MAE420 Applied Fluid Mechanics

MAE420 Applied Fluid Mechanics
Homework #1 (6 problems)
Due: 10:00pm on March 23, 2015.
-----------------------------------------------------------------------------------------------1. Show that the viscous force term of the momentum conservation equation, ∇ ∙ 𝝉𝝉 , is equal
to 𝜇𝜇∇2 𝑽𝑽 in an incompressible flow with constant viscosity. Use the constitutive relation
presented in the lecture note.
∂τ
∂τ
∂τ 
∂τ
 ∂τ
 ∂τ
∂τ   ∂τ
∂τ 
=
∇ ⋅ τ i  xx + yx + zx  + j  xy + yy + zy  + k  xz + yz + zz 
∂y
∂z   ∂x
∂y
∂z 
∂y
∂z 
 ∂x
 ∂x
2. Let the vortex/sink flow of Eq. (8.16) simulate a tornado as in the figure below. Suppose that
the circulation about the tornado is Γ = 8000 m2/s and that the pressure at r = 40 m is 2200 Pa
less than the far-field pressure (pressure at infinite distance from the sink). Assuming potential
flow at sea-level density (1.225 kg/m3), estimate (a) the appropriate sink strength –m, (b) the
pressure at r = 15 m, and (c) the angle β at which the streamlines cross the circle at r = 40 m.
Assume that the far-field pressure is 101 kPa.
3. A Rankine half-body is formed as shown in the figure below. For the stream velocity and
body dimension shown, compute (a) the source strength m in m2/s, (b) the distance a, (c) the
distance h, and (d) the total velocity at point A.
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MAE420 Applied Fluid Mechanics
Homework #1 (6 problems)
Due: 10:00pm on March 23, 2015.
-----------------------------------------------------------------------------------------------4. Wind at U∞ and p∞ flows past a Quonset hut which is a half-cylinder of radius a and length
L (see the figure below). The internal pressure is pi. Using potential flow theory, derive an
expression for the upward force on the hut due to the difference between pi and ps.
5. It is desired to simulate flow past a two-dimensional ridge or bump by using a streamline
that passes above the flow over a cylinder, as in the figure below. The bump is to be a/2 high,
where a is the cylinder radius. What is the elevation h of this streamline? What is Umax on the
bump compared with stream velocity U?
6. A positive line vortex K is trapped in a corner, as in the figure below (+x and +y axes are
walls). Compute the total induced velocity vector at point B, (x, y) = (2a, a), and compare with
the induced velocity when no walls are present.
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