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24th Symposium on Naval Hydrodynamics
Fukuoka, JAPAN, 8-13 July 2002
Unsteady RANS Simulation of a Surface Combatant
with Roll Motion
Robert Wilson and Fred Stern
(University of Towa, USA)
ABSTRACT
An unsteady RANS method is developed to
compute the flow and wave field for surface ships with
general non-linear 6DOF motions. The method is based
on an extension of CFDSHIP-IOWA (a general-purpose
code for computational ship hydrodynamic) to predict
ship motions with large amplitude and non-slender
geometry. The flow solver uses hi~her-order upwind
discretization, PISO method for pressure-velocity
coupling, k-m two-equation turbulence model, free-
surface tracking approach, and structured multi-block
grid systems. As an initial step, unsteady simulations of
a surface combatant with prescribed sinusoidal roll
motion are performed over a range of three frequencies.
The response of the boundary layer and free-surface is
described and quantified using instantaneous results and
Fourier analysis. The hydrodynamic roll moment is
post-processed to give added moment of inertia and roll
damping coefficient, which is compared to previous
experimental measurements and computational
predictions for 2D and 3D rolling bodies. Preliminary
results for the surface combatant without forward speed
and with free roll decay motion are given where the
model is released from an initial angular displacement
and the resulting roll motion predicted. The final paper
will include simulations with larger roll angles where
non-linear effects are important. Also, free-roll decay
simulations with forward speed will be performed and
the effect of ship appendages on roll damping will be
investigated by performing simulations of the 5512
geometry with bilge keels.
1 INTRODUCTION
Recent progress in RANS CFD code
development and application is making simulation
based design an imminent reality. Development of such
a tool will allow the merging of traditionally separate
naval architecture sub-disciplines for resistance and
propulsion, maneuvering, and seakeeping, and when
combined with CFD-based optimization, will likely
revolutionize the ship design process.
Of the three sub-discipline areas, application
of RANS methods to resistance and propulsion is the
most advanced with nearly two decades of
experience. Existing approaches are able to predict
ship resistance with reasonable accuracy as shown
from results for three steady flow test cases at the
recent Gothenburg 2000 Workshop on CFD in Ship
Hydrodynamics, Larsson et al., (2000~. Methods
have recently been applied to optimize hull forms for
a variety of objective functions for ships with steady
forward speed in calm seas (Tahara et al., 2000~.
In comparison, application of RANS methods
to maneuvering and seakeeping is less mature due to
obstacles from unsteady flows, ship motions, complex
environment (e.g., incident waves, wave breaking,
bubbly flow) and increased required computer
resources. Also, methods developed for resistance
and propulsion may not be easily extended into these
areas. Application of RANS methods for steady
maneuvers (e.g., off-design yaw, steady turn) can be
found in Tahara et al. (1998~; Alessandrini and
Delhommeau (19981; and Di Mascio and Campana
(19991. However, application for more complex
unsteady maneuvers is rare and most investigations
rely on motion simulation programs with empirically
derived coefficients.
Typical seakeeping solution techniques are
based on assumptions of small amplitude motions and
potential flow so that the general 6DOF non-linear
equations of motions are reduced to two separate sets
of linear equations (i.e., vertical plane motions
become uncoupled from horizontal plane motions)
and are solved in the frequency domain. Within those
assumptions, predictions show good agreement for
vertical plane motions. For horizontal plane motions,
seakeeping codes based on potential flow methods
model viscous effects by incorporating empirically
derived roll damping data. For example, Taz Ul
Mulk and Falzarano (1994) used a linear frequency-
dependent hydrodynamics model augmented by
empirically derived linear and non-linear roll
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damping data from Himeno (1982) and non-linear roll
and heave/pitch restoring forces. Predictions with these
methods are limited to the range of geometry,
frequency, and operating parameters from the empirical
data and suffer from scale effects. Simulations in the
time-domain have been performed to predict larger non-
linear roll motions, but most suffer from use of potential
flow methods with empirical damping data (Tanizawa
and Naito, 1998~. Thus, there is a critical need for
development of numerical methods for viscous flows
and prediction of large amplitude motions for surface
ships with appendages.
In an effort to develop a physics-based approach,
RANS methods have recently been applied to prediction
of roll motion of oscillating bodies. These methods
have the potential to produce superior results since
effects due to viscosity, creation of vorticity in the
boundary layer, vortex shedding, and turbulence are
naturally included. RANS methods were used to study
the flow around 2D oscillating cylinders by Korpus and
Falzarano (1997), Sarkar and Vassalos (2000), and
Yeung et al. (1998~. Accurate prediction of forces and
moments on a 3D submerged cylinder fitted with bilge
keels and with prescribed roll motion was demonstrated
in Kim (2001~. Prediction of pitch and heave motions
for the Wigley hull and Series 60 cargo ship advancing
in regular head waves with a RANS method was
demonstrated by Sato et al. (1999~. Free roll decay
motion of a barge in calm sea and roll motion in
incident waves were performed using a Chimera RANS
method by Chen et al. (2001~.
Development of unsteady RANS methods within
the ship hydrodynamics group at IIHR Hydroscience
and Engineering Lab (IIHR) enables extensions from
previous applications for resistance and propulsion to
applications for seakeeping and maneuvering. A step-
by-step approach was followed towards this goal by
initially performing unsteady RANS simulations for the
forward speed diffraction (ship advancing in waves but
constrained from motions) and radiation (prescribed
ship motions in calm water) problems. In the former
case, simulations were performed for the Wigley hull
for a wide range of conditions (Rhee and Stern, 2001)
and for the surface combatant, DTMB model 5512
(5512), for medium speed/long wave and high
speed/short wave conditions (Wilson and Stern, 2002)
including comparisons with IIHR towing tank data (Gui
et al., 2002; Longo et al., 2002~. In the latter case,
simulations were performed for a double body 5512
model with prescribed vertical (pitch and heave) and
horizontal (roll) plane motions as part of a DoD
challenge project on 6DOF motions and maneuvering
for surface ships (Kim, 20011.
