An insight on the fundamentals of fluid mechanics through this excellent, comprehensive textbook. I hope the notes would be of great interest for instructors, students and researchers.
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Munson, Young and Okiishi's Fundamentals of Fluid
Mechanics, 8th Edition (2016) – Review
Gur Mittelman, gur.mittelman@gmail.com
Synopsis
The fundamental principles of fluid mechanics were developed for centuries now. Thus, if
we ever get a chance to challenge the very basic laws of this discipline, well, it could be
quite exciting. In the excellent textbook by Munson et al., this material is delivered with
great detail and patience, while uncompromising the degree of clarity. However, fluid
mechanics is a very cunning field, and the deep observations provided in this book give an
opportunity to think it over again. The current review comes across some of the
fundamental concepts, not just in the current textbook, but in the field as general (see for
example note 4). The following annotations are definitely not recommended for the faint-
hearted readers.
Review
1. Reynolds transport theorem, Section 4.4.1 equation (4.19) p. 182.
sys
CV CS
DB bd bV ndA
Dt t
It seems that the ( partial) time derivative of the first term on the right side could be
replaced with ordinary derivative i.e.
sys
CV CS
DB d bd bV ndA
Dt dt
because any integration over the entire space of the control volume (fixed or moving,
nondeformed or deformed) will remove the spatial dependence, resulting in an
expression which is only time dependent.
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2. Finite control volume analysis – linear momentum. Pressure forces Example 5.10
p.212.
It seems that the vertical atmospheric pressure forces on the control volume do not
cancel out since part of the vane is attached to the ground (no atmospheric pressure
force there).
Figure E5.10.
3. Finite control volume analysis – angular momentum equation. Section 5.2.3
equation (5.38) p. 227.
sys sys
DD
(r V) d (r V) d
Dt Dt
Replacing the sequential order of differentiation and integration in this equation could
be not trivial because of the Leibnitz rule:
where the system boundaries could be deforming and time dependent.
Section 5.2.3 (derivation of the moment of momentum equation) is presented in a
different manner compared to section 5.2.1 (linear momentum), beginning the
analysis at the particle level, rather than using the integration of the entire system
directly, such as in equation (5.19).
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4. Finite control volume analysis – the Pelton wheel.
The following comment refers to incompatibilities in the finite control volume analysis
(integral form equations), while considering the Pelton wheel.
See the technical drawing on page 6 below, Figures 12.24-12.27 in Munson's.
Using the moment of momentum equation (equation 12.50), the torque on the Pelton
wheel is
shaft m 1
T mr (U V)(1 cos )
where
and
.
So the force exerted on the blade is
shaft
blade 1
m
( )T
F m(V U)(1 cos )
r
(1)
This is the reference equation for the force.
Using the stationary C.V. (highlighted red), the power transferred to the blade by the jet
is,
Moving C.V.
The moving control volume is highlighted in green. This C.V. is moving in the tangential
speed of the cup, U.
The jet speeds are as following (equations 12.48-12.49):
CV
11
22
W V V
W V U
W cos V U
Where V refers to the jet speed relative to earth and W refers to a jet speed relative to
the blade. From mass conservation we get (equation 5.16),
CS
22
22
W ndA 0
Q 2 W A 0
2W A Q
With agreement with the water sprinkler example on page 208.
If friction and gravity can be neglected along the jet streamline, th en from the Bernoulli's
equation in the moving C.V we have:
1
P
1
gz 22
1P
1W
2
2
gz 2
2
12
1W
2
WW
Note that no work is done on the cup in the moving C.V. as the cup speed is zero in such
coordinates.
From the linear momentum conservation in the moving C.V. we have (equation 5.29):
contents of the
control volume
CS
W W ndA F
And we get,
2
CV 1 1 2 2 2
F W A 2 (W cos )W A
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Substituting:
and
, we obtain:
2
blade CV 1 1 2 2 2
2
1 1 1 1 1
1
1 1 1 1
1
11
F F W A 2 (W cos )W A
W A VA W cos
W
A V W ( cos )
V
W
m(V U)( cos )
V
(2)
Which is not identical to equation (1).
Stationary C.V.
If take the Bernoulli along the jet streamline in the stationary C.V., we have:
1
P
1
gz 22
1P
1V
2
2
gz 2
2 shaft
1Vw
2
where
.
