This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 208 CHAPTER 5 I ntrsonucnou T0 otmnamst ANALYSIS or FLUID M0110" 6r Navier—Stokes equations (incom
pressible. constant viscosity): REFERENCES 1. Li, W. 14.. and S. H. Lam, Principles rngluid Mechanics.
Reading, MA: AddisonWesley. 1964. 2. Daily. .l. W.. and D. R. F. Harleman. Fluid Dynamics. Reading.
MA: AddisonWesley, [966. 3. Schlichting, H.. Boundary~Layer Theory. 7th ed. New York:
McGrawHill, I979. 4. White, F. M., Viscous Fluid Flow. 3rd ed. New York:
McGrawHill, 2000. 2
Particle acceleration components in = 1/ 6V, + EBV, ._ E + mfg—m. + (iv. Page 177
an, r .. (5.123)
cylindrical coordinates: 65 V rVagV 1" V V 3?. 8V (‘3!
an,,=V—"+—"—"+ "WV. "
8r 1' 66 r 8:
+5VH
+81“ (5.1213)
_ —V g: V9617 0V 0V. (5.12c} r36 iriiiriiiifliilill’lll'”l‘lllllllllllllllllllljll‘ililllll?'il:llillllillllllllli'UiflllilllTil Zaz Page 189 iTEiltll‘lllil l illii 5. Sabersky, R. H., A. J. Acosta. E. G. Hauptmann. and E. M.
Gates, Fluid Flow—A First Course in Fluid Mechanics, 4th ed.
New Jersey: Prentice Hall. I999. 6. Fluent. Fluent incorporated. Centerra Resources Park. 10
Cavendish Court. Lebanon, NH 03766 (wwwﬂuentcom). 7. STARCD. Adapco. 60 Broadhollow Road. Melville. NY
1 I747 (www.cdadapco.com). PROBLEMS 5.1 Which of the following sets of equations represent possible
twodimensional incompressible ﬂow cases? a. u : 2.x:2 + y2 — ray; 0 = x1 + x03  23;) b. u=2xy.rz+y;u=2xy—yz+x2 c. u = x: + 2y; 1: = xrz — yr d. u = {x + Zytrr: v =etlx + ytyt 5.2 Which of the following sets of equations represent possible
three—dimensional incompressible ﬂow cases?
a.u=y2 +2xz;u= —2yz+.rzvz'uvg:.—.r'.:2 +x3y4
b u = xyzr; v =—xyzt2: w— = (272)00‘2  W)
C. u=x 2+y+z2;v=xy+z:w=H1xz+y2+z 5.3 The three components of Velocity in a velocity ﬁeld are given
byu=Ax+ By + Car) = Dx+ Ey + Faandw Gx+Htv+Ja
Determine the relationship among the coefﬁcients A through J that
is necessary if this is to be a possible incompressible flow ﬁeld. 5.4 For a ﬂow in the xy plane. the x component of velocity is gi
ven byu = Arty — B), where/l =lft‘1s_]_. B = 6 ft. andxand
y are measured in feet. Find a possible y component for steady.
incompressible How. Is it also valid for unsteady. incompressible
ﬂow? Why? How many y components are possible? 5.5 For a ﬂow in the xy plane. the x component of velocity is
given by u x" — 3xy2. Determine a possible y component for
steady. incompressible ﬂow. Is it also valid for unsteady. incom
pressible ﬂow? Why? How many possible y components are there? 5.6 The x component of velocity in a steady, incompressible
ﬂow ﬁeld in the xy plane is u = A/x. where A = 2 It'll/S, and xix
measured in meters. Find the simplest y component of velocity for
this ﬂow ﬁeld. 5. 7 The 3: component of velocity in a steady. incompressible
ﬂow ﬁeld in the 3y plane is U f ‘iixyly2 x“) where A— “ 2 component of velocity for this ﬂow ﬁeld. 5.8 The x component of velocity in a steady incompressible ﬂow
lieldEn the xy plane is u = A?“ costar/b). where A : 10 [1115. b =
5m, and x and _v are measured in meters. Find the simplest y com
ponent of velocity for this ﬂow ﬁeld. an in the xy plane is
2.13? u = —1 (x3 + 3’2)“ iShow that the Simplest expression for the .r component of velocity is l 2 r2 (x3 +5.2)? "‘nnni 5.10 A crude approximation for the .r component of velocity in
u incompressible laminar boundary layer is a linear variation
3‘; m u = 0 at the surface (_v = 0 )to the freeSIream velocity, U. at
{r boundarylayer edge [v d). The equation for the proﬁle is
. = Uy/d, where 6 2 ex”: and c is a constant. Show that the sim
ivst expression for the y component of velocity is u = try/4x. .. a = 0 at the surface (y = 0) to the freestream velocity. U. at
edge of the boundary layer (y : 6). The equation for Ihe pro
1: is m‘U = 20:10.) — 0785):. where d = arm and c is a constant.
