On a Refinement of the Transonic Approximation Theory

抄録

In this paper it is proposed that the basic equation for the two-dimensional transonic flow of a compressible fluid should be
\frac∂²ψ∂τ²+\frac∂²ψ∂θ²=−\frac536\frac1τ²ψ (A)
rather than the commonly used one
\frac∂²Ψ∂η²−η\frac∂²Ψ∂θ²=0, η=(γ+1)^1⁄3\fracq−q_*q_* (B)
or its equivalent. Here τ=−∫μq⁻¹dq, μ=(1−M²)^1⁄2, K=(μρ₀⁄ρ)², and Ψ=K^−1⁄4ψ is the stream function, (q, θ) are the magnitude and direction of the velocity, q_* the critical velocity, M the local Mach number, ρ the density, ρ₀ the stagnation density, and γ the ratio of specific heats.
A set of fundamental solutions of (A), which have singularity at the point (τ=τ₁, θ=0), is obtained in a natural and elementary way in the form
ψ_m=τ^msin^mα{C₁Q^m_−1⁄6(cosα)+C₂Q^m_−5⁄6(cosα)},
where τ₁−iθ=τcosα, Q_n^m being the associated Legendre function. In particular, flow through a Laval nozzle and past a lenticular-shaped body can be constructed by employing the solutions with m=±1⁄2. Finally, discussion is given on the advantage of (A) over (B) as the fundamental equation for the transonic flow.