Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Rank — where it appears

The number of independent rows in a matrix, and in dimensional analysis the number of dimensions the problem's quantities actually use. Subtracting it from the number of quantities gives the number of dimensionless groups.

Named by 4 essays across 2 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 1.139 asks for a Francis. The specific-speed axis, with the four machine types on it and one duty marked: 3 m³/s at 60 m, on a shaft turning at 750 rev/min. The number is 1.1387, and the choice of runner follows from it before any blade has been drawn. What the number contains is a ratio of flow to head; what it does not contain is any size at all, which is why one axis serves a garden pump and a gigawatt turbine.

One number picks the machine

A flow rate, a head and a shaft speed contain exactly one dimensionless combination with no size in it. That combination decides whether a duty wants an impulse wheel, a Francis runner or a propeller — before anything has been drawn, sized, or costed.

applied · Specific speed Two ways to move a duty, along one axis, in opposite directions. A duty at a specific speed of 0.01 and what splitting it does. Dividing the head between stages in series multiplies each stage's specific speed by the number of stages to the three-quarter power, because the group carries (gH)^(−3/4); dividing the flow between units in parallel divides it by the square root of the number of units, because the group carries √Q. Both exponents are read off the group rather than remembered, and they are why the two operations are not interchangeable: three stages buy a factor of 2.28 and three units cost a factor of 0.58. The bands are drawn in the colour this site reserves for a borrowed claim, because where a Francis runner stops is practice rather than a result.

The duty that had no machine

Some duties have a specific speed outside every band, and no runner will do them at any size because the number contains no size. That is true and it is not the end. The same group says how to split the duty until it fits, and the two ways of splitting move it along the same axis in opposite directions, by exponents read straight off it.

applied · Specific speed 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

Named alongside it

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

Dimensional analysisScalingBuckingham's pi theoremCorrelationDimensionlessSimilarityAffinity lawsDrag coefficientOptimisationReynolds numberSpecific speedDimensionless number

All concepts