STREAMFUNCTION COORDINATE FORMULATION FOR ONE-DIMENSIONAL UNSTEADY FLOW

Abstract

In this paper, the streamfunction-as-a-coordinate concept has been extended to 1-D unsteady flow. The governing equation in streamfunction coordinates in this case is a second order nonlinear hyperbolic partial differential equation or a wave equation with variable propagation speed for the Cartesian coordinate x=x(ψ, τ) where ψ is streamfunction and τ is time. The equation has been solved numerically by the leapfrog scheme. A piston-driven expansion wave and a density disturbance propagation have been successfully simulated and the agreement between the present method and exact solution is excellent. The streamfunction coordinate formulation for 1-D unsteady flow is particularly advantageous due to the fact that the Jacobian is always a positive finite quantity so that there is no singularity in the transformation. An additional benefit of the streamfunction formulation for 1-D unsteady flow is the ability to easily draw particle path lines.