Numerical simulation of viscous flows with free surface around realistic hull forms with transom

Abstract

SUMMARYThis paper describes a method for simulation of viscous flows with a free surface around realistic hull forms with a transom, which has been developed based on a FINFLO RANS solver with a moving mesh. A dry-transom model is proposed and implemented for the treatment of flows off the transom. The bulk RANS flow with the artificial compressibility is solved by a cell-centred finite volume multigrid scheme and the free surface deformed by wave motions is tracked by satisfying the kinematic and dynamic free-surface boundary conditions on the actual location of the surface. The effects of turbulence on flows are evaluated with the Baldwin-Lomax turbulence model without a wall function. A test case is modern container ship model with a transom, the Hamburg Test Case. The calculated results are validated and they agree well with the measured results in terms of the free-surface waves and the total resistance coefficient. Furthermore, the numerical solutions successfully captured many important features of the complicated interaction of the free surface with viscous flows around transom stern ships. In addition, the convergence performance and the grid refinement studies are also investigated.

Wave overtopping over a sea dike

“…(8). In this study, the variation of the unknown fluxes at the ðn þ 1Þth time step is obtained with the introduction of the so-called Delta form [25,34,44], indicating that a local linearization of the fluxes created by the convective and diffusion terms, respectively, is enforced. As a result, the fluxes at the ðn þ 1Þth time step is formulated by a Taylor series extension to first-order accuracy in terms of the time t. Namely…”

“…Evaluation of the derivatives oF i ou and oF v ou at a certain face In our implicit stage, the derivative oF ou is decomposed as two terms oF i ou and oF v ou , respectively. For the evaluation of the corresponding values, we adopt the two different approximations: one [25,44] is to use an one-order upwind scheme for achievement of oF i ou ; one [25,34] is to neglect the cross derivatives related with the viscid fluxes, when calculating oF v ou with a central-difference scheme. The former maintains the bandwidth of tridiagonal block in the linear equation, resulting in lower memory for the solution of Eq.…”

Computation of turbulent free‐surface flows around modern ships

“…This is the hybrid Cartesian=curvilinear approach for the bulk RANS ow and the free surface, respectively. It has been successfully applied to our viscous free-surface calculations [2,22,32]. Therefore, an outline with additional features associated with the application of the Cartesian co-ordinate system is given using a cell-centred ÿnite-volume (FV) method.…”

“…The inviscid free-surface boundary conditions (IFSBC): If the free-surface boundary layer is neglected and gradients of the velocity are assumed to be zero, thus, this leads to the inviscid dynamic boundary conditions that have been widely applied in the literature [4,5,22] due to its simplicity. In this way, three components of the Cartesian velocities are extrapolated with gradients of zero-normal velocity from the interior of the ow by @u @n = @v @n = @w @n = 0 (27) and the pressure is determined from…”

“…The former can be applied to resolve a discontinuous jump of the density when passing through the interface, especially for large density ratios; the latter is used for interpolated region of the wave elevation in order to cope with a grid that is highly clustered in the near-wall spacing. For transom stern ships concerned, the socalled dry-transom model could be implemented [20][21][22]. The relevant work can be found in References [23][24][25].…”