Pressure has no speed

How does the air ahead of a wing know the wing is coming?

The question sounds naive and it has a sharp answer that changes depending on which set of equations is being used. In the compressible equations the news travels outwards as sound, at a finite speed, and there are places it has not reached. In the incompressible equations there is no travelling at all: the pressure field adjusts everywhere at the same instant, including infinitely far away, and the mathematics that says so is worth looking at directly because it is the deepest assumption in half of this site.

Where the equation comes from #

Start with the momentum equation for a constant-density fluid,

$$\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{u} = -\frac{1}{\rho}\nabla p + \nu\nabla^2\mathbf{u}$$

and take the divergence of every term.

The first term becomes $\partial(\nabla\cdot\mathbf{u})/\partial t$, and continuity says $\nabla\cdot\mathbf{u} = 0$, so it is zero. The viscous term becomes $\nu\nabla^2(\nabla\cdot\mathbf{u})$, which is zero for the same reason. What is left is

$$\nabla^2 p = -\rho\,\nabla\cdot\left[(\mathbf{u}\cdot\nabla)\mathbf{u}\right] = -\rho\left(u_x^2 + 2u_y v_x + v_y^2\right)$$

Poisson’s equation. Three things about it are worth stating one at a time, because each is surprising.

There is no time in it. The pressure at this instant depends on the velocity field at this instant and on nothing earlier. It is not integrated forward; it is solved for, afresh, at every moment.

There is no viscosity in it. The source term contains only velocity gradients. Viscosity affects the pressure — by changing the velocity field, whose gradients are the source — but it appears nowhere in the equation that determines the pressure from that field.

There is no pressure on the right-hand side. The whole of the source is kinematic, and a flow field that has been measured, or computed some other way, determines its own pressure completely once the boundary values are given.

The source term is the strain against the spin #

The right-hand side looks like an accident of algebra and is not. Written in terms of the two halves the velocity gradient splits into, it is

$$\nabla^2 p = 2\rho\left(\tfrac{1}{2}|\boldsymbol{\Omega}|^2 - \tfrac{1}{2}|\mathbf{S}|^2\right)$$

— the spin’s magnitude against the strain’s, and nothing else. So the sign of the source at a point says which of the two is winning there.

Where rotation dominates, the Laplacian of the pressure is positive, and a function whose Laplacian is positive cannot have a local maximum. Vortex cores are pressure minima, necessarily, in every incompressible flow, with no dynamics invoked and no model chosen. It is why a bathtub vortex dimples the surface, why the core of a tip vortex condenses water out of humid air on a damp day, and why cavitation begins in a vortex before it begins anywhere else.

Where straining dominates the sign reverses and the pressure has no minimum to offer. That is the same quantity — the second invariant of the velocity gradient — that a modern flow visualisation uses to decide what to call a vortex at all, and it arrives here as the source term of an equation about pressure.

Elliptic, and what that costs #

Poisson’s equation is elliptic, and the classification is not a technicality. A hyperbolic equation — the wave equation, the supersonic small-disturbance equation — carries information along characteristics at a finite speed, so a solution at a point needs only what is inside its own domain of dependence. An elliptic equation has no characteristics at all. It cannot be solved anywhere without a boundary condition everywhere on the edge.

That is the property the figures are built to show, and it is why the boundary values in the relaxation are imposed from the closed form rather than guessed. Doing so is not cheating: the interior is what is being tested, and it was told nothing but the source term. Removing the boundary condition does not make the problem harder, it makes it insoluble, and the solver refuses it rather than returning something plausible.

What the solver computed, and how it was checked #

The relaxation is a successive over-relaxation sweep on a polar lattice fitted to the cylinder, run until the largest correction anywhere falls below $10^{-11}$.

Then the check: at every interior node, the relaxed pressure against $\tfrac{1}{2}\rho(U^2 - q^2)$ from Bernoulli’s equation. One is a lattice relaxation given gradients and edges; the other is an algebraic relation applied at a point. They share no arithmetic, and the worst departure anywhere in the domain is 0.018 per cent of the dynamic pressure.

A Cartesian grid was tried first and is worth recording, because it converged beautifully to the wrong answer. Cutting a circular hole out of a square lattice leaves a staircase, and the first few cells off the surface came out 22 per cent of the dynamic pressure wrong while the relaxation itself reached $10^{-10}$. The defect was not in the solver. It is that a circle drawn on square paper is not a circle — and an elliptic equation carries a bad boundary value into the whole interior rather than downstream of it, which is precisely the property this essay is about, arriving as a bug.

