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

Hydrodynamic impulse — where it appears

The impulse an external agency must supply to set a body moving in an ideal fluid from rest, computed as a surface integral over the body. It is finite and unique where the fluid's own momentum is neither, it equals the added mass times the speed, and its rate of change is the force on the body.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The momentum of the fluid, against the shape of the region it is added up over. Momentum of the fluid around a cylinder moving through it, divided by the body's hydrodynamic impulse, against the aspect ratio of the rectangle the integral was taken over. Every rectangle has the same area and contains the same body. A tall region gives minus the impulse, a long one gives plus it, a square gives exactly zero, and the limit of a large region is whichever of those the region was shaped like. The momentum of an unbounded ideal flow is not a number.

The momentum with no value

A cylinder is pushed from rest to a steady speed. Work was done, energy went into the fluid, something was pushed. How much momentum does the fluid carry? The integral converges, the answer is finite, and it is a different finite number for every shape of region it is summed over.

inviscid · Impulse A tube opened at the foot of a reservoir. Speed in a two-metre tube fed by a one-metre head, from the instant it is opened. The quasi-steady answer — Torricelli's √(2gh), which is what dropping the time derivative from Bernoulli's equation gives — is reached at the instant of opening, before any fluid has moved. The true answer is a hyperbolic tangent with a time constant of 2L/√(2gh), and it takes 2.4 seconds to come within one per cent. The whole of that transient is one term.

The pressure that depends on the past

Bernoulli's equation for an unsteady flow has a term nobody writes down and a right-hand side that is a function of time rather than a constant. The term is exactly zero once a flow has started and is the whole of the flow while it is starting, which is why it never appears in an answer and is never negligible in getting to one.

inviscid · Unsteady bernoulli

Named alongside it

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

Added massMomentumPotential flowBernoulli's equationControl volumeConvergenceDoubletFar fieldKinetic energyMeasurementModel limitPressure impulse

All concepts