流体力学答桉(刘鹤年)


? ?
 ? > X? [\
?

?ü


?ü>? &??   ax =

1 u y = ? y3 3

&? ?&E? 

? &

X? [\

?

?&E

  ?   

 

?ux ?u ?u ?u 1 16 + ux x + u y x + uz x = xy 4 = ?t ?x ?y ?z 3 3

? ?  E?

3 1 32 a y = ? y 2u y = y 5 = 3 3 3 1 2 16 3 a z = yux + xu y = xy ? xy3 = xy3 = 3 3 3 2 2 2 a = a x + a y + a z = 13.06m/s2

1 4 xy 3

 ? ?E ?>??? u x = ? % 
dx dy = ? ? ux uy dx cy 2 x + y
?

?? ?/???' ?
=
2

cy cx  ? u y = 2 2 x + y2 x +y
2

F 

?/ ? X? X?

?? ?? [ \ F
?

dy cx 2 x + y

? -x d x = y d y
2

???

 ? ????????? ?? ? ?  X? D\ X? D[ X? 


ux = ?

cy cx ,uy = 2 , uz = 0, ??a c% 2 x +y x + y2 cy cx ,uy = 2 , uz = 0, ??a c% ux = ? 2 2 x +y x + y2 1 ?u ?u 1 ω z = ( y ? x ) = (a + a) = a ?? ?   ω x = ω y = 0 2 ?x ?y 2 1 ?u ?u 1 ε xy = ( y + x ) = (a ? a) = 0 ε xy = ε zx ? ? ?y 2 ?x 2
2



1 ?u y ?ux 1 ? ( x2 + y 2 )c ? 2cx2 ( x 2 + y 2 )c ? 2cy 2 ? ωz = ( ? )= ? + ? 2 ?x ?y 2 ? ( x2 + y 2 )2 ( x2 + y 2 )2 ? 1 ? 2c( x 2 + y 2 ) ? 2c( x2 + y 2 ) ? = ? ?=0 2? ( x 2 + y 2 )2 ?

ωx = ω y = 0

?? ? ? ?

1 ?u y ?u x 1 ? ?2c( x 2 ? y 2 ) ? c( x 2 ? y 2 ) ε xy = ( + )= ? =? 2 ≠0 2 ?x ?y 2 ? ( x2 + y 2 ) ? ( x + y 2 )2 ?

S¤ N3DS? N3D% ? ?E? > Y? PV  ?  ? ?? 

  ???

$%G¤ P G? P ?
K P ?

υ A = υB

dB2 0.4 = 1.5 × ( ) 2 = 6m/s 2 dA 0.2

H A = zA +

pA υ A2 30 62 + = 0+ + = 4.90m 9.8 2 g ρ g 2g

pB υ B 2 40 1.52 HB = ZB + + = 1.5 + + = 5.69m 9.8 19.6 ρ g 2g

HB>HA,

/B

 A;

#

5.69-4.90=0.79m 
K PP 

  / ??? ? >

?

?? > X 

 u = 2 g ×12.6?h = 19.6 ×12.6 × 0.06 = 3.85m/s

G ? PP ? ? ?? G? PP ?  ?4  ? ? ? %  ? ? ??? 

  ? ?

>? ?

?? ? >?? ?
§

% K¨ PP?

NJP 

p1 υ12 p υ2 p p υ2 υ2 + = Z 2 + 2 + 2 + hl ? ( Z1 + 1 ) ? ( Z 2 + 2 ) = 2 + hl ? 1 ρ g 2g ρ g 2g ρg ρg 2g 2g ?  ρ ?ρ p p d1 2 ( Z1 + 1 ) ? ( Z 2 + 2 ) = ( hp = 15hp v2 = v  ) = 4v1 1 d2 ρg ρg ρ 19.6 ×15 × 0.15 ∴ = 1.715m/s υ1 = 15 Z1 +

Q = ?υ A = 0.95 ×1.715 ×
1 ? K = π d12 2 g 4
2

π (0.02)2
4

= 0.051m3 /s



G PP 




??>

NJP 
§
2

? + PP

??

