Joukowsky transform

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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.^[1]

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.

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

z = x + i y = ζ + 1 ζ = χ + i η + 1 χ + i η = χ + i η + χ − i η χ 2 + η 2 = χ ( 1 + 1 χ 2 + η 2 ) + i η ( 1 − 1 χ 2 + η 2 ) . {\displaystyle {\begin{aligned}z&=x+iy=\zeta +{\frac {1}{\zeta }}\\&=\chi +i\eta +{\frac {1}{\chi +i\eta }}\\[2pt]&=\chi +i\eta +{\frac {\chi -i\eta }{\chi ^{2}+\eta ^{2}}}\\[2pt]&=\chi \left(1+{\frac {1}{\chi ^{2}+\eta ^{2}}}\right)+i\eta \left(1-{\frac {1}{\chi ^{2}+\eta ^{2}}}\right).\end{aligned}}}

So the real ( x {\displaystyle x} ) and imaginary ( y {\displaystyle y} ) components are:

x = χ ( 1 + 1 χ 2 + η 2 ) , y = η ( 1 − 1 χ 2 + η 2 ) . {\displaystyle {\begin{aligned}x&=\chi \left(1+{\frac {1}{\chi ^{2}+\eta ^{2}}}\right),\\[2pt]y&=\eta \left(1-{\frac {1}{\chi ^{2}+\eta ^{2}}}\right).\end{aligned}}}

The transformation of all complex numbers on the unit circle is a special case.

| ζ | = χ 2 + η 2 = 1 , {\displaystyle |\zeta |={\sqrt {\chi ^{2}+\eta ^{2}}}=1,}

which gives

χ 2 + η 2 = 1. {\displaystyle \chi ^{2}+\eta ^{2}=1.}

So the real component becomes x = χ ( 1 + 1 ) = 2 χ {\textstyle x=\chi (1+1)=2\chi } and the imaginary component becomes y = η ( 1 − 1 ) = 0 {\textstyle y=\eta (1-1)=0} .

Thus the complex unit circle maps to a flat plate on the real-number line from −2 to +2.

Transformations from other circles make a wide range of airfoil shapes.

The solution to potential flow around a circular cylinder is analytic and well known. It is the superposition of uniform flow, a doublet, and a vortex.

The complex conjugate velocity W ~ = u ~ x − i u ~ y , {\displaystyle {\widetilde {W}}={\widetilde {u}}_{x}-i{\widetilde {u}}_{y},} around the circle in the ζ {\displaystyle \zeta } -plane is W ~ = V ∞ e − i α + i Γ 2 π ( ζ − μ ) − V ∞ R 2 e i α ( ζ − μ ) 2 , {\displaystyle {\widetilde {W}}=V_{\infty }e^{-i\alpha }+{\frac {i\Gamma }{2\pi (\zeta -\mu )}}-{\frac {V_{\infty }R^{2}e^{i\alpha }}{(\zeta -\mu )^{2}}},}

where

• μ = μ x + i μ y {\displaystyle \mu =\mu _{x}+i\mu _{y}} is the complex coordinate of the centre of the circle,
• V ∞ {\displaystyle V_{\infty }} is the freestream velocity of the fluid,

α {\displaystyle \alpha } is the angle of attack of the airfoil with respect to the freestream flow,

• R {\displaystyle R} is the radius of the circle, calculated using R = ( 1 − μ x ) 2 + μ y 2 {\textstyle R={\sqrt {\left(1-\mu _{x}\right)^{2}+\mu _{y}^{2}}}} ,
• Γ {\displaystyle \Gamma } is the circulation, found using the Kutta condition, which reduces in this case to Γ = 4 π V ∞ R sin ⁡ ( α + sin − 1 ⁡ μ y R ) . {\displaystyle \Gamma =4\pi V_{\infty }R\sin \left(\alpha +\sin ^{-1}{\frac {\mu _{y}}{R}}\right).}

The complex velocity W {\displaystyle W} around the airfoil in the z {\displaystyle z} -plane is, according to the rules of conformal mapping and using the Joukowsky transformation, W = W ~ d z d ζ = W ~ 1 − 1 ζ 2 . {\displaystyle W={\frac {\widetilde {W}}{\frac {dz}{d\zeta }}}={\frac {\widetilde {W}}{1-{\frac {1}{\zeta ^{2}}}}}.}

Here W = u x − i u y , {\displaystyle W=u_{x}-iu_{y},} with u x {\displaystyle u_{x}} and u y {\displaystyle u_{y}} the velocity components in the x {\displaystyle x} and y {\displaystyle y} directions respectively ( z = x + i y , {\displaystyle z=x+iy,} with x {\displaystyle x} and y {\displaystyle y} real-valued). From this velocity, other properties of interest of the flow, such as the coefficient of pressure and lift per unit of span can be calculated.

