Sep 22, 2006 - N62306-01-D-7110 is also acknowledged. Contribution of figures by Claus. Simonsen and solutions by Ron Miller are deeply acknowledged.
26th Symposium on Naval Hydrodynamics Rome, Italy, 17-22 September 2006
A Dynamic Overset, Single-Phase Level Set Approach for Viscous Ship Flows and Large Amplitude Motions and Maneuvering
P. M. Carrica1, R.V. Wilson1,2, R. Noack3, T. Xing1, M. Kandasamy1, J. Shao1, N. Sakamoto1 and F. Stern1 (1The University of Iowa, 2currently The University of Tennessee at Chattanooga, 3 The University of Alabama at Birmingham) ABSTRACT A method to handle large amplitude ship motions in free surface viscous flows is presented. The method relies on dynamic overset grids to accommodate motions and single-phase level set to compute the free surface flow. Overset grid connectivity is found at run time using the code SUGGAR, allowing implicit coupling between fluid flow and body motions. Higher order solvers are used to compute mass and momentum conservation, as well as the level set function used to detect the air/water interface. Geometric/implicit reinitialization methods are used to maintain the level set function as a distance function. A six degree of freedom (6DOF) module was implemented that allows for explicit or implicit computation of predicted motions. Prescribed/predicted motions combinations are also possible, allowing for the simulation of towing tank experiments such as Planar Motion Mechanism (PMM) tests. These methods have been implemented in the code CFDShip-Iowa, version 4. We present code results for several problems including examples from the traditional areas of naval architecture of resistance and propulsion, seakeeping and maneuvering. INTRODUCTION The simulation of the viscous flow for ships under large-amplitude motions is a challenging problem that is attracting increasing interest. Experiments are expensive and difficult, and often not feasible. Seakeeping and maneuvering at high speeds, mainly in rough seas, lead to motions conditions that are important for ship performance and structural design. The concept of Simulation Based Design (SBD), in which Computational Fluid Dynamics (CFD) complements experiments and overall uncertainty analysis, is becoming more feasible as CFD tools become faster and more reliable. The objective of a
“numerical towing tank,” that would complement experimental towing tanks, has been pursued over the past years with mixed success. Of the traditional areas of naval arquitecture, resistance can in general be appropriately tackled using current standard RANS capabilities. Static, steady-state computations, in which the ship is steady throughout the computation either in static or dynamic orientation, have been performed by a significant number of researchers, with a notable improvement on the computational capabilities over the past 10 years (Larsson et al., 2003; Hino, 2005). In general, resistance and powering characteristics require the simultaneous computation of the sinkage and trim of the ship, which requires the computation of the ship’s motions. Seakeeping and maneuvering problems are more complex, and, with the exception of prescribed steady turn and drift, and the forward speed diffraction problem, require motions capabilities. Seakeeping and ship response in a seaway require 6DOF capabilities to predict the ship rigid body motion. The same can be said for maneuvers, in which case some degrees of freedom may be imposed and others predicted. This has been recognized early by designers and researchers, and a variety of computational tools are available to compute loads and responses in waves. However, as the 2005 Seakeeping Committee of the International Towing Tank Conference (ITTC, 2005) states in the Final Report and Recommendations, “seakeeping computations are still far from a state of mature engineering science.” Most of the computational tools available today for ship motions are based on potential flow solvers (Beck and Reed, 2001), but there have been a few successful efforts in solving viscous ship hydrodynamics with motions. Most of these methods suffer from limitations restricting the applications to small amplitude motions.
Examples of computations of viscous free surface flows with ship motions include roll decay computations (Wilson et al., 2006a) and pitch and heave response in regular head waves (Weymouth et al., 2005) using free surface tracking approaches, and pitch and heave in regular head waves (Hochbaum and Vogt, 2002; Klemt, 2005; Sato et al., 1999; Orihara and Miyata, 2003) using surface capturing approaches. Other than the present authors only Orihara and Miyata (2003) and Klemt (2005) use dynamic overset technology for ship motions, though in the aerospace community codes that handle motions with overset grids are more common. The computational fluid dynamics code CFDShipIowa, developed at the University of Iowa for support of student thesis and project research at IIHR as well as transition to Navy laboratories, industry, and other universities, relied on its previous versions on a surface tracking technique to compute the free surface elevation (CFDShip-Iowa versions 3 and earlier, Paterson et al., 2003). This limited the computation capabilities to situations in which the free surface remained smooth, and consequently to small amplitude motions. To overcome these limitations and to increase the robustness necessary to handle large-amplitude motions a totally new approach has been implemented to model the free surface (Carrica et al., 2006a; 2006b) with static overset grids (Wilson et al., 2006b) and to simulate motions with dynamic overset grids (Carrica et al., 2006c). Details of such implementations and a description of the numerical methods are presented in this paper, along to other unpublished methods, including irregular waves and turbulence models. The resulting code is CFDShip-Iowa version 4. MATHEMATICAL MODELING Details of the mathematical and numerical methods used in CFDShip-Iowa have been published elsewhere. We summarize here the most important aspects of the mathematical and numerical models and add also some previously unpublished details. The mass and momentum conservation equations for an incompressible fluid using the RANS approximations with isotropic turbulence are written as: ⎡ 1 ⎤ ∂v + v ⋅∇v = −∇p + ∇ ⋅ ⎢ ∇v + ∇vT ) ⎥ + S ( ∂t ⎣⎢ Reeff ⎦⎥
(1)
∇⋅v = 0
(2)
where the dimensionless piezometric pressure is: p=
pabs z + ρ U 02 Fr 2
(3)
and the Reynolds and Froude numbers are defined with the free stream velocity U 0 and the ship length L as: Reeff =
