# Title: Planar capillary waves # # Description: # # A small amplitude sinusoidal wave oscillates under surface # tension. Prosperetti \cite{prosperetti81} found a solution to this # initial value problem in the limit of a vanishingly small initial # amplitude. # # The domain size is 1x3 units, large enough to minimise the effect of # boundaries (Prosperetti's theory is valid for infinite domains). # # Table \ref{convergence} shows the convergence of various solvers as # a function of resolution: Gerris, the marker technique of # \cite{popinet99}, Surfer \cite{gueyffier98}, PROST and CLSVOF # \cite{gerlach2006}. The same data is represented on Figure \ref{fig-convergence}. # # The time-evolution of the amplitude given by Prosperetti's theory # and Gerris ($64^2$) is given on Figure \ref{amplitude}. # # \begin{table}[htbp] # \caption{\label{convergence}Convergence of the relative error between the analytical # solution and simulation results from various solvers.} # \begin{center} # \begin{tabular}{|l|ccccc|} \hline # Method & $8^2$ & $16^2$ & $32^2$ & $64^2$ & $128^2$ \\ \hline # \input{convergence.tex} & 0.000545 \\ # \input{markers.tex} \\ # \input{surfer.tex} \\ # \input{prost.tex} \\ # \input{clsvof.tex} \\ \hline # \end{tabular} # \end{center} # \end{table} # # \begin{figure}[htbp] # \caption{\label{fig-convergence}Convergence of the RMS error as a # function of resolution (number of grid points per wavelength) for # the methods indicated in the legend.} # \begin{center} # \includegraphics[width=\hsize]{convergence.eps} # \end{center} # \end{figure} # # \begin{figure}[htbp] # \caption{\label{amplitude}Evolution of the amplitude of the capillary wave as a # function of non-dimensional time $\tau=\omega_0 t$.} # \begin{center} # \includegraphics[width=\hsize]{amplitude.eps} # \end{center} # \end{figure} # # Author: St\'ephane Popinet # Command: sh capwave.sh capwave.gfs # Version: 1.1.0 # Required files: capwave.sh convergence.ref prosperetti markers.tex surfer.tex prost.tex clsvof.tex # Generated files: convergence.tex amplitude.eps convergence.eps markers.tex surfer.tex prost.tex clsvof.tex # 3 5 GfsSimulation GfsBox GfsGEdge {} { Time { end = 2.2426211256 } ApproxProjectionParams { tolerance = 1e-6 } ProjectionParams { tolerance = 1e-6 } # Decrease the resolution linearly down to 3 levels close to the # bottom and top boundaries Refine floor(LEVEL + 1 - (LEVEL - 2)*fabs(y)/1.5) VariableTracerVOF T VariableCurvature K T SourceTension T 1 K VariablePosition Y T y SourceDiffusion U 0.0182571749236 SourceDiffusion V 0.0182571749236 InitFraction T (y - 0.01*cos (2.*M_PI*x)) OutputScalarNorm { step = 3.04290519077e-3 } { awk '{printf ("%g %g\n", $3*11.1366559937,$9); fflush(stdout); }' > wave-LEVEL } { v = (T > 0. && T < 1. ? Y : 0.) } } GfsBox {} GfsBox {} GfsBox {} 1 1 right 2 2 right 3 3 right 1 2 top 1 3 bottom