstructured

grid, 77M

grid refinement

around the ship

hull and in the

wake area

Table 4: Overview of generated grids for all CFD solvers.

IIHR

CFDShip-

Iowa

Grid 1 2 3 4

Ship 144×88×35=

443,520

203×122×49=

1,213,534

287×174×69=

3,445,722

406×244×98=

9,708,272

Background 76×47×41=

146,452

107×66×58=

409,596

152×93×82=

1,159,152

214×132×116=

3,276,768

Total 589,972 1,623,130 4,604,874 12,985,040

y+ (max.) 1.13 0.80 0.56 0.40

MARIN

ReFRESCO

Grid 1 2 3 4 5 6 7

Number

of cells 121,330 1,590,144 2269674 3,340,098 5,388,408 8,454,880 12,721,152

Elements

on hull 2904 17216 21978 27740 38904 52528 68864

y+ (max.) 2.82 1.45 1.29 1.13 1.01 0.86 0.74

TUHH

FreSCo+

Grid 1 2 3

Number of cells 2.97 M 4.15 M 11.0 M.

Typical cell size in

refinement area (mm) 5.0 × 5.0 × 5.0 3.54 × 3.54 × 3.54 2.5 × 2.5× 2.5

y+ (max.) 1.2 1.2 1.2

FOI

Edge

Grid 1

Number of nodes 17.4 M

Number of elements 73.5 M

Elements on hull 524408

y+ (mean, URANS + hyb0) 0.49

FOI

OpenFOAM

Grid 1

Number of nodes 36.5 M

Number of elements 202.0 M

Elements on hull 611403

y+ (mean) 10.7

NSWCCD

NavyFOAM

Grid 1

Number of cells 77 M

y+ (90% of all values) 40-100


Table 5: Overview of normalized force coefficients for the different

calculations. Only light blue marked computations are considered for

RMS value calculation.

Solver Grid size

in M.

Turbulence model

Steady / unsteady

calculation

X' [-] ×10-2

E% RMS

Y’ [-] ×10-2

E% RMS

N’ [-] ×10-2

E% RMS

RMS value of all

highlighted CFD results

- - - -0.311 - 27.836 - 6.515 -

CFDShip-Iowa (IIHR)

0.59 ARS-DES unsteady -0.481 54.79 32.736 17.61 7.418 13.87

1.6 ARS-DES unsteady -0.614 97.60 31.325 12.54 7.142 9.63

4.6 ARS-DES unsteady -0.629 102.42 32.096 15.31 7.202 10.55

13.0 ARS-DES unsteady -0.581 86.98 31.713 13.93 7.272 11.63

ReFRESCO (MARIN)

12.7 SST1994 steady -0.154 -50.44 26.023 -6.51 5.488 -15.76

12.7 SST2003 steady -0.063 -79.73 25.695 -7.69 5.552 -14.78

FreSCo+

(TUHH)

2.97 standard

k-ω steady -0.094 -69.75 26.698 -4.09 6.866 5.39

4.15 standard

k-ω steady -0.088 -71.68 26.691 -4.11 6.789 4.21

11 standard

k-ω steady -0.079 -74.58 26.275 -5.61 6.883 5.65

2.97 SST k-ω

2003 steady -0.159 -48.83 27.988 0.55 6.817 4.64

4.15 SST k-ω

2003 steady -0.143 -53.98 27.562 -0.98 6.745 3.54

11 SST k-ω

2003 steady -0.134 -56.88 27.533 -1.09 6.818 4.66

11 SST k-ω

2003 unsteady -0.131 -41.16 27.416 2.91 6.884 11.19

Edge (FOI) 17.4 EARSM unsteady -0.518 -75.22 24.83 -15.18 5.457 -9.28

17.4 HYB0 unsteady -0.234 -24.69 23.92 -14.07 5.508 -15.45

OpenFOAM (FOI)

36.5 LES-MM unsteady -0.03 -90.35 25.2 -9.47 5.09 -21.87

NavyFOAM (NSWCCD)

77 Wilcox’ k-

model steady -0.166 -46.58 28.1 0.95 6.09 -6.52

Table 6: Axial vortex core prediction errors for FSV.