The objective of the present work is to extend
previous work to unsteady RANS simulations of
general 6DOF ship motions and maneuvering, but
with focus on applications for prescribed and
predicted roll motions for which viscous effects are
predominant. As an initial step towards this goal,
simulations are performed for 5512 with and without
bilge keels for prescribed and predicted roll motions.
Simulations with prescribed motions are used to
investigate the unsteady response of the turbulent
boundary layer, wave, and wake fields to roll motion.
Simulations with free roll decay are used to predict
roll damping and resonant frequencies for the surface
combatant. Simulations are performed using the
RANS CFD code, CFDSHIP-IOWA, which was
shown to be one of the better codes at the recent
Gothenburg 2000 Workshop on CFD in Ship
Hydrodynamics (2000) for the surface combatant test
case (Wilson et al, 2000~. The CFD results will be
used to guide IIHR towing tank measurements, which
are then used to validate simulations in a
complementary fashion. Verification and validation
of simulations follow Stern et al. (2001) and Wilson
et al. (2001) and are based on the detailed study for
steady flow simulations of 5415 (Wilson and Stern,
2002~.
This paper is organized as follows. In Section
2, the RANS solution methodology is presented,
while capabilities for prescribed and predicted ship
motions are discussed in Section 3. In Section 4 and
5, a description of geometry, data, conditions, and
grids is given. In Section 6, verification and
validation of the simulations is addressed and results
are presented in Section 7. Finally, concluding
remarks are presented in Section 8.
2 RANS SOLUTION METHODOLOGY
Unsteady RANS simulations are performed
with CFDSHIP-IOWA, which is a general-purpose,
multi-block, high-performance parallel computing
code developed for computational ship
hydrodynamics and applied to surface ships and
complex propulsors without ship motions. The basic
methodology is given in detail in Paterson et al.
(2002), and the reader is referred to this paper when
solution details are omitted. CFDSHIP-IOWA was
recently extended to predict general 6DOF ship
motions with large amplitude, non-linear, non-slender
geometry for viscous flows.
In a mathematical framework, the ship
motions problem is cast as an initial boundary value
problem (IBVP), which is solved using a RANS
solution methodology. Details of the methodology are
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presented in this section, including governing equations,
initial and boundary conditions, and numerical method.
2.1 Governing Equations
In non-dimensional and Cartesian tensor
notation form, the unsteady incompressible RANS and
continuity equations are written for an inertial reference
frame as
computational domain in non-orthogonal curvilinear
coordinates if, it, (,4. A partial transformation is
used in which only the independent variables are
transformed, leaving the velocity components Ui in
Cartesian coordinates. The continuity (1) and
momentum (2) equations in the transformed space are
given by
aUi =0 t~y J by, (6iiUj)=o (5)
axi
-O (1)
at + Uj Oxj = - Ox; + Re Oxjaxj - Oxj a juj + fib; (2)
where Ui = (U. V, W) are the Reynolds-averaged
velocity components, xi = (XY,ZJ are the Cartesian
coordinates, p is the piezometric pressure
~ p+Z/Fr2 ), ujuj are the Reynolds stresses, fb; are
the body-force terms, which represent the effects of the
propeller, Fr = U0/~/~; is the Froude number, and Re
= UoL/v is the Reynolds number. Equations are
normalized by reference velocity UO, ship length L, and
density, p. The Reynolds stresses are related to the
mean rate of strain through an isotropic eddy viscosity
vt
; (aUi aUj ) 2 ,~ k (3)
where did is the Kronecker delta, v' is the isotropic
eddy viscosity, and k is the turbulent kinetic energy.
Substituting (3) for the Reynolds-stress term in
(2), the momentum equations become
auj +U auj _ aP + ~ abut
at ~axj axi Ree~axjaxj
+axj (axj + axj )+fbi
where
P=p+3k
1 1
= _+V.
Reek Re
(4)
Eddy viscosity is calculated using Menter's blended k-
^-e model with the standard model.
The equations are transformed from the physical
domain in Cartesian coordinates (x,y,z,t) into the
aui+ak aUi_ ~ gii 32Ui =
an u ark Red Ii
1 k aP
--hi k +SU
where
(6)
k I bk (U l by a v, ax ) f <7'
sum = R2 (gl2 a{ta;2 + gl3 a3'a¢3
+ 23 a a + - b k
~ Go) ( A)
(8)
. . .
and bt', gel, and J are the geometric coefficients,
conjugate metric tensor, and Jacobian, respectively.
2.2 Initial and Boundary Conditions
Solution of the IBVP requires specification of
initial and boundary conditions. For the unsteady
simulations with prescribed roll motion, initial
conditions are specified using a fully converged
steady state solution for the 5512 geometry with
uniform forward speed and zero roll angle. Details of
the steady state simulation are given in Wilson and
Stern (2002) where detailed verification and
validation is performed to estimate simulation errors.
The steady state solution provides a consistent initial
condition at t=0, after which the prescribed roll
motion is gradually increased to full magnitude over
the first period, thus reducing the time required to
damp initial transients and reach a periodic response.
For simulations with free roll decay, the ship is given
an angular displacement as initial condition and then
released and allowed to freely rotate about the roll
axis.
