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Problem 5.C1
Estimating pipe wall friction is one of the most common tasks in fluids engineering. For long
circular, rough pipes in turbulent flow, wall shear
w is a function of density
, viscosity
, average
velocity V, pipe diameter d, and wall roughness height
. Thus, functionally, we can write
w = fcn(
,
, V, d,
). (a) Using dimensional analysis, rewrite this function in dimensionless form.
(b) A certain pipe has d = 5 cm and
= 0.25 mm. For flow of water at 20C, measurements show
the following values of wall shear stress:
Q (in gal/min)
~
1.5
3.0
6.0
9.0
12.0
14.0
w (in Pa)
~
0.05
0.18
0.37
0.64
0.86
1.25
Plot this data in the dimensionless form obtained in part (a) and suggest a curve-fit formula. Does
your plot reveal the entire functional relation obtained in part (a)?
Solution 5.C1
(a) There are 6 variables and 3 primary dimensions, therefore we expect 3 Pi groups. The
traditional choices are:
Problem 5.C2
When the fluid exiting a nozzle, as in Fig. P3.49, is a gas, instead of water, compressibility may be
important, especially if upstream pressure p1 is large and exit diameter d2 is small. In this case,
the difference p1 − p2 is no longer controlling, and the gas mass flow,
m,
reaches a maximum
value that depends upon p1 and d2 and also upon the absolute upstream temperature, T1, and the
gas constant, R. Thus, functionally,
=1 2 1
m fcn(p , d , T , R)
.
(a) Using dimensional analysis, rewrite this
function in dimensionless form. (b) A certain pipe has d2 = 1 cm. For flow of air, measurements
show the following values of mass flow through the nozzle:
T1 (in K)
300
300
300
500
800
p1 (in kPa)
200
250
300
300
300
m
(in kg/s)
0.037
0.046
0.055
0.043
0.034
Plot this data in the dimensionless form obtained in part (a). Does your plot reveal the entire
functional relation obtained in part (a)?
Solution 5.C2
(a) There are n = 5 variables and j = 4 dimensions (M, L, T, ), hence we expect only
n
−
j = 5 − 4 = 1 Pi group, which turns out to be
Problem 5.C3
Reconsider the fully developed draining vertical oil film problem (see Fig. P4.80) as an exercise in
dimensional analysis. Let the vertical velocity be a function only of distance from the plate, fluid
properties, gravity, and film thickness. That is, w = fcn(x,
,
, g,
).
(a) Use the pi theorem to rewrite this function in terms of dimensionless parameters.
(b) Verify that the exact solution from Prob. 4.80 is consistent with your result in part (a).
Solution 5.C3
There are n = 6 variables and j = 3 dimensions (M, L, T), hence we expect only
Problem 5.C4
The Taco Inc. Model 4013 centrifugal pump has an impeller of diameter D = 12.95 in. When pumping
20C water at = 1160 rev/min, the measured flow rate Q and pressure rise p are given by the
manufacturer as follows:
Q (gal/min)
~
200
300
400
500
600
700
p (lb/in2)
~
36
35
34
32
29
23
(a) Assuming that p = fcn(
, Q, D, ), use the pi theorem to rewrite this function in terms of
dimensionless parameters and then plot the given data in dimensionless form. (b) It is desired to
use the same pump, running at 900 rev/min, to pump 20C gasoline at 400 gal/min. According to
your dimensionless correlation, what pressure rise p is expected, in lbf/in2?
Solution 5.C4
There are n = 5 variables and j = 3 dimensions (M, L, T), hence we expect
Problem 5.C5
Does an automobile radio antenna vibrate in resonance due to vortex shedding? Consider an antenna
of length L and diameter D. According to beam-vibration theory [see [34] or [35, p.401]] , the first
mode natural frequency of a solid circular cantilever beam is
n = 3.516[EI/(
AL4)]1/2, where E is
the modulus of elasticity, I is the area moment of inertia,
is the beam material density, and A is the
beam cross-section area. (a) Show that
n is proportional to the antenna radius R. (b) If the antenna
is steel, with L = 60 cm and D = 4 mm, estimate the natural vibration frequency, in Hz. (c)
Compare with the shedding frequency if the car moves at 65 mi/h.
Solution 5.C5
(a) From Fig. 2.13 for a circular cross-section, A =
R2 and I =
R4/4. Then the natural
frequency is predicted to be:
Problem 5.W1
In 98 percent of data analysis cases, the “reducing factor” j , which lowers the number n of
dimensional variables to n − j dimensionless groups, exactly equals the number of relevant
dimensions ( M , L , T , Q ). In one case (Example 5.5) this was not so. Explain in words why
this situation happens.
Solution 5.W1
Problem 5.W2
Consider the following equation: 1 dollar bill ≈ 6 in. Is this relation dimensionally inconsistent?
Does it satisfy the PDH? Why?
Solution 5.W2
Problem 5.W3
In making a dimensional analysis, what rules do you follow for choosing your scaling variables?
Solution 5.W3
Problem 5.W4
In an earlier edition, the writer asked the following question about Fig. 5.1: “Which of the three
graphs is a more effective presentation?” Why was this a dumb question?
