Minimizing undesired wave reflections at the domain boundaries in flow simulations with free-surface waves based on the Navier-Stokes equations
Minimierung unerwünschter Wellenreflexionen an den Gebietsrändern in Strömungssimulationen mit freier Oberfläche basierend auf den Navier-Stokes Gleichungen
Wave reflections at the boundaries of the computational domain can cause significant errors in flow simulations, and must therefore be reduced. This can be achieved via forcing-zone-type approaches such as absorbing layers, sponge layers, damping zones, or relaxation zones. These approaches gradually force the solution in the vicinity of the boundary towards some prescribed reference solution. Forcing zones are versatile since they can be applied near both wave-generating and wave-absorbing boundaries and can be used for flows with superposed currents. However, the key problem was that forcing zones contain case-dependent parameters (such as the zone thickness or the source term magnitude). Prior to this project work it was not known how to optimally select these parameters. Although forcing-zone-type approaches are widely used in industry and research, it was common practice to use the default parameters or tune by trial and error.
The aim of the project was to develop an analytical approach to estimate the forcing zone's reflection coefficient. For this, an analytical solution for the forcing zone's behavior was derived and verified via flow simulation results with long-crested regular and irregular waves in 2D- and 3D-flow simulations. Computer programs to evaluate the theory were published as free software to facilitate evaluation of the theory for other researchers; they can be downloaded via the following links:
The theory predictions were found to be of satisfactory accuracy for engineering practice, and also applied to 3D-flow simulations with strongly reflecting bodies in waves. It was found that for practical discretizations, forcing zones behave with good approximation independent of the discretization, the order of the underlying schemes and the used flow solver. In all simulations that were performed, including those with highly nonlinear waves, when the forcing zone was tuned using the developed theory, the simulation results for the reflection coefficient were in most cases smaller or nearly equal to those predicted by theory, but never more than 3.4% larger.
The figure below shows results for one forcing zone setup of reflection coefficient C_R versus source term magnitude γ for different values zone thickness x_d.
A forcing zone setup for 3D-flow simulations with wave-reflecting bodies subjected to long-crested far-field waves was devised, and was demonstrated to minimize undesired wave reflections while enabling the use of smaller computational domain sizes than are commonly used. Thus a substantial decrease in computational effort of over 72% was achieved in the investigated cases.
The figure below shows the free-surface elevation for a strongly wave-reflecting body (gray pontoon) with optimally tuned forcing zones (shaded gray domain parts) for a large and small domain; the surface elevations in the vicinity of the body are nearly identical, which demonstrates that already on the small domain reflections were satisfactorily minimized.
Further investigations, including the tuning of forcing zones for 3D-flow simulations with short-crested waves, will be carried out in the follow-up research project.
The project was funded by the Deutsche Forschungsgemeinschaft (DFG) with grant AB 112/11-1.
Prof. Dr.-Ing. Moustafa Abdel-Maksoud
M.Sc. Robinson Peric