Download White FM, “Viscous Fluid Flow”

White F.M., “Viscous Fluid Flow”
Chapter 4, Exercise 8
8. Homework. Due date: 20.11.2015
HW1 (3p): Flow straighteners consist of arrays of narrow ducts placed in a flow
to remove swirl and other transverse (secondary) velocities. One element can be
idealized as a box with thin sides as shown in Fig. 1. Using laminar flat-plate
theory, derive a formula for the pressure loss ∆p across an N × N bundle of such
boxes, and apply it to compute ∆p, if the dimension of each element are given
as: a = 10mm and L = 100mm; and that the properties of the fluid are given as:
ρ = 1.23kg/m3 and µ = 0.0183 · 10−3 Pa · s. Use U∞ = 10 m/s.
Figure 1: An idealized flow straightener element
HW2 (3p): A thin equilateral triangle plate is immersed parallel to a 6 m/s
stream of air at +20o C and at 1 atm, as shown in the figure below. Assuming
laminar flow, estimate the friction drag of this plate.
1
White F.M., “Viscous Fluid Flow”
Chapter 4, Exercise 8
Demo: Air flows over a smooth 1m × 1m flat plate affecting only one side of
the plate. The free-stream velocity of the flow is 10m/s, temperature 20o C and
pressure 1bar. Calculate the boundary layer thickness at the trailing edge and
find an expression for the shear stress at the plate surface. What is magnitude
and direction of the force that the flow exerts on the plate?
Solution: Using U = 10m/s, L = 1m and ν = 15 × 10−6 m2 /s, the Reynolds
number at the trailing edge is Re = UνL ≈ 666 667. The limit of transition is
Re ≈ 106 , so the flow is laminar and we can apply the Blasius solution. The
boundary layer thickness at the trailing edge can be calculated from eq. 4-49 in
White
5.0 x
5.0 × 1
δ=√
=q
= 6.1 mm
10×1
Rex
−5
(1)
1.5×10
Using equation 4-52, the shear stress can be written as
τW =
0.332ρU 2
√
.
Rex
(2)
The force exerted on the plate can be calculated by integrating the shear stress
over the plate
Z 1
Z 1
1/2 3/2 1/2
Fx = W
τW dx = 0.332W ρ U µ
x−1/2 dx
0
0
= 0.664W ρ
= 0.098 N.
The force acts on the direction of the flow.
2
1/2
U
3/2 1/2
µ
(3)