The Joukowski Map, the Kutta Condition and the $2\pi$ Lift Slope

Contents

Flow past a circle with circulation
The Joukowski map and the cusped edge
The Kutta condition and the lift

This appendix proves Proposition 35.70 of Fluid Dynamics — that the Kutta condition selects Equation (35.60) from the one-parameter family of irrotational flows past a thin profile, giving a lift-curve slope of exactly $2\pi$ per radian. The chapter proves Theorem 35.68, which converts a circulation into a force and says nothing about what the circulation is; Remark 35.69 names the physical selection principle and defers the computation here. Three things are established below: the flow past a circle with arbitrary circulation (Flow past a circle with circulation), the conformal transfer to the profile with the trailing edge as the distinguished point (The Joukowski map and the cusped edge), and the value of $\Gamma$ that the finiteness of the velocity there enforces (The Kutta condition and the lift).

The sign convention is the chapter's throughout: $\Gamma$ is the circulation measured counterclockwise, Equation (35.55), and the transverse force per unit span is $Y=-\rho U\Gamma$, Equation (35.59). The complex velocity is Equation (35.57), $\dd w/\dd z=v_{x}-\ii v_{y}$.

Flow past a circle with circulation

Lemma A62.1 (The circle with incidence and circulation).

Let $U>0$ in $\mathrm{m}/\mathrm{s}$, $R>0$ in $\mathrm{m}$, $\alpha\in\left(-\pi/2,\pi/2\right)$ and $\Gamma\in\R$ in $\mathrm{m}^{2}/\mathrm{s}$. The function

\begin{equation}\tag{A62.1} w(\zeta)=U\left(\zeta\,\ee^{-\ii\alpha} +\frac{R^{2}\ee^{\ii\alpha}}{\zeta}\right) -\frac{\ii\Gamma}{2\pi}\log\zeta \end{equation}

is holomorphic on $\abs{\zeta}>R$ apart from the branch cut of the logarithm, has $\abs{\zeta}=R$ as a streamline, tends to the uniform stream $U\ee^{\ii\alpha}$ at infinity, and carries circulation $\Gamma$ around the circle. Its derivative on the circle is

\begin{equation}\tag{A62.2} \left.\dv{w}{\zeta}\right|_{\zeta=R\ee^{\ii\theta}} =\ii\,\ee^{-\ii\theta}\left[2U\sin\left(\theta-\alpha\right) -\frac{\Gamma}{2\pi R}\right]\ec \end{equation}

so the stagnation points on the circle are at the angles $\theta$ with

\begin{equation}\tag{A62.3} \sin\left(\theta-\alpha\right)=\frac{\Gamma}{4\pi UR}\ep \end{equation}

Rests on Proposition 35.28 and Equation (35.57).

Proof.

Derives Lemma A62.1. On $\zeta=R\ee^{\ii\theta}$ the first bracket of Equation (A62.1) is $UR\left(\ee^{\ii\left(\theta-\alpha\right)} +\ee^{-\ii\left(\theta-\alpha\right)}\right) =2UR\cos\left(\theta-\alpha\right)$, which is real, and $\log\zeta=\log R+\ii\theta$, so $\operatorname{Im}w=-\left(\Gamma/2\pi\right)\log R$, a constant: the circle is a streamline. As $\zeta\rightarrow\infty$, $\dd w/\dd\zeta\rightarrow U\ee^{-\ii\alpha}$, which by Equation (35.57) is the stream $\vect{v}=U\left(\cos\alpha,\sin\alpha\right)$. The circulation is $\oint\left(\dd w/\dd\zeta\right)\dd\zeta$ around the circle; the first bracket is single valued and contributes nothing, and $-\left(\ii\Gamma/2\pi\right)\oint\dd\zeta/\zeta =-\left(\ii\Gamma/2\pi\right)2\pi\ii=\Gamma$ by Theorem 12.24.