Vortex CFD Vortex

location x/Lpp

Flow variable prediction error, ABS (E%D)

Overall USN UD UV

y/Lpp z/Lpp Ux/U 𝝎x*Lpp

/U

FSV

ReFRESCO, 12.7M,

SST2003

0.3765 3.3 2.2 4.8 35.4 11.4

5.5

0.4326 3.6 0.7 14.5 9.7 7.1

0.4866 0.6 1.0 20.0 13.1 8.7

0.7029 6.4 17.0 39.0 54.7 29.3

0.7472 7.3 15.4 36.0 66.5 31.3

AVG 4,3 7,3 22,9 35,9 17,6

FreSCo+, 11.0M, SST k-ω 2003

0.3765 4.1 4.1 0.2 57.0 16.4

5.5

0.4326 4.2 1.1 6.5 25.7 9.4

0.4866 0.8 0.9 14.8 4.5 5.3

0.7029 5.2 17.9 28.4 56.2 26.9

AVG 3,6 6,0 12,5 35,8 14,5

CFDShip-Iowa,

13.0M, ARS-DES

0.3765 9.1 1.3 0.5 1.3 3.0

5.5

0.4326 9.5 2.4 3.3 33.8 12.2

0.4866 5.3 11.0 4.3 41.0 15.4

0.6840 3.2 2.9 10.1 70.7 21.7

0.7029 2.7 3.3 8.9 74.8 22.4

0.7316 5.1 1.6 3.3

0.7472 3.9 2.7 3.3

AVG 5,5 3,6 5,4 44,3 11,6

Edge, 17.4M, EARSM

0.3765 3.7 5.4 18.5 20.8 12.1

5.5

0.4326 3.6 5.2 22.1 33.1 16.0

0.4866 0.4 6.5 27.2 47.5 20.4

0.6840 7.8 3.3 38.1 64.7 28.5

0.7029 6.9 9.1 39.3 69.4 31.2

0.7316 8.7 5.6 37.1 74.8 31.6

0.7472 7.6 5.9 36.2 78.3 32.0

AVG 5,5 5,9 31,2 55,5 24,5

Edge, 17.4M, HYB0

0.3765 1.7 5.9 9.6 10.6 7.0

5.5

0.4326 2.6 2.6 13.5 7.3 6.5

0.4866 0.4 5.3 20.5 25.3 12.9

0.6840 8.6 4.1 33.6 46.7 23.2

0.7029 8.2 10.3 35.5 53.3 26.8

0.7316 10.6 6.9 33.2 60.1 27.7

0.7472 9.9 6.7 33.5 67.6 29.4

AVG 6,0 6,0 25,6 38,7 19,1


Table 7: Axial vortex core prediction errors for ASV.

