the past few years (References 1–5). IFLOW incorporates most of the recently developed numerical techniques, including multiblock, multigrid, local refinement, preconditioning, and adaptive artificial dissipation model. The code is designed for general-purpose applications. It can be used for arbitrary 2D and 3D complex geometries and it can also be used to solve both steady and unsteady problems. At present, only the k-ω and the modified Baldwin-Lomax turbulence model are actively being used. These two simple models are quite adequate for attached turbulent boundary layer flows. Modifications of Baldwin-Lomax model are necessary because of some unique flow characteristics often encountered in ship hydrodynamic applications. These include the so-called thick boundary layers near ship sterns and the cross-flow separations typically occurring on maneuvering vehicles. In this paper, the numerical schemes implemented will be described and the modifications necessary to overcome the difficulties mentioned above will be presented. Practical definitions of a converged solution, a grid-independent solution and two measures of errors are given. The computed results on the 4 grids are then analyzed based on the definitions given.

The three-dimensional, incompressible, Reynolds-averaged Navier-Stokes equation is solved using the artificial compressibility approach first proposed by Chorin^{6} and subsequently generalized and improved by Turkel.^{7}^{–}^{8} This approach has been successfully used by Chang and Kwak^{9} and many others. The origin of the body coordinate system is fixed at the center of gravity of the vehicle as shown in Fig.1. The x_{b}-axis is along the longitudinal axis of the vehicle with the positive x_{b} pointing forward, y_{b} in lateral direction with the y_{b} positive pointing to the starboard, and z_{b} in normal direction with the positive z_{b} pointing downward. Fig.2 shows the velocity and angle of inclination of an axisymmetric body undergoing steady right turn. The turning radius is denoted by R _{turn} and the body with a total length L is rotating around a center point C with an angular velocity Ω in the positive z direction (pointing downward). The velocity components [u, v, w] and angular velocity components [p, q, r] relative to the body fixed axes [x_{b}, y_{b}, z_{b}]. The coordinate system attached to the rotating center C is [x, y, z]. The angles of attack and drift α and β are related to the velocity components by sin β=–v/U and tan α= +w/u, where U^{2}=u^{2}+v^{2}+w^{2}. For a constant radius steady turn we have x=x_{b}, z=z_{b} y=y_{b}-R_{turn}, when α=β=0. In hydrodynamics the orientation of a body in space is usually described in terms of angles of yaw (ψ), then of pitch (θ), and finally of roll () and has the following relationship:

When the axisymmetric body executes a constant radius steady turn, the heading angular velocity is equal to the angular velocity of turn and the angles of pitch and roll (=0) are constant. Then:

Let *τ*_{c.f.} be the local resultant cross flow angle positive clockwise (looking forward) with respect to the body coordinate y_{b} (positive starboard). Then:

In this case α=θ and β=ψ. The nondimensional turning angular velocity for small angles of pitch and yaw, where R_{turn} is the turning radius and L is the body length.

The formulation of IFLOW developed at the David Taylor Model Basin is based on the following conservative formulation

(1)

where the subscripts indicate partial differentiations with respect to time t, and the three Cartesian coordinates x, y, and z. The preconditioned matrix P_{o} in the conservative form and the column vectors of the dependent variable q and fluxes F, G, H and the source term S are defined as

(2)