For Equation (A62.2), differentiate Equation (A62.1) and set $\zeta=R\ee^{\ii\theta}$:

\[ \dv{w}{\zeta} =U\left(\ee^{-\ii\alpha} -\frac{R^{2}\ee^{\ii\alpha}}{\zeta^{2}}\right) -\frac{\ii\Gamma}{2\pi\zeta} =U\ee^{-\ii\theta}\left(\ee^{\ii\left(\theta-\alpha\right)} -\ee^{-\ii\left(\theta-\alpha\right)}\right) -\frac{\ii\Gamma\ee^{-\ii\theta}}{2\pi R}\ec \]

and the bracket is $2\ii\sin\left(\theta-\alpha\right)$ by Equation (12.3). Factoring out $\ii\ee^{-\ii\theta}$ gives Equation (A62.2), which vanishes exactly at the angles Equation (A62.3).

Uniqueness among flows with the stated data is the Neumann uniqueness of Proposition 35.28: two such potentials differ by a function whose velocity vanishes at infinity, whose normal derivative vanishes on the circle, and around which the circulation is zero, hence by the maximum principle (Theorem 14.77) a constant.

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Note that Equation (A62.3) has a solution only for $\abs{\Gamma}\le4\pi UR$; larger circulations lift the stagnation points off the circle entirely. Nothing in the potential problem picks a value — every $\Gamma$ gives a legitimate irrotational flow that does not penetrate the body, which is exactly the indeterminacy stated at Remark 35.69.

The Joukowski map and the cusped edge

Definition A62.2 (The Joukowski map).

For $b>0$ in $\mathrm{m}$, the map

\begin{equation}\tag{A62.4} z=Z(\zeta):=\zeta+\frac{b^{2}}{\zeta} \end{equation}

carries $\abs{\zeta}>b$ bijectively onto the complement of the segment $\left[-2b,2b\right]$ of the real axis, with inverse

\begin{equation}\tag{A62.5} \zeta=\tfrac{1}{2} \left(z+\sqrt{z^{2}-4b^{2}}\right)\ec \end{equation}

the branch of the square root behaving as $z$ at infinity. Rests on Theorem 12.6.

That Equation (A62.5) inverts Equation (A62.4) is the quadratic formula applied to $\zeta^{2}-z\zeta+b^{2}=0$, whose two roots have product $b^{2}$; the branch named is the one of modulus greater than $b$, and it is holomorphic off the segment because $z^{2}-4b^{2}$ vanishes only at $z=\pm2b$. The circle $\abs{\zeta}=b$ itself maps two-to-one onto the segment: $\zeta=b\ee^{\ii\theta}$ gives $z=2b\cos\theta$. The chord is therefore $c=4b$, and the point $\zeta=b$ maps to the trailing edge $z=2b$.

Lemma A62.3 (Transfer of the flow).

Let $w$ be as in Equation (A62.1) with $R=b$, and put $W(z):=w\bigl(\zeta(z)\bigr)$ with Equation (A62.5). Then $W$ is a complex potential for the flow past the segment: it is holomorphic off the segment, the segment is a streamline, the free stream at infinity is $U\ee^{\ii\alpha}$, the circulation around the segment is $\Gamma$, and the physical velocity is

\begin{equation}\tag{A62.6} \dv{W}{z}=\frac{\dd w/\dd\zeta}{\dd z/\dd\zeta} =\frac{\dd w/\dd\zeta}{1-b^{2}/\zeta^{2}}\ep \end{equation}

Rests on Definition A62.2 and Lemma A62.1.

Proof.

