The lowest pressure is on the body

Fast means low pressure is the site’s account of why the two go together, and it is a statement about points on a streamline. This essay is about a statement over a whole region, which is stronger and less well known: in an ideal flow the lowest pressure anywhere is on a surface. Not typically, not for well-behaved shapes — always, whatever the body, at whatever incidence, steady or not.

And the proof says exactly what has to be true for it to hold, which turns out to be the interesting part.

The pressure has a source, and the source is Q #

Take the divergence of the momentum equation for an incompressible fluid. The unsteady term drops because $\nabla\cdot\mathbf{u} = 0$; the viscous term drops for the same reason. What is left is

$$\nabla^2 p = -\rho\,\frac{\partial u_i}{\partial x_j}\frac{\partial u_j}{\partial x_i} = 2\rho Q,$$

with

$$Q = \tfrac12\left(|\boldsymbol{\Omega}|^2 - |\mathbf{S}|^2\right)$$

the excess of rotation over strain that where a vortex stops is about. Note what this equation does not contain: viscosity, time, or any property of the fluid but its density. It holds for Navier–Stokes as much as for Euler, and for unsteady flow as much as for steady.

If the flow is irrotational then $\boldsymbol{\Omega} = 0$, so $Q = -\tfrac12|\mathbf{S}|^2$ is never positive, so $\nabla^2 p \le 0$ everywhere. A function with a non-positive Laplacian is superharmonic, and a superharmonic function attains its minimum on the boundary of every region it is defined in.

That is the whole theorem, and it has taken four lines.

What a maximum principle actually says #

The phrase is worth unpacking, because it is stronger than “the biggest value is at the edge”.

A harmonic function — one with zero Laplacian — attains both its maximum and its minimum on the boundary, and attains neither in the interior unless it is constant. That is the mean value property: the value at a point is the average over any circle round it, so no point can beat all of its neighbours. The velocity potential and the stream function of an ideal flow are harmonic, so one function instead of two is already a statement of this kind about $\phi$ and $\psi$ themselves.

A superharmonic function — non-positive Laplacian — has the one-sided version: the value at a point is at least the average over any circle round it, so a point can beat its neighbours from above but never from below. Its minimum is on the boundary; its maximum may be anywhere.

That asymmetry is exactly right for pressure. A potential flow may have an interior pressure maximum — it does, at every stagnation point, where the pressure reaches its stagnation value in the middle of the fluid if the stagnation point is not on a wall. What it may not have is an interior minimum. The theorem is one-sided because $Q$ has a sign, and $Q$ has a sign because the flow is irrotational.

The identity, checked rather than trusted #

In two dimensions the same statement can be made about the speed directly, and there it is an identity rather than an inequality. For an irrotational, solenoidal plane field,

$$\nabla^2\!\left(q^2\right) = 4\left(u_x^2 + u_y^2\right),$$

which cannot be negative because it is a sum of squares.

Across four solved flows — a cylinder, a cylinder with circulation, a Joukowski section at six degrees and a Rankine oval — the worst departure is eight parts in a hundred thousand, and that is the five-point stencil’s own truncation error rather than anything about the fields. Not one sampled point has a negative Laplacian.

The distinction between an identity and an inequality matters here. An inequality of this kind is usually established by an estimate, and an estimate can be sharp in some places and slack in others. A sum of squares cannot be negative anywhere at all, so the conclusion holds pointwise, with no room anywhere for a special case.

Four flows, and the surface wins every time #

The theorem is about a region, so the honest test is to sample the whole region.

For each of the four flows the exterior is sampled on a grid of 240 by 240, with points near the surface excluded so that “interior” means the fluid rather than the wall, and the surface is marched at 720 stations. The surface minimum is lower in every case, by a margin between 0.18 and 0.56 in pressure coefficient. That is not a near thing; there is no competition.

The cylinder’s surface minimum comes out at $-2.9999992$, which is the closed-form $-3$ at the shoulder to seven figures, and that is the check that the sampling is finding what it claims to find. The cylinder with circulation goes to $-5.14$ because the circulation adds to the speed on one side, and the Joukowski section at six degrees to $-2.22$ near the nose.

Looking along a single ray makes the reason visible. From the shoulder outwards, the pressure rises monotonically from $-3$ to zero. There is nowhere for a minimum to hide, and the same is true along every other ray, in every one of the four flows.

Where the margin is smallest, and why #

The four margins are not equal, and the pattern in them is informative.

