NOTULES HYDRAULIQUES
APPLICATION OF MOMENTUM EQUATION IN THE HYDRAULIC JUMP
SV N.s. GOVINDA RAO * AND RAMAPRASAD
* *
which simplifies to :
Substituting these values in eg. (1) we get :
. ') ( 1 1 \) 1 ( 9 9)
. . q- - - - = ' ) g YI- - !h-
!12 !11 -
(2) is very small compared to the rate of change of momentum of the water mass and so has little efIeet on it. In eq. (1), III and 1I2 denote the velo- cities of water at sections I and II respectively at a height y from the bottom of the channel.
The most usual procedure is to assume 1I1 and 1I2
have constant values. If the discharge pel' unit width of the channel is q, then :
This paper concerns the use of a weIl known method in solving the hydraulic jump. This method can be recognized as a particular case of the solution of the momentum integral equation used extensively in boundary layer problems. In solving the momentum integral equation for any boundary layer, a velocity distribution in the boun- dary layer satisfying aIl or some boundary condi- tions is assumed and is substituted in the equation.
If the boundary layer is turbulent an addition al assumption has to be made about the shear stress at the wall.
1'0 apply the method to the problem of the hydraulic jump, we consider for simplicity a broad channel of rectangular section with a horizontal bed (Fig. 1). The momentum integral equation for this case can be written as :
(1) (3)
While deriving this equation, the shear stresses at the bottom and sides of the channel have been neglected because their contribution to the resulting force on the water mass between sections I and II
vVe shall denote the quantity q2/mh:l henceforth in this paper by F1 2. Solving (:3), '\ve get the weIl known equation :
Secfion Il
1
(4)
Section I
u,
Il
• Professor of Civil and Hydraulic Eng., Indian Institute of Science, Bangalore-12.
** Lecturer, Department of Civil and Hydraulic Eng., In- di an Institute of Science, Bangalore-12.
The assumption represented by eq. (2) is only one of many which can be made regarding the velocity distribution. The basis for the assumption of any distribution is commonly taken in boundary layer problems to be the boundary conditions which have to be satisfied by the distribution. The distribu- tion (2), however, does not satisfy what is consider- ed to be the most important condition of zero slip at the boundary (in this case at the channel bed).
vVe shall now examine the consequences of as- suming velocity distributions satisfying various conceivable boundary conditions. \\7'e may, for example, seek to satisfy several or aIl of the follo- wing conditions :
451
Article published by SHF and available at http://www.shf-lhb.org or http://dx.doi.org/10.1051/lhb/1966029
N. S. GOVINDA RAO and RAMAPRASAD
(5)
112dy = q,
at U---Ih: dU2
- 0 at y- !h dy
al U- 0: (/211 2ily2 - 0 at y--,. 0
al y- !h: 1/2112 0 at y !h
dy2 -
at y -
o·
, ~Jy~-=dll.) 0 at y- 0Ilis to be remarked here that no boundary condi- tion written above, except for condition (a), is assumed to be a priori true in the hydraulic jump.
The object is only to investigate the result of satis- fying some or aIl of the conditions set forth above.
Condition (a) ensures that there is no slip at the channel bed. Conditions (b), Cc) and (d) are ana- logous to those satisfied by the velocity distribution in a laminaI' boundary layer on a Hat plate in the absence of pressure gradient. The boundary condi- tion (e) should be satisfied if we consider that the boundary layer on the channel bed separates at the beginning of the jump due to the adverse pressure gradient (induced by the sudden rise in the water level) and re-attaches at its end.
The table above gives the difIerent velocity dis- tributions assumed, the boundary conditions which they satisfy and the resulting formula for the ratio of the water depths of the jump. The distribution of u1 only is given, that for 112 being obtained by substiluting 1/2 for !h in it.
The numerical coefficients of the distribution have been so adjusted that aIl the distributions satisfy in addition to the respective boundary condi- tions the equation :
the discharge pel' unit width of the channel.
Il will be noted that a Il distributions give rise ta formulae of the type:
The formulae (A) to (J) l'rom the table are shown plolted in Figure 2. Experimental dala l'rom difIe- rent sources are also shown in the figure. The figure shows that the formula CA), derived l'rom a distributionwhich does not satisfy the most important zero slip boundary condition, gives the most satisfactory result whereas distributions satis- fying the zero slip condition give formulae which are in varying degrees of error. Next to distribution (A), the most satisfactory result is given by distri- bution CC), which is the ,vell known one-seventh power la\\' extensively used in turbulent boundary layer problems. But even this distribution which satisfies the zero slip condition leads to a formula more in error than CA). Thus, although for prac- tical purposes eq. (4) is admirably suiled, ils suc- cess, cou pIed with the l'ail ure of other formulae derived l'rom apparently more suitable velocily distributions, calls for expIa nation. The answer to
(a) Il]
=
0 and 11 2=
0 at U=
0(b) dll].
=
0du
1/2111 (c) ... ----:;--
=
0dU- (d)!!.~ll= 0
dy2 (e) dll]
=
0dy
-
1 - - - - . --1'1'1 +8.13 F 2_12 - . ]
-
• Bakhmeteff EX Matzke
• Gibson +Ramamurthy
(a)
(a)
(b), (c)
11.