_:' ow that the simplest expression for the 3; component of velocity ”—i‘fl‘fl U X 2 r5 3 r3 1.. UlU versus y/ri to ﬁnd the location of the maximum value of
=' ratio ulU. Evaluate the ratio where 6 = 5 mm and x 2 0.5 m. t 712 A useful approximation for the .r component of velocity in
l_ incompressible laminar boundary layer is a sinusoidal variation
'  u = 0 at the surface (y = O) to the freestream velocity. U. at u Id rtv av . nv —=—r .' —'—. + 7:. .‘ 4: ,1 U ax [mini (admin) l g) ulU and till] versus 3215. and ﬁnd the location of the maximum  of the ratio UIU. Evaluate the ratio where x = 0.5 tn and
r5mm. A useful approximation for the .r component of velocity in
mcompressible laminar boundary layer is a cubic variation
= u = 0 at the surface (y = 0) to the freestream velocity. U,
'illlc edge of the boundary layer 0' = :5). The equation for the
jr e is ulU = i (yr/6) — {7016)}. where (‘i = or”2 and r is a con~
.Derive the simplest expreszsion for vIU. the y component of
ity ratio. Plot u/U and MW versus _v/r5. and ﬁnd the location
maximum value of the ratio v/U. Evaluate the ratio where
'_.'=5mm andx = 0.5 m. in—3  s" and .r and ,v are measured in meters. Find the Simplestx 6.65.14 The y component of velocity in a steady. incompressible 56 suumv AND USEFUL enunnoss 209 ﬂow ﬁeld in the .ry plane is u :—B.r_v“. where B = 0.2 mTJ s" '. and .r and ,v are measured in meters. Find the simplest 3: com
ponent of velocity for this ﬂow ﬁeld. Find the equation of the
streamlines for this ﬂow. Plot the streamlines through points ( I. 4)
and (2. 4). 5.9 They component of velocity in a steady incompressible flow 36515 For 11 ﬂow in the xy plane. the .1' component of velocity is given by u : stray“. where A 2 0.3 m’3 s7'. and x and _v are
measured in meters. Find a possible _v component for steady. in
compressible ﬂow. is it also valid for unsteady. incompressible
flow"? Why? How many possible _v components are there? Deter
mine the equation of the streamline for the simplest y component
of velocity. Plot the streamlines through points ( l. 4) and (2, 4). 5. l6 Derive the differential form of conservation of mass in rec
tangular coordinates by expanding the products of density and the
velocity components. on, pt). and pw. in a Taylor series about a
point 0. Show that the result is identical to Eq. 5.1a. 5.17 Consider a water stream from a jet of an oscillating lawn
sprinkler. Describe the corresponding pathline and strealdine. 5.18 Which of the following sets of equations represent possible
incompressible ﬂow cases? a. V, = U cos t); V" = —U sin if b. V, =q/2m“. V" = Kertr c. Vr = Ucos 6 [1 r (an)2]; v” : com 011 + (rt/r)2] 5.19 For an incompressible ﬂow in the r6 plane. the r com
ponent of velocity is given as V, = —A cos Ulrz. Determine a poss
ible ll component of velocity. How many possible 0 components
are there? 5.20 A viscous liquid is sheared between two parallel disks of
radius R. one of which rotates while the other is ﬁxed. The vet
ocity ﬁeld is purely tangential, and the velocity varies linearly
with z from V” = 0 at z = 0 (the ﬁxed disk) to the velocity of the
rotating disk at its surface it. = h). Derive an expression for the
velocity ﬁeld between the disks. 5.21 Evaluate V  p? in cylindrical coordinates. Use the deﬁ
nition of V in cylindrical coordinates. Substitute the velocity vec
tor and perform the indicated operations. using the hint in
footnote 1 on page 169. Collect terms and simplify; shew that the
result is identical to Eq. 5.2c. 5.22 A velocity ﬁeld in cylindrical coordinates is given as
l7 = é,Alr+ énBlr. where A and B are constants with dimensions
of mzfs. Does this represent a possible incompressible ﬂow?
