The picture belongs to whoever is watching

Every figure on this site is drawn in some frame, and until now none of them has said which. For most of the results that is harmless — a drag, a lift, a pressure coefficient are the same numbers whatever the camera is doing — and for the pictures themselves it is not harmless at all.

Take the flow this collection opens with: ideal flow past a circular cylinder. In the wind tunnel the body is at rest, the air arrives from the left, and the streamline pattern has a dividing streamline that splits at a stagnation point on the nose and closes at another on the tail. That picture is in every textbook.

Now stand in the air instead. The cylinder is moving through still fluid, and the field is the same field minus one constant vector. The streamlines are closed loops. There is no stagnation point anywhere — not on the body, not in the fluid, not at infinity.

The three transformations, and what each leaves alone #

A change of observer is one of three things, and they have very different consequences.

A Galilean boost — a frame translating at constant $\mathbf V$ — gives $\mathbf u' = \mathbf u - \mathbf V$. The velocity changes everywhere by a constant, so the gradient of the velocity does not change at all. Every quantity built from $\nabla\mathbf u$ survives: the vorticity, the rate of strain, the divergence, and everything computed from them.

A rigid rotation of the frame at rate $\Omega$ gives $\nabla\mathbf u' = \mathsf Q^{\mathsf T}(\nabla\mathbf u)\mathsf Q - \tilde\Omega$, and the extra term is antisymmetric. So the symmetric part — the rate of strain — merely rotates, and its eigenvalues are unchanged, while the antisymmetric part picks up a constant. The vorticity shifts by exactly $-2\Omega$.

An accelerating frame adds body forces and is a different subject; nothing here needs one.

The split into strain and rotation is the one deformation made by an argument that never asked who was watching. This is what that argument bought: one half of the tensor is a property of the fluid and the other half is a property of the observer.

The surface speed, which is uniform in one frame and not in the other #

The cylinder’s surface is where this stops being a curiosity, because that surface carries the pressure distribution every force on the body is integrated from.

In the tunnel the tangential speed is $2U\sin\theta$: zero at the nose and tail, twice the free stream at the shoulders. In the frame of the still air the same surface velocity is

$$\mathbf u - (U,0) = (-U\cos 2\theta,\ -U\sin 2\theta),$$

whose magnitude is $U$ for every $\theta$. The surface on which the speed runs from nothing to twice the free stream in one frame is a surface of uniform speed in the other, exactly, and the computed departure from uniformity is two parts in a thousand million — which is the offset at which the surface is sampled and nothing else.

The pressure, of course, is the same in both. It is recovered from neither column alone: Bernoulli’s constant is not frame-independent either, and the two observers write down two different constants and two different speed distributions whose difference is the same pressure field. This is one more entry on the list of things Bernoulli’s equation needs stated before it means anything.

Steadiness belongs to the observer too #

The flow in the tunnel is steady: photograph it twice and the two pictures are identical, which is exactly the observation acceleration starts from. In the frame of the air it is unsteady, because the body is moving through the field and the velocity at any fixed point changes as it passes.

That matters more than it sounds. Steady is the hypothesis behind Bernoulli’s equation along a streamline, behind the statement that streamlines and pathlines coincide, and behind half the simplifications in the subject. All of them are conditional on a frame, and the frame is almost never named.

The consequence for the pictures is immediate. In a steady frame the streamlines are the pathlines, and drawing one is drawing the other. In the moving frame they are as different as a photograph and a long exposure, and the closed loops in the right-hand panel of the first figure are instantaneous streamlines of a field that has moved on by the time a particle has gone round one of them.

What both observers do agree about #

There is one curve neither can dispute, and it is the material one.

A particle either passes through a given piece of fluid or it does not. Integrate its path in the tunnel frame, integrate it independently in the moving frame through the unsteady field there, and translate the second by the frame’s own displacement: the two land on top of each other. The two integrations share no arithmetic — the second is a fourth-order Runge–Kutta pass through a time-dependent field with the body’s position moving inside it — and they agree to eight parts in a thousand million million.

