Function flare_reduction_turns_rad 

pub fn flare_reduction_turns_rad(
    aft: &AftFlow,
) -> Result<ReductionTurns, AeroError>

Where the second-order shock-expansion method’s exponential form fails at a corner behind aft, the flow a body delivers to its aft end (ShockExpansionBody::aft_flow): the two turns between which the march reduces the element behind that corner to the generalized method.

An element is reduced exactly when its turn lies strictly between the two, in whichever order they come. Eq. 9’s rate is η/(x − x₂) = (∂p/∂s)₂ / ((p_c − p₂) cos δ₂), and TN 3527 p. 13 keeps the exponential form only where η ≥ 0, so a reduced element is one whose gradient behind the corner and whose gap to its tangent cone have opposite signs. Each of those two is a continuous function of the turn, and (on every corner state measured for ADR-050, an observation rather than a proof) each has a single zero, so the signs disagree on exactly the open interval between them and nowhere else.

Both zeros are properties of the corner’s own state. With δ₁ the angle ahead, r the radius at the corner, B = γpM²/(2(M² − 1)) (eq. 6) and Ω = A/A* (eq. 7), all read from aft:

• the balance solves sin(δ₁ + θ) = (Ω₁/Ω₂(θ)) (sin δ₁ + r (∂p/∂s)₁ / B₁), which is eq. 4 set to zero and rearranged. Ω₁/Ω₂ is 1 + O(θ), so iterating on it contracts;
• the crossing solves p₂(θ) = p_c(δ₁ + θ), the isentropic turn’s pressure against its tangent cone’s (cone_flow), by false position from the turn that would bring p₂ back to the free stream’s pressure, θ ≈ (1/p₁ − 1)√(M₁² − 1)/(γM₁²).

Neither is a search over the march’s own refusal, which is a sign test on two pressures within a thousandth of each other and so carries about nine significant digits (issue #117). What is left is the accuracy of aft and of the tangent cone, and each solution reports what it left behind, in ReductionTurns::crossing_residual_p0 and ReductionTurns::balance_residual_p0_per_m, because neither is promised to be zero. Differentiating the crossing’s equation, a change Δp₁ in the pressure the body delivers moves the crossing by about Δp₁ √(M₁² − 1) / (γ p₁ M₁²).

Errors

• AeroError::Unsupported where the corner’s state can’t carry a turn (a surface that isn’t supersonic, no radius, a free-stream Mach number at or below 1), or where either root lies outside the turns a widening corner can make: between zero surface angle and the shallower of the isentropic turn’s end and the cone tables’ 30°.