A numerical method for studying impulsively generated convection from heated tubes

Abstract

This study presents a numerical method for solving the two-dimensional unsteady problem of laminar free convection from a heated tube in an otherwise quiescent fluid. The governing Navier-Stokes and energy equations are formulated in terms of the streamfunction and vorticity. The numerical scheme is designed to handle a large range of Grashof numbers and to capture the physical behaviour inherent in the initial flow. To numerically solve the governing equations a spectral finitedifference method is proposed. The temperature and vorticity are advanced in time using an implicit scheme of Crank-Nicholson type. The streamfunction, on the other hand, is expanded in a truncated Fourier series. To determine the surface vorticity exact integral conditions are derived and incorporated into the numerical method. The numerical results have been verified against derived analytical solutions which are valid for small times and large Grashof numbers. The numerical and analytical results are found to be in good agreement.