Fundamentals
BRUCE R. MUNSON DONALD F. YOUNG Department of Aerospace Engineering and Engineering Mechanics
John Wiley & Sons, Inc.
THEODORE H. OKIISHI Department of Mechanical Engineering Iowa State University Ames, Iowa, USA
New York Chichester Brisbane Toronto Singapore
TABLE OF CONTENTS
SOLUTIONS
Fluid Kinematics... ...... .......... ......... ..... ................... .......................... 4-1
Differential Analysis of Fluid Flow ................................................. 6-1
Similitude, Dimensional Analysis, and Modeling ............ ............... 7-1
Viscous Pipe Flow............................................ ................................ 8-1
Open-Channel Flow...... ...... ......... ....... ..................................... ...... 10-1
Compressible Flow ......................................................................... 11-1
INTRODUCTION
This manual contains solutions to the problems presented at the end of the chapters in the Fourth Edition of FUNDAMENTALS OF FLUID MECHANICS. It is our intention that the material in this manual be used as an aid in the teaching of the course. We feel quite strongly that problem solving is an essential ingredient in the process of understanding the variety of interesting concepts involved in fluid mechanics. This solutions manual is structured to enhance the learning process.
Approximately 1220 problems are solved in a complete, detailed fashion with (in most cases) one problem per page. The problem statements and figures are included with the problem solutions to provide an easier and clearer understanding of the solution procedure. Except where a greater accuracy is warranted, all intermediate calculations and answers are given to three significant figures.
Unless otherwise indicated in the problem statement, values of fluid properties used in the solutions are those given in the tables on the inside of the front cover of the text. Other fluid properties and necessary conversion factors are found in the tables of Chapter I or in the appendices.
Some of the problems [those designed with an (*)] are intended to be solved with the aid of a programmable calculator or a computer. The solutions for each of these problems are presented in essentially the same format as for the non-computer problems. Where appropriate a graph of the results is also included. Further details concerning the computer and their solutions can be found in the following section entitled Computer Problems.
In most chapters there are several problems [those designated with a (t)] that are "open­ ended" problems and require critical thinking in that to work them one must make various assumptions and provide necessary data. There is not a unique answer to these problems. Since there are various ways that one may approach many of these problems and since specific values of data need to be assumed, looked up, or approximated, we have not included solutions to these problems in the manual. Providing solutions, we feel, would be counter to the rational for having these problems-we want students to realize that in the real world problems are not necessarily uniquely formulated to a have a specific answer.
One of the new features of the Fourth Edition of FUNDAMENTALS OF FLUID MECHANICS is the inclusion of new problems which refer to the fluid video segments contained in the E-book CD. These problems are clearly identified in the problem statement. Although it is not necessary to use the CD to solve these "video­ related" problems, it is hoped that the use of the CD will help students relate the analysis and solution of the problem to actual fluid mechanics phenomena.
Another new feature of the Fourth Edition is the inclusion of laboratory-related problems. In most chapters the last few problems are based on actual data from simple laboratory experiments. These problems are clearly identified by the "click here" words in the problem statement. This allows the user of the E-book CD to link to the complete problem statement and the EXCEL data for the problem. Copies of the problem statement, the original data, the EXCEL spread sheet calculations, and the resulting graphs are given in this solution manual.
Considerable effort has been put forth to develop appropriate problems and to present their solutions in a manner that we feel is helpful to both instructors and students. Any comments or suggestions as to how we can improve this material are most welcome.
COMPUTER PROBLEMS
As noted, problems designated with an (*) in the text are intended to be solved with the aid of a programmable calculator or computer. These problems typically involve solutions requiring repetitive calculations, iterative procedures, curve fitting, numerical integration, etc. Knowledge of advanced numerical techniques is not required. Solutions to all computer problems are included in the solutions manual. Although programs for many of these problems are written in the BASIC programming language, there are obviously several other math-solver or spreadsheet programs that can be used.
A number of the solutions require the use of the same program, such as a program 'for curve fitting, or a numerical integration program, and these "standard" programs are included. For those requiring use of one of the standard programs, there is a statement in the problem solution which simply indicates the standard program used to solve the problem. A list of these standard programs, with their file names, follow. The actual programs are given in the appendix. Most of the standard programs are, of course, readily available in other math-solver or spreadsheet programs, and the student can simply use the programs with which they are most familiar.
