moving towards high-fidelity rans calculations of...
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8th International Conference on Numerical Ship HydrodynamicsBusan, Korea, Sept.22-25, 2003
Moving Towards High-Fidelity RANS Calculations Of ManeuveringSurface Vessels Using Unstructured Grids
Clarence O. E. Burg David L. MarcumComputational Simulation and Design Center ComputationalSimulation and Design Center
Mississippi State University Mississippi State [email protected] [email protected]
ABSTRACTSimulations of maneuvering surface ships us-
ing high-fidelity viscous computational fluid dynam-ics requires the capability to model the conservationof mass and momentum via the unsteady Reynolds-averaged Navier-Stokes, along with appropriate tur-bulence modeling, the capability to rotate and de-flect appendages, accurate force and moment pre-dictions and robust and efficient unsteady nonlinearfree surface capabilities. Simulations of flow aroundtransom sterns for naval combatants introduces addi-tional complexities due to grid generation issues aris-ing from wetted or unwetted conditions. Finally, tosimulate flow for fully-appended configurations withstruts, rudders, shafts and propellers, unstructuredgrids offer much promise due to the geometric com-plexities. In this research, a three-dimensional un-structured mixed-element flow solver of the incom-pressible Reynolds-averaged Navier-Stokes equationswith various turbulence models and nonlinear freesurface capabilities is studied with the goal of address-ing some preliminary issues for rudder-induced, pro-peller driven, six-degree-of-freedom maneuvers. Theresearch presented herein is divided into three topics,each focused on numerical simulations of the DTMBModel 5415 series hullform. The first topic concernsthe accuracy of the free surface and of the forces andmoments generated by the unstructured flow solver.Results from a grid refinement study is used to de-termine the level of grid refinement that are requiredfor accurate predictions of the forces and moments forthis code. The second topic covers two different typesof steady-state maneuvers, the constant drift case andthe constant turn case. Flow around the Model 5415hullform at drift angles from 1 to 5 degrees are pre-
sented, as well as flow for turns of constant radiusranging from 6 to 24 ship lengths. The third topicis a prescribed maneuver for the model 5415 hull-form with a linear free surface. Simulation data froma rudder-induced, propeller-driven maneuver with nofree surface (i.e., the free surface was a symmetryplane) is used to prescribe a maneuver for the Model5415 hullform. Due to the complexities of moving thegrid at each time step, only a linear free surface wasused, so that the grid movement algorithm is ignored.
NOMENCLATUREY Free surface elevation~u Velocity (u,v,w)at Grid speed (velocity) −(Vx,Vy,Vz)χ U-MUSCL parameterP PressureFr Froude numberU∞√
gLβ Artificial compressibility coefficientQ Vector of flow variables~F Inviscid flux vector~G Viscous flux vectorτi j Viscous stressesn Outward pointing unit normal to facenf s Normal at free surface~r Vector along edge between two nodesA Inviscid flux Jacobian matrixR Matrix of right eigenvectors ofAΛ Diagonal matrix of eigenvalues ofAW Characteristic variablesφ j Finite element weight functionγ Free surface velocity restriction coeff.
INTRODUCTIONModeling the free surface generated by surface
ships and by submerged vessels near the water sur-face is important for proper understanding of the flowaround these vessels. In particular, the free surfacechanges the resistance of the vessels through the wa-ter by changing the pressure distribution and the wet-ted surface area. The free surface affects the loca-tion and magnitude of the vortices that originate fromvarious locations, including the bow, appendages andpropulsors, which can greatly affect the performanceof the propulsors. The wave signatures of ships andsubmarines is also of importance. Surface vessels un-der investigation by ship designers attempt to mitigatethe negative influences of the free surface interactionon cavitation, power requirements and wave signatureand attempt to use the free surface to improve the per-formance in other ways. In addition to the ship cruis-ing straight and level, the effects on the ship’s perfor-mance caused by maneuvering through a change indirection, or through various sea states, especially inregards to wave signature, sea-keeping and cavitationin littoral water are important in the designs and cannot be readily tested by experiment.
A long-term goal of the research into numericalsimulations is to develop the ability to study the per-formance of a full ship design, including the interac-tions of the various appendages, sonar domes, rud-ders, shafts and propulsors, as the ship maneuvers inresponse to changes in the settings of the rudders andpropulsors. To accomplish this goal, efficient and ac-curate free surface simulations are needed. Other nec-essary components to achieve this goal include capa-bilities to monitor the onset of incipient cavitation, toperform simulation of several vessels in motion rel-ative to one another and to adapt the grid to capturevortices in the water.
Currently, the implicit unstructured code beingdeveloped by researchers in the Computational Sim-ulation and Design Center at Mississippi State Uni-versity solves the three-dimensional, incompressibleNavier-Stokes equations, using an edge-based, flux-differencing finite volume method with Roe-averagedvariables. The unstructured flow solver was success-fully parallelized (Hyams, 2000a), and the code hasbeen ported to a wide variety of high-performancemachines. The unstructured grid generator (Marcum,1998) has the capability of building highly stretchedhigh-aspect ratio grids near viscous surfaces, whichinclude higher order elements such as pyramids andprisms, and allows the flow solver to fully resolveboundary layer features. The unstructured flow solveris also capable of rotating the propulsors and deflect-ing the rudder appendages. And in conjunction withthe grid generator, the flow solver can simulate flowat full-scale Reynolds number, withy+ values on the
order of 1.0.At each time step, the nonlinear free surface al-
gorithm solves the kinematic free surface equation
∂Y∂t
+(u−Vx)∂Y∂x
− (v−Vy)+ (w−Vz)∂Y∂z
= 0 (1)
whereY = Y(x,z,t) is the free surface defined as asingle-valued function over thexz plane,(u,v,w) arethe velocity components in the coordinate directionsand(Vx,Vy,Vz) are the grid velocities. A flow-throughboundary condition for the free surface based on char-acteristic variable boundary conditions is used, withthe pressure on the free surface set toP = Y
Fr2 whereP is the pressure andFr is the Froude number. Af-ter several time steps, the grid is moved to match thefree surface while conforming to the geometry, us-ing a three dimensional extension of Farhat’s torsionalspring method (Farhat, 1998). This grid movement al-gorithm is quite robust, allowing for moderate to ex-treme distortions as required by the free surface simu-lations. As the solution converges, the flow along thefree surface becomes tangent to the free surface, andthe nonlinear free surface solution is obtained.