Vortex CFD

Vortex location

x/Lpp


Overall USN UD UV

y/Lpp z/Lpp Ux/U 𝝎x*Lpp/

U

ASV

ReFRESCO, 12.7M,

SST2003

0.5309 95.3 0.2 12.6 102.5 52.7

5.5

0.5818 18.9 3.7 28.4 7.0 14.5

0.6156 2.0 0.1 30.3 29.0 15.4

0.7472 41.1 18.3 28.2 225.1 78.2

AVG 39.3 5.6 24.9 90.9 40.2


0.5309 52.1 6.7 12.1 54.2 31.3

5.5 0.5818 32.4 1.8 24.2 19.0 19.4

0.6156 6.1 8.7 30.8 1.1 11.7

AVG 30.2 5.8 22.4 24.8 20.8

CFDShip-Iowa,

13.0M, ARS-DES

0.5309 161.5 6.1 46.8 64.8 69.8

5.5

0.5648 27.1 4.0 43.7 49.6 31.1

0.5818 1.7 12.6 38.5 60.4 28.3

0.6156 40.3 9.7 29.0 43.3 30.6

0.6326 88.9 17.9 23.9 32.1 40.7

0.6495 67.1 18.2 26.6 40.8 38.2

0.6840 136.8 17.6 25.4 53.2 58.3

0.7166 63.6 11.4 9.7 60.0 36.1

0.7472 51.0 14.7 16.6 32.6 28.7

AVG 70.9 12.5 28.9 48.5 40.2

Edge, 17.4M, EARSM

0.5309 66.4 1.2 24.4 32.5 31.1

5.5

0.5648 12.3 2.5 34.7 23.0 18.2

0.5818 24.9 5.8 33.5 35.8 25.0

0.6156 1.9 3.5 25.6 15.7 11.7

0.6326 11.3 5.3 19.6 0.2 9.1

0.6495 40.7 7.7 21.3 6.1 19.0

0.6840 25.2 9.8 21.8 17.7 18.6

0.7166 40.4 23.0 11.6 17.2 23.0

0.7472 29.4 21.9 23.7 62.8 34.4

AVG 28.1 9.0 24.0 23.4 21.1

Edge, 17.4M, HYB0

0.5309 79.9 1.7 19.2 157.8 64.6

5.5

0.5648 3.3 1.5 24.9 167.5 49.3

0.5818 26.7 7.5 33.9 9.3 19.4

0.6156 11.9 1.9 39.8 23.5 19.3

0.6326 1.7 12.7 39.6 42.6 24.2

0.6495 18.0 14.0 41.2 26.6 25.0

0.6840 47.8 12.7 35.4 15.4 27.8

0.7166 38.1 14.1 24.4 29.7 26.6

0.7472 28.5 15.3 38.1 168.7 62.6

AVG 28.4 9.0 32.9 71.2 35.4

Table 8: Axial vortex core prediction errors for ABV.

Vortex CFD Vortex

location x/Lpp


Overall USN UD UV

y/Lpp z/Lpp Ux/U 𝝎x*Lpp/

U

ABV

ReFRESCO, 12.7M,

SST2003

0.9368 16.9 1.1 48.6 198.8 66.4

5.5

1.0345 3.9 9.5 60.9 43.0 29.3

1.2332 15.2 10.0 2.7 8.4 9.1

1.3303 16.1 8.8 2.5 16.8 11.1

1.4241 9.4 2.0 8.0 16.9 9.1

1.5362 8.9 20.6 19.4 10.8 14.9

AVG 11.7 8.7 23.7 49.1 23.3


0.9368 19.8 2.7 4.1 164.9 47.9

5.5

1.0345 6.9 11.0 10.6 17.1 11.4

1.2332 9.7 21.6 14.8 25.7 18.0

1.3303 5.9 31.9 13.0 17.7 17.1

1.4241 1.3 20.5 0.5 31.0 13.3

1.5362 2.9 13.0 10.8 64.5 22.8

AVG 7.8 16.8 9.0 53.5 21.7

CFDShip-Iowa,

13.0M, ARS-DES

0.9368 45.5 6.2 98.2 243.9 98.4

5.5

1.0345 23.6 16.0 72.3 11.0 30.7

1.2332 21.3 8.6 29.9 100.4 40.0

1.3303 19.1 7.7 30.9 96.5 38.5

1.4241 12.2 22.6 17.4

1.5362 14.9 38.5 26.7

AVG 22.8 16.6 57.8 112.9 42.0

Edge, 17.4M, EARSM

0.9368 16.9 4.0 15.5 81.8 29.5

5.5

1.0345 3.8 17.4 19.2 35.1 18.9

1.2332 12.2 4.7 7.2 50.0 18.5

1.3303 10.7 11.9 6.7 38.7 17.0

AVG 10.9 9.5 12.2 51.4 21.0

Edge, 17.4M, HYB0

0.9368 8.3 1.2 59.0 151.4 55.0

5.5

1.0345 1.2 16.1 45.5 2.5 16.3

1.2332 20.3 19.5 4.3 23.4 16.9

1.3303 22.3 3.2 2.4 8.1 9.0

AVG 13.1 10.0 27.8 46.3 24.3

Experimental and Numerical Investigations on Flow ... · The viscous flow on ship hulls at 0°...

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