Numerical Analysis on Nonlinear Evolution of Pressure Waves in Bubbly LiquidsBased on KdV-Burgers Equation

Abstract

Oscillation of gas bubbles in a bubbly liquid induces dissipation and dispersion effectsof waves into a nonlinear evolution of pressure waves. Long-range propagation of pressure waves with a moderately small amplitude is described by the KdV-Burgers (KdVB) equation. This paper numerically solves the KdVB equation via a spectral method to predict the nonlinear evolution of waves in bubbly liquids. Focusing on the waveform, and the nonlinear, dissipation and dispersion terms, the following results are obtained: (i) An initially sinusoidal waveform satisfying a periodic boundary condition is firstly distorted due to the nonlinear effect; (ii) Wave distortion is suppressed by increasing the dissipation and dispersion effects; (iii) A break-up due to the dispersion effect appears; (iv) A balance between the nonlinear and dispersion effects is accomplished and then a pulse wave satisfying a feature of soliton is formed. As a result, the initial bubble radius and the initial void fraction strongly contribute the dissipation and dispersion effects, respectively.