ArticleslgStudy

engineering

Joukowsky transform

Joukowsky transform is a engineering topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Joukowsky transform rather than just read about it. In short: In applied mathematics, the Joukowsky transform (sometimes transliterated Joukovsky, Joukowski or Zhukovsky) is a conformal map historically used to understand some principles of airfoil design. It is named after Nikolai Zhukovsky, who published it in 1910.

Joukowsky transform — main illustration
Joukowsky transform — illustration

Key takeaways

  • Joukowsky transform belongs to engineering; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Joukowsky transform to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Joukowsky transform from memory before moving on to harder problems.

Reference excerpt

In applied mathematics, the Joukowsky transform (sometimes transliterated Joukovsky, Joukowski or Zhukovsky) is a conformal map historically used to understand some principles of airfoil design. It is named after Nikolai Zhukovsky, who published it in 1910. The transform and its right-inverse are

z = ζ + 1 ζ , ζ = 1 2 z ± ( 1 2 z ) 2 − 1 = 1 1 2 z ∓ ( 1 2 z ) 2 − 1 , {\displaystyle z=\zeta +{\frac {1}{\zeta }},\qquad \zeta ={\tfrac {1}{2}}z\pm {\sqrt {{\bigl (}{\tfrac {1}{2}}z{\bigr )}^{2}-1}}={\frac {1}{{\tfrac {1}{2}}z\mp {\sqrt {{\bigl (}{\tfrac {1}{2}}z{\bigr )}^{2}-1}}}},}

where z = x + i y {\displaystyle z=x+iy} is a complex variable in the new space and ζ = χ + i η {\displaystyle \zeta =\chi +i\eta } is a complex variable in the original space. The right-inverse is not a global left-inverse because ζ ↦ z {\displaystyle \zeta \mapsto z} is 2-to-1; but a local left-inverse is always one of the right-inverse branches. In aerodynamics, the transform is used to solve for the two-dimensional potential flow around a class of airfoils known as Joukowsky airfoils. A Joukowsky airfoil is generated in the complex plane ( z {\displaystyle z} -plane) by applying the Joukowsky transform to a circle in the ζ {\displaystyle \zeta } -plane. The coordinates of the centre of the circle are variables, and varying them modifies the shape of the resulting airfoil. The circle encloses the point ζ = − 1 {\displaystyle \zeta =-1} (where the derivative is zero) and intersects the point ζ = 1. {\displaystyle \zeta =1.} This can be achieved for any allowable centre position μ x + i μ y {\displaystyle \mu _{x}+i\mu _{y}} by varying the radius of the circle. Joukowsky airfoils have a cusp at their trailing edge. A closely related conformal mapping, the Kármán–Trefftz transform, generates the broader class of Kármán–Trefftz airfoils by controlling the trailing edge angle. When a trailing edge angle of zero is specified, the Kármán–Trefftz transform reduces to the Joukowsky transform.

General Joukowsky transform The Joukowsky transform of any complex number ζ {\displaystyle \zeta } to z {\displaystyle z} is as follows:

… excerpt ends here. Continue reading the full article.

Illustrations

Joukowsky transform: Example of a Joukowsky transform. The circle above is transformed into the Joukowsky airfoil below.
Example of a Joukowsky transform. The circle above is transformed into the Joukowsky airfoil below.
Joukowsky transform: Example of a Kármán–Trefftz transform. The circle above in the ζ {\displaystyle \zeta } -plane is transformed into the Kármán–Trefftz airfoil below, in the z {\displaystyle z} -plane. The parameters used are: μ x = − 0.08 , {\displaystyle \mu _{x}=-0.08,} μ y = + 0.08 {\displaystyle \mu _{y}=+0.08} and n = 1.94. {\displaystyle n=1.94.} Note that the airfoil in the z {\displaystyle z} -plane has been normalised using the chord length.
Example of a Kármán–Trefftz transform. The circle above in the ζ {\displaystyle \zeta } -plane is transformed into the Kármán–Trefftz airfoil below, in the z {\displaystyle z} -plane. The parameters used are: μ x = − 0.08 , {\displaystyle \mu _{x}=-0.08,} μ y = + 0.08 {\displaystyle \mu _{y}=+0.08} and n = 1.94. {\displaystyle n=1.94.} Note that the airfoil in the z {\displaystyle z} -plane has been normalised using the chord length.

Worked examples

Example 1 — a first encounter with Joukowsky transform

Start with the simplest possible case. Write down what Joukowsky transform claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In engineering, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Joukowsky transform before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Joukowsky transform ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Joukowsky transform

In research
Joukowsky transform appears in engineering research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Joukowsky transform in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Joukowsky transform is common in secondary-school and first-year university syllabi. It links to neighbouring topics Aircraft aerodynamics, Aircraft wing design, Conformal mappings, so understanding it makes those chapters shorter.
In everyday life
Look for Joukowsky transform outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Joukowsky transform” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Joukowsky transform in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Joukowsky transform means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Joukowsky transform out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Joukowsky transform in simple terms?

In applied mathematics, the Joukowsky transform (sometimes transliterated Joukovsky, Joukowski or Zhukovsky) is a conformal map historically used to understand some principles of airfoil design. It is named after Nikolai Zhukovsky, who published it in 1910.

Why does Joukowsky transform matter?

Because it connects several engineering ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Joukowsky transform?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Joukowsky transform.

Tags

  • Aircraft aerodynamics
  • Aircraft wing design
  • Conformal mappings

Keep exploring