Solution 5.W4
Problem 5.W5
This chapter discusses the difficulty of scaling Mach and Reynolds numbers together (an
airplane) and Froude and Reynolds numbers together (a ship). Give an example of a flow that
would combine Mach and Froude numbers. Would there be scaling problems for common
fluids?
Solution 5.W5
Problem 5.W6
What is different about a very small model of a weir or dam (Fig. P5.32) that would make the
test results difficult to relate to the prototype?
Solution 5.W6
Problem 5.W7
What else are you studying this term? Give an example of a popular equation or formula from
another course (thermodynamics, strength of materials, or the like) that does not satisfy the
principle of dimensional homogeneity. Explain what is wrong and whether it can be modified to
be homogeneous.
Solution 5.W7
Problem 5.W8
Some colleges (such as Colorado State University) have environmental wind tunnels that can be
used to study phenomena like wind flow over city buildings. What details of scaling might be
important in such studies?
Solution 5.W8
The solution to this word problem is not provided.
Problem 5.W9
If the model scale ratio is α = Lm/Lp , as in Eq. (5.31), and the Weber number is important, how
must the model and prototype surface tension be related to α for dynamic similarity?
Solution 5.W9
Problem 5.W10
For a typical incompressible velocity potential analysis in Chap. 8 we solve
20
=
, subject to
known values of
/n
on the boundaries. What dimensionless parameters govern this type of
motion?
Solution 5.W10
Problem 5.1
For axial flow through a circular tube, the Reynolds number for transition to turbulence is
approximately 2300 [see Eq. (6.2)], based upon the diameter and average velocity. If d = 5 cm and
the fluid is kerosene at 20C, find the volume flow rate in m3/h which causes transition.
Solution 5.1
For kerosene at 20C, take
= 804 kg/m3 and
= 0.00192 kg/ms. The only unknown in the
Problem 5.2
A prototype automobile is designed for cold weather in Denver, CO (-10C, 83 kPa). Its drag
force is to be tested in on a one-seventh-scale model in a wind tunnel at 150 mi/h and at 20C
and 1 atm. If model and prototype satisfy dynamic similarity, what prototype velocity, in mi/h,
is matched? Comment on your result.
Solution 5.2
First assemble the necessary air density and viscosity data:
Problem 5.3
The transfer of energy by viscous dissipation is dependent upon viscosity
, thermal conductivity
k, stream velocity U, and stream temperature To. Group these quantities, if possible, into the
dimensionless Brinkman number, which is proportional to
.
Solution 5.3
Here we have only a single dimensionless group. List the dimensions, from Table 5.1:
Problem 5.4
The transfer of energy by viscous dissipation is dependent upon viscosity
, thermal conductivity
k, stream velocity U, and stream temperature To. Group these quantities, if possible, into the
dimensionless Brinkman number, which is proportional to
.
Solution 5.4
For water at 20C take
998 kg/m3 and
0.001 kg/ms. For sea-level standard air take
Problem 5.5
An automobile has a characteristic length and area of 8 ft and 60 ft2, respectively. When tested in
sea-level standard air, it has the following measured drag force versus speed:
V, mi/h: 20 40 60
Drag, lbf: 31 115 249
The same car travels in Colorado at 65 mi/h at an altitude of 3500 m. Using dimensional analysis,
estimate (a) its drag force and (b) the horsepower required to overcome air drag.
Solution 5.5
For sea-level air in BG units, take
0.00238 slug/ft3 and
3.72E−7 slug/ft·s. Convert
the raw drag and velocity data into dimensionless form:
Problem 5.6
The disk-gap-band parachute in the chapter-opener photo had a drag of 1600 lbf when tested at
15 mi/h in air at 20ºC and 1 atm. (a) What was its drag coefficient? (b) If, as stated, the drag on
Mars is 65,000 lbf and the velocity is 375 mi/h in the thin Mars atmosphere, ρ ≈ 0.020 kg/m3,
what is the drag coefficient on Mars? (c) Can you explain the difference between (a) and (b)?
Solution 5.6
(a) Convert to SI units: 1600 lbf = 7117 N, 15 mi/h = 6.71 m/s, 51 ft = 15.5 m. For air at 20ºC
and 1 atm, take ρ ≈ 1.20 kg/m3. Compute the drag coefficient in the wind tunnel:
Problem 5.7
A body is dropped on the moon (g = 1.62 m/s2) with an initial velocity of 12 m/s. By using option-
2 variables, Eq. (5.11), the ground impact occurs at
** 0.34t=
and
** 0.84.S=
Estimate (a) the
initial displacement, (b) the final displacement, and (c) the time of impact.
Solution 5.7
(a) The initial displacement follows from the “option 2” formula, Eq. (5.12):
Problem 5.8
The Archimedes number, Ar, used to analyze flow of stratified fluids, is a dimensionless
combination of gravity g, density difference Δρ, fluid width H, and viscosity μ. Find the form of
this number if it is proportional to g.
