Treatise of Physics

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The Free-Streamline Jet: Kirchhoff's Contraction Coefficient

Contents
  1. What is quoted here
  2. Formulation
  3. The hodograph region is a half-strip
  4. Two maps onto the same half plane
  5. Integrating back along the free streamline
  6. What the theorem does not cover

This appendix proves Proposition 40.3 of Experiment: Fluid Flow and Turbulence: the two-dimensional efflux of an ideal incompressible liquid through a sharp-edged slit of width \(a\) in a plane wall contracts to an asymptotic jet width \(a_{j}\) with

\begin{equation}\tag{A67.1} C_{c}=\frac{a_{j}}{a}=\frac{\pi}{\pi+2}=0.611\ec \end{equation}

which is Equation (40.6), and does so with no empirical input whatever [Helmholtz:1868] [Kirchhoff:1869].

The chapter's paragraph after the prooflink explains why the problem is hard and what the hodograph does about it; that discussion is not repeated here. This section begins where it stops: it builds the hodograph region explicitly, maps it, and integrates back.

What is quoted here

One theorem is imported, and it is the only one.

Theorem A67.1 (Riemann mapping with boundary correspondence, quoted).

Let \(D\subsetneq\C\) be a simply connected domain whose boundary on the Riemann sphere is a Jordan curve. Then there is a holomorphic bijection of \(D\) onto the upper half plane \(\mathbb{H}:=\set{t\in\C:\Im t>0}\), it extends to a homeomorphism of the closures, and it is unique once three boundary points and their cyclic order are prescribed. Consequently two such maps that agree on three boundary points agree everywhere. Rests on Definition 12.5 and Theorem 12.6.

Remark A67.2 (What is quoted here).

Complex Analysis builds holomorphy (Definition 12.5), the Cauchy–Riemann equations (Theorem 12.6), Cauchy's theorem and the residue calculus, but no theory of conformal mapping: it has no Riemann mapping theorem, no boundary-correspondence theorem and no Schwarz–Christoffel formula. Theorem A67.1 is therefore imported, and it is used for exactly one purpose — to identify the parameter \(t\) reached from the hodograph plane with the parameter \(t\) reached from the potential plane. Everything else below is elementary and is verified here: the mapping properties of \(\exp\), of \(\chi\mapsto \chi^{2}\) and of the Joukowski map are established by tracking their boundary values corner by corner, so no general mapping theorem is needed for them, and the Schwarz–Christoffel formula, which the standard treatments reach for at this point, is not used at all.

The imported theorem has no key in references.bib — the only Riemann and Carathéodory entries there are a paper on electrodynamics and one on the foundations of thermodynamics — so its attribution is made in words, on the footing of Darboux's memoir in Remark A12.15: Riemann's inaugural dissertation of 1851 for the mapping, Carathéodory's papers of 1913 for the extension to the boundary. A reader who grants Theorem A67.1 is granted everything else in this section.

Two further limitations are honest to state here rather than at the end. The construction below is a verification: it exhibits a flow satisfying every condition of the problem and computes its contraction. That the free-boundary problem has no other solution is not proved, and the treatise does not carry the machinery to prove it. And gravity is neglected within the jet, so that the speed on the free streamline is the single constant \(q_{0}\); this is exact only in the limit \(a\ll h\), and the chapter's measurements are made in that regime.

Formulation

Let the wall occupy the plane \(x=0\) with a slit \(\abs{y}<a/2\), the reservoir the region \(x<0\), and the jet issue into \(x>0\); the flow is two-dimensional, steady, incompressible and irrotational, so by Proposition 35.28 it has a harmonic velocity potential Equation (35.25) and, with the stream function Equation (35.12), a holomorphic complex potential \(w(z)=\phi+\ii\psi\) of \(z=x+\ii y\), whose derivative is the complex velocity Equation (35.57),

\begin{equation}\tag{A67.2} \Omega:=\dv{w}{z}=v_{x}-\ii v_{y}=q\,\ee^{-\ii\theta}\ec \end{equation}

with \(q\ge0\) the speed in \(\mathrm{m}/\mathrm{s}\) and \(\theta\) the direction of the velocity.

