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ashfordeOU (@ashfordeou) published this skill. Their other Agent Skills are listed on their SkillMD profile.

Transonic Similarity by ashfordeou — Agent Skill

Name: Transonic Similarity
Author: ashfordeOU

Transonic Similarity

Use when you must apply compressibility corrections to subsonic aerodynamic coefficients: compute the Prandtl-Glauert factor and the refined Karman-Tsien correction for the pressure coefficient at a given Mach number, evaluate the transonic similarity parameter linking thickness and sweep effects, and estimate the critical Mach number at which local flow first reaches sonic speed. Produces corrected pressure coefficients and corrected lift slope, the critical Mach estimate, and drag-divergence Mach guidance for airfoil selection and high-subsonic wing design. Trigger: compressibility correction, Prandtl-Glauert, Karman-Tsien, transonic similarity, critical Mach number, pressure coefficient.

ashfordeOU Updated Sep 17, 2026

File contents

Transonic Similarity Corrections (aerodynamics/high-speed/transonic-similarity)

Use when the task is compressibility corrections for high-subsonic flows: the Prandtl-Glauert and Karman-Tsien pressure coefficient corrections, the transonic similarity parameter, and critical Mach estimation.

Domain quick reference

Prandtl-Glauert (linearized thin-airfoil theory, valid below M ~ 0.7): perturbation quantities scale with the factor 1 / sqrt(1 - M^2). Pressure coefficient C_p = C_p0 / sqrt(1 - M^2), lift coefficient C_L = C_L0 / sqrt(1 - M^2), and section lift-curve slope a = a0 / sqrt(1 - M^2), where subscript 0 marks the incompressible value.
Karman-Tsien (extended, usable toward M ~ 0.85): C_p = C_p0 / (sqrt(1 - M^2) + (M^2 / (1 + sqrt(1 - M^2))) * C_p0 / 2). The denominator shrinks less than the Prandtl-Glauert factor alone, so the correction stays finite closer to M = 1.
Transonic similarity parameter: K = (1 - M^2) / tau^(2/3), with tau the thickness ratio (sweep enters through the effective Mach M * cos(Lambda)). Two thin configurations with equal K have similar pressure fields near M = 1.
Critical pressure coefficient (isentropic sonic limit at freestream Mach M, gamma = 1.4 default): C_p* = (2 / (gamma * M^2)) * (((1 + (gamma - 1) / 2 * M^2) / (1 + (gamma - 1) / 2))^(gamma / (gamma - 1)) - 1). Local flow is sonic where C_p equals C_p*.
Critical Mach number M_cr: solve C_p0 / sqrt(1 - M^2) = C_p*(M) for the smallest M; the peak-suction point is the first to reach sonic speed. Drag-divergence Mach M_DD sits roughly 0.05 to 0.08 above M_cr for typical sections.
Rule of thumb: thinner sections and weaker peak suction raise M_cr; typical transport sections fall near M_cr 0.70 to 0.78.

Workflow

Obtain the incompressible peak (or local) C_p0 and section slope a0 from a panel code, XFOIL, or published data.
Below M 0.7 apply the Prandtl-Glauert factor; from 0.7 to 0.85 prefer the Karman-Tsien correction.
Estimate M_cr with critical_mach_number on the peak C_p0; keep the cruise Mach below M_cr for attached subsonic flow.
Use the transonic similarity parameter to scale thickness or sweep effects between configurations.
Cross-check with drag-divergence rules of thumb and wind-tunnel data when available.

Pitfalls

Applying Prandtl-Glauert past M ~ 0.7; linearized theory overpredicts suction near M = 1.
Correcting a pressure coefficient that was already measured at a high subsonic Mach; the corrections apply to the incompressible reference value.
Forgetting that C_p* depends on freestream Mach; it is not a fixed number.
Confusing critical Mach with drag-divergence Mach; M_DD is higher and depends on thickness ratio and Reynolds number.
Comparing signed C_p values instead of magnitudes when checking the sonic limit; C_p* is negative.
Applying the similarity parameter to thick or blunt bodies; it is a thin-airfoil, small-disturbance result.
Using gamma = 1.4 without checking the gas; hot or real-gas flows shift C_p*.

Behavior contract (gate 3)

The correction logic is exercised by the gate 3 contract test: scripts/test_transonic_similarity.py against scripts/transonic_similarity_logic.py (stdlib unittest, offline). Run: python3 scripts/test_transonic_similarity.py

Compliance

Formulas are standard compressible-flow theory (Anderson; Houghton and Carpenter); paraphrase and computed values only, no verbatim excerpts of any standard.
Standards reference: NACA TR 824 (airfoil section data, reference-only) per standards-map.yaml.
compliance: STANDARDS-REF, gated: false.

