An approximate Riemann solver for shallow water equations and heat advection in horizontal centrifugal casting

An approximate Riemann solver was developed for solving modified shallow water equations (SWE) and energy transport describing the average flow dynamics of a single layer spreading inside a horizontally rotating cylinder. The numerical model was particularly developed for simulating the horizontal centrifugal casting (HSC) of the outer shell of a work roll. The SWE were derived in the rotating frame of reference; therefore, fictitious forces (the centrifugal force and the Coriolis force) were considered. In addition, other forces such as the bed shear force, the force of gravity, the wind shear force and forces arising from the variable liquid/solid interface were taken into account. The Jacobian matrix of the nonlinear hyperbolic system of PDEs was decomposed into a set of eigenvalues and corresponding eigenvectors using standard and corrected Roe averages. A Harten–Hyman entropy fix was used to prevent expansion shocks (entropy violating solutions) typically occurring during transonic rarefactions. Source terms were applied as a stationary discontinuity and were physically bounded and well-balanced for steady states (producing non-oscillatory solutions). Each wave was upwinded using the explicit Godunov’s method. The high resolution corrections with flux limiters were used to achieve second order of accuracy and dispersion free solutions at discontinuities. In addition to the Riemann solver, a central scheme FV model was used to solve the heat diffusion inside the cylinder (mold) and partially solidified liquid layer, coupled with the solidification model. Several simulations were performed, results were analyzed and discussed.