Author Hoekstra, M. Title Numerical simulation of ship stern flows with a space-marching Navier-Stokes method Conference/Journal PhD-thesis, Delft University of Technology Month October Year 1999

Abstract
This thesis is about a numerical method for the simulation of the flow around a ship, moving steadily on a straight course in still water, the focus being on the prediction of the water motion near the ship's aft end. The purpose is a simulation model that can fruitfully be used in the design process (as far as concerned with the hydrodynamic optimization of the hull shape and its propeller), and in the interpretation of model test results.

Even if a ship operates in still water, and not in a wind-disturbed sea, the complexity of the physical problem to be modelled is considerable. This is in the first place caused by the occurrence of turbulence in a part of the flow field. Although there is an adequate mathematical model describing the fluid motion including the turbulence, it is necessary for practical reasons to work with a time-averaged form of this model. The pertinent equations of motion are referred to as the Reynolds averaged Navier-Stokes (RANS) equations. They require to be supplemented with a turbulence model, because the time-averaging has introduced new unknowns, the Reynolds stresses.

A second complication is that the ship moves at the interface of two media, water and air, with widely different density, so that a wave pattern is created. In the method described in this thesis, the wave formation is neglected by treating the undisturbed free surface as a symmetry plane. So we consider effectively the flow around a body deeply submerged in water, where the shape of the body is determined by the composition of the underwater part of the ship and its mirror image (double-body flow).

The flow field to be simulated can thus be classified as an external flow around a more or less streamlined body in a fluid of small viscosity. For normal operating conditions of a ship, the characteristic Reynolds number, based on the length and the speed of the ship, is in the range of, say, 10e6 < Re < 10e9 , if the ship is to be considered both at model scale and in its true size. The flow is steady in a ship-fixed reference frame, while the fluid is assumed to be incompressible.

Because we are concerned with an external flow, a division of the fluid domain into zones is for reasons of computational efficiency a natural thing to do. Viscous effects are noticeable only in a thin layer around the hull and in the wake. Further away from the hull, the fluid behaves as being essentially inviscid, which allows a considerable reduction of the complexity of the mathematical model. On the forward part of the ship the viscous layer is so thin that the boundary-layer equations are an adequate description. The solution of the RANS equations can therefore be restricted to a relatively small part of the fluid domain, enclosing the aft half of the ship and a part of its wake. For the required turbulence model we use the concept of the isotropic eddy viscosity. Two versions are used in this thesis, one based on an algebraic formulation, and the other on a transport equation for the eddy viscosity. The action of the propeller, if included, is modelled by a specified body force distribution (actuator disk).

Analytical solution being beyond reach, a discrete analogue of the RANS equations is established. The discretisation is based on the finite-volume technique. This requires first of all that the selected domain is overlayed with a 3D grid. A boundaryfitted, structured, mono-block, H-O grid is used. Two methods are described by which such a grid can be constructed. The first connects a set of 2D grids, essentially generated by a conformal mapping technique based on the Schwarz-Christoffel transformation. In the second, the grid is found via the solution of a Poisson equation for the grid coordinates.

All flow variables are defined in the cent er of the grid cells (collocated, cellcentered variable arrangement). The equations are integrated for each grid cell and Gauss theorem is applied to yield an integral over the cell faces. These cell face integrals are then approximated, using interpolations of the cell-centered data.

The properties of the continuum equations should as far as possible be carried over to the discrete analogue. Thus, the maintenance of the elliptic character of the equation system in the discrete approximation is discussed in some detail. It leads the way to a proper choice for the discretisation of the continuity equation and the pressure gradient in the momentum equation. Furthermore, an account is given of how flux-limiting properties are added to the discretisation of the convection terms. The resulting discretisation is conservative for mass and momentum, formally second-order accurate, and uniform-flow preserving.

The most distinguishing feature of the method is the solution strategy. While the majority of computation methods for the flow around a ship employs either the pressure-correction or the artificial-compressibility method, our approach is a spacemarching scheme, in which the coupling between the momentum and the continuity equations is maintained in the solution process. Three convergence-accelerating techniques are introduced: grid sequencing, a predictor-corrector method for the pressure and an approximate multigrid strategy. The solution of the system of algebraic equations is accomplished with a coupled incomplete LV decomposition or with GMRES; in the latter case the Ll.I-factorisation is employed as a preconditioner.

Verification of the computational method is carried through in three test cases: the laminar wake of a flat plate, the flow around the aft end of a modified prolate spheroid and the same for the Wigley hull. Convergence of the solution on grids of different density is examined and the effect of the Reynolds number on the convergence rate of the solution process is demonstrated to be practically nil.

Finally, results of application are shown and discussed, and validated against experimental data or other sources. Both the laminar and the turbulent flow along a flat plate are considered; in addition to the modified spheroid, the true spheroid with flow separation at its tail is dealt with. The 3D applications are completely focused on the HSVA tanker. Results are given for the model test condition (Re = 5 x 10e6) and the full scale case (Re = 2 x 10e9 ) , while in a third case the effects of the propeller action are included.

The code is currently in use at MARIN as a tool for quality assessment of hull designs on request of ship yards, navies and other customers.