68
Channel Tranistions
A channel transition is defined as change in channel cross section e.g. change in channel width and/or channel bottom slope.
May be accomplished over a long distance or it may be sudden
A transition is designed such that the losses at the transition are small.
Provide a smooth change in channel cross section.
Provide a smooth change (possibly linear) in water surface elevation.
Gradually accelerate flow at pipe inlets and decelerate them at point outlets.
Avoid unnecessary head loss through change in cross section.
Transitions are normally made of concrete or earth, the latter often having some sort of riprap protection.
Earthen transitions between open channels and pipe flow (culverts and siphon) are often acceptable when the flow velocity is less than 3.5 fps (1m/s).
Can have both lateral and vertical (bed) contraction and expansion
The optimum angle of lateral convergence (at contractions) is given by 12.5o by Chow (1959), corresponding to a 4.5 : 1 ratio.
The optimum angle of lateral divergence (at expansions) is often taken as approximately 9.5o or at 6:1.
Consider a small hump in a rectangular channel with subcritical upstream flow
A slight depression is caused in the water surface over the hump.
The hump normally causes a drop in the specific energy.
The minimum height that causes the critical depth is known as critical hump height.
A further increase in height results in an increase in depth upstream of hump with critical depth maintained over the hump.
This is known as damming action.
Flow through a contraction behaves in a similar manner to that of flow over a hump.
A small contraction causes a slight depression of water surface for subcritical flow.
The minimum contraction (maximum width) at which critical depth occurs is the critical contracted width.
An increase in contraction results in an increase in depth upstream of contraction even though the depth is still critical.
Damming action occurs.
Consider a constant width rectangular channel having a bottom step.
It is to be determined whether the water surface rises or drops downstream of the transition for a specified flow depth and flow velocity upstream of transition.
Because of constant channel width, q is same on both sides of transition and the same specific energy curve is applicable to both u/s and d/s sides.
If it is considered that the energy losses are negligible, then total head H1 should be equal to H2.
1 and 2 refer to u/s and d/s
From the figure,
11 HE = zHE Δ−= 22
The flow corresponding to flow conditions at 1 is marked as 1.
For 2, a vertical line is drawn such that E = E2
Flow depth corresponding to points where the vertical line intersects the specific energy curve are the depths of flow.
In this case, there are three such points marked as 2, 2’ and 2’’.
2” corresponds to negative depth which is not physically possible.
Of the two points, 2 and 2’, it has to be determined, which one is practically possible.
One can easily move from 1-2 along the specific energy curve.
To go from 2 to 2’, two different paths can be followed.
For the path along the vertical line 2-2’, we will have to move off the specific energy curve and pass through the curves corresponding to higher unit discharges.
Higher discharges are possible only when channel width is reduced at the transition.
Since there is no such contraction for this case, so this path is not feasible.
For the second path, 2-C-2’, a decrease in E is necessary.
E only decrease if the channel bottom rises until E= Ec and then drops again until E=E2.
There is no such rise or drop at the bottom of the channel under consideration.
So the second path 2-C-2’ is not possible either.
So only one depth is possible which corresponds to point 2.
In a similar manner, one can also show that if the flow upstream of the transition is supercritical, then the downstream flow depth will be the one corresponding to 2 and not 2’.
It can also be concluded that for a step rise in channel bottom, the flow depth
◦ Decreases downstream of the step, if the flow upstream of the transition is subcritical.
◦ Increases if the upstream flow is supercritical.
Can be derived mathematically.
Total head H at a channel section is given by
gVyzH2
2
++=
2
2
2gAQyzH ++=
We have to determine the sign of variation of ‘y’ with respect to elevation of channel bottom ‘z’.
Assume d/s flow direction to be +ve for distance ‘x’ measured along the channel bottom.
Flow depth increases if dy/dx is positive and decreases if dy/dx is negative.
Differentiating the first equation with respect to ‘x’ we get
⎟⎠⎞
⎜⎝⎛++= 2
2 12 Adx
dg
Qdxdy
dxdz
dxdH
And
dxdA
AAdxd
32
21 −=⎟
⎠⎞
⎜⎝⎛
dxdy
dydA
dxdA
=
For a small change in flow depth, Δy, change in the flow area is ΔA~BΔy.
B is the top surface width so for we may write
thenBdydA =
dxdyB
dxdA
=
Froude’s number is defined by
Therefore
3
222
/ gABQ
BgAVFr ==
( )dxdyF
dxdz
dxdH
r21−+=
If there are no losses then dH/dx=0 and
The above equation describes the variation of flow depth for any variation in bottom elevation.
( )dxdyF
dxdz
r 12 −=
For a step rise, dz/dx>0,
For the RHS to be positive, there are two possibilities.
(Fr2 -1) and dy/dx are both positive or both
negative.