The Rankine oval’s margin is 0.18, the smallest of the four; the cylinder with circulation’s is 0.56, the largest. The margin measures how much lower the surface pressure is than the best the interior can manage, and a body that accelerates the flow hard over a short stretch of surface — a circulating cylinder, a section at incidence — makes a deep, localised suction that no interior point can approach. A body that accelerates the flow gently, like the oval built out of a source and a sink in a stream, makes a shallow one that a point just off the surface nearly matches.

So the theorem’s strength is a property of the shape even though its truth is not. That is worth knowing when the theorem is being used as a numerical check: on a slender, gently loaded body the margin can be small enough that a coarse grid’s interpolation error crosses it, and a violation reported at that level is a statement about the grid.

Which is why an ideal flow cannot cavitate away from a wall #

The practical reading of the theorem is about cavitation. Water tears where the pressure falls to its vapour pressure, so the theorem says that in an irrotational flow that happens on a surface first — and therefore that the cavitation number a designer computes from a body’s minimum surface pressure coefficient is not merely the first place it happens to be checked, but the first place it can happen at all.

It also says something about the unsteady case that is easy to get wrong. The unsteady Bernoulli equation puts $\partial\phi/\partial t$ in the pressure, and $\phi$ is harmonic, so $\nabla^2(\partial\phi/\partial t) = 0$: the unsteady term is harmonic rather than subharmonic, and adding a harmonic function to a superharmonic one leaves it superharmonic. So the theorem survives unsteadiness intact. An accelerating body still has its lowest pressure on its own surface, however violently it is accelerating, which is not obvious from the pressure that depends on the past and follows immediately here.

And exactly where it fails #

The hypothesis is that $Q \le 0$, and the moment there is vorticity that stops being automatic.

The pressure here is not from Bernoulli, which does not hold across a core; it is the radial momentum balance $\partial p/\partial r = \rho v_\theta^2/r$ integrated inwards, which is exact for a steady circular vortex. The lowest pressure is on the axis — as far from any boundary as a flow can put it.

$Q$ is positive inside 1.08 core radii and negative outside. The pressure minimum is inside that circle, which is not a coincidence: a region of positive $Q$ is precisely a region where the pressure is subharmonic and an interior minimum is permitted.

The theorem’s exemption and the site’s own definition of a vortex are the same condition. That is worth stating as a fact about the subject rather than about this calculation: the one place in a real flow that can hold a pressure minimum away from every surface is a vortex core, in the technical sense of the word. Which is exactly where cavitation is observed to start in a propeller wake — several diameters downstream of the blade that made the vortex, with clear water between the bubble and any solid surface.

The identity itself is worth watching break. Inside the core the vortex is nearly solid-body rotation; far outside it is nearly irrotational; both of those satisfy $\nabla^2(q^2) = 4(u_x^2 + u_y^2)$, so a test point chosen anywhere but the shear annulus between them would prove nothing at all. In the annulus the Laplacian of the squared speed is negative — about $-41$ where the identity asks for $+62$ — which a sum of squares cannot produce. That is where the vorticity is, and it is the only place the flow is neither of the two things that satisfy the identity.

The same statement, read as a statement about the wall #

There is a reformulation that makes the theorem feel less like an accident of the algebra.

The pressure gradient normal to a wall in an ideal flow is set by the curvature of the streamline there: $\partial p/\partial n = \rho q^2/R$, with $R$ the radius of curvature and $n$ pointing away from the centre of curvature. Over a convex body the streamlines curve round it, so the pressure increases outwards from the surface — which is the local version of the global statement, and it is why the surface value is the extreme one.

Read the other way, the theorem says an ideal flow cannot arrange its streamlines so that the curvature reverses in a way that would trap a low-pressure pocket in mid-fluid. It cannot, because a closed low-pressure pocket would need streamlines curving towards it from all sides, which is a region of net rotation, which is $Q > 0$, which is vorticity. Every route to the conclusion arrives at the same condition. The condition is worth more than the theorem: the interesting object here is $Q$, and the maximum principle is one of the things it decides — the others being where a vortex is, which is the criterion this collection already argued about, and whether a tracer particle in the flow will be centrifuged out of a core.

Which forces how a cavity has to be modelled #

The theorem does more than say where cavitation starts. It decides how the flow afterwards can be represented, and the constraint is severe.

A cavity is a region at the vapour pressure, which is lower than anything around it. The theorem forbids exactly that: an irrotational flow cannot hold a pressure minimum in its interior. So a potential-flow model cannot describe a cavity as a low-pressure pocket sitting in the fluid — the description is not merely inaccurate, it is inconsistent with the equations being solved.