'/?

? d12 ? ? d 2 ? ? 1 = 0.036 ? 2 ?

Q = ? K ?h = 51.3 l/s

?ǐ?

 $ × 

??

??

?



? 4

 ? ?

?? ?

??  ?E  ?ǐ?
2

?E  ?? 

ρυ2 1 p1 + ρυ + ( ρ ? ρ ) g ( z ? z ) = p + a 2 1 2 2 2 + pl 2 2 0 + 0 + 0 = ?0.15 ×103 × 9.8 + ρυ υ =47.74m/s Q = 1.5m3 /s 2 +0

? K?  ? ? 4 P V ? $?Dü
§

 ??? ? ??

% P ? ? K ? P ?? ?? ? ?

/ü???E ??? ?  # ? Q = VA ? V1 =

???

Q 8 = = 1m / s A1 2 × 4

 /?? ? ? v 2 = v 1

???

A1 4× 2 )=v  ) = 8m / s 1 A2 0.5 × 2

P1 ? P2 ? R ' = ρ Q(v 2 ? v1 )
4 P 1 = ρ ghc1 Bh1 = 9.8 × 2 × 2 × 4 = 156.8 KN

P2 = ρ ghc 2 Bh2 = 9.8 × 0.5 2 × 2 × 0.5 = 2.45 KN

??"??ü 5 5’

R' = P 1?P 2 ? ρ Q (v2 ? v1 ) = 98.35KN

.1

?E ?ü ??ü  1 1 2 P = ρ gh 2 B = ×1000 × 9.8 × ( 4 ? 0.5) × 2 = 120.05KN 2 2
4-21 ?ψ ?ψ  dψ = dx + dy
?x ?y



/
?x

? ux = 20 y =

ψ = 10 y 2 + f ( x )  ? u y = 0 = ? ?ψ  ? f ( x) = c  %ψ = 10 y 2 + c

?ψ ?y



?ψ ?y

??

%?? ? ωz = ? y ? x ? ≠ 0  2 ? ?x ?y ?
1 ? ?u ?u ?

??>?

 ?

?E??

??

?% ψ = xy + 2x ? 3y + 10 
?ψ = ?y ? 2 ?x

>?

/ ?%? ? u x =

?ψ = x ?3 ?y

uy = ?

? = ∫ u x dx+f ( y ) =
1 f ( y ) = ? y 2 ? 2y + c 2

1 2 x ? 3x+f ( y ) 2

?? = f ′ ( y) = ?y ? 2 ?y

? q 

1 2 1 x ? 3x+ ? y 2 ? 2y + c 2 2  ? ? ??>? U 0 ????'  ?? ?? ?=
? ?  ??/??  ??ü?? ?% ?3? 

>? u r =
uθ =

? ?? ?% q ψ = u 0 rcosθ ? ln r 2π

q ?ψ = u 0 cos θ ? ln r 2π ?r

θ = 0, r ? / u θ = 0 ? θ=0 "
u r = 0 ? rs =

?ψ = ? u 0 r sin θ ?θ

s

=

q 2πu 0

θ =π

? ?? ?? ??/??  ? ?%? ψ = 0 

?  θ = 0, rs =

q  θ = π ?? rs < 0 "??? 2πu 0 cos θ

q 2πu 0

u 0 rsinθ ?


? d1 ? ?? d2 ?? ? ? ρ ??? ? % ν ?? /π  ? >? ?

? ?? ? ?

q θ=0 2π

> v ? ? ?? ?p  ?

? v d ρ  '?n=6, m = 3, n- m = 3

f ( v, ?p, d1 , d 2 , ρ ,ν ) = 0

π1 =

d1 ?p ν = = , π , π 3 2 b b b v a1 d 2 1 ρ c1 v a2 d 2 2 ρ c2 v a3 d 2 3 ρ c3

π 1 : [?p] = [v] a [d 2 ]b [ ρ ]c
1 1

1

ML?1T ?2 = ( LT ?1 ) a1 ( L) b1 ( ML?3 ) c1

M : 1 = c1 L : ? 1 = a1 + b1 ? 3c1 T : ? 2 = ?a1

? a1 = 2 ? ∴ ? b1 = 0 ?c = 1 ? 1

?p π1 = 2 v ρ
d1 π2 = d2

Q a2 = 0 b2 = 1 c2 = 0
3 3

?

