A pseudospectral Fourier method for a 1D incompressible two‐fluid model

Abstract

AbstractThis paper presents an accurate and efficient pseudospectral (PS) Fourier method for a standard 1D incompressible two‐fluid model. To the knowledge of the authors, it is the first PS method developed for the purpose of modelling waves in multiphase pipe flow. Contrary to conventional numerical methods, the PS method combines high accuracy and low computational costs with flexibility in terms of handling higher order derivatives and different types of partial differential equations. In an effort to improve the description of the stratified wavy flow regime, it can thus serve as a valuable tool for testing out new two‐fluid model formulations. The main part of the algorithm is based on mathematical reformulations of the governing equations combined with extensive use of fast Fourier transforms. All the linear operations, including differentiations, are performed in Fourier space, whereas the nonlinear computations are performed in physical space. Furthermore, by exploiting the concept of an integrating factor, all linear parts of the problem are integrated analytically. The remaining nonlinear parts are advanced in time using a Runge–Kutta solver with an adaptive time step control. As demonstrated in the results section, these steps in sum yield a very accurate, fast and stable numerical method. A grid refinement analysis is used to compare the spatial convergence with the convergence rates of finite difference (FD) methods of up to order six. It is clear that the exponential convergence of the PS method is by far superior to the algebraic convergence of the FD schemes. Combined with the fact that the scheme is unconditionally linearly stable, the resulting increase in accuracy opens for several orders of magnitude savings in computational time. Finally, simulations of small amplitude, long wavelength sinusoidal waves are presented to illustrate the remarkable ability of the PS method to reproduce the linear stability properties of the two‐fluid model. Copyright © 2008 John Wiley & Sons, Ltd.