Derives Lemma A62.3. A composition of holomorphic functions is holomorphic and its derivative is given by the chain rule; both statements follow from Theorem 12.6 and Definition 12.5 exactly as for real functions, and $\dd z/\dd\zeta=1-b^{2}/\zeta^{2}$ is non-zero on $\abs{\zeta}>b$, which gives Equation (A62.6). Since $W$ takes the same values as $w$, its imaginary part is constant on the image of the circle, i.e. on the segment, and its behaviour at infinity is that of $w$ because $\zeta(z)=z+O\left(z^{-1}\right)$ there. The circulation is unchanged for the same reason: it is the real part of $\oint_{C_{z}}\dd W$, and $\dd W=\dd w$ under the substitution $z\mapsto\zeta(z)$, which carries a circuit around the segment to a circuit around the circle.

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Remark A62.4 (What conformal mapping buys, and what it is).

The step just taken is the only one where a general theorem might have been invoked, and it has been avoided. What is usually quoted here is that a holomorphic bijection carries harmonic functions to harmonic functions and streamlines to streamlines, so a solved problem in one plane is a solved problem in the other; the general statement needs the holomorphic inverse function theorem, and beyond it, to know that a suitable map exists at all, the Riemann mapping theorem. Part II's complex-analysis chapter carries neither: Complex Analysis ends at Theorem 12.24 and has no conformal-mapping material, which is a real gap and is recorded as such. Nothing above needs it, because the map Equation (A62.4) and its inverse Equation (A62.5) are both written down explicitly, and the only property used is that a composition of holomorphic functions is holomorphic — which is Theorem 12.6 and the chain rule.

The transformation is degenerate at $\zeta=\pm b$, where $\dd z/\dd\zeta=0$. This is not an accident of the map but the definition of a sharp edge: the interior angle of the profile at $z=2b$ is zero, and a smooth curve is being folded onto itself. Equation (A62.6) then says that the physical velocity at the trailing edge is infinite — unless the numerator vanishes there too.

The Kutta condition and the lift

Theorem A62.5 (The lift-curve slope of the flat plate).

Of the one-parameter family of flows Equation (A62.1) with $R=b$, exactly one has a finite velocity at the trailing edge $z=2b$, namely that with

\begin{equation}\tag{A62.7} \Gamma=-4\pi Ub\sin\alpha=-\pi cU\sin\alpha\ec \qquad c=4b\ec \end{equation}

and for it Equation (35.59) gives the lift per unit span and lift coefficient

\begin{equation}\tag{A62.8} Y=\pi\rho U^{2}c\sin\alpha\ec \qquad C_{\mathrm{L}}=\frac{Y}{\tfrac{1}{2}\rho U^{2}c} =2\pi\sin\alpha\approx2\pi\alpha\ec \end{equation}

which is Equation (35.60). Rests on Lemma A62.3 and Theorem 35.68.

Proof.

Derives Theorem A62.5. By Equation (A62.6) the velocity at $z=2b$ is the quotient of $\dd w/\dd\zeta$ and $1-b^{2}/\zeta^{2}$ as $\zeta\rightarrow b$ along the circle. The denominator has a simple zero there: with $\zeta=b\ee^{\ii\theta}$, $1-\ee^{-2\ii\theta}=2\ii\ee^{-\ii\theta}\sin\theta$, which vanishes linearly in $\theta$. The numerator is Equation (A62.2) with $R=b$. The quotient is therefore

\[ \dv{W}{z} =\frac{\ii\ee^{-\ii\theta}\left[2U\sin\left(\theta-\alpha\right) -\Gamma/2\pi b\right]} {2\ii\ee^{-\ii\theta}\sin\theta} =\frac{2U\sin\left(\theta-\alpha\right)-\Gamma/2\pi b} {2\sin\theta}\ec \]

and its limit as $\theta\rightarrow0$ is finite if and only if the numerator vanishes at $\theta=0$, that is $-2U\sin\alpha=\Gamma/2\pi b$, which is Equation (A62.7); every other value of $\Gamma$ gives a velocity diverging like $1/\theta$. (When the condition holds, l'Hôpital's rule gives the finite edge velocity $U\cos\alpha$, the component of the stream along the plate — the flow leaves the edge smoothly, along the chord.)

