Tip Clearance Loss: Method

Tip Clearance and the Duct's Suction Peak

A duct earns its keep by carrying part of the thrust on its lip: the inflow accelerating around the leading edge creates a suction peak whose axial component is thrust. That peak depends on the rotor pulling the flow through the duct close to the wall. Air leaking through the gap between the blade tips and the wall short-circuits the pressure difference across the rotor near the tip, the suction peak collapses, and the duct's share of the thrust falls away. Martin and Tung's potential-flow pictures (their Figure 9) show the lip pressure distribution shrinking as the gap grows from 1 to 5 percent of the radius.

The data

Two public test series are used. Martin and Tung (NASA Ames, 2004) measured a 10-inch ducted rotor at 9,000 rpm with the tip gap varied from 1.0 to 4.2 percent of the radius; the hover figure of merit fell from 0.67 to 0.47 toward the isolated rotor's 0.44, most of the loss in the first percent. Pereira (University of Maryland, 2008) tested seventeen 16-cm shrouded-rotor models; his Figure 3.17 gives, for four shroud series, the ratio of the shrouded rotor's maximum thrust-to-power to the open rotor's against tip clearance in percent of the tip diameter. The series with the smaller lip radius and the shorter diffuser lose fastest.

How the tool uses them

Each curve is normalized by its value at the tightest clearance tested and treated as the fraction of the zero-gap figure of merit kept. The Martin and Tung curve is extrapolated linearly from 1 percent to zero gap; Pereira's series are averaged over those that cover the entered clearance. Both fractions are scaled to the zero-gap figure of merit entered on the form, and the power penalty at constant thrust (power scales with 1/FM) and the thrust penalty at constant power (thrust with FM^2/3) follow. The break-even clearance is where the scaled Martin and Tung figure of merit falls to the entered open-rotor value.

Limits

Both rotors are small (tip Reynolds numbers of order 10⁵), where the lip boundary layer is thick and the loss is severe; a large fan at the same percentage clearance should fare better, so the estimate is conservative for it. The gap under load is what matters: add the centrifugal and thermal growth of the blade and the ovality of the duct to the static clearance. Designs that keep the gap below about half a percent of the radius use a stiff duct and a tip-path controlled by a rigid hub; abradable liners allow a running clearance near zero.