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Unformatted text preview: 7. 3 LI The drag characteristics of a torpedlo are to be studied in a water tunnel using a 1:5 scale model. The tunnel operates . with freshwater at 20 °C, whereas the prototype torpedo Is to be used in seawater at 15.6 °C. To correctly simulate the be havior of the prototype moving with a velocity of 30 m/s, what
velocity is required in the water tunnel? F29;— dynamic, 61m; /4r/ 7’3 #72 fawn/d3 numbfr musf be 725
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”’4 (//7xw“ 41") N905) 5 791 7. 43 Water ﬂowing under the obstacle shown 111 Fig P7. 43 puts
a vertical force, Fv, on the obstacle This force 15 as$umedl to be a
function of the flowrate Q, the density of the water, p, the accel
eration of gravity, g, and a length, 6, that characterizes the size of
the obstacle. A 1/20 scale model 18 to be used to predict the ver~
tical force on the prototype. (3) Perform a dimensional analysis
for this problem ('3) If the prototype flowrate IS 1000 ft3/s, deter
mine the water ﬂowrate for the model if the ﬂows are to be sim~ 11211. (c) If the model force' is measured as (Fv)m = 20 lb predict
the corresponding force on the prototype. (a) Fv=f(6?,ﬂng/Q) _ (0
1:2: 1W .091", :1 C9 Thu; 11/5771 50:; acts
0.557“; (C) 7716 Precltcl'mn 93144.50” Ls
Fix 1% ”£31 131 £35,». “I; So ’hmi F}, :3 F 49 2' I (0 :1“ FL
From 77% P6 The» rem} .513 2 ﬁg Jt’rm: regwrepl
an d d; dimenJ/ona/ ﬁrmly»; 51e/d4r IFIGURE P7.43 “2 ‘{547' 15L (1,) For JJm,'/ar;+, 111W“ mac/cl am! 1311543,“; 1 WW1 WW(~1%) ”31ft? The drag, ‘22), on a sphere located in a pipe through
which a ﬂuid is ﬂowing is to be determined experimentally (see
Fig. P750). Assume that the drag is a function of the sphere
diameter, d, the pipe diameter, D, the ﬂuid velocity, V, and the
ﬂuid density, p. (a) What dimensionless parameters Would you
use for this problem? (b) Some experiments using Water indi
cate that for d = 0.2 in., D = 0.5 in., and V = 2 ft/sj, the drag
is LS X 10‘3 lb. If possible, estimate the drag on a sphere
located in a 2—ft—diameter pipe through which water is ﬂowing
with a velocity of 6 ft/s. The sphere diameter is such that ge—
ometric similarity is maintained. If it is not possible, explain
why not. (a) 08:76 [91,0] Vl/o)
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and A: 0. 3 74:5 (raga/it’d 074ml”). Thus, 7716 PY€JIC51L1031 geium/z/aiv LI: ,8 g .29 ,‘ 444 M201 xii/4:04: Wind blowing past a ﬂag causes it to “ﬂutter in the
breeze.” The frequency of this ﬂuttering, w, is assumed to be
a function of the wind speed, V, the air density, p, the acceler—
ation of gravity, 3, the length of the ﬂag, 6, and the “area den—
sity,” pA, (with dimensions of ML”) of the ﬂag material. It is
desired to predict the ﬂutter frequency of a large 6 = 40 ft ﬂag
in a V = 30 ft/s wind. To do this a model ﬂag with 6 = 4 ft
is to be tested in a wind tunnel. (a) Determine the required area
density of the model ﬂag material if the large ﬂag has pA =
O 006 slugs/ft? (b) What wind tunnel velocity 1s required for testing the model? (c) If the model ﬂag ﬂutters at 6 Hz, predict
the frequency for the large ﬂag. ‘ w=£(l/Jipi?i’e2ﬂ4)
40:57", ViLTUlf’fML‘? g5/v7‘z ﬂéL ﬁi/‘(LZ‘ From #76 [2/ Theorem, é‘g ’L' 3 PI +€rms Vega/Val 4144 4
dimeHJIa/M/ dim/751.: 7/2/43 QJV:——= 7L(V;/—7) 72) (a) Fzr J/bnlar/rf ~ rfmlm
41M 5114.0: ﬂ”: $1];
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4'an win 5 :5 MW: @V = V560?)= 9.47% ct) W’h 7‘7“ similarrl‘y rézliiremé’nis 54111.56?! We Predfcéfm €Xuaﬁno'n 15 Cu V? : f £04m V3312: 50 ﬂﬂ/‘E F V—E—ﬁw — 7— (96 7 &5 7.6.5' The pressure rise, Ap, across a centrifugal pump of a
given shape (see Fig. P1653) can be expressed as 8 AP = f(D. w. p, Q) where D is the impeller diameter, to the angular velocity of the " _________
impeller, p the fluid density, and Q the volume rate of ﬂow' 6 ”WWW—imam“? mmmmmmmmm a """""""" through the pump. A model pump having a diameter of 8 in. is i ' I
tested in the laboratory using water. When operated at an an
gular velocity of 4077' rad/s the model pressure rise as a function
of Q is shown in Fig. P7.6§b. Use this curve to predict the
pressure rise across a geometrically similar pump (prototype)
for a prototype ﬂowrate of 6 ft3/s. The prototype has a diameter
of 12 in. and operates at an angular velocity of 6071' rad/s. The 2 t s
prototype ﬂuid is also water. ‘ E 2 _,
! Model data
; (um = 401: rad/s Ap=pzepr o 0.5 1.0 1.5 ' "2.0
Qm(ft3/s) I FIGURE P7.65
Centrifugal pump ~  ,,,,, (a) AP: £{D,43/,4)>
Ape/1'” 125 L A): T" g: FNT‘ Q ; 1.37"
Ham 1%: Pi 'theorrm/ 5‘3 = 2 PL £0712: ”gt/Wed, 41,14 4 d,,,',m5m'”’/
ﬁrm/gm} g; 614/5 ‘
Ag =¢LQJ . . My
771(15/ ’fhe Similar/5‘7 mil/Ikmﬂti IL: 4% =.Q
QMDB 0D I»!
_ M D 3
mm 74/ ﬁre 44%: j/Vf/I
4”": (W $73502 #3) = mi”
(W :3!) mg.
P7941 7'21? jot/é (Fig, i016?!) 4,9,” 2.5750123 743V 43., ”/73" . ”ms, AP Ag, KW” 8:04:12.)
ﬂ urn: ﬂ‘ﬂ ( 4):
4/1 I "rm 2 r4 ‘
e 77» , /z . I
t ésafpeyzsf = 41—5—— ., —.”' ) (5750px)
13%" Wm 1%” ”" ’7‘” " 9'” (%7?
= 27.8 gsé 59772419 7".7/ 7. 47‘ i in? A viscous ﬂuid is contained between wide, parallel
plates spaced a distance it apart as shown in Fig.1 P7.67. The
upper plate is ﬁxed, and the bottom plate oscillates harmonically
with a velocity amplitude U and frequency w. The differential
equation for the velocity distribution between the plates is Bu  32”
p _ [‘1' ayz Fixed plate at where u is the velocity, t is time, and p and ,u. are fluid density
and viscosity, respectively. Rewrite this equation in a suitable
nondimensional form using h, (J, and a) as reference parameters. l a My" 1: are" &
9‘ g, if , 20.1f
&_‘j 33/ a3 51 IL: 5 or a: 3
ﬂaw] W ~ 2W" ...
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 Spring '09
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