The only move left is to take the cavity out of the domain and make its surface a boundary: a free streamline along which the pressure, and therefore by Bernoulli the speed, is constant. That is Helmholtz and Kirchhoff’s free-streamline theory, and it is not a stylistic choice.

The price is that the boundary’s position is unknown. The condition imposed there is on the pressure rather than on where the surface is, so the shape is part of the answer — which is what makes these problems hard although the governing equation is still Laplace’s.

And such a cavity cannot close. A constant-pressure boundary has no way to recover to ambient, so every model of this kind ends in an invented termination — a re-entrant jet, a fictitious plate — chosen for tractability rather than found.

Three ways the theorem is quietly used #

Panel methods and cavitation inception. Every low-order potential-flow code reports a minimum pressure coefficient by scanning the panels. That scan is a scan of the boundary, and the theorem is the reason it is enough — otherwise the code would have to search the field.

Grid checks. A potential-flow solve that produces an interior pressure below every boundary value has a bug, and this is a check that costs one pass over the field and needs no reference solution. The same reasoning gives a check on any incompressible solve: an interior minimum should coincide with a region of positive $Q$, and one that does not is a numerical artefact.

And the reason a diffuser’s worst point is its throat. The minimum pressure in an internal potential flow is on a wall, so a duct’s worst adverse gradient begins at a wall station rather than somewhere in the core — which is what makes wall-bounded diffuser design a one-dimensional problem in the first place.

What it costs to give the hypothesis up #

It is tempting to read the exemption as a small correction — vorticity is confined to thin layers and cores, so surely the theorem holds nearly everywhere and the exceptions are local.

The first half of that is true and the second does not follow. Vorticity is confined, and the pressure is not: the Poisson equation is elliptic, so a compact patch of positive $Q$ lowers the pressure over a region far larger than itself, decaying only algebraically. A vortex core a millimetre across sets the pressure over centimetres, which is why a tip vortex is visible in a humid sky as a condensation trail orders of magnitude wider than the vorticity that made it.

So the honest summary is not “the theorem holds except in cores”. It is that the theorem holds exactly when the flow is irrotational, that real flows are irrotational almost everywhere and not everywhere, and that the almost is doing more work than its measure suggests. This is the same asymmetry that makes the exact theory predict no drag: the region where the hypothesis fails is small, and the quantity that depends on it is not.

One number that is not exempt #

There is a quantity in this essay that survives everything, and it is the cylinder’s $-3$.

The minimum pressure coefficient on a circular cylinder in ideal flow is exactly $-3$, at exactly the shoulder, and it follows from two lines: the surface speed is $2U\sin\theta$, so $C_p = 1 - 4\sin^2\theta$, whose least value is $1 - 4$. It contains no radius, no speed, no fluid, and no approximation. Every one of the twenty-two figures in the four flows above was checked against it, and the sampled value came back at $-2.9999992$ — where the residual is the surface march’s angular resolution and nothing else.

That number is why the sampling above can be trusted at all. A search over a region reports the smallest thing it found, and a search that is looking in the wrong place reports a wrong answer with complete confidence; the only defence is one case whose answer is known in closed form and checked.

What is not claimed #

The bodies are two-dimensional and the sampling is finite. A grid of 240 by 240 with a clearance band around each surface can miss a minimum in a region smaller than a cell, and nothing here rules that out by measurement — what rules it out is the identity, and the sampling is a check on the identity rather than a substitute for it.

Compressibility is not treated. The Poisson equation above assumes $\nabla\cdot\mathbf{u} = 0$, and in a compressible flow the dilatation contributes its own source term whose sign is not fixed. Nothing in this essay applies above about Mach 0.3.

The vortex is a model. A Lamb–Oseen profile is an exact solution of the diffusion equation for vorticity and is not what any particular real vortex looks like; the essay’s claim about it is that $Q > 0$ permits an interior minimum, not that this profile is the one a propeller makes.

And the four flows are all exterior problems. The theorem holds in any region, interior or exterior, and nothing here demonstrates it on an internal flow — the diffuser paragraph above is an inference from the identity rather than a computation, and the eddies nobody stirs is a corner flow where the boundary is harder to sample than any of these.

And a free surface is a boundary. A pressure minimum on a free surface is on a boundary in the theorem’s sense, so the result says nothing about where a wave breaks — that is a different question with a different answer.