Uniform flows and gradually varied flows
5.2 NON-UNIFORM FLOWS .1 Introduction
5.2.3 Discussion
Singularity of tlie energy equation
The energy equations (5.3), (5.4) or (5.6) can be applied to non-uniform flow (GVF) situations as long as the friction slope differs fi-om the bed slope. Indeed equation (5.6) may be trans- formed as:
— - ^o~^f (5.7) ds l-Fr^
where ^f and Fr are functions of the flow depth d for a given value of Q. Equation (5.7) empha- sizes three possible singularities of the energy equation. Their physical meaning is summarized as follows:
Case Singularity condition Physical meaning [1]
[2]
[3]
f = »
Fr= 1
—- = 0 and Fr = 1
Uniform equilibrium flow: i.e. Sf = SQ Critical flow conditions: i.e. Sf = ^c
Uniform equilibrium flow conditions and critical slope:
i.e. Sf= So = Sc. This case is seldom encountered because critical (and near-critical) flows are unstable
Free-surface profiles
With aid of equations (5.6) it is possible to establish the behaviour of (dd/ds) as a function of the magnitudes of the flow depth d, the critical depth d^ and the normal depth d^. The sign of (dd/ds) and the direction of depth change can be determined by the actual depth relative to both the normal and critical flow depths. Altogether, there are 12 different types of free-surface profiles (of non-uniform GVFs), excluding uniform flow. They are listed in the following table:
^ 0
(1)
> 0
= 0
< 0
d/d.
(2)
< 1
> 1
< 1
> 1
< 1
> 1
d^dc (3)
< 1
> 1
< 1
> 1 N/A N/A N/A N/A
d/d.
(4)
< 1
> 1
< 1
> 1
< 1
> 1
< 1
> 1 N/A N/A N/A N/A
Fr (5)
> 1
> 1
> 1
> 1
< 1
< 1
< 1
< 1
> 1
< 1
> 1
< 1
SfIS, (6)
> 1
< 1
> 1
< 1
> 1
< 1
> 1
< 1 N/A N/A N/A N/A
d, do, dc (7) d<do<d.
do<d<d.
d < df.< dQ Not possible Not possible do<d,<d dc < d < do dc < do < d d<dc d>dc d<dc d>dc
Name^
(8) S3 S2 M3 SI M2 M l H3 H2 A3 A2 Remarks: ^Name of the free-surface profile where the letter is descriptive for the slope: H for horizontal, M for mil4 S for steep and A for adverse (negative slope) (e.g. Chow, 1973).
For a given free-surface profile, the relationships between the flow depth d, the normal depth d^
and the critical depth d^ give the shape of the longitudinal profile. The relationship between d and dc enables to predict the Froude number and the relationship between d and d^ gives the sign of (5*0 — Sf). Combining with the differential form of the mean specific energy (i.e. equation (5.6)), we can determine how the behaviour of(^d//!is) (i.e. longitudinal variation of flow depth)
5.2 Non-uniform flows 105 is affected by the relative magnitude ofd, d^ and d^ (see table below):
Name^ Case (1) (2)
SQ - Sf
(3)
Fr (4)
Ad/As (5)
Remarks (6) M l
M2 M3 SI S2 S3 H2 H3 A2 A3
d>do>d.
dQ> d > dc do>d,>d d>d,>d^
dc> d> do dc> dQ> d d>dc d^>d d>d.
d,>d
So>Sf So<Sf So<Sf So>Sf So>Sf So<Sf So = 0<Sf So = 0<Sf So<0<Sf So<0<Sf
Fr< 1 Fr< 1 Fr> 1 Fr< 1 Fr> 1 Fr> 1 Fr< 1 Fr> 1 Fr< 1 Fr> 1
Positive Negative Positive Positive Negative Positive Negative Positive Negative Positive
The water surface is asymptotic to a hori- zontal line (backwater curve behind a dam).
Sf tends to SQ and we reach a transition from subcritical to critical flow (i.e. overfall).
Supercritical flow that tends to a hydraulic jump.
Remarks: ^Name of the free-surface profile where the letter is descriptive for the slope: H for horizontal, M for mild, S for steep and A for adverse. A^/Ay is negative for gradually accelerating flow. For Ad/Ay > 0 the flow is decelerating by application of the continuity principle.
Notes
1. The classification of backwater curves was developed first by the French professor J.A.C. Bresse (Bresse, 1860). Bresse originally considered only wide rectangular channels.
2. The friction slope may be re-formulated as:
Expression Comments
Friction slope _ Q'P.f
o _ >^ ^ w (Qhezy) ^
/ i s the Darcy fiiction factor
Cchezy is the Chezy coefiicient
3. The Froude number can be re-formulated as:
Expression Comments
Froude number Fr =
Fr-- Q
General case (i.e. any cross-sectional shape)
Rectangular channel (channel width B)
Application
Considering a mild slope channel, discuss all possible cases of free-surface profiles (non-uniform flows). Give some practical examples.
Solution
Considering a GVF down a mild slope channel, the normal depth must be large than the critical depth (i.e. definition of mild slope). There are three cases of non-uniform flows:
Case djd^ dld^ Fr Sf M/As Remarks
d>do>dc >1 >1 <1 <SQ > 0 The water surface is asymptotic to a horizontal line (e.g. backwater curve behind a dam) dQ> d> dc >1 >1 <1 >SQ < 0 Sf tends to SQ and we reach a transition from
subcritical to critical flow (e.g. overfall) do> dc> d >1 <1 >1 >So >0 Supercritical flow that tends to a hydraulic
jump (e.g. flow downstream of sluice gate with a downstream control)
Discussion
The Ml-profile is a subcritical flow. It occurs when the downstream of a long mild channel is sub- merged in a reservoir of greater depth than the normal depth J^ of the flow in the channel.
The M2-profile is also a subcritical flow {d > d^. It can occur when the bottom of the channel at the downstream end is submerged in a reservoir to a depth less than the normal depth d^.
The M3-profile is a supercritical flow. This profile occurs usually when a supercritical flow enters a mild channel (e.g. downstream of a sluice gate and downstream of a steep channel).
The SI-profile is a subcritical flow in a steep channel.
The S2-profile is a supercritical flow. Examples are the profiles formed on a channel as the slope changes from steep to steeper and downstream of an enlargement of a step channel.
The S3-profile is a supercritical flow. An example is the flow downstream of a sluice gate in a steep slope when the gate opening is less than the normal depth.