Upload
ab
View
218
Download
0
Embed Size (px)
Citation preview
7/25/2019 Assignment 6 With Solutions[2] Fm white
1/9
MECH 3492 Fluid Mechanics and Applications Univ. of Manitoba Winter Term, 2015
Assignment 6
Please google hotwire velocimetry and wind-tunnel for information on their structures and func-
tions.
Problem 1 (Purpose: understanding Reynolds averaging, viscous and turbulent shear stresses, andeddy viscosity)
In a wind-tunnel experiment, hotwire velocimetry was used for measuring a turbulent wind field. The
recorded time series of instantaneous velocity data for u, v and w (in m/s) at two close points are
given in the excel book Turbulence.xlsx. The measurements were taken at 1000 Hz. The spatial
distances between the two points are x= x2 x1 = 2.12 mm, y = y2 y1 = 3.91 mm and z =
z2 z1= 1.88 mm. The density and dynamic viscosity of the air are 1.25 kg/m3 and 1.5 105 Pa s,
respectively.
(1) Determine time-averaged velocities u, v and w at both points;
(2) Plot uand uw.r.t. time in the same diagram for point 1;
(3) Using the measured data to verify the assumption of incompressibility (for the mean flow) at the
midpoint (between the two points);(4) Determine the value of the viscous shear stress tensor based on Stokess hypothesis at the midpoint;
(5) Determine the value of the Reynolds stress tensor at the midpoint;
(6) Determine the value of the eddy viscosity at the midpoint based on Boussinesqs assumption.
Hints:
(1) you would need to pay attention to the units;
(2) this represents a much simplified hotwire experiment, as only two measurement points are involved.
When you calculate the derivatives, use the first-order accurate scheme (e.g., du/dx u/x).
Problem 2(Purpose: understanding BL thickness, displacement thickness, momentum thickness, and
integral equation)
For a zero-pressure-gradient laminar BL flow over a flat plate, assume that the velocity profile is of a
polynomial form: u= a+by+cy3. Derive all the equations and parameters listed in row 3 of Table 9.2.
Specifically, determine: (1) uU
; (2) w; (3) non-dimensionalized BL thickness x
(as a function ofRex);
(4) Cf(as a function ofRex); (5)
; (6)
; and (7) shape factor H=
.
Problem 3(Purpose: understanding turbulent BL and wind-tunnel design)
9.21
1
7/25/2019 Assignment 6 With Solutions[2] Fm white
2/9
7/25/2019 Assignment 6 With Solutions[2] Fm white
3/9
7/25/2019 Assignment 6 With Solutions[2] Fm white
4/9
7/25/2019 Assignment 6 With Solutions[2] Fm white
5/9
7/25/2019 Assignment 6 With Solutions[2] Fm white
6/9
7/25/2019 Assignment 6 With Solutions[2] Fm white
7/9
7/25/2019 Assignment 6 With Solutions[2] Fm white
8/9
Problem 9.21 [Difficulty: 2]
Given: Data on wind tunnel and boundary layers
Find: Displacement thickness at exit; Percent change in uniform velocity through test section
Solution
:The solution involves using mass conservation in the inviscid core, allowing for the fact that as the boundary
layer grows it reduces the size of the core. One approach would be to integrate the 1/7 law velocity profile tocompute the mass flow in the boundary layer; an easier approach is to simply use the displacement thickness!
Basic
equations
(4.12) disp
0
y1u
U
d
Assumptions: 1) Steady flow 2) Incompressible 3) No friction outside boundary layer 4) Flow along streamline 5) Horizontal
For this flow U A const andu
U
y
17
The design data is Udesign 160ft
s w 1 ft h 1 ft Adesign w h Adesign 1 ft
2
The volume flow rate is Q Udesign Adesign Q 160ft
3
s
We also have in 0.4 in exit 1 in
Hence disp
0
y1u
U
d
0
y1y
1
7
d 0
1
1
1
7
d where y
disp
8
Hence at the inlet and exit
dispin
in
8 dispin 0.05 in dispexit
exit
8 dispexit 0.125 in
7/25/2019 Assignment 6 With Solutions[2] Fm white
9/9
Hence the areas are Ain w 2 dispin h 2 dispin Ain 0.9834 ft2
Aexit w 2 dispexit h 2 dispexit Aexit 0.9588 ft2
Applying mass conservation between "design" conditions and the inlet
Udesign
Adesign
Uin
Ain
0
or Uin Udesign
Adesign
Ain
Uin 162.7ft
s
Also Uexit Udesign
Adesign
Aexit
Uexit 166.9ft
s
The percent change in uniform velocity is thenUexit Uin
Uin
2.57 % The exit displacement thickness is dispexit 0.125 in