Theoretical and numerical investigations of wind effects on wave growth and shape variation

Abstract

Abstract Content of image described in text. An improved model for two-dimensional wind-driven water waves is developed by incorporating both wind pressure and tangential viscous stress. These aerodynamic forces are determined via numerical simulations of turbulent flow over a slowly moving, wavy boundary, and are validated against established experimental and numerical data. The model accounts for wind input and viscous dissipation simultaneously, bypassing the oversimplifications of irrotational and inviscid flow assumptions. Utilising a multi-scale asymptotic expansion, we derive a theoretical solution for the wave profile and introduce a complex, time-dependent parameter that quantifies the effects of wind on wave evolution. Its real part governs the wave growth or decay rate, while the imaginary part modulates the phase shift. This formulation explicitly captures the often-overlooked role of tangential viscous stress. Furthermore, unsteady numerical simulations using a conformal mapping technique confirm the theoretical growth rates. Finally, the theoretical solution enables parametrising wave skewness and asymmetry. Incorporating an empirical relationship between wave slope and inverse wave age, the model yields predictions that closely align with laboratory and field observations and provides an improved description of nonlinear wave profiles.