Much of this work is an extrapolation of the freesurface algorithm used within the three-dimensionalstructured code UNCLE also developed at MississippiState University. Beddhu (1994) developed the struc-tured free surface solver, using a modified artificialcompressibility formulation. This algorithm was ap-plied to steady and unsteady flow around the Wigleyhull (Beddhu, 1998a), to the barehull model 5415 se-ries hull (Beddhu, 1998b), to the Series 60CB = 0.6ship (Beddhu, 1998c) and to a more detailed sternanalysis for the DTMB Model 5415 series hull (Bed-dhu, 1999), and further results from simulations ofthis hullform were presented at the Gothenburg con-ference (Beddhu, 2000). Initial verification and val-idation exercises of the unstructured nonlinear freesurface algorithm has been presented (Burg, 2002a),which includes a grid refinement study for flow overa submerged NACA0012 hydrofoil, flow around in-viscid and viscous Wigley hulls, and flow around thebarehull DTMB Model 5415 series hull. More com-plicated free surface simulations around the DTMBModel 5415 series hull with rudders, shafts, struts andpropulsors have been attempted (Burg, 2002b), demon-strating the need for complicated gridding technologyand the successful simulation of a rotating propulsornear a nonlinear free surface.
In this paper, the barehull DTMB Model 5415 se-ries hullform will be analyzed exclusively. The pri-mary goal of this work is demonstrate that the non-linear free surface capabilities within the unstructuredflow solver are adequate for simulations of moder-ate turns into the flow of approximately 5 degrees.
Force comparisons for several different angles of yawand for several different turning radii are presented,as well as force comparisons and free surface com-parisons for a grid refinement study. Finally, a pre-scribed maneuver with a quasi-linear free surface isperformed, using the grid that was moved to matchthe free surface for the converged steady-state straightlevel simulation.
In the next section, the Navier-Stokes solution al-gorithm is reviewed. Then, the free surface algorithmis presented, which includes the method for solvingthe kinematic free surface equation, the imposition ofthe hydrostatic pressure on the Navier-Stokes equa-tions and the grid movement algorithm. Finally, theresults for the DTMB Model 5415 series simulationsare presented.
NAVIER-STOKES SOLUTION ALGORITHMThree different sets of equations are solved in the
process of simulating the flow around surface vesselsand submerged vessels near the water surface. Theunsteady incompressible Reynolds-averaged Navier-Stokes equations, which are presented here in Carte-sian coordinates and in conservative form, are solvedto determine the velocity and pressure within the com-putational domain. Several turbulence models are avail-able including the one-equation Spalart-Allmaras tur-bulence model, andq−ω, k−ε andk−ω two-equationturbulence models, to simulate the turbulent viscos-ity primarily within the boundary layer, and the kine-matic free surface equation is solved to advance thefree surface in time.
Governing EquationsAssuming that gravity acts in the y-direction (i.e.,
the vertical direction), the incompressible Navier-Stokesequations can be expressed in dimensional form, de-noted with superscript∗, as
∇ · (~u∗) = 0
∂u∗
∂t∗+ ∇ · (u∗~u∗)+
∂∂x∗
(
P∗
ρ∞
)
= µ∗∇2u∗
∂v∗
∂t∗+ ∇ · (v∗~u∗)+
∂∂y∗
(
P∗
ρ∞+gy∗
)
= µ∗∇2v∗
∂w∗
∂t∗+ ∇ · (w∗~u∗)+
∂∂z∗
(
P∗
ρ∞
)
= µ∗∇2w∗
(2)
The variables in the preceding equation are normal-ized with respect to a characteristic length scale (L)and free stream values of velocity (U∞), density (ρ∞),and viscosity (µ∞). Thus, the Reynolds number is de-fined asRe= U∞L/ν∞. Pressure is normalized with
P = (P∗ + ρ∞gy∗ − P∞)/ρ∞U2∞, whereP∗ is the lo-
cal dimensional static pressure. Following (Chorin,1967), an artificial time derivative term (∂ρa/∂t, whereρa = P/β) has been added to the continuity equationto cast the complete set of governing equations into atime-marching form. The nondimensionalized equa-tions can be written in integral form as
∂∂t
Z
ΩQdV+
Z
∂Ω~F ·~ndA=
1Re
Z
∂Ω~G ·~ndA (3)
where~n is the outward pointing unit normal to thecontrol volumeV. The vector of dependent variablesand the components of the inviscid and viscous fluxvectors are given as
Q =
Puvw
(4)
F ·~n =
β(Θ−at)uΘ + nxPvΘ + nyPwΘ + nzP
(5)
G ·~n =
0nxτxx+ nyτxy+ nzτxz
nxτyx+ nyτyy+ nzτyz
nxτzx+ nyτzy+ nzτzz
(6)
whereβ is the artificial compressibility parameter (typ-ically 15 in this work),u, v, andw are the Cartesianvelocity components in thex, y, andz directions, andnx, ny, andnz are the components of the normalizedcontrol volume face vector.Θ is the velocity normalto a control volume face:
Θ = nxu+ nyv+ nzw+at (7)
where the grid speedat =−(Vxnx +Vyny +Vznz). Thecontrol volume face velocity is given by~Vs = Vxi +Vy j +Vzk. The viscous stresses given in Equation (6)are
τi j = (µ+µt)
(
∂ui
∂x j+
∂u j
∂xi
)
(8)
whereµandµt are the molecular and eddy viscosities,respectively.
Numerical Approach for Navier-Stokes Equa-tions
The baseline flow solver is a node-centered, fi-nite volume, implicit scheme applied to general un-structured grids with nonsimplical elements, such asprisms and pyramids. The flow variables are storedat the vertices, and surface integrals are evaluated onthe median dual surrounding each of these vertices.The nonoverlapping control volumes formed by themedian dual completely cover the domain, and form amesh that is dual to the elemental grid. Thus, a one-to-one mapping exists between the edges of the originalgrid and the faces of the control volumes. The solu-tion algorithm consists of the following basic steps:reconstruction of the solution states at the control vol-ume faces, evaluation of the flux integrals for eachcontrol volume, and the evolution of the solution ineach control volume in time.
ReconstructionA higher order spatial method is constructed by
extrapolating the solution at the vertices to the facesof the surrounding control volume. The unweightedleast squares method (solved via QR factorization (An-derson, 1994)) is used to compute the gradients at thevertices for the extrapolation, which is similar to acentral difference approximation to the gradient. Withthese gradients known, the variables at the interfaceare computed using the newly-developed UnstructuredMUSCL (U-MUSCL) scheme,
Qf = Qi +χ2
(Q j −Qi)+ (1−χ)∇Qi ·~r2
(9)
whereQf is the value at nodei side of the face be-tween nodesi and j, χ is a parameter that works thesame as the parameter within the MUSCL-approach,and~r is a vector that extends from nodei to nodej associated with the control volume face in ques-tion. Whenχ = 0, the more common formulation forthe variable extrapolation within unstructured codes isachieved. Whenχ = 1/2, the error of the extrapola-tion at the face is minimized for uniform, one-dimen-sional flow. In all of the simulations reported herein,this parameter is set to 1/2. For many simulations,this choice ofχ allows for a more stable simulationwhich does not need any limiters, and produces a so-lution whose residual has been converged to machinezero.