U0 L
ν +ν t
; Fr =
U0 gL
(4)
with ν t the turbulent viscosity. The level set function φ models the free surface. Since it is a passive scalar: ∂φ + v ⋅ ∇φ = 0 ∂t
(5)
φ is a distance function positive in water and negative in air, and therefore requires reinitialization since convective transport will not maintain φ as the distance to the interface (Adalsteinsson and Sethian 1999). The isosurface φ = 0 defines the free surface. The reinitialization is done geometrically on grid points that are first neighbors to the interface, and by solving n ⋅∇φ = sign (φ0 )
(6)
on the rest of the points, with φ0 the level set function before reinitialization and n the unit normal to the free surface, computed as n = ∇φ ∇φ . Dynamic free surface boundary conditions to Eq. (1) are enforced by setting zero normal velocity gradient and atmospheric pressure at the free surface: ∇v ⋅ n = 0 z p= 2 Fr
(7) (8)
The turbulent kinetic energy is computed using a blended k-ω / k-ε model (Menter 1994), where the turbulent kinetic energy k and the specific dissipation rate ω are computed from: ∂k 1 + ( v − σ k ∇ν t ) ⋅∇k − ∇ 2 k + sk = 0 ∂t Pk ∂ω 1 + ( v − σ ω ∇ν t ) ⋅ ∇ω − ∇ 2ω + sω = 0 ∂t Pω
(9) (10)
where turbulent viscosity, effective Peclet numbers and sources for k and ω are:
ν t = k / ω , Pk =
1 1 (11) , Pω = 1/ Re+ σ kν t 1/ Re+ σ ων t
sk = −G + β *ω k
(12)
sω = −γ
ω
G + β *ω 2 − 2 (1 − F1 ) σ w 2
k G = ν t τ : ∇v
1
ω
∇k ⋅∇ω
(13) (14)
Details are given in Shao (2006). The momentum equations remain the same with one additional term ex ∂ aij( ) k ∂x j included in the source to account for the
(
)
( ex )
In Eq. (13) F1 is the blending function, that switches between the k-ω model near the wall to the kε model on the free-stream region. The constants of the model are also a function of F1 . This model keeps the advantages of the k-ε (simple boundary conditions at solid walls) and of the k- ω (independence of the freestream conditions) models. The reader is referred to Paterson et al. (2003) for details on the model and its implementation within the framework of CFDShipIowa. At the free surface, it is customary to use zero normal gradient for both k and ω, which translates in the following conditions: ∇k ⋅ n = ∇ ω ⋅ n = 0
(15)
Eqs. (7) and (15) are solved in air to transport the velocity, k and ω from the interface into the air. This is called an extension with the normal and enforces the zero normal gradient of these quantities. The k- ε/k- ω model is extended to detached eddy simulation (DES). The dissipative term of the ktransport equation is modified as (Travin et al. 2002): k DRANS = ρβ * kω =
k DDES =ρ
k
lk −ω =
ρk3 2
(16)
lk − ω
32
1 ⎛ ⎞ = β 3 ⎜ Ωik Ω kj − II Ωδ ij ⎟ + β 4 Sik Ω kj − Ωik Skj 3 ⎝ ⎠ 2 ⎛ ⎞ + β 6 ⎜ Sik Ω kl Ωlj + Ωik Ω kl Slj − II Ω Sij − IV δ ij ⎟ 3 ⎝ ⎠
(
aij(
ex )
(
+ β 9 Ωik S kl Ωlm Ω mj − Ωik Ω kl Slm Ω mj
1 ⎛ ∂U ∂U j Sij ≡ τ ⎜ i + 2 ⎜⎝ ∂x j ∂xi
II S = S kl Slk ,
)
⎞ 1 ⎛ ∂U ∂U j ⎟⎟ , Ωij ≡ τ ⎜⎜ i − 2 ⎝ ∂x j ∂xi ⎠
⎞ ⎟⎟ ⎠
(18)
III S = S kl Slm Smk
(23)
(
N 2 N 2 − 7 II Ω Q
(19)
with CDES = 0.65 is the DES constant, adopted from typical homogeneous turbulence problems. An anisotropic algebraic Reynolds stress model (ASM) is also available, which is based on a modified version of Menter’s k-ε/k-ω turbulence model as the scale determining model, and an explicit algebraic Reynolds stress model as the constitutive relation in place of the Boussinesq hypothesis: ⎞ 2 ( ex ) ⎟ + kδ ij + aij k ⎟ 3 ⎠
Q
(
),β
6
3
=−
=−
)(
(
12 IV NQ
6N 6 , β9 = Q Q
)
)
(
(
)
)
⎛ A′2 9 ⎞ ⎛ A′2 9 ⎞ 2 2 P1 = ⎜ 3 + II S − II Ω ⎟ A3′ , P2 = P12 − ⎜ 3 + II S + II Ω ⎟ 9 10 3 27 20 3 ⎝ ⎠ ⎝ ⎠ 9 9 ( eq ) A3′ = + CDiff max 1 + β1 II S , 0 5 4
(
β1( eq ) = −
(20)
(
2 N 2 − 2 II Ω
), β
5 2 N − 2 II Ω 2 N 2 − II Ω 6 13 13 ⎧ A3′ + sign P1 − P2 P1 − P2 ( P2 ≥ 0 ) ⎪ 3 + P1 + P2 ⎪ N =⎨ ⎡ ⎛ ⎞⎤ 16 P1 1 ⎪ A3′ + 2 P 2 − P ⎟ ⎥ ( P2 < 0 ) cos ⎢ arccos ⎜ 1 2 ⎪3 ⎢3 ⎜ P2 − P ⎟⎥ 1 2 ⎝ ⎠⎦ ⎣ ⎩
Q=
l% = min(lk −ω , CDES ∆ )
(22)
The model coefficients are function of the invariants in Eq. (23):
β4 = −
β *ω
(21)
⎛ 1 ν ⎞ The time scale is τ = max ⎜⎜ * ; Cτ ⎟ and the * β kω ⎟⎠ ⎝β ω invariants for the strain rate and vorticity tensors are:
β1 = −
12
)
where the non-dimensional strain-rate and vorticity tensors are defined by:
(17)
l%
⎛ ∂U ∂U j ui u j = −ν T ⎜ i + ⎜ ∂x j ∂xi ⎝
:
II Ω = Ω kl Ωlk , IV = Skl Ωlm Ω mk
where the length scale is: k
effect of the extra anisotropic tensor aij
N( ) 5 , 6 N ( eq ) 2 − 2 II Ω eq
(
)
3
)
N(
eq )
= A3 + A4 =
81 20
(24)
NUMERICAL MODELING
The equations were first transformed from the physical ( x, y, z, t ) domain to the curvilinear non-orthogonal computational domain
( ξ ,η , ζ , τ ) .
Details of this
transformation can be found in Thompson et al. (1992). As an example, the resulting transformed level set equation is ∂ x j ⎞ ∂φ ∂φ 1 k ⎛ + bj ⎜U j − =0 ⎟ ∂τ J ⎝ ∂τ ⎠ ∂ξ k
(25)
where x& j = ∂x j ∂τ is the grid velocity and b kj are the metric coefficients. Other equations were transformed similarly. Convective terms were discretized using a secondorder upwind scheme for RANS computations, while third-order upwind biased is used for DES. Diffusion terms were discretized with a second-order central scheme. For the time derivatives a second-order backward scheme was used. Time derivatives near the interface require an special treatment since grid points can change from air to water from one time step to the next and then the time derivative in Eulerian form is not properly defined. CFDShip-Iowa uses a Lagrangian-Eulerian approach for these points, as discussed in Carrica et al. (2006a). na air
pint
water
interpolating along the line joining the points p and na the interfacial pressure is
pna = ( pint − ph )
f bx = Ax r * 1 − r * f bθ = Aθ
(28)
(29)
r 1− r (1 − RH ) r * + RH *
*
(30)
where r* =
r RP − RH 1 − RH
r=
( y − Y _ prop _ center )
Aθ =
h
dist (rna ,int) + pint dist (rp ,int) + dist (rp , rh )
The source term S in Eq. (1) is used to include a propeller model without resolving the detailed flow structures generated by the blades. CFDShip-Iowa includes a prescribed axisymmetric body force with axial and tangential components. The radial distribution of forces is based on the Hough an Ordway circulation distribution (Hough and Ordway, 1964) that has zero loading at the root and tip.
Ax =
p
(27)
Fr 2
and thus the neighbor pressure that satisfies the desired pressure at the interface is
φ =0
η
(1 − η ) z p + η zna
pint =
2
+ ( z − Z _ prop _ center )
2
CT 105 DX 16(4 + 3RH )(1 − RH ) KQ 105 DXJ 2 π (4 + 3RH )(1 − RH )
(31) nw
Figure 1: Interpolation to enforce pressure at the interface.