Three conditions define the problem.

  1. Impermeability on the wall: \(\psi\) is constant there.

  2. Free surface: the jet is bounded by two streamlines on which the pressure equals the atmospheric pressure, so by Theorem 35.24 and Equation (40.2) the speed on them is the constant

    \begin{equation}\tag{A67.3} q_{0}=\sqrt{2gh}\ec \end{equation}

    \(h\) being the head in \(\mathrm{m}\) and \(g\) the acceleration of free fall.

  3. Position of the free surface: unknown. It is a streamline whose location is part of the answer.

The third condition is what makes this a free-boundary problem, and it is the reason no direct solution of Equation (35.25) is available: the domain on which the equation is to be solved is not given.

The configuration is symmetric about \(y=0\), so it suffices to treat the upper half. Its boundary consists of three arcs, on each of which the stream function is constant:

Here \(m\) is half the volume flux per unit span; evaluating it far downstream, where the jet is uniform at speed \(q_{0}\) and width \(a_{j}\),

\begin{equation}\tag{A67.4} m=\tfrac{1}{2}q_{0}a_{j}\ec \end{equation}

in \(\mathrm{m}^{2}/\mathrm{s}\). The image of the flow region in the \(w\) plane is therefore the strip \(0<\psi<m\): the reservoir at infinity is the end \(\phi\to-\infty\), the jet at infinity the end \(\phi\to+\infty\), and \(\phi\) is monotone along every streamline.

The hodograph region is a half-strip

Lemma A67.3 (The hodograph image).

Put

\begin{equation}\tag{A67.5} \zeta:=\log\frac{q_{0}}{\Omega} =\log\frac{q_{0}}{q}+\ii\theta=:\sigma+\ii\theta\ep \end{equation}

Then the flow region maps into the semi-infinite strip

\begin{equation}\tag{A67.6} S:=\set{\sigma+\ii\theta\ :\ \sigma>0\ec\ -\tfrac{\pi}{2}<\theta<0}\ec \end{equation}

with the three boundary arcs going to the three sides as follows: the free streamline to \(\sigma=0\), the wall to \(\theta=-\pi/2\), the axis to \(\theta=0\), and the reservoir at infinity to \(\sigma=+\infty\). Rests on Equation (A67.2), Equation (A67.3) and Definition 12.3.

Proof.

Derives Lemma A67.3. Take the boundary arcs in turn. On the free streamline \(q=q_{0}\) by Equation (A67.3), so \(\sigma=0\) exactly; the direction turns from \(\theta=-\pi/2\) at the edge, where the fluid is still moving down the inner face of the wall, to \(\theta=0\) far downstream, where the jet is parallel to the axis. On the wall \(x=0\), \(y>a/2\), the velocity is tangent to the wall and directed towards the slit, that is in the \(-y\) direction, so \(\theta=-\pi/2\) throughout, while the speed rises from \(0\) deep in the reservoir to \(q_{0}\) at the edge — so \(\sigma\) falls from \(+\infty\) to \(0\). On the axis \(\theta=0\) by symmetry, and the speed rises from \(0\) far upstream to \(q_{0}\) far downstream, so \(\sigma\) falls from \(+\infty\) to \(0\).

Everywhere in the flow \(0<q<q_{0}\), since by Theorem 35.24 the speed is greatest where the pressure is least and the least pressure in the field is the atmospheric pressure on the free surface; and \(-\pi/2<\theta<0\), since the flow turns monotonically from the wall direction to the axis direction between the two. Hence the image lies in \(S\), and the three sides of \(S\) carry the three boundary arcs as listed. The corner \(\zeta=0\) is the edge \(E\), and the corner \(\zeta=-\ii\pi/2\) at \(\sigma=0\) does not occur: the wall and the free streamline meet only at \(E\).