Spatial ResidualThe evaluation of the discrete residual is performed
separately for the inviscid and viscous terms given inEquation (3). The Roe scheme (Roe, 1981) is used toevaluate the inviscid fluxes at each face of the control
volumes. The algebraic flux vector is replaced by anumerical flux function, which depends on the recon-structed data on each side of the control volume face:
Φ =12
(F(QL)+F(QR))− 12
A(QR−QL) (10)
whereA = RΛR−1. The matrixR is constructed fromthe right eigenvectors of the flux Jacobian, andΛ isa diagonal matrix whose entries contain the absolutevalues of the eigenvalues of the flux Jacobian (Taylor,1991). The matrixA is evaluated with Roe-averagedvariables, which is simply the arithmetic average be-tween left and right solution states in the case of in-compressible flows.
For general element grids, it is expedient to useonly edge-local information to compute the viscousfluxes, which allows the evaluation of viscous fluxeson each face of the control volume without regard tothe varying element types of the mesh. An algorithmin which no element information is used outside ofmetric computations is termed a “grid transparent” al-gorithm (Haselbacher, 1999). To this end, the viscousfluxes are evaluated directly at each edge midpoint us-ing separate approximations for the normal and tan-gential components of the gradient vector to constructthe velocity derivatives . Using a directional deriva-tive along the edge to approximate the normal compo-nent of the gradient and the average of the nodal gra-dients to approximate the tangential component of thegradient leads to the following expression (Hyams,2000b):
∇Qi j ≈ ∇Q+[
Q j −Qi −∇Q·~r] ~r|~r|2 (11)
where∇Q is the average of∇Qi and∇Q j . As statedabove, the weighted least squares method is used toevaluate the nodal gradients in the preceding formula.
Solution ProcedureThe temporal derivative is discretized so that ei-
ther a first or second-order time discretization is avail-able. The resulting system of discretized governingequations is solved via a bi-directional Gauss-Seidelalgorithm, which splits the matrix into a diagonal, up-per triangular and lower triangular parts and using subit-erations to converge the solution at each time step(Hyams, 2000b) (Burg, 2002b).
Boundary ConditionsThree different boundary conditions are used with-
in the free surface simulations: viscous wall (i.e., no-slip) boundary conditions, farfield boundary condi-tions and free surface boundary conditions, each of
which are handled implicitly. Viscous conditions areenforced by modifying the linear system such that nochange is allowed in the velocity, and the pressure isdriven according to the imbalance in the continuityequation in the boundary control volume (Anderson,1994). Farfield conditions are handled via a character-istic variable reconstruction. The free surface bound-ary conditions will be presented below - in essence, atsteady-state, the velocity is tangent to the free surface(i.e.,~u · nf s = 0) and the pressure isP = Y
Fr2 , whereY is the elevation of the free surface above the undis-turbed waterline and the Froude number isFr = U∞√
gL.
Turbulence ModelingBoth the one-equation turbulence model of Spalart
and Allmaras (Spalart, etal, 1992) and theq−ω (Coak-ley, 1985), thek− ε and thek−ω two-equation tur-bulence models are available within the solver. Con-stants are taken from the version IIq−ω model givenin (Coakley, 1985). Thek− ε andk−ω turbulencemodels have been newly incorporated within the un-structured flow solver. For the simulations presentedherein, the one-equation turbulence model of Spalart-Allmaras is used.
The diffusive terms in both turbulence models arediscretized in the same manner as the viscous termsfor the mean flow, and the convective terms are com-puted via pure upwinding. Appropriate considerationis given to maintain positive operators in the forma-tion of the Jacobian matrix for the implicit solutionof the transport equation(s). The respective turbu-lence models are incorporated with the mean flow so-lution in a “loosely-coupled” procedure; that is, thecore governing equations are solved first, then the tur-bulence model is solved independently. This proce-dure allows for easy interchange of the turbulence mod-els.
NONLINEAR FREE SURFACE ALGORITHMA nonlinear free surface is obtained for a steady-
state simulation by solving the kinematic free surfaceboundary condition at each time level and imposingthe hydrostatic pressure distribution based on the newfree surface elevation onto the free surface boundarywithin the mean flow. After several time steps, typi-cally on the order of 500, the grid is moved to matchthe free surface elevations while conforming to thesurfaces that intersect the free surface, such as thehull of a ship or the sail of a submarine; and as theloosely coupled interaction between the free surface,the Navier-Stokes equations and the grid movementalgorithm converges, the solution approaches the non-linear free surface solution.
Governing EquationsIn deriving the kinematic free surface boundary
condition, the free surfaceη(x,y,z,t) = 0 is consid-ered as a material surface (i.e., no flow travels throughthe surface), so
DηDt
= 0 (12)
or
∂η∂t
+(u−Vx)∂η∂x
+(v−Vy)∂η∂y
+(w−Vz)∂η∂z
= 0
(13)By making the assumption that the free surfaceη canbe expressed asη(x,y,z,t) = y−Y(x,z,t) = 0, thisequation reduces to
∂Y∂t
+(u−Vx)∂Y∂x
− (v−Vy)+ (w−Vz)∂Y∂z
= 0 (14)
At steady-state, the kinematic boundary condition be-comes
(~u+at) ·(
∂Y∂x
,−1,∂Y∂z
)
= 0
(~u+at) ·αnf s = 0(15)
whereα =
√
1+(
∂Y∂x
)2+
(
∂Y∂z
)2, so the flow is tan-
gent to the free surface at steady-state.The pressure in the numerical simulation at the
free surface is set based on the free surface elevation.For the original, dimensional equations, the pressureat the free surface is the atmospheric pressureP∞. Us-ing the nondimensionalization,P= P∗+ρ∞gy∗−P∞
ρ∞U2∞
with
P∗ = P∞ and y∗L = Y, the pressure is clearly set to
P = YFr2 , where the Froude number isU∞√
gL.