Pressure and velocity are coupled using a PISO algorithm (Issa 1985). The enforcement of the pressure condition at the free surface, Eq. (8), requires interpolation since the interface generally does not lay on the grid points. This is done following Figure. 1. The relative distance between a grid point in water and the interface is
φp η= φ p − φna
(26)
with CT and KQ the thrust and torque coefficients, J the advance coefficient, RP the non-dimensional propeller radius, RH is the hub radius as fraction of RP. DX is the thickness of the propeller disk. Integration of body forces over the propeller disk will recover the prescribed thrust and torque: 2π
∫ ∫ ∫ ∫ ∫ ∫ RP
T = ρ L2U 02
RH
0
Q = ρ L3U 02
RH
0
fbx rdxdθ dr
(32)
f bθ r 2 dxdθ dr
(33)
xp
2π
RP
xs
xs
xp
The longitudinal and tangential forces are then projected into the ship coordinate system to result in a net force and moment to be used in the computation of the motions. The location of the propeller is defined in the static condition of the ship. When motions are involved, the propeller will move accordingly with the ship’s motions and possibly will intersect different cell from different blocks as the ship evolves. MOTIONS
0 ⎡1 ⎢ ν i 2 = ⎢0 cos φi ⎣⎢0 − sin φi
− sin θi
⎤ cos θ i sin φi ⎥⎥ η& i 2 = J i−21 η& i 2 cos θ i cos φi ⎦⎥
(36)
and the linear velocities are related by: ν i1 = J i−11 η& i1 cψ i c θ i ⎡ ⎢ − sψ c φ + s φ s θ cψ i i i i i ⎢ ⎢⎣ s θ i sψ i + c φi s θ i cψ i
sψ i c θ i cψ i c φi + s φi s θ i sψ i − s φi cψ i + c φi s θ i sψ i
− s θi ⎤ s φi c θ i ⎥⎥ η& i1 c θ i c φi ⎥⎦
(37)
The motions are described in terms of an earth-fixed inertial reference coordinate system and a ship-fixed system. Figure. 2 shows two ships each with its own reference system and the earth fixed.
where c and s symbolize cosine and sine respectively. The rigid-body equations of motion are written for each ship independently. It is assumed that the principal axes of inertia are coincident with the ship coordinate system. In this case the inertia tensor is diagonal with components ( I xcg , I ycg , I zcg ) respect to the center of gravity. If the center of rotation of the ship is different than the center of gravity, as in the case of a model ship allowed to roll about an axis that does not pass through the center of gravity, then the moments of inertia respect to the center of rotation are: I x = I xcg + m ( yG2 + zG2 ) I y = I ycg + m ( xG2 + zG2 ) I z = I zcg + m ( x + y 2 G
Figure 2: Earth and ship fixed reference systems.
The motion of each of the ships considered is described by the translations and rotations with respect to the inertial frame, the linear and angular velocity with respect to the ship-fixed frame, and the forces and moments with respect to the ship-fixed frame. The position and orientation of ship-I is given by the linear translations and the Euler angles of roll, pitch and yaw:
2 G
(38)
)
where xG = xrot − xcg and similarly for yG and zG . Under the discussed assumption the rigid-body equations for ship-I are: mi ⎡⎣u& − v r + w q − xG ( q 2 + r 2 ) + yG ( pq − r& ) + zG ( pr + q& ) ⎤⎦ = X i i
mi ⎡⎣v& − w p + u r − yG ( r 2 + p 2 ) + zG ( qr − p& ) + xG ( qp + r& ) ⎤⎦ = Yi i
ηi = ( ηi1 , ηi 2 ) = ( x1,i , x2,i , z3,i , φi ,θ i ,ψ i )
(34)
The linear and angular velocities are computed in the ship reference system and are: ν i = ( ν i1 , ν i 2 ) = ( ui , vi , wi , pi , qi , ri )
mi ⎡⎣ w& − u q + v p − zG ( p 2 + q 2 ) + xG ( rp − q& ) + yG ( rq + p& ) ⎤⎦ = Z i i
⎡ I x p& + ( I z − I y ) q r + m { yG ( w& − u q + v p ) − zG ( v& − w p + u r )}⎤ = K i ⎣ ⎦i
⎡⎣ I y q& + ( I x − I z ) r p + m { zG ( u& − v r + w q ) − xG ( w& − u q + v p )}⎤⎦ = M i i ⎡ I z r& + ( I y − I x ) p q + m { xG ( v& − w p + u r ) − yG ( u& − v r + w q )}⎤ = N i ⎣ ⎦i
(35)
(39)
where ui, vi and wi are the surge, sway and heave velocities for the i-ship. pi, qi and ri are the roll, pitch and yaw angular velocities in the ship system. These relate to the time rate of change of the Euler angles by (see for instance Fossen 1994):
In Eq. (39) the forces and moments are computed by integration of the forces on the ship hull, plus the forces and moments caused by the propellers and the gravity force. The fluid and gravity forces and moments are first computed in the earth system, where the flow solution is available and the gravity is vertical:
Fe ,i =
⎡ ⎛ ∇u + ∇uT ⎢⎜ ⎝ 2 Re ship − i ⎣
∫
⎞ ⎛ z ⎟−⎜ p − 2 Fr ⎠ ⎝
⎞ ⎤ ⎟ I ⎥ ⋅ da e + mi g (40) ⎠ ⎦
CFDShip-Iowa supports imposed motions on the earth system and aligns these to the ship coordinate system. INCIDENT WAVES IMPLEMENTATION
L e,i
T z ⎞ ⎤ ⎪⎧ ⎡⎛ ∇u + ∇u ⎞ ⎛ ⎪⎫ = r × ⎨ ⎢⎜ ⎟ − ⎜ p − 2 ⎟ I ⎥ ⋅ da e ⎬ + rGi × mi g Fr ⎠ ⎦ ⎩⎪ ⎣⎝ 2 Re ⎠ ⎝ ⎭⎪ ship − i
∫
and then translated into the ship system where the propeller forces and moments are added:
Regular, irregular unidirectional and irregular multidirectional linear incident waves with arbitrary heading are available in CFDShip-Iowa version 4. The general implementation follows a Bretschneider spectrum as recommended by ITTC (1978). The elevation of the wave at any time t is
Fi = J i−11 Fei + Fprop ,i = ( X i , Yi , Z i )
(42)
ξ ( x, y , t ) =
Li = J L ei + L prop ,i = ( K i , M i , N i )
(43)
(41)
−1 i1
∑∑ a cos ⎡⎣k ( x cos µ − y sin µ ) − ω ij
i
Equations (39) are integrated numerically using a predictor-corrector scheme (Wilson et al. 2006a). The predictor scheme uses the forces and moments at the end of the current time step to guess the next time step 6DOF solution. For any of the degrees of freedom ϕ the predictor step solves:
ϕ& n = ϕ& n −1 + ∆t ( c1 ϕ&&n −1 + c2 ϕ&&n − 2 + c3 ϕ&&n − 3 )
(44)
ϕ n = ϕ n −1 + ∆t ( c1 ϕ& n −1 + c2 ϕ& n − 2 + c3 ϕ& n − 3 )
where the accelerations ϕ&& = u&, v&, w& , p& , q& , r& are computed by solving Eqs. (39). The corrector step is implicit since it uses forces and moments computed from the current time step:
ϕ& n = ϕ& n −1 + ∆t ( c1 ϕ&&n + c2 ϕ&&n −1 + c3 ϕ&&n − 2 )
(45)
ϕ n = ϕ n −1 + ∆t ( c1 ϕ& n + c2 ϕ& n −1 + c3 ϕ& n − 2 )
In Eqs. (44) and (45) the constants c1, c2 and c3 define the order of accuracy of the integration. These are user selectable and are indicated below: Explicit averaged
c1
c2
c3
1/3
1/3
1/3
st
1
0
0
nd
1/2
1/2
0
rd
5/12
8/12
-1/12
nd
7/12
4/12
1/12
1 order implicit 2 order implicit 3 order implicit 2 order least squares implicit
j
j
e ,ij
j
where φij
t + φij ⎤⎦
(46)
is a random phase, aij is the wave
amplitude, ωe ,ij is the encounter frequency, ki is the wavenumber and µ j is the angle of incidence, all for the wave component with wavelength i and angle j. The angle of incidence is composed of the dispersion angle and the heading angle α 0 of the ship:
µ j = α j + α0
(47)
The encounter frequency (depending on the ship speed) and the wave amplitude are computed from:
ωe,ij = ki cos µ j +
1 ki Fr
aij = 2 S (ωi ) M (α j ) δω δα
(48) (49)
In a Bretschneider spectrum the directional spectrum M and the frequency spectrum S are given by: S (ω ) =
A
ω5
e− B / ω
A = 484.54 B=
(α
j
4
H1/2 3 Tm4
1935.35 Tm4
M (α j ) =
The explicit mode is useful for cases in which a final steady-state is seek, like sinkage and trim computations, or when a transient is very slow, as in full resistance curves obtained through cuasi-static acceleration of the model. In model testing it is usual to impose motions (or lack of) in some degrees of freedom. At this point
