456
Solutions Manual x Fluid Mechanics, Fifth Edition
where h f
32 P LV
Ugd 2
32(0.29)(25)(4.76)
140.5 m, Solve for h pump
891(9.81)(0.03)2
118.9 m
The pump power is then given by
U gQh p
Power
p
mgh
m·
§ kg · §
¨© 3
¸¹ ¨© 9.81 2 ¸¹ (118.9 m )
s
s
3500 watts
Ans.
6.34 Derive the time-averaged x-momentum equation (6.21) by direct substitution of
Eqs. (6.19) into the momentum equation (6.14). It is convenient to write the convective
acceleration as
du w 2
w
w
(u ) (uv) (uw)
dt w x
wy
wz
which is valid because of the continuity relation, Eq. (6.14).
Solution: Into the x-momentum eqn. substitute u
u u’, v
v v’, etc., to obtain
ªw 2
º
w
w
(u 2uu’ u’2 ) (v u vu’ v’u v’u’) (wu wu’ w’u w’u’) »
wy
wz
¬wx
¼
U«
w
(p p’) Ug x P[’ 2 (u u’)]
wx
Now take the time-average of the entire equation to obtain Eq. (6.21) of the text:
ª du w
º
w
w
u’2 u’v’ u’w’ »
wy
wz
¬ dt w x
¼
U«
wp
U g x P’ 2 u Ans.
wx
6.35 By analogy with Eq. (6.21) write the turbulent mean-momentum differential
equation for (a) the y direction and (b) the z direction. How many turbulent stress terms
appear in each equation? How many unique turbulent stresses are there for the total of
three directions?
Solution: You can re-derive, as in Prob. 6.34, or just permute the axes:
(a) y: U
dv
dt
·
wp
w § wv
· w § wv
U gy U u’v’¸ U v’v’¸
¨P
¨P
wy
wx© wx
¹ wy© wy
¹
w § wv
·
U v’w’¸
¨P
wz© wz
¹