π 3 : [ν ] = [ v] a [d 2 ]b [ ρ ]c a3 = 1 b3 = 1 c3 = 0
π3 = ν
vd 2

3

?p d 1 ν f( 2 , , )=0 v ρ d 2 vd 2 d 2 vd 2 ?p ) = f1 ( , 2 d1 ν v ρ
d v2 ρ = Φ1 (Re, 2 ) ?p d1
υ=


?p

ρ

Φ(Re,

d2 ) d1
H ??? d ?

? ρ ??ü ? % ??üü>? J ?

? ?? ? > v ?/

H
d

? / π 

??

? ??

? v H ρ  '?n=6, m = 3, n- m = 3

f ( v, H , d , ρ , ? , g ) = 0

π1 =

d g ? , π 2 = a 2 b2 c 2 , π 3 = a3 b3 c3 H v H ρ v H ρ
f( d ? gH , , )=0 H vHρ v2 gH d vHρ = ( , ) f 1 2 ? v H d vHρ ) v = gH Φ ( , ? H

Q=

d vHρ , ) 4 H ? d Q = A 2 gH Φ1 ( , Re) H Q = ? 0 A 2 gH gH Φ(

πd 2

?0 = Φ1 (
 ? PP ?????/




?

>??? PP ???Dü? ?/?  % ? P V
 

d , Re) H

? ? % ? P V
 

? P V ??? ? ??? ?

? ?

>?

?

?

??

υm dm υ p d p = νm νp

υm ν m d p = ? υ p ν p dm
Qm υ m d m 2 ν m d m = ? = ? Qp υ p d p 2 ν p d p

υm dm 1.6 × 10?5 300 Qm = Q p = 0.283 × × = 2.26m 3 /s ?6 1× 10 600 υp dp

hp
 = ?  =??ü?  ?

vm h m

pm
?

hp=1.5P =>? vp  NPK =?Dü Pm=1.4N1



=? ? hm ? =??

> vm PV

??? ?

υ m hm υ p hp = νm νp
Q
hm = h p ×

νm =ν p

υp 108000 = 1.5 × = 1.0 m υm 3600 × 45

??

pp

ρ pvp2

=

pm ρ m vm 2

ρm = ρ p


Pp Pm
 

pp pm
vp vm
2 2

=

vp vm
2 2

2 2

=

pp pm

×

hp hm

2 2

=

×

hp hm

=

hm hp

2 2

×

hp hm

2 2

=1

∴ Pp = Pm = 1.4 KN
? ,??ē

 > PV ? ?? E?  3D? E? 3D (??? > ? PV ? ? E?? E?? ?? ??

?



υ p2 = 2 pm υ m
pp

p p = pm × (

vp vm

)2



 

12 2 ) = 74.67 N / m 2 9 12 p = ?20 × ( ) 2 = ?35.56 N / m 2 9 p = 42 × (
?? G    > ? W  ? ?

? ? 3

> Y PV (   ?/?
!

? P V

υd 1× 0.1 = = 7.6 × 104 > 2300 ?6 ν 1.31× 10 υc d ν 1.31× 10 ?6 Rec = = 2300 ? υ c = 2300 × = 2300 × = 0.03m/s d 0 .1 ν
Re =

>  PV

 ??

(  

?E?

?





?

?  ?

 



R=

A

χ

=

bh 0.2 × 0.15 = = 0.06m b + 2h 0.2 + 2 × 0.15

Re =

  ? NJP" ?? 

 ?/? ?

υ R 0.15 × 0.06 = = 6870>575 1.31× 10?6 ν

 K  ? ???

? O  P

?

?

?ν ??ü

? ρ 

?

?

