Generation and Evolution of Cnoidal Waves in a Two-Dimensional Numerical Viscous Wave Flume

The generation and propagation of water waves in a numerical wave flume with Ursell numbers (Ur) ranging from 0.67 to 43.81 were investigated using the wave generation theory of Goring and Raichlen and a two-dimensional numerical viscous wave flume model. The unsteady Navier–Stokes equations, along with nonlinear free surface boundary conditions and upstream boundary conditions at the wavemaker, were solved to build the numerical wave flume. The generated waves included small-amplitude, finite-amplitude, cnoidal, and solitary waves. For computational efficiency, the Jacobi elliptic function representing the surface elevation of a cnoidal wave was expressed as a Fourier series expansion. The accuracy of the generated waveforms and associated flow fields was validated through comparison with theoretical solutions. For Ur<26.32, small-amplitude waves generated using Goring and Raichlen’s wave generation theory matched those obtained from linear wave theory, while finite-amplitude waves matched those obtained using Madsen’s wave generation theory. For Ur>26.32, nonlinear wave generated using Goring and Raichlen’s theory remained permanent, whereas that generated using Madsen’s theory did not. The evolution of a cnoidal wave train with Ur=43.81 was examined, and it was found that, after an extended propagation period, the leading waves in the wave train evolved into a series of solitary waves, with the tallest wave positioned at the front.