i
2
π
(50)
cos 2 α j
between -π / 2 and π / 2 , 0 otherwise )
where Tm is the modal wave period and H1/ 3 is the significant wave height, both available from direct sea state measurements. The velocities and pressures are then given by:
8
ϕint = ∑ ad ϕd U ( x, y , z , t ) = U 0 + aij
∑∑ Fr i
ki e ki z cos ⎡⎣ ki ( x cos µ j − y sin µ j ) − ωe,ij t + φij ⎤⎦ cos µ j
j
V ( x, y, z , t ) = V0 + aij
∑∑ Fr i
ki e ki z cos ⎡⎣ ki ( x cos µ j − y sin µ j ) − ωe,ij t + φij ⎤⎦ sin µ j
j
W ( x, y, z , t ) = W0 + aij
∑∑ Fr i
ki e ki z sin ⎡⎣ ki ( x cos µ j − y sin µ j ) − ωe,ij t + φij ⎤⎦
j
p ( x, y , z , t ) = aij
∑∑ Fr i
j
2
(52)
d =1
aij ki z ⎤ ⎡ e ki z cos ⎢ ki ( x cos µ j − y sin µ j ) − ωe,ij t + φij − e ⎥ 2 ki ⎣ ⎦
(51) DYNAMIC OVERSET APPROACH
Overset grids may be static, which means that they either do not move or that they move as a rigid system so that the grid connectivity can be computed as a preprocessing step. The grid connectivity needs to be computed at run time when the grids experience relative motion between them during the simulation. Grids are classified in terms of their ability to move. Static grids will not move at all. These grids are usually used to refine the free surface and to prescribe the outer boundary conditions. Predicted motion grids are grids in which the predicted/prescribed motions will be applied. These grids include the ship hull and all the appendages, plus refinement grids that are designed to resolve flow phenomena that move with the ship. Hybrid grids will be allowed to move in some degree of freedom but not in others. For instance, a free surface refinement grid will follow the ship in the water plane, so we will allow surge, sway and yaw but not heave, roll or pitch. As in static overset approaches, every grid point in the computational domain is marked as active, interpolated or hole. Hole points lay inside objects or outside the computational domain and are therefore “blanked out” and excluded from the computations. Between the active and hole points a set of intergrid boundary points are identified where values interpolated from other grids are used to link the solution on the different overset grids. Every time a grid is moved, new interpolation coefficients need to be computed to link these moving grids with the static grids and between each other. For any variable ϕ the trilinear interpolation from the donor cell is expressed as:
where ad are the interpolation coefficients. The SUGGAR code (Noack 2005) is used to obtain the overset domain connectivity between the set of overlapping grids. SUGGAR was designed for moving body problems and communication with a motion controller such as the 6DOF module within CFDShip-Iowa. Data structures associated with moving bodies are fixed to their original coordinate system and internal transformations are used to properly position points between the different coordinate systems. SUGGAR runs as a separate process from the flow solver. Communication and synchronization between SUGGAR and the flow solver are accomplished through the standard Unix facility of named pipes. The motion controller sends translation and rotation positioning information to SUGGAR via one named pipe. When the overset domain connectivity is complete SUGGAR will signal CFDShip-Iowa via a second named pipe. A general hierarchical organization and motion of objects is allowed where a child object is moved relative to its parent such as a control surface attached to a moving ship that is also moving relative to the ship. SUGGAR also has the capability to reduce the amount of overlap between the components grids by blanking out unneeded grid points in the overlap region. This feature minimizes the numerical inaccuracies introduced by overset by interpolating cells of dissimilar sizes. Interpolation in the region of the single-phase free surface is complicated by the fact that the pressure is not defined for grid points in air. The standard interpolation of points in water from donor cells with grid points in air must be modified to prevent interpolating from the point in air. The present approach removes the grid points in air from the interpolation stencil so that only grid points in water are used along with points defined in the free surface by linear interpolation between the interpolant and the donors in air, see Figure. 3. Details of the method for static grids are discussed in Wilson et al. (2006b), and only the necessary information to follow the paper will be included here. For any interpolant point P in water for which some of the donor points are in air, we define a line joining each of the donors in air with the interpolant, and find the location of the free surface along each of these lines. At those free surface locations we know the pressure, and therefore it is a valid donor point. The location of the interface is found from linear interpolation along the line, where the weight for the interpolant is:
η=
φd
(53)
φd − φ p
and therefore rint = rp η + rd (1 − η )
(54)
where r = ( x, y, z ) are the coordinates of any spatial point. The pressure at the new interfacial point is: pint =
zint Fr 2
(55)
The donor cell is then deformed to conform to the location of the new points in the interface, and the interpolation coefficients recomputed. This process is only necessary and performed for interpolant points in which one or more of the donor points are in air. Since the free surface location, affecting Eq. (53), and the grids, affecting Eq. (54), are moving in time, the cells that need to be conformed to the free surface change in time.
d7
d6 air η d8
d5 P
Modified donor cell
∇ 2ϕ = 0
(56)
where ϕ is any solution variable. The boundary conditions for Eq. (56) are of the Dirichlet type at the interpolated points, and zero gradient everywhere else. This guarantees extension of the solution from the interpolated points into the hole points, providing a reasonable estimate to compute the time derivative. Since the fluid flow equations are solved in the inertial coordinate system, the no-slip boundary condition requires that the fluid at all solid surfaces move with the ship velocity, which in our formulation is the grid velocity. The grid velocity is computed using a second-order backward approximation: x& ti , j , k =
(1.5 x
t i , j ,k
− 2 xti ,−j1, k + 0.5 xti ,−j2, k ) ∆t
(57)
and the velocity on the non-slip boundaries is:
rint
Original donor cell
between two time steps as compared to the grid size, mainly near the boundary layer where the grid is small. In order to have reasonable values for every time step on the hole points close to the interpolated points a Laplace equation is solved within the hole points:
φ =0
water d2
d3
v ti , j , k
no − slip
= x& ti , j , k
(58)
When grids overlap on boundaries where the forces need to be computed, weights are assigned to each overset cell surface and then the integrals Eqs. (40) and (41) are computed. Those weights are evaluated using the code USURP (Boger and Dreyer, 2006) in a pre-processing step. SOLUTION STRATEGY
d1
d4
Figure 3: Donor cell with grid points in air and conformation to the free surface.