1'1'1+8 F 2-1Jand (d) 1
(a), (c) 1..!..[vI'=tTC["(nn-11
and (d) 2 - ] -
(a), (b) 1.!.[vT+~f9-I<'2_11
and (c) 2 1 .
. . . - - - - -
tlvl+10.1 F12-1]
BOUNDAHY COND- ITIONS SATIS- FIED
TABLE
q il]
2q iJ-~- ·iI
.
]VELOCITY DISTRIBUTION
U,
B.
A.
.J.
5 6
4 7
G. - - ( j
4q H. - - - IJ--
Ih2 •
~.5L IJ2
+ .._'0L
li)Ih:)' 31hI
10 11 12
c. ~!L
(
Il\]/71 Ih \iI] )
_._-- - - ---'---- - - ' ... - - . - ...•._---,---.
D.
a
Cl .IJ-_:~_(L
.IP "'I,«.la)([) ,
t) [vl=t9]rF;~-1
]Ih2 ' 2 Ih:) ' " _
E. ~(L sin ""iL
1
2 Ih 2 Ih!
~----
') "(1
F ....• ~-q100'. ::-'c;r.·
)J)
ill \ Ih
3
2
1
1 2 3 4 5 6 7 8 9 10
FI 2/
452
LA HOUILLE BLANCHE/N° 4-1966
where chas the same value as for
(7)
Section 1
In the same way,
Io
Y2 1I22Y d!]also reduces to (7) and hence expression (6) again vanishes identically.
This result Ineans that the algebraic sum of the moments of the external forces on the water mass about P lllUSt be zero. On assuming hydrostatic pressure distribution on sections 1 and II, it can he easily shown that Fl and 1"2 give rise to an anti- elockwise moment equal to :
Sectionil
--L(j g(1] :) -, l 1]":))• _
where p is the density of water. Therefore the weight v\T and the resultant hed reaction R must touether uive rise to a clockwise moment equal to
b b . d
this magnitude. Renee R must be sItuate some distance ta the left of VV. This means that the pressure on the hed of the channel must be more than that due to hydrostatic pressure in the begin- ning of the jump and less towards the end. When this is the case, the centre of gravity of any clement contained between two vertical sections separated hy an infinitesimal distance must be accelerated up- wards in the beginning of the jump and downwards towards the end. This means that the mean profile of the water surface in the .lump must he as shown in Figures 2 and 3 and must not have a continuous- ly rising form as is usually assumed.
where IIr is the friction velocity and F is an arbitrary function. IIr depends, again, on q, Yl and v, so that III ultimately depends on q, Yl and v.
Rence, the integral
Io
Y1 Zl12 Y dyalso depends on q, Yl and v. Sa, by dimensional analysis, the integral is readily seen to be the fonn : 3/
(6)
1
- .
f _._-)
IJYl2
dyIII / _
IJ
\-1
2 (IJI,1/;)_
d \-;;;)Î!h
Jo
his question may have a be~U'ing on the utilityand reliability of the momentum integral equation me- thod in solving many problems in hydraulics.
The momentum integral equation is not the only one by which a solution is likely ta be obtained.
The energy equation is difIicult ta derive because of the importance of turbulent friction as a con tri- butor to the loss of energy. Rowever, we can start with the moment of momentum equation, which expresses the fact that the rate of change of mo- ment of momentum of the water mass between sections 1 and II (Fig. 3) about a traverse axis lying in the bed of the channel is equal to the moment of the external forces acting on this water mass.
The rate of change of 1I1Oment of momentum of the water mass considered about the point P can be easily proved ta be :
ÎlIo
;'1/1
Jo -
U22Y dy -Jo'
U 12Y dywhere c is a constant.
Similarly :
The external forces acting on the water mass are:
1. the force 1"1 due ta water pressure on section 1;
2. the force 1"2 due ta water pressure on section II;
3. the weight VV of the ,vater mass acting at its centre of gravity;
4. the reaclion R from the bed acting at the centre of pressure at the base of the water mass.
For each of the distributions given in the pre- vious table the expression (6) is identically zero.
This is true for any distribution of the type given in the table. For, in aIl distributions of III of that type, the coefIicients of the powers of y depend only on q and Ill, due to the conditions to be satisfied.
Thus q, y and Yl are the only variables afIeeting
Ill' and it follows from dimensional analysis that a relation of the type:
( ,
q /' / li \
III = - - 1_'L!
Yl \ Yl /
should hold,
f
being a function depending to some extent on the houndary conditions.Rence,
References Thus the expression (6) vanishes identically.
This is still the case if one supposes that Il] is affeeted not only by q, Ih and y, but by the kine- matk viscosity and the hed shear stress as weI!.
For, in that case, th cre will he a relation of the form:
The experimental data ploUed in Figure 2 have heen tal,en from the following references :
[11 BAJ,]DIETEFF and MATZJŒ. - Trans. ASCE, lOI (1936), p. 630.
[21 GIIlSON (A. H.). _.- Proe. Insfn. C.E., 197 (1913-14) pt. III, p. 233.
[;l] nAMA~IURTHY (A. S.). - Tllesis M. Sc., (II Sc.) (1960), p. 155.
453