∎

This is the step the chapter's prose promises, and it is worth naming what it has achieved. The unknown of the problem was the shape of the free surface. In the \(\zeta\) plane that surface is the segment \(\sigma=0\), \(-\pi/2\le\theta\le0\) — a known set, fixed before anything has been solved. The price is that the map from the physical plane to the hodograph plane is itself unknown; but that map is what the potential will supply.

Two maps onto the same half plane

Lemma A67.4 (From the half-strip to the half plane).

Define

\begin{equation}\tag{A67.7} \chi=\ee^{-\zeta}=\frac{\Omega}{q_{0}}\ec\qquad s=\chi^{2}\ec\qquad t=-\frac{1}{2}\left(s+\frac{1}{s}\right)\ep \end{equation}

Then \(\zeta\mapsto t\) carries \(S\) holomorphically and bijectively onto \(\mathbb{H}\), with the boundary correspondence

\begin{equation}\tag{A67.8} \text{axis}\mapsto t\le-1\ec\qquad \text{free streamline}\mapsto -1\le t\le 1\ec\qquad \text{wall}\mapsto t\ge 1\ec \end{equation}

the edge \(E\) going to \(t=1\), the jet at infinity to \(t=-1\) and the reservoir at infinity to \(t=\infty\). On the free streamline the parametrization is

\begin{equation}\tag{A67.9} \theta=-\alpha\ec\qquad t=-\cos2\alpha\ec\qquad 0\le\alpha\le\frac{\pi}{2}\ep \end{equation}

Rests on Lemma A67.3, Proposition 12.4 and Definition 12.3.

Proof.

Derives Lemma A67.4. Each map is holomorphic where used, and each is checked on the boundary.

The exponential. \(\chi=\ee^{-\zeta}=\ee^{-\sigma}\ee^{-\ii\theta}\) has \(\abs{\chi}=\ee^{-\sigma}\le1\) and \(\arg\chi=-\theta\in[0,\pi/2]\), and \(\zeta\mapsto\chi\) is injective on \(S\) because the imaginary part of \(\zeta\) spans an interval of length \(\pi/2<2\pi\). So \(\chi\) ranges over the open quarter disc \(\set{0<\abs{\chi}<1,\ 0<\arg\chi<\pi/2}\), and the three sides of \(S\) go to the two radii and the arc.

The square. On the quarter disc \(\chi\mapsto s=\chi^{2}\) is injective, since \(\arg\chi\) spans an interval of length \(\pi/2\) and doubling it spans \(\pi<2\pi\). Its image is the open half disc \(\set{0<\abs{s}<1,\ 0<\arg s<\pi}\); the arc \(\abs{\chi}=1\) goes to the arc \(\abs{s}=1\), the radius \(\arg\chi=0\) to \(s\in(0,1)\) and the radius \(\arg\chi=\pi/2\) to \(s\in(-1,0)\).

The Joukowski map. On the half disc, write \(s=\varrho\ee^{\ii \vartheta}\) and

\begin{equation}\tag{A67.10} t=-\frac{1}{2}\left(s+\frac{1}{s}\right) =-\frac{1}{2}\left(\varrho+\frac{1}{\varrho}\right)\cos\vartheta -\frac{\ii}{2}\left(\varrho-\frac{1}{\varrho}\right)\sin\vartheta\ep \end{equation}

For \(0<\varrho<1\) the bracket \(\varrho-1/\varrho\) is negative and \(\sin\vartheta>0\), so \(\Im t>0\): the half disc goes into \(\mathbb{H}\). Injectivity is the observation that \(s\) and \(1/s\) are the two preimages of a given \(t\) and that exactly one of them has modulus less than one. Surjectivity onto \(\mathbb{H}\) follows because for each \(t\in\mathbb{H}\) the quadratic \(s^{2}+2ts+1=0\) has roots with product \(1\) and neither on the unit circle, since a root of modulus one would make \(t\) real by Equation (A67.10).