Standard Programs-File Names and Use
EXPFIT.BAS
Curve Fitting
Determines the least squares fit for a function of the form y=ae bx
Determines the least squares fit for a function of the form y=bx
Determines the least squares fit for a function of the form y=a+bx
Determines the least squares fit for a function of the form y = do + d JX + d 2x2 + d 3x3 + ... Determines the least squares fit for a function of the form y=axb
SIMPSON.BAS
TRAPEZOLBAS
Numerical Integration
Calculates the value of a definite integral over an odd num­ ber of equally spaced points using Simpson's rule
Calculates the value of a definite integral using the Trapezoidal Rule
Miscellaneous
COLEBROO.BAS Determines the friction factor for laminar or turbulent pipe flow with the Reynolds number and relative roughness specified (for turbulent flow the Colebrook formula, Eq. 8.35, is used)
CUBIC.BAS Determines the real roots of a cubic equation
FAN_RA Y.BAS Calculates Fanno or Ray leigh flow parameters for an ideal gas with constant specific heat ratio (k> 1) for entered Mach number
ISENTROP.BAS Calculates one-dimensional isentropic flow parameters for an ideal gas with constant specific heat ration (k> 1) for entered Mach number
SHOCK.BAS Calculates normal-shock flow parameters for an ideal gas with constant specific heat ratio (k> 1) for entered upstream Mach number (Ma)
3
t. t I
1..1 Detennine the dimensions. in both the FLT system and the MLT system, for (a) the product of mass times velocity, (b) the product of force times volume. and (c:) kinetic energy divided by area,
mASS
Sinee.
F .:. M L T-.2
Fr =
/'2
( 0.)
1.2 Verify the dim~nsions, in both the FLT and MLT~ystems .. ofthe folioWing quantities which appear in Table 1.1: (a) angular velocity, (b) en­ ergy, (c) moment of inertia (area), (d) power, and (e) pressure.
= a 1'19 tI //1 r c/'spkce /?'J()~';' ..!.
-time
(.b) e he 1'"1:J ~ C.a.;aci +!J 01 b~cJ!1 1-0 do w()rk
Since. Wt?/'"K = I()rce;( d/sl-tll1tt:..)
tJr ~if;, F _' /11 L T- 2
e. n e rj tj ~ (M I- T -2) (L) == M L 2 T -2
cc) /7l{pmfl1t 0/ inerlltt.~V'ea.) = sec~l?d /nl'Jme/}f D/ t:lff?l
. (1.:2-)(L~) =. L If
LZ.
J..---------- -----------------------
/-2.
1.3 \. ~ Verify the dimensions, in both the FLT system and the
MLT system, of the following quantities which appear in Table 1.1: (a) acceleration, (b) stress, (c) moment of a force, (d) vol­ ume, and (e) work.
a c c-e/e ro.:tt'tJl1 :::: Ve.JDC.I+~ .:=
+/me
0. rea..
=f/1LT-VL ...: I1L2 T- Z
(e) Work - !=L
/.5 I
1.4 If P is a force and x a length, what are the dimensions (in the FLT system) of (a) dPI dx, (b) tf'Pldx\ and (c) JP dx?
dP . p- . != L- 2 - -- -- - - dJC L
d 3.f F 1= L-3 . . -:::r - - dx:. 3 L3
jPdx . PL --"'
1.5 If p is a pressure, V a velocity, and p a fluid density, what are the dimensions (in the MLT system) of (a) pip, (b) pVp, and (c) p/pV2?
1> _ --f (a. ) . --
) Z -
'--_________ ._ ........... _____________________ ......J
I-~
/. ID I 1.6 If V is a velocity, fa length, and \I a fluid property having dimensions of UT-I, which of the fo llowing combinations are dimensionless: (a) vr", (b) VC/', (e) V'" (d) VIM
(L T -'j(L)f1. zr) , L ~ T-1 mol dlm.nsienle,s) (a.) V J. -zJ .:.. -
(1:, ) v.R (Lr')(L) . LOr" ( dimension /ess) - - V (L'2. T I)
(C! ) V 2 -z) - (L T-) "(L • r - I
) ~ L~r3 (oof dimfnsl'oIl!ess) - (d) V (LT- 1) . -l. (not dlfnen sion!e>s ) - - L ).11 {L )(L' r ')
j· 7 I 1.7 Dimensionless combinations of quan- tities (commonly called dimensionless parame- ters) play an important role in flu id mechanics. Make up five possible dimensionless parameters by using combinations of some of the quantities listed in Table 1.1.
