Displacement Thickness ,Momentum Thickness and Energy Thickness

Q. For air flow over a flat plate, velocity (U) and boundary layer thickness ($\delta$) can be expressed respectively, as

$\frac{u}{u_{\infty }}=\frac{3}{2}\frac{y}{\delta }-\frac{1}{2}{\left(\frac{y}{\delta }\right)}^3;$ $\delta =\frac{4.64x}{\sqrt{R{e}_x}}$

If the free stream velocity is 2 m/s and air has kinematic viscosity of 1.5$\times {10}^{-5}{m}^2/s$ and density of 1.23 kg/$m^3$, the wall shear stress at x = 1 m, is

1. $2.36\times {10}^{2}N/m^2$
2. $43.6\times {10}^{-3}N/m^2$
3. $4.36\times {10}^{-3}N/m^2$
4. $2.18\times {10}^{-3}N/m^2$

Q. The thickness of the laminar boundary layer over a flat plate at two different sections P and Q are 0.8 cm and 2.4 cm respectively. If the section Q is 3.6 m downstream of P, the distance of section P from the leading edge of the plate is

1. 0.32 m
2. 0.22 m
3. 0.40 m
4. 0.53 m

Q. The velocity profile inside the boundary layer for flow over a plate is given as${\frac{u}{U}}_{\infty }=sin\left(\frac{\pi }{2}\frac{y}{\delta }\right)$ where $U_{\infty }$ is the free stream velocity and $\delta$ is the local boundary layer thickness. If $\delta \ast$ is the local displacement thickness, the value of $\frac{{\delta }^{\ast }}{\delta }$ is

1. $\frac{2}{\pi }$
2. $1-\frac{2}{\pi }$
3. $1+\frac{2}{\pi }$
   
4. 0

Q. The displacement thickness of a boundary layer is

1. The distance to the point where $\frac{v}{V}=0.99$ .
2. Distance where the velocity 'v' is equal to the shear velocity V', that is, where v = V.
3. The distance by which the main flow is to be shifted from the boundary to maintain the continuity equation.
4. One half the actual thickness of the boundary layer.

Q. The velocity distribution in the boundary layer is given by $\frac{u}{U}=\frac{y}{\delta },$ where u is the velocity at a distance of y from the boundary and u=U at y=$\delta ,\delta$ being boundary layer thickness. Then the value of momentum thickness will be

1. $\frac{\delta }{2}$
2. $\frac{\delta }{4}$
3. $\frac{\delta }{6}$
4. $\frac{\delta }{8}$

Q. The viscous laminar flow of air over a flat plate results in the formation of a boundary layer. The boundary layer thickness at the end of the plate of length L is ${\delta }_L$. When the plate length is increased to twice of its original length, the percentage change in laminar boundary layer thickness at the end of the plate (with respect to ${\delta }_L$) is_____(correct to two decimal places).

1. 41.42

Q. What is the momentum thickness for the boundary layer with velocity distribution $\frac{u}{U}=\frac{y}{\delta }$?

1. $\delta /6$
2. $\delta /2$
3. $3\delta /2$
4. $2\delta$

Q. For a turbulent boundary layer (under zero pressure gradient), the velocity profile is described by the one-fifth power law. What is the ratio of displacement thickness to boundary layer thickness?

1. 1/7
2. 1/6
3. 1/5
4. 1/4

Q. In a laminar boundary layer, the velocity distribution can be assumed to be given, in usual notations, as
$\frac{u}{v}=\frac{y}{\delta }$
Which one of the following is the correct experession for the displacement thickness $\left({\delta }^{\ast }\right)$ for this boundary layer?

1. $({\delta }^{\ast }=\delta )$
2. $({\delta }^{\ast }=\delta /2)$
3. $({\delta }^{\ast }=\delta /4)$
4. $({\delta }^{\ast }=\delta /6)$

Q. The following are the velocity profiles of a flow over a boundary. Match the following
A. $\frac{u}{U}=\frac{2y}{\delta }-{\left(\frac{y}{\delta }\right)}^2$

B. $\frac{u}{U}=-\frac{2y}{\delta }+{\left(\frac{y}{\delta }\right)}^3+{\left(\frac{y}{\delta }\right)}^4$

C. $\frac{u}{U}=2{\left(\frac{y}{\delta }\right)}^2+{\left(\frac{y}{\delta }\right)}^3-2{\left(\frac{y}{\delta }\right)}^4$