On the boundary: \(\abs{s}=1\), \(s=\ee^{\ii\vartheta}\), gives \(t=-\cos\vartheta\in[-1,1]\); \(s\in(0,1)\) gives \(t=-\tfrac{1}{2}\left(s+1/s\right)\le-1\); and \(s\in(-1,0)\) gives \(t\ge1\). Tracing back through the two earlier maps, \(s\in(0,1)\) is the axis and \(s\in(-1,0)\) is the wall, which is Equation (A67.8). On the free streamline \(\Omega=q_{0}\ee^{-\ii\theta}\), so \(\chi=\ee^{\ii\alpha}\) with \(\alpha=-\theta\), \(s=\ee^{2\ii\alpha}\) and \(t=-\cos2\alpha\), which is Equation (A67.9). Its endpoints are \(t=1\) at \(\alpha=\pi/2\) (the edge) and \(t=-1\) at \(\alpha=0\) (downstream).

∎
Lemma A67.5 (From the potential strip to the half plane).

Fix the origin of the velocity potential at the edge \(E\) and define

\begin{equation}\tag{A67.11} t=-1-2\,\ee^{-\pi w/m}\ep \end{equation}

Then \(w\mapsto t\) carries the strip \(0<\psi<m\) holomorphically and bijectively onto \(\mathbb{H}\), with the same boundary correspondence Equation (A67.8), and

\begin{equation}\tag{A67.12} \dd w=-\frac{m}{\pi}\,\frac{\dd t}{t+1}\ep \end{equation}

Rests on Equation (A67.4), Definition 12.3 and Lemma A67.4.

Proof.

Derives Lemma A67.5. For \(0<\psi<m\) the argument of \(\ee^{-\pi w/m}\) is \(-\pi\psi/m\in(-\pi,0)\), so the argument of \(-2\ee^{-\pi w/m}\) lies in \((0,\pi)\) and \(\Im t>0\); the map is injective because the imaginary part of \(-\pi w/m\) spans an interval of length \(\pi<2\pi\), and it is onto \(\mathbb{H}\) because \(\ee^{-\pi w/m}\) covers the lower half plane.

On \(\psi=0\), \(t=-1-2\ee^{-\pi\phi/m}\) decreases through \((-\infty,-1)\) as \(\phi\) runs from \(-\infty\) to \(+\infty\): the reservoir goes to \(t=\infty\) and the jet at infinity to \(t=-1\). On \(\psi=m\), \(\ee^{-\pi\left(\phi+\ii m\right)/m}=-\ee^{-\pi\phi/m}\), so \(t=-1+2\ee^{-\pi\phi/m}\), which is \(+\infty\) deep in the reservoir, equals \(1\) at \(\phi=0\), and tends to \(-1\) far downstream. The choice of origin therefore puts the edge at \(t=1\), matching Lemma A67.4 arc for arc.

Differentiating \(t+1=-2\ee^{-\pi w/m}\) gives \(\dd t=-\left(\pi/m\right)\left(t+1\right)\dd w\), which is Equation (A67.12).

∎
Corollary A67.6 (The two parameters are the same parameter).

Lemmas A67.4 and A67.5 are two conformal maps of the flow region onto \(\mathbb{H}\) agreeing at the three boundary points \(E\), the jet at infinity, and the reservoir at infinity. By Theorem A67.1 they are the same map. Hence \(\Omega\) and \(w\) are both explicit functions of one parameter \(t\), and

\begin{equation}\tag{A67.13} \dd z=\frac{\dd w}{\Omega} =-\frac{m}{\pi}\,\frac{1}{\Omega\left(t\right)}\, \frac{\dd t}{t+1}\ep \end{equation}

Rests on Theorem A67.1, Lemma A67.4 and Lemma A67.5.