A second common drawback of dynamic overset approaches arises from points that in any of the two previous time steps (for second order in time) were holes and in the current time step are active. For these points the time derivative is ill defined since they were excluded from the computations and the solution variables were therefore not computed. Since in our second order approach there are at least two interpolated points between the active points and the hole points, this problem seldom happens if the motions between time steps are small, which can be achieved using small time increments between time steps. However, for large amplitude motions in an implicit approach the motions can be significant
The overall solution strategy is presented in reference to Figure. 4. Detailed discussion on different aspects of the solution procedures for static problems with the single-phase level set approach is provided in Carrica et al. (2006a, 2006b) and Wilson et al. (2006b). We concentrate our discussion in issues related to dynamic overset and motions. The grids are read and split according to user directives for domain decomposition parallelization. The grids that will move along with their associated degrees of freedom are defined at run time. Motions can be prescribed or predicted on each degree of freedom. Grids are classified in terms of to what object they belong to, and any object can move independently from others or remain static. Each moving object is split into separate processors, and one of those processors is in charge of
computing the 6DOF equations for that object. The prescribed motions for each object are then read by the corresponding processor. Once all the variables are initialized, SUGGAR is called for the first time to obtain the initial overset interpolation information.
Once the updated flow field is obtained, the forces and moments are computed. Then the global residuals are evaluated. If the time step is converged, showing a drop in residuals to 10−3 in all variables, then the motions are predicted for next time step according to Eq. (44), and SUGGAR is called to compute the interpolation given the new location of the moving grids. If the nonlinear iteration is not converged, then the motions are corrected using Eq. (45), SUGGAR is called, and a new nonlinear iteration started. Once the time step is finished, a convective extension is done in air to assign the velocity and turbulence quantities in air for the current time step (Carrica et al 2006a), and Eq. (56) solved inside the hole points to approximate variables in those regions. The overhead that SUGGAR takes to compute the overset interpolation coefficients varies with the number of grid points. In small problems (less than about a million grid points) the percentage of CPU time consumed by SUGGAR is very small, less than 5 %. For larger problems (over six million grid points) the overhead approaches 30 %. EXAMPLE APPLICATIONS
Figure 4: Solution strategy.
A non-linear loop is used to converge the flow field and motions within each time step. At the beginning of each nonlinear iteration the overset information is read from a binary file produced by SUGGAR, the grids are moved according to the motions resulting from the 6DOF predictor or corrector steps, and the transformation metrics and grid velocity are computed. Then the flow solver solution starts. First, the k − ω equations are solved implicitly and the turbulent viscosity computed. Next the level set function is transported and reinitialized sequentially. With the new location of the free surface the pressure gradient is computed, and then the velocity is solved implicitly. This is the first step on the PISO (Issa 1985) algorithm to enforce continuity. Next the pressure is obtained by solving a Poisson equation. This is the single most computationally expensive operation of the solution algorithm. The pressure gradient is then computed and the divergence free velocity field obtained explicitly on the last step of the PISO iteration. Since we use a collocated method, the divergence of the velocity field does not converge to zero but to a small value. Increasing the number of PISO iterations beyond 3 to 5 does not drop the divergence further below a minimum value.
The purpose of this section is to show different capabilities of CFDShip-Iowa. We present example problems solved with CFDShip-Iowa by several users. The examples are classified accordingly with the main areas of naval architecture: resistance/propulsion, seakeeping and manoeuvring. Resistance and propulsion
CFDShip-Iowa was initially designed as a resistance code, and tested for a wide range of geometries. These include from simpler geometries like surface piercing flat plates and foils, Wigley hull and Series 60, to surface combatants DTMB 1052, DTMB 5415 and ONR tumblehome, the container KCS, the tankers Esso-Osaka and KVLCC2, the high speed Athena R/V, and recently the multihull geometries DTMB 5594 and HSSL catamarans and trimarans. In addition submerged objects interacting with a free surface, like torpedoes and a dolphin, have been computed using CFDShip-Iowa version 4 (Hyman 2005). Free surface: Two of the most important requirements for modern free surface solvers is to properly predict the free surface elevation and the capability to handle breaking waves. The proper prediction of the wave field is important to compute the residuary resistance. Ship breaking waves are typically computed with CFDShip-Iowa by using overset refinement grids to resolve the small scales involved in the wave formation and plunging process. Wilson et al. (2006b, 2006c)
computed breaking waves for the surface combatant model DTMB 5415 and for Athena R/V at several Froude numbers. A typical overset grid design used to run Athena R/V at Fr = 0.43 is shown in Figure 5. In this case the grid modelled half domain taking advantage of the problem symmetry, and consisted of some 7 Million grid points, with refinement blocks at the bow and transom to capture overturning waves present in those locations. Figure 6 shows a comparison of wave elevation against experimental data. The bow and transom breaking waves were properly captured by the CFD computation. As a result, the comparison with experimental data is excellent. In Figure 7 we show the computation of the breaking wave for a surface combatant model DTMB 5415 at Fr = 0.35 . In this case the plunging bow wave splashes up twice causing pairs of counter-rotating vortices that show up as “scars” on the free surface.
state. In this type of runs overset grids can be used to refine regions of interest, as discussed before for free surface computations, but the overset interpolation coefficients are static and therefore computed as a preprocessing step. Table 1 summarizes the resistance coefficients predicted by CFDShip-Iowa and corresponding experimental data for several ships.
scars
Figure 7: Breaking bow wave and scars for DTMB 5415 at Fr = 0.35 .
Figure 5: Grid design and boundary conditions for Athena R/V at Fr = 0.43 .
Figure 6: Free surface for Athena R/V at Fr = 0.43 .
