In this expression, is the Jacobian of the numerical flux vector, with the first subscript representing the position of the cell face of the numerical flux vector and the second subscript representing the position of the dependent variable vector that the numerical flux vector is differentiated with respect to. I_a is an identity matrix, except the first diagonal element is zero in order to satisfy the true incompressible continuity equation.

A linear system of equations must be solved at each iteration of Newton's method. For three-dimensional problems a direct solution seems to be impractical [17] and in this work symmetric Gauss-Seidel relaxation is used. Because the flux Jacobian of the flux vector based on Roe's formulation is difficult to obtain analytically in three dimensions, and also in the interest of simplicity, the flux Jacobian is obtained numerically [18]. The solution scheme is referred to as discretized Newton-relaxation [16], or the DNR scheme [17]. Multigrid is used to accelerate the numerical solutions [19]. This multigrid scheme has been extended to multiblock [20] and unsteady flow [21]. The solution process is, therefore, a multigrid scheme for three-dimensional unsteady viscous flow on dynamic relative motion multiblock grids.

From the beginning of the program it was determined that there needed to be two methods of handling the propulsor. One was to be a simulation of the propulsor using a relatively simple model that was computationally efficient, and the other was to incorporate the capability of handling the actual rotating propulsor. Both methods were developed, have been included in the code, and are discussed below.

The model selected that satisfies the conditions of simplicity and efficiency was one that has been used for a number of years for similar simulations such as open propellers [22] and ducted fans [23]. The basic approach is explained in [22] and consists of including body forces in Eq. (1) which operate on the fluid in a manner similar to the way an actual propulsor operates on the fluid. All three components of the body force vector were taken into account and consequently thrust, swirl, and their radial distributions can be included in the computations. In any given situation this force data may be obtained from conventional propulsor design tools or data bases. Here this information is based on thrust and torque coefficient data which were obtained from an actual marine propulsor. A description of the experiment and the measured results are reported in [24]. This coefficient information was used to determine the components of the body force vector and then distribute these components to the center of the cells in this cell-centered finite volume scheme in the region where the propulsor was located.

The method used to include an actual rotating propulsor is one that has been continually developed and also used for a number of years, primarily for compressible flows [25–31]. The approach is to use relative motion blocks with structured grids, whereby the blocks that include the rotating blades move relative to adjacent blocks with a region of the blocks near the relative motion interface being treated using the localized grid distortion technique introduced by Janus [31]. This method of handling relative motion blocks insures the continuity of grid lines (although they do change partners periodically, or “click”) and restricts the maximum distortion of the grid to be of the order of the distance between grid points which for viscous grids is, of course, small. This approach eliminates the need to interpolate the solution vector from one grid to another. The cell volumes do change in time, however, and the geometric conservation law must be satisfied [31].

The trajectory of the body at any instant of time is described by its linear velocities u, v, w, and by its angular velocities p, q, r, in the body fixed frame of reference and its position and orientation in an inertial frame of reference. The governing six-degree-of-freedom (6-DOF) Equations Of Motion (EOM) can be written as:

Axial Force

Lateral Force

Normal Force