Application
of momentum equation
Hydraulic jump
K141 HYAE
Application of momentum eq.
1
Hydraulic jump
- transition from supercritical to subcritical flow
type of h.j. is given by Fr of supercritical flow
presumptions::
•small longitudinal slope of channel G of water can be neglected
•small length of h.j.  Zt can be neglected
•in both profiles (1 and 2) – hydrostatic pressure distribution
 
 

F

F

ρQ
v
 from momentum equation:
1
2
2  v1 
F1  ρgz1S1
F2  ρgz2S 2
v1 
Q
Q
, v2 
S1
S2
K141 HYAE
Zh.j.
a2v22
2g
EL
1
a1v1
2g
2
y1
2
z2
v1
z1
Application of momentum eq.
v2
y2
2
Q Q

ρgz
S

ρgz
S

ρQ
 

1 1
2 2
 S2 S1 
1

g
Q2
Q2
 z1S1 
 z 2S 2

gS1
gS2
Q2
Function of h. j.:
 zS   y  , where S  f y , z  f y 
gS
 y1   y 2  , where y1 and y2 ... conjugate depths of h. j..
y  0  y   ,
y    y   
y  yk  y   min
It can be proved that for
 shape of function of h. j.
y
a1v
2g
2
1
EL
1
Zh.j.
2
a2v22
2g
y2
y1
(y)
y2
Zh.j.
yk
Ed(y)
y1
min
K141 HYAE
Application of momentum eq.
Edmin
,Ed
3
Force effects of stream and jet
K141 HYAE
Application of momentum eq.
4
PRINCIPLE OF SOLUTION
Change of direction or change of flow velocity
application of momentum equation
force effects of stream – jet on walls and pipeline fittings,
on rotating (blades) of turbines and pumps,
on surfaces of flowed-round solid bodies
transition between supercritical and subcritical flow
application for dimensioning and shape optimalization of
flowed-round elements, for determination of hydraulic
resistances and for calculation of capacity of
hydrodynamic machines, stilling pool resolution
K141 HYAE
Application of momentum eq.
5
FORCE EFFECT OF STREAM ON PIPELINE
(BEND, KNEEPIPE, CURVED CANAL)
 
 


F1  F2  G  FA  Qv 2  v1
F1 – pressure force in entrance profile
(F1 = p1S1)
F2 – pressure force in exit profile
(F2 = p2S2)
G – weight of liquid
Q1v1 – discharge force in entrance profile
Q2v2 – discharge force in exit profile
FA – force of pipe wall acting on liquid
FR – reaction, force effect of liquid


  
 
FR   FA  F1  F2  G  ρQv1  v 2 
K141 HYAE
Application of momentum eq.
6
• vector equation - in components  appropriate coordinate system
• large pressures and small dimensions of pipe (canal), horizontal
arrangement  neglect G
Horizontal arrangement of curved element:
FRx  F1  cos a  F2  cos  Qv1  cos a  v 2  cos
FRy  F1  sin a  F2  sin   Qv1  sin a  v 2  sin 
2
2
FR  FRx  FRy , tg 
K141 HYAE
Application of momentum eq.
FRy
FRx
7
Division and connection of flow (axis in horizontal plane)

   




FR  F1  F2  F3  F4  ρQ1v1  ρQ2v 2  ρQ3 v 3  ρQ4 v 4
FRx  F1  cos a1  F2  cos a2   F3  cos a3    F4  cos a 4  
 Qv1  cos a1  Qv 2  cos a2  Qv 3  cos a3  Qv 4  cos a 4
FRy  F1  sin a1   F2  sin a 2   F3  sin a3   F4  sin a 4  
 Qv1  sin a1   Qv 2  sin a 2    Qv 3  sin a3   Qv 4  sin a 4
F1  p1  S1
F2  p2  S2
F3  p3  S3
F4  p 4  S 4
FR  FRx 2  FRy 2
tg 
K141 HYAE
Application of momentum eq.
FRy
FRx
8
HYDRODYNAMIC FORCE ON SLIDE GATE
αv 02
αv c2
h
 yc 
Z
2g
2g

Q  μvba 2gE  y c 
y c  εa

 
 
FR  F1  F2  Qv1  v 2 
v1  v 0,
v 2  vc
2
h
F1    g  b 
2
yc 2
F2    g  b 
2
sections with
paralel flow
K141 HYAE


1
2
FR    g  b  h  yc 2  Qv 0  v c 
2
Application of momentum eq.
9
FORCE EFFECT OF FREE JET ON
SURFACES (BODIES)
assumptions: axially symmetric flow round a surface (body)
p1 = p2 (generally), G  0



FR  Q1v1  Q2 v 2 , practically v1  v2
FR  Qv1  Qv1  cos   Qv11  cos 
losses by eddies at strike
and by friction along surface
change of v
correction coefficient   1
FR  Qv11  cos 
K141 HYAE
Application of momentum eq.
10
In case of  = 90° and sufficient dimension of surface (plate):
D  4 till 6   D0 , l  2D0
cos   0
FR  Qv 1
(  0,95)
v1

2
2
pmax
For semi-spherical blade
( = 180° )
cos = -1
FR  2Qv1
K141 HYAE
maximum FR
  0,94 
Application of momentum eq.
11
Note:
for surface (body) moving
with velocity u
 relative velocity: v1 – u
 relative discharge: Q = S1(v1 – u)
for system of blades (canals)
rotating with peripheral velocity u
 relative velocity: v1 - u
 discharge: Q = S1v1
K141 HYAE
Application of momentum eq.
12
Pelton-turbine
wheel
Historical
water wheels
Installation of Pelton
turbine with more
injection nozzles
K141 HYAE
Application of momentum eq.
13