? 4  FP(V

?

% ?

?

?? G

??

υ=

4Q 4 × 70 = = 139cm / s 2 2 πd π × ( 0.8 )

hf =

p1 ? p2 ρ# ? ρ$ 13600 ? 901 = × 30 = 4.228m h= $ $ 901 ρ g ρ

l υ 2 64ν l υ 2 = d 2 g υd d 2 g 2 gd 2 1960 × 0.82 ν = hf = 422.8 × = 2.98 × 10?5 m 2 /s 64lυ 64 × 200 × 139 ? Re = υ d = 1.39 × 0.008 = 373 < 2300 2.98 × 10?5 ν hf = λ

 

? 



?3 ?? ?E

??   ? ? KS %??

 

???

? NJP' ?? ??" ?



?

?

%  FP)V

?h =

ρ% ? ρ & 13600 ? 901 hp = × 0.02 = 0.282m & 901 ρ
u = 2 g?h = 19.6 × 0.282 = 2.35m / s



Y X

Q=

u πd 2 2.35 π (0.075) = × = 5.19l / s 2 4 2 4
2

υd ud 2.35 × 0.075 = = = 979 ν 2ν 2 × 0.9 ×10?4  ? > ????
Re =

>??  >?

?  OV ???  ??? ?Dü

%

 K 

#

?



υ=
 / ?

4Q 4 × 0.00334 = = 1.70m / s πd 2 π (0.05)2
? >?? ? >/ Y ? Y??? ? Y ? G

hm = 12.6?h = 12.6 × 0.15 = 1.89m

G ? > Y ? & ?

+" 

K 

?

?E? K ??

p1 υ12 p2 υ22 (υ1 ?υ2 ) Z1 + + = Z2 + + + 2g ρ g 2g ρ g 2g

2

? p2 ? ? p1 ? υ12 ? υ22 (υ1 ? υ2 ) h = ? Z2 + ? ? ? ? Z1 + ?= ρg ? ρg ? 2g 2g ? ? =?

2

υ22 υ1υ2
g + g

2υ υ dh =? 2 + 1 =0 dυ 2 g g

υ2 =

υ1
2

d2 =


2 d1

hm ax =

υ 12
4g
  

?  P>?  >?

???



? 4 lV %

??    ?E??  P ???

υ=

4Q 4 × 0.006 = = 3.06m / s πd 2 π (0.05)2

2 p1 υ12 p2 υ2 Z1 + + = Z2 + + +h ρ g 2g ρ g 2g f

? p1 ? ? p2 ? Z Z + ? + ? 1 ? ? 2 ? = hf ρg ? ? ρg ? ? = 12.6hp = 12.6 × 0.25 = 3.15m
l υ2 = 3.15m hf = λ d 2g

λ=


3.15 = 0.022 15 3.062 × 0.05 19.6
? ? ?

? G PP

K 0 P K1 P%& ??=? 1 ?

$% ?? # ?





???

%& ??? O P ? $% %ζ ? Dü % λ  ?& '  %

 ? ?

>??

?

? ?& ?

???? P ??? ?/?E? % 

?Dü

?

>

?

?E?? ù 
> $4



? Z1 ? Z2 = h1 ?% ?E? &

2 p1 υ12 p2 υ2 Z1 + + = Z2 + + +h ? ρ g 2g ρ g 2g

"? v2 PV

p1



p1

 ρ g

v1

?E??



? Z2 ? Z3 = h2 "? h = 18m

2 p3 υ32 p2 υ2 Z2 + + = Z3 + + +h ρ g 2g ρ g 2g

v2

v3

p3



l v2 h = (ζ + λ ) d 2g h × 2g l 18 × 2 × 9.8 50 ? 0.035 × = 19.498 ζ = 2 ?λ = 2 v d 0.1 ( 3.088 )

? G P ??? ??"?  3 ü



3.088 × π × ( 0.1 )2 = = 24.25l / s ?Q = v 4 4

πd2

>  PV ?? ?

?  NJP(?   ?23P4V



?

+ P?

