The displacement thickness delta represents the distance that streamlines are di
ID: 2326134 • Letter: T
Question
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.Explanation / Answer
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.
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