Specification of boundary conditions are also
required for t > 0. Twenty-seven different boundary
condition types are available in CFDSHIP-IOWA and
can be organized into physical (e.g., no-slip, free-
surface), computational (e.g., inflow, far-field, exit),
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Representative terms from entire chapter:
roll damping
and multi-block (e.g., patched and overset). Only the
free-surface boundary condition and modifications to
boundary conditions required for the ship motions
problem will be discussed in detail here. Figure 1
shows the starboard side of the full computational
domain including coordinate system, grid, and boundary
conditions (body SB; free-surface SFS; far-field SFF; exit
SE; and multi-block SMB) for the simulations of the 5512
geometry.
The free surface boundary conditions are based
upon the exact nonlinear kinematic and approximate
dynamic free-surface boundary conditions, both of
which are applied on the actual free surface. The
kinematic equation is a 2D hyperbolic wave equation
for the wave elevation ~
(~+U(x+V
Applying the temporal and spatial discretizations
given by Eqns. (15) and (16) to the continuous
momentum equations (6) gives
treatment to avoid odd-even decoupling. Fourth-
order artificial dissipation is implicitly added by using
a half-cell operator
YOU, + ~ AnbU, nb = SU ——b, ~e P (17) EP + NP = age Bali (23)
where Aijk and Anb denotes the central and neighboring
coefficients of the discretized momentum equations,
respectively. The source term, Sue, contains the
velocity at the previous time step and the mixed
derivative terms originating from the viscous terms
which are lagged to the previous iteration.
2.3.2 Pressure-Velocity Coupling
The PISO method is used for the pressure-
velocity coupling, which uses a predictor-corrector
approach to advance the momentum equation in time
while enforcing the continuity equation.
In the predictor step, the momentum equation
(17) is advanced implicitly using the pressure field from
the previous time step A'
~ ~ Dn-1
Ai U * ~ A U * S ~ bk or ( 1 8)
nb J ~
where superscript '*' is used to denote advancement to
an intermediate time level. In the corrector step, the
velocity is updated explicitly
Ui = Hi JO bi auk (19)
using a pressure obtained from a derived Poisson
equation and where the psuedo-velocity is defined as
d = _(S —~ ~ U ) (20)
A
Ink nb
A pressure-Poisson equation is derived by taking
the divergence of equation (19)
~ and bi'U,** =-aS~j bind
.~ a; j ( JO i ask )
(21)
and by realizing that the LHS of equation (21) goes to
zero upon convergence
a2;j(~4,,~k ask) ask, ~ i (22)
Because a regular, or collocated, grid approach
is used, solution of equation (22) requires special
where L is the half-cell operator and N is the
operator containing mixed-derivative terms
~ {aid (a §~t ) + 3~2 (a §~2 ) + 3~3 (a333~3 )} (24)
N = J God, (a]23~2 + ai33~3 ) + 342 (2, + a233~3 )
+~3 (3 + a323~2 )}
with 3;i (I) = ((t)i+y2 - (t)i-~/2 ) ' §(i (A = ((hi+ - ¢)i-~ ~ / 2, and
aij = JgiiI47k .
2.3.3 Kinematic Free-Surface Boundary Condition
Solver (KFSBC)
The KFSBC Eqn. (9) is discretized using
Eqns. (15) and (16) and solved on the faces of the
free-surface blocks. Given the solution for wave
elevation A, the volume grid is conformed to the new
wave elevation for each free-surface block through
the use of linear interpolation.
For stability reasons, however, several
numerical concerns must be addressed. First, a
combination of highly clustered near-wall spacing
(i.e., 10-6 or less) and lack of either physical or
numerical dissipation (i.e., for the higher-order
schemes) results in an unstable numerical system.
Secondly, the KFSBC becomes singular at the no-
slip/free-surface intersection due to the "contact line"
problem. At the intersection point, the fluid velocity
is zero due to the no-slip condition and at the same
time the wave elevation on the hull is in motion: an
obvious contradiction.
To address both of these problems, near-wall
blanking and solution filtering are used. An
approximate contact-line model is implemented
wherein the highly-clustered near-wall region is
blanked out for solution of A. Values of ~ in this
region are obtained using linear extrapolation. To
maintain stability and eliminate spurious oscillations,
{is filtered with a sixth-order filter after each
iteration.
Solution procedures for the KFSBC equation
have been shown to be stable and accurate for
simulation of the surface combatant without motions
In calm sea and in regular head waves. However, for
simulations with ship motions, additional damping of
the free-surface is required to stabilize the solution in
the region behind the transom face. In this localized
region, a weighted average of the wave elevation from
the steady solution is used to stabilize the simulation.
2.3.4 Summary
With a description of the numerical method
given, the overall solution procedure is summarized
below for a given time step:
1. Define the six ship motion components as
described in Section 3.
2. Translate and rotate all computational blocks using
components from step 1 and conform the free
surface blocks to the ship hull and current wave
elevation.
3. Solve the KFSBC Eqn. (9) and conform grid to the
wave field.
4. Calculate transformation metrics including grid
velocity terms mica .
5. Solve two-equation turbulence model equations and
calculate eddy viscosity.
6. Execute the predictor step by solving the
momentum Eqn. (181.
7. Execute the corrector step by solving the pressure
Eqn. (23) and correcting the velocity field using
Eqn. (19~. The corrector step is repeated until
velocity corrections are negligible. For the
unsteady roll simulations, five corrections were
used to reach convergence for the corrector step.
8. Post-process results and output to file for
diagnostic and visualization purposes.