1. On the verge of separation

2. Attached flow

3. Detached flow

1. $A-1,B-2,C-3$
2. $A-2,B-3,C-1$
3. $A-3,B-1,C-2$
4. None of these

Q. Air$(\rho =1.2kg/m^3$ and kinematic viscosity, $v=2\times {10}^{-5}{m}^2/s)$ with a velocity of 2 m/s flows over the top surface of a flat plate of length 2.5 m. If the average value of friction coefficient is $C_f=\frac{1.328}{{\sqrt{Re}}_x}$, the total drag force (in N) per unit width of the plate is_________

1. 0.0159

Q. A flat with a sharp leading edge is placed along a free stream of fluid flow. Local Reynolds number at 3 cm from the leading edge is ${10}^5$. What is the thickness of the boundary layer?

1. 0.47 mm
2. 0.35 mm
3. 0.23 mm
4. 0.12 mm

Q. If the diameter of the pipe is given as D, what is the maximum thickness of the boundary layer?

1. 0
2. $\frac{D}{2}$
3. D
4. 2D

Q.

A layer of glycerine of thickness $1mm$ is enclosed between a big plate and another plane of area ${10}^{-2}{m}^2$. If the coefficient of viscosity of glycerine is $1kg·m^{-1}·s^{-1}$, then the force in $N$ required to move the plate with a velocity of $0.07m·s^{-1}$.

1. $7$

2. $0.7$

3. $70$

4. $0.07$

Q. A flat plate is kept in an infinite fluid medium. The fluid has a uniform free-stream velocity parallel to the plate. For the laminar boundary layer formed on the plate, pick the correct option matching List-I and List-II.

List - I	List - II
A. Boundary layer thickness	1. Decrease in the flow direction
B. Shear stress at the plate	2. Increase in the flow direction
C. Pressure gradient along the plate	3. Remains unchanged

1. A - 1, B - 2, C - 3
2. A - 2, B - 2, C - 2
3. A - 1, B - 1, C - 2
4. A - 2, B - 1, C - 3

Q. Air at ${20}^{\circ }C$ forms a boundary layer near a solid wall in which velocity distribution is given by:
$u=u_0\sin \left(\frac{\pi y}{2\delta }\right)$
The boundary layer thickness is 7 mm and the peak velocity is 9 m/sec. What will be the shear stress in the boundary layer at y = 3.5 mm ? $({\mu }_{air,{20}^{\ast }}=1.81\times {10}^{-5})$

1. $2.59\times {10}^{-3}Pa$
2. $25.9\times {10}^{-3}Pa$
3. $3.66\times {10}^{-3}Pa$
4. $36.6\times {10}^{-3}Pa$

Q. If in a turbulent boundary layer the velocity distribution is assumed to be given by $u\propto y^{1/4}$, then the boundary layer thickness varies with longitudinal distance X as

1. $X^{1/2}$
2. $X^{1/7}$
3. $X^{1/3}$
4. $X^{4/5}$

Q. Match List-1 with List-2 and select the correct answer using the codes given below the lists:
List-1
A. ${\left(\frac{\partial u}{\partial y}\right)}_{y=0}$ is zero
B. ${\left(\frac{\partial u}{\partial y}\right)}_{y=0}$ is +ve
C. Displacement thickness
D. Momentum thickness

List-2
1. The flow is attached flow
2. The flow is on the verge of separation
3. ${\int }_0^{\delta }\frac{u}{U}(1-\frac{u}{U})dy$
4. ${\int }_0^{\delta }(1-\frac{u}{U})dy$

1. A-1, B-2, C-3, D-4
2. A-2, B-1, C-3, D-4
3. A-1, B-2, C-4, D-3
4. A-2, B-1, C-4, D-3

Q. A steady laminar boundary layer is formed over a flat plate as shown in the figure. The free stream velocity of the fluid is $U_0$. The velocity profile at the inlet a-b is uniform, while that at a downstream location c-d is given by $u=U_0[2\left(\frac{y}{\delta }\right)-{\left(\frac{y}{\delta }\right)}^2]$.

The ratio of the mass flow rate, ${\dot{m}}_{bd}$ , leaving through the horizontal section b-d to that entering through the vertical section a-b is________

1. 0.333

Q. The displacement thickness of a boundary layer is

1. the distance to the point where (v/V) =0.99
2. The distance where the velocity 'v' is equal to the shear velocity V, that is, where v=V.
3. the distance by which the main flow is to be shifted from the boundary to maintain the continuity equation
4. one half the actual thickness of the boundary layer