Prepared by Professor J. M. Cimbala, Penn State University
Latest revision: 11 January 2012
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(5) 1. Blasius (in 1908) obtained a numerical solution for the laminar flat plate boundary layer (see Reference 1). His solution is in terms of nondimensional variables
U u

as a function of y


U

 x


1
2
. Because of this clever nondimensionalization, any laminar flat plate boundary layer for any Newtonian fluid at any Reynolds number should collapse onto the same universal Blasius profile! Fill in the blank spaces to complete this predicted laminar flat plate boundary layer velocity profile, and show where you obtained your data:
Table 1 . Blasius velocity profile in nondimensional form (valid for all flat plate laminar boundary layers). y


U

 x


1
2 u
U

1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
0.0
0.2
0.4
0.6
0.8
1.0
(at the wall) 0.0
0.06641
0.13277
0.19894
________
0.32979
0.39378
________
0.51676
________
________
0.68132
0.72899
0.77246
(no slip condition)
2.8
3.0
3.2
3.4
3.6
3.8
4.0
4.2
________
0.84605
0.87609
0.90177
________
0.94112
0.95552
0.96696
4.4
4.6
4.8
5.0

0.97587
0.98269
0.98779
0.99155
(far outside the boundary layer) 1.00000 (freestream velocity)
Data are from the following Reference (author, title, publisher, copyright date, page number(s), table number, etc.):
(2) 2. The maximum speed of the small laboratory water channel is about 1.3 m/s. The flat plate available for this lab experiment has a total length of L = 21.0 inches, but the measurements with the LV probe are conducted at x =
18.0 inches. Assuming the water is at 20 o C, estimate the Reynolds number Re x
at this measurement location when the water channel is running at full speed. (Show your calculations in the space provided below):
Re x
= ________________________.

(5) 3. For the conditions above, and assuming the boundary layer remains laminar, estimate the laminar boundary layer thickness at the measurement location. (Show your calculations in the space provided below):
 laminar
= ___________________ mm.
(5) 4. As can be seen from the above calculations, the laminar boundary layer is extremely thin, even near the end of the plate. It would be difficult to accurately measure such a thin velocity profile! A boundary layer thickness of around 10 mm would be preferable. A thicker boundary layer is obtainable by decreasing the water channel speed. Estimate the freestream velocity which would produce a laminar boundary layer 10 mm thick at the measurement location near the end of the flat plate. (Show your calculations in the space provided below):
U

= _______________________ m/s.
5. For the case of a turbulent flat plate boundary layer, the flow is three-dimensional, unsteady, and random.
Therefore an exact analytical solution is not available. There are several empirical curve fits available, however, that match the mean velocity profile of a turbulent flat plate boundary layer quite well. The two most popular ones are the “log law” and the “1/7 th power law”. For the purposes of this lab, the 1/7 th power law curve fit is sufficiently accurate. Specifically, the mean streamwise velocity component in a turbulent boundary layer is approximated by the relationship u
U

(3) 6. At maximum water channel speed (approx. 1.3 m/s), estimate the boundary layer thickness at the measurement
(5) location if the boundary layer were turbulent rather than laminar. (Show your calculations in the space provided below):
 turbulent
= __________________ mm.
(5) 1. The half angle of the converging laser beams is determined by the focusing lens used. This angle (

) can be calculated from simple trigonometry. Namely, the focal length of the focusing lens is provided by the manufacturer as 160.2 mm, and the distance d between the two laser beams just before the focusing lens is 38.0 mm. From Figure 1 of the Introduction, it is seen that

= arctan [ d / (2
 f ) ] = 6.764

. A helium-neon laser has a wavelength,

, of 632.8 nm (one nanometer is equal to 10 -9 meter). Using Equation (1) of the Introduction, calculate the fringe spacing, d f
, of our LV measurement volume. (Show your calculations in the space provided below): d f
= ___________________ m.
(5) 2. Doppler frequency, F , is measured by the LV electronics, and Equation (2) can then be used to obtain velocity.
For a Doppler frequency of 2.43

10 5 Hz, calculate the flow velocity, showing all your work in the space below.
(Give your answer to three significant digits): u = __________________ m/s.

These calculations are performed automatically by the software supplied with the LV equipment. Thus, while the
LV user sees a measured velocity on the computer screen, the LV system is actually measuring the frequency of scattered light as a particle passes through the fringe pattern in the focal volume, and this frequency is converted into velocity using Eqs. (1) and (2) of the Introduction, just as was done above.
a.
Familiarize yourself with operation of the small water channel. b.
Measure boundary layer profiles on a flat plate (both laminar and turbulent). c.
Measure the boundary layer profile on the hull of a model submarine.