9. Advance to the next time step.
The method is fully implicit and therefore
requires iterative solvers. Currently, a line-ADI scheme
with pen/a-diagonal solvers and under relaxation is used
to solve the algebraic equations that arise from
discretization of the momentum, pressure Poisson, and
KFSBC equations. To achieve time-accurate
simulations, the iterative solvers are run until a pre-
selected convergence criterion is met. The tolerances
for solution of the KFSBC in step 3, the momentum
equation in step 6, and the pressure equation in step 7
are 1x10-6, 1x10-5, and 1x10-5, respectively. On average
5, 10, and as many as 500 iterations are required to
reach convergence for solution of the KFSBC,
momentum, and pressure equations, respectively. The
simulations were performed using 24 processors of the
SGI Origin 3000 parallel machine at ARL and
typically required 800 total CPU hours to simulate
three periods of rolling motion.
3 SHIP MOTIONS
Ship motions are either prescribed from an
input file or predicted by integrating the equations of
rigid body motions as described in this section.
3.1 Prescribed Motions
For simulations with prescribed 6DOF ship
motions, the time history of the translation of the
center of rotation (OR' YR, ZR) and the ship orientation
is specified from an input file. For simulations
presented here, only pure roll motion is considered so
that the grid coordinates at time level 'n+1'
(xn+~,yn+7,zn+~) are computed from the base
coordinates (XB,YB,ZB) using a solid body rotation
inn+! = XB
Y YR + ~ YB YR ~ COS (,1?')—(\ ZB—ZR ~ Sin (~)
zn+l = z + (I YB—YR ~ sin (mat) + (\ZB ZR N) ~ ~
A sinusoidal function is used to specify the roll angle
O(t) = A sin (2~ f t + ~) (26)
where A, f = 1/:, a, and ~ are the amplitude, non-
dimensional frequency, non-dimensional period, and
phase angle of the roll motion, respectively.
3.2 Predicted Motions
Prediction of ship motions is accomplished
through development of software modules to solve
the full, non-linear rigid body equations of motion for
the ship trajectory and orientation. Prediction of fully
coupled motions will require flow solution in a ship-
fixed, non-inertial reference frame where ship
geometric properties (e.g., moments and products of
inertia) are independent of time. For this case, the
dynamic rigid body equations in the ship-fixed
reference frame must be supplemented with kinematic
equations which relate velocities and angular rates in
the body-fixed frame with ship trajectory and
orientation is the earth fixed frame. However, for
pure roll simulations considered here, these issues do
not arise so that an inertial coordinate system is used
(i.e., the grid coordinates are a function of time).
Roll motion is predicted by numerically
integrating the rigid body equation of motion where
the time rate of change of angular momentum about
the roll axis is balanced by external roll moments L
acting on the ship
..
L=Ix~
where Ix is the moment of inertial about the roll axis and
the roll moment L is composed of contributions from
hydrodynamic pressure and friction buoYancY. and shin
weight.
art---,, A-- - -- r
4 GEOMETRY, DATA, AND CONDITIONS
4.1 Geometry
Unsteady RANS simulations are performed for
the Model 5512 geometry which is representative of a
modern full-scale naval combatant with sonar dome and
transom stern (Fig. 21. Geometric properties for this
model are given in Table 1. Currently, the moment of
inertia about the roll axis Ix has not been measured
experimentally. To perform free roll decay simulations
a value of Ix = 5.75x10-5 is used until an experimentally
measured value is available.
Table 1. Geometric properties of Model 5512.
Parameter Model 5512
Beam/length, B/L 0.1266
Draft/length, T/L 0.0702
Wetted surface area, S/L2 0.1475
Displacement, S/L3 4.502x10-3
Block coefficient
CB = V I (LBT) 0.5060
Fig. 2 Model 5512 surface combatant geometry.
4.2 Data
For the steady flow over the 5512 without ship
motions, a large experimental database exists for the
surface combatant, which was procured as part of an
international collaboration between IIHR, Istituto
Nazionale per Studi ed Esperienze di Architettura
Navale (INSEAN), and DTMB. Facility and scale
effect biases were evaluated for Fr=0.28 (design
condition) through overlapping tests at all three
institutes as documented in Stern et al. (20001.
Recently, this collaboration has been extended to
perform experiments for the surface combatant with roll
motion where forces and moments, wave elevation,
mean velocity, turbulence quantities, and ship motions
will be measured. Most of the roll experiments are in
the planning phase and currently there is little
available data to compare with the present CFD
simulations. For the final paper, simulation results
will be compared with all available experimental
data.
4.3 Conditions
Unsteady simulations are performed in calm
water with roll motion. Flow conditions for the
unsteady simulations are based on EFD steady flow
experiments for the 5512 geometry with Fr=0.28 and
Re=4.65xl06. Simulations are performed with the
ship in the static orientation.
Simulations are performed over a range of
three rolling frequencies where the roll center of the
model was fixed at the design waterline and ship
symmetry plane, i.e., (OR' OR, ZR) = (0,0,01. Initially,
two frequencies were selected with the lower non-
dimensional frequency JO having a roll period equal
to the ship time scale T (i.e., T=L/UO and based on
ship length L and forward speed UOJ and the higher
frequency f=2 selected to be twice that of the lower
frequency. Subsequently, the roll natural frequency
of the 5512 geometry was estimated to be roughly
f~1.18 by scaling the full-scale appended ship value
reported in Bishop (1983), which is intermediate
between the originally selected lower and higher
frequencies. As a result, simulations performed at
these three frequencies will be referred to as the sub-
resonant, resonant, and super-resonant cases as shown
in Table 2. All simulations are performed with roll
amplitude and phase angle, A=5 degrees and B=0,
respectively.
Selection of final frequencies for simulations
with prescribed roll motion will be based on CFD
predictions and EFD measurements of the resonant
frequency for 5512 geometry with free roll decay
motion.
Table 2. Conditions for unsteady roll
simulations for Model 5512.