Some possible e"Qmple~ :
(L r2)(T) u C( e Ie r,,-/-'M " f 1m e • . L"T" - - ve /OCI f '1 (L rlJ -
frefllenc'j ;( hme - (rl){r) ..:. TO
(ve!oci+!j) 2. • (LT - I)'" ,
L"T = ",
(1'7 zr:J(Lrj /771Y/n en rum (11 LT-~
deMif-') " velocil-j " len-P'4 --' (Mr3)(LT - I}(d • M'L"T ' - = d'fnllr>1i< visUJ~if:J Mr' 7-1
1- 5
/.~ I
118 The force, P, that is exerted on a spher­ ical particle moving slowly through a liquid is given by the equation
P = 37CJlDV
where Jl is a fluid property (viscosity) having di­ mensions of FL -2T, D is the particle diameter, and V is the particle velocity. What are the di­ mensions of the constant, 37C? Would you classify this equation as a general homogeneous equa­ tion?
.p =- 37T;<D V
[F] == [:?7TJ [p J ,
/- ~
t. /0 I
According to information found in an old hydraulics book, the energy loss per unit weight of fluid flowing through a nozzle connected to a hose can be estimated by the formula
h = (0.04 to 0.09){D / d)4V2 /2g
where h is the energy loss per unit weight, D the hose diameter, d the nozzle tip diameter, V the fluid velocity in the hose, and g the acceleration of gravity. Do you think this equation is valid in any system of units? Explain.
~ = (O.OLf 1-0 ('). {) 9) (.!J ) If ~~ 2.J
gn= [D.O~ 1-. O.O~ [tJ[i] [~:J[ t] [L J == [O.OLf -1-0 0,07] [LJ
Since eac.h hrf}z li-t the e$tt.a..f./~h must: n4t1e the :Slll71e d;'mel1$/tJh5 -the Cf!)I1~"'lfi I-erm (~. ~'f ~ ~. ~tj) rnusf
I /;~ climfns/f.,hless. Thus1 the e$ti/{,t/~H /.5 a. !J~n(lY~ I h~mo1enet!Jvs €lttA..-6I4;;' .fh{(.i: IS 1I11//c/ IH CiI1.!! ~!:f5Iem ~f Un ,·f..:5. Yes.
1.10 The pressure difference, Ap, across a partial blockage in an artery (called a stenosis) is approximated by the equation
cosity (FL -~T), p the blood density (ML -3), D the artery diameter, Ao the area of the unob­ structed artery. and A I the area of the stenosis. Determine the dimensions of the constants K,. and K". Would this equation be valid in any sys­ tem of units?
pV (All )2 1
where V is the blood velocity, Jl the blood vis-
Since eac.h -terM mv.st h~lJe. the same dimensions; k'v Cll'ld Ku are dirnen5ionJe-:'5. Thu~.1 fhe efuafltJJI/ IS (;( ttener~1 h()f71~jel1eO"s e~ ua.l-;tJv, -tnCI'/- w{)uld be va/ic/ t'n Cfn!! C()tJ5isffnt sfjsl-em of U)1jf5. yes.
/-7
I. / / J I . II Assume that the speed of sound, c, in a fluid depends on an elastic modulus, Eu, with dimensions FL ~2, and the fluid density, p, in the form c = (Eu)"(p)h. If this is to be a dimen­ sionally homogeneous equation, what are the values for a and h? Is your result consistent with the standard formula for the speed of sound? (See Eq. 1.19.)
0)
FPr ~ d)J11eY1~/Of1tt/I'1 h(!)mt1ef1eDIJ5 -€$ad'{!)'J1 ea.ch +erm In the etua.t,bJ-f fntlS1- haf/(. -fJu 5f1/)/e dlmeY15JO#.s, Thtl5, -/ne Y"'9Jtf hand ~/de ()f. P~l OJ mus+ h~ve the dlmenslPA,s of- L 7-'. There /dYe)
a-tb==o 2.},=-1
.ta -f If b = - I (i:1> sa -/-1 's.f." C6"t/, ',,,()~ "n r)
(.£. :!iJ 1-, ~ I-y ~Y1 dJ/o'" "" L)
a. = L tlnt! /:; = - ). Z. 2.
