Concept

Drag coefficient — where it appears

Drag divided by the dynamic pressure and a reference area, in which form two bodies of different size may be compared. Its near-constancy over decades of Reynolds number for a bluff body is the dissipation anomaly seen as a force.

Named by 6 essays across 4 fields — each of them below, with the objects they name alongside it.

5 quantities, 3 rows, 2 left over. The dimension matrix for the drag on a sphere: one column per quantity, one row per base dimension, and every entry an exponent. Buckingham's theorem is a statement about this matrix and nothing else — the number of independent dimensionless groups is the number of columns minus the rank, computed here by elimination. Nothing about fluids enters until somebody decides which columns to write down.

Counting what matters

Five quantities decide the drag on a sphere, and the experiment that measures it has one curve in it rather than a five-dimensional table. The reason is a rank: the matrix of dimensions has three independent rows, and what is left over is the number of dimensionless groups the answer can possibly depend on.

regimes · Dimensional

The tunnel makes the stream 7.4% faster. The flow past a cylinder between two walls, solved from the closed form of an infinite image row rather than from a truncated sum. The walls are streamlines exactly — that is what the images are for, and the normal velocity on them is zero to the last bit — and the flow beside the body is squeezed between the body and the wall, which is the whole of the blockage effect. The stream at the model is 7.40% faster than the speed the tunnel's own instruments report far upstream.

The instrument in the answer

A model in a wind tunnel is not a model in the sky. The walls are supplied by an infinite row of reflections, the stream at the model is faster than the tunnel's own instruments report, and the correction is not an empirical fudge — it is a series with a closed form.

applied · Metering

The Kirchhoff flow past a flat plate. A uniform stream meeting a flat plate held across it, with two streamlines leaving the edges and never returning. Between them is a wake of fluid at rest at a constant pressure. The equations solved are the same equations that give d'Alembert's paradox for a closed body, and this flow has a drag coefficient of 0.8798.

Drag in the theory that forbids it

d'Alembert's paradox is a theorem about flows that close behind the body. Stop requiring that, let two streamlines leave the edges and never come back, and the same equations — no viscosity, no vorticity — produce a drag coefficient of 0.8798.

inviscid · Free-streamline

Morison's two terms over one cycle, at KC = 10. The drag term, in phase with the velocity and going as its square; the inertia term, in phase with the acceleration and ninety degrees ahead of it; and their sum, which is what a load cell records. The peak of the total is not the peak of either, and its position in the cycle is the only thing in the record that says how the two are divided.

Long enough to make a wake

A Reynolds number cannot ask whether an oscillating flow gets round a body before it turns and comes back, because it has no time in it. The Keulegan–Carpenter number can, and it decides which of Morison's two terms is the force. What it discards is the phase — and a peak force measurement cannot recover it.

regimes · Keulegan–Carpenter number

The same data, in a basis the theorem allows just as much. The identical two hundred and forty points, plotted as F/(mu U d) — which is the drag coefficient times the Reynolds number, a perfectly legitimate pi group forming a complete pair with the abscissa. The result is a straight line of slope 0.9985 through five decades with an r-squared of 0.9985, and it contains no physics: the ordinate contains the abscissa.

The groups are not the only groups

Buckingham's theorem fixes how many dimensionless groups an answer can depend on and says nothing about which. Two of the infinitely many legitimate choices are used here on the same data: one manufactures a straight line through five decades out of a constant, and the other erases Stokes' law completely.

regimes · Dimensional

In clean water a bubble rises nearly three times as fast as the same bubble in tap water. The terminal rise speed of an air bubble in water at 20 °C against its radius, from buoyancy balanced against drag: with a clean, shear-free surface using Moore's law, and with a surface immobilised by contamination using the rigid-sphere correlation. At 0.3 mm the clean bubble rises at 13.0 cm/s against 6.7; at 0.5 mm at 31.0 against 11.2, a factor of 2.76. Beyond a radius of 0.47 mm the clean bubble's Weber number passes one, its shape flattens, and a spherical calculation stops describing it; that region is shaded. Nothing about the bubble's size, gas or liquid changes between the two curves — only whether its surface can move.

The vorticity a clean surface cannot refuse

A clean bubble's surface cannot hold a shear stress, and it is easy to conclude that it makes no vorticity. On a curved surface it must carry exactly 2κu — three times the speed over the radius at a sphere's equator, whatever the Reynolds number. That is so much weaker than a rigid wall's that the flow stays irrotational to leading order, and the bubble's drag is the dissipation of that irrotational flow: 48/Re, three to ten times below a rigid sphere's.

kinematics · Boundary conditions

Named alongside it

The objects these essays reach for when they reach for this one.

Measurement Model limit Potential flow Boundary condition Buckingham's pi theorem Correlation d'Alembert's paradox Dimensional analysis Dimensionless number Dissipation Rank Reynolds number

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