Static resistance: Basic resistance predictions are performed by fixing the sinkage and trim to the dynamic condition, and then run the case to steady-
Resistance, sinkage and trim: Simultaneous prediction of sinkage and trim requires prediction of pitch and heave motions until the steady state is reached. This is done in CFDShip-Iowa by allowing the ship grids to move in accordance to the motions predictions, while keeping the background and refinement grids fixed. Figure 8 shows an example of a grid prepared for sinkage and trim or pitch and heave in waves computations. The background (blue) and refinement (red) grids are static, while the boundary layer grid (black) can move freely. The hole cutting performed by SUGGAR as well as the overlap optimization can be clearly observed. Steady-state sinkage, trim and resistance predictions for a DTMB 5512 model were performed by Carrica et al. (2006c) for medium ( Fr = 0.28 ) and high speed ( Fr = 0.41 ). Figure 9 shows convergence curves for sinkage and trim, which shows maximum differences with experiments of 0.014 degrees for trim (10.4 %) and 0.14 10−3 for sinkage (7.37 %), both for Fr = 0.28 . Fr = 0.41 sinkage and trim results are much closer to experiments with differences of 1.1 % and 1.7 % respectively. Resistance predictions differ
from experimental values 4.3 % for Fr = 0.28 and 1.5 % for Fr = 0.41 . Fr
CFD
EFD
CFD − EFD EFD
Reference
0.28 0.35 0.41 0.26
4.55 4.74 6.53 3.7
4.23 4.84 6.67 3.56
+7.5 % -2.0 % -2.1 % + 3.9 %
Wilson et al. (2006b)
0
4.00
4.13
-3.1 %
5594
0.51
3.52
3.89
-9.5 %
HSSL trimaran
0.54
6.33
6.87
-7.8 %
0.28 0.35 0.48 0.65 0.8 1.0
5.54 5.11 5.57 4.85 4.475 3.75
5.5 5.0 5.7 4.8 4.3 4.0
-0.8 % +2.2 % -2.2 % +1 % +4.1 % -6.2 %
Ship 5415 KCS KVLCC2
Athena
Wilson et al. (2005) Simonsen and Stern (2005) Stern et al. (2006) Stern et al. (2006)
Miller et al. (2006)
Table 1: Steady-state resistance coefficients (times 1000) for some static computations.
negligibility of the time derivatives is necessary. The physical characteristic times involved in the process of reaching steady-state are the times necessary to form the boundary layer and to develop the Kelvin wave. These two times are in the order of the time needed to cover one ship length. So during this time we would like the change of ship speed to be small, to obtain essentially constant speed values of resistance, sinkage and trim. If the acceleration to full speed ( U 0 = 1 corresponds to Frmax ) is linear, t Fr = T Frmax
U=
(59)
where T is the nondimensional time to reach maximum speed. To cover one ship length the ship needs to travel a time ∆t obtained from: 1= ∫
t + ∆t / 2
t −∆t / 2
t T dt =
t ∆t T
(60)
and using Eqs. (59) and (60) we obtain the change of Froude number to cover one ship length: ∆Fr =
Figure 8: Typical grid designed for computations of sinkage and trim in calm water or pitch and heave in waves. 1.E-03
Fr=0.41
0.4
0.E+00
0.3
-1.E-03
0.2
Fr=0.28
Sinkage
Trim angle (degrees)
0.5
-2.E-03
0.1
-3.E-03
0.0
Fr=0.28
-0.1
-4.E-03
Fr=0.41
-5.E-03
-0.2 -0.3
-6.E-03
0.0
5.0
10.0
15.0
Time
20.0
25.0
30.0
0.0
5.0
10.0
15.0
20.0
25.0
30.0
Time
Figure 9: Sinkage and trim convergence for DTMB 5512 and target experimental values.
It is possible by performing a slow acceleration to obtain a resistance, sinkage and trim curve for a full range of speeds. This is done in CFDShip-Iowa by imposing a forward motion and predicting heave and pitch which, in a pseudo steady-state computation, will result in the resistance, sinkage and trim for each instantaneous speed. This slow acceleration essentially seeks to zero the time derivatives in Eqs. (1), (5), (9) and (10). The level of accuracy in this pseudo steadystate computation depends on the acceleration imposed. A trade off between computation speed and
2 Frmax T Fr
(61)
which shows that the change in the Froude number while the ship travels a ship length is proportional to the inverse of the period and the instantaneous Froude number. This means that at lower speeds the error on the Froude number evaluation is larger. For instance, with Frmax = 1 , T = 100 s * and Fr = 0.3 we get ∆Fr = 0.033 , while at Fr = 0.8 we obtain ∆Fr = 0.0125 . Notice that, since we are accelerating slowly, the solution at any speed starts from a solution that is very close, and thus the one ship length necessary to develop the boundary layer and wave field may be largely overestimated. This appears to be confirmed by Figure 10, which shows predictions and experimental data of resistance, sinkage and trim for Athena R/V. We can see that for an acceleration time T = 100 s * the solution is almost the same as that obtained through steady-state computations, and very close to experimental data. Appended ships: Appendages are also treated using overset grids. In SUGGAR, overset grids over viscous surfaces are treated independently by running a threestep process: first the hole-cutting information is performed with SUGGAR, then the overset information on the solid surface is obtained using the
accompanying code SURFASM, and then SUGGAR is run to provide the final connectivity information. 0.006
Resistance
Fixed sinkage and trim
Full curve, Acceleration time: 50 s*
0.004
EFD
Full curve, Acceleration time: 100 s*
0.2*Trim (radians) 0.002
0.000
Steady-state, predicted sinkage and trim
-0.002
Sinkage
-0.004 0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
Froude number
Figure 10: Results of resistance, sinkage and trim for a slow acceleration of Athena R/V barehull with skeg. Solid circles: experimental data; open triangles: resistance predictions for fixed experimental sinkage and trim; open squares: steadystate computation of resistance, sinkage and trim; fine line: acceleration in 50 s*; thick line: acceleration in 100 s*. From Miller et al. (2006).
Fr = 0.336 and Re = 14.2 106 , with fixed dynamic sinkage and trim. The experimental resistance coefficient of 0.00728 was matched within 4.1 % by the computed resistance coefficient of 0.00758. This case was also run using DES, resulting in a resistance coefficient of 0.00742, only 1.9 % off the experimental resistance coefficient.
Propulsion computations: Propulsor effects are added to the flow field as body forces, as discussed before. Also the resulting thrust and torque are applied to the forces and moments that act on the 6DOF solver. Multiple propellers per body are allowed, each with different thrust, torque and advance coefficients, to simulate diverse operational problems. The propeller model, though designed for screw propellers, can be applied also to water jet propulsion. An example of such a calculation is shown in Figure 12, where a water jet propelled Athena R/V at fixed sinkage and trim is depicted. The free surface, boundary layer contours and streamlines are colored with axial velocity. Two circular jets are issued at the transom face of the ship, with significant effect on the free surface and momentum wake. To be noticed is the thinning of the boundary layer downstream of the water jet intake. Self propulsion computations are supported through unsteady acceleration to steady state, in a similar fashion to the resistance, sinkage and trim procedure. In this case the code will predict surge, heave and pitch for a given set of thrust, torque and advance coefficients imposed on the propeller, corresponding to an operational condition at given RPM. As a result, ship velocity (and Froude number), resistance, sinkage and trim are predicted.
Figure 11: Isocontours of axial velocity showing the boundary layer of a fully appended Athena R/V.
As an example of resistance computations in a fully appended ship, a fully appended Athena R/V has been computed, see Figure 11. In this case the grid consisted of 13 base grids with a total of 5.5 Million grid points, covering the ship hull, skeg, rudders, stabilizers, shafts and struts. The conditions were
Figure 12: Water jet computations for Athena R/V. From Miller et al. (2006)
A self-propulsion computation of a fully appended Athena R/V is shown in Figure 13. The propeller constant coefficients were imposed for an experimental self-propulsion point at Fr = 0.336 . The resistance, sinkage, trim and final equilibrium speed were predicted. As can be seen from the figure, the Froude number and trim are predicted very well, while the error on the sinkage is around 20 %. It should be noted that since the vertical location of the center of gravity was not available from the experiment conditions, it was guessed from geometry information. This possible disagreement on the location of the center of gravity between experiment and CFD could have a large impact on the sinkage and trim values.
zeroth and first harmonic and first harmonic phase of the free surface elevation. At high speed with short wavelength the encounter frequency increases significantly and a breaking bow wave formed and the response became non-linear. A view of the breaking wave is provided in Figure 15.
