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automatic design techniques based on control theory [10] as well as the extension of a time accurate multigrid driven, implicit scheme [11] for the analysis of “seakeeping”, and maneuvering.

2
Mathematical Models

For a Viscous incompressible fluid moving under the influence of gravity, the differential form of the continuity equation and the Reynolds Averaged Navier-Stokes equations (RANS) in a Cartesian coordinate system can be cast, using tensor notation, in the form,

Here, Ūi is the mean velocity components in the xi direction, the mean pressure, and the gravity force acting in the i-th direction, and is the Reynolds stress which requires an additional model for closure. For implementation in a computer code, it is more convenient to use a dimensionless form of the equation which is obtained by dividing all lengths by the ship (body) length L and all velocity by the free stream velocity U. Moreover, one can define a new variable Ψ as the sum of the mean static pressure P minus the hydrostatic component –xkFr–2. Thus the dimensionless form of the RANS becomes:

where is the Froude number and the Reynolds number Re is defined by where v is the kinematic viscosity, and is a dimensionless form of the Reynolds stress.

Figure 1 shows the reference frame and ship location used in this work. A right-handed coordinate system Oxyz, with the origin fixed at the intersection of the bow and the mean free surface is established. The z direction is positive upwards, y is positive towards the starboard side and x is positive in the aft direction. The free stream velocity vector is parallel to the x axis and points in the same direction. The ship hull pierces the uniform flow and is held fixed in place, ie. the ship is not allowed to sink (translate in z direction) or trim (rotate in x–z plane).

It is well known that the closure of the Reynolds averaged system of equation requires a model for the Reynolds stress. There are several alternatives of increasing complexity. Generally speaking, when the flow remains attached to the body, a simple turbulence model based on the Boussinesq hypothesis and the mixing length concept yields predictions which are in good agreement with experimental evidence. For this

Figure 1: Reference Frame and Ship Location

reason a Baldwin and Lomax turbulence model has been initially implemented and tested [14]. On the other hand, more sophisticated models based on the solution of additional differential equations for the component of the Reynolds stress may be required. Notice that when the Reynolds stress vanishes, the form of the equation is identical to that of the Navier Stokes equations. Also, the inviscid form of the Euler equations is recovered in the limit of high Reynolds numbers. Thus, a hierarchy of mathematical model can be easily implemented on a single computer code, allowing study of the controlling mechanisms of the flow. For example, it has been shown in reference [18] that realistic prediction of the wave pattern about an advancing ship can be obtained by using the Euler equations as the mathematical model of the bulk flow, provided that a non-linear evolution of the free surface is accounted for. This is not surprising, since the typical Reynolds number of an advancing vessel is of the order of 108.

Free Surface Boundary Conditions

When the effects of surface tension and viscosity are neglected, the boundary condition on the free surface consists of two equations. The first, the dynamic condition, states that the pressure acting on the free surface is constant. The second, the kinematic condition, states that the free surface is a material surface: once a fluid particle is on the free surface, it forever remains on the surface. The dynamic and kinematic boundary conditions may be expressed as

(1)

where z=β(x,y,t) is the free surface location.

Hull and Farfield Boundary Conditions

The remaining boundaries consist of the ship hull, the meridian, or symmetry plane, and the far field of the computational



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