---name: transonic-similaritydescription: Use when you must apply compressibility corrections to subsonic aerodynamic coefficients: compute the Prandtl-Glauert factor and the refined Karman-Tsien correction for the pressure coefficient at a given Mach number, evaluate the transonic similarity parameter linking thickness and sweep effects, and estimate the critical Mach number at which local flow first reaches sonic speed. Produces corrected pressure coefficients and corrected lift slope, the critical Mach estimate, and drag-divergence Mach guidance for airfoil selection and high-subsonic wing design. Trigger: compressibility correction, Prandtl-Glauert, Karman-Tsien, transonic similarity, critical Mach number, pressure coefficient.license: Apache-2.0---# Transonic Similarity Corrections (aerodynamics/high-speed/transonic-similarity)Use when the task is compressibility corrections for high-subsonicflows: the Prandtl-Glauert and Karman-Tsien pressure coefficientcorrections, the transonic similarity parameter, and critical Machestimation.## Domain quick reference- Prandtl-Glauert (linearized thin-airfoil theory, valid below M ~ 0.7): perturbation quantities scale with the factor 1 / sqrt(1 - M^2). Pressure coefficient C_p = C_p0 / sqrt(1 - M^2), lift coefficient C_L = C_L0 / sqrt(1 - M^2), and section lift-curve slope a = a0 / sqrt(1 - M^2), where subscript 0 marks the incompressible value.- Karman-Tsien (extended, usable toward M ~ 0.85): C_p = C_p0 / (sqrt(1 - M^2) + (M^2 / (1 + sqrt(1 - M^2))) * C_p0 / 2). The denominator shrinks less than the Prandtl-Glauert factor alone, so the correction stays finite closer to M = 1.- Transonic similarity parameter: K = (1 - M^2) / tau^(2/3), with tau the thickness ratio (sweep enters through the effective Mach M * cos(Lambda)). Two thin configurations with equal K have similar pressure fields near M = 1.- Critical pressure coefficient (isentropic sonic limit at freestream Mach M, gamma = 1.4 default): C_p* = (2 / (gamma * M^2)) * (((1 + (gamma - 1) / 2 * M^2) / (1 + (gamma - 1) / 2))^(gamma / (gamma - 1)) - 1). Local flow is sonic where C_p equals C_p*.- Critical Mach number M_cr: solve C_p0 / sqrt(1 - M^2) = C_p*(M) for the smallest M; the peak-suction point is the first to reach sonic speed. Drag-divergence Mach M_DD sits roughly 0.05 to 0.08 above M_cr for typical sections.- Rule of thumb: thinner sections and weaker peak suction raise M_cr; typical transport sections fall near M_cr 0.70 to 0.78.## Workflow1. Obtain the incompressible peak (or local) C_p0 and section slope a0 from a panel code, XFOIL, or published data.2. Below M 0.7 apply the Prandtl-Glauert factor; from 0.7 to 0.85 prefer the Karman-Tsien correction.3. Estimate M_cr with critical_mach_number on the peak C_p0; keep the cruise Mach below M_cr for attached subsonic flow.4. Use the transonic similarity parameter to scale thickness or sweep effects between configurations.5. Cross-check with drag-divergence rules of thumb and wind-tunnel data when available.## Pitfalls- Applying Prandtl-Glauert past M ~ 0.7; linearized theory overpredicts suction near M = 1.- Correcting a pressure coefficient that was already measured at a high subsonic Mach; the corrections apply to the incompressible reference value.- Forgetting that C_p* depends on freestream Mach; it is not a fixed number.- Confusing critical Mach with drag-divergence Mach; M_DD is higher and depends on thickness ratio and Reynolds number.- Comparing signed C_p values instead of magnitudes when checking the sonic limit; C_p* is negative.- Applying the similarity parameter to thick or blunt bodies; it is a thin-airfoil, small-disturbance result.- Using gamma = 1.4 without checking the gas; hot or real-gas flows shift C_p*.## Behavior contract (gate 3)The correction logic is exercised by the gate 3 contract test:scripts/test_transonic_similarity.py againstscripts/transonic_similarity_logic.py (stdlib unittest, offline). Run:python3 scripts/test_transonic_similarity.py## Compliance- Formulas are standard compressible-flow theory (Anderson; Houghton and Carpenter); paraphrase and computed values only, no verbatim excerpts of any standard.- Standards reference: NACA TR 824 (airfoil section data, reference-only) per standards-map.yaml.- compliance: STANDARDS-REF, gated: false.

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