The Method of Complex Characteristics

Abstract

We have seen that physically realistic transonic flow computations can be based on partial differential equations for the velocity potential and stream function that presuppose conservation of entropy. In terms of characteristic coordinates ξ and η we have

$$ {\varphi_{\xi }}(\xi, {\eta_0}) = g(\xi ),\quad {\varphi_{\xi }}({\xi_0},\eta ) = \overline {g(\xi )} $$

The coordinates ξ and n can be specified in terms of the speed q and the flow angle θ by the formulas

$$ {\varphi_{\xi }} = {\tau_{ + }}{\psi_{\xi }},\quad {\varphi_{\eta }} = {\tau_{{_{ - }}}}{\psi_{{\eta.}}} $$

where f is any complex analytic function. Prescription of a second arbitrary function g serves to determine φ and ψ as solutions of the characteristic initial value problem

$$ ({c^2} - {u^2}){\phi_{{xx}}} - 2uv{\phi_{{xy}}} + ({c^2} - {v^2}){\phi_{{xy}}} = 0 $$

where ξ₀ = η̄₀ is a fixed subsonic point in the complex plane. With these conventions it turns out that <Inline>1</Inline> as can be seen from the uniqueness of the solution. Hence for subsonic flow the real hodograph plane corresponds to points in the complex domain where ξ₀ = η̄₀. To calculate φ and ψ paths of integration are laid down in the complex plane, and then a stable finite difference scheme is applied to solve the characteristic initial value problem (see Volume I).