Sub-
resonant
Resonant
Super-
resonant
1.0
1.0
1.18
0.85
2.0
0.5
5 COMPUTATIONAL GRIDS AND TIME STEP
The computational grid for the unsteady roll
simulations is based on the coarsest of three grids used
in a study to assess simulation errors and uncertainties
for steady free-surface flow around the surface
combatant. Details of the verification and validation
study and generation of the three grids are given in
Wilson and Stern (2002~. A brief description of the
three grids and relationship to the grid used for the
unsteady roll simulations is provided here.
5.1 Verification and Validation of Steady Flow
Steady flow solutions for the surface combatant
were performed in Wilson and Stem (2002) so that the
flow was symmetric about the ship centerline (i.e., only
the starboard side of the ship was simulated). The finest
grid was generated using the commercial code
GRIDGEN and consists of a hyperbolically generated
near-hull grid and algebraic far-field grid (Fig. 11. Grid
topology was selected so that the block fixed to the
transom face could be conformed around the bottom
edge of the transom face. A coarse-grain parallel
approach was used where the base grid is decomposed
into 24 blocks of varying sizes and each block was
mapped to a separate processor on an Origin 2000/3000
machine. The decomposed grid contained 12 free-
surface blocks that are dynamically conformed to the
wave elevation and ship hull and 12 keel blocks that do
not require conforming. For systematic grid refinement
with rG = ~/i, the coarsest grid is obtained simply by
removing every other point from the fine grid.
5.2 Unsteady Roll Simulations
The grids used for the current unsteady roll
simulations and the coarsest of the three-grid study
contain the same grid number and distribution in the
axial and transverse computational directions. For the
keel to free-surface direction, the keel and free-surface
blocks of the steady coarse grid are joined to improve
the grid quality during the conforming process while the
ship is rolling. The grid is then mirrored about the free-
surface plane to provide an adequate amount of grid for
the conforming process as described in Section 2.3.3.
The grid is also mirrored about the ship centerline since
both port and starboard sides are required to simulate
the unsteady asymmetric flow due to the roll motion.
Modifications to the grid number in the keel to free-
surface direction and mirroring processes results in a
24-block grid system with 0.8M total grid points.
5.3 Time Step
The time step is selected so that the temporal
evolution of each period of unsteady motion is resolved
with 100 time steps, i.e., At = a/ 100, where ~ is the
period of the roll motion.
6 VERIFICATION AND VALIDATION
Verification and validation of simulations
follow Stern et al. (2001) and Wilson et al. (2001)
and will be based on the detailed study for steady
flow simulations of 5415 (Wilson and Stern, 2002~.
7 RESULTS
Discussion and analysis of unsteady results for
the surface combatant with prescribed sinusoidal roll
motion are provided in this section for boundary layer
(Sect. 7.1), free-surface (Sect. 7.2), and forces and
moments (Sect. 7.3~. Preliminary results for 5512
with free roll decay are given in Sect. 7.4.
Figure 3 shows the time history of the
prescribed roll angle and angular velocity normalized
by maximum values. During the first period and a
half (0 < t < 1.5~), the amplitude of the roll angle is
gradually increased to its maximum value of A=5
degrees. The resulting solution undergoes a transient
response for the first two periods of prescribed
motion, after which the transients decay and a
periodic response is achieved for t > 2T. Results are
analyzed and presented for one typically cycle of the
periodic response (i.e., 2T < t < 3r ). Time sequences
at each quarter-phase are used to describe the
unsteady results with quarter-phases indicated in Fig.
3. Time sequences of instantaneous results and
harmonic analysis are used to quantify the unsteady
response of the boundary layer and wave-field to the
roll motion. The effect of roll frequency is
investigated by performing simulations at three
frequencies, although most of the focus in this section
is on the sub- and super-resonant cases.
In
._
ct
a)
._
by
o
._ O
._
ct
~ -1
A
i Roll Angle, 4)/(P'MAX
-1~ Angular Velocity, dO/dt
Time, t/:
Fig. 3 Motion kinematics for unsteady 5512
simulation with quarter-phase indicated by symbols.
7.1 Boundary Layer
Since the volume grid is undergoing both solid-
body rotation about the roll axis and deformation (due
to grid conforming to free-surface and ship), the
solution is transformed from the inertial reference frame
to the ship-fixed reference frame and interpolated onto a
grid that is fixed in time for analysis purposes. This
procedure provides a time history for the flow field at
fixed spatial locations so that the harmonic content can
be analyzed using Fast Fourier Transforms.
Figures 4 and S show a time sequence for axial
velocity and vorticity with roll motion for the super-
resonant case. In the figure, the ship is advancing from
bottom to top with under free-surface perspective. The
rolling motion of the ship induces cross flow velocity
resulting in unsteady asymmetric axial velocity contours
(Fig. 4) and a serpentine motion of vorticies emanating
from the sonar dome at ~c/L=O.1 (Fig. S). These
unsteady features are in contrast to the steady state
solution without ship motions, which shows symmetric
and axial velocity and sonar dome vortices with straight
vortex cores.
The wavelength Jv and period tv for the motion
of the sonar dome vortices is estimated from the product
of the forward speed of the ship UO and period of the
roll motion ~ (i.e., by = UOT ). The motion is
analogous to a vibrating string with fixed ends (i.e., an
oscillating standing wave with zero displacement at
nodal points and maximum displacement between nodal
points). The wavelength Jv corresponds to the distance
between nodal points, as shown below.
The effect of roll frequency is examined by
comparing instantaneous results for the super-resonant
case (Fig 4a and Sa) with that from resonant and sub-
resonant cases at t=~/4 as shown in Fig. 6. The
results show the presence of nodal points at x/L=0.5,
0.85, and 1.0 for super-resonant, resonant, and sub-
resonant cases, respectively which correspond to
wavelength estimates, As = UOT .