For the force, Lemma A62.3 places a circulation $\Gamma$ around a body in a uniform stream of speed $U$, so Theorem 35.68 applies in axes aligned with the stream: the drag vanishes and the transverse force is $-\rho U\Gamma=\pi\rho U^{2}c\sin\alpha$, directed perpendicular to the stream and, since $\Gamma<0$ for $\alpha>0$, upwards. Dividing by $\tfrac{1}{2}\rho U^{2}c$ gives Equation (A62.8), and for small incidence $\sin\alpha=\alpha+O\left(\alpha^{3}\right)$, so $\Gamma\rightarrow-\pi cU\alpha$ and $C_{\mathrm{L}}\rightarrow2\pi \alpha$ as in Equation (35.60).

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Remark A62.6 (Why the circulation may be read off in either plane).

The step that makes the calculation possible is that $\Gamma$ is the same number in the two planes. It is worth isolating, because it is what licenses computing the circulation where the geometry is easy — a circle — and applying Theorem 35.68 where the geometry is the one that matters. The reason is that the circulation is a contour integral of an exact differential, $\oint\dd W$, and a change of variable in a contour integral is a substitution and nothing more: no property of the map enters beyond its being holomorphic and one-to-one. The same remark explains why the Blasius formula Equation (35.58) may be evaluated on a large circle rather than on the profile — Corollary 12.15, used already in the chapter's proof of Lemma 35.67.

Remark A62.7 (What the $2\pi$ is worth).

Equation (A62.8) is a genuine prediction with no adjustable content, and measured slopes for thin two-dimensional sections at small incidence do come within a few percent of $2\pi$ per radian. Four qualifications are owed.

Thickness and camber. The circle of Lemma A62.1 was concentric with the origin, which is what degenerated the profile to a segment. Displacing its centre gives thickness (a shift along the real axis) and camber (a shift along the imaginary axis), and the same argument then yields $C_{\mathrm{L}}=2\pi\sin\left(\alpha-\alpha_{0}\right)$: camber moves the zero-lift angle $\alpha_{0}$ and leaves the slope alone, which is why the number $2\pi$ is so much more robust than anything else in aerofoil theory.

Finite span. A real wing sheds trailing vorticity, whose downwash reduces the effective incidence, and the slope falls with aspect ratio. This is a three-dimensional effect and nothing in a plane flow can see it.

The Kutta condition is a viscous fact. It has been imposed here, not derived. Its justification is that the boundary layer of Section 35.7.2 cannot negotiate the infinite adverse pressure gradient implied by an infinite edge velocity and separates there instead — Proposition 35.73 — and its consistency with the conservation of circulation Corollary 35.65 is secured by the starting vortex described at Remark 35.69. An inviscid theory that selects its solution by a viscous criterion is not a closed theory, and it should not be presented as one.

No stall. Equation (A62.8) rises without bound with $\alpha$, which is false: beyond ten to fifteen degrees the flow separates from the upper surface and the lift collapses. The theory contains no mechanism for this whatever, because it contains no separation. The measured lift curve, its linear range and its breakdown are in Experiment: Fluid Flow and Turbulence.

Remark A62.8.

The Joukowski Map, the Kutta Condition and the $2\pi$ Lift Slope discharges Proposition 35.70 of Fluid Dynamics, whose statement Equation (35.60) is proved here as Equation (A62.8), and completes Remark 35.69 by supplying the value of $\Gamma$ that the chapter's discussion of the Kutta condition leaves open. It should be read against D'Alembert's Paradox for a Body of Arbitrary Shape: the circulation is the first of the two escapes from d'Alembert's theorem catalogued at Remark 35.31, and what this section computes is precisely how much of it a sharp trailing edge is worth. The drag is still exactly zero, here as there — Theorem 35.68 gives $X=0$ — so the escape buys lift and nothing else. Kutta's own computation is [Kutta:1902]; the general circulation theorem is Joukowski's [Joukowski:1910].