ME 306 Fluid Mechanics II
Spring 2022
Homework 1
Due Date: April 1st, 2022, Friday, 23:59 hours
SOLUTION
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1. Consider a liquid in a cylindrical container in which both the container and the liquid are to be rotated. The
container and the liquid are initially at rest. At t = 0, the container
begins to rotate at w. The liquid is sheared until steady-state conditions
are reached but once the process becomes steady, the liquid rotation
becomes a rigid-body rotation (in which the fluid does not shear and
acts like a solid, i.e. solid-body rotation). During the transient state,
time t is an influential process parameter. Other parameters that may
influence the process are the fluid density r, gravitational acceleration
g, and the container radius R. The elevation difference h between the
,µ
center and the rim of the liquid surface is to be expressed as a function
of problem parameters.
a) Using dimensional analysis, derive a dimensionless relationship for
the development of height h during the transient state as a function of
other problem parameters. Identify any established nondimensional parameters that appear in your result.
Hint: In deciding on the repeating kinematic variable, choose angular velocity rather than other problem
parameters.
b) Ethylene glycol at 60°C is to be rotated at a speed of 80 rpm in a large tank (glycol density and kinematic
viscosity at the given temperature are 1.019 kg/m3 and 4.75 x 10-6 m2/s, respectively). A one-quarter scale
model container is built for experimental testing in the laboratory, and water is used as the test liquid.
(i) Determine the water temperature and the model rotational speed,
(ii) make an estimate of the time it will take the ethylene glycol to reach rigid-body rotation if it takes
water 5 s to reach steady-state conditions, and
(iii) determine the ratio of the power required to rotate the prototype tank to that required to rotate the
model container.
2. A 1/20th scale hydraulic model is built to study the periodic flow conditions of the spillway of a dam in a
laboratory. A spillway is an open channel for the controlled release of fresh water from a dam. The laboratory
tests also use fresh water.
a) If the wave period in the model spillway is about 2 s, estimate the
wave period in the actual dam spillway by neglecting the viscous
effects.
flow
b) Suppose the model spillway is very shallow (h, in the figure, is quite small). Explain why the applicability
of model experiment results to the prediction of prototype flow behavior becomes questionable as the spillway
becomes shallow. Then, suggest how the experiment can be modified so that the model test results can be used
to predict prototype conditions.
Problem -1
:
Step -1
Assuming
Determine
:
the
given
h
Together
Step -2
:
Express
all the
t
R
[ LI
ITI
IL 't
Step -3
:
Determine
In step -2
that
are
n1<=7
Since
according
Step -4
required
to
:
r=3
Buckingham
required
Considering
and
Select
the
describe
IT
set
a
,
parameter
then
theorem
of
the
Tv
f
,
,
involves
11
"
,
parameters
equal
L
M
,
to
L
kinematic
→
of
nk
-
dimensions
3
is
r
dynamic
var
var
.
var
.
.
the
reference
repeating
geometric
→
,
)
-
,
/ primary
where
,
number
three
→
T
-
[n' i'T -1
groups
be
.
µ
variables
will
are
,
r =3
i. e,
4
IT
number
of
=
terms
.
dimensional independence
R
above
there
repeating
is
reference
of
number
the
IT
of
number
,
( M.LI
dimensions
Liii :|
"
required
that
to
[ IT ]
IT -1
ones
.
.
s
g
w
seen
basic
of
terms
-
the
is
it
,
in
problem
there
,
problem
this
in
the
)
( h)
variable
variables
variables
h
dependent
the
g. 9. µ
,
in
relevant
only
the
are
flt.R.tv
n1<=7 )
(
7-
with
parameters
=
involved
variables
the
}
parameters
dimensions
are
dimensionally
,
r
chosen
indep
=3
as
✓
.
.
,
Step -5
Form
:
b
a
-
IT,
IT
terms
,
[ i][LILI'THE]
c
h.R.ws
=
3
the
→
c
IT
=
↳
so
length
,
It
:[ I ] =[ L' Lai
time
:[To]
mass
:[MI
follows
that
a
-112
=gÑwj
mass
length
0
:
time
Tk
-
b
[M ]
:
=
Tj
-1
,
10=0
,
c
'
=
0
-
→
[
LT
a
-3C
b
c
hence
h
=
R
][L]a[TIÉmL→ ]
-2
2=0
C
-
b
b
a
=
=
-2
-1
gR"wJ°
=
=
and
,
IT ,
"Z=w[gR
Tlz
=
o
→
0=1+9 -3A
:
0
→
+
'
=
=
=
0=1
→
IT ]
=
0=-2
:
IT,
modify
]
=hR"w%
IT ,
-
"
Fr
g
II
=WR=
GR
=
war
him
=
Fr
'
dimensionless
IT,
-
Rdurbg
=µ Raw 's
Tk,
-
t
=
mass
time
,
1
=
0=-1
:
length
Tk
0
:
'
'
→
+
→
b
→
0=-1+9 -3C
:
[Mi 'T
c
-
=µR"wi's
Tls
→
b
a
Tla
→
modify
Write
:
¥=f( Fr
,
wt
=
-1
-2
gwrz
rvrim
=stwR)R
relationship
.