Numerical Approach for Free SurfaceThe kinematic equation for the linear free surface
is solved via a Galerkin finite element approach. Inthis approach, an algebraic equation is obtained foreach nodej by multiplying the governing equation bya weight functionϕ j(x,z), approximating the terms inthe governing equation by using linear and bilinear in-terpolating functionsφi(x,z) and integrating over thecomputational domain, or
Z
Ωϕ j(x,z)
(
∂Y∂t
+(
u− Vx) ∂Y
∂x−
(
v− Vy)
+(
w− Vz) ∂Y
∂z
)
= 0
(16)
where the interpolated free surface elevation isY(x,z)=
∑i Yiφi(x,z) and the velocities are of the form ˜u(x,z) =
∑i uiφi(x,z).For the triangular elements, the resulting equa-
tions are integrated using 7 point Gauss quadrature,and for the quadrilateral elements, the integrals aresolved using 6 point Gauss quadrature in both direc-tions. A first-order backward time discretization ofEquation (16) is used for the temporal derivative, where∂Y∂t = Yn+1−Yn
∆t and the spatial terms evaluated at timelevel n+ 1. This discretization results in a linear al-gebraic system of the formℜ f s(Yn+1,Yn) = 0. Sincethese equations are linear in the unknown variableYn+1
iat each time level (i.e., the velocities are frozen), theJacobian matrix is calculated only once per time level.A Gauss-Seidel iterative solver similar to the one usedto solve the discretized Navier-Stokes equations is usedto identify the solution of the kinematic free surfaceequation by solving
[
∂ℜ f s
∂Yn+1
]
∆yn+1,m = −ℜ f s(Yn+1,m,Yn) (17)
whereYn+1,m+1 =Yn+1,m+∆yn+1,m andYn+1 =Yn+1,m+1
when‖∆yn+1,m‖ < tolerance.Within the viscous boundary layer, special care is
needed in solving the kinematic free surface equation.On viscous surfaces, the flow velocity plus the gridvelocity (i.e.,~u+ at) is set to zero, which preventsthe free surface from moving at the viscous surface.Physically, however, surface tension effects force thefree surface to rise and fall along the viscous surfaces.Since the solver does not simulate surface tension, an-other method must be employed to move the free sur-face near the viscous surfaces. Following the methodused in the structured solver, the free surface at a cer-tain distance from the viscous surface is extended at aconstant height to the viscous surface. This distanceis typically on the order of 2×10−4 times the charac-teristic lengthL, whereas the point spacing off of theviscous surface is on the order 10−6 for model scalesimulations and 10−8 for full scale simulations.
Free Surface Boundary ConditionAt the free surface boundary, the pressure is set to
the hydrostatic pressure ofP = YFr2 . Determining the
velocity to impose on the boundary is more difficult.If the velocity were forced to be tangent to the freesurface (i.e.,(~u+at) · nf s = 0), then the free surfaceequation would reduce to∂Y
∂t = 0, and the free surfacecould not evolve. Thus, this tangency condition is re-laxed to allow the free surface to evolve.
This boundary condition is implemented using acharacteristic variable boundary condition, similar to
the derivation used for the farfield boundary condi-tion. For hyperbolic systems, flow information travelsalong characteristics; and for three-dimensional in-compressible flow, three characteristics originate up-stream, and one originates downstream. If the flow istraveling out of the domain (i.e,(~u+at) · n > 0), thenonly one characteristic needs to be specified from theoutside, which is derived from the hydrostatic pres-sure. If the flow is traveling into the domain, threecharacteristics need to be specified from the outside.The characteristic variables are derived from consid-ering the inviscid fluxes across the the boundary face,via
∂Q∂t
+ ∇F ·~n = 0 (18)
where~n is the normal to the grid for each boundaryface. The inviscid flux term can be rewritten as
∇F =∂F∂Q
∇Q = RΛR−1∇Q (19)
whereΛ is a diagonal matrix with the eigenvaluesΘ,Θ, Θ + c andΘ− c, whereΘ = unx + vny + wnz+ at
andc =
√
(
Θ− at2
)2+ β. Premultiplying byR−1,
R−1 ∂Q∂t
+ ΛR−1∇Q ·~n = 0 (20)
Freezing the matrixR−1o = R−1(Qin),
∂R−1o Q∂t
+ Λ∇(
R−1o Q
)
·~n = 0 (21)
Using the definitionW(Q) = R−1o Q, Equation (18) de-
couples into four equations for the four characteristicvariablesW as
∂W∂t
+ Λ∇W ·~n = 0 (22)
Three of the characteristics are obtained from upstreamvalues, and one of the characteristics is obtained fromdownstream values. In the case of a boundary face,there are a set of characteristics associated with theinside flowWin(Qin) and a set of characteristics as-sociated with the outside flowWout(Qout). The flowvariables on the boundary face are determined fromthe appropriate characteristic variables viaQ = RoW.For flow traveling out of the domain, the characteristic
variables are chosen to be
W =
win1
win2
win3
wout4
(23)
because the fourth eigenvalueΘ−c is negative, whilethe other eigenvalues are positive. For flow travelinginto the domain, only the third eigenvalue is positive,so the characteristic variables are chosen as
W =
wout1
wout2
win3
wout4
(24)
For farfield boundary conditions, the characteristic vari-ables associated with the outside flow are taken asthe freestream variables. For free surface boundaryconditions, the characteristic variables for the outsideflow are determined from the hydrostatic pressure andthe only available velocity information, which is thevelocity on the inside. However, this velocity is mod-ified by removing a portion of the velocity componentthat is not tangent to the grid, via
~uout =~u− γ((~u+at) · n) n (25)
where
γ =
−(~u+at) · n if −1≤ (~u+at) · n < 0
1 if (~u+at) · n < −1(26)
Whenγ is 1, the velocity imposed from the outside istangent to the grid (i.e.,~uout · n = 0). When the ve-locity was not modified or constrained in some fash-ion, the free surface algorithm became unstable forflow into the domain due to the inconsistency of us-ing downstream information. This instability was asevere difficulty for the transom stern of Model 5415,but was not a difficulty for previous cases such as theWigley Hull and Series 60. This modification pro-vides enough control over the velocity to maintain sta-bility of the algorithm, and if the grid is allowed tomove to match the free surface, then this modificationensures that the flow will be tangent to the free surfaceat convergence. When using the grid speed termsat ,such as for the constant turning case, the grid speedterms are not restricted via equation (25), only the ve-locity ~uout is restricted.