One blob, felt everywhere #

The sharpest demonstration of the ellipticity is to put a single compact patch of acceleration into an otherwise quiet box and measure the response.

A logarithm is barely a decay at all. Compare it with the alternatives a reader might expect: an inverse square would fall by a factor of a hundred over the same range, and an exponential would fall to nothing. The elliptic response has no length scale of its own — nothing in the equation sets a distance at which influence stops — and the only length in the answer is the size of the box, which is to say the position of the boundary.

That is why an incompressible calculation is global. Change the geometry anywhere and the pressure changes everywhere, and any solver that does not enforce this is not solving the equations.

The price, in one number #

The incompressible assumption is the statement that a pressure signal is infinitely fast compared with the flow. The ratio of the two speeds has a name.

$$\frac{\text{signal speed}}{\text{flow speed}} = \frac{a}{U} = \frac{1}{M}$$

So the assumption is exactly the statement that $1/M$ is infinite, and how good it is depends on how large $1/M$ actually is.

The conventional cut-off at $M = 0.3$ is where the density error reaches about five per cent, and it can now be read a second way: it is where a signal only manages three lengths per length of flight, so “instantaneous” has stopped being a fair description of it.

Where the assumption breaks completely #

In supersonic flow the ratio inverts and the geometry of the problem changes kind. A body moving faster than sound cannot send any news forward at all: there is a cone behind it that hears, and everything ahead is silent until the body arrives.

This is why the zone of silence is the most important structural fact about supersonic flow, and why the equations governing it are of a completely different type. The transition is not a matter of accuracy — an incompressible solver run at Mach 2 does not give a poor answer, it gives an answer to a different question.

What the pressure is actually for #

There is a way of reading all of this that makes the strangeness go away, and it is worth having.

In a compressible fluid the pressure is a thermodynamic variable: it is related to the density and the temperature by an equation of state, and it carries energy about as sound. In an incompressible fluid none of that is true. There is no equation of state — the density is a constant, and no relation ties it to the pressure — and the pressure has no independent existence of its own.

What it is instead is the field that enforces the constraint. The velocity is required to be divergence-free at every instant, and the pressure gradient is exactly whatever force field is needed to keep it so. In the language of constrained mechanics it is a Lagrange multiplier: it does no work, it stores no energy, it has no evolution equation, and its value at any moment is fixed by the requirement rather than by a history.

That is why it satisfies an elliptic equation with no time in it, and why the equation contains no viscosity. A constraint is enforced everywhere at once or it is not enforced. And it is why an incompressible pressure is only ever defined up to a constant: adding a constant changes no gradient, so it changes no force — a fact this site takes further in an argument about the word “suction”.

The compressible field on this site is where the pressure gets its thermodynamics back. There it has an equation of state, it carries sound at a finite speed, and the elegant global instantaneity above disappears — replaced by characteristics, zones of silence and shocks.

What the picture cannot show #

The boundary values had to come from somewhere. Every figure here imposes the exact pressure on the edges, which is available because the flow is one this site can solve in closed form. A general calculation cannot do that, and what it does instead — imposing a normal-derivative condition from the momentum equation at the wall — is one of the fiddliest parts of writing an incompressible solver.

Two dimensions flatter the argument. The logarithm becomes $1/r$ in three dimensions, which is a genuine decay, so the “felt everywhere” claim is weaker in three dimensions than these pictures make it look. It is still infinitely fast, and still global.

Nothing here is unsteady. The equation has no time in it, but the velocity field does, and in an unsteady flow the pressure is re-solved at every instant. A figure of a steady field cannot show the thing that makes the instantaneity strange, which is that a wing that changes its incidence alters the pressure a mile away with no delay at all — while the circulation that decides its lift takes tens of chords to settle. The pressure is instantaneous and the flow it is a pressure of is not, and keeping those two straight is most of what unsteady aerodynamics is.

The relaxation is not how a real solver does it. Six thousand Gauss–Seidel sweeps is the slowest respectable method there is, chosen here because it is transparent; a production code uses a multigrid or a spectral solve and reaches the same answer in a fraction of the work. What does not change is that the solve is global, which is the whole point.