Solution 5.8
Write the dimensions of the variables, from Table 5.1:
Problem 5.9
The Richardson number, Ri, which correlates the production of turbulence by buoyancy, is a
dimensionless combination of the acceleration of gravity g, the fluid temperature To, the local
temperature gradient T/z, and the local velocity gradient u/z. Determine the form of the
Richardson number if it is proportional to g.
Solution 5.9
In the {MLT} system, these variables have the dimensions {g} = {L/T2}, {To} = {},
Problem 5.10
Determine the dimension {MLT} of the following quantities:
22
0
1
(a) (b) ( ) (c) (d)
p
u T u
u p p dA c dx dy dz
x x y t
−
All quantities have their standard meanings; for example,
is density, etc.
Solution 5.10
Problem 5.11
During World War II, Sir Geoffrey Taylor, a British fluid dynamicist, used dimensional analysis
to estimate the wave speed of an atomic bomb explosion. He assumed that the blast wave radius
R was a function of energy released E, air density
, and time t. Use dimensional analysis to
show how wave radius must vary with time.
Solution 5.11
The proposed function is R = f(E,
, t). There are four variables (n = 4) and three primary
dimensions (MLT, or j = 3), thus we expect n-j = 4-3 = 1 pi group. List the dimensions:
Problem 5.12
The Stokes number, St, used in particle dynamics studies, is a dimensionless combination of five
variables: acceleration of gravity g , viscosity μ , density ρ , particle velocity U , and particle
diameter D . ( a ) If St is proportional to μ and inversely proportional to g , find its form .
(b) Show that St is actually the quotient of two more traditional dimensionless groups.
Solution 5.12
Problem 5.13
The speed of propagation C of a capillary (very small) wave in deep water is known to be a function
only of density
, wavelength
, and surface tension Y. Find the proper functional relationship,
completing it with a dimensionless constant. For a given density and wavelength, how does the
propagation speed change if the surface tension is doubled?
Solution 5.13
The “function” of
and Y must have velocity units. Thus
Problem 5.14
Flow in a pipe is often measured with an orifice plate, as in Fig, P5.14. The volume flow Q is a
function of the pressure drop Δp across the plate, the fluid density ρ, the pipe diameter D, and the
orifice diameter d. Rewrite this functional relationship in dimensionless form
.
Solution 5.14
Write out the dimensions of the variables:
Q p D d
Problem 5.15
The wall shear stress
w in a boundary layer is assumed to be a function of stream velocity U,
boundary layer thickness
, local turbulence velocity u, density
, and local pressure gradient
dp/dx. Using (
, U,
) as repeating variables, rewrite this relationship as a dimensionless function.
Solution 5.15
The relevant dimensions are {
w} = {ML–1T–2}, {U} = {LT–1}, {
} = {L}, {u} = {LT–1},
{
} = {ML–3}, and {dp/dx} = {ML–2T–2}. With n = 6 and j = 3, we expect n − j = k = 3 pi
Problem 5.16
Convection heat-transfer data are often reported as a heat-transfer coefficient h, defined by
Q h A T
=
where
Q
= heat flow, J/s
A = surface area, m2
T = temperature difference, K
The dimensionless form of h, called the Stanton number, is a combination of h, fluid density
, specific
heat cp, and flow velocity V. Derive the Stanton number if it is proportional to h. What are the units
of h?
Solution 5.16
22
33
ML M
If {Q} {hA T}, then {h}{L }{ }, or: {h}
TT
= = =
Problem 5.17
If you disturb a tank of length L and water depth h, the surface will oscillate back and forth at
frequency , assumed here to depend also upon water density
and the acceleration of gravity
g. (a) Rewrite this as a dimensionless function. (b) If a tank of water sloshes at 2.0 Hz on earth,
how fast would it oscillate on Mars (g 3.7 m/s2)?
Solution 5.17
Write out the dimensions of the five variables. We hardly even need Table 5.1:
Problem 5.18
Under laminar conditions, the volume flow Q through a small triangular-section pore of side length
b and length L is a function of viscosity
, pressure drop per unit length p/L, and b. Using the pi
theorem, rewrite this relation in dimensionless form. How does the volume flow change if the pore
size b is doubled?
Solution 5.18
Establish the variables and their dimensions:
Q = fcn(p/L ,
, b )
{L3/T} {M/L2T2} {M/LT} {L}
Problem 5.19
The period of oscillation T of a water surface wave is assumed to be a function of density
,
wavelength
, depth h, gravity g, and surface tension Y. Rewrite this relationship in dimensionless
form. What results if Y is negligible?
Solution 5.19
Establish the variables and their dimensions:
Problem 5.20
A fixed cylinder of diameter D and length L, immersed in a stream flowing normal to its axis at
velocity U, will experience zero average lift. However, if the cylinder is rotating at angular
velocity , a lift force F will arise. The fluid density
is important, but viscosity is secondary
and can be neglected. Formulate this lift behavior as a dimensionless function.
Solution 5.20
No suggestion was given for the repeating variables, but for this type of problem (force
coefficient, lift coefficient), we normally choose (
, U, D) for the task. List the dimensions:
D L U F