Thus, the power transfer to the cup is
From classical mechanics, the force applied to the blade is equal to the difference
between the jet inlet and outlet momentum,
Substituting:
11
2 2 1
V W U
V W cos U W cos U
We have:
blade 1 2 1 1
1
F m(V V ) m[W U (W cos U)]
mW (1 cos )
(3)
Which is identical to equation (1).
From the linear momentum conservation on the stationary C.V. (equation 5.22) we have,
contents of the
control volume
CS
V V ndA F
22
CV 1 1 2 2
22
blade 1 1 2 2
F V A 2 V A
F V A 2 V A
Substituting:
2 2 1 1
1 2 1 1
11
21
Q 2W A V A
2W A V A
VA
2A W
We get:
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22
blade 1 1 2 2
22
11
1 1 2 1
2
2
1 1 1 1
F V A 2 V A
VA
V A V W
V
VA (V )
W
(4)
Substituting:
11
21
V W U
V W cos U
2
2
11
2
1
11
2 2 2 2
1 1 1 1
1
V
VW
(W cos U)
WU W
W UW (W cos 2W cos U U )
W
Which together with equation (4), does not reduce to equation 1.
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5. Finite control volume analysis – energy equation. Normal stress power, Section
5.3.1 p. 234.
According to equation (5.61), the normal stress used in the evaluation of the normal
stress power transfer is:
However, from the stress-deformation relationships, the normal stress includes a
viscosity term as well [equation (6.125a)]:
where both of the terms on the right side of the equation are further considered for
the derivation of the Navier-Stokes equations.
6. Finite control volume analysis – energy equation. Viscous dissipation Section 5.3.2
p. 238.
The one-dimensional energy equation for steady-in-the-mean flow is given in
equation (5.67):
22
out in
out in out in out in net
in
VV
pp
m[u u ( ) ( ) g(z z )] Q W 0
2
Where the work rate term includes both shaft and viscous (shear, tangential) effects
e.g.
shaft
W W Wtangential stress
.
Now, consider a fully developed, incompressible flow in an adiabatic, horizontal pipe.
If we select the control volume to be the pipe surface, we have no viscous power
transfer,
because the velocity at the pipe (solid) surface is zero. The
energy equation then becomes:
22
out in
out in
out in
VV
pp
m[u u
out in
g( z z
2 net
in
)] Q W 0
or
in out
out in v out in pp
u u c (T T )
Thus, the fluid is heated due to the pressure drop, which is directly related to the wall
shear stress (friction),
from the momentum balance [equation (8.5)].
Hence, fluid heating is related to friction despite the fact that viscous power transfer
is obscured. This is quite tricky.
A similar argument may valid for example 5.22 : temperature change in a waterfall.
Viscous power transfer is neglected but yet, the water in section 2 is heated due to
friction.
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7. Similitude based on governing differential equations. Section 7.10 p.385.
It is argued that dimensional analysis may go wrong when important variables are
omitted. However, it seems that this can also go the other way around as dimensional
analysis often yields more dimensionless groups than required due to lack of
information, which is available in the governing equations. Also, in boundary layer
problems, it looks like nondimensionalization could yield false prediction for the
functional dependence of local parameters.
Consider the following dimensionless boundary layer equations for a flat plate in
steady, laminar, incompressible 2D parallel flow:
2
2
L
u* v* 0
x* y*
u* u* 1 u *
u* v*
x* y* Re y*
where
L
uv
uVV
xy
xLL
VL
Re
* v* =
* y* =
Hence,
and
Now, the wall shear stress is
w
y 0 y* 0
u V u*
y L y*
And finally, the local friction coefficient is
w
f2 L y* 0
2 u*
c1 Re y*
V
2
or
i.e. 3 groups,
Which is quite different than the analytical solution e.g. Blasius:
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i.e. 2 groups only
Thus, the result obtained by nondimensionalizing the governing equations is probably
false, providing one extra group.
A possible explanation could be that the dimension L, which is apparent in the
governing equations normalization (x*=x/L, etc.), does not really have an influence on
the local wall shear stress or solution cf (x) .
The local wall shear stress scales as
(equation 9.28) where
is the
boundary layer thickness which develops from the leading edge [say
] and
further the downstream. The boundary layer problem is similar to initial value
problems, where the solution is affected only from the past, but not by the future.
Thus, any local solution can't be dependent on information available downstream
such as the plate length, L.
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