Solution 7.13
The Navier-Stokes equations for cylindrical coordinates are given in Appendix D, with “x” in
the Fig. P7.13 denoting the axial coordinate “z.” Assume “axisymmetric” flow, that is, v
= 0
and
/
= 0 everywhere. The boundary layer assumptions are:
Problem 7.14
Show that the two-dimensional laminar-flow pattern with dp/dx = 0,
Cy
o0
u U (1 e ) 0vv= − =
is an exact solution to the boundary-layer equations (7.19). Find the value of the constant C in
terms of the flow parameters. Are the boundary conditions satisfied? What might this flow
represent?
Solution 7.14
Substitute these (u,v) into the x-momentum equation (7.19b) with
u/
x = 0:
Problem 7.15
Discuss whether fully developed laminar incompressible flow between parallel plates,
Eq. (4.134) and Fig. 4.14b, represents an exact solution to the boundary-layer equations (7.19)
and the boundary conditions (7.20). In what sense, if any, are duct flows also boundary-layer
flows?
Solution 7.15
The analysis for flow between parallel plates leads to Eq. (4.134):
Problem 7.16
A thin flat plate 55 by 110 cm is immersed in a 6-m/s stream of SAE 10 oil at 20C. Compute the total
friction drag if the stream is parallel to (a) the long side and (b) the short side.
Solution 7.16
For SAE 30 oil at 20C, take
= 891 kg/m3 and
= 0.29 kg/ms.
Problem 7.17
Consider laminar flow past a flat plate of width b and length L. What percentage of the friction
drag on the plate is carried by the rear half of the plate?
Solution 7.17
The formula for laminar boundary drag on a plate is Eq. (7.26):
Problem 7.18
Air at 20ºC and 1 atm flows at 5 m/s past a flat plate. At x = 60 cm and y = 2.95 mm, use the
Blasius solution, Table 7.1, to find (a) the velocity u; and (b) the wall shear stress. (c) For extra
credit, find a Blasius formula for the shear stress away from the wall.
Solution 7.18
For air, take ρ = 1.2 kg/m3, μ = 1.8E-5 kg/m·s, and ν ≈ 1.5E-5 m2/s. Is it laminar?
Problem 7.19
Air at 20C and 1 atm flows at 50 ft/s past a thin flat plate whose area (bL) is 24 ft2. If the total
friction drag is 0.3 lbf, what are the length and width of the plate?
Solution 7.19
For air at 20C and 1 atm, take
= 0.00238 slug/ft3 and
= 3.76E-7 slug/ft-s. Low speed air,
not too big a plate: Guess laminar flow and check this later. Use Eq. (7.27):
Problem 7.20
Air at 20C and 1 atm flows at 20 m/s past the flat plate in Fig. P7.20. A Pitot stagnation tube,
placed 2 mm from the wall, develops a manometer head h = 16 mm of Meriam red oil, SG = 0.827.
Use this information to estimate the downstream position x of the Pitot tube. Assume laminar
flow.
Solution 7.20
For air at 20C, take
= 1.2 kg/m3 and
= 1.8E−5 kg/ms. Assume constant stream pressure,
then the manometer can be used to estimate the local velocity u at the position of the pitot inlet:
Problem 7.21
For the experimental set-up of Fig. P7.20, suppose the stream velocity is unknown and the pitot
stagnation tube is traversed across the boundary layer of air at 1 atm and 20C. The manometer
fluid is Meriam red oil, and the following readings are made:
y, mm: 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
h, mm: 1.2 4.6 9.8 15.8 21.2 25.3 27.8 29.0 29.7 29.7
Using this data only (not the Blasius theory) estimate (a) the stream velocity, (b) the boundary
layer thickness, (c) the wall shear stress, and (d) the total friction drag between the leading edge
and the position of the pitot tube.