W  

Re =

U0d

3 ?  ?

ν

=

20 × 0.6 5 = 7.64 × 10 15.7 × 10?6
C D = 0.4

D = CD
 ?

ρU 02
2

1.20 × 202 A = 0.4 × × 40 × 0.6 = 2304 N 2

 ? 

?? ?? G PP ? " 

 ?

%3?E?1  

 ? ??  ? ? "5
π ( 0.02 )
2



+ P







G 6 FP %

/?

4 2 π ( 0.02 ) Qn = ?n A 2 gH = 0.82 × 2 g × 2 = 1.61 m3 /s 4 pv = 0.75H = 0.75 × 2 = 1.5 m ρg

Q =? A 2 gH = 0.62 ×

2 g × 2 = 1.22 m3 /s

+ P K8 P

?? ?? 

? $ % 

?

 K 6 K7  ?

??? G7 FP ?

???

????

? ? 4

  ?  ( ?% 

Q=?
0.62 ×

π d1
4
4

2 gh1 = ?n

π d22
4

2 g ( H ? h1 ) = Qn
π (0.03)2
4 2 g (H ? h1 )

π (0.04 )2

2 gh1 = 0.82 ×

K 6 P
Q=?

K7 + K 6 P
π ( 0.04 )
4
2

π d1
4

2 gh1 = 0.62 ×

2 g ×1.07 = 3.57 l/s

?>?  

? P

P ? P ? ??? FP % "????

V = 25 × 10 × 1.5 = 375m3 1 Qmax = ? A 2 gH 0 = 0.62 × π d 2 2 gH 0 = 2.64 × 10?2 m3 4  2V t= = 7.89h Qmax
$ ? ? % E ? + P? ?

 ? Dü %

?? G PP

? ?? l 6 P l7 P K P

? A

 ?Dü %?? 9= @? ?? >? ? ?E?1 ? ? ??

??(

H + 0 + 0 = 0 + 0 + 0 + hl

2 2 8 υ2 ? l1 + l2 ?υ ? ?υ hl = ? λ +ζa +ζb +ζc ? = ? 0.02 × + 0.5 + 0.2 + 1? = 3.83 d 0.075 2g ? ? 2g ? ? 2g

υ=

?(

2 gH 19.6 × 2 = = 3.20 m s 3.83 3.83 2 π × ( 0.075 ) Q = υ A = 3.20 × = 14.13 l/s 4

?? ?

? ??

pa pc υ 2 +0 = h+ + + h l1 H+ ρg ρ g 2g



υ2 ? l1 ? h l1 = ? λ + ζ a + ζ b ? = 1.5 + 2.5 × 2g ? d ? Pv Pa ? Pc 3.202 υ2 = = h+ = 3.11m (1 + 1.5) = 1.8 + 2.5 × 2g 19.6 ρg ρg
× ?  $ ?? G PP? l $ Dü %  E?? ? p 6 DW E ?Dü P ? > + 6 P

+ 6 P +7 P D 

  %?

? l 2×98000 υ2 1+ = 5+16.4 9800 2g ?υ =

E ? p1 H + + 0 = H 2 + 0 + 0 + hl  1 ρg 2 2 10 υ2 ? d ?υ ? ?υ + 0.5+ 4 +3×0.3+1? =16.4 hl = ?λ +ζe +ζv +3ζb +ζ B? = ?0.025×
? 2g ? 0.025 ? 2g

%??

2g

? ?ü  ? ? ? ? >?  Dü %
6 7

19.6×16 = 4.373 m s 16.4

Q =υA = 2.15 l s

l7 P ?? G 6 PP G7 PP + P ? l 6 P  ??? ?? ?
pM υ 2 + 0 = 0 + 0 + 2 + hl ρg 2g
2

 pm DW ?



H +

υ 1 = 4υ 2

l υ2 l υ2 υ2 h f = λ1 1 1 + λ2 2 2 = 51 2 d1 2 g d2 2 g 2g

?? d ?2 ? υ 2 υ2 hm = 0.5 + ? ? 2 ? ? 1? 2 = 17 2 2 g ? ? d1 ? 2g 2g ? ? ?