Sketch the streamline that passes through the point r0 = l m.
a = 90“ ifA = a 2 lmzis,ifA :1m2/s and B = 0, and if
B =1m1/sandA = o. *5.23 The velocity ﬁeld for the viscometric ﬂow of Example 5.7
is l/ 2 U(ylh)i. Find the stream function for this ﬂow. Locate the
streamline that divides the total ﬂow rate into two equal parts. *5.24 Detemiine the family of stream functions Ill that will yield
the velocity ﬁeld V = _v(2x + l)f+ [.r(x + l)  yzlj'. 5“5.25 The stream function for a certain incompressible ﬂow ﬁeld
is given by the expression ill =—Ur sin F) + 40/21:. Obtain an
expression for the velocity ﬁeld. Find the stagnation poinlts)
where l7 = 0, and show that I}: = 0 there.  problems require material from sections that may be omitted without loss ol‘continuity in the text material. 210 CHAPTER 5 I imouucnon T0 DIFFEﬂEﬂ'I’lAL sumsrs or FLUID nonon ”“526 Does the velocity ﬁeld of Problem 5.22 represent a possible
incompressible ﬂow case? If so. evaluate and sketch the stream
function for the ﬂow. If not. evaluate the rate of change of density
in the ﬂow ﬁeld. *527 Consider a ﬂow with velocity components ti = 0. v =
y(y2 — 3:2). and w = z z2 — 3y2).
a. ls this a one. two—_. or threedimensional ﬂow?
b. Demonstrate whether this is an incompressible or com
pressible ﬂow.
c. If possible. derive a stream function for this ﬂow. *528 An incompressible frictionless ﬂow ﬁeld is speciﬁed by the
stream function i]; =—2Ax — SAy. where A = 1 m/s. and x and y
are coordinates in meters. Sketch the streamlines ti} = 0 and t!) =
5. Indicate the direction of the velocity vector at the point (0. 0)
on the sketch. Determine the magnitude of the ﬂow rate between
the streamlines passing through the points (2. 2) and (4. l). *5.29 In a parallel onedimensional flow in the positive x direc—
tion. the velocity varies linearly from zero at y = 0 to 30 mls at
y = [.5 m. Determine an expression for the stream function. Ill.
Also determine the y coordinate above which the volume ﬂow rate
is halfthe total between y = 0 and y 2 1.5 m. *5.30 A linear velocity proﬁle was used to model ﬂow in a laminar
incompressible boundary layer in Problem 5. IO. Derive the stream
function for this ﬂow ﬁeld. Locate streamlines at onequarter and
onehalf the total volume ﬂow rate in the boundary layer. $5.31 A parabolic velocity proﬁle was used to model ﬂow in a
laminar incompressible boundary layer in Problem 5 1. Derive
the stream function for this flow ﬁeld. Locate streamlines at one—
quarter and onehalf the total volume ﬂow rate in the boundary
layer. $5.32 Derive the stream function that represents the sinusoidal
approximation used to model the x component of velocity for the
boundary layer of Problem 5.12. Locate streamlines at one—quarter
and onehalf the total volume ﬂow rate in the boundary layer. $5.33 A cubic velocity proﬁle was used to model flow in a laminar
incompressible boundary layer in Problem 5. l3. Derive the stream
function for this ﬂow ﬁeld. Locate streamlines at onequarter and
onehalf the total volume ﬂow rate in the boundary layer. $5.34 A rigidbody motion was modeled in Example 5.6 by the
velocity ﬁeld 17 = rote“u. Find the stream function for this ﬂow.
Evaluate the volume ﬂow rate per unit depth between 1'. = 0. I0 m
and r3 = 0.]2 m. if m = 0.5 radfs. Sketch the velocity proﬁle
along a line of constant 0. Check the ﬂow rate calculated from the stream function by integrating the velocity proﬁle along this line. \9‘5.“ An incompressible liquid with negligible viscosity ll‘_=. $5.35 Example 5.6 showed that the velocity ﬁeld for a free vertex
in the r0 plane is ii" = éuC/r. Find the stream function for this ﬂow.
