312
Solutions Manual x Fluid Mechanics, Fifth Edition
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.2
O
-0.1
0.0
0.1
0.2
0.3
0.4
The streamlines are logarithmic spirals moving out from the origin. [They have axisymmetry
about O.] This simple distribution is often used to simulate a swirling flow such as a tornado.
4.17 A reasonable approximation for the
two-dimensional incompressible laminar
boundary layer on the flat surface in
Fig. P4.17 is
§ 2 y y2 ·
u U¨ 2¸
©G G ¹
for y d G
Fig. P4.17
where G | Cx1/2 , C const
(a) Assuming a no-slip condition at the wall, find an expression for the velocity component
v(x, y) for y d G. (b) Then find the maximum value of v at the station x 1 m, for the
particular case of airflow, when U 3 m/s and G 1.1 cm.
Solution: The two-dimensional incompressible continuity equation yields
wv
wy
wu
wx
or: v
§ 2 y dG 2 y 2 dG ·
U ¨ 2
, or: v
© G dx G 3 dx ¸¹
2U
dG § y2
y3 ·
dG
, where
2
3¸
¨
dx © 2G
dx
3G ¹
2U
y
dG § y y 2 ·
dy_x
dx ³0 ¨© G 2 G 3 ¸¹
C
G
2 x
2x
(b) We see that v increases monotonically with y, thus vmax occurs at y
Ans. (a)
G:
const
313
Chapter 4 x Differential Relations for a Fluid Particle
vmax
v_y
UG
6x
G
(3 m/s )(0.011 m )
6(1 m )
m
s
0.0055
Ans. (b)
This estimate is within 4% of the exact vmax computed from boundary layer theory.
4.18 A piston compresses gas in a cylinder
by moving at constant speed V, as in
Fig. P4.18. Let the gas density and length at
t 0 be Uo and Lo, respectively. Let the gas
velocity vary linearly from u
V at the
piston face to u 0 at x L. If the gas density
varies only with time, find an expression
for U(t).
V = constant
p(t)
x
x = L(t)
x=0
Fig. P4.18
Solution: The one-dimensional unsteady continuity equation reduces to
wU w
(Uu)
wt wx
Enter
dU
wu
U , where u
dt
wx
wu
wx
V
L
x·
§
V ¨1 ¸ , L
© L¹
and separate variables:
The solution is ln(U/Uo )
U
³
Uo
ln(1 Vt/L o ), or: U
L o Vt, U
dU
U
t
V³
o
U(t) only
dt
L o Vt
§ Lo ·
© Lo Vt ¸¹
Uo ¨
Ans.
4.19 An incompressible flow field has the cylindrical velocity components XT Cr, Xz
K(R2 – r2), Xr 0, where C and K are constants and r d R, z d L. Does this flow satisfy
continuity? What might it represent physically?
Solution: We check the incompressible continuity relation in cylindrical coordinates:
1w
1 w vT w vz
(rv r ) r wr
r wT
wz
0 0 0 0 satisfied identically
Ans.
This flow also satisfies (cylindrical) momentum and could represent laminar flow inside a
tube of radius R whose outer wall (r R) is rotating at uniform angular velocity.
4.20 A two-dimensional incompressible velocity field has u K(1 – e–ay), for x d L and
0 d y d f. What is the most general form of v(x, y) for which continuity is satisfied and
v vo at y 0? What are the proper dimensions for constants K and a?