Contours of first and second harmonic
amplitudes from a Fourier analysis of the axial velocity
are shown in Fig. 7 for super- and sub-resonant cases.
For both cases, the results show minimum amplitudes
near nodal points and maximum amplitudes midway
between nodal points. The analysis shows that higher
harmonics (i.e., third and higher) are at least one order
of magnitude smaller than first harmonic amplitudes
and thus negligible.
The unsteady evolution of the sonar dome
vorticies is analyzed by tracing the port and starboard
vortex core locations during one period of motion as
shown in Fig. 8. A harmonic analysis of the
transverse vortex core locations shows axial locations
for amplitude maximums and minimums (Fig. 9),
which are consistent with those for axial velocity.
Results for first harmonic phase angle shown in Fig. 9
are used to quantify phase lags in the boundary layer
vortex and is consistent with wavelength estimates
(i.e., the phase angle completes one cycle from O to
2~ within one spatial wavelength).
7.2 Free-surface
Instantaneous total g and fluctuating
~ - {M~ wave elevation at quarter phases are shown
in Figs. 10 and 11, respectively. Recall that at
t = if/ 4 ~ t = 3~/ 4 ), the ship has rolled to maximum
positive (negative) roll angle as shown in the Fig. 11
insert. Generation and propagation of fluctuating
wave contours are described in terms of viscous and
pressure effects at the hull surface due to the roll
motion.
Due to the viscous no-slip condition, the
rolling motion of the ship induces a cross-plane
velocity VNO_S,IP at the ship hull, which is in phase
with angular velocity d ~?/dt (Fig. 3)
-
VNO_SLIP= d~ xr (28)
where r is the position vector from roll axis to the
point on the hull. The magnitude |VNO-5LU| is given
|VNO-SLIP | = | al |~|r| = A0)|r|cos (cot) (29)
with roll angle ~ from Eqn. (26~. Therefore,
viscous effects generate a vertical velocity WNO_SLIP at
the intersection of the ship hull and free-surface that
is a wave elevation source on one side of the ship and
a sink on the other, depending on the phase of the
motion. For example, a source exists on the port side
for theinterval, r14
~ `: - A ~ (d) t=t |
~ ' -,,)_ .,
a_ '~_
~ ,! _
_ I'',,.
_ ;~ ''
I,.. 1'—~
Fig. 4 Time sequence of axial velocity contours U for 5512 simulations with prescribed roll motion (f=2J:
(a) t=~4; (b) t=~2; (c) t=3~4; and (d) t=r
_
Few
Fin
~ :
-
-
he
[ (d) to= |
it. ...
he, .. ..
- ....
.....
as:
.
5.00
9.00
3.00
3.00
-9.00
5.00
Fig. 5 Time sequence of axial vorticity contours (x for 5512 simulations with prescribed roll motion (f=2):
(a) t=~4; (b) t=~2; (c) t=3~4; and (d) t=z:
| (a) U(,~l
~ \W'~
.1&
~ ': ;~
~ - 1 ~ ~
~ l_ ;~ ~
tic' u t=T/4
~ i,
i' v_...:
en'
be.;
_
~,.J
in'
Fig. 6 Instantaneous axial velocity U and vorticity {x at t= ~4 for
(a), (b) resonant case (f=1.18) and (c), (d) sub-resonant case ~1~.
-
, ~
I(c)W
-
-
a
~ 0.1`
0.1(
O.O'
~ 0.0,
En o.o~
0.0'
~ 0.0
— 0.0
. ~ ., _ .
~ (d)(X,t=~/41
...,..,,,_
''' - 'I
I'''
me,.
- 1'
z
_ r
- ~
. _
_~U(2)
A_ Z
_]
_
l_
E_
_.
_ .
_ _
At.. ..........
0.12
0.10
~ o.os
EN 0.07
~ 0.06
1~1 0.04
0.03
0.01
Fig. 7 Harmonic analysis of axial velocity for (a), (b) super-resonant case ~2) and (c), (d) sub-resonant case (f=11.
(a), (c) first harmonic and (b), (d) second harmonic amplitudes.
0.04 _
~0
t=0.2T
.— — — · t=0.4T
t=0.6T
t=0.8T .—- _
t=T ,~ .: _
S'~A
- (a?
~ MA
-u us O2 ORAL 0.6 0 8
0.04 _
:
~,0
On4 , , ,
-.- 0.2 04xIL 06 o 8
Fig. 8 Time sequence for transverse location of sonar
dome vortex cores for (a) super-resonant case ~-2)
and (b) sub-resonant case ~1~.
0.02
0.01~
O v
0 0.2 0.4 0.6
Axial Distance, xlL
· Amplitude (f=2)
- - - - - - Phase Angle (f=2)
· Amplitude, (f=1)
- - - - - - Phase Angle (f=1 )
In<
- _ ~ ~ /
540
450
360
270
180
90
Fig. 9 Harmonic analysis of transverse location
of sonar dome vortex cores.
Examination of a time sequence at one typical
ship cross-section shows generation of waves due to
frictional and pressure effects. Focusing on the
starboard side, a wave crest is generated during the
first quarter period due to viscous effects since the
starboard side is rolling up and generating positive
vertical velocity at the free-surface. During the
second and third quarter period, the roll motion
reverses and pressure effects cause the crest to
propagate away from the hull in the transverse
direction, during which time viscous effects act to
create a wave trough at the free-surface, which
propagates in the transverse direction during the
fourth quarter period. Waves are generated in a
similar manner on the port side, but are 180 degrees
out of phase with the starboard side. The importance
of viscous effects is mainly determined by Re and roll
speed |VNO-SIU | which depends on roll amplitude A,
frequency lo, and cross-sectional size through r as
shown by Eqn. (29~. Importance of pressure effects is
determined by the transverse displacement of fluid at
the free-surface, which depends on roll frequency and
shape of the cross-section at the free-surface.
