# Simulation of unsteady free surface flows - code verification and discretisation error

Tutkimustuotos: Doctoral ThesisMonograph

## Abstrakti

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.
Alkuperäiskieli Englanti Tohtorintutkinto Helsinki University of Technology Espoo Helsinki University of Technology 978-952-248-136-8 978-952-248-137-5 Julkaistu - 2009 G4 Tohtorinväitöskirja (monografia)