The Kármán–Trefftz transform is a conformal map closely related to the Joukowsky transform. While a Joukowsky airfoil has a cusped trailing edge, a Kármán–Trefftz airfoil—which is the result of the transform of a circle in the ζ {\displaystyle \zeta } -plane to the physical z {\displaystyle z} -plane, analogue to the definition of the Joukowsky airfoil—has a non-zero angle at the trailing edge, between the upper and lower airfoil surface. The Kármán–Trefftz transform therefore requires an additional parameter: the trailing-edge angle α . {\displaystyle \alpha .} This transform is^[2][3]

z = n b ( ζ + b ) n + ( ζ − b ) n ( ζ + b ) n − ( ζ − b ) n , {\displaystyle z=nb{\frac {(\zeta +b)^{n}+(\zeta -b)^{n}}{(\zeta +b)^{n}-(\zeta -b)^{n}}},}

A

where b {\displaystyle b} is a real constant that determines the positions where d z / d ζ = 0 {\displaystyle dz/d\zeta =0} , and n {\displaystyle n} is slightly smaller than 2. The angle α {\displaystyle \alpha } between the tangents of the upper and lower airfoil surfaces at the trailing edge is related to n {\displaystyle n} as^[2]

α = 2 π − n π , n = 2 − α π . {\displaystyle \alpha =2\pi -n\pi ,\quad n=2-{\frac {\alpha }{\pi }}.}

The derivative d z / d ζ {\displaystyle dz/d\zeta } , required to compute the velocity field, is

d z d ζ = 4 n 2 ζ 2 − 1 ( 1 + 1 ζ ) n ( 1 − 1 ζ ) n [ ( 1 + 1 ζ ) n − ( 1 − 1 ζ ) n ] 2 . {\displaystyle {\frac {dz}{d\zeta }}={\frac {4n^{2}}{\zeta ^{2}-1}}{\frac {\left(1+{\frac {1}{\zeta }}\right)^{n}\left(1-{\frac {1}{\zeta }}\right)^{n}}{\left[\left(1+{\frac {1}{\zeta }}\right)^{n}-\left(1-{\frac {1}{\zeta }}\right)^{n}\right]^{2}}}.}

First, add and subtract 2 from the Joukowsky transform, as given above:

z + 2 = ζ + 2 + 1 ζ = 1 ζ ( ζ + 1 ) 2 , z − 2 = ζ − 2 + 1 ζ = 1 ζ ( ζ − 1 ) 2 . {\displaystyle {\begin{aligned}z+2&=\zeta +2+{\frac {1}{\zeta }}={\frac {1}{\zeta }}(\zeta +1)^{2},\\[3pt]z-2&=\zeta -2+{\frac {1}{\zeta }}={\frac {1}{\zeta }}(\zeta -1)^{2}.\end{aligned}}}

Dividing the left and right hand sides gives

z − 2 z + 2 = ( ζ − 1 ζ + 1 ) 2 . {\displaystyle {\frac {z-2}{z+2}}=\left({\frac {\zeta -1}{\zeta +1}}\right)^{2}.}

The right hand side contains (as a factor) the simple second-power law from potential flow theory, applied at the trailing edge near ζ = + 1. {\displaystyle \zeta =+1.} From conformal mapping theory, this quadratic map is known to change a half plane in the ζ {\displaystyle \zeta } -space into potential flow around a semi-infinite straight line. Further, values of the power less than 2 will result in flow around a finite angle. So, by changing the power in the Joukowsky transform to a value slightly less than 2, the result is a finite angle instead of a cusp. Replacing 2 by n {\displaystyle n} in the previous equation gives^[2]

z − n z + n = ( ζ − 1 ζ + 1 ) n , {\displaystyle {\frac {z-n}{z+n}}=\left({\frac {\zeta -1}{\zeta +1}}\right)^{n},}

which is the Kármán–Trefftz transform. Solving for z {\displaystyle z} gives it in the form of equation A.

In 1943 Hsue-shen Tsien published a transform of a circle of radius a {\displaystyle a} into a symmetrical airfoil that depends on parameter ϵ {\displaystyle \epsilon } and angle of inclination α {\displaystyle \alpha } :^[4]

z = e i α ( ζ − ϵ + 1 ζ − ϵ + 2 ϵ 2 a + ϵ ) . {\displaystyle z=e^{i\alpha }\left(\zeta -\epsilon +{\frac {1}{\zeta -\epsilon }}+{\frac {2\epsilon ^{2}}{a+\epsilon }}\right).}

The parameter ϵ {\displaystyle \epsilon } yields a flat plate when zero, and a circle when infinite; thus it corresponds to the thickness of the airfoil. Furthermore the radius of the cylinder a = 1 + ϵ {\displaystyle a=1+\epsilon } .

• Joukowski Transform NASA Applet
• Joukowsky Transform Interactive WebApp