Class-Shape Transformation (CST) for Airfoil Parameterization: A Practical Guide

Class sets the family. Shape sets the member.

The parameterization lesson placed CST among the other generators. Here is the arithmetic you actually code. On the unit chord, a surface is

y(x) = C(x) S(x) + x yte

The class function C(x) = x^N1 (1 − x)^N2 fixes the roundness of the nose and the closure of the tail. For a conventional round-nose, sharp-trailing-edge airfoil, N1 = 1/2 and N2 = 1. The square root makes a finite leading-edge radius. The factor (1 − x) drives y to the trailing-edge value. A different class, for example a larger N1, is a sharper nose and a different family, not a small change of coefficient.

The shape function is a Bernstein polynomial

S(x) = Σ Ai · [n! / (i! (n − i)!)] · x^i (1 − x)^(n − i)

from i = 0 to n. Upper and lower surfaces have their own coefficients Ai. The trailing-edge term x yte is how you keep a blunt edge instead of forcing both surfaces through zero. If yte is zero and N2 = 1, the surface ends on the chord.

What the first coefficient is doing

For the round-nose class N1 = 1/2, the leading-edge radius is set by the value of the shape function at x = 0, which is A0. Upper and lower A0 should match if you want a single circular nose sitting on the chord line. If they differ, the geometric nose walks off the chord and the radius is no longer one number. Later coefficients sculpt crest, camber and the recovery. They should not be asked to repair a nose that A0 already got wrong.

A practical budget

• Start from a fit to a known section, not from a random coefficient vector. Four to six coefficients per surface is enough for a large family of smooth airfoils. More coefficients will fit a wavy scan and then “optimize” the waves.
• After every new vector, regenerate cosine-spaced points and plot curvature. A legal CST formula can still be a useless airfoil if the coefficients are huge.
• Put thickness at the spar, trailing-edge angle and a bound on A0 in the constraint list before an optimizer sees the vector.
• Store the class exponents and the coefficient list with the DAT file. The points are a sample. The coefficients are the design.

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Where this sits in the series

In simple words: what this page is for

CST writes each surface as a class function, which sets the nose and the trailing edge, times a shape function, which is a short polynomial. A handful of coefficients replaces a cloud of points.

This lesson exists because section decisions fail when geometry, method labels and selection reasons are scattered. Class-Shape Transformation (CST) for Airfoil Parameterization: A Practical Guide helps you keep a clean trail from shape to polar to choice.

Why the topic matters

Engineers lose time when a title sounds clear but the page only sketches the idea. This article expands Class-Shape Transformation (CST) for Airfoil Parameterization: A Practical Guide so you can answer three questions without jargon: what is the idea, when do you use it, and how do you check it in a HENDOI product.

Read the sections above for the core definitions. Use the steps below to move from reading into a real project trail you can share with a teammate.

How to apply this in HENDOI

In practice: create a HENDOI account or log in, open HENFOIL, bring a section in (create, import DAT, or digitize), pass the geometry gate, climb the analysis ladder only as far as the decision needs, then compare and archive. Every calculated number should keep its method label.

When the section choice is ready for the aircraft, continue in HENSYNTH for mission, compliance and trades. Same account. Same plans.

What this page will not claim: a teaching polar or geometry note is not a flight-Re certificate. HENFOIL labels methods so you know what you bought. Tunnel data and professional judgement still sit outside any single software run.

Checklist before you leave this page

• Can you state the idea of Class-Shape Transformation (CST) for Airfoil Parameterization: A Practical Guide in one sentence?
• Do you know which product owns the next action — HENFOIL or HENSYNTH?
• Will every number you keep carry a method label or a stored-run source?
• If you are new here, have you opened Register or Login?

Related reading on this site

• Active Flow Control on Airfoils: Suction, Blowing and Synthetic Jets
• How Adverse Pressure Gradients Affect Airfoil Boundary Layer Separation
• What Is the Aerodynamic Center of an Airfoil? Location, Meaning and Applications
• What Is Airfoil Aerodynamic Efficiency Across a Range of Flight Conditions?
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• Previous: Surface coordinates from NACA equations
• Next: Bézier curves vs splines
• Index: all HENFOIL airfoil lessons

These pages teach the ideas. They are not wind-tunnel data and not a certification analysis. When a number matters, compute it at a stated Reynolds number, Mach number and transition setting, and keep the method label on the result. In HENFOIL that production polar is a VIC polar — see the analysis ladder and the polar field guide.