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

Material lines — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Two flows with the same rate of strain, doing different things to a blob. A circle of fluid carried by two flows chosen to have exactly the same rate-of-strain magnitude, drawn at four times. Pure strain pulls it into an ellipse whose axes stay put; simple shear pulls it into an ellipse whose axes rotate as fast as they stretch. Both have zero divergence, so both preserve the area. The difference between them is not the strength of the straining but what the rotation does to the direction being stretched.

    Longer, with nothing pulling it

    Two flows with exactly the same rate of strain. In one a line of fluid grows by a factor of 148 in five time units; in the other it grows by 10. Turn the straining axes faster than the strain rate and no line grows at all, however hard the fluid is being strained.

    part 1 · kinematics
  2. One rate per moment, and none of them the same. lambda_p = ln<l^p>/(p t) against p. As p goes to zero it is the Lyapunov exponent, the rate of the typical element; at p = 1 it is the rate of the average length, which is nearly twice as large. If ln l were exactly Gaussian this would be a straight line with the Lyapunov exponent as its intercept, and the departure from that line is the same multifractality the velocity increments have.

    The stretching rate that is not one number

    A material line in a flow gets longer, and there is a theorem saying its length grows at a definite exponential rate. There is also a rate at which the average length grows, and it is nearly twice as large — and a different rate for every moment of the distribution.

    part 2 · kinematics
  3. One pair of strainings, two orders, two lengths. The stretch of the most-stretched material direction against time, for a simple shear followed by a pure strain and for the same two in the other order. The two curves are identical until the swap and separate afterwards, ending a factor of 2.16 apart.

    Two strainings, and the order they came in

    A material line is stretched by a shear and then by a pure strain, and then by the same two in the other order. Every instantaneous measure of how hard the fluid was being worked is identical in the two cases. The lengths at the end differ by a factor of 2.16.

    part 3 · kinematics

All series