So -tn..-f. c = ~i0: 1
Thb re.5u 1+ /s ~nsisl-f"r /AI;-!/1 the, sblltlt/J'p ~rIl1U/A -kr 17te :5peed ()j2- 5DUJlJd. YeS.
1- 'j
I, /2. I 1.12 A formula for estimating the volume rate of flow, Q. over the spillway of a dam is
Q = C v28 B (H + V2/2g)3/2
where C is a constant. g the acceleration of gravity. B the spillway width. H the depth of water passing over the spillway. and V the velocity of water just upstream of the dam. Would this equation be valid in any system of units? Explain.
5/~ce ea.c;" I:errn ,i1 ~e .e.Su.Lf/~H rnus-t- ha.ve +he SQ/7Ie dimellsi{)l/s -the ~11.sb1l/i C VI must:- he cilmeI15/!)/J )e~s. Thtls; -tnt!.. .et(f~tltJH is a ~-ene r-a I htPl1IP ,e/ledJ t(J eg Ua.,tIOJl -1'n¢,f WOf,{ /~ be. v t).. //d I»
411'1 e4)A~/sl:ent Set: of (,Iilif.s. Ye~.
/. / if I 1.14- Make use of Table 1.3 to express the following quantities in SI units: (a) 10.2 in.lmin, (b) 4.81 slugs, (c) 3.02lb, (d) 73.1 ft/s2, (e) 0.0234 lb·s/ft2•
(c>-) 1t),2 :;;'1 - (;0. 2 ;,;J (Z,S*;t/O-",:'.) ( ~;;n) -3 /W1 - i-. 'a2. .;c It) s = tf. 32. T
[ h) If. 9/ S /fA l' = ('I:?/ sill!> ) (;. 'f$f' ;< I () sju~) = 70, 2 ). ff
( ~ ) 3. tJ:L / b::: (3. ~ Z / b ) ( If. If 'If f1 ).=: /3. If AI
Cd) 73. J :Efi :
ce) CJ, tJ23'1 Ib·s (0. ~Z3'f ITt.) ('/,7.?1;tIO N· -': ) ~ ",.,1-
ff~ lb. s -ft'l-
1-/0
/./.5' I 1.15 Make use of Table 1.4 to express the following quantities in BG units: (a) 14.2 km, (b) 8.14 N/m3 , (c) 1.61 kg/m\ (d) 0.0320 N·m/s, (e) 5.67 mm/hr.
(b) o !!.. o.llf. ,11'I'f 3 " (g. 'If ~ ) (~3U;(/O·3 ':3 ) = 5'. IF)( 10'2 Pt.
,,",,3
( -3 SJUjS) l I. Cf Iff) )(. /0 W = ~ ~~
(d) 0.0320 N-1'H1 (~, 0 j 20 N ~ I1f1 ) (7, 371P;( / V-I
il-·Ib ) - - -- oS S
2.3b)(JD - - oS
/-11
/. /(0 I 1.lG Make use of Appendix A to express the following quantities in SI units: (a) 160 acre, (b) 742 Btu, (c) 240 miles, (d) 79.1 hp, (e) 60.3 OF.
IfpO a. ere
(6) 7tf2 137U = 6'1-2 sru) (.°£,;</0 3 J.)= 7.g3X/~5J BTU
C~) .2LjO int.' = (;'''10 tni ) (;'''Oq;(./(;.3 1"YY1,)::: 38iDX/oS" t?11 I'n1L
Cd) 71. / h p 0: (7'i'./ hp ) (7.'f5"7 X /02. (;{;) '"
(e) Tc = l' ~1).3 - 32) '= /5.7 "C::
k = /5",7 f) ( -r 273 ::::), gr 1<
1-/2
/./7 I 1.17 Clouds can weigh thousands of pounds due to their liquid water content. Often this content is measured in grams per cubic meter (glm3). Assume that a cumulus cloud occupies a volume of one cubic kilometer, and its liquid water content is 0.2 glm3. (a) What is the volume of this cloud in cubic miles? (b) How much does the water in the cloud weigh in pounds?
1M1= 3.281 U (;0'/111.1) (g, Z8'1 ~ )
J
(h) %J == 0 X -Vol"rn~
d' =: jJ d = {0.2 ;'3 ) {!D- l ;g. )(r.8/ ;) = f. UU/iJ-;;J
"lJ =- (I. '( (,,2 ;( JD -3 ;;', ) (10 1;m3) = /. '( ~2 X I D I, N
= (I. "t,z X /D (. N ) (:1., 2tf8 x/D- 1
-J& ) :::: ~, If! X JO S
f h
1- 13
1.18
(a)
1.18 For Table 1.3 verify the conversion re­ lationships for: (a) area, (b) density, (c) velocity, and (d) specific weight. Use the basic conversion relationships: 1 ft = 0.3048 m; lib = 4.4482 N; and 1 slug = 14.594 kg.