This is the whole solution. The physical plane has been eliminated in favour of \(t\), and Equation (A67.13) recovers it by quadrature.

Integrating back along the free streamline

Theorem A67.7 (Kirchhoff's contraction coefficient).

The free streamline leaves the edge of the slit at height \(a/2\) and approaches the height \(a_{j}/2\) downstream, with

\begin{equation}\tag{A67.14} \frac{a_{j}}{2}-\frac{a}{2}=-\frac{a_{j}}{\pi}\ec \end{equation}

so that \(a=a_{j}\left(\pi+2\right)/\pi\) and Equation (A67.1) holds. Rests on Corollary A67.6, Equation (A67.9) and Equation (A67.4).

Proof.

Derives Theorem A67.7. Parametrize the free streamline by \(\alpha\) as in Equation (A67.9), so that \(\alpha=\pi/2\) at the edge and \(\alpha=0\) downstream. Then

\begin{equation}\tag{A67.15} t+1=1-\cos2\alpha=2\sin^{2}\alpha\ec\qquad \dd t=2\sin2\alpha\,\dd\alpha\ec \end{equation}

and Equation (A67.12) gives

\begin{equation}\tag{A67.16} \dd w=-\frac{m}{\pi}\, \frac{2\sin2\alpha\,\dd\alpha}{2\sin^{2}\alpha} =-\frac{2m}{\pi}\,\cot\alpha\,\dd\alpha\ec \end{equation}

using \(\sin2\alpha=2\sin\alpha\cos\alpha\). The sign is right: along the streamline \(\alpha\) decreases and \(\cot\alpha>0\), so \(\dd\phi>0\) and the potential increases downstream.

On the free streamline \(\Omega=q_{0}\ee^{-\ii\theta} =q_{0}\ee^{\ii\alpha}\), so

\begin{equation}\tag{A67.17} \dd z=\frac{\dd w}{\Omega} =-\frac{2m}{\pi q_{0}}\,\cot\alpha\,\ee^{-\ii\alpha}\,\dd\alpha\ec \end{equation}

whose imaginary part is

\begin{equation}\tag{A67.18} \dd y=-\frac{2m}{\pi q_{0}}\cot\alpha \left(-\sin\alpha\right)\dd\alpha =\frac{2m}{\pi q_{0}}\cos\alpha\,\dd\alpha\ep \end{equation}

Integrating from the edge to the far downstream station,

\begin{equation}\tag{A67.19} \Delta y=\frac{2m}{\pi q_{0}}\int_{\pi/2}^{0}\cos\alpha\,\dd\alpha =\frac{2m}{\pi q_{0}} \Bigl[\sin\alpha\Bigr]_{\pi/2}^{0} =-\frac{2m}{\pi q_{0}}\ec \end{equation}

and Equation (A67.4) turns \(2m/q_{0}\) into \(a_{j}\), giving Equation (A67.14). Since the streamline starts at \(y=a/2\) and ends at \(y=a_{j}/2\),

\begin{equation}\tag{A67.20} \frac{a}{2}=\frac{a_{j}}{2}+\frac{a_{j}}{\pi} =a_{j}\,\frac{\pi+2}{2\pi} \quad\Longrightarrow\quad \frac{a_{j}}{a}=\frac{\pi}{\pi+2}\ec \end{equation}

which is Equation (A67.1). Numerically \(\pi/\left(\pi+2\right)=0.61101\).

Note what has and has not entered. The head \(h\) appears only through \(q_{0}\), and \(q_{0}\) cancels between Equation (A67.4) and Equation (A67.19); so does the density, which never appeared at all. The result is a pure number, as Proposition 40.3 claims, and it is the same number at every head.

∎
Remark A67.8 (Where the two constants come from).

