# What is the ratio of displacement thickness to boundary layer thickness for a linear distribution of velocity $\frac{u}{{{u_\infty }}} = \frac{y}{\de 0 votes 324 views in General closed 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 ​ = $\frac{{\frac{δ }{2}}}{δ }=\frac{1}{2}=0.5$