I it 1..: (/ .ft'")f( a 301f.>') 2/1?1 ,,-] = 0, () q 29{) /H1 ~ L I-i ~
Thus) rnu//-'/0 -ft2 bJ 9. '2'i{) £ - 2. +0 t!trJnvfrf
fo /ffI :2..
II;) /
Thus) mu/fipJ'j slugs/ .ft.3 b!:J 57 IS-If E of 2. ;'0 CtJl'Jtlfrl
-to Ie? / /I'n ~
Thus.) muillpl!) Ills bIJ 3.0'le f - / -1-0 cOl1vert
-I: 0 /t11 /s.
(d) I JIz - (I !l ') (If. 't'l12 !!..) [ I Ii 3 l If 3 - l' -It 3 ) l ~. / j, ( 0, "3 () Iff) 3 /W1 3 J
IV -= /57, / ;;;;
TfJlAS) m IA If/pI:; / b/ R ~ b!:J /. 5'7/ }; -t 2 -10 t'e>ntlfY't
fo #/;m3 4
/,/9 .J -- -
1..1 q For Table 1.4 verify the conversion re- lationships for: (a) acceleration, (b) density. (c) pressure. and (d) volume f1owrate. Use the basic conversion relationships: 1 m = 3.2808 ft; 1 N = 0.22481 lb; and 1 kg = 0.068521 slug.
(a)
(b)
Thus) m""/+ipllj tt/ .J.t / .5 J..
I ~ ~ = (I ~3 ') (0. oft> f/5:L/ slugs) [ 1m,3 J 1111 ~ "" \ ( T; (3. ZFO~)3 -f1:: 3
- I 040 x /0- 3 S l u ~~ . 1 f-t3
Th ~S.i m ul.f.i pJ'1 ~J/tt113 h,!j /. qLfo E-3 to ~J1t/fri.
-1:0 S /u~/.ft 3.
(C) I Ji :: (I !:!. ) (O,2.2lfgl ~)f I (M1. l /'I't1 ? tn1 2. N l (3. lfOg) 2. ft 1. J
-.2. Ik "=' '2. () g r i. I D f.t1-
Thu5) m/,.{lfip/~ N/rrn l b~ ;;'.Ogq E-l fo ~~n()fYt
1::-0 / h / f.t :L,
(d) / 73 == (I ~) [cg, 1.KOS/ ~:l= 35". 3/ fr'
T h US) rn f.,( I t ifl':J 1»1 3 /5 b~ 3. 531 E+ I -1:.0 rlOl1Vfyt
+(/ ft 3/s.
( ()...)
1.20 Water flows from a large drainage pipe at a rate of 1200 gal/min. What is this volume rate of flow in (a) m3/s. (b) liters/min. and (c) ft3/s?
f./owrat e =
/lowrfLte= (7.57 ;'/6- 2 ~.3)(/o3///.er.5)({Po.s) S /H1 3 /'1?1/11
(C ) I I () W r (I. +. e. = (7 S 7 )( J ()- ~ if 3 ) (3 S 3 I X J 0
-I'tJ
I-/~
1,,;2 / 1.2 , A tank of oil has a mass of 3 0 slugs. (a) Determine its weight in pounds and in new­ tons at the earth's surface. (b) What would be its mass (in slugs) and its weight (in pounds) if lo­ cated on the moon's surface where the gravita­ tional attraction is approximately one-sixth that at the earth's surface?
( t(.) w.e i9h i- .: ~. as.5 )(. 3
;,:2 2
= (3 0 5 /uqs ) ( 32.2 ;:)== _o/~r;, 16
- (30 shillS) ('t. Sf 14 ) ("I.E! -f,,)-= ,/Z'foN
( b) /h') 4 s.s = 3 () 5 J /A 9 S ( /n1 ASS dtJts t}IJt- de p~;1d t!)1'1
JY'~ vihfitJl1ll / a ffrtu..J-if!)11 )
w.eijhi = (30 s/uqS ) (32.~:Ef.. ) / fa/ /b…