Abstract

This paper shows the solution to the problem of seismic wave propagation in 2-D using generalized finite difference (GFD) explicit schemes. Regular and irregular meshes can be used with this method. As we are using an explicit method, it is necessary to obtain the stability condition by using the von Neumann analysis. We also obtained the star dispersion formulas for the phase velocities for the P and S waves, as well as the ones for the group velocities. As the control over the irregularity in the mesh is very important in the application of this method, we have defined an index of irregularity for the star (IIS) and another for the cloud (IIC), analyzing its relationship with the dispersion and time step used in the calculations.

Abstract

Through complementary experimental and numerical efforts, the present paper aims to make a significant contribution to the overall understanding of backfilling processes beneath submarine pipelines. For this purpose, we aim to simplify the experimental backfilling process to an elementary two-stage process: (1) initial scour induced by a pure current, followed by: (2) backfilling induced by pure waves. A steady current is introduced via a re-circulating pump, and is kept constant with a cross-sectional velocity of V = 0.48 m/s until an initial equilibrium scour depth, S0, is reached. Then, the current is stopped and waves (characterized by their Keulegan-Carpener number KC and Shields parameter θ)are introduced to initiate the backfilling process, which is maintained until a new equilibrium scour depth, Sf, is reached. The time at which waves are introduced will be denoted as t = 0. For the backfilling process both regular and irregular waves are used during the experiments. As a demonstration of the initiated two-stage (scour followed by backfilling) process, bed profiles based on video recordings from a case having KC = 9.7 and ϴ = 0.195,are depicted at selected stages in Figure 1. Figure 1(upper left) depicts the current-induced equilibrium scour hole in the near vicinity of the pipe at t = 0, with the profile approximated as the dashed red line. Similarly, Figure 1(upper right) depicts the new equilibrium scour profile (approximated as the full blue line) that has developed under wave-induced backfilling, corresponding to t = 60min. To ease comparison, the dashed red and full bluelines from these plots are additionally combined onto Figure 1 (bottom).The experimental campaign has additionally been complemented with similar numerical simulations (using regular waves), based on a fully-coupled hydrodynamicand morphodynamic CFD model (Jacobsen et al., 2014),extending previous pipeline scour-related applications ofFuhrman et al. (2014) and Larsen et al. (2016). Comparison of the numerical and experimental results demonstrate the ability of the CFD model to reasonably simulate the current-to-wave backfilling process, both interms of the achieved new wave induced equilibriumscour depths as well as the corresponding backfilling timescales. Figure 2 depicts a summary of both experimental and numerical backfilling time scale Tb versus Shields parameter θ. As can be seen, both experimental as well as numerical results match the regression equation:T"sub"b"/sub"=0.3 θ-"sup"5/3 "/sup"quite closely (solid line in Figure 2).