Grid Movement AlgorithmAfter several time steps, the grid is moved to match
the free surface while conforming to any solid sur-
faces intersecting the free surface, with displacementson the surface being propagated into the volume grid.Several methods are available for this grid movement,including the use of a Laplacian solver to propagatethe surface perturbations into the volume grid, the useof a linear spring analogy where each edge is a springwhose stiffness is determined by the length of the edge,a torsional/linear spring analogy (Farhat, 1998) wherethe angle between the edges affects the stiffness ofthe springs in the mesh, or solving the linear elas-ticity equations to propagate the perturbations withinthe grid. The spring analogy is computationally effi-cient but is not robust. Both the spring analogy andthe Laplacian solver generate elements with negativevolumes for moderate amounts of movement, on theorder of the size of 3 to 4 elements. Solving the linearelasticity equations to move the grid is robust, but thecomputational cost associated with this method is theprimary drawback.
Thus, following the work of Farhat and extend-ing it to three dimensional grids, the torsional/linearspring analogy has been developed and applied to theproblem of moving the grid to match the linear freesurface elevations. For tetrahedral grids, this methodis quite robust, allowing severe distortions on the sur-face while maintaining positive volume elements. Inpractice, this algorithm allows up to approximately80% compression of elements. Within the boundarylayer, (i.e., near the viscous surfaces), the grid movesthe same as the points on the boundary. This gridmovement method is computationally costly but pro-vides excellent robustness for the free surface solver.
The linear spring method for moving an unstruc-tured grid is presented in (Batina, 1989) where eachedge in the grid is replaced by a linear spring whosestiffness is inversely proportional to the length of theedge. Unfortunately, this method often fails for com-plicated geometries and for large changes in the bound-ary, because negative areas are generated when a nodecrosses over an edge in the grid. The creation of neg-ative areas is illustrated in Figure 1, where node 1 ispushed downwards and node 4 is forced to cross theedge between nodes 2 and 3.
4
1
1
1
4
4
2 3 2 3 2 3
Figure 1. Negative Areas Produced by Linear SpringMethod.
The reason for this failure is that the stiffness in thelinear spring method prevents two nodes from collid-ing but does not prevent a node from crossing over an
edge. As two nodes get closer together, the stiffnessincreases without bound preventing the collision; butthere is no mechanism to prevent a node from crossingan edge, because these crossovers can occur withoutthe stiffness increasing without bound.
To provide a more robust movement algorithmfor unstructured grid, Farhat developed an algorithmto prevent a node from crossing an edge by using tor-sional springs around each node, as shown in Figure2.
θ iijk
Torsional SpringLinear Spring
Linear Spring
k
ji
Figure 2. Placement of Linear and Torsion Springs.
The stiffness of the torsion springCi jk is inverselyproportional to the sine of the angle, so that the stiff-ness grows without bound as the angle decreases to-wards zero or increases towards 180o, or
Ci jk =1
sin2 θi jki
(27)
whereθi jki is the angle centered at nodei formed from
the edgesi j and ik. Thus, as a node moves towardsan edge, the angle goes towards zero, and the stiff-ness of the torsion spring grows. The sine of the an-gle is squared to prevent a negative stiffness. In addi-tion, multiplying the stiffness by term that is inverselyproportional to the distance to the viscous surfaces(Murayama, 2002) further improves robustness and isused here as well.
Farhat has derived the equations for the torsionalspring methods for two-dimensional grids and pro-vides an iterative solution method. In the present work,these methods have been extended into three dimen-sions, which provides an adequate level of robustnessfor the grid movement algorithm. The derivations ofthe three-dimensional torsional spring equations arebeyond the scope of this paper and will be presentedelsewhere.
The grid movement algorithm is implemented viaseveral steps, with the goal of moving the grid to matchthe free surface while conforming to any geometrythat intersects the free surface, such as a hull. Thesesteps are described below:
1. Move nodes along surface to surface edges. Thereare several different types of edges, dependingon each surface’s relationship to the free surface.The nodal locations are determined based on thetype of edge. For instance, an edge that lies fullywithin the free surface has its y-component de-termined by the free surface value, but its x andz-components are determined by the surface’s re-lationship to the free surface. If the surface is pro-jected in the z-direction, then the z-component isdetermined by this projection, while its x-compo-nent is determined by any stretching at the end-points. When this process is attempted acrossseveral processor blocks, moving nodes in the edgesbecomes quite complicated.
2. Move nodes in the surface. Based on the dis-placements at the edges of each surface, the nodesin the surface are moved via torsional springs.Before this is attempted, however, the nodes mustbe projected onto a planar surface, requiring thateach surface be “projectible”, such that the sur-face can be uniquely projected onto a plane insome constant direction.
3. Once the nodes are moved in the projected sur-face, they must be projected onto a backgroundsurface representing the geometry. This back-ground surface is significantly more resolved thanthe grid, and this background surface must extendabove the original waterline, so as to allow formotion of the free surface above the waterline.
4. Move surface displacements into the prismatic,boundary layer grid, along boundary layer linesoriginating at each node in the viscous surface.
5. Move nodes in the volume based on surface dis-placements in all coordinate directions
EXAMPLESThese examples demonstrate the changes in the
free surface profiles and the forces on the hull forchanges in the grid resolution as well as for changesin incoming flow angles and for changes in the turn-ing radius. The results from the grid refinement studyindicate that the general features of the flow are ob-tained for each grid, but that the level of grid refine-ment has not reached the “grid independent” level,in that changes in grid refinement generates a notice-able change in the free surface from one grid to thenext. Furthermore, the vortical structures under thehull change even more dramatically from one grid tothe next, which is beyond the scope of this report.
These simulations were completed on an in-houseLINUX cluster consisting of 1,024 processors, whichconsists of 1.0 GHz and 1.266 GHz Pentium III pro-cessors, with approximately 1.0 Gigabytes of RAMfor each pair of processors. The grids are partitioned
such that approximately 100,000 nodes are in eachpartition. Hence, the largest grid is partitioned into50 pieces to run on 50 processors.
Much of the work in this paper is based on lessonslearned from simulations completed through the De-partment of Defense High Performance Computing(HPC) Challenge Project, including computational timeon the IBM-SP3 at the Naval Oceanographic OfficeMSRC, the IBM-SP3 at the Maui High PerformanceComputing Center and the Cray T3E at U. S. ArmyEngineer Research and Development Center MSRC(Kim,2003)
Grid Refinement StudyThe free surface flow around the DTMB Model
5415 Series hullform was simulated for six viscousgrids with different point spacings, in order to deter-mine the appropriate level of grid refinement for ac-curate free surface simulations and force and momentpredictions. No attempt was made in the grid refine-ment study to insure that the grids were geometricallysimilar, in that control volumes were consistently di-vided from grid level to the next. In each grid level,the overall grid spacing was reduced by a factor of 0.8.By reducing the point spacing at the controlling pointswithin the geometry, the point spacing on the surfacesand within the volume were also reduced by a simi-lar amount. In addition, the point spacing in the sternregion and along the hull behind the bow wave werefurther reduced, because of the sensitivity of these re-gions to changes in the point spacings. The size ofeach grid is shown in Table 1. The prisms are gen-erated off of the viscous surfaces, which consist oftriangular faces. The point spacing off of the viscoussurfaces is selected to produce ay+ of approximately1 on the viscous surfaces. The stack of prisms origi-nating from each triangular face terminates when theaspect ratio of the prisms is approximately 1.0, so asthe grid on the viscous surface is refined, the thick-ness of the prismatic region within the volume griddecreases. The effects of this boundary layer growthare outside the scope of this paper.