What the ellipticity costs a computer, and the two ways round it #

The instantaneity is elegant on paper and it is the single most expensive property an incompressible calculation has, because a global solve is exactly the thing a parallel machine is worst at. A compressible code advances every cell from its own neighbours and nothing else; an incompressible one must, at every time step, solve a problem in which every cell depends on every other. The architecture of most of computational fluid dynamics is a response to that one sentence.

Two families of answer exist and they make opposite bargains.

Solve the elliptic problem. The projection method takes a provisional velocity ignoring the pressure, then finds the pressure whose gradient projects it back onto the divergence-free fields — which is a Poisson solve. It is exact, it respects the physics of the limit, and it costs a global communication at every step. Multigrid and spectral methods make it as cheap as it can be made, and most of the run time of most incompressible codes is still inside it.

Or put the sound speed back, deliberately reduced. Artificial compressibility restores a $\partial p/\partial t$ term with a fabricated coefficient, which makes the system hyperbolic again and therefore marchable with a purely local stencil. The fabricated wave speed is chosen large enough that pressure crosses the domain in a few iterations — so it is effectively instantaneous — and small enough that the time step it forces is affordable. The price is that the answer means nothing until the artificial transient has died, so the method is only valid at steady state, and the whole family of unsteady variants exists to work around that by iterating to convergence inside every physical time step.

The same bargain appears in a third guise wherever a flow is slow but its density is not constant. A flame, a heated cavity, a boiling channel: the Mach number is tiny, so resolving sound would cost everything, and the density varies by factors of several, so the incompressible equations do not apply. The standard resolution is a low-Mach splitting: the pressure is written as a large part that is uniform in space and varies only in time — the thermodynamic pressure, which is what feeds the equation of state — plus a small part that varies in space and satisfies an elliptic equation, and is what drives the flow. Acoustics has been removed by hand, the thermodynamics has been kept, and the elliptic solve is back.

The pattern across all three is worth naming, because it is the same one this essay is about read from the other side. The incompressible limit trades a finite signal speed for a global constraint, and every numerical method for it is a decision about which of those two to pay for. Keeping the constraint costs a global solve. Restoring a signal speed costs a transient that has to be waited out. There is no third option, because the two are the same fact.

And the reason the trade is worth making at all is the number this essay ends on. Resolving genuine acoustics in a flow at Mach 0.01 would mean a time step a hundred times smaller than the flow itself requires, for a wave whose amplitude is a ten-thousandth of the pressures being computed. The elliptic pressure is what is left of the sound after its speed has been taken to infinity, and paying for a global solve is what it costs not to resolve a wave nobody wanted.

Where the model stops #

Incompressible flow is not a fluid. It is a limit — the limit of the compressible equations as the Mach number goes to zero — and the elliptic pressure equation is what that limit does to the wave equation the pressure really obeys. Sound waves are still there in the real fluid; the model has taken their speed to infinity, so they arrive before anything else can happen, and their dynamics has been discarded.

Everything downstream of that decision inherits it. Bernoulli’s equation along a streamline is an instantaneous relation between pressure and speed; the pressure integral that gives lift assumes the whole surface pressure is in equilibrium with the whole flow field; d’Alembert’s paradox is derived on a field that had no time in it. None of those is wrong within the limit, and all of them are silent about how the fluid got there.

The limit is excellent for a very wide range of the subject. Below $M = 0.3$ the error is a few per cent, and for water at any speed a laboratory can reach it is negligible — which is why naval hydrodynamics never mentions it and aeronautics cannot stop mentioning it.

Who found it, and when #

Euler wrote the momentum equation for an inviscid fluid in 1757 and the incompressible constraint with it, so the pressure Poisson equation is implicit in his work. Poisson’s equation itself dates from 1813, in gravitation rather than fluids, and Lamb’s Hydrodynamics treats the pressure as an elliptic problem throughout.

The practical consequence — that solving an incompressible flow means solving an elliptic equation for the pressure at every time step — became the central difficulty of computational fluid dynamics in the 1960s. Chorin’s projection method of 1968 is the standard answer: advance the velocity ignoring the pressure, then solve a Poisson equation to project it back onto the divergence-free fields. Most of the run time of most incompressible solvers is spent inside that one elliptic solve, which is the computational shadow of everything above.

Where the ladder goes next #

Pressure has now been established as a field with an equation of its own, determined everywhere at once and holding no memory. What has not been asked is what it can do: whether a low pressure over a wing is a pull, and whether the sign of a number that depends on where zero was put can be a mechanism at all. It cannot, and the proof is a line of vector calculus in a later field.