Solution 7.21
As in Prob. 7.20, the air velocity u = [2(
oil −
air)gh/
air]1/2. For the oil, take
oil = 0.827(998) = 825 kg/m3. For air,
= 1.2 kg/m3 and
= 1.8E−5 kg/ms. (a, b) We see that h
levels out to 29.7 mm at y = 4.5 mm. Thus
Problem 7.22
In the Blasius equation (7.22), f is a dimensionless plane stream function:
( , )
() xy
fUx
=
Values of f are not given in Table 7.1, but one published value is f(2.0) = 0.6500. Consider
airflow at 6 m/s, 20º and 1 atm past a flat plate. At x = 1 m, estimate (a) the height y; (b) the
velocity, and (b) the stream function at η = 2.0.
Solution 7.22
From Table A.4 for air, ν ≈ 1.5E-5 m2/s. All of the requested values follow from the definitions of
Problem 7.23
Suppose you buy a 4 8-ft sheet of plywood and put it on your roof rack. (See Fig. P7.23) You
drive home at 35 mi/h. (a) If the board is perfectly aligned with the airflow, how thick is the
boundary layer at the end of the board? (b) Estimate the drag on the sheet of plywood if the
boundary layer remains laminar. (c) Estimate the drag on the sheet of plywood if the boundary
layer is turbulent (assume the wood is smooth), and compare the result to that of the laminar
boundary layer case.
Solution 7.23
For air take
= 1.2 kg/m3 and
= 1.8E−5 kg/ms. Convert L = 8 ft = 2.44 m and
U = 35 mi/h = 15.6 m/s. Evaluate the Reynolds number, is it laminar or turbulent?
Problem 7.24*
Air at 20°C and 1 atm flows past the flat plate in Fig. P7.24 under laminar conditions. There are two
equally spaced pitot stagnation tubes, each placed 2 mm from the wall. The manometer fluid is water
at 20°C. If U = 15 m/s and L = 50 cm, determine the values of the manometer readings h1 and h2, in
mm.
Solution 7.24
For air at 20C, take
= 1.2 kg/m3 and
= 1.8E−5 kg/ms. The velocities u at each pitot inlet can
be estimated from the Blasius solution:
Problem 7.25*
Consider the smooth square 10 by 10 cm duct in Fig. P7.25. The fluid is air at 20C and 1 atm,
flowing at Vavg = 24 m/s. It is desired to increase the pressure drop over the 1-m length by
adding sharp 8-mm-long flat plates across the duct, as shown.(a) Estimate the pressure drop if
there are no plates. (b) Estimate how many plates are needed to generate an additional 100 Pa of
pressure drop.
Solution 7.25
For air, take
= 1.2 kg/m3 and
= 1.8E−5 kg/ms. (a) Compute the duct Reynolds number and hence
the Moody–type pressure drop. The hydraulic diameter is 10 cm, thus
Problem 7.26
Consider laminar boundary layer flow past the square-plate arrangements in Fig. P7.26.
Compared to the friction drag of a single plate 1, how much larger is the drag of four plates
together as in configurations (a) and (b)? Explain your results.
Solution 7.26
The laminar formula CD = 1.328/ReL1/2 means that CD L−1/2. Thus:
Problem 7.27
Air at 20C and 1 atm flows at 3 m/s past a sharp flat plate 2 m wide and 1 m long.
(a) What is the wall shear stress at the end of the plate? (b) What is the air velocity at a point
4.5 mm normal to the end of the plate? (c) What is the total friction drag on the plate?
Solution 7.27
For at 20C and 1 atm, take
= 1.2 kg/m3 and
= 1.8E-5 kg/m-s. Check the Reynolds number
to see if the flow is laminar or turbulent:
Problem 7.28
Flow straighteners are arrays of narrow ducts placed in wind tunnels to remove swirl and other in–
plane secondary velocities. They can be idealized as square boxes constructed by vertical and
horizontal plates, as in Fig. P7.28. The cross section is a by a, and the box length is L. Assuming
laminar flat-plate flow and an array of N N boxes, derive a formula for (a) the total drag on the
bundle of boxes and (b) the effective pressure drop across the bundle.
Solution 7.28
For laminar flow over any one wall of size a by L, we estimate
Problem 7.29
Let the flow straighteners in Fig. P7.28 form an array of 20 20 boxes of size a = 4 cm and
L = 25 cm. If the approach velocity is Uo = 12 m/s and the fluid is sea-level standard air, estimate
(a) the total array drag and (b) the pressure drop across the array. Compare with Sec. 6.8.