The \(\pi\) and the \(2\) of Equation (A67.1) have different origins, and the calculation is worth reading once with that in mind. The \(\pi\) is the width of the potential strip: it enters through the exponential map Equation (A67.11), whose exponent is \(\pi w/m\) precisely because the strip has height \(m\) and must be opened into a half plane. It is therefore a statement about the flux. The \(2\) is \(\int_{0}^{\pi/2}\cos\alpha\,\dd\alpha=1\) counted for the two halves of the jet — a statement about the turning of the velocity through a right angle at the edge. Nothing else survives. (No arctangent occurs anywhere in the evaluation, and no Schwarz–Christoffel integral is needed: the one quadrature to be done is Equation (A67.19), and it is elementary.)

Remark A67.9 (Two checks on the solution).

The edge. At \(\alpha=\pi/2\) the speed is \(q_{0}\), finite. A free-streamline solution is admissible only if the velocity is finite where the free surface leaves the solid boundary, and here that condition holds automatically, because \(\abs{\Omega}=q_{0}\) on the whole free streamline including its endpoint. Nothing had to be imposed to secure it, which is a feature of the sharp-edged geometry and is not true of a rounded mouthpiece.

The asymptote. The real part of Equation (A67.17) is \(\dd x=-\left(2m/\pi q_{0}\right) \left(\cos^{2}\alpha/\sin\alpha\right)\dd\alpha\), whose integrand behaves as \(1/\alpha\) as \(\alpha\to0\). The streamline therefore runs off to \(x=+\infty\) logarithmically in \(\alpha\), while Equation (A67.18) shows the remaining drop in \(y\) is of order \(\alpha^{2}\): the jet approaches its asymptotic width exponentially in \(x\), so “the” contraction is reached within a few slit widths and is a measurable quantity rather than a limit that is never attained.

What the theorem does not cover

Remark A67.10 (The round hole is a different problem).

The hodograph is a two-dimensional device. It works because the velocity of a plane irrotational flow is a holomorphic function of position, so that the map \(z\mapsto\Omega\) can be treated as a conformal change of variable and an unknown boundary in one plane becomes a known boundary in another. In three dimensions there is no such structure: the velocity is a harmonic vector field, the “hodograph” is a map from a three-dimensional region to another and carries no conformal group beyond the Möbius transformations, and no closed solution of the axisymmetric free-boundary problem is known. The measured contraction coefficient for a sharp-edged circular orifice is near \(0.61\) — close enough to Equation (A67.1) that the coincidence is often quoted as though the theorem covered it. It does not, and Proposition 40.3 says so.

Remark A67.11 (Why Borda's mouthpiece needs none of this).

Proposition 40.2 obtains \(C_{c}=1/2\) exactly for the re-entrant mouthpiece, and it does so in half a page with no complex analysis at all. The contrast is instructive rather than embarrassing. What made the flush slit hard was that the pressure over the wall around the orifice is unknown, so the momentum balance carries an undetermined term; the whole apparatus above exists to determine it by finding the flow. The re-entrant geometry removes the term instead: the tube is wetted on both faces, so it transmits no net force, and every other wetted surface carries a hydrostatic pressure. The unknown was arranged out of the problem rather than computed.

The two results are consistent and they bracket the observations: \(\tfrac{1}{2}\) for the re-entrant mouthpiece, \(\pi/(\pi+2)\) for the flush slit, and measurements in between for intermediate geometries. Neither is an approximation to the other.

Remark A67.12.

The Free-Streamline Jet: Kirchhoff's Contraction Coefficient discharges the derivation owed at Proposition 40.3 of Experiment: Fluid Flow and Turbulence, in the efflux experiment Section 40.1. The reader returning there should carry back the division the chapter draws in its Interpretation: Bernoulli's theorem gives the speed Equation (40.2) exactly and gives the discharge not at all, because it says nothing about the area of the bundle of streamlines that reaches the jet. This section supplies that area for one geometry — and the reason it can is that the geometry is two-dimensional and sharp-edged, so the free surface has a known image in the hodograph plane. That is a narrower statement than “the ideal theory predicts the discharge”, and the narrowness is the point.