The above discussion applies to 2D rolling
bodies, while the problem considered here is more
complicated due to the effects of varying cross-
sectional shape with axial distance, presence of
transom stern, and forward speed, which generates a
steady wave system. Excluding the transom region,
the roll motion and variation in cross-sectional shape
results in a transverse propagating wave system
traveling upstream and originating on the forebody
(0.3 < C/L < 0.5) as shown in Fig. 11.
To facilitate a harmonic analysis of the free-
surface, the wave elevation is interpolated at every
time step onto a 2D grid which is fixed in time. A
Fourier analysis is performed on time histories at
each point to quantify the response of the wave-field
to the roll motion. Results from the harmonic
analysis of the wave elevation are shown in Fig. 12
where first harmonic amplitude and phase angle are
given for the super- and sub-resonant cases. First
harmonic amplitude in Fig. 12a shows maximums on
the forebody due to the upstream traveling wave
system as discussed above. The gradient of phase
angle contours V(3 in Fig. 12b is used to indicate
wave propagation direction and speed, in the same
manner that the temperature gradient is used to
visualize heat flux vectors. Since the gradient is
mainly increasing in the transverse direction, the
figure confirms that waves propagate from the ship
Fig. 10 Wave elevation contours ~
for 5512 simulations with prescribed
Fig. 11 Fluctuating wave contours,
~ - (M~ J=21: (a) t=~4; (b) t= ~2;
roll motion (f=2): (a) t=~4; (b) t=~2; (c) t=3~4; and (d) to=: Cross-section
(c) t=3~4; and (d) t= ~ through sonar dome showing ship
orientation (insert).
1 ~
1 -360-300-240-180-120 60 0 60
-0-~
- 1 (b) 7(1) 1
l l l
Fig. 12 Harmonic analysis of wave
elevation for (a),~b) super-resonant
case ~2) and (c),gd) sub-resonant
case (f=1~. First harmonic amplitude
A(1J (a),~c) and phase angle 71J (b),td).
hull towards the far-field, except at the forebody
(~/L=0.25), where phase angle contours are curved
and more closely spaced indicating upstream
propagation at larger speed. Examination of higher
harmonics shows that the free-surface response is
largely first harmonic at the roll frequency. Comparison
of first harmonic amplitude contours for super- (Fig
12a) and sub-resonant (Fig 12c) cases show similar
peak values at the forebody (x/L=0.251. In comparison
with the super-resonant case, the second region of peak
amplitude (0.3 < C/L < 0.5) is absent for the sub-
resonant case and a large region of near-zero amplitude
exists on the afterbody (0.5 < C/L < 1.0) near the hull.
Comparison of phase angle contours show straight
contours for the sub-resonant case (Fig. 12d), indicating
uni-directional wave propagation at an angle of 23
degrees to the ship centerline, in contrast to curved
contours for the super-harmonic case (Fig. 12b). Also,
phase angle contours indicate wavelengths of 0.2L and
0.4L for super- and sub-resonant cases, respectively.
7.3 Forces and Moments
Elemental hydrodynamic pressure and frictional
forces and moments about the roll axis are integrated
over the hull surface at each time step to give a time
history of resultant forces and moments. Total roll
moment MT is shown in Fig. 13 as well as frictional and
pressure components (e.g., MT=MF+MP). Figure 13
shows that the total roll moment is dominated by the
pressure component for both cases (i.e., MF < 5%MT)
and a Fourier analysis of MT shows first harmonic
response. Larger roll angles will be required to excite
higher harmonics with non-linear response. For the
super-resonant case, the pressure and frictional
components lag the roll motion by 22 and 76 degrees,
respectively, since the maximum roll angle occurs at 90
degrees of phase. Trends for the sub-resonant case are
similar with a roll moment amplitude of 45% of the
super-resonant value.
-
c
Velocity and pressure are interpolated onto
equally spaced y-z planes so that the cross-sectional roll `~
moment MT can be computed to show the contribution O
of each station to the total roll moment MT. This
procedure gives a time history for the roll moment at
each axial station which is analyzed to yield Fourier
amplitude and phase angle as shown in Figs. 14a and
14b, respectively. The first harmonic amplitude shows
large peaks at the front (~=0.01) and rear ('c/L=0.08)
of the sonar dome, with contributions from the sonar
dome (O
Although the approach presented here is
applicable to fully non-linear ship motions, traditional
solution procedures for seakeeping problems use linear
assumptions so that the problem can be split into the
two separate problems of forward speed diffraction and
radiation. For the radiation problem with linear
assumptions, the response of forces and moments are
assumed to be purely first harmonic and proportional to
the motion and its first and second time derivative, with
the expression for pure roll given by
.. .
MT = -A44~? - B44~ (30)
The constants of proportionality for the first and second
derivative terms in Eqn. (30) are referred to as the roll
damping B44 and added roll moment of inertia A44,
respectively. To facilitate comparison with EFD and
CFD results for 2D and 3D rolling bodies, the total
hydrodynamic roll moment is transformed to the
frequency domain where the imaginary MT, and real
MTR parts are used to compute the A44 and B44 values,
respectively
A44 = ~ MT
B. - ~ M
Normalized coefficients are given by
(32)
a = A44 `33'
b = B44 ~ `34>
where V and B are the ship displacement and beam,
respectively. Normalized values are shown in Fig. 15
for the three frequencies along with comparison of the
current results with those from EFD and CFD for 2D
and 3D rolling bodies. In Fig. 15a, added moment
values from the present CFD results and experimental
measurements from Tanaka, et al. (1983) for a 3D
fishing vessel show large differences when compared to
results for 2D rolling bodies. Present values for the
surface combatant and measurements for the fishing
vessel show a similar trend of decreasing am with
frequency. Roll damping coefficients for the fishing
vessel, cargo ship, and surface combatant all show
increased damping with frequency. However, large
differences in value and slope exist between the three
geometries. Experimental measurements for the 5512
geometry will be required to quantitatively validate the
present results.