Grid Level Nodes Tetrahedra Prisms
Viscous 1 477 K 891 K 603 KViscous 2 789 K 1,356 K 1,049 KViscous 3 1,184 K 2,383 K 1,459 KViscous 4 1,878 K 3,734 K 2,333 KViscous 5 2,563 K 6,377 K 2,759 KViscous 6 4,114 K 11,282 K 4,082 K
Table 1. Grid Sizes for Grid Refinement Study.
The force in the x-direction (i.e., drag) for eachgrid are tabulated below and are divided into three dif-ferent parts, the force based on the dynamic pressure(Dyn. P), the force based on the hydrostatic pressure
(Hydro. P) and the viscous force (Visc. Drag). Thehydrostatic pressure force is defined herein as the sur-face integral of the vertical locationy divided by thesquare of the Froude number dotted into the normaldirection, over the viscous surface∂Ω, or
Z
∂ΩyndS (28)
The hydrostatic pressure force is similar to the pres-sure attributable to the weight of the water on the sur-face. For fully-submerged objects, such as a subma-rine, this integral is equivalent to the weight of thedisplaced fluid and points in the downward direction.However, for surface vessels, the free surface maycause a non-zero component in the other directions,as well.
The viscous component is the primary influencein the drag, but the hydrostatic pressure contributionrepresents approximately 10% of the drag, which areshown in Table 2. In order to predict this component,the free surface must be simulated. Furthermore, thefree surface simulation also directly affects the dy-namic pressure by changing the amount of the wet-ted surface area over which the dynamic pressure acts,and by changing the pressure distribution on these sur-faces. The effects of the free surface on the viscous ef-fects is much more subtle, especially in understandinghow the free surface interacts with the vortical struc-tures, changing their strength and location. The vis-cous drag drops slightly as the grid is refined, whilethe hydrostatic pressure contribution stays almost con-stant as would be expected since it only depends onthe free surface profile at the hull. The dynamic pres-sure changes more, which is probably a result of changesof the free surface in the stern. These results are com-pared with the experimental results as published in theGothenburg 2000 Proceedings.
Grid Dyn. P Hydro. P Visc. Drag Total
1 2.160 0.887 4.214 7.2612 1.889 0.812 4.105 6.8063 1.643 0.816 4.088 6.5474 2.222 0.850 3.820 6.8925 2.340 0.810 4.028 7.1786 2.520 0.882 3.824 7.226
Exp. 1.36 na 4.93 6.29
Table 2. Force in x-Direction (i.e., drag)×104.
The free surface for this sequence of grids is shownin Figures 3 and 4, which highlight the bow wave peakand trough and the stern. The height of the bow waveis approximately the same for each grid, although itdoes vary within a banded range, similar to the errorbars for the experimental solution. The profile in the
bow trough also varies from one grid to the next, in-dicating that the grids are not in the “grid asymptotic”range, where changes in the grid do not change the re-sult significantly. In the bow region, the free surfaceis reasonably symmetric, since the port and starboardsides are almost indistinguishable.
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Grid 1Grid 2Grid 3Grid 4Grid 5Grid 6Structured ResultExperimental Results
Grid Refinement StudyFree Surface Profile Along Port Side
Figure 3. Comparison of Free Surface Profiles AlongPort Side.
But in the stern, the free surface is not symmet-ric, even though every effort has been made to insuresymmetry. The reason for this asymmetry is the in-teraction between the free surface, the grid movementalgorithm and the turbulence effects. Slight asymme-tries in the free surface cause the free surface grid tobecome asymmetric once the grid is moved to matchthe free surface. This asymmetry in the grid causesthe free surface to become more asymmetric, whichbegins to affect the vortical structures underneath theship. Once these asymmetries exist in the vorticalstructures, the free surface is completely altered, align-ing with the vortical structures. Returning the freesurface to a symmetric flow state after this asymme-try has set up is quite challenging. Also, in the sternprofile, the elevations for the first three grids is sig-nificantly higher than for the last three grids. As canbe seen from the roughness of the curve, the resolu-tion in the stern region for the first three grids is notenough to capture the flow features. For the last threegrids, much effort has been made to ensure that thegrid along the free surface is symmetric, which affectsthe wave elevations. These simulations have not yetconverged, due to asymmetries in the vortical struc-tures. The drag is very sensitive to the free surfaceprofile in the stern region, and since the free surfacein the stern is not yet converged, the forces fluctuategreatly.
-0.04 -0.02 0 0.02 0.04Distance Along Stern (Z/L)
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StructuredGrid 1Grid 2Grid 3Grid 4Grid 5Grid 6
Grid Refinement StudyFree Surface Elevation Along Stern
Figure 4. Comparison of Free Surface Profiles AlongStern.
In Figure 5, the vortical structure underneath theModel 5415 and the free surface contours showing thebow and stern waves are shown. These results arefrom the most refined grid, Grid 6. The vortex shedfrom the bulbous bow can be clearly seen for much ofthe length of the hull. In the stern, the vortical struc-ture is quite complicated and is not symmetric, whichadversely affects the symmetry of the free surface. Asstated above, the interaction between the free surface,the grid movement algorithm and the vortical struc-tures is quite complicated and prevents truly symmet-ric flow. The free surface contours show that the sternwave and the bow wave continue downstream of theiroriginations.
Figure 5. Free Surface Contours and VorticalStructure for Grid 6.