Spin, restated as a statement about observers #

The site’s founding distinction is that spin is not the same as going round: a whirlpool with perfectly circular streamlines can have no rotation in it anywhere, and a flow with dead straight parallel streamlines can be rotating everywhere. That essay’s two examples are solid-body rotation and the free vortex, and they are worth measuring again from a moving observer.

Solid-body rotation has $\omega = 2\Omega$ in the laboratory. In the frame that turns with it the vorticity is zero — to machine precision, because $2\Omega$ is exactly the shift — and it is irrotational. Its principal rates of strain are zero in both frames, which is what rigid means.

A free vortex is irrotational in the laboratory. In a frame rotating at the fluid’s own local angular velocity it has a vorticity of $-0.377$ at the sample radius. Its strain rates are the same non-zero pair in both frames.

So the distinction survives, and what survives it is the strain. That is the honest form of the rule: spin is not the same as going round, and neither of them is a property of the fluid alone. What is a property of the fluid alone is how fast a fluid element is being pulled apart, which is the subject of the next essay in this field.

Why the stress law could only ever have contained the strain #

There is a consequence of the classification above that is not about pictures at all, and it decides the form of the equations this whole collection solves.

A constitutive law is a statement about what a material does — how hard it resists being deformed — and a material cannot know who is watching it. So the relation between stress and motion must take the same form for every observer, which is the principle of material frame indifference, and it immediately forbids the law from containing anything on the essay’s third list.

The stress in a fluid is objective: it is a real force per unit area transmitted across a real surface, and two observers agree about it. The rate of strain is objective. The vorticity is not. So a linear relation between stress and the velocity gradient can contain the symmetric part of that gradient and cannot contain the antisymmetric part, and the Newtonian law

$$\sigma_{ij} = -p\,\delta_{ij} + 2\mu\,e_{ij} + \zeta\,(\nabla\!\cdot\!\mathbf{u})\,\delta_{ij}$$

is not one modelling choice among several. It is the only linear isotropic form available, and the absence of the vorticity from it is required rather than observed.

That also settles, from a different direction, a result this collection derives by algebra elsewhere. The dissipation function contains no rotation because contracting a symmetric stress with an antisymmetric velocity gradient gives zero — which is true, and reads as a happy cancellation. The deeper statement is that the stress could not have contained a rotation in the first place, so a fluid in solid-body rotation dissipating nothing is not an accident of the algebra. A fluid cannot be charged for a motion an observer could remove by turning round.

The same principle explains an asymmetry that otherwise looks arbitrary. Write the equations in a rotating frame and the momentum equation gains Coriolis and centrifugal terms while the constitutive law gains nothing at all. That is exactly right: acceleration is not objective, so a frame change must show up in the term that contains it, and the stress is objective, so it must not show up in the term that contains that.

And it constrains every more elaborate model the subject has. A viscoelastic fluid carries a stress with a memory, so its law involves a time derivative of the stress — and the ordinary partial derivative of a tensor is not objective, so a model built on it predicts different behaviour for two observers of the same experiment. The repair is to use a derivative that is objective, and the upper-convected and corotational derivatives that fill the polymer literature exist for precisely that reason rather than for any reason about polymers.

Which turns this essay’s classification from a caution about figures into a design rule for theories. The list of quantities that survive a change of observer is the list of ingredients a constitutive law is allowed to be made of, and the fact that so short a list forces so much of the Navier–Stokes equations is the strongest thing the argument does.

The criterion that inherits the problem #

The Q-criterion — the excess of rotation over strain — is how most of computational fluid dynamics decides where a vortex is. It is built from the antisymmetric part of the velocity gradient, so it is not objective, and this is what that costs:

Take pure straining flow, $\mathbf u = (\alpha x, -\alpha y)$. It has no vorticity anywhere, so $Q$ is negative everywhere and no part of it is a vortex by any reckoning. An observer rotating at $\Omega$ measures $Q = \tfrac12\big((2\Omega - \omega)^2/2\ -\ |\mathsf D|^2\big)$, which turns positive as soon as $\Omega$ exceeds the strain rate. The same flow, at the same point, at the same instant, contains a vortex or does not, according to how fast the person looking is spinning.

This is not a hypothetical objection. Vortex identification in a turbomachine is routinely done in the rotating frame of the blade row, and the Q-field there is a different field from the one in the stationary frame. Which of them is the answer is a question with no answer, and what to do about it is a whole essay.

