The displacement thickness delta represents the distance tha

The displacement thickness delta represents the distance that streamlines are displaced due to the development of a boundary layer. For a given duct, this thickness acts as a downstream area contraction of the core flow. Consider a wind tunnel test section that has a constant 80 cm times 80 cm cross-section, and is 3 m in length. If the freestream velocity is 30 m/s at the start of the test section (x = 0), estimate a) the magnitude of the core velocity gradient (du/dx) between the start and end of the test section, b) If du/dx = 0 is desired along the entire test section length, approximate the angle that the test section walls should be set at.

Solution

This problem required additional information, namely the velocity of entry flow:

Assuming it is such that the flow is laminar ( found by comouting the Reynolds number), then regardless of the thickness of the boundary layer, the CORE flow retains the same velocity, by definition.

As such du/dx = 0

b) if du/dx is zero throughout the section ( this is true regardless for the core), the angle of the wall should be such that the boundary layer one each wall should stay separate and not merge For a 3 m length, we need to calculate what the thickness of the BL becomes at 3 m.. To do this one needs the Reynolds number which depends on the velocity.

In general for laminar flow L/d =.06Re, approx 140

so d/L =1/140

from the trig tables or calculator, arcTan(1/ 140 ) =.409 degrees,

for a length of 3 metres and radial thickness of 80/2 cm =.4 metres, there is no effect of this angle meaning the core remains inviscid throughout.

One could answer part (b) starightaway, but I guess they want this kind of explanation in the course.