Mat-1.3656 Seminar on
numerical analysis and computational science
Monday, November 30, 2009, room
U322 at 14.15, Eirola & Stenberg
Tommi Mikkola, TKK Marine Technology
Simulation of unsteady free surface flows - code
verification and discretisation error
In
this work a numerical method for the solution of unsteady, inviscid
free surface flows is developed. The method is verified and the
behaviour of the error related to the numerical method, as the
discretisation is refined, is studied in detail. The work divides into
two distinct parts. The first one focuses on the development of the
solution method. The method is based on unstructured, two dimensional
finite volume method. The free surface boundary conditions are
satisfied on the instantaneous free surface and the computational grid
tracks the deformation of the surface. Typically, in comparable methods
the flow and free surface solutions are solved by time integrating the
governing equations in two separate stages, which are iterated. The
decoupling of the solutions limits the allowable time step in the
integration, which makes the approach computationally expensive. In
this work two different approaches are presented for the coupling of
the solutions, which relax the time step restriction. The approaches
that are proposed differ significantly from the coupling approaches
presented previously in the literature in that the implementation into
the existing pressure correction type solvers is straightforward. The
second part concentrates on the verification of the implementation of
the numerical method, i.e. on code verification, and on the
investigation of the error related to the discretisation of the
continuous problem. In both cases, the analysis is based on the method
of manufactured solutions (MMS), in which the governing equations are
modified, so that the modified equations have a desired analytical
solution. The difference to previous studies is that here the technique
has been applied for the verification of an unsteady free surface
solution method. The verification of such methods has typically been
based on i.a. the use of approximate, high order solutions. MMS has the
advantage that the numerical solution can be compared with an exact,
analytical solution. It is demonstrated in the work that the governing
equations were implemented correctly into the developed method and that
the method is of second order of accuracy. In addition to the code
verification, MMS is used to study the influence of different
discretisations and grid refinement strategies on the local error and
its convergence. In case of the verification of the free surface
solution method the investigation based on a global error norm is
extended with an analysis of the Fourier components of the error. A two
parameter, approximate model is presented for the temporal variation of
the primary component of the solution, with which it is possible to
deepen the verification. The model is also used for an uncertainty
estimation.