=
Re
µ
,
'Re )
b)
Part
glycol
N,
=
S
=
,
80
water
V2.52
rpm
1019
kg / m
"
V,
R
=
-1
↳
functional
final
the
=
'
a
-_
Tk,
Step -6
][ LIFT "I[me ]
c
→
"
"
=wt
=
4,75×10
Rz=
'
Ri
,
V2
/4
mys
,
IT,
=
-2
→
hi
=
R,
#
=
hz=hi/4
,
=Frz
Fr,
YÑ =¥¥ri
→
(can use N instead of w since w = 2*pi*N/60)
☒
4
Nik
✗
Re
=
,
Rez
*
→
Nz
V2
=
HÉ=KÑzRi
☒
Hip
,
¥
V2
k=¥r
;
=
,
I
V2
From
is
Table B. 2
found
by
I
Munson 's
of
interpolation
linear
160 rpm
=
FOFD
it
and
is
8th ed
=
5,94
corresponding
.
✗
107m21s
water
temperature
,
46.1°C
=
=
-
ji )
Tlz
=
Wt
Nit
→
t,
Nztz
=
,
5s
NHN
N2tz
=
,
=L
t,
>
Ni
=
10s
r
iii )
I
=
wR
T.us
=/ F. RIW
1st ) / Yt )
=m.aR-w=
Rw
=gtw2R2E
I ~fR5_w2E
*
☒
§:( r.H.us/
Ri
=
r=p→
Wi
2
N,
R =4Rz
,
Wi
'
1019 kg/m3
(E)
>
2
\
~
V-~R3
'
=N4z
-
=
Kiwi
t,
In
5'
=
sa
=
-_
Ztz
45-(1/2)<(1/2)
128-51
52
From
→
Table
Ir
=
B. 2
131,8
water
density
@ 46,1°C
is
52=989,7 kg / m3
Problem 2
:
free
surface
under the action of gravity
flow
with
periodic
no
¥m
=
20
,
→
effects
viscous
Frp
→
Vm
→
=
Stp
=
grai
Stm
Wplp
→
Tm
=
'
IT
2s
→
→
Tp
Tp
V
need
for Re
Y÷=(¥1
"
Vm
=
-wm
~
=
Wmlm
=
Vp
Wp
W
Fe
Wh
St
no
=
Frm
=
Tei
✓
Fr
→
-4dm
Vm
=
=
Tm
lp
20T
8,94s
=
To
do
=
¥0
-
Roz
Part (b):
As spillway becomes shallow, the influence of viscous effects on the flow, coming from “edge” effects (from
the spillway bottom surface) increases and the negligence of viscous forces no longer becomes a valid
assumption. In addition, if the spillway gets very shallow, the lowered volume of fluid compared to its free
surface that is in contact with the spillway walls can lead to a higher influence of capillary effects (surface
tension) which may not be negligible (if approaching thin film-like behavior), though the viscous effects would
become prominent long before the capillary effects as the spillway gets shallower. Therefore, Reynolds
number needs to be taken into account in model experiments design for reliable transfer of model experiment
results to the prediction of prototype conditions. In that case, the equality of the model and prototype Reynolds
numbers must be enforced for similitude. As such, if the same fluid is used in the model experiments, this
causes a conflict with the velocity ratio obtained from Froude number, i.e.
"
!! = #$! from Froude number but !! = # from Reynolds number (for the same fluid; kinematic viscosity
ratio, %! = 1).
!
One way to overcome this could be to change the model fluid so that both Reynolds and Froude numbers are
satisfied, i.e.
From Froude number equality, !! = #$! . Then,
'($
%$
%$
!! $! #$! $!
=1=
=
→ %! = +$&! =
→ %% =
'(%
%!
%!
%%
#$&!
Therefore, in the case when viscous effects become nonnegligble, a model fluid that has a kinematic viscosity
1/#$&! times the fresh water kinematic viscosity must be utilized for similitude to be achieved.
If, in addition to viscous effects, the capillary effects are also important, the Weber number must be taken into
account for similitude. In that case, an additional condition must be imposed on the density and the surface
tension of the model fluid as
-($
/$ ⁄.$
.! !!' $! .! (#$! )' $! .! $'! /!
/% /$ ⁄.$
=1=
=
=
→
= $'! =
→
=
-(%
/!
/!
/!
.!
/% ⁄.% .%
$'!
In many instances, it is not easy to find just the right fluid. If that’s the case, another option could be to
reconsider the model scale in designing the model experiments so that the edge effects (viscous effects) can
be negligible (capillary effects will be negligible if spillway height is large enough to neglect viscous effects).
In the case these effects are not negligible, with the appropriate adjustment of model scale, the right fluid may
be found for Reynolds number (and Weber number, if very shallow spillway) similarity.
Comment: In thin-film applications (where fluid height gets so small that the fluid “covers” the surface as a
thin film and surface tension effects are prominent), the Froude number (related to wave drag) would not be
an important parameter as surface tension effects would take over, closely followed by viscous effects.
However, a dam spillway should never be modeled through a thin-film fluid model (a very shallow model) –
this does not represent the actual (prototype) spillway accurately.