Froude number
0.35
0.25
Trim (degrees) 0.15
0.05
-0.05
-0.15
Sinkage*100 -0.25 0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
Time
Figure 13: Self-propulsion computations for fully appended Athena R/V. Experimental values shown in constant lines.
Seakeeping
Figure 14: Free surface harmonic analysis for the FSD problem at medium speed.
Seakeeping computations performed with CFDShipIowa include forward speed diffraction (FSD), pitch and heave computations in regular and irregular waves, and pitch, heave and roll computations in irregular waves. Forward speed diffraction: Carrica et al. (2006b) performed computations of the FSD problem for a surface combatant model DTMB 5512, including verification and validation. Medium speed ( Fr = 0.28 ) with long wavelength ( λ = 1.5 ) and high speed ( Fr = 0.41 ) with short wavelength ( λ = 0.5 ) were computed, and comparison with experimental data performed for unsteady free surface, forces and moment, and nominal wake velocities. The boundary layer response to the incident waves was discussed in detail. At medium speed the response proved to be linear. Figure 14 shows a comparison with experiments of the
Figure 15: Breaking bow wave observed for the FSD problem at high speed.
Pitch and heave in regular waves: Pitch and heave response to regular head waves is of great importance on evaluating seakeeping performance of ships. In CFDShip-Iowa these calculations are done by imposing a wave through the boundary conditions and allowing the ship grids to pitch and heave, restricting all other motions. Stern et al. (2006) performed computations of the response of HSSL (High Speed Sea Lift) catamarans and trimarans to regular head
waves simulating sea states up to 7 near the resonant condition, see Figure. 16. In these cases the motion responses are very large, causing the ship’s bow to submerge. One of the conclusions of these computations was that for large amplitude waves the motions response decreases respect to the linear behavior.
Figure 18: Free surface at t/T=0, ¼, ½ and ¾ for DTMB 5512 Fr = 0.41 , ak = 0.075 free to pitch and heave. Figure 16: Pitch and heave of HSSL trimaran (top) and catamaran (bottom).
Similar conclusions were drawn by Carrica et al. (2006c), who computed pitch and heave motions for a surface combatant. For moderate wave amplitude ( ak = 0.025 ) the motions response was very close to the experimental data, and was essentially linear. For large amplitude waves ( ak = 0.075 ) the transfer functions for both pitch and heave decrease. In contrast with the FSD case, the bow wave does not break for the high speed case with large amplitude motions, but a transom breaking wave forms, see Figures 17 and 18.
Figure 19: Two surface combatants following each other at Fr = 0.41 , ak = 0.025 free to pitch and heave. 0.016
Ship 2 (back ship)
Resistance
0.014 0.012 0.010 0.008 0.006
Figure 17: Free surface for DTMB 5512 Fr = 0.41 , ak = 0.075 free to pitch and heave.
The case of two ships following each other is also demonstrated in Carrica et al (2006c), in which two surface combatants advance in regular waves free to pitch and heave (see Figure 19). In this type of computations the interaction effects between ships can be studied. As an example, the unsteady resistance of both ships is shown in Figure 20, where a higher resistance on the second ship is caused by the rooster tail of the first ship.
Ship 1 (lead ship)
0.004 0.0
1.0
2.0
3.0
4.0
5.0
t/T
Figure 20: Resistance for two surface combatants following each other in regular waves free to pitch and heave.
Motions in irregular waves: The Bretschneider model for linear irregular multidirectional waves has been exercised for a DTMB 5512 advancing at Fr = 0.41 in a sea state 6 with 45 degrees heading. In this case the model was free to pitch, heave and roll, but the forward speed was constant and the sway and yaw set to zero. Figure 21 shows an instantaneous free surface field colored with free surface elevation, and the history of
motions. Notice that the roll motion responds to the excitation by the waves by oscillating with the natural rolling frequency of the ship. Stern et al. (2006) computed the response of a HSSL trimaran in following (135 degrees heading, Figure 22) and head seas (45 degrees heading, Figure 23), and for a DTMB 5594 trimaran, and computed slamming pressures on the hull, all free to pitch, heave and roll. In all cases the ships advanced at high speed ( Fr 0.5 ). For these hull shapes, motions in irregular waves are significantly less extreme than in regular waves in the resonant condition.
Manoeuvring
Manoeuvring computations have been restricted to captive models. Imposed trajectory computations can be performed statically and therefore require only a pre-processing step to compute the overset grid connectivity. Examples of these computations include pure drift and steady turn tests. Planar motion mechanism (PMM) tests in an unconstrained sea also belong to this category. If the walls of the towing tank need to be modelled or motions are to be predicted, then a dynamic overset computation is necessary.
Figure 21: DTMB 5512 free to heave, pitch and roll advancing in irregular seas ( Fr = 0.41 , 45 degrees heading)
Figure 24: Free surface computation of KVLCC2 advancing with a 4 degree drift in shallow water.
Figure 22: HSSL trimaran free to heave, pitch and roll advancing in irregular seas (sea state 6, 135 degrees heading)
Figure 25: DES computation of the vortex structures of a KVLCC2 advancing with a 30 degree drift in deep water.
Figure 23: HSSL trimaran free to heave, pitch and roll advancing in irregular seas (sea state 6, 45 degrees heading)
Pure drift: Pure drift computations with no motions in deep water have been performed for several ships using CFDShip-Iowa version 4, including Wigley hull, DTMB 5512 and KVLCC2. A complex free surface
computation was performed for a KVLCC2 tanker advancing in shallow water (Simonsen and Stern 2006). The axial velocity contours for the tanker advancing with 4 degree drift at Fr = 0.1424 with a water depth 1.2 times the ship’s draft is shown in Figure 24. An extreme drift angle of 30 degrees for the tanker KVLCC2 has been computed by Shao (2006) using the EASM/DES turbulence model, for Fr = 0 . Figure 25 shows the vortex structure obtained for this case. Shao (2006) also performs a thorough analysis of the frequency response of the detached vortices and a physical interpretation of their origin. Steady turn: Steady turn computations are performed in the earth inertial reference system prescribing a circular trajectory, or using a body force in the ship system. CFDShip-Iowa version 4 favors computations in an inertial reference system, which allows the computations of several objects and the easy addition of predicted motions. Figure 26 shows an example computation of a steady turn computation of a DTMB 5415 with bilge keels advancing at Fr = 0.28 . The boundary layer is shown colored with axial velocity. Notice the vortices detached from the bilge keels and the sonar dome.
motions were imposed and all the other motions were disabled. Figure 27 shows the wave field and the boundary layer (contour lines of axial velocity). Notice strong vortices shed at the bow and stern. PMM computations of a surface combatant, model 5512 in bare hull with bilge keels configuration at Fr = 0.28 , have been performed by Sakamoto et al. (2006). The computations include pure sway and pure yaw captive model tests, and pure sway and pure yaw free to pitch, heave and roll. CFD results have been compared with the experiments by Longo et al. (2006). Also pure drift and steady turn cases were computed. An example of the free surface field in a pure sway computation is shown in Figure 28. Refinement blocks have been added to capture the breaking wave and the very deep trough on the port side as the ship moves toward the starboard side. Pure sway CFD motions, forces and moments results are compared against experimental data for the free to heave, pitch and roll case. The predicted motions, shown in Figure 29, agree with the experimental data very well. However, a nonsymmetric behavior is observed mostly for EFD. The heave motion shows a remarkable difference in amplitude when moving to the starboard direction respect to when it moves to port. Since this is a long computation (each period requires 3.75 ship lengths) probably more periods are needed both in CFD and experiments to reach a periodic solution. Also, it must be noted that the centers of gravity and rotation
Figure 26: Boundary layer for a surface combatant in a steady turn.