The cross-sectional roll moment MT was also
post-processed to give cross-sectional added moment
A A
A44 and roll damping B44 coefficient with
normalized values given by a44 and b44 . Normalized
values are shown in Fig. 16 for the super- and sub-
resonant cases. The results show positive roll
damping over most of the ship with cross-sectional
values consistent with those for total roll damping
except at the rear of the sonar dome. Negative
damping occurs at this location since the pressure
component leads the roll angle by 90 degrees as
discussed above.
~ G ~ t::1
O Young (1975), 2D Square Body (Expt)
~ Young (1975), 2D Square Body (Inviscid)
_ Sarkar (2000), 2D Square Body (RANS)
, ~: Tanaka (1983), 3D Fish. Vessel (Expt)
_~ Present, 3D Surface Combalam (PANS)
OCR for page 125
7.4 Free Roll Decay
Preliminary results for simulation of the surface
combatant without forward speed and with free roll
decay are given in this section. To simulate and predict
roll decay motion, the ship is given an angular
displacement as initial condition and then released and
allowed to freely rotate about the roll axis. The ship
restoring moment damps the resulting sinusoidal
motion, which eventually decays to a steady state
condition. Figure 17 shows the time history for the total
roll moment and predicted roll angle after the model is
released from an initial angular displacement of 5
degrees at t=0.
~ 2
-
-
<5:
o
~ -2
_A
6
1 _
Roll Angle
O.- ~ Roll Moment _ 0.001
O E
_
0
_
o
it, ,vv~~ ~ .
T! _ ~
\'v'' ':.,'jl
3;" I'
'' I '--
_ ·,', \ ~
ail+.+ , ~
'hi
V, rj ~ '~
·'~'~,, of%<,<
.~ %~' _
:~
!< ~ ~~
-0.001
-6, ~ O Time, VT 1 5
Fig. 17 Time history of predicted roll angle
and moment from free roll simulation.
8 CONCLUDING REMARKS
A RANS based method for prediction of flow
around ships with rolling motion was presented in this
paper. Unsteady simulations are used to investigate the
response of the boundary layer and wave-field to rolling
motion. The effect of roll frequency was demonstrated
by performing simulations at three frequencies
corresponding to estimates of sub-resonant, resonant,
and super-resonant natural roll frequencies for the 5512
geometry.
It was shown that the rolling motion of the ship
resulted in an unsteady oscillatory motion of the
boundary layer dependent upon position along the axial
direction. The unsteady motion was described using a
standing wave analogy with reduced transverse
displacement at nodal points and maximum
displacement between nodal points. The distance
between nodal points (i.e., the wavelength) was found to
be dependent on the product of the forward speed and
roll period. Analysis of the free-surface showed that the
fluctuating wave pattern was dominated by upstream
traveling waves with amplitude, direction, and velocity
dependent on rolling frequency. Time histories of axial
force and roll motion show linear response to roll
motion at this modest roll amplitude (A=5 degrees).
Analysis of the hydrodynamic rolling moment was
used to compute added moment of inertia and
damping from a linear theory allowing the present
results to be compared with computational and
experimental values for other 2D and 3D rolling
bodies.
Preliminary results were presented which
demonstrated prediction of free roll decay motion of
the surface combatant without forward speed. The
final paper will include simulations with larger roll
angles where non-linear effects should be important.
Free-roll decay simulations with forward speed will
be performed and the effect of ship appendages on
roll damping will be investigated by performing
simulations of the 5512 geometry with bilge keels.
9 ACKNOWLEDGEMENTS
This research was sponsored by Office of
Naval Research grant N00014-01-1-0073 under the
administration of Dr. Patrick Purtell. The authors
would like to acknowledge the DoD High
Performance Computing Modernization Office and
the DoD Challenge Program. Simulations were
performed at the Army Research Lab Major Shared
Resource Center using both the SGI Origin 2000 and
3000 machines.
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DISCUSSION
Ernie O. Tuck
University of Adelaide, Australia
In your forced-roll cartoon of the fluctuating part
of the disturbance, the waves just seem to
disappear as they approach the edge of the
computational domain. Should they not continue
until they reach (and pass across or be reflected
by) the edge of the computational domain? This
would surely be so unless the viscosity is being
overestimated relative to the real-world value, so
that too large a damping of waves occurs. In real
life such waves would not be so strongly
damped.
AUTHORS' REPLY
The focus of the current paper is on accurate
prediction of near field viscous and pressure
forces, which is required for accurate prediction
of general 6DOF ship motions. Resolution of the
turbulent boundary layer at the ship hull requires
fine grid spacing in the wall normal direction
with expansion of the grid towards the far-field
(see Fig. 1), leading to some damping of the
waves as they approach the far-field. In Wilson
and Stern (2002b), verification and validation is
performed for the current application to access
numerical error and uncertainty by performing
simulations on three systematically refined grids.
The results show that the current grid accurately
resolves the free-surface in the near-field, with
uncertainties for the wave profile on the hull of
less than 1% of the bow wave height and 5.5% at
a transverse wave cut at 0.1 7L from the ship hull.
Thus, grid resolution in the far-field must be
sacrificed to accurately resolve the viscous
forces at the ship hull and to predict general
6DOF ship motions.