Angles of DriftsTwo different steady-state maneuvers were simu-
lated for the DTMB Model 5415 series hullform. The
first maneuver was flow coming at the hull at differ-ent angles of yaw, simulating the effects of the shipdrifting into the flow. The sink and trim were dif-ferent for these simulations and hence the forces willnot agree with those listed above. For this simulation,the grid was a symmetric grid, consisting of 4.8 mil-lion nodes, with 4.6 million prisms and 14.1 milliontetrahedra, which is more resolved that Grid 6 in therefinement study. The goal of these simulations wasto determine whether the free surface algorithm, andespecially the grid movement algorithm, could satis-factorily capture the free surface in the bow and sternregions. In the bow region, the bow wave grows toan excessive height, whereas in the stern region, thefree surface is quite complicated. Six different an-gles of drift were simulated, from 0 degrees to 5 de-grees of yaw. As the angle of drift increases, theside force grows dramatically. At 0 degree angle ofdrift, the flow should be symmetric, so that the sideforce should be zero. Special effort has been made toenforce symmetry for the 0 degree case, which ad-versely affected the dynamic pressure for the drag,and even with this special effort, symmetric flow hasnot quite been achieved. The forces in the x-direction(i.e., drag) and in the z-direction (i.e., side-force) areshown in Tables 3 and 4. The change in the forcesin Table 3, especially the viscous drag which comesfrom the skin friction, varies monotonically with achange in the angle. The slight drop in the dynamicpressure for Angle 4 may be a result of an under con-verged solution.
Angle Dyn. P Hydro. P Visc. Drag Total
0 2.252 0.821 4.406 7.4791 2.742 0.907 4.423 8.0722 2.811 0.956 4.440 8.2073 2.876 1.045 4.462 8.3834 2.823 1.172 4.479 8.4745 3.122 1.335 4.507 8.964
Table 3. Force in x-Direction (i.e., drag).
The contributions to the side force are primarilyfrom the difference in the dynamic pressure and hy-drostatic pressure on the starboard and port sides ofthe hull caused by the flow approaching the ship atan angle. These forces scale linearly with the anglefor the smaller angles and then they grow faster thanlinearly as the angle increases. As the flow angle in-creases, the ship’s silhouette changes and the flow fea-tures change. For small angles, this change can berepresented by a perturbation from flow at a 0 degreeangle of yaw, but for larger angles, the flow patternsare significantly different, resulting in greater forces.
Angle Dyn. P Hydro. P Vis. Drag Total
0 0.062 0.006 0.007 00751 2.382 0.757 0.033 3.1722 4.832 1.527 0.075 6.4343 7.272 2.334 0.133 9.7394 9.891 3.162 0.167 13.2205 12.934 4.079 0.199 17.212
Table 4. Force in z-Direction (i.e., Side-Force).
The free surface along the hull is shown in the fol-lowing figures along the port (Figure 7) and starboardside (Figure 8) and along the stern (Figure 9). Theflow is approaching from the starboard side, which re-sults in highly bow waves on the starboard side. Thechange in the free surface profiles for the bow waveare almost linear on both the port and starboard sides,not only at the bow but also along the entire length ofthe ship. The deflections in the stern are also as ex-pected, showing a gradual change from the symmet-ric flow. The image for the stern region shows resultsfrom a previous simulation, which is more convergedthan the solution on the current grid, but used a dif-ferent version of the free surface algorithm. For thesesimulations, the free surface algorithm and the gridmovement algorithm are adequate for the free surfacedistortions.
0 0.2 0.4 0.6 0.8 1Distance Along Ship (X/L)
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Angle = 0Angle = 1Angle = 2Angle = 3Angle = 4Angle = 5
Free Surface Profiles on Port SideConstant Angles of Drift
Figure 7. Free Surface Profiles Along Port Side.
0 0.2 0.4 0.6 0.8 1Distance Along Ship (X/L)
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Angle = 0Angle = 1Angle = 2Angle = 3Angle = 4Angle = 5
Free Surface Profile For Starboard SideConstant Angles of Drift
Figure 8. Free Surface Profiles Along Starboard Side.
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Angle 1Angle 2Angle 3Angle 4Angle 5
Free Surface Along SternConstant Drift Angles
Figure 9. Free Surface Profiles Along Stern.
Finally, the vortex originating from the bulbousbow is shown in Figure 10, as well as the free surfacecontours for the bow wave and the stern wave. Thisimage is for the case of the 5 degree turn.
Figure 10. Free Surface Contours and VorticalStructure for Angle 5.
Constant Turning Radius
The second steady-state maneuvering case involvedsimulating the flow around the DTMB Model 5415 se-ries hullform in a constant turn, with turning radii of 6,12 and 24 times the hull length. For the tightest case,involving the turn of radius 6 times the hull length,the flow at the bow has an angle of approximately 5degrees with the axis of the ship. For this simulation,the grid speed terms are non-zero while the farfieldconditions are set to zero, so different aspects of thecode are tested. The grid used for this simulation hada similar number of nodes as Grid 4 in the grid refine-ment study, but it differed in the placement of thesepoints.
Radius Dyn. P Hydro. P Visc. Drag Total
24 L 2.481 1.057 4.135 7.67312 L 2.581 1.113 4.146 7.846 L 2.951 1.452 4.220 8.623
Table 5. Force in x-Direction (i.e., drag).
The forces generated by these simulations are givenin Table 5 and 6, showing the force in the x-direction,which is akin to drag, and the force in the z-direction,which is the side force. For the force in the x-direction,the viscous drag varies only slightly as the turningradius gets tighter, but the dynamic and hydrostaticpressure contributions increase as the turning radiusdecreases, indicating that the ship experiences greaterdrag through a tighter turn. The two pressure contri-butions to the side force approximately double as theturning radius halves, and the viscous force contribu-tion to the side force is minimal.
Radius Dyn. P Hydro. P Visc. Total
24 L -2.249 -1.113 0.002 3.36012 L -4.482 -2.234 0.016 6.7006 L -9.457 -4.703 0.035 -14.125
Table 5. Force in z-Direction (i.e., Side-Force).
The free surface profile along the port and star-board sides are shown in Figure 11. As would beexpected, the free surface rises on the starboard side,which is the side turning into the flow and drops onthe port side. As was the case with the angles of drift,the free surface profiles vary continuously with thechange in the radius of the turn.
0 0.2 0.4 0.6 0.8 1Distance Along Hull (X/L)
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Radius = 6 L (Starboard)Radius = 6 L (Port)Radius = 12 L (Starboard)Radius = 12 L (Port)Radius = 24 L (Starboard)Radius = 24 L (Port)
Free Surface Profiles for Constant Turns
Figure 11. Free Surface Profiles along Port andStarboard.
Finally, the vortical structure and the free surfacecontours are shown in Figure 12. The vortical struc-ture arising from the bulbous bow is quite strong andcontinues far downstream of the bow as it curves withthe rotation of the ship. There is a strong vortex thatoriginates at one corner of the stern and curves in re-sponse to the rise and fall of the free surface. The freesurface also curves with the rotation of the ship.
Figure 12. Free Surface Contours and VorticalStructure for Radius 6 L Turn.
