ME331-430 Advanced Fluid Mechanics

Prof. A. Banerjee
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Solutions to Home Work Problem Set 1

 
t
x 
1. Consider a one-dimensional flow whose velocity is u   U 0   e  iˆ , where Uo and τ are constant
 
velocity and time scales. Compute the unsteady, convective and total accelerations. (2 points)

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 ME331-430 Advanced Fluid Mechanics
Prof. A. Banerjee
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 4U ˆ
2. For flow near the stagnation point of a cylinder, the velocity is u 
D
 
x i  y ˆj , where D is the
cylinder’s diameter and U is the speed of the incident flow. Determine the Lagrangian description of

the fluid-particle position vector, r  x iˆ  y ˆj , in terms of U, D, t and the initial values of the
coordinates, xo and yo. (3 points)

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 ME331-430 Advanced Fluid Mechanics
Prof. A. Banerjee
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 U
3. A flow field has u   x iˆ   z ˆj   y  z  kˆ  , where U and R are constant velocity and length
R 
scales, respectively. Is there any value of the constant  for which the flow is irrotational? (3 points)

Solution: The vorticity is given by the following expression:

iˆ ˆj kˆ
   

x y z
x z yz
   ˆj    x     z    kˆ    z    x 
 iˆ   y  z    z     x
 y z 
 z x 
 y 
 iˆ 1     ˆj  0  kˆ  0
 1    iˆ

Thus, the velocity field is irrotational when  = 1.

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 ME331-430 Advanced Fluid Mechanics
Prof. A. Banerjee
-------------------------------------------------------------------------------------------------------------------------------

4. The velocity for incompressible flow above an infinite flat wall at y = 0 oscillating with velocity
u (0, t )  U cos  t  (Stokes’ Second Problem) is given by:
u ( y, t )  Ue  ky cos  t  ky  , v( y, t )  w( y, t )  0 ;
where t is time, Ω is frequency, y is distance normal to the surface and k is a constant.

a. Determine the Lagrangian description of the fluid-particle co-ordinates, x and y, in terms of

U, Ω , k and t and the initial values of the co-ordinates, xo and yo. (3 points)

b. Compute the acceleration vector and the vorticity vector. (3 points)

Solution (4a):

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 ME331-430 Advanced Fluid Mechanics
Prof. A. Banerjee
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(4b):

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 ME331-430 Advanced Fluid Mechanics
Prof. A. Banerjee
-------------------------------------------------------------------------------------------------------------------------------

5. The velocity for a two-dimensional flow is

  xy 2 x2 y 
u  U  3 iˆ  3 ˆj  , where U and h are
h h 
constants of dimensions L/T and L, respectively.
Compute the circulation, Γ, on the rectangular
contour shown in the figure. Verify that your result
is consistent with finding the circulation using the
expression     kˆ dA . (3 points)
A

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 ME331-430 Advanced Fluid Mechanics
Prof. A. Banerjee
-------------------------------------------------------------------------------------------------------------------------------

6. Derive an equation defining the streamlines for the incompressible flow whose velocity field is:
 UR 2
u  2  cos  sin  eˆr  cos 2  eˆ  , where U is freestream velocity and R is a constant length scale.
r
HINT: The differential equation in two dimensions for streamlines in cylindrical coordinates is:
dr ur  r d u . (3 points)

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