What is not in the velocity field #

The rule the essays around this one are written to is that each names the piece of information its answer needs which the velocity field does not contain. Here it is the plainest it will be: the observer.

A velocity field is a set of vectors attached to points, and every vector in it is measured relative to something. Change that something and every vector changes, most of the derived pictures change, and a short list of quantities does not:

• survive a boost: the whole velocity gradient — vorticity, strain, divergence — and therefore the pressure field, the forces, the circulation round a material loop, and the sequence of events each parcel undergoes;
• survive a rotation as well: the principal rates of strain and everything built only from the symmetric part;
• survive neither: the streamline pattern, the stagnation points, whether the flow is steady, the Q-criterion, the Bernoulli constant, and the answer to “is there a recirculating region here”.

The two experiments that are the same experiment #

The practical form of all this is old and is usually stated without the reasoning.

A wind tunnel holds a model still and blows air past it. A towing tank pulls a model through still water. A flight test flies the aeroplane. These are held to be equivalent, and they are — the forces are identical, because forces are integrals of pressure and pressure survives a boost — but the data they produce are not the same data at all.

A tunnel’s hot wire sits at a fixed point in the tunnel frame and records a signal that is steady. The same instrument towed alongside a model records the same physical fluid and returns a steady signal too, because it is at a fixed point in the model’s frame. An instrument fixed to the tank, which is the natural thing to build, records the passage of the whole disturbance and returns something unsteady with a duration set by how long the model takes to go by. Three records, one flow, and only the last of them looks like an unsteady problem.

Particle image velocimetry makes this concrete in a way a pitot tube does not. It returns a velocity field, and that field is in the frame of the camera. A camera bolted to the tank floor and a camera riding the carriage produce fields differing by a constant vector — which is to say, produce completely different streamline plots of the same water, with the vorticity fields identical and the recirculation regions not.

Where the convention came from, and where it leaks #

Aerodynamics adopted the tunnel frame because that is where the measurements were made, and it named the free stream the relative wind — a phrase that quietly asserts that the body’s frame is the natural one. For a steady flight condition it is the right choice and nothing goes wrong.

It leaks in three places, all of which this collection has already met without saying so.

A wing near the ground. The image system that makes the ground work is written in the frame of the ground, and the wing is moving in it; the same problem in the wing’s frame has a moving wall, which is a different boundary condition with a different mathematical status. Both are solved routinely and the second is harder for no physical reason.

A rotor blade. The blade sees a steady flow in its own rotating frame, and that frame is not inertial, so the tidy picture is bought with centrifugal and Coriolis terms in the momentum equation. The Coriolis term is a frame term and its size relative to the others is the whole of what the Rossby number measures.

An unsteady aerofoil. The lift that arrives late is computed in the frame of the aerofoil, where the shed wake streams away downstream. In the frame of the still air the wake sits where it was made and the aerofoil moves away from it, which is the picture that makes Kelvin’s theorem obvious and the one nobody draws.

None of these is an error. All three are places where a reader who has been shown one picture will find the other unrecognisable, and where the reason is not physics.

What to do about it #

Three habits follow, and this collection now keeps all three.

Name the frame on any figure whose content is a pattern. A streamline plot, a stagnation-point count or a recirculation bubble is a statement about an observer. A pressure distribution, a drag coefficient or a vorticity contour under a boost is not.

Prefer the invariant quantity when there is a choice. If a question can be asked about the rate of strain rather than about the streamline topology, ask it that way: the answer will then be a property of the flow. The stretching of material lines is such a question, and it turns out to be the one that decides mixing.

And do not read a pattern as a mechanism. The closed loops in the moving frame are not eddies and nothing is recirculating: they are the level sets of a scalar that is different in every frame. The count of critical points in a plane flow obeys an integer constraint whichever frame it is drawn in, and the constraint is satisfied by different counts in different frames — two on the body here, none there, and no contradiction.

The apparatus of this subject was built by people standing in wind tunnels, and it inherited the wind tunnel’s frame without ever saying so. Most of the time that costs nothing. It costs something exactly where the picture is the argument, and this collection is built on pictures.