Prescribed Linear Free Surface ManeuverFor the final case, the Model 5415 hullform is
forced through a prescribed maneuver to study thefree surface through a maneuver. The data for themaneuver was generated by using a rudder-induced,propulsor-driven3 DOF maneuver for the Model 5415without a free surface. In this simulation, the freesurface was simulated as a symmetry plane, so thatthe free surface effects could be ignored. A fully-appended geometry of the Model 5415 was used, withthe propulsors rotating to power the ship and with therudders deflecting through the turn. The time step
was 0.0002068 seconds per iteration, with each itera-tion taking approximately 60 seconds, and there were3230 iterations in the maneuver. The ship turns toapproximately 8 degrees from the original angle andreaches an angle with the incoming flow direction ofslightly more than 6 degrees.
For the simulation presented herein, a steady-statesimulation is performed with the ship traveling straightand level before the maneuver begins. This simulationdoes not have the rudders, propulsors or appendages,but the free surface is activated. The grid is not movedthrough the maneuver due to the computational costand the probable instabilities associated with movingthe grid frequently, but the grid at the beginning ofthe simulation is moved to match the steady-state freesurface for the incoming flow.
For the case of the constant angles of drift, thesimulation was performed by changing the flow di-rection at the farfield boundaries, hence imposing theangle of drift into the simulation. For the case ofconstant turning radii, a different aspect of the codewas used, which involved the grid speed termsat =−(Vxnx +Vyny +Vznz) while setting the farfield ve-locities to zero. These grid speed terms were con-stant throughout the simulation. For the case of theprescribed maneuver, the grid speed terms again wereenabled, but they varied through the simulation as theship performed its maneuver. Hence, this test case ex-ercised a different portion of the code.
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2
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Port/Starboard (Before Maneuver)
Stern (Before Maneuver)
Port (After Maneuver)
Starboard (After Maneuver)
Stern (After Maneuver)
Figure 13. Free Surface along Hull for Maneuver.
One of the goals of this simulation was to deter-mine how high the free surface would be along thehull. In Figure 13, the free surface along the hull be-fore the maneuver and after the maneuver is shown.The free surface on the starboard side rises dramati-cally as the ship turns to starboard, while the free sur-face on the port drops off quite dramatically. Further-more, the free surface profile along the ship’s hull is
quite different, inducing a significant force imbalancefrom the starboard to the port side. And even thoughthe ship reaches only an angle of only slightly morethan 6 degrees with the incoming flow, the free sur-face rises to a height significantly higher than for the5 degree drift case or the case with the tightest turningradius. For this case, the grid was fixed and was not al-lowed to conform to the free surface. If the grid weremoved, then the free surface would probably becomeeven more extreme, and may approach the limit of thegrid movement algorithm to match the free surface.
Figure 14. Free Surface Contours Before Maneuver.
The primary goal of this simulation, beyond determin-ing whether the free surface elevations would growtoo large for the grid movement algorithm, was to seewhether the free surface would respond properly asthe grid was rotated to match the ship’s maneuver. Inthis maneuver, the entire grid was rotated and trans-lated with the ship, as it moved. However, the freesurface behind the ship should not rotate with the shipor the grid because it should not feel the effects of therotation, but rather, the free surface should stay rela-tively fixed in relation to the original location of theship. In other words, once the wave has left the ship,the behavior of the ship should not affect the wave.The free surfaces at the beginning and the end of themaneuver are shown in Figures 14 and 15, respec-tively. The free surface in Figure 14 is nearly symmet-ric about the centerline, but it does have some level ofasymmetry, as discussed above. In Figure 15, the freesurface aft of the ship again has some asymmetries,due to the asymmetries in the original free surface,but the free surface has not rotated with the ship orthe grid, but remains in its original orientation.
Figure 15. Free Surface Contours Before Maneuver.
CONCLUSIONSA robust nonlinear free surface algorithm has been
implemented within an unstructured finite volume flowsolver of the the turbulent three-dimensional Reynolds-averaged Navier-Stokes equations. The free surfacealgorithm is a surface tracking algorithm which movesthe grid to match the free surface while conforming tothe hull and is loosely coupled to the flow solver via acharacteristic variable boundary condition impositionof the free surface pressure distribution. This bound-ary condition implementation also enforces flow tan-gency along the free surface at convergence. Currentsimulations also indicate that the free surface algo-rithm is applicable to rotating and translating grids.
Results from a grid refinement study for the DTMBModel 5415 barehull form are provided which indi-cate that even more resolution is needed for a grid in-dependent solution to be obtained. A discussion of thehydrostatic pressure force is provided which is an ad-ditional force not observed for fully-submerged sim-ulations such as for a submerged submarine or for anaircraft in flight.
Several initial simulations progressing towards atrue 6-degree-of-freedom(6-DOF) rudder-induced, pro-peller-driven maneuver in unsteady seas were com-pleted, including various angles of drift up to 5 de-grees and constant turns of various turning radius, withturns into the flow of up to 5 degrees, and one pre-scribed maneuver. The goal of these simulations wasto determine whether the free surface algorithm couldmodel such moderate maneuvers, in that the free sur-face would stay attached to the stern and that the gridmovement algorithm could move the grid for such ex-treme grid motions, especially in the bow region. Forthe angle of drift and the constant turn cases, the freesurface profiles gradually change over the spectrumof runs, as do the force predictions. In addition, thevortical structures respond to the changes in the flowconditions as expected. For the prescribed maneuver,
with fixed grid, the free surface contours that are aftof the ship do not rotate as the ship turns, which isphysically realistic, and the vortical structures againbehave appropriately.
These successful steady-state and linear free sur-face maneuvering simulations indicate that this ap-proach to modeling the free surface can be useful forfully-appended, powered and self-directed 6-DOF ma-neuvers. Previous work has shown that the grid gener-ation algorithm and the flow solver along with the freesurface can build grids and simulate flow for fully-appended cases with rotating propulsors and at bothmodel and full-scale Reynolds numbers. Continuedresearch is needed to analyze, implement and test afully-unsteady nonlinear free surface boundary con-dition implementation as well as to improve the com-putational efficiency of the grid movement algorithmso that the grid can be moved more frequently. Stud-ies of the impact of moving the grid at each time stepare also needed. Finally, a better understanding of thechoice of turbulence model is needed, with the hopesthat changing the turbulence model will improve theforce predictions.
ACKNOWLEDGMENTThis research was sponsored by the Office of Naval
Research with Dr. L. Patrick Purtell grant monitor andwith Ki-Han Kim as the project leader for the HPCChallenge Project. This support is gratefully acknowl-edged.
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