The first condition implies that Fr >1, flow is supercritical, then dy/dx>0, i.e flow depth increases at the step.
Second condition implies that Fr<1, flow is subcritical, dy/dx<0, i.e. flow depth decreases at the step.
Similarly, one can also say that for a drop in channel bottom, the flow depth decreases if the flow u/s is supercritical, and increases if the flow u/s is subcritical.
Depth of flow does not remain constant along a given length of channel.
If the changing conditions extend over a long distance, the flow is known as gradually varied flow (GVF) while if they change abruptly or extend over a short distance, then the flow is known as rapidly varied flow (RVF)
RVF usually occurs at channel exit, at change in X- section, at bends and at obstruction such as dams, weirs or bridge piers.
Consider and element between section (1) and section (2) of an open channel
(z+y) is the potential energy head while
is the kinetic energy head
gvyzH2
2
++=
gv2
2
v is the mean velocity in the section.
Since bed slope is very small, so α= 1
Differentiating with respect to x,
dxdv
gdxdy
dxdz
dxdh 2
21
++=
Previous eq was the general equation for GVF with
dxdhS −= Slope of the energy line (-ve
sign shows decreasing)
dxdzSo −= Slope of channel bed
Slope of hydraulic grade line or water surface)(
dxdy
dxdzSw +−=
Energy equation between sections (1) and (2) is given by
Lhg
vyZ
gv
yZ +++=++22
22
22
21
11
xzzSo Δ
−= 21 And
xh
S L
Δ=
)2
()2
(2
11
22
2 gv
yg
vyxSxSo +−+=Δ−Δ
( ) xSSg
vyg
vy o Δ−++=+22
22
2
21
1
sosg
vyg
vyx
−
+−+=Δ
)2
()2
(2
22
21
1
3/249.1 m
m
RnV
S =3/2
m
m
RnV
S =
Since V=q/y,
dxdy
ygq
yq
dxd
gdxvd
g 3
2
2
22 121)(
21
−=⎟⎟⎠
⎞⎜⎜⎝
⎛=
Substituting this and the S and So terms in the differentiated equation
2232
3
2
111
1
r
ooo
o
FSS
gyVSS
gyqSS
dxdyor
gyq
dxdySS
−−
=−
−=
−−
=
⎟⎟⎠
⎞⎜⎜⎝
⎛−+−=−
If dy/dx is positive, the depth of flow will be increasing along the channel and vice versa.
Finite difference form of equation
meanr
o
FSS
xy
⎟⎟⎠
⎞⎜⎜⎝
⎛−−
=ΔΔ
21
The analysis of the numerator and denominator of eq(1) result in formation of a series of water surface profiles.
Also, S, the slope of energy line is given by
3/42
22
hRAQnS =
2
3/42
22
1 r
ho
FRAQnS
dxdy
−
−=
For a specified Q, both Fr and S are functions of depth y.
Both decrease as y increases.
oSS <
Recalling the definitions of yn and yc, following inequalities can be illustrated
oSS > when nyy <
when nyy >
cyy <1>F
1<F yy >
when
whenc
Gradually varied flow profile is classified based on channel slope and flow depth y in comparison with yn and yc.
Classifications of the channel slope based upon normal depth yn and critical depth yc are given below
The space above the channel bed can be divided into three zones depending upon the inequalities defined above. These zones are
Flow profiles are classified based upon
Channel slope
Zone in which they occur
For example if the water surface lies in zone 1in a channel with mild slop, it will be designated as M1 profile
Water surface profiles can be sketched without any computations
Can be achieved by considering the signs of numerator and denominator in the general equation.
Helps to know whether the depth increases or decreases with the distance
How the profile approaches the upstream and downstream limits
The important considerations are:
y>yc; flow is subcritical; F<1; denominator is positive
y<yc; flow is supercritical; F>1; denominator is negative
y=yn; flow is uniform; S=S0; numerator is zero
y>yn; S<S0; numerator is positive
y<yn; S>S0; numerator is negative
as y yn ; S S0 numerator approaches zero, dy/dx approaches zero asymptotically.
as y approaches yc, flow approaches critical conditions; F approaches unity (1); denominator approaches zero; dy/dx approaches infinity; water surface profile approaches critical depth vertically.
not possible to have a vertical water surface profile.
it is assumed that the water surface profile approaches CDL at very steep slope.
for a steep slope, one cannot assume that acceleration in vertical direction is negligible. So GVF theory doesn’t exist in such cases as pressure is no longer hydrostatic in those regions.
as y approaches infinity; S approaches zero; F also approaches zero; dy/dx approaches S0; water surface profile becomes very large as flow depth becomes very large.
for a wide rectangular channel, Hydraulic radius R~h, and
3
22
gyqF = then
( )3/10
223/100
3
ynqySgy
dxdy −
=
water surface profile tends to become vertical as the flow tends to become zero.
