On Some Developments and Evaluation of an Eulerian-Lagrangian Method for the Transport Equation
by
Frank Bierbrauer
Department of Mechanical Engineering, University of Wollongong, NSW, 2522, Australia
Coauthors: W.K. Soh (Department of Mechanical Engineering, University of Wollongong, NSW, 2522, Australia), W.Y.D. Yuen (BHP Steel Research Laboratories, P.O. Box 202, Port Kembla, NSW 2505, Australia)

The modelling of typical engineering problems such as water-jet cooling of hot-rolled steel strip products in industry, directly involves the solution of a transport (convection-diffusion) equation for the cooling characteristics of the strip. The non-linear nature of the heat conduction involved, aggravates the difficulty of the problem.

Traditional Finite Difference techniques for the solution of this advection dominated transport equation incur severe Courant number stability restrictions as well as instabilities in the presence of temperature discontinuities. Eulerian-Lagrangian Methods (ELM's) solve the transport equation in Lagrangian form `along' backward characteristics effectively decoupling the advection and diffusion terms but retaining the convenience of fixed computational grids. Typical interpolation methods used to obtain the values at the foot of characteristic lines lead to spurious oscillations, numerical diffusion, peak clipping and phase errors.

Through the use of `peak tracking', by the forward tracking of extreme points, this paper attempts to alleviate these errors. A comparison of both 1-D and 2-D benchmark tests from the Convection-Diffusion Forum as well as appropriate error measures, are shown to produce appreciable improvements over the standard methods. It is shown that improved ELM's provide accurate solutions to the transport equation in the presence of non-constant velocity fields and discontinuous boundary conditions.

Date received: July 28, 1999