υ12

69

υ 22
2g

= 12

? υ 2 = 1.846 m s

Q = υ 2 A2 = 58 l s

 ?  l?? G ? >??4? ????? l ?% ?:? ? % ?? ? ?4 H = 2alQ1
2

+ ???

?# 

?4

5 ? Q2 ? 2 H alQ al = alQ2 = + ? ? 2 ?4? 4 ? 2 ? 5 2 2 2alQ 2 = alQ 2 ? 4
2

2



? L (>?

?E( ?





E P ? ??

?

Q2 8 = = 1.26 5 Q1

% P  ? K P >? ?

 ????(E? 4  m 3 / s ? L   ?E ü? ?

A = ( b + mh ) h = ( 3 + 2 × 1.2 ) × 1.2 = 6.48m2 6.48 6.48 A R= = = = 0.775m x b + 2h 1 + m2 3 + 2.4 1 + 22 ?n = 0.025 1 16  C = R = 38.24m0.5 / s n Q = AC Ri = 6.48 × 38.34 0.775 × 0.0002 = 3.09m3 / s
? ü??E'? ? (>

EK  A 2h 2 2 = 0.5h A = bh = 2h x = b + 2h = 4h R = = x 4h 1 1 1 1 1 ?n = 0.017 C = R 6 = ( 0.5h ) 6 = 52.41h 6 0.017 n 5 1 1 1 2 Q = AC Ri = 2h × 52.41 h 6 ( 0.5h ) 2 ( 0.001) 2 = 2.34h 3 = 9.7
C

?E

>?

?



( )

 ?? ??

? 9.7 ? h=? ? ? 2.34 ?

3

8

= 1.70m

b = 2h = 3.4m

% P ? L  >>YEFG@ PV

?E(>

>? ?E

% Q &??

? 4 PDV(>? 

>

A=

Q



υ max

= ( b + mh ) h = 1.25 m 2
2 1 2

R = 0.366 m
2 3

υ max

1 = C Ri = R 3 i n

? 1? 1.25 = ? ? n ? b + 2h 1 + m 2 ?

i

1

2

= 0.8

17 ? ?E(> ? / ? b=2h

? ? h ( b + h ) = 1 .2 5 ? h 2 ? 1 .8 6 8 h + 0 .6 8 4 = 0 ? ? ? b + 2 2 h = 3 .4 1 5 8 ? h1 = 0 .5 m ? h 2 = 1 .3 6 8 m ? ( ) ? ? = 2 b m ? 1 ? b 2 = ? 0 .4 5 4 m
 900>? ? Q=0.8m3/s,

/

αQ2
g Ac = =

Ac 3 Bc

1 bhc = hc 2 2

//

?? B = 2h c c
h 2α Q 2 5 = g = 0 .666m

 ??

E P?E?

(>?E

c

K’???

?

 A = h ( b + mh ) = h(12 + 1.5h)

% Q  >?

% P 

?

? 4  PDV /

K

$

??/? ?

%

$D%

B = b + 2mh = 12 + 3h

' 2 2 ?E(> hc = 3 aQ gb = 0.612m "?? K ?

?/??

??? $D% /? $D%K ??

α Q2

??

Ac3 = = 33.06 g Bc
KF P
2

 

     

                 ?

Rc = 0.589m ? Q?n ? ic = ? = 0.0069 2/3 ? ? AR ?

 /

KH P

? '

?E??(> ?? ? % Q 

(>?

?(> E?? ?? bh 0.5 A = 0.25m (1) A = bh0 = 1× 0.5 = 0.5m2 R = = 0 = x b + 2h0 1 + 2 × 0.5  1 1 1 1 C= R 6 = ( 0.25) 6 = 56.69m0.5/s Q = AC Ri = 0.283m3/s n 0.014 ???E/
ag 2 = ? hc = 3 q

???E?

(>?

E P ? L  ?

 (>'

 (>'

( 0.283) 3
9.8

2

= 0.202m


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