PMM computations: The simulation of PMM tests is challenging because it involves forced motions that result in imposed accelerations, in some cases significantly stronger than those expected in a free ship. When some degrees of freedom are predicted this may cause significant stress on the 6DOF solvers. Stern et al. (2006) computed pure yaw PMM cases for the HSSL catamaran. In this case sway and yaw
Figure 27: Boundary layer for a HSSL catamaran on PMM pure yaw.
occur at phase 180 and 0 degrees, respectively. The lateral force, however, has a lead of about 35 degrees, which means that this force is maximum before the ship reaches maximum sway velocity. Similarly, the longitudinal force has a maximum around 10 degrees before the lateral force reaches its maximum or minimum. It has been noticed by comparing captive with free computations that forces are not strongly affected by pitch, heave or roll. This trend has also been observed in the experiments.
Figure 28: Free surface colored by axial velocity for model 5512 in pure sway motion. Notice the breaking bow wave. 2.5 2.0
EFD CFD
Roll (degrees)
1.5 1.0
Pitch (degrees)
0.5 0.0 -0.5 -1.0 -1.5 Heave * 1000
-2.0 -2.5 -3.0 -3.5 0.0
0.2
0.4
0.6
0.8
1.0
t/T
Figure 29: Motions history over one period for model 5512 in pure sway motion, free to heave, pitch and roll. EFD CFD
0.02
Cy 0.01
Cx
Figure 31: Axial velocity contours for CFD and EFD of a prescribed pure sway PMM test for model 5512.
0.00
Mz -0.01
-0.02 0.0
0.2
0.4
0.6
0.8
1.0
t/T
Figure 30: Forces history over one period for model 5512 in pure sway motion, free to heave, pitch and roll.
The longitudinal and lateral force coefficients Cx and C y , and the yaw moment coefficient M z , shown in Figure 30, show an excellent agreement with the EFD data. The maximum and minimum moments
Velocities at four axial cross sections have been compared against 3D-PIV data from Longo et al. (2006) for a pure sway captive model case, at eight different phases of the PMM sway motion. Figure 31 shows axial velocity U contours for the cross sections x = 0.135, 0.235, 0.735 and 0.935 , and for the phases φ = 0, 90, 180 and 270 degrees. The lateral V and vertical W velocities are shown in Figure 32, for cross sections x = 0.135 and 0.735. These figures show in general a good agreement between CFD and experiments. However, at the most downstream
sections some numerical diffusion can be observed, most likely due to insufficient grid refinement.
The comparison of longitudinal and lateral forces and yaw moment on the ship coordinate system is presented in Figure 34. Experimental data includes IIHR data (Longo et al., 2006) and DMI data free to heave and pitch but not to roll (Simonsen, 2004). The yaw moment compares well against the data, though is slightly underpredicted. The underprediction of the lateral force is more remarkable, in the order of 25 % against IIHR data. The reasons for this discrepancy is not yet known, and work is under way to understand the phenomena involved. The excellent agreement between EFD and CFD for pure sway contrasts with the results for pure yaw. Velocities at the nominal wake plane are compared against PIV measurements by Longo et al. (2006). Two phases, φ = 90o and φ = 270o were selected. The agreement with experimental data is good and the general trend is properly captured. Notice that the data was taken for a captive test while the CFD was done for a free to pitch, heave and roll case, so some differences can be expected due to this fact. However, the grid used for the tests free to pitch, heave and roll tests is relatively coarse and refinement should improve the CFD results. 2.00 CFD EFD (IIHR)
1.00
Roll (degrees)
Pitch (degrees)
0.00 -1.00 -2.00
Figure 32: Lateral and vertical velocity contours for CFD and EFD of a prescribed pure sway PMM test for model 5512.
Figures 31 and 32 show that the bilge keels play an important role on the structure of the boundary layer when lateral motions are present. This effect was discussed by Wilson et al. (2006a) for a roll decay study on the same geometry and in the case presented here is even more dramatic. Vortical structures are also generated by the sonar dome and the ship keel during the sway motions. Pure yaw PMM computations have also been performed and compared with experimental data. The conditions were the same as those of the pure sway test. In this case the model was free to pitch, heave and roll. The CFD predicted motions are compared against experimental data in Figure 33. The general agreement is good, though the experimental data is significantly deviated from the periodic condition. This is mostly noticeably for the heave motion, but also for the pitching angle.
Heave * 1000 -3.00 -4.00 0.0
0.2
0.4
0.6
t/T
0.8
1.0
Figure 33: Motions history over one period for model 5512 in pure yaw motion, free to heave, pitch and roll. 0.010 0.008
Cx
0.006 0.004 0.002 0.000 -0.002
Mz
Cy
-0.004 CFD
-0.006
EFD (IIHR)
-0.008
EFD (DMI)
-0.010 0.0
0.2
0.4
t/T
0.6
0.8
1.0
Figure 34: Forces history over one period for model 5512 in pure yaw motion, free to heave, pitch and roll.
ACKNOWLEDGEMENTS
This work has been sponsored under ONR Grants N00014-01-1-0073 and N00014-05-1-0723 under administration Dr.Patrick L. Purtell. The authors would like to acknowledge the DoD High Performance Computing Modernization Program (HPCMP) for providing computer time for the simulations. Support for the development of the SUGGAR code by the HPCMP Programming Environment and Training (PET) activities through Mississippi State University under the terms of Contract No. N62306-01-D-7110 is also acknowledged. Contribution of figures by Claus Simonsen and solutions by Ron Miller are deeply acknowledged. REFERENCES
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Figure 35: Velocity contours for CFD and EFD at the nominal wake plane of a pure yaw PMM test for model 5512.
CONCLUSIONS AND FUTURE WORK
The state of development of the code CFDShip-Iowa version 4 has been presented. Example cases showing the capabilities of the code have been included. Though significant progress has been made over the past two years in improving robustness and adding capabilities to the code, there is still much work to do to take the code to the level of capabilities that the users require. These further developments can be classified into accuracy/robustness improvements and capability additions. In terms of new capabilities, the following tasks can be mentioned: addition of new wave models, including linear and nonlinear short and long crested; addition of parent/child object capabilities to model moving rudders, stabilizers, etc., with independent motion equations; addition of dynamic controllers for implementation of course keeping auto pilots, forward speed control, turning controllers, track-keeping systems, rudder/roll stabilization, etc.; improved propeller models, including two-way propeller flow coupling and propeller dynamics for speed control; etc. Numerical improvements include higher-order convection schemes, more accurate level set transport and reinitialization algorithms, and diverse new implementations to speed up calculations and turn around time.
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