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What is the ratio of displacement thickness to boundary layer thickness for a linear distribution of velocity \(\frac{u}{{{u_\infty }}} = \frac{y}{\delta }\) in the boundary layer on a flat plate, where δ is the boundary layer thickness and uis the free stream velocity? 
1. 0.5
2. 0.67
3. 0.75
4. 0.8

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Correct Answer - Option 1 : 0.5

 

Concept:

Nominal boundary thickness as δ 

Displacement thickness is given as

\({δ ^*} = \mathop \smallint \limits_o^δ \left( {1 - \frac{u}{U_\infty}} \right)dy\)

Calculation:

Given, \(\frac{u}{{{u_\infty }}} = \frac{y}{\delta }\)

\({δ ^*} = \mathop \smallint \limits_o^δ \left( {1 - \frac{y }{δ}} \right)dy\)

\({δ ^*} =δ - \frac{δ }{2}\)\(\frac{δ }{2}\)

∴ The ratio of displacement thickness to nominal thickness will be ​ = \(\frac{{\frac{δ }{2}}}{δ }=\frac{1}{2}=0.5\) 

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