Evaluate the volume ﬂow rate per unit depth between r1 = 0.10 m
and r2 = 0.  2 m. if C = 0.5 mgi’s. Sketch the velocity proﬁle along
a line of constant 0. Check the ﬂow rate calculated from the stream
function by integrating the velocity proﬁle along this line. 5.36 Consider the velocity ﬁeld ‘7 =1’t(ir4 — 6r2y2 + y‘)i +
A(4x_v3 —4.r3_v)f in the xy plane. where A = 0.25 m'As 1. and
the coordinates are measured in meters. Is this a possible incom
pressible ﬂow ﬁeld? Calculate the acceleration of a ﬂuid particle
at point (.t,y) = (2. I). *These problems require material from sections that may be omitted without loss of continuity in the text material. 5.37 Consider the ﬂow ﬁeld given by t7 =2:ny — W} + iyt. r..
terrnine (a) the number of dimensions of the ﬂow. (b) if it is a n m 
ible incompressible ﬂow. and (c) the acceleration of a ﬂuid panic at point (x. y. z) = {1.2. 3)
s.3s Consider the ﬂow ﬁeld given by t7 =ax3yf—byjml '
wherea = . m‘2¢s".b = 3 s", and c = 2 ru‘I s". Detenn'
(a) the number of dimensions of the ﬂow. (b) if it is a possible ' 
compressible ﬂow, and (c) the acceleration of a ﬂuid particle ._
point (x, y. z) = (3. l. 2).
5.39 The velocity ﬁeld within a laminar boundary layer
approximated by the expression In this expression, A = 141 m‘ ”2. and U = 0.240 m/s is the ..
stream velocity. Show that this velocity ﬁeld represents a possl
incompressible flow. Calculate the acceleration of a ﬂuid parti‘
at point (.r. y) = (0.5 m. 5 mm). Determine the slope of :
streamline through the point. 5.40 The x component of velocity in a steady. incompressin,
ﬂow ﬁeld in the xy plane is u = A(x5—10x3y2+5xy4). where A 
2 m"4 s’ ' and x is measured in meters. Find the simplest y i .
ponent of velocity for this ﬂow ﬁeld. Evaluate the acceleration .' a ﬂuid particle at point (x. y) : {1, 3). 5.41 Considerthe velocity ﬁeld I? = Axi'(x2 + f)? + Ayiol + y‘
in the xy plane. where A = ID mails, and x and y are measured'
meters. Is this an incompressible ﬂow ﬁeld? Derive an expressii.’
for the ﬂuid acceleration. Evaluate the velocity and accelerati
along the x axis. the y axis. and along a line deﬁned by y = x. '
can you conclude about this ﬂow ﬁeld? 5.42 The y component of velocity in a twodimensional. inconr.
pressible ﬂow ﬁeld is given by U = —Axy. where v is in m/s, r and
are in meters. and A is a dimensional constant. There is no velocii;
component or variation in the z direction. Determine the di .ue‘.
sions of the constant. A. Find the simplest 1: component of veloc'
in this ﬂow ﬁeld. Calculate the acceleration of a ﬂuid particle. point (x, y) = (l. 2) 5.43 An incompressible liquid with negligible viscosity ll 
steadily through a horizontal pipe of constant diameter. In a n _
ous section of length L = 0.3 m. liquid is removed at a cons '
rate per unit length. so the uniform axial velocity in the pipe 5'
ri(.r) = Utl — xl2L), where U = 5 mls. Develop an expres'
for the acceleration of a ﬂuid particle along the centerline of i
porous section. steadily through a horizontal pipe. The pipe diameter linearly ..
ies from a diameter of IO cm to a diameter of 2.5 cm over a lett
of 2 m. Develop an expression for the acceleration of a ﬂuid in
ticle along the pipe centerline. Plot the centerline velocity and
celeration versus position along the pipe. if the inlet center‘
velocity is 1 m/s. 5.45 Solve Problem 4.] 18 to show that the radial velocity in r.
narrow gap is V, = QIanIi. Derive an expression for the accel
tion of a ﬂuid particle in the gap.
5.46 Consider the lowspeed ﬂow of air between parallel as.
as shown. Assume that the ﬂow is incompressible and invim ...
View
Full Document
 Fall '08
 ZOHAR

Click to edit the document details