Solution 7.29
For sea-level air, take
= 1.205 kg/m3 and
= 1.78E−5 kg/ms. The analytical formulas for
array drag and pressure drop are given above. Hence
Problem 7.30
In Ref. 56 of Ch. 6, McKeon et al. propose new, supposedly more accurate values for the
turbulent log-law constants,
= 0.421 and B = 5.62. Use these constants, and the one-seventh
power-law, to repeat the analysis that led to the formula for turbulent boundary layer thickness,
Eq. (7.42). By what percent is δ/x in your new formula different from that in Eq. (7.42)?
Comment.
Solution 7.30
We can start with Eq. (7.37), modified for the new constants:
Problem 7.31
The centerboard on a sailboat is 3 ft long parallel to the flow and protrudes 7 ft down below the
hull into seawater at 20C. Using flat-plate theory for a smooth surface, estimate its drag if the
boat moves at 10 knots. Assume Rex,tr = 5E5.
Solution 7.31
For seawater, take
=
1.99 slug/ft3 and
= 2.23E−5 slug/fts. Evaluate ReL and the drag.
Convert 10 knots to 16.9 ft/s.
Problem 7.32
A flat plate of length L and height
is placed at a wall and is parallel to an approaching boundary
layer, as in Fig. P7.32. Assume that the flow over the plate is fully turbulent and that the approaching
flow is a one-seventh-power law
1/7
o
() y
u y U
=
Using strip theory, derive a formula for the drag coefficient of this plate. Compare this result
with the drag of the same plate immersed in a uniform stream Uo.
Solution 7.32
For a ‘strip’ of plate dy high and L long, subjected to flow u(y), the force is
Problem 7.33
An alternate analysis of turbulent flat-plate flow was given by Prandtl in 1927, using a wall
shear-stress formula from pipe flow
1/4
2
0.0225
wUU
=
Show that this formula can be combined with Eqs. (7.33) and (7.40) to derive the following
relations for turbulent flat-plate flow.
1/5 1/5 1/5
0.37 0.0577 0.072
Re Re Re
fD
x x L
cC
x
= = =
These formulas are limited to Rex between 5 105 and 107.
Solution 7.33
Use Prandtl’s correlation for the left hand side of Eq. (7.32) in the text:
Problem 7.34
Consider turbulent flow past a flat smooth plate of width b and length L. What percentage of the
friction drag on the plate is carried by the rear half of the plate?
Solution 7.34
The formula for turbulent boundary drag on a plate is Eq. (7.45):
Problem 7.35
Water at 20ºC flows at 5 m/s past a 2-m-wide sharp flat plate. (a) Estimate the boundary layer
thickness at x = 1.2 m. (b) If the total drag (on both sides of the plate) is 310 N, estimate the
length of the plate using, for simplicity, Eq. (7.45).
Solution 7.35
For water take ρ = 998 kg/m3 and μ = 0.0010 kg/m·s. (a) Find the local Reynolds number:
Problem 7.36
A ship is 125 m long and has a wetted area of 3500 m2. Its propellers can deliver a maximum
power of 1.1 MW to seawater at 20C. If all drag is due to friction, estimate the maximum ship
speed, in kn.
Solution 7.36
For seawater at 20C, take
= 1025 kg/m3 and
= 0.00107 kg/ms. Evaluate
Problem 7.37
Air at 20C and 1 atm flows past a long flat plate, at the end of which is placed a narrow scoop,
as shown in Fig. P7.37. (a) Estimate the height h of the scoop if it is to extract 4 kg/s per meter of
width into the paper. (b) Find the drag on the plate up to the inlet of the scoop, per meter of
width.
Solution 7.37
For air, take
= 1.2 kg/m3 and
= 1.8E−5 kg/ms. We assume that the scoop does not alter the
boundary layer at its entrance